Phase Transitions and Self-Organization in Electronic and Molecular Networks [1 ed.] 030646568X, 9780306465680, 9780306471131

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Phase Transitions and Self-Organization in Electronic and Molecular Networks

FUNDAMENTAL MATERIALS RESEARCH Series Editor: M. F. Thorpe, Michigan State University East Lansing, Michigan

ACCESS IN NANOPOROUS MATERIALS Edited by Thomas J. Pinnavaia and M. F. Thorpe DYNAMICS OF CRYSTAL SURFACES AND INTERFACES Edited by P. M. Duxbury and T. J. Pence ELECTRONIC PROPERTIES OF SOLIDS USING CLUSTER METHODS Edited by T. A. Kaplan and S. D. Mahanti LOCAL STRUCTURE FROM DIFFRACTION Edited by S. J. L. Billinge and M. F. Thorpe PHASE TRANSITIONS AND SELF-ORGANIZATION IN ELECTRONIC AND MOLECULAR NETWORKS Edited by J. C. Phillips and M. F. Thorpe PHYSICS OF MANGANITES Edited by T. A. Kaplan and S. D. Mahanti RIGIDITY THEORY AND APPLICATIONS Edited by M. F. Thorpe and P. M. Duxbury SCIENCE AND APPLICATION OF NANOTUBES Edited by D. Tománek and R. J. Enbody

A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher.

Phase Transitions and Self-Organization in Electronic and Molecular Networks Edited by

J. C. Phillips Lucent Technologies Bell Labs Innovations Murray Hill, New Jersey

and

M. F. Thorpe Michigan State University

East Lansing, Michigan

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SERIES PREFACE

This series of books, which is published at the rate of about one per year, addresses fundamental problems in materials science. The contents cover a broad range of topics from small clusters of atoms to engineering materials and involve chemistry, physics, materials science, and engineering, with length scales ranging from Ångstroms up to millimeters. The emphasis is on basic science rather than on applications. Each book focuses on a single area of current interest and brings together leading experts to give an up-to-date discussion of their work and the work of others. Each article contains enough references that the interested reader can access the relevant literature. Thanks are given to the Center for Fundamental Materials Research at Michigan State University for supporting this series. M.F. Thorpe, Series Editor E-mail: [email protected] East Lansing, Michigan, September 2000

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PREFACE

The problem of phase transitions in disordered materials is quite old, but until recently it has seemed too complex a subject for formal study. The advent of computers has changed matters in two important ways. First, it has become possible to implement formal methods for microscopic study of phase transitions in ordered materials, even in the quantum limit, in great detail. This work has been so successful that few qualitative

mysteries remain, and many microscopic details have been measured experimentally and derived theoretically from first principles. The second radical change brought about by computers is that scientists have been forced to recognize that even today phase transitions in disordered materials are very poorly understood. Apart from the inherent statistical problems raised by disorder, it is becoming clear that new fundamental concepts are needed to explain qualitatively new phenomena that arise in disordered materials that were absent in ordered crystalline materials, or even in such materials with disordered sublattices. This workshop addresses the need for fundamentally new concepts in three areas of physical science. The first is network glasses, simple mechanical systems in which important new phenomena (the intermediate phases, the reversibility window) have been discovered as a result of exploring stiffness transitions both experimentally and in

numerical simulations made possible by new computer algorithms. The considerable progress made here is most encouraging, but surprisingly it has turned out that these new mechanical phenomena are closely paralleled by new electronic phenomena. These are discussed for the second area, the metal-insulator transition in semiconductor impurity bands, in which an intermediate phase has also been identified. The third area is (mostly cuprate) perovskites, where an intermediate phase occurs which can have superconductive transition temperatures well above 100K. It appears very likely

that the electronic intermediate phases exist because of disorder, and that the electronic phase diagrams closely parallel the mechanical phase diagrams of network glasses. On a microscopic level, minimization of the free energy of a disordered system at moderate temperatures, followed by some kind of (mild) quenching, can produce selforganization. There are many indications of this in network glasses, but of course life itself is self-organized. Proteins can be described as self-organized disordered networks, and they are discussed briefly here, and in a special issue of Journal of Molecular Graphics and Modelling (edited by L.A. Kuhn and M.F. Thorpe, to appear early 2001). It turns out that several constraint-based concepts that have been developed for network glasses apply equally well to the apparently unrelated subject of protein folding. This focused workshop was held at Hughes Hall, Cambridge, England, July 10-14, 2000. We are grateful to Dr. Martin Dove for assistance with local arrangements, and Ms. Janet King and Mr. Mykyta Chubynsky for extensive editorial assistance. J.C. Phillips M.F. Thorpe

East Lansing, Michigan, September 2000

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CONTENTS

I. Some Mathematics Mathematical Principles of Intermediate Phases in Disordered Systems................................................................................................ 1 J.C. Phillips Reduced Density Matrices and Correlation Matrix.............................................................. 23 A. John Coleman The Sixteen-Percent Solution: Critical Volume Fraction for Percolation................................................................. 37 Richard Zallen The Intermediate Phase and Self-Organization in Network Glasses................................................................................................... 43 M.F. Thorpe and M.V. Chubynsky

II. Glasses and Supercooled Liquids Evidence for the Intermediate Phase in Chalcogenide Glasses........................................................................................... 65 P. Boolchand, W.J. Bresser, D.G. Georgiev, Y. Wang, and J. Wells Thermal Relaxation and Criticality of the Stiffness Transition....................................................................................................... 85 Y. Wang, T. Nakaoka, and K. Murase Solidity of Viscous Liquids................................................................................................ 101 J.C. Dyre Non-Ergodic Dynamics in Supercooled Liquids................................................................ 111 M. Dzugutov, S. Simdyankin, and F. Zetterling Network Stiffening and Chemical Ordering in Chalcogenide Glasses: Compositional Trends of Tg in Relation to Structural Information from Solid and Liquid State NMR ........................ 123 Carsten Rosenhahn, Sophia Hayes, Gunther Brunklaus, and Hellmut Eckert

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Glass Transition Temperature Variation as a Probe for Network Connectivity............................................................................................ 143 M. Micoulaut

Floppy Modes Effects in the Thermodynamical Properties of Chalcogenide Glasses....................................................................... 161 Gerardo G. Naumis The Dalton-Maxwell-Pauling Recipe for Window Glass .................................................. 171

Richard Kerner Local Bonding, Phase Stability and Interface Properties of Replacement Gate Dielectrics, Including Silicon Oxynitride Alloys

and Nitrides, and Film ‘Amphoteric’ Elemental Oxides and Silicates ...... 189 G. Lucovsky

Experimental Methods for Local Structure Determination on the Atomic Scale ....................................................................... 209 E.A. Stern

Zeolite Instability and Collapse.......................................................................................... 225 G.N. Greaves III. Metal-Insulator Transitions

Thermodynamics and Transport Properties of Interacting Systems with Localized Electrons.......................................................................... 247 A.L. Efros The Metal-Insulator Transition in Doped Semiconductors: Transport Properties and Critical Behavior............................................................ 263 Theodore G. Castner

Metal-Insulator Transition in Homogeneously Doped Germanium.................................................................. 291 Michio Watanabe IV. High Temperature Superconductors Experimental Evidence for Ferroelastic Nanodomains in

HTSC Cuprates and Related Oxides...................................................................... 311 J. Jung Role of Sr Dopants in the Inhomogeneous Ground State of La2-xSrxCuO4....................................................................................................... 323 D. Haskel, E.A. Stern, and F. Dogan

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Universal Phase Diagrams and “Ideal” High Temperature Superconductors: ..............................................................................................................331 J.L. Wagner, T.M. Clemens, D.C. Mathew, O. Chmaissem B. Dabrowski, J.D. Jorgensen, and D.G. Hinks Coexistence of Superconductivity and Weak Ferromagnetism in Eu1.5Ce0.5RuSr2Cu2O10......................................................................................... 341 I. Felner

Quantum Percolation in High Tc Superconductors............................................................ 357 V. Dallacasa Superstripes: Self Organization of Quantum Wires in High Tc Superconductors.................................................................................... 375 A. Bianconi, D. DiCastro, N.L. Saini, and G. Bianconi Electron Strings in Oxides.................................................................................................. 389 F.V. Kusmartsev

High-Temperature Superconductivity is Charge-Reservoir Superconductivity.................................................................. 403 John D. Dow, Howard A. Blackstead, and Dale R. Harshman

Electronic Inhomogeneities in High-Tc Superconductors Observed by NMR.................................................................................................. 413

J. Haase, C.P. Slichter, R. Stern, C.T. Milling, and D.G. Hinks Tailoring the Properties of High-Tc and Related Oxides: From Fundamentals to Gap Nanoengineering........................................................ 431 Davor Pavuna

V. Self-Organization in Proteins

Designing Protein Structures.............................................................................................. 441 Hao Li, Chao Tang, and Ned S. Wingreen

List of Participants.............................................................................................................. 447 Index................................................................................................................................... 451

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MATHEMATICAL PRINCIPLES OF INTERMEDIATE PHASES IN DISORDERED SYSTEMS

J. C. PHILLIPS Bell Laboratories, Lucent Technologies (Retired) Murray Hill, N. J. 07974-0636 ([email protected])

INTRODUCTION Intermediate phases are found in disordered systems that for a long time were supposed to exhibit simple connectivity transitions, similar to dilute magnetic transitions. The latter can be modeled by percolation on a lattice. The paradigmatic disordered offlattice systems that exhibit intermediate phases are network glasses, impurity bands in semiconductors [the metal-insulator transition (MIT)], and high-temperature doped (pseudo-)perovskite superconductors. The first two (relatively simple) examples show that self-organization of the flexibility inherent in disorder is what creates intermediate phases, and that these must be described by finite-size scaling methods. The third (very complex) example shows that high temperature superconductivity (HTSC) itself depends on glassy dopant disorder, and only indirectly on the crystalline matrix with its long-range order. The mathematical principles underlying the filamentary or percolative theory of such internally organized systems are fundamentally different from those of theories based on the effective medium approximation (EMA) or fully disorganized (randomly) diluted lattice connectivity transitions. These principles have been developed only in the last hundred years and are little known to most scientists. The counting methods used in the filamentary theory bear a striking resemblance to those used to prove Fermat’s Last Theorem or to factor efficiently large numbers using quantum computers. Examples of the intermediate phase for these three classes of materials are given that specifically identify the internal self-organized complexity that is responsible for the remarkable physical properties of each case. There is a growing realization that the physics of complex disordered systems differs qualitatively from that of simple crystalline systems with long-range order, especially in the vicinity of connectivity transitions. In this workshop both experimental and theoretical work illustrating this theme are discussed for a wide range of subjects, with special emphasis on three topics: network glasses, impurity band MITs, and HTSC. In each case we find that the single connectivity transition to which we are accustomed in simple

Phase Transitions And Self-Organization in Electronic and Molecular Networks Edited by J. C. Phillips and M. F Thorpe, Kluwer Academic/Plenum Publishers, 2001

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systems is replaced by two transitions of quite different character. The first resembles the continuous transition expected from percolation theory, but with much simpler exponents,

while the second is a first-order transition with catastrophic character. Between these two transitions we find an intermediate phase which always has novel properties that are indeed qualitatively different from those of simple dilute lattice systems. In some cases these novel properties are of enormous technological value (window glass), and the study of intermediate phases has for the first time enabled us to understand quantitatively why these properties occur. In other cases, such as HTSC, the intermediate phase has properties that are so novel and so unexpected that so far almost all theories have failed to develop beyond the macroscopic or phenomenological stage. Within the physical sciences the level of interest in cuprate HTSC has greatly surpassed that of any other subject, with the sole exception of semiconductive materials (such as Si) basic to modern electronics. Initially the interest was stimulated by amazingly large values of the superconductive transition temperatures Tc, typically five (ten) times larger than the highest values found in compound (elemental) metals [1]. As expected, there were other anomalies as well: high sensitivity to doping by non-magnetic oxygen, and very little sensitivity to the presence of magnetic rare earths [2], both anomalies reversing the situation found in metallic superconductors. In the normal-state to superconductive phase transitions of metals, the superconductive properties are generally rather insensitive to the normal-state behavior, but in the cuprates the normal-state transport at high temperatures itself is anomalous. The anomalous behavior becomes most

characteristic at just those compositions that maximize Tc, even in cases where This tells us that a new electronic theory is needed to describe such “strange metals” [3].

This new theory must be very different from the Fermi liquid or Landau-Ginzburg theories used to describe normal metals, which are based on the effective medium approximation (EMA). The EMA cannot be even qualitatively correct here [4], as the Fermi liquid phase is separated from the intermediate phase by a first-order phase transition. Unfortunately, although the need for an alternative to Fermi liquid or LandauGinzburg theory is widely recognized [2,3], only the author’s own filamentary or percolative theory [5] avoids the EMA. This theory relies essentially on set-theoretic methods derived from number theory to establish quantitative results, and these methods are largely unknown to physical scientists. These methods have long been regarded as rather esoteric, even by most mathematicians, but their true significance, as a way of unifying results from algebra, analysis, geometry and topology, has become apparent recently from the proof [6] of Fermat’s Last Theorem (FLT). Several popular discussions

of set theory and FLT are available, but the connections with network glasses, impurity bands, and perovskite superconductivity are so simple and so direct that this paper will provide them as a matter of convenience to busy readers. We will then show how these novel mathematical ideas match the results of several recent decisive experiments in great detail. DISCRETE INTEGER AND CONTINUOUS REAL NUMBER FIELDS: FLT Physical scientists without a strong background in modern mathematics will find an excellent introduction to the subject, which carries them from its beginnings right through to an outline of the steps that led to the proof of FLT, in [7], amusing, anecdotal and thoroughly entertaining. For a long time number theory was regarded as a collection of

strange and rather accidental results of no general significance, but in the late 1800’s Cantor invented set theory and established an essential difference between integers and real numbers. Although both sets are infinitely large, the number (or order or cardinality) of

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real (irrational) numbers is larger than that of the integers and rational numbers, which have the same cardinality. He also hypothesized that there is no cardinality intermediate between those of the rationals and irrationals (continuum hypothesis); later Godel established a series of equivalent statements, including the axiom of choice, which showed that this axiom is independent of the rest of mathematics. These ideas are important in the present context because they highlight the idea that the methods of effective medium theories are fundamentally limited because they apply only to simple continuum systems

represented by real number fields. If the number field of interest is the integers or rationals, or a combination of these with the reals, then special methods need to be developed to prove theorems or derive results that do not exist for real number fields only.

The nature of these special methods is dramatically illustrated in the proof of FLT, which states that integer triples exist which satisfy the Pythagorean or Euclidean metric only for n = 2. Physical scientists, familiar with the example find Fermat’s conjecture quite plausible, especially as it has been confirmed by computer searches up to n = four million, but of course these searches do not constitute a proof. The proof involves two abstract mathematical tools, elliptic curves and modular functions.

An elliptic curve is not an ellipse: it is the set of solutions to a cubic polynomial in two variables, usually written in the form y2 = x3 + Ax2 + Bx + C. For number theory x and

y are integers. Modular functions are periodic and are a kind of integral generalization of sines and cosines. One can conjecture that all elliptic curves are modular. It then turns out that if this conjecture is valid, FLT follows. The proof of the latter began with a counter-example (Frey’s curve), which shows that should such an exist, it would generate an elliptic function with anomalous properties, in the sense that it would not be modular, as it is for integer triples with n = 2. To prove that this relation between elliptic and modular functions is necessary, Wiles counted the

number of both and showed that the two numbers were the same; thus the essential step was this counting [7].

Counting is a set-theoretic integral process. It is essential to our filamentary model of network glasses and the semiconductor impurity band transition [8-10] and to our

filamentary model of cuprate superconductivity [5]. In all cases the number of basis functions associated with cyclical vibrational states, or current-carrying states (or Cooperpaired current-carrying states in the superconductive case) is actually counted, as part of their separation from localized states in the neighborhood of the stiffness or metal-insulator transitions. Within the EMA and real number fields only, so far counting methods appear not to be feasible, and have not been used to discuss either the metal-insulator transition (MIT) or HTSC. All the EMA results that have been obtained are based on analytic (continuum) methods alone, which we believe are not well suited even to impurity band metal-insulator transitions and to the anomalous electronic properties of cuprates in the normal or superconductive states. It is obvious that in the network glass case continuum methods cannot identify floppy modes, which are obtained only by numerical solutions of matrix equations. BROKEN SYMMETRY, QUANTUM COMPUTERS, AND SHOR’S ALGORITHM The essential idea of our filamentary or percolative theory of random metals near the metal-insulator transition (MIT) is that in the limit such metals develop a new kind of broken symmetry even in the normal state. Electronic motion tangential ( ) to percolative filaments is phase-coherent, just as in normal metals, but normal to the filaments the motion is diffusive, as it is on the insulating side of the transition. This

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fractal behavior is what makes it possible for the MIT to be continuous, even in the presence of long-range Coulomb interactions, which could render the MIT first order in the

EMA, for example the MIT or Wigner transition of electrons in a box.

The presence of limited filamentary phase coherence makes many kinds of novel effects possible. Consider, for example, hypothetical quantum computers, which have attracted great interest recently among computer scientists [11]. These process complex integers (amplitude and phase) rather than merely real integers. Such hypothetical computers consist of quantum cells (cubits) connected by quantum wires which transmit

amplitude and phase information and thus are exactly the same as the quantum filaments discussed above. With such hypothetical computers Shor showed [11] how to factor large numbers in polynomial rather than exponential times by making use of fast Fourier

transforms and matrix methods, which rely on taking advantage of interference effects which occur conveniently for complex numbers but not for real numbers. COUNTING IN NETWORK GLASSES: AN EXAMPLE Counting is essential to our understanding of the remarkable properties of network glasses. Unlike the electronic cases, where the analysis is greatly complicated by both long-range Coulomb interactions and phase effects associated with complex wave functions, the properties of network glasses can be modeled by simple point mass-andspring systems. On the one hand, these matrix models with short-range forces and no quantum effects have proved to be (relatively) easily solved, compared to the electronic models (apparently insoluble). On the other hand, all the effects predicted by the network glass models are gradually being observed experimentally, and especially the properties of the intermediate phase are astonishingly similar to those observed for electronic materials. Thus, while it would have been easy to be skeptical of these mathematical analogies alone, it is apparent that they capture most, if not all, of the essential properties of intermediate phases. Recent work on intermediate phases in network glasses is discussed here by Boolchand, Thorpe, and Kerner, and the details can be found in their papers. Apart from the pivotal importance of counting in understanding the properties of network glasses, there is a second, and equally important, analogy between the methods Wiles used to prove FLT and the constraint theory of network glasses. The proof relies on establishing the connection between two sets, modular functions and elliptic curves, that at first seem to be unrelated, except that their numbers are the same. In constraint theory one compares the number of spatial degrees of freedom of the system to the apparently unrelated number of Lagrangian constraints associated with bonding interactions with localized vibrational frequencies (intact constraints). These constraints may involve n-body forces with (such as bond-bending forces, n = 3). In conventional continuum treatments, the relevant number is the number of interparticle forces (off-diagonal elements of the dynamical matrix, each of which contributes a different “interaction line” in diagrammatic perturbation theory). Constraint theory has shown that the relevant number is the number of intact potential interactions in potential space, not the number of forces implied by real-space derivatives of these potentials. In other words, spatial coordinates and interaction potential coordinates are treated as separate and distinct sets. The mean-field condition for the glass stiffness transition is that the numbers of elements in the two (apparently unrelated) sets are equal. This point is illustrated in Fig. 1, where the number of vibrational modes with zero

frequencies (cyclical modes) is plotted [12] against average coordination number r in as glassy network with bond-stretching and bending forces. At r = 2.40 the number of constraints equals the number of degrees of freedom, and the extrapolated number of

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Figure 1. The number of floppy modes as a function of r in bond-diluted models of three-dimensional glassy networks with stretching and bending forces [12]. The inset shows a blowup of the critical region. The second derivative of these curves shows a peak. This peak resembles the specific heat of a second-order phase transition, which shows that incomplete relaxation of the models generates the largest number of hidden linear

dependencies of constraints very near the connectivity transition.

cyclical modes is zero. (The smoothing of the kink at r = 2.40 is probably due to the fact that in the numerically simulated “random” network the topology is not ideally random.

This leads to redundancies among the constraints.) INTERMEDIATE PHASES: THERE ARE TWO STIFFNESS TRANSITIONS! For a long time we have all believed that in percolative problems there is only one connectivity transition. The first doubts began to appear in Boolchand’s measurements of

the critical coordination number, which was nearly always close to 2.40. However, even in cases where there was no evidence for nanoscale phase separation in the critical region, in other words, in cases where the theory should have worked, there were small discrepancies. Indeed, the numerical simulations shown in Fig. 1 predict small discrepancies, with a shift of the critical coordination number to 2.38 or 2.39. Experimentally, the shifts were in the other direction, to Those not familiar with constraint theory would probably say that such small discrepancies are insignificant - after all, in non-equilibrated glasses one should not expect better agreement between theory and experiment. But to us these discrepancies seemed significant. In particular, Boolchand’s ultraprecise Raman data also began to show evidence for a first-order transition, whereas all percolative models predict a continuous transition. The problems took definite form in 1998 workshop papers, where Boolchand et al. showed (see Fig. 2) that an apparent jump in the Raman frequency associated with corner-sharing tetrahedra in (Ge, Se) glassy alloys occurs at r = 2.46. This is the same critical value of r as occurs in the density and non-reversible part of the glass-transition enthalpy (discussed in more detail below). Yet still other Mossbauer experiments showed that some kind of transition, probably continuous, was happening at r = 2.40. I also found 5

Figure 2. Composition dependence in glasses (r = 2 + 2x) of corner-sharing Raman frequency, nonreversible enthalpy of glass transition, and molar volume.

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some marginal evidence for two transitions in elasticity data, with the compressibility transition occurring at r = 2.40, and the Young’s modulus transition occurring at r = 2.46. Today we are quite certain that these two transitions mark the boundaries of a new kind of phase in disordered systems, that we call the intermediate phase. To see what causes the intermediate phase, suppose we prepare an underconstrained network by bond dilution. Next we add bonds to the network at random until we reach the first connectivity transition. At this point the backbone begins to percolate from one face of the sample to the opposite face. It percolates as a “pure” filament that neither branches nor intersects itself. As we continue to add more bonds, two things can happen: we may get new “pure” filaments, or one of the old filaments can branch or cross. At the branching or crossing points locally the network is overconstrained and this increases the strain energy anharmonically compared to growing new filaments. Therefore the enthalpy, and initially the free energy, can be reduced by adding bonds selectively to avoid branching and crossing (“smart bonds”), and creating new filaments. However, as we add more and more bonds, and more and more filaments, at a certain point adding one more bond will lead to crossing or branching, no matter where it is added. This is the upper density limit for the second transition. It is intuitively plausible, and it is confirmed by numerical simulations, that the first transition is continuous, and the second is first-order (M. F. Thorpe et al., this volume). [It should be remarked that it is not surprising that the intermediate phase was overlooked for so long. It occurs only because the glassy network is not confined to a lattice. Whenever percolation occurs on a randomly diluted lattice with short hops (shortrange forces), there is only one transition, and it is continuous. It is the off-lattice selective relaxation character of the glassy network that makes “smart bonds” and a first-order transition possible.] COUNTING IN QUANTUM PERCOLATION THEORY: ANOTHER EXAMPLE In a d-dimensional sample with Nd randomly distributed impurities the formation of phase-coherent ballistic states is blocked (in the sense of Lagrange) by constraints [13,14]; note that from a counting viewpoint the microscopic nature of these constraints, orbital or spin, external or electron-electron interactions, is irrelevant; all that matters is their number. The central result of the filamentary theory of the MIT is the existence theorem, which states that these filaments can exist providing that the following condition on the log of this number is satisfied (Eqn. (7) of [8]):

This is a very amusing equation because of the way that it combines real and complex numbers. On the left hand side we have the exponent that represents the number of transverse degrees of freedom of our complex, current carrying basis states. Had these states been real standing waves, the factor 2 would have been absent. The first term on the right hand side is the number of real constraints generated by randomness. The second term is the (transverse) areal density of current-carrying filaments, an observable which of course is also real. Thus the left hand side measures complex quantum dimensionalities, while the right hand side measures real observable dimensionalities. In a simple but very fundamental way this equation describes all the implications of the quantum theory of measurement for the transport properties of random metals [14]. Quantum percolation theory explains all the experimentally observed critical exponents and prefactor sign reversals which are observed [15,16] in uncompensated random metals near the MIT such as Si:P and Ge:Ga. It applies to the scaling phase, which exhibits power-law behavior over

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a range 100 times larger than that found for magnetic critical phenomena. Very close to the continuous MIT, even the purest materials exhibit the effects of compensation, and the residual resistivity associated with scattering from compensating impurities dominates the transport properties, which revert to the conventional behavior predicted by the EMA.

COMPARISON WITH SCALING THEORY The dependence of (1) on dimensionality is strongly suggestive of scaling theories that have been developed to describe critical behavior of magnetic phase transitions [17]. It seems, however, that the results given in [8,9] differ fundamentally from those in the magnetic literature in two important respects. The latter rely on Bose, not Fermi, statistics, and hence contain no destructive interference effects. The interparticle forces of the latter are of short range, while electron-electron interactions are long range. Whether these two differences are necessary or sufficient to explain the large qualitative differences between the properties of random magnets and random metals has been unclear for a long time. It was even suggested [18] that the experimental data [15] may have been in error, a

suggestion which has recently been laid to rest [16]. Detailed comparison of filamentary quantum percolation theory with magnetic lattice percolation theory [17,18] has shown [10] that both Fermionic destructive interference and

long-range forces are necessary and sufficient to produce a consistent and successful theory of impurity band random metals such as Si:P and Ge:Ga. The destructive interference suppresses the divergence of the specific heat which is otherwise a characteristic of quasione dimensional ( d* = 1 ) Fermionic systems embedded in a d-dimensional matrix. This destructive interference is represented mathematically by a non-crossing condition for

semiclasical percolative paths similar to one that is already known for the integral quantum Hall effect at large n. (The elliptic curves which play an essential part in the proof of Fermat’s Last Theorem also satisfy a non-crossing condition [19], which suggests that

“arithmetic algebraic geometry” [briefly, “modern arithmetic”] may have a lot to offer in treating problems involving many Fermions.) In the limit Fermion statistics combined with long-range interactions cause d to be replaced by d + d* = d + 1. One can identify d* with the fluctuations of the component of the internal electric field that is locally tangential to the filament. Or one can introduce local times for Fermi-energy electrons moving along the filament. Then Newton is replaced by Einstein, and because of internal fields the filamentary paths fluctuate dynamically not only in space, but also in their local time It is amusing that the concepts of special relativity, originally developed to explain non-linear aspects of the Doppler effect, should reappear in the context of critical fluctuations in random metals. If the intermediate phase has a distinct topological character that is associated with off-lattice disorder, then one can immediately infer that it can occur in any disordered system that either has long-range forces, or can self-anneal. So I reasoned, and this led me to re-examine all the experimental data on strongly disordered systems near a connectivity

transition. Two such systems immediately spring to mind: the simpler system is the metalinsulator transition in semiconductor impurity bands, such as Si:P; following Shockley’s rule, we discuss it first.

There are two kinds of data on the metal-insulator transition in Si:P, both taken some 20 years ago. At that time everyone assumed that there was only one transition. This transition was supposed to occur continuously in both the conductivity and the coefficient of the linear term in the specific heat at the same value of uncompensated dopant density n and with related exponents, as both observables were supposed to be continuous functions of n.

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The trouble with this assumption was that when it was applied to the experimental data, the two sets of data, transport and thermal, did not lie on smooth, power-law curves (see Fig. 3 ). However, as everyone at that time was certain that there was only one transition, this problem passed unnoticed. Naturally, when I re-examined the by-now-allbut-forgotten thermal data some twenty years later [14], I immediately saw that the critical density for the thermal transition was actually much larger than that for the transport transition (by almost a factor of 2). All the effects of the intermediate phase that we have been discussing are connected with filamentary coherence and finite-size scaling. When a few minority dopants are present in the sample, two things happen, both based on interruption of coherence. Very close to the low-density continuous MIT, the conventional incoherent MIT occurs, which is smoother and has larger density exponents than the coherent transition. This conventional transition is of little interest here, except that it shows how different the intermediate phase is from a Fermi liquid. The second point of interest is that the presence of the minority dopants creates a natural length scale, associated with the average minority-minority spacing, that is larger than the majority-majority spacing. Physically this larger length scale is that associated with the residual resistance [8], a quantity that does not arise in many “modern” scaling theories of metallic behavior. The residual resistance is, of course, a very important quantity, as it eliminates divergences in the conductivity as It also provides a natural platform for filamentary counting. One constructs Voronoi polyhedra around each minority impurity, and counts the number of filaments crossing such polyhedra in various limits. Thus here counting shows up as a very basic operation. The omission of this construction has led some authors to the conclusion that the residual resistance of metals is not associated with impurities at all, but depends on “interactions of the electromagnetic field with the environment”, which is nonsense [8]. Counting has important implications for scaling in general. In continuum scaling

theory critical densities are irrelevant constants, and only the critical exponents are universal for a given class of interactions, independent of coupling strength, at least over limited ranges. Moreover, these critical exponents are in general irrational numbers. In

Figure 3. The electronic specific heat coefficient

for Si:P, showing both the continuous transport

transition at nc and the first-order thermal transition at ncb. The dashed line shows the value expected for a Fermi liquid.

The transition from the Fermi liquid to the filamentary metal occurs at n = n cb , and this

transition is first-order [14].

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network glasses, for simple alloys, there are only a few classes of intact constraints, and the condition that their average number/atom be integral automatically leads to average coordination numbers that are simple small fractions (such as 12/5). The existence of such “magic” fractions is a direct consequence of treating the bonding interactions as a set separate from the set of spatial coordinates – in other words, the bonding interactions form a “space” different from real space. By connecting these two separate spaces we can identify one, and possibly even both, of the connectivity transition compositions. The process of connecting two apparently different spaces to prove a certain arithmetic result is much the same as in the proof of FLT. It has often been conjectured that the results obtained from continuum lattice scaling theories are “universal”. Specifically for given classes of interactions, for example, the critical densities for different diluted lattices – cubic, hexagonal, etc., will differ, but the critical exponents will be the same. This is no doubt correct, but it does not include the effects associated with intermediate phases in disordered solids, which are a new phenomenon that lies entirely outside the framework of continuum lattice scaling theory. The new phase transitions and the intermediate phase cannot even be described properly in terms of diluted lattices with their single connectivity transitions. A more flexible and more abstract description is required, that uses the methods and concepts of modem mathematics. In particular, one must be satisfied to describe the properties of sets, as the presence of disorder makes it impossible to describe fully the properties of the individual elements of the sets.

BASIS FUNCTIONS IN FILAMENTARY METALS

Suppose we have an impurity band in the intermediate phase. In this phase the metallic states are centered on arrays of filamentary, non-crossing paths that extend from electrode to electrode. These paths are similar in some respects to the Self-Avoiding random Walks (SAW’s) that are used in statistical mechanics to describe the mathematical properties of diluted magnetic lattices. There are also important differences. In the magnetic case the SAW’s are closed loops with pseudovector symmetry, whereas the electrical paths have

vector symmetry. In the magnetic case we are concerned with minimizing the free energy associated with magnetic susceptibility, essentially an equilibrium property. In the electrical case, the metallic conductivity contributes to dielectric screening of internal electric fields, thus it also can be varied to minimize the free energy. Because of quantum mechanics, the kinetic energy associated with transverse localization of charge carriers on filaments increases as the filaments become more closely packed, eventually delocalizing the electrons and leading to the transition to the Fermi liquid state at higher densities. This does not occur in the spin case, as spins are already localized objects with no intrinsic kinetic energy. The generalization of Fermi liquid wave functions, indexed by the continuous momentum p and represented by the EMA wave functions to the discrete case of filamentary arrays, is not difficult. One assumes that the real-space centers of density of each filament j are known, and denotes the corresponding path by Longitudinal wave vectors are oriented parallel to the local tangent to the path. There are no transverse wave vectors, only a local transverse localization length. In the normal state in the absence of a magnetic field these wave vectors can be used to construct basis functions for each filament. The actual wave functions near the Fermi energy will be time-dependent linear combinations of individual filamentary wave functions that minimize the free energy by screening internal electric fields.

10

BROKEN SYMMETRY IN THE SUPERCONDUCTIVE AND NORMAL STATES

The key feature of the BCS theory of metallic superconductivity is the formation of Cooper pairs, which become the Landau-Ginzburg order parameter with 1 = and As the volume of the system tends to the number of possible

choices for 2 (even when these are restricted by the isoenergetic constraint ) also tends to but from this infinitely large set only one state, the time-reversed one, is used to form the Cooper pair. In other words, the cardinality of the set of states consisting of the time-reversed state is lower than that of the set consisting of all the isoenergetic states. This is a characteristic feature of continuum or effective medium models in which scattering by some kind of disorder is added after the basis states have been chosen. In strongly disordered systems, such as random metals near the MIT, the situation is quite different. In order to explain the critical properties of such systems one must select the correct filamentary basis states at the outset. This is done variationally, and it affects both normal-state and superconductive properties in many ways that are radically different from normal metals, where all the isoenergetic states are essentially equivalent, as in Landau Fermi liquid theory. The special properties of filamentary metals are the result of atomic relaxation that leads to preselection of basis states even in the normal state, where there are only two states per filament, and with In other words,

Figure 4. There are two components in the filamentary model, the CuO2 planes (A), and the impurity bridges combined with secondary metallic planes (B). This Figure shows the density of electronic states near the Fermi energy, for with the energy scale set by the resonant bridging impurity width WR. There is a strong peak in due to the impurity resonance pinned by electron-ion polarization energies and the anti-Jahn-Teller effect, and a strong dip in NA (E) due to electron-electron Coulomb interactions. The product has peaks near At the optimal composition there are only extended states for The scattering rates are also shown for the optimal

composition. They are much larger for the localized states,

than for the extended states,

11

for the one-dimensional filaments the Fermi surface collapses to two points, and this has happened in the normal state because of quantum percolation. Another extremely unexpected feature of filamentary states is that they maximize (minimize) the conductivity (resistivity). This variational property means that the dynamically optimized filamentary states already include all many-electron interaction effects, including those of electron-electron scattering which give rise to T2 resistivities in Landau Fermi liquid theory. That such many-electron interactions are absent in impurity band random metals has been shown by a filamentary analysis [8,9] of critical exponents in Si:P [15] and neutron-transmutation-doped, isotopically pure Ge:Ga [16]. The dominant remaining source of scattering in the cuprates is thermal excitation into localized states with energies outside the WR resonance region [5] shown in Fig. 4. (This disappearance of the Coulomb interaction between discrete and dynamically optimized filamentary currentcarrying states is analogous to the existence of zero-frequency floppy modes in the intermediate phase of network glasses.)

FILAMENTARY MODEL FOR HTSC

Even without measurements of the properties of the cuprates it was clear to crystal chemists and materials scientists that these multinary compounds would be extremely unusual from a structural point of view [1]. In addition to containing rare earths and

oxygen, these pseudoperovskites nearly always contain Cu, an element that in one oxidation state shows a greater diversity in its stereochemical behavior than any other element. This observation, together with the extremely anomalous transport properties, certainly bodes ill for any microscopic theory of the cuprates which is based on the EMA, as that approximation ignores the flexible material properties of Cu, and the ferroelastic properties of the perovskite family, altogether. (All materials with unusual properties, from Si to conjugated hydrocarbons to DNA, conform to the central principle of organic chemistry, “structure is function”.) Undeterred by these inescapable ground rules, almost all the theories developed so far, such as [2] and [3], are based on the EMA, augmented

only by good intentions and wishful thinking; given the richness and complexity of materials science, this is unlikely to suffice. The filamentary theory of HTSC diverges from EMA theories in two basic ways: it incorporates an extensive knowledge of the experimental data [1], and it has a sound mathematical foundation in the filamentary theory of the MIT in semiconductor impurity bands [14], which supersedes inadequate EMA theories of the Fermi liquid or LandauGinzburg type [2,3]. Because of interlayer ferroelastic interactions the “metallic” CuO2 planes are partitioned into metallic nanodomains separated by semiconductive domain walls. A specific filamentary path was envisaged [20] that connects metallic CuO 2 planes with secondary metallic planes (CuO chains or BiO planes) via resonant impurity states located in semiconductive planes (such as BaO) sandwiched between the metallic planes, as shown in Fig. 5. In addition to the bridging impurity points most samples contain two kinds of extensive defects which act as blocking lines or layers. Blocking macroscopic ab planar layers explain the usually semiconductive c-axis resistance; this aspect of the data has received too much attention [3], as these blocking layers are essentially extrinsic and can be avoided in some cases, by relieving interlayer strain energies [21], or by overdoping [22]. The interesting extensive defects are intraplanar semiconductive nanodomain lines in the metallic planes; these form grids, and each cell of a grid is connected to a square in an adjacent metallic plane by resonant (metallic) tunneling through a bridging impurity. The experimental evidence for the existence of such buckled cellular grids is discussed here by Jung. It is difficult to obtain evidence for spatial inhomogeneities on this scale, but the evidence available has been growing steadily if slowly. 12

Figure 5. The variationally optimized percolative filaments, shown in cross section, follow planar locally metallic CuO2 layers until they approach a domain wall which is locally insulating. The zigzag metallic path is continued by resonant tunneling through a state pinned at the Fermi energy associated with a defect, such as an oxygen vacancy. The next segment of the path is that of a chain, and this segment terminates at a chain O vacancy, where the zigzag path is continued by resonant tunneling back to a CuO2 layer, and so on. This model is designed for YBCO; in LSCO the tunnel paths simply connect CuO2 layers. Such filamentary paths should never be confused with “stripe phases” or pinned charge density waves, which are incidental

minority insulating phases.

There is also dramatic evidence for the existence of nanoscale spatial inhomogeneities in Debye-Waller factors measured by ion channeling, which are very sensitive to a few large out-of-plane atomic displacements and show striking precursive anomalies at Tc; these are absent from neutron diffraction data which measure EMA properties [23,4]. It is the electronic structure associated with Fermi-level pinning defects which experimentalists tune when they adjust oxygen concentrations, refine by annealing or observe as aging or quenching effects. To understand transport properties one must understand the topological

connectivity of these states, which is scarcely possible within the EMA. Within a single-particle picture the Fermi-level pinning metallic states can be represented as a narrow band of resonant states of width few meV the valence band width as shown in Fig. 4. Ordinarily one might expect that such states would be unstable against a Jahn-Teller distortion, and indeed it has been stated [3] without proof or citation that this is always the case. In fact, it is easy to find exceptions where such peaks are located self-consistently (with respect to both electronic and atomic coordinates) at EF, for example, in many total energy calculations. Moreover, Fermi-level pinning by impurity, surface or interfacial states at metal-insulator junctions (Schottky barriers) is one of the basic principles of semiconductor device physics. The error [3] arose from simplistic inclusion of only one-electron interactions and neglect of both core-core

and non-local electron-electron interactions. An amusingly similar error (sometimes called the Wentzel mistake), also on the subject of instabilities and superconductivity, led Wentzel to suggest [24] that the Bardeen-Frohlich attractive electron-phonon interaction was not the correct mechanism for simple metallic superconductivity. What is needed for the cuprates is a general mechanism for frustrating the Jahn-Teller effect. This is provided by a self-screening atomic relaxation mechanism which involves 13

long-range Coulomb interactions not representable as local single-particle energies [25]. The attractive self-screening energy of the already narrow band of resonant states is maximized by further narrowing the band and centering it on EF. This “anti-Jahn-Teller

effect” has the congenial feature that it is expected to be especially effective in strongly ionic materials where the long-range Coulomb forces are only weakly screened by a few metallic electrons. This is exactly the situation in the cuprates, which are close to a metalinsulator transition; it is also the case for impurity band random metals [8-10], where this

weak screening is responsible for the anomalously small exponent [15,16] which lies below the limit expected from scaling theory [18] with Boson statistics and shortrange forces only, and far below the value of 3/2 predicted by some one-electron EMA theories. The narrow resonance region of width WR was previously portrayed [5] as a peak in the density of electronic states centered on EF, but this need not be the case. All that is necessary is that in this region the scattering rate be extremely low compared to that at higher energies outside this region. Thus the picture we have now is that shown in Fig.4. The density of extended states is depressed relative to that of the localized states by Coulomb interactions, as happens for the random metal on the insulating side of the MIT [16,26]. This density of states would go to zero at E = EF and T = 0 were it not for the antiJahn-Teller effect, which leaves a residue of carriers at T = Tc which is about half that for T = T0 [27]. The pinning of the most polarizable filamentary states to EF by Coulomb interactions is similar to the energy-level reordering responsible for the pseudogap in

random metals [26]. NORMAL-STATE TRANSPORT

In a normal metal electron-phonon interactions typically contribute a temperaturedependent term to the resistivity proportional to with For large crystalline disorder, as in metallic glasses and thin films quenched at low temperatures, electronelectron scattering is strong and In certain cuprates, notably those without secondary metallic planes involving metallic elements other than Cu (Bi or Hg), the ab planar normalstate resisitivity is linear in T approximately from T0 to Th, where T0 is close to Tc and Th is the high-temperature limit of compositional stability [5]. This is a very remarkable result, as in cases where Tc is low, the ratio Th/T0 has been observed to be as large as 100. However, it holds only for those samples whose composition corresponds to a maximum in Tc; increasing doping causes to cross over to the Fermi liquid (strong electron-electron scattering) value of 2. This means that a satisfactory theory should contain some continuously tunable factor which will alter both anomalies at the same time, and this factor should be responsible for the MIT as well. As the reader will realize, these demands are very severe. He will probably not be surprised to learn that they are met by the author’s filamentary theory [5], but not by any theory based on the EMA, such as [2] or [3]. The limitations of EMA models becomes obvious when one examines the field theories developed by various authors [28] to implement Anderson’s suggestion that electric (holon) and magnetic (spinon) effects are somehow separated in the cuprates [3]. It is clear that such a separation is essential if the magnetic moments of the rare earths are not to quench superconductivity, but Anderson gives no microscopic explanation of how this can happen; it is merely one of his axioms, or, as he prefers, dogmas. Given this dogma, one is able to explain microscopically why normal-state transport anomalies exist and are loosely correlated with the optimization of Tc. However, one is unable to derive any functional form for the temperature dependence of the resistivity, much less to explain why 14

without assuming what was to be derived. Even the temperature scale ratio Th/T0 is merely the ratio of an inelastic high-temperature scattering rate to Tc, which is yet another assumption, which turns out to be incorrect, as we shall see. The separation (“spin bags” [2]) of magnetic and electrical effects, moreover, need not be axiomatic. It is derivable in the filamentary model simply by observing that the magnetic states are all localized, and that the separation of the extended current-carrying states from the localized states [14,5] automatically separates spin and Cooper-pair-forming states. Note, however, that this separation cannot be carried out correctly within the EMA because in that picture one is unable to count [6,7,19] the states that are being separated. The importance of counting is illustrated convincingly by the much simpler case of random impurity band metals, where the EMA has failed in calculating critical exponents [8,9], By contrast, the dogmatic holon-spinon separation [3] becomes, in the filamentary model, what one would naturally expect in optimized HTSC because of the success of the filamentary model for the closely related impurity band MIT. It is nothing more than intralayer nanoscale phase separation, driven by interlayer ferroelastic misfit forces.

Because the CuO2 planes are divided into an irregular checkerboard or grid of nanodomains by intraplanar domain lines which have semiconductive gaps currents can flow only along filamentary paths passing through interplanar resonant tunneling centers (impurity bridges). In YBCO, for example, such centers might be represented by the much-studied apical oxygen sites between Cu atoms in the CuO2 planes and the chains, selectively associated with vacancies on the later. For optimal doping there are two such centers per CuO2 planar domain, one source and one drain. When there are fewer than two, the sample is underdoped, and when there are more than two, it is overdoped (see Fig. 2 of [5]). Thus the average integral (bridge/domain) ratio is the continuously tunable factor mentioned above; there it was shown that when this factor is two all the normal-state anomalies are explained, as is the optimization of Tc. It was also

explained why overdoping depresses Tc and increases

from 1 to 2, at the same time producing the observed anomaly in the Hall resistance [29]. An important historical point is that the fact that all the normal-state transport anomalies can be explained by the existence of a narrow, high-mobility band pinned to EF was first explained in [29]. At that time the explanation was not generally accepted because it was not accompanied by a specific structural model that explained the origin of

the narrow band. Such a model is shown in Fig. 5, and it is the only such model that has been advanced. The narrow high-mobility band itself is the only way of explaining the normal-state transport anomalies, so that together with Fig. 5 it may be taken as the only satisfactory, perovskite-specific model for HTSC. What happens to underdoped samples? In the YBCO case, underdoping produces

more O vacancies on the chains that generate the crystalline orthorhombic symmetry. These chains are almost surely responsible for the phase shifts at twin boundaries where the chain orientations rotate by which experimentalists and many EMA theorists often like to describe as “d-wave superconductivity”. This is an EMA (or Fermi liquid) misnomer that is entirely inappropriate for the non-Fermi liquid intermediate phase. It implies a fundamental significance of what amounts to a non-bulk edge or surface

effect, which is seen to be trivial as soon as one realizes that the b-axis chains are essentially involved in constructing filamentary paths, so that the observed phase shift is unavoidable. The chain segments become shorter as x increases, and although probably only short chain segments are needed to bypass intraplanar CuO2 domain walls, it is clear that the ab planar resistance will increase as the chain segments shorten. Aside. Many experiments have shown that residual states exist within the pseudogap; these residual states are also often described as the result of “d-wave superconductivity”. In fact, the observed residual states are very similar to those predicted by [26]. Thus if

15

there are dopants in semiconductive domain walls that generate a pseudogap, then this would easily account for the experimental observations, without nonsensically using Fermi liquid terminology to describe the non-Fermi liquid intermediate phase. In fact, we expect increasing phonon-assisted currents across oxygen vacancies within the chains. The importance of these will increase with x and (the orthorhombic plateau in Tc(x)), it is possible that virtual phonon exchange at these vacancies will provide a stronger attractive interaction for forming Cooper pairs than phonon exchange at the caxis impurity bridge resonances does. The width and strength of the density-of-states peak of the latter may not depend on the oxygen mass, as they may well be associated with collective relaxation and optimization of many internal coordinates. This would explain the disappearance of the oxygen isotope effect for small x [30]. On the other hand, the electron-phonon interactions at chain vacancies promote phonon-enhanced coherent currents across these micro-weak links, enhancing the local energy gap. When this effect is linearized with respect to vibrational amplitudes, it may still be equivalent to a local Bardeen-Frohlich interaction and may thus give rise to what resembles a normal isotope effect for large After the above was written, a very important paper [31] appeared concerning the relaxation of Tc in YBCO after abrupt release of pressure. The relaxation was found to follow the form of a “stretched exponential”, with The key parameter of interest is the dimensionless stretching fraction which turns out to be highly informative. The stretched exponential can be derived from a microscopic model. The model involves diffusion of excitations in a configuration space of dimension d*p to

randomly distributed traps.

As time passes, all the excitations near the traps have disappeared, and only excitations distant from the traps remain. The latter must diffuse further and further. This leads to the stretched exponential and to

At first, it might appear that all that has been done is to replace one empirical parameter, with another, d*p. In fact,

for homogeneous glasses. (The dopants in a well-annealed and homogeneous HTSC presumably form a glassy array.) Here d = 3 is the dimensionality of Euclidean space. The key factor now is p. Comparison with experiment and several very accurate MDS showed that for homogeneous glasses p is nearly always 1 or 2; it measures the number pd of interaction channels involved in diffusion of excitations in d dimensions. In metals where phonon scattering dominates the resistivity, one of these channels is always e-p interactions. However, if other classes of interactions are present, there may be other diffusive interaction channels as well. It is easy to see that adding channels increases the stretching factor, which is Mathematically the simplest and most rigorous example with p = 2 is provided by quasicrystals, where the Euclidean coordinates r become and the Penrose projective coordinates are Motion in space (the first d channels) involves phonons and

produces relaxation, while motion in space (the second channels) involves phasons, which only rearrange particles without diffusion or relaxation. In the ideally random quasicrystal a given hop may tale place along either or Thus

16

where f p measures the effectiveness of hopping in pd channels, only d of which is associated with relaxation. For an axial quasicrystal, which is quasi-periodic only in the plane normal to the axis, the calculation is somewhat more complex. There are five channels, three in space, and two in space, so that fp = 3/5, and d*p = 9/5. Thus in excellent agreement with MDS15 which give From the value one can rigorously infer that p = 1, and thus only electronphonon interactions can cause HTSC. The proof is based on grouping the interactions involved in diffusive relaxation into classes of interactions that are effective (such as electron-phonon interactions) and ineffective (such as electron-magnon interactions) [32]. Because only the electron-phonon interaction is needed to explain all other interactions (such as electron-[magnon, plasmon, any-old-whaton] interactions) are excluded by experiment [31,33]. The remarkable aspect of this experiment and theory is that the conclusion transcends all the details of structure and large-scale relaxation around

dopants in these complex materials. Note that once again, the success of this approach rests on identifying two different sets, interaction space and real space, and then connecting them, just as in constraint theory and the proof of FLT.

CHAIN LENGTHS AND LOW TEMPERATURE CUFOFF T0 In the filamentary theory there is a close relation between average chain lengths L and the lower cutoff (or pseudogap) temperature T0 in for optimally and

underdoped samples:

(see (1,2) of [5]); here This relationship gave good results for the YBCO T0(x)], which are linear in x with T0 (0.1) 150K for Tc optimized 90K. It should be mentioned here that many samples appear to give T0 less than Tc, but these are probably inhomogeneously overdoped. For optimally doped (x = 0.1) samples the value T0 (0.1) 150K has been confirmed for single crystals and for thin films grown by several methods, and even fine-tuned with low-energy electron irradiation [27]. In a large magnetic field Tc is suppressed, “unmasking” or exposing the normal-state

resistivity at temperatures lower than Tc(H) for H = 0 in relatively “low Tc” cuprates ( T c

40K), such as [35], The central “unmasked” single-crystal results stressed by [34] are that for underdoped compositions pab(T) increases and becomes semiconductive below T0. Even more significant, however, is the disappearance of the pseudogap (the dip in normal-state resistivity between Tc and T0) in large magnetic fields. This disappearance is not discussed at all, as it is virtually inexplicable within the EMA. In terms of our topological model of the intermediate state, the explanation is immediate. The large magnetic field replaces the self-organized, non-crossing coherent filamentary basis states with vector symmetry by quasi-circular orbital states with pseudovector symmetry, thereby restoring Fermi liquid character, including strong electron-electron scattering, to states near EF.

Because chains represent the ideal local structure for filamentary currents, microscopic probes of the local chain structure in untwinned samples of YBCO are of great interest. Recently two experiments have done this, with results that cannot be explained by EMA models based on bulk energy bands and bulk phonons. The first experiment [36] revealed systematic changes in scattering strength with doping of longitudinal optic (LO) phonons

17

propagating in the basal plane normal to the chain direction. These phonon spectra contain a pseudogap which can be explained [33] as the result of short-range ordering of chain segments that alternate in oxygen filling factors. The changes in scattering strength are more interesting, as they turn out to be direct measures of phonon coherence along filamentary paths, and they change abruptly in as the composition passes

through the metal-insulator transition near x = 0.4. There is also a second abrupt change in scattering strengths centered on x = ¾ [the transition between the Tc = 60K and 90K plateaus] as the smaller filling factor passes through ½; this change is just what one would expect from percolation theory, and from it one can successfully predict the change in the Tc ratios of the two plateaus. In the second experiment [22], the longitudinal magnetoresistance in slightly overdoped untwinned YBCO was studied; in these samples the c-axis resistivity is linear in T, just as the ab planar resistivity is, which means that the coherent percolative paths are 3dimensional. Again the results show strong anisotropy that is connected with coherent current flow along the chains.

NMR RELAXATION AND THE SPIN PSEUDOGAP

Anderson has discussed (see [3], Fig. 3.27) the observation of anomalous nonKorringa planar Cu relaxation in various cuprates, and states that “this anomalous relaxation seems to be one of the common features of the high-Tc. state”, but “it is actually relatively more pronounced for somewhat lower Tc materials., so is not closely related to superconductivty.” This author concurs, and would add that in his opinion in most cases (LSCO is an exception) all that spin-scattering experiments are measuring is spin relaxation in pockets of insulating material with compositions which can be quite different from those of the superconductive bulk. For example, this effect is quite small in optimal with but is much larger in optimal Note that the oxygen diffusivity is very high in YBCO, but not the Sr diffusivity in LSCO, so that while it is possible to make very nearly homogeneous samples of the former, this is not possible for the latter. Thus in general there is no connection between T0 and the spin pseudogap Tsg, nor should we expect to find one, except for materials like LSCO, where magnetic microphase inclusions may be absent. In that case both T0 and the spin pseudogap Tsg can be related to the resonance width WR. COMPOSITION DEPENDENCE OF THE ENTROPY OF THE VORTEX PHASE TRANSITION

The physics of magnetic vortices in the mixed state is extremely complex, and it is nearly always treated from the point of view of the EMA, although it is clear that if the sample is spatially inhomogeneous the vortices will nearly always localize preferentially in regions of lower Tc. In the author’s view this is the most natural explanation of the “step kink - peak” phenomena in vortex lattice melting which have attracted much attention from experimentalists [37]. However, two-phase models have many adjustable parameters and it would appear that this greatly limits what can be gained from the analysis of such phenomena. Thus most of the discussion of these phenomena by experimentalists has focused on the fact that the entropy of vortex lattice melting is much larger than would be expected if the vortices are treated merely as point objects [38]. This can be easily explained by taking account of changes in the nonlocal structure [39] of the vortices near Tc. There is, however, 18

one very puzzling feature of these data which is explained quite easily by the present theory. This theory is a counting theory, and thus it is well-suited to studying the entropy of the vortex phase transition. In Bi2Sr2CaCu2Oy single crystals it is observed [37] that this entropy is several times larger for overdoped than for optimally doped samples, especially

at low T. Ordinarily in the EMA one would expect that Tc would reach its maximum value at the composition where N(EF), the density of electronic states in the normal state, has its maximum value, that is, at optimal doping EF coincides with a peak in N(E). In such a case

the entropy should reach its maximum at the same optimally doped composition. However, in the present model N(EF) is larger in the overdoped state than in the optimally doped state, so giving a larger entropy of melting. The transition temperature decreases in the overdoped state because the electrons at the Fermi energy spend more time in Fermi-liquidlike states in the CuO2 planes, where the electron-phonon coupling is weak, and less time at the discrete resonant tunneling centers, where it is very strong. See Eqn. [3] and Fig. 2(c) of [5].

INTERMEDIATE PHASES AND FIRST-ORDER PHASE TRANSITIONS

The mathematical character of intermediate phases is characterized by some discrete features (the filaments) and some continuous features (the off-lattice space in which both the network glasses and the spatially disordered impurities of the electronic examples are embedded). One felicitous consequence of mixing discrete and continuous features is that the lower-density (or first) transition from the disconnected (or insulating) phase to the intermediate phase is continuous, while the higher-density (second) transition from the filamentary phase to the effective medium (overconstrained, or Fermi liquid) phase is first order. This asymmetry is quite striking, and it cannot be explained by an entirely discrete

lattice model, or by an entirely continuous effective medium model. Both of the latter contain only one transition, and it was the similarity in this respect that has led many people to suppose (mistakenly) that there is a “universal” character of phase transitions that can be independent of the discrete/continuous dichotomy. In this workshop both Boolchand and Thorpe discuss these two phase transitions in convincing detail. In Figs. 3 and 6 the two transitions are sketched for impurity bands in Si:P. The two critical densities are separated by a factor of 2, and there is no doubt that the first one is continuous, and the second is not. The layered cuprates that form HTSC are complex multinary compounds, and

preparing samples that are microscopically homogeneous is not easy. Of course, unless such homogeneity is achieved, the second transition will be greatly broadened, and it will be difficult to show that the second transition is first-order. So far, three successful studies have reported first-order phase transitions: (1) near x = 0.21 in (after annealing for several months [41] at high T and constant O partial pressure), (2) near = 0.19 in

(also after annealing at constant composition [42]), and near x =

0.95 in (carefully designed chemical and thermal history, including slow cooling [43]). Normally ones observes only parabolic Tc(x)’s. Self-organization is not easily achieved, that is why only in a few experiments are the two HTSC transitions

separated to give trapezoidal Tc(x)’s.

19

Figure 6. A sketch of the thermal data on Si:P, showing both transition [40].

and

in relation to the transport

GENERAL CONCLUSIONS: ANALYTICITY AND CARDINALITY The essence of filamentary percolation theory is that it replaces the analyticity of field theory, which has been an excellent guide to the physics of the “old” metallic superconductors, by the “countability”, or cardinality of modern set theory. The justification for this in HTSC is an internally consistent theory of the crucial experimental properties, notably the normal state transport properties, as functions of both temperature and composition. Analytic models [3] can be constructed which account qualitatively for the temperature dependence, for example, but when these are examined in detail it soon becomes apparent that they are not capable of accounting precisely for the observed functional forms, either the resistivity linearity in T or the linearity and magnitude of the composition dependence of the low-temperature pseudogap linear resistivity cutoff T0(x) in YBCO7-x. The filamentary theory exhibits many similarities between the MIT in impurity bands and that in the cuprates, especially YBCO, which is the most homogeneous and macroscopic-defect free of the cuprates, because of its chains. In this sense the theory is self-proving (self-testing), because one would expect those similarities to be most pronounced for the best material. The disappearance of many of these properties from LSCO and (Y,Ca)BCO alloys shows that the theory is both selective and incisive, for it

successfully differentiates those properties which EMA models cannot explain because they are microscopic, from those which it cannot explain because they are macroscopic. Both electronic theories are reduced to their simplest form in the intermediate phases of network glasses. The overall similarity of the three systems shows that it is their shared

20

(discrete/continuous) topology that is responsible for their remarkable properties, from the reversibility window to HTSC itself.

THE BIG, BIG PICTURE

The differences between pure continuum models (or the effective medium approximation, EMA) and discrete off-lattice models (embedded in a continuum) are huge, not only conceptually, but also psychologically. Scientists who have been educated to think only in terms of continuum models, and who have developed their own concepts in that framework, often find themselves enslaved by that framework. (A concrete analogy, close to home, is that of the experimenter married to his equipment, or the computer scientist married to his software.) In this volume one can find many examples of problems and principles that certainly go beyond the limits of the continuum approach. Perhaps it is not surprising that many of these examples occur in the context of network glasses, as these materials are obviously unsuited to continuum treatments. On the other hand, that nanodomains should exist in perovskites and pseudoperovskites should come as no surprise, as this family of materials has been known to be ferroelastic (and prototypically so) for more than 50 years. Yet Jan Jung’s elegant survey in this volume of these nanodomains reaches conclusions that are

politically very unpopular, as anyone who has attended one of the numerous conferences on

HTSC can testify. One can ask oneself just why such an obvious consequence of one of the most general

principles in the materials properties of oxides should be deemed “politically incorrect”. More than 10 years ago, when the subject of HTSC was still in an embryonic stage, I believed that most people were still being conservative; perhaps there was not enough direct evidence for the existence of nanodomains, and their dimensions remained to be determined. As Jung shows, this is certainly not the case today. Another explanation is that most people feel that if their own experiments do not

directly exhibit nanodomain features, then those features are not needed to explain their results. This is not so naïve as it sounds: in the theory of critical exponents of continuous phase transitions, some very sophisticated theorists have postulated that (loosely speaking), all such transitions are equivalent. (This is the concept of universality.) Such on-lattice transitions never exhibit an intermediate phase. Thus, it is the existence of intermediate phases in self-organized disordered systems that causes the breakdown of universality. We are now at the crucial point, both conceptually and psychologically. It is widely admitted, even by almost all those who still adhere to continuum descriptions of HTSC, that the intermediate (often called “non-Fermi liquid”) phase is responsible for HTSC. If this is the case (and all the experimental evidence so indicates), then the fact that no microscopic continuum model is known that produces an intermediate phase between the insulating and Fermi liquid phase, becomes decisive. There is such a model in discrete theories, and it is very successful in describing intermediate phases in network glasses and semiconductor impurity bands, as discussed elsewhere in this volume. It follows that a discrete network model is the only practical model for HTSC. Of course, a successful continuum model based on the EMA may be developed someday for HTSC, about the time that there is pie in the sky.

REFERENCES 1.

J. C Phillips, Physics of High-Tc Superconductors, Academic Press, Boston 1989.

2.

J. R. Schrieffer, X. G Wen and S. C Zhang, Phys. Rev. B 39, 11663 (1989).

21

3.

P. W. Anderson, Theory of Superconductivity in the High-Tc Cuprates, Princeton Univ. Press, Princeton 1997.

4.

J. C. Phillips, Physica C252, 188 (1995).

5.

J. C. Phillips, Proc. Nat. Acad. Sci. 94, 12771 (1997).

6.

A. Wiles, Ann. Math. 141, 443 (1995).

7.

A. D. Aczel, Fermat’s Last Theorem, Four Walls Eight Windows, New York, 1996; S. Singh and K. A. Ribet, Scien. Am. (11): 68 (1997); D. Mackenzie, Science 285, 178 (1999). 8. J. C. Phillips, Proc. Nat. Acad. Sci. 94, 10528 (1997). 9. J. C. Phillips, Proc. Nat. Acad. Sci. 94, 10532 (1997). 10. J. C. Phillips (unpublished). 11. A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, 733 (1996); C. H. Bennett, Physics Today 48 (10), 24 (1995). 12. H. He and M. F. Thorpe, Phys. Rev. Lett. 54, 2107 (1985); M. F. Thorpe, D. J. Jacobs, N. V. Chubynsky and A. J. Rader, Rigidity Theory and Applications (Ed. M. F. Thorpe and P. Duxbury Kluwer Academic / Plenum Publishers, New York, 1999), p. 239. 13. Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35, 1399 (1975). 14. J. C. Phillips, Solid State Commun. 47, 191 (1983). 15. G. A. Thomas, M. A Paalanen,. and T. F. Rosenbaum, Phys. Rev. B 27, 3897 (1983). 16. K.M.Itoh, E. E.Haller, J. W. Beeman, W. L. Hansen, J.Emes, L.A. Reichertz, E. Kreysa, T. Shutt, A. Cummings, W. Stockwell, B. Sadoulet, J. Muto, J. W. Farmer, and V. I. Ozhogin, Phys. Rev. Lett. 77, 4058 (1996). 17. M. F. Collins, Magnetic Critical Scattering, Oxford Univ. Press, Oxford (1989). 18. J. T. Chayes, L. Chayes, D. S. Fisher and T. Spencer, Phys. Rev. Lett. 57, 2999 (1986). 19.

K. A. Ribet, and B. Hayes, American Scientist 82, 144 (1994).

20.

J. C. Phillips, Phys. Rev. B 41, 8968 (1990).

21.

X. D. Xiang, W. A. Vareka, A. Zettl, J. L. Corkill, M. L. Cohen, N. Kijima, and R. Gronsky, Phys. Rev. Lett., 68,530(1992). N. E. Hussey, H. Takagi, Y. Iye, S. Tajima, A. I. Rykov, and K. Yoshida, Phys. Rev. B 61, R64 (2000). R. P. Sharma, F. J. Rotella, J. D Jorgensen,. and L. E. Rehn, Physica C 174, 409 (1991). G. Wentzel, Phys. Rev. 83, 168 (1951). J. C. Phillips, Phys. Rev. B 47, 11615 (1993). A. L. Efros and B. I. Shklovski, J. Phys. C 8, L49( 1975). S. K. Tolpygo, J.-Y. Lin, M. Gurvitch, S. Y. Hou and J. M. Phillips, Physica C 269, 207 (1996). N. Nagaosa and P. A. Lee, Phys. Rev. B 45, 966 (1992). H. L. Stormer, A. F. J. Levi, K. W. Baldwin, M. Anzlowar, and G. S. Boebinger, Phys. Rev. B 38, 2472 (1988).

22. 23. 24. 25. 26. 27. 28. 29. 30.

J. P. Franck, Physica C 282-287, 198 (1997); Phys. Scrip. T66, 220 (1996).

31. 32.

S. Sadewasser, J. S. Schilling, A. P. Paulikas and B. W. Veal, Phys. Rev. B 61, 741 (2000). J. C. Phillips, Rep. Prog. Phys. 59, 1133 (1996); J. C. Phillips and J. M. Vandenberg, J. Phys.: Condens. Matter 9, L251-L258 (1997).

33. 34.

J. C. Phillips (unpublished). G. S. Boebinger, Y. Ando, A. Passner, T. Kimura, M. Okuya, J. Shimoyama, K. Kishio, K. Tamasaku, N. Ichikawa, and S. Uchida, Phys. Rev. Lett. 77, 5417 (1996). T. Ito, K. Takenaka, and S. Uchida, Phys. Rev. Lett. 70, 3995 (1993). Y. Petrov, T. Egami, R. J. McQueeney, M. Yethiraj, H. A. Mook, and F. Dogan, LANL CondMat/0003414 (2000).

35. 36.

37. 38. 39. 40. 41. 42.

T. Hanaguri et al., Physica C 256, 111 (1996). A. I. M. Rae, E. M. Forgan, and R. A. Doyle, Physica C 301, 301 (1998). H. Darhmaoui and J. Jung, Phys. Rev. B 53, 14621 (1996). J. C. Phillips, Solid State Commun. 109, 301 (1999). H. Takagi, R. J. Cava, B. Batlogg, J. J. Krajewski, W. F. Peck, P. Bordet, and D. E. Cox, Phys. Rev. Lett. 68, 3777 (1996); H. Y. Hwang, B. Batlogg, H. Takagi, J. Kao, R. J. Cava, J. J. Krajewski, and W. F. Peck, Phys. Rev. Lett. 72, 2636 (1994). J. Wagner (this workshop).

43.

E. Kaldis, J. Rohler, E. Liarokapis, N. Poulakis, K. Conder, and P. W. Loeffen, Phys. Rev. Lett. 79,

4894 (1997).

22

REDUCED DENSITY MATRICES AND CORRELATION MATRIX

A. JOHN COLEMAN Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada [email protected]

I tackle a herculean task - attempting to wean our imagination from the 1-particle picture which, implicitly, we have all been using since our youth. I shall try to entice you to join a crusade for the creation of new concepts and images needed for problems in which interaction between 3 or more electrons is significant and which are appropriate for describing the information encapsulated in the second order reduced density matrix. (“2-matrix” for short) Perhaps the difficult part of our task is changing our language and mental images. It was to this task that we were called by Charles Coulson in private conversation, and in his speech [1] in Boulder in June 1959, urging us to look in the 2-matrix for correlation. Also, by H. Froehlich [2] when he bemoaned the fact that we have failed to exploit the deep import of the results [3] of C.N Yang on the 2-matrix. It is to this task that I have devoted much of my time and interest since 1952 culminating in the publication of REDUCED DENSITY MATRICES - Coulson’s Challenge [4]. I shall refer to this book, by Coleman and Yukalov, as “CY”. When we wrote CY, although it was known that the order parameter of one of the phases of He3 had p-symmetry, we were unaware that the existence of s,p and d symmetry has appeared in some of the high temperature superconductors. As a result there is only a brief reference in CY to the correlation matrix. I have made a modest effort to redress this lacuna in the present paper. ENERGY AND N-REPRESENTABILITY A reduced density operator (RDO), for a normalized pure state, (123... N), of a system of N identical fermions or bosons can be represented as an integral operator. For example, the kernel of a 1-RDO, D1 , is the first order reduced density matrix (1-RDM):

Phase Transitions And Self-Organization in Electronic and Molecular Networks Edited by J. C. Phillips and M. F. Thorpe, Kluwer Academic/Plenum Publishers, 2001

23

whereas, the 2-RDM is

Thus the operator, D 1 , acting on a symmetric or antisymmetric function f of N particles,

defines a function D1 f such that

More generally, the p-RDO, Dp, is an operator with unit trace such that

In the context of Second quantization, it is usual to employ RDO’s with somewhat different normalization introduced by Dirac and defined by

and

Thus, and

As far as I am aware, it was Dirac [5] who first made effective use of RDM’s. But he considered only states described by a single Slater determinant formed from N orthonormal

spin-orbitals

in which case

As can be easily verified, this operator is merely the Identity operator on the linear space, spanned by the N functions Dirac showed that all the physical properties of the Slater state, including the p-matrices, can be obtained from a knowledge of or,equivalently, of It is astounding that so much physics, including our understanding of the Periodic Table, has been built with what would seem to be a trivial tool - the identity operator on a linear space of dimension N. As we all know, the physics consists of a skillful choice of the spin-orbitals, or rather of It is precisely the purpose of Hartree-Fock theory to lead us to the “best possible” choice. A large thriving industry and much of the wealth of the pharmaceutical companies is based on the simple equation (7). Such is the power of mathematics! It was Husimi [6] who, apparently, first discussed the more general RDM’s in (1) and (2) .Indeed, he considered p-RDM’s for arbitrary p. When the hamiltonian is represented in the form

as a sum of N one-particle terms and two-particles terms, it is easy to see that, for both fermions and bosons, the exact energy, E, of the state is given by

where the reduced hamiltonian is defined by

24

In my view, these last two formulas are absolutely basic for understanding the quantum mechanics of many-particle systems in which interaction among the particles plays a significant role. From the form of (10) it appears that as N increases the relative importance of interaction becomes increasingly significant! Unfortunately, hitherto little attention has been given to the eigenstates of K and the role of N in determining its eigenvalues. I regard this as a key challenge for any analyst who is interested in making a significant contribution to the N-body problem. - cf. pp. 11 and 257 of CY. I discovered (9) in 1952 while trying to understand Frenkel’s exposition of so-called Second Quantization. Husimi had seen it at least ten years earlier! I immediately applied (9) to calculate the ground-state energy of Li by assuming a simple ansatz A for D2 such that

I did extremely well, indeed too well! The result was about 20% below the observed value! This was impossible and forced me to realize that imposing fermion statistics was more subtle than I had imagined. This led me to invent the concept of N-representability: The 2-matrix of a pure state,

must be representable in the form (2) in order to satisfy

fermion or boson statistics. Analogously for a p-matrix.

So, in 1952 I proudly announced to a group of able physicists at Chalk River that I had reduced the N-body problem to a body problem - we now merely had to solve the N-representability problem, which I assumed would be child’s play, and using (9) find the ground states using Rayleigh-Ritz. After 48 years there is no easy practical way of doing this in general. However, Carmela Valdemoro made a big break-through in 1992 which was quickly followed up Hiroshi Nakatsuji and then by David Mazziotti. They have devised an

effective method of calculating the energy levels, which I have dubbed the VNM method and which has been described [7] as “wave mechanics without wave functions”. For atoms and molecules with as many as 20 electrons, the VNM method competes favourably with FCI calculations of equal accuracy. Since RDM’s are initial values of Green’s Functions, a similar condition must be satisfied by GF’s. This has been generally unnoticed until quite recently and still has not been really absorbed by main-line physicists. However, the book [8] by Parr and Yang about Density Functional Theory (DFT) contains an early Section pointing out that N-representability is a dark cloud hovering over the validity of DFT. The usual methods of dealing with this problem is either not to be aware of it or to hope that it will go away! The latter method is not satisfactory. For small N, my experience with Li shows it is risky; whereas, for large N, the theorem of Hugenholtz [9] that, for an interacting system in the limit as N gets arbitrarily large, a single Slater determinant is orthogonal to the true wavefunction is rather dramatic. Perhaps this was in Froehlich’s mind [2] when he spoke to David Peat! BCS, MATTHIAS AND ALL THAT

I would be the first to admit that the BCS theory has been extraordinarily successful, making a contribution of immense value to Condensed Matter theory. Even so, as with the remarkable success of a single Slater determinant, I have always been amazed how the original BCS simple theory, managed to change and persist so long. In the cold light of current knowledge we now realize that the simple BCS theory had only two essential ingredients (i) The choice of a trial wavefunction formed with the same material as is needed to characterize one antisymmetric 2-particle function, or geminal. 25

(ii) An extremely simple ansatz for the potential as a step function exercising a positive attraction between electrons with energy close to the fermi energy.

Of these, I certainly consider (i) as more important. The BCS wavefunction is a Fock space equivalent of the wavefunction considered by Schafroth [10] and which, by a stroke

of luck, is as Yang proved [3], the type of wavefunction most likely to give rise to a large eigenvalue of the 2-matrix. To my mind this is the explanation of the success of the BCS

model. As for the nature of the force involved, we were told that the positive isotope effect definitely proved that it was phonon-mediated. So in my innocence, as a naive mathematician when a negative isotope effect was observed, I immediately inferred that this proved that the force could not be phonon mediated. But no! Since by this time the idea that the force was phonon-mediated had become firmly implanted in our collective consciousness, it was soon “proved” by an able theoretician that a negative isotope gave us even added evidence of our - by now - blind faith that the force was phonon-mediated! Also the myth was firmly established that “Cooper pairs” consist of two electrons with opposite spins.

Apparently

many phyicists still believe that this is essential to BCS theory. As suggested in Chapter 4 of

CY, this is not necessarily the case. For every new observation that contradicted the currently accepted theory our faith was saved by a small add-on or by a major or minor modification of the current formulation. The evolving BCS theory became more and more complex and subtle. But at that period, during

which I had the rare privilege of meeting and challenging Bernd Matthias every winter at Sanibel until his death, I developed the feeling that BCS theory had become, like Ptolemaic astronomy, a system of epicycles piled on epicycles! Bernd proudly proclaimed that he was anathematized by all theoretical physicists because for every new version of the theory proposed, he would go into the Bell Lab and emerge with a counter-example! Perhaps because he was a polite Swiss being kind to a Canadian or perhaps because he took pity on me as an innocent mathematician wandering among chemists and physicists, he carefully stroked my ego by stating that the ideas re. superconductivity that

I advanced were not contradicted by any known observation. I will pursue this below! However, in private conversation and in his lectures [11] at McGill in 1968, Matthias insisted that the truly interesting theoretical question is why do nearly all substances manifest

a form of Long Range Order (LRO) at sufficiently low temperature. He asserted that even gold would become a superconductor! I very much regret that he died before the discovery of HTSC. He would have so much fun bating theoreticians re. “anyons”, “RVB” and the other exotic ideas that have been bruited! When I asked John Harrison, the former Editor of JLTP and my colleague in Physics at Queen’s, to explain Matthias’s observation, his response was immediate. “It’s really not so surprising. At absolute Zero the entropy will vanish so we should expect total order”. Indeed, this is true and proves that my knowledge of thermodynamics is almost nil or I would have made this point to Matthias. So the interesting question becomes, not why there is LRO, but rather why is the LRO of the nature that actually occurs in a particular substance? I do not pretend to have a detailed answer to this question. I do think that I offer the basic set of the ideas essential for its answer. Another beef that I have with current physics practice is the error which Whitehead [12] calls “The Fallacy of Misplaced Concreteness” exhibited in such terms as p-electrons or Cooper pairs. If you understand the meaning of the word fermion or if you believe in democracy you know that all electrons are equal. They do not live in George Orwell’s Animal Farm

in which “all electrons are equal but some are more equal that others”. Only occasionally, have I noticed momentary indications of a bad conscience by chemists or physicists about this misuse of language. If challenged, as I am doing now, they excuse themselves with the same remark Bourbaki often uses “This is merely an innocent abus de language”. Whereas, I regard it as a noxious avoidance of our proper task of instilling in the minds of students a

26

set of valid concepts with which to explore the inner riches of Quantum Theory. It is not an electron which has p-symmety but a spin-orbital. In fact, it is a partially occupied eigenfunction of the 1-matrix! There are no such things as Cooper pairs, even if we think of them in the charming image, due I understand to Schrieffer, as partners dancing to Rock so that they can be at far ends of the floor yet fully synchronized, rather than breastto-breast in a gentle Strauss waltz. To even propose such an image almost makes the concept absurd. The functional unit is not a pair of electrons it is a spin-geminal. In fact, the key concept which we must learn to deploy is that of a partially occupied eigengeminal of the 2-matrix. CORRELATION MATRIX AND ORDER INDICES

I read somewhere that the nobellist, C.N. Yang, regarded the paper [3], in which he associated the onset of superconductivity with the appearance of a “large” eigenvalue in the 2-

matrix, as the most important paper of his distinguished career. My initial conjecture [13] was the obvious generalization of his observation and asserts that every type of LRO is associated with a large eigenvalue ot the 2-matrix. This was refined [14] by the definition of order indices and the correlation matrix. It is known that the least upper bound for the eigenvalues of the 2-matrix of a boson system is N(N – 1) and for a fermion system [15] the unattainable such bound is N. If we call the occupants of geminals pairons then we can say that the l.u.b. for the pairon occupation of a natural geminal is N(N – 1) for a system in which the constituent identical particles are bosons and N for a system if they are fermions. If a “Cooper pair” is anything it is a “pairon”. But the term pairon is more general and is not necessarily associated with superconductivity. Yang argued that superconductivity in a metal is triggered when

has an eigenvalue

of order N. Bloch [16] connected such an eigenvalue with flux quantization related to carriers with charge 2e, confirming Yang’s theory. From energy considerations sketched below, I inferred that eigenvalues of proportional to N were associated with eigengeminals describing a correlation which extends throughout the substance.- in other words, a Long Range correlation. The order index was then defined [17] as the largest value of such that has a finite non-zero value, in the thermodynamic limit. The correlation matrix is defined as By the above-mentioned [14] result, for systems of fermions For one or more eigenvalues of is proportional to N so LRO is present. For systems of bosons, could be as large as 2. As long as we conjecture that some form of mesoscopic [17] or local order is present. If we compare this with a percolation model for the onset of a new

phase of matter, cluster is infinite,

corresponds to the critical value of p when the diameter of an open close to 0 corresponds to the first moments at which nuclei of the new

phase are present. As

increases these nuclei become more widespread and larger. I am

thinking of the small bubbles becoming more widespread and larger which appear in water as it approaches the boiling point, or the complex systems of “fjords” of superconducting phase penetrating the whole of a cylindrical block of material which I had the privilege of viewing via polarized light as it was cooled by Martin Edwards in his Low Temperatre Lab at the Royal Military College of Canada many years ago. CONJECTURE. For all mono-particle systems, the appropriate order parameter (OP) is the correlation matrix From the structure of the 2-matrix it immediately follows that the order parameter can have spin character s, p or d and any combination of these. This is consistent with recent observations [18] that the order parameters of some HTSC’s exhibit s, p or d spin-symmetry or a combination of these - a phenomenon, which, apparently, BCS has difficulty accommo-

27

dating. My conjecture is that is the appropriate order parameter for all types of order in many-particle systems of one type of identical particles. Thus, I am making a bold generalization of Yang’s observation from superconducting to many other order transitions. I am encouraged by the fact that this conjecture is consistent with observations on the symmetry of the OP for HTSC and also for He3 in which p-type order occurs in at least one phase. I assume that for helical magnetism and many other types of order it will be necessary to study not only spin-symmetry but the total symmetry of the eigengeminals. I have called the above a conjecture rather than a theorem because a “proof” has not been obtained. This is because we do not yet have a sufficient understanding of the relation of the eigenvalues of to the occupation numbers of eigenstates of the reduced hamiltonian K. I regard this as an important urgent issue for theoretical research. Another is to explore the

dependence on the Order Index, of the physical properties of substances near the critical point. Note that if fermi-pairons were bosons, their occupation numbers could go to N(N – 1). This differs from the actual limit of N by a factor of (N – 1) which is infinite in the thermodynamic limit. Thus the universal practice in text-books, and in articles by writers who should know better, of saying that superconductivity arises as a result of a bose condensation of pairs is misleading talk which brings comfort, by creating the illusion that we know what we are talking about, but prevents us from coping with the real task of forging a set of meaningful concepts with which to understand condensed physics.

It is known that when Fock space is displayed with respect to a basis of a finite number, r, of orthonormal orbitals (i.e. 1-particle functions), the highest possible value for the eigenvalues n2i of is This is attained(CY,Chapter 3) only if the wavefunction is an antisymmetrized power of a single geminal - an AGP function and if the eigenvalues n1i of are equal. In this case(CY, p. 137) there is one large eigenvalue and the rest are equal to 2 N ( N – 2 ) / r(r–2). If we relax the condition that n1i be equal, it is possible to arrange that for an AGP function has several eigenvalues which are proportional to N. and thus model a variety of other situations including the co-existence of superconductivity and magnetic ordering. It is perhaps worth recalling here that r = N is a necessary and sufficent condition that

the wave function be a Slater determinant. In this case all the eigenvalues of are equal to 2 which is a long way from N. This corresponds to the fact that HF is accurate if and only if the effective hamiltonian has no 2-particle terms. We introduce some essential notation by recalling that in CY. Denote the eigenfunctions of Dp by

with corresponding eigenvalues

so

Setting

we obtain

and

28

Further, if the reduced hamiltonian (10) has eigengeminals, gi, such that then the total internal energy

where

Suppose that the are so numbered that they increase monotonically with i, and the numbering of n2i so that they decrease. Then in the ground state the system will choose so that pi for small i, and especially for i = 1, are as large as possible consistent with N-representability. The largest occupation of a natural geminal is n21. By the familiar theory of separation of eigenvalues of hermitian operators, 2 (N – 1)

In particular,

2(N – 1) p1 = n21 if and only if the eigenfunction g1 of K coincides with the first natural geminal, of the state. Exact coincidence is highly unlikely, but there will be a strong tendency towards this so it is possible that p1 will be of order N. In this case, we would expect that g1 describes a 2-particle correlation which extends throughout the sample, that is a LRO. For N electrons in a lattice if we neglect spin, we are led to study the hamiltonian

where i and j refer to electrons and k to nuclei; Z k is the charge on an ion at sk. For neutral systems This implies, in the notation of (8), that

Notice that, though we mentioned electrons in a lattice, if k assumes only one value, (23) would describe the hamiltonian of an N-electron atom with nuclear charge Z k = N, whereas, if k takes two values, a diatomic molecule. And so on. In fact, almost anything. By (10), associated with (23) is the Reduced Density Operator, K. However, for reasons which will become apparent, we introduce an additional parameter, t, and define K (t) by

If we divide by N2, set then (25) takes the form

and N2U (t) = K(t), and replace Nr i by ri and Nsk by sk,

29

For a neutral system,

For fixed N the spectrum of U(t) will depend continuously

on t. A famous theorem [19] of Zhislin assures us that when

the operator

(26) has an infinite number of bound states with energy levels crowding up to the limit of the continuous spectrum.

U (0) is a two-electron hamiltonian which approaches the hamiltonian of H – as N increases to infinity. On the other hand, for t = 0, and N = 2, (25) is the hamiltonian of the helium atom. According to (18) and (22) it is the spectrum of K = K(1) which is of real

interest in the study of energy levels of N-particle systems. Since K = N2U (l), it follows that the spectrum of K is obtained from that of U (1) by scaling by the factor N2.

It is known [20] that H– has only one bound state. It is a 1S state slightly below the continuum which accounts for an absorption line in the solar spectrum. The two lowest states of the helium atom are a 1 S and a 3S state. For a fixed system, (25) depends continuously on t so we expect that as t varies from 1 to 0 a correspondence will be established between the spectra of U (1) and U (0). However, while U (0) is an atomic hamiltonian, we shall expect the spectrum of U ( t ) , when t > 0, to be a series of bands, possibly narrow, each of which collapses, when and which could be named by an energy level of the atomic system which U (0) describes. If spin is neglected then it would be reasonable to expect that all levels of the lowest band would be 1 S.

In this case, for a system manifesting long-range order at low temperature, we would anticipate that the correlation matrix will depend on the eigenfunctions corresponding to the levels of the lowest band weighted by a distribution function depending on the inverse temperature, Unfortunately, little study has been made of the spectrum of K for solids or other condensed matter even though the fact that it must play a key role in understanding the energetics of condensed matter has been obvious for forty or fifty years. We noted above that the late Bernd Matthias, who probably discovered more superconductors than any three other experimentalists together, constantly insisted that an important

task for theoretical physics was to explain why nearly all fermion systems manifest longrange order of some type at sufficiently low temperatures - superfluidity, superconductivity, ferro- or antiferro-magnetism, charge density waves, coexistence of superconductivity and helical spin density waves, etc. To properly describe the electrons in condensed matter, our hamiltonian (23) would need to be supplemented by terms describing L·S coupling, spinspin effects, motion of the ions etc. However, the electric forces described by (18) would probably dominate the energy. If in fact the spectrum of K is similar to that of H— in having one eigenvalue, or a band of eigenvalues significantly below all others, then that level would tend to be occupied as fully as possible consistent with the statistics, the inter-particle forces and the temperature. For fermions, n21 could be of order N which, if it occurred, would manifest itself as long-range order. The nature of the particular LRO would be characterized by the correlation matrix. ANTISYMMETRIZED GEMINAL POWER

A theorem attributed [21] to Zumino states that a fermion geminal, in other words, an antisymmetric two-particle function, can be transformed by a unitary transformation into a canonical form in which each orbital is a member of a unique pair of orbitals. The reader should be aware that it is my custom to denote by the word orbital a function of a single particle including all relevant coordinates. Thus, depending on context, the word may denote the classical meaning of a chemist(if the particle is without spin), or what a chemist means by spin-orbital, or a function of spatial co-ordinates and two dichotomic variables for spin and isotopic spin. Thus if

30

is such that the normalized function g (12) = – g (21), with then r = 2s is even, and by a unitary transformation it is possible to find an orthonormal basis αi with respect to which

In (27), r is the rank of g and also the rank of the matrix c ij. It is well-known that the rank of an antisymmetric matrix is even. The antisymmetrized power of an orbital is always zero. Thus f(1)f(2) – f ( 2 ) f ( 1 ) = 0. However, the antisymmetrized power of a geminal, g, to obtain a function of N particles will vanish if and only if the rank, r, of g is less than N. When N = r, this N-particle function is a single Slater determinant formed with a basis of g. Perhaps inadvisedly, I have adopted the symbol gN to denote a normalized N – particle function obtained by antisymmetrizing an appropriate power of g. Several persons, of whom Nakamura [22] may have been the first, showed that the projection of the BCS function(which is a coherent ensemble in Fock space of functions of all possible particle number) onto a subspace of Fock space of particle number N produces an AGP function of rather special type. The importance of the AGP function is signalled by the fact that it has appeared in a variety of contexts with different names such as: Schaftroth condensed pair function; projected BCS function; correlated pair function; pairiing function; Generalized Hartree-Fock function. The mathematical concept goes back to Hermann Grassmann in the 1840’s since it arises naturally in Grassmann algebra. I prefer to use a name which suggests its mathematical nature and does not place it in a misleading context. For applications to physics and chemistry there is no need to insist that the occupants of a geminal are particles with opposite spin. Any kind of fermion geminal forces a natural pairing. Therefore it can be cogently argued that the apparent “pairing” in BCS is not forced by the physics but rather appears as a mathematical artifact forced by the assumption that the wavefunction is AGP. I realize that there is such a widespread commitment to the religious

belief that Cooper pairs are “real” that there is a high probability that I shall be accused of blasphemy, tried, condemned and burned at the stake!! Since the whole of Chapter 4 of CY is devoted to AGP, here I shall restrict myself to quickly mentioning what every young person should know about Grassmann algebra and fermions. 1) If is an N-particle fermion function and is an orbital such that the Grassmann product then there is an (N – l)-particle function, such that Further, these equivalent conditions are necessary and sufficient that be a natural orbital of with occupation unity.

2) Suppose that an AGP function formed from an arbitrary geminal, g, then if r is the rank of a) r < N implies that b) r = N implies that is a Slater determinant, c) r > N implies that can be expressed as a linear combination of ( ) Slater determinants consisting only of paired orbitals, where r = 2s, and N = 2m. 3) If N is even and the natural orbitals of

are evenly degenerate, with occupation

strictly less than unity, then there exists a geminal g such that This remarkable result, proved around 1965 by Erdahl, Kummer and myself, implies that for a manyparticle fermion state satisfying these conditions, all one-particle properties can be exactly described by an AGP function.

4) Further, suppose that N = p + q where q is even and precisely p natural orbitals have

occupation unity, then where S is a Slater determinant containing the indicated p natural orbitals and g is a geminal. Such functions have been called Generalized AGP functions.

31

5) On p. 139 of CY we indicate that an AGP function might have the possibility of having a 2-matrix with a finite number of “large” eigenvalues. It is therefore conceivable that observed co-existence of superconductivity and helical magnetism could be modelled by GAGP. 6) If N is even and if is of rank N + 2, then by making use of the so-called Hodge Correspondence between subspaces of dimension 2 and those of dimension N in a space of dimension N + 2, we can prove that is an AGP function. As the “cranking model”, AGP proved useful in nuclear theory. At first a theory with the fanciful sobriquet “superconducting nuclei” was introduced using the BCS coherent ensemble

equation subject to a condition that the expected value of the number operator be N. However, Nogami and others soon noticed that it was more accurate to use a projected BCS function, that is an AGP function. Chemists also found that AGP, as an ansatz for the wave-function, was more successful in modelling the dissociation of diatomic molecules than Haertree-Fock. It was observed that the Random Phase Approximation is self-contradictory i f , as is common, a single Slater is taken as the initial ground state, whereas the most obvious contradictions are avoided if the ground state is assumed to be a Generalized AGP.(For this, cf. p. 140 of CY). In view of these properties and the fact that GAGP can be a single Slater modeling a fermion system with no correlation or, on the other hand, a system with a with the largest possible eigenvalue and therefore modelling the highest possible correlation, it is apparent that the GAGP ansatz is of great scope and could be used to provide insight into a wide variety of fermion systems.

GRASSMANN AND THE FERMI SURFACE I come now to a little-known theorem of Grassmann for which I will present a proof, partly because I do not want to disappoint your expectation that proving theorems is my main

purpose in life, as a mathematician, but also because the result is unexpected and may be the “real” reason why we must replace “fermions” by “fermi pairons” in our thinking and

therefore the ultimate reason that “Cooper pairs” proved so serviceable. I announced this result [23] without giving the proof in 1961. In fact the proof was rather easy making use of the Hodge correspondence between sub-spaces of dimension p and those of dimension n – p in a linear space of dimension n. With a more complicated proof, the same result was proved later in the RMP by a theoretical physicist, but I have lost the reference. In fact, Whitehead [24] provides an almost trivial proof, attributing it to Grassmann - presumably from the 1840’s! Here in two equivalent forms, first Grassmann’s and secondly mine, is the

Theorem (i) A homogeneous element of order n in a Grassmann algebra of rank n + 1, is elementary. (ii) If is a pure N-particle state, then the rank of is not N + 1. Proof. I shall state the argument in the language of antisymmetric wavefunctions. Suppose the basis has N + 1 orbitals. Associated with a Slater determinant of order N is a unique subspace of dimension N spanned by the N vectors of the determinant or by any N linearly independent vectors in the same subspace. Changing these vectors does not change the subspace but may multiply the Slater by a constant. Suppose that is a linear combination of two Slaters. By a basic theory about subspaces, the dimension of the intersection of the subspaces associated to the two Slaters is N + N – (N + l ) = N – l. This intersection is common to both subspaces and is characterized by a Slater S. of order N – 1. Adjoin vectors and 32

to the intersection so that S and are N-th order Slaters respectively characterizing the two subspaces Then there are constants a and b such that

which is a single Slater of rank N. By induction we see that any linear combination of Slaters

of order N, in a space spanned by N + 1 orbitals, is again a Slater(i.e , in the language of Grassmann algebra, “elementary”) of N orbitals. It follows easily that version (ii) is implied by version (i) of the statement of the theorem. Hence if is not a Slater it must have rank at least N + 2. But it could have that rank as follows from Item 6 of the previous Section. The discussion of the energy of an AGP state in Section 4.6 of CY was used to estimate the change in energy if a Slater state of rank N is changed to a state of rank N + 2 by replacing two orbitals each of occupancy 1 by four orbitals each with occupancy 1/2. It was found (CY, p.155) that the change in energy of the state was

where κ and name distinct “pairs” of orbitals. The number denotes the interaction energy where and and has fixed phase. Whereas, denotes the interaction energy and has adjustable phase which was used to arrange the negative contribution in (29). Thus the Fermi surface is unstable with respect to pair formation unless is positive and numerically greater than If the so-called “pairing hamiltonian” is used, automatically so that for the pairing hamiltonian, the Fermi surface is always unstable. This seems to contradict the commonly expressed view that the

existence of superconductivity requires an attractive force which was part of the rational for the existence of Cooper pairs.

It is now widely recognized that HTSC is usually associated with phase separation. In the next section we find that a sufficiently strong repulsive Coulomb force is required to account for phase separation. PHASE SEPARATION AND SUPERCONDUCTIVITY

In his well-known survey [25], published in 1989, of the properties of HTSC, Phillips has 11 references to “Lattice instabilities”, a topic to which he devotes several pages at various points in the book. He even went so far as to suggest that lattice instability is the only factor that causes HTSC. I do not admit this since in 1991 Yukalov [26] surveyed 508 experimental and theoretical papers which dealt with evidence bearing on the incidence of phase transitions of what he called “heterophase fluctuations”. Yukalov, who is a Senior theoretical physicst in the Joint Institute for Nuclear Research in Dubna, is a remarkably competent and careful authority on Quantum Statistics . He agrees that instability of the lattice can be important but there are other significant factors. During the past ten years more evidence [26] has accumulated similar to the impressive collection which he assembled. In his basic paper referenced above, are foreshadowed many ideas that have recently become current. Chapters 5 and 6 of our book were laregly due to Yukalov since I know so little of the nitty-gritty of physics. In particular this is the case for Section 6.2 of CY in which we attempt to work out in a form relevant to HTSC the theory developed in his paper [26] when there are only two phases interpenetrating. Here I merely sketch the course of our argument directing the interested reader to Section 6.2 of CY and the references in Notes 2 and 3. We posit a situation which can be thought of as microsopic or mesoscopic nuclei of one phase (e.g. “superconducting”) scattered randomly through a host phase (e.g. “normal”) Experimental observation of this possibility was recently provided by a group from Dubna 33

with associates in a paper [28] entitled “Microscopic phase separation in

induced

by the superconducting transition”. We propose a simple model which takes into account the three interrelated factors: Coulomb interaction, phase separation, and lattice softening. We give a detailed analysis of the dependence of the critical temperature on parameters related to the attractive and repulsive interactions and to the superconducting phase fraction w.

Since the Hartree-Fock-Bogolubov - essentially AGP - approximation is used, our formulas look very much like those in usual presentations of BCS theory.. However, they have quite different meaning because they involve the parameter w in an intricate manner.We assume that the interaction between electrons is the sum of two components

- a direct part which is taken as a Debye-type shielded Coulomb force. and - an indirect part for which we assume the conventional Froehlich phonon term. Additional parameters are introduced by which it is possible to vary (i) the phonon

frequency, (ii) electron-phonon coupling, and (iii) the strength of the direct interaction. The resulting equations were solved numerically by Dr. E. Yukalova. The results are exhibited in CY as twelve graphs portraying the superconducting critical temperature against w, in three

groups corresponding to weak, moderate and strong softening of the lattice. We were pleasantly surprised by the wide variety of shapes of these graphs. Despite the rough approximations assumed for our model, the behaviour of the critical temperature in Figs. 3,4,7 and 8, has striking similarity to some corresponding experimental curves observed for HTSC. Even though one might expect the relation between doping intensity and w to be monotone, the actual relation is not known so a detailed comparison of our results with those of experiment is not possible. We were led to the following conclusions: - The presence of repulsive interaction is a necessary condition for mesoscopic phase separation. - Phase separation favours superconductiviy making it possible in certain het-

erophase samples when it would not occur in a pure sample. - The critical temperature as a function of the relative fraction of superconductive phase can exhibit the nonmonotonic behaviour characteristic of HTSC. FINAL REMARKS

1) I conclude that we need to develop a habit of thinking more comfortably about the second order reduced density matrix its eigenvalues and its eigengeminals. 2) For a system of a large number of identical particles which is all that I discussed,

the “large” component of the 2-RMD, denoted by parameter. If

is proposed as the appropriate order

there is no “order” present so this could correspond to what we usually

call “normal”. For fermion systems Long Range Order corresponds to

and for bosons,

to I have not expatiated on my conviction that neither bose nor fermi condensation, as normally understood, in the simple-minded sense derived from London(whose memory I honour!), actually occur in an interacting system and that we poison the innocent minds of our students if we persist in suggesting that they do. 3) More imporrtant, it is my view that because for the earliest discovered superconductors, Tc was so low and the isotope effect had a simple explanation, we were misled into thinking that the origin of superconductivity is exotic and/or subtle. However, I take seriously Matthias’s observation that LRO at sufficiently low temperature is universal and therefore should have a robust explanation.. Further the behaviour of various substances near the 34

critical temperature seems to have much in common. When charged particles are involved, I conclude that Coulomb forces are the real culprit. So I claim that the secret for this universal phenomenon is to be found in the second order reduced hamiltonian. The isotope effect implies that interaction with the lattice must play a role. My musings at the end of Section 3, suggest that contributions by the static coulomb interaction, specified by Zhislin’s theorem, involve quite minute energy differences for K jbetween the continuum level and a narrow energy band which is almost at the continuum limit. This means that lattice dynamics, L. S coupling and other spin-effects could also play a significant role accounting for the known variation of Tc across the Periodic Table which was noted by Matthias [11] in his McGill lectures. 4) I fully realize that some of my heterodox opinions are anathema to many. I shall try to face this with the equanimity of old age, welcoming all comments, questions and counterexamples at my email address on the title-page.

NOTES 1. 2.

Charles Coulson, in conversation with graduate students and Coleman in Oxford June 1975. Also in a speech in Boulder, Colorado: Rev. Mod. Phys. 32, 175 (1960). H. Froehlich, shortly before his death, in private conversation with David Peat.

3.

C.N. Yang, Rev. Mod. Phys. 34, 694 (1962).

4. 5. 6.

By A.J. Coleman and V.I. Yukalov, Vol 72 in Series published by Springer in Lecture Notes in Chemistry, April, 2000. P.A.M. Dirac, Proc. Cam. Ph. Soc. 26, 376 (1930); 27, 240 (1931). K. Husimi, Proc. Phys. Math. Soc. Japan 22, 264 (1940).

7.

Section 7.3 of CY.

8.

9.

R.G. Parr and W. Yang, Density Theory of Atoms and Molecules, Oxford University Press, 1980.

N.M. Hugenholtz, Physica 23, 481 (1957); L. van Hove, Physica 25, 849 (1958).

10.

M.R. Schafroth, Phys. Rev. 96, 1149, 1442 (1954); 100, 463, 502 (1955); 111, 72 (1958).

11.

B. Matthias, Three Lectures, in Superconductivity, Proc. Ad. Summer Study Institute, June, 1968, at

12.

McGill University, ed. P.R. Wallace, Gordon and Breach, New York. A.N. Whitehead, p. 64 Science and the Modern World, Cambridge U.P., 1933; p.11 Process and Reality,

13. 14. 15.

Macmillan Comp.,1929. A.J. Coleman, Can. J. Phys. 42, 226 (1964). A.J. Coleman, V.I.Yukalov, Nuovo Cimento B 108, 1377 (1993). A.J. Coleman, Rev. Mod. Phys. 35, 668 (1963).

16.

F. Bloch, Phys. Rev. A 137, 787 (1962).

17. 18.

A.J. Coleman, Jl. Low Temp. Phys. 74, 1 (1989). H. Srikanth et al., Phys. Rev. B 55, R14 733 (1997); K.A. Kouznetsov et al., Phys. Rev. Lett. 79, 3050 (1997); in Physica C 317-318, 410 (1999), van Hartington claims “unambiguous determination” of d-wave symmetry in HTSC cuprates; in Nature 396, 658 (1998), Ikeda et al. observe p-wave symmetry in a second HTSC. G.M. Zhislin, Trudi Mosk.Mat. Obsc. 9, 81 (1960), Th.III, p.84. R.N. Hill, J. Math. Phys. 18, 2316(1977).

19. 20.

21.

B. Zumino, J. Math. Phys. 3, 1055 (1963); see also Thm.6, Coleman, Bull. Can. Math. Soc. 4, 209 (1961).

22. 23. 24. 25.

K. Nakamura, Progr. Theor. Phys. (Kyoto) 21, 273 (1959). A.J. Coleman, Can. Math. Bull. 4, 209 (1961), Thm.7. A.N. Whitehead, Universal Algebra, Cambridge University Press, 1898. J.C. Phillips, Physics of High-Tc Superconductors, Academic Press, 1989.

26.

V.I. Yukalov, Phase Transitions and Hetrophase Fluctuations, Physics Reports 206, 395–488(1991).

27. 28.

Phys. Rev. B 54, 9054 (1996); Phys. Rev. Lett. 76, 439 (1996). V.Yu. Pomjakushin et al., Phys. Rev. B 58, 12 350 (1998).

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THE SIXTEEN-PERCENT SOLUTION: CRITICAL VOLUME FRACTION FOR PERCOLATION

RICHARD ZALLEN Department of Physics, Virginia Tech Blacksburg, VA 24061

INTRODUCTION The English call it “value for money” (vfm). The American equivalent is “bang for the buck”. The idea is simple: to provide a rough measure of the ratio of benefit to cost. For an author of scientific papers, one possibility for a vfm-type measure of “benefit” (impact) to “cost” (time and effort) is this: vfm = (number of citations)/(paper’s length in printed pages). In my case, the vfm winner is clear. It is a two-page paper by Harvey Scher and myself, published quietly as a note in J. Chem. Phys. [1], which has been cited over 350 times. Later work related to the central idea of that paper has also been widely cited [2, 3]. That idea is the concept of a critical volume fraction for site-percolation processes. NOSTALGIA One afternoon in mid-May of 1970, at my desk in the research building of the Xerox complex near Rochester, NY, I was poring over experimental Raman spectra, searching for significant peaks with my “spectroscopist’s eye” [4]. I was not having much luck, and I needed a break. So I left my office, walked down the hall, and went into the office of a colleague, Harvey Scher. Harvey was, as usual, good-natured and patient about the interruption of his own work, and he took the opportunity to describe an interesting problem that he was working on. A very approachable resident theorist, Harvey had been consulted by a technology group working on photosensitive layers in which photoconductor particles were dispersed in a resin. Their measurements had shown a dramatic threshold in the dependence of photosensitivity on photoconductor concentration. Elliott Montroll, then a frequent visitor to Xerox, had suggested to Harvey that he look at the literature on percolation theory. Harvey had assimilated that literature and made use of it, and he introduced me to percolation theory that afternoon. I was fascinated by this stuff,

Phase Transitions And Self-Organization in Electronic and Molecular Networks Edited by J. C. Phillips and M. F. Thorpe, Kluwer Academic/Plenum Publishers, 2001

37

and when I got back to my office I did not return to the strip-chart recordings (no PC’s in 1970). Instead, I worked on some geometry problems related to ideas that we had kicked around, and I became enthusiastic about writing up a short paper reporting what we had found. Ten days later the paper was circulated internally within Xerox, and it was submitted for publication in mid-June. This speed was then, and is now, uncharacteristic of both authors. The reason for the choice of J. Chem. Phys. is somewhat obscure. We did not want to send it to a math or math-phys journal, and we had seen a short paper in J. Chem. Phys. that mentioned percolation. It turns out that the referee for our paper was almost certainly a mathematician! He (or she) chided us for the empirical and approximate nature of our

critical density. (We knew it was approximate, and we were proud of “empirical”!) But he (or she) nevertheless pointed out to us an additional result for (an exact value for the two-dimensional Kagomé lattice) which fit our ideas very well. We added it to the Table in

our paper. The anecdote described above, dealing with the fruitfulness of an afternoon schmooze session at the Xerox lab in Webster, NY, was characteristic of a period now remembered by some as a “golden age” of industrial research [5]. The scientific issue arose in the context of a technological setting, which is of course a familiar tradition in condensedmatter physics [6]. The atmosphere was one in which it was OK to spend time on scientific

issues as well as on product-development and engineering ones. In the year 2000, that era is history and opportunities to do science are rare in present-day corporations. Globalization is sometimes given as the reason (or excuse) for this, but human herd-instinct considerations also enter: Everybody did it then (corporations supported research) because everybody else did it; nobody does it now because nobody else does it. This is a

cooperative phenomenon, so perhaps we can hope that a phase transition can happen again. CRITICAL VOLUME FRACTION In three dimensions, the percolation threshold

for site-percolation processes varies

from lattice to lattice by more than a factor of two [7]. For two-dimensional lattices,

varies by more than a factor of 1.5 [7, 8]. The Scher-Zallen construction for the critical volume fraction associates with each site a sphere (or circle, in 2d) of diameter equal to the nearest-neighbor separation. Spheres surrounding filled sites are taken to be filled. At the critical value of the site-occupation probability p , the fraction of space occupied by the filled spheres is taken to be the critical volume fraction The key point is this: From lattice to lattice (in a given dimensionality), is nearly constant, varying by just a

few percent. It is an approximate dimensional invariant. In three dimensions, 0.16; in two dimensions, is close to 0.45 [1, 3]. The relationship between and is

is close to

where f is the filling factor of the

lattice when viewed as a sphere packing. The values forming the basis of the 1970 paper correspond to familiar crystal structures. A structure that is not crystalline but is experimentally well defined is random close packing (rcp). The rcp structure corresponds to the atomic-scale structure of simple amorphous metals [3]. Since f is known for the rcp structure, predicts the value for this structure. Experiments carried out to determine the conductivity threshold (insulator-to-metal transition) of rcp mixtures of insulating and metallic spheres are in good agreement with this prediction [9, 10, 11]. One way to view is as an expression connecting the ease-of-percolation with the connectivity of the underlying structure. For bond percolation, such an

38

(approximate) connection had been found earlier, in 1960 [12, 13]. A reasonable measure

for the ease-of-percolation for a given structure is (1/ p c) , the reciprocal of the percolation threshold. For bond percolation,

(1/p c) is very close to (2/3)z in three dimensions and

(l/2)z in two dimensions. Here z is the average coordination number of the lattice. The proportionality between ease-of-percolation and coordination number shows that, for bondpercolation processes, the coordination number is the appropriate measure of the connectivity of the lattice. This, of course, makes sense. But for site-percolation processes, z does not work. Instead, shows that (1/p c) is proportional to f. This reveals that, for site-percolation processes, the sphere-packing filling factor is the appropriate measure of the connectivity of the underlying structure. This insight is a byproduct of the work on the critical volume fraction.

LIMITATIONS

Thanks to the piece of information provided by the unknown referee, we knew immediately that is only approximately invariant. The site-percolation threshold is known exactly for two two-dimensional lattices, the triangular lattice (2d close packing) and the Kagomé lattice, so that

is exactly determined for each. The two values differ by

2%. A few people in the critical-phenomena community took an instant dislike to It wasn’t exact. It wasn’t rigorous. It wasn’t even an exponent, so why care about it? [One can imagine one of them having the following reaction to the experimental discovery of a new superconductor: “So Tc is 450 K, so what? What are the exponents?” But maybe that’s unfair.] The value of

can be estimated from a plot of (1/p c) versus f [3]; the slope is

Here a question arises at the low end of the plot, where the proportionality

between the ease-of-percolation and the filling factor has to eventually fail because (1/ p c) cannot be less than 1. This consideration is unimportant in three dimensions in which (1 / pc ) does not closely approach unity; the values cluster in the region from about 2.3 to 5.0. In two dimensions, typical (1/p c) values are closer to 1.0, lying between 1.4 and 2.0. Within this region, the proportionality of (1/p c) to f holds very well [1]. However, Suding and Ziff [8] have recently considered very-low-connectivity two-dimensional lattices with (1/p c) values down to 1.24. Their results show that at these very low connectivities, the deviation from becomes appreciable. Suding and Ziff offer a revised, nonlinear relation between pc and f that improves the fit in the very-lowconnectivity region. Most structures of physical interest are far from this region. APPLICATIONS The notion of a critical volume fraction insensitive to the details of local structure, as

suggested in the 1970 paper, is an attractive one. But it is heuristic, empirical, approximate. It had been my original plan for this paper to review its success (or failure) in relation to experimental literature on metal/insulator composites. This has turned out to be too

mammoth an undertaking for the presently available space and time, and will have to be deferred. The experimental literature is vast; one extensive compilation can be found in a 1993 article by Ce-Wen Nan [14]. The experimental studies span an enormous variety of systems and differ greatly in depth and quality.

39

Figure 1. The conductivity threshold in graphite/boron-nitride composites [19].

At a later time I may attempt a plot of frequency-of-occurrence versus value, but here only some less-than-satisfactory observations will be offered. For three-dimensional

composites a value close to 0.16 is very often encountered, and it is interesting that this occurs for some of the most carefully studied systems. Examples are the carbonblack/polymer composites studied by Heaney and co-workers [15, 16, 17] and the graphite/boron-nitride composites studied by Wu and McLachlan [18, 19]. Figure 1 displays the very clean experimental results of Wu and McLachlan, showing a conductivity threshold spanning many orders of magnitude. Graphite and boron nitride are structural

and mechanical isomorphs, but differ in conductivity by a factor of 1018. The points are

measured values; the curves are scaling-law fits that closely determine

(0.15 for this

system). But there are many systems for which is quite different from 0.16; this value is not universal. The reason is unclear, though different classes of topology have been suggested. One of these is the “Swiss-cheese” void-percolation topology analyzed by Halperin and coworkers [20] and studied experimentally by Lee et al. [11] ACKNOWLEDGMENTS

I wish to thank Wantana Songprakob for crucial help in preparing this paper. I also wish to thank Harvey Scher for thirty years of friendly interaction.

40

REFERENCES 1. 2.

3. 4.

Scher, H. and Zallen, R. (1970) Critical density in percolation processes, J. Chem. Phys. 53, 3759. Zallen, R. and Scher, H. (1971) Percolation on a continuum and the localization-delocalization transition in amorphous semiconductors, Phys. Rev. B. 4, 4471. Zallen, R. (1998) The Physics of Amorphous Solids, John Wiley and Sons, New York. pp. 183-191. I first heard this apt term mentioned in a talk given by Manuel Cardona.

5.

In the seventies, the Xerox lab in Palo Alto was the site of some now-famous computer-science

6.

Harvey Scher, now at the Weizmann Institute, has commented on technology as a rich source of

examples: Hiltzik, M.A. (1999) Dealers of Lighting, Harper, New York.

7. 8.

9. 10.

scientific questions in his recent Festschrift article: Scher, H. (2000) Reminiscences, J. Phys. Chem. B 104, 3768. Reference [3], p. 170. Suding, P.N. and Ziff, R.M. (1999) Site percolation thresholds for Archimedean lattices, Phys. Rev. E 60, 275. Fitzpatrick, J.P., Malt, R.B., and Spaepen, F. (1974) Percolation theory and the conductivity of random close packed mixtures of hard spheres, Physics Letters 47A, 207. Ottavi, H., Clerc, J.P., Giraud, G., Roussenq, J., Guyon, E., and Mitescu, C.D. (1978) Electrical conductivity of conducting and insulating spheres: an application of some percolation concepts, J. Phys. C: Solid State Phys. 11, 1311.

11. 12.

13. 14. 15. 16.

17. 18.

19. 20.

Lee, S.I., Song, Y., Noh, T.W., Chen, X.D., and Gaines, J.R. (1986) Experimental observation of nonuniversal behavior of the conductivity exponent for three-dimensional continuum percolation systems, Phys. Rev. B 34, 6719. Domb, C. and Sykes, M.F. (1960) Cluster size in random mixtures and percolation processes, Phys. Rev. 122, 170. Shklovskii, B.I. and Efros, A.L. (1984) Electrical Properties of Doped Semiconductors, SpringerVerlag, Berlin, p. 106. Nan, C.W. (1993) Physics of inhomogeneous inorganic materials, Prog. Mater. Sci. 37, 1. Viswanathan, R. and Heaney, M.B. (1995) Direct imaging of the percolation network in a threedimensional disordered conductor-insulator composite, Phys. Rev. Letters 75, 4433. Heaney, M.B. (1995) Measurement and interpretation of nonuniversal critical exponents in disordered conductor/insulator composites, Phys. Rev. B 52, 12477.

Heaney, M.B. (1997) Electrical transport measurements of a carbon-black/polymer composite, Physica A 241, 296. Wu, J., and McLachlan, D.S. (1997) Percolation exponents and thresholds obtained from the nearly ideal continuum percolation system graphite/boron-nitride, Phys. Rev. B 56, 1236. Wu, J., and McLachlan, D.S. (1997) Percolation exponents and thresholds in two nearly ideal anisotropic continuum systems, Physica A 241, 360. Halperin, B.I., Feng, S., and Sen, P.N. (1985) Differences between lattice and continuum percolation transport exponents, Phys. Rev. Letters 54, 2391.

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THE INTERMEDIATE PHASE AND SELF-ORGANIZATION IN NETWORK GLASSES

M.F. THORPE and M.V.CHUBYNSKY Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824

INTRODUCTION The study of the structure of covalent glasses has progressed steadily since the initial work of Zachariasen [1] in 1932 that introduced the idea of the Continuous Random Network (CRN). Zachariasen envisaged such networks maintaining local chemical order,

but by incorporating small structural distortions, having a topology that is non-crystalline. This seminal idea has met some opposition over the years from proponents of various microcrystalline models, but today is widely accepted, mainly as a result of careful diffraction experiments from which the radial distribution function can be determined. The CRN has been established as the basis for most modem discussions of covalent glasses, and this has occurred because of the interplay between diffraction experiments and model building. The early model building involved building networks with ~ 500 atoms from a seed with free boundaries in a roughly spherical shape [2]. Subsequent efforts have refined this approach and made it less subjective by using a computer to make the decisions and incorporating periodic boundary conditions. The best of these approaches was introduced by Wooten, Winer and Weaire [3] and consists of restructuring a crystalline lattice with a designated large unit supercell, until the supercell becomes amorphous. The large supercell contains typically ~5000 atoms. Both the hand built models and the Wooten, Winer and Weaire model are relaxed during the building process using a potential. The final structure is rather insensitive to the exact form of the potential and a Kirkwood [4] or Keating [5] potential is typically used. Despite this success in understanding the structure, some concerns remain. Perhaps the most serious of these is that the network cannot be truly random. Even though bulk glasses form at high temperatures where entropic effects are dominant, it is clearly not correct to completely ignore energy considerations that can favor particular local structural arrangements over others. A simple example of this is local chemical ordering, where, for example, bonding between certain same-type atoms is unfavorable. This can lead to chemical thresholds that appear at certain concentrations, at which unfavorable bonding can no longer be avoided. A more interesting and subtle effect of interest to us here is how the structure itself can incorporate non-random features in order to minimize the free

Phase Transitions And Self-Organization in Electronic and Molecular Networks Edited by J. C. Phillips and M. F. Thorpe, Kluwer Academic/Plenum Publishers, 2001

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energy at the temperature of formation. Such subtle structural correlations, which we refer to as self-organization, will almost certainly not show up in diffraction experiments, but may have other manifestations, as discussed in the paper of P. Boolchand in this volume. Here we focus on the mechanical properties and critical mechanical thresholds, as this is where it is easiest to make theoretical progress at this time. How can such an idea be developed theoretically? A proper procedure might be to consider a very large supercell and use a first principles quantum approach, like that of Car and Parrinello [6], to form the glass. The problem with this is that the relaxation times at the appropriate temperatures are very large, so full equilibration is impossible. The structure thus obtained would be unreasonably strained. This situation is made worse as only small supercells with about 100 atoms can be used at present and in these the periodic boundary conditions produce unacceptably large internal strains. Using the fastest linearscaling electronic structure methods or even molecular dynamics with empirical potentials is still much too slow. We therefore need to look at other ways of generating selforganizing networks. One promising approach is that of Mousseau and Barkema [7] who explore the energy landscape of a glass by moving over saddle points. In network terms, this corresponds to selective (thus non-random) bond switching. In these lecture notes, we look at even more simplified approaches that show what kinds of effects self-organization, and the resulting non-randomness, can lead to. The layout of this paper is as follows. In the next section we review ideas of rigidity percolation that lead to a mechanical threshold in networks as the number of bonds per atom in them (related to the mean coordination number) changes. Then we describe our

model of self-organization of glassy networks, which has two thresholds instead of one and

thus exhibits an intermediate phase, and study the properties of this model. We also consider a similar model for random resistor networks, based on the analogy between the usual (connectivity) percolation and rigidity percolation. Throughout much of this paper we focus on both central force networks in two dimensions and bond-bending networks in three dimensions that have central and noncentral forces as these are more relevant to glasses. The reader should be aware that we do flip back and forth between these model systems, as appropriate, in order to illustrate various points.

RIGIDITY PERCOLATION Going back more than a century, Maxwell was intrigued with the conditions under which mechanical structures made out of struts, joined together at their ends, would be stable (or unstable) [8]. To determine the stability, without doing any detailed calculations (that would have been impossible then except for the simplest structures), Maxwell used the approximate method of constraint counting. The idea of a constraint in a mechanical system goes back to Lagrange [9] who used the concept of holonomic constraints to reduce the effective dimensionality of the space.

The problem under consideration is a static one – given a mechanical system, how many independent deformations are possible without any cost in energy? These are the zero frequency modes, which we prefer to refer to as floppy modes because in any real system there will usually be some weak restoring force associated with the motion. Sometimes it is convenient to look at the system as a dynamical one, and assign potentials or spring constants to deformations involving the various struts (bonds) and

angles. It does not matter whether these potentials are harmonic or not, as the displacements are virtual. However it is convenient to use harmonic potentials so that the system is linear. It is then possible to set up a Lagrangian for the system and hence define a dynamical matrix, which is a real symmetric matrix having real eigenvalues. These eigenvalues are either positive or zero. The number of finite (non-zero) eigenvalues defines 44

the rank of the matrix. Thus our counting problem is rigorously reduced to finding the rank of the dynamical matrix. The rank of a matrix is also the number of linearly independent rows or columns in the matrix. Neither of these definitions is of much practical help, and a

numerical determination of the rank of a large matrix is difficult and of course requires a particular realization of the network to be constructed in the computer. Nevertheless the rank is a useful notion as it defines the mathematical framework within which the problem is well posed.

The rigidity of a network glass is related to how amenable the glass is to continuous deformations that require very little cost in energy. A small energy cost will arise from weak forces, which are always present in addition to the hard covalent forces that involve bond lengths and bond angles. These small energies can be ignored because the degree to which the network deforms is well quantified by just the number of floppy modes [10] within the system. This picture of floppy and rigid regions within the network has led to the idea of rigidity percolation [11,12]. When new constraints are added to an initially floppy network and it crosses the rigidity percolation threshold, a single rigid region percolates through the network and it becomes stable against external straining (elastic moduli become non-zero). There are two important differences between rigidity and connectivity percolation. The first difference is that rigidity percolation is a vector (not a scalar) problem, and secondly, there is an inherent long-range aspect to rigidity percolation. These differences make the rigidity problem become successively more difficult as the dimensionality of the network increases. In two dimensions, Figure 1(a) shows four distinct rigid clusters consisting of two rigid bodies attached together by two rods connecting at pivot joints. Now the placement of one additional rod, as shown in Figure l(b), locks the previous four clusters into a single rigid cluster. This non-local character allows a single rod (or bond) on one end of the network to affect the rigidity all across the network from one side to the other.

Using concepts from graph theory, we have set up generic networks where the connectivity or topology is uniquely defined but the bond lengths and bond angles are

arbitrary. A generic network does not contain any geometric singularities [13], which occur when certain geometries lead to null projections of reaction forces. Null projections are caused by special symmetries, such as, the presence of parallel bonds or connected collinear bonds. Rather than these atypical cases, their generic counterparts as shown in Figures 1(b) and (c) will be present. This ensures that all infinitesimal floppy motions carry over to finite motions [13-15].

Figure 1. The shaded regions represent 2D rigid bodies. The (closed, open) circles denote pivot-joints that are members of (one, more than one) rigid body. (a) A floppy piece of network with four distinct rigid clusters. (b) Three generic cross links between two rigid bodies make the whole structure rigid. If the bonds were parallel, the structure would not be rigid to shear. (c) A set of three non-collinear connected rods connecting across a rigid body is generic and contains one internal floppy mode. If they were collinear (along

the dashed line), then there would be two infinitesimal (not finite) floppy motions, and under a horizontal compression buckling would occur.

By considering generic networks, the problematic geometric singularities are completely eliminated. Therefore, the problem of rigidity percolation on generic networks leads to many conceptual advantages because all geometrical properties are robust.

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Moreover, real glass networks have local distortions, and are modeled better by generic networks.

Constraint Counting The genius of Maxwell [8] was to devise the simple constraint counting method that allows us to estimate the rank of the dynamical matrix and hence the number of floppy modes.

The number of floppy modes in d dimensions is given by the total number of degrees of freedom for N sites (equal to dN ) minus the number of independent constraints. A dependent (redundant) constraint does not change the number of floppy modes. It can only add additional reinforcement and it cannot be accommodated without changing the natural bond lengths and angles of the network, so stressed (over-constrained) regions would be created. A key quantity is the number of floppy modes, F , in the network, or normalized per degree of freedom, f = F/dN. By defining the total number of constraints per degree of freedom as nc and the number of redundant constraints per degree of freedom as nr , we can write quite generally,

It is straightforward to find the total number of constraints (and consequently nc) for each

given network. Neglecting redundant constraints [n r in Eq. (1)] as first done by Maxwell [8], we come to Maxwell counting:

Now the idea is to associate the rigidity percolation transition with the point where fM goes to zero. The Maxwell approximation gives a good account of the location of the phase transition and the number of floppy modes, but it ultimately fails, because some constraints are redundant and also because, as we will see soon, there are still some floppy pockets inside an overall rigid network. We now describe Maxwell counting for specific cases. Central Force Network in Two Dimensions. The elastic properties of random networks of Hooke springs have been studied over the past 15 years [11,16-20]. This system can be viewed as a network of Hooke springs in 2 dimensions, which is built from a regular (say, triangular) lattice, whose bonds are represented by springs, by removing the bonds at random, so each one is present with probability p (bond dilution). The site diluted version of the problem was also considered (see, e.g., [21]). For constraint counting it is convenient to introduce the mean coordination as an average number of bonds stemming from a site. It is given by where p is the probability of the bond being present, z is the coordination of the underlying regular lattice (6 for the triangular lattice, for example). If the total number of sites is N , the number of bonds is Each of these bonds represents one constraint, as always in central force networks, and therefore the number of constraints per degree of freedom is given by

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Therefore, according to Eq. (2), Maxwell counting gives

This quantity goes to zero at

which we associate with the rigidity percolation transition. For the triangular lattice this corresponds to pc =2/3. Bond-Bending Glassy Networks in Three Dimensions. We start by examining a large covalent network that contains no dangling bonds or singly coordinated atoms. We can describe such a network by the chemical formula GexAsySe1–x–y , where the chemical element, Ge, stands for any fourfold bonded atom, As for any threefold bonded atom and Se for any twofold bonded atom. Each atom has its full complement of nearest neighbors and we consider the system in the thermodynamic limit, where the number of atoms There are no surfaces or voids and the chemical distribution of the elements is not

relevant, except that we assume there are no isolated pieces, like a ring of Se atoms. The total number of atoms is N and there are nr atoms with coordination r (r = 2, 3 or 4), then

and we can define the mean coordination

We note that (where ) gives a partial but very important description of the network. Indeed, when questions of connectivity are involved the average coordination is the key quantity. In covalent networks like GexAsySe1–x–y , the bond lengths and angles are well defined. Small displacements from the equilibrium structure can be described by a Kirkwood [4] or Keating [5] potential, which we can write schematically as

The mean bond length is l,

is the change in the bond length and is the change in the bond angle. The bond-bending force is essential to the constraint counting approach for stability, in addition to the bond stretching term The other terms in the potential are assumed to be much smaller and can be neglected at this stage. If floppy modes are present in the system, then these smaller terms in the potential will give the floppy modes a small finite frequency. For more details see Ref. [16]. If the modes already have a finite frequency, these extra small terms will produce a small, and rather uninteresting, shift in the frequency. This division into strong and weak forces is essential if the constraint counting approach is to be useful. It is for this reason that it is of little, if any, use in metals

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and ionic solids. It is fortunate that this approach provides a very reasonable starting point in many covalent glasses.

To estimate the total number of zero-frequency modes, Maxwell counting was first applied by Thorpe [16], following the work of J.C. Phillips [22,23] on ideal coordinations for glass formation. It proceeds as follows. There are a total of 3N degrees of freedom. There is a single central-force constraint associated with each bond. We assign r/2 constraints associated with each r-coordinated atom. In addition there are constraints associated with the angular forces in Eq. (7). For a twofold coordinated atom there is a single angular constraint; for an r-fold coordinated atom there are a total of 2r–3 angular constraints. The total number of constraints is therefore

Using Eqs. (5) and (6), their fraction nc can be rewritten as

thus, according to Eq. (2),

Note that this result only depends upon the combination

which is the relevant

variable. When (e.g. Se chains), then fM = 1/3 ; that is, one third of all the modes are floppy. As atoms with higher coordination than two are added to the network as crosslinks, fM drops and goes to zero at and network goes through the rigidity percolation transition. This mean field approach has been quite successful in covalent glasses and helps explain a number of experiments. Also in later sections, we discuss the results of computer experiments and show that they are rather well described by the results of this subsection.

We note that Eq. (8) holds only when there are no 1-fold coordinated atoms. Their presence leads to the threshold being shifted down [24-26].

The Pebble Game

Until recently it has not been possible to improve on the approximate Maxwell constraint counting method, except on small systems with up to ~ 104 sites using brute force numerical methods. Now a powerful exact combinatorial algorithm, called the Pebble Game, has become available. This algorithm, first suggested by Hendrickson [13] and implemented by Jacobs and Thorpe [12,27,28], allows systems containing more than 106 sites to be analyzed in two-dimensional generic central-force networks and in threedimensional networks with both central forces and bond-bending forces. The crux of the Pebble Game algorithm in two dimensions is based on a theorem by Laman [14] from graph theory. We note first that if for a two dimensional network Maxwell counting gives less than 3 floppy modes (3 modes are always there, as they correspond to rigid motions of the network), the counting cannot be exact and thus a redundant bond (or bonds) are present. One says that the Laman condition for the network is violated in this case. But if the opposite is true (the Laman condition is satisfied), this is not sufficient for redundant bonds to be absent, as the network can have more than 3 floppy modes and redundant bonds simultaneously. The statement of the theorem is that non48

violation of the Laman condition for every subnetwork is sufficient for not having redundant bonds. This statement does not generalize to dimensions higher than two. We do not go into details of the algorithm, which can be found elsewhere [12,27,29]. For this consideration it is enough to know that one starts from an “empty” lattice (having no bonds, only sites) and adds bonds one at a time. Each newly added bond is tested for independence and each independent bond decreases the number of floppy modes by one. Besides providing exact constraint counting, the algorithm is able to identify all rigid

clusters (and thus whether or not rigidity percolation occurs) and find all the regions, in which redundant bonds introduced stress (over-constrained regions).

Figure 3. The topology of a typical section from a bond-diluted generic network at p = 0.62 (below percolation) and at p = 0.70 (above percolation). A particular realization would have local distortions (not shown), thus making the network generic. The heavy dark lines correspond to over-constrained regions. The open circles correspond to sites that are acting as pivots between two or more rigid bodies.

Sections of a large network on the bond-diluted generic triangular lattice are shown in Figure 2 after the pebble game was applied. Below the transition the network can be macroscopically deformed as the floppy region percolates across the sample. Above the rigidity transition, stress will propagate across the sample. However, below the transition there are clearly pockets of large rigid clusters and over-constrained regions, while above the transition there are pockets of floppy inclusions within the network. While Laman’s theorem does not generally apply to three dimensions, it is possible to generalize the Pebble Game algorithm for a particular class of networks, namely, the bondbending networks with angular forces included as in a Kirkwood or Keating potential [Eq. (7)]. Fortunately, the bond-bending model is precisely the class of models that is applicable to the study of many covalent glass networks. A longer discussion of the three dimensional Pebble Game is given in Refs. [28,30].

Two Dimensional Central Force Network. In this subsection, we review some results for central-force generic rigidity percolation on the triangular net. A more detailed account can be found in Ref. [27]. We begin by finding the number of floppy modes and comparing it to the Maxwell counting result. The exact value of f is very close to fM far enough below the mean-field estimate for the rigidity transition

but then starts to deviate significantly and does not reach zero (until full coordination, is reached). The quantity f looks quite smooth, but the second derivative of it with respect to (shown in the insert) does in fact have a singularity. This singularity corresponds to the rigidity percolation threshold, as can be checked by detecting the percolating rigid cluster directly. Using finite-size

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scaling, the position of the transition was found to be This is amazingly close to the mean-field value of 4. The behavior of the second derivative suggests that the number of floppy modes is an analogous quantity for rigidity and connectivity percolation. In the case of connectivity percolation, the number of floppy modes is simply equal to the total number of clusters, which corresponds to the free energy [16,31-33]. It would be nice if a similar result holds for rigidity percolation. It turns out that the second derivative of the total number of clusters changes sign across the transition, thus violating convexity requirements. Noting that typically rigid clusters are not disconnected, it was suggested that the number of floppy modes generalizes as an appropriate free energy [16,31-33]. With this assumption, the exponent is estimated in the usual context of a heat capacity critical exponent, even though no temperature is involved here. Again analogously to connectivity percolation, the fraction of bonds in the percolating rigid cluster serves as the order parameter for this system. The critical exponent is

defined as the rigid cluster size critical exponent. Another order parameter is also possible, namely, the fraction of bonds in the percolating stressed cluster, which is defined as a percolating stressed subset of the percolating rigid cluster. It was found (and this is an important point) that both and go to zero at the same point – the percolation transition. This will be different in the next chapter on self-organization and will lead to the existence of the intermediate phase, and two phase transitions. The results of study of this model [27] lead to the conclusion that the rigidity transition in this system is second order, but in a different universality class than connectivity percolation. It has been suggested by Duxbury and co-workers [21] that the rigidity transition might be weakly first order on triangular networks. While we think this is unlikely, it

cannot be completely ruled out at the present time. Three Dimensional Bond Bending Networks. It can be shown [28] that the only floppy element in a three dimensional bond-bending network is a hinge joint. Hinge joints can only occur through a central-force (CF) bond and are always shared by two rigid clusters – allowing one degree of freedom of rotation through a dihedral angle. Note that in two dimensional central force generic networks, sites that belong to more than one cluster act as a pivot joint, and more than two rigid clusters can share a pivot joint. Because of this difference between CF and bond-bending networks, the order parameters analogous to and of the previous subsection, have to be defined as a fraction of sites in respective percolating clusters and not bonds, as bonds can be shared between a percolating and a non-percolating clusters. For purposes of testing rigidity in generic three-dimensional bond-bending networks, it is only necessary to specify the network topology or connectivity of the CF bonds, since the second nearest neighbors via CF bonds define the associated bond-bending constraints. Here, we have considered two test models. In the first model, a unit cell is defined from our realistic computer generated network of amorphous silicon [34] consisting of 4,096 atoms having periodic boundary conditions. Larger completely four-coordinated periodic networks containing 32,768, 262,144 and 884,736 atoms are then constructed from the amorphous 4,096-atom unit cell. The four-coordinated network is randomly diluted by removing CF bonds one at a

time with the constraint that no site can be less than two-coordinated. That is, a CF bond is randomly selected to be removed. If upon removal either of its incident atoms becomes less than two coordinated, then it is not removed and another CF bond is randomly selected from the remaining pool of possibilities. The order of removing CF bonds is recorded. This process is carried out until all remaining CF bonds cannot be removed, leading to as low an

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average coordination number as possible. All CF bonds that were successfully removed are marked. This method of bond dilution gives a simple prescription for generating a very large model of a continuous random GexAsySe1-x-y type of network. For comparison, a second test model, a diamond lattice, was diluted in the same way and contained 32,768, 262,144 and 106 atoms. The results of simulations of both models are qualitatively similar to those for 2D central-force networks. Both have a rigidity transition slightly below the Maxwell counting estimate of 2.4. Again, the rigidity transition can be accurately found from the sharp peak in the second derivative of the fraction of floppy modes. In particular, for the diamond lattice and for a-Si, Remarkably, the Maxwell counting estimate is accurate to about 1% in locating the threshold in both cases. A more detailed account of the results can be found in Ref. [35]. SELF-ORGANIZATION AND INTERMEDIATE PHASE Self-Organization in Rigidity Percolation

Description of the Model. We have mentioned that starting from an “empty” lattice

(without bonds) and adding one bond at a time, we can use the pebble game to analyze whether the bond we are adding is independent of those already in the network or redundant. We also know that redundant bonds create stressed (over-constrained) regions. Thus within the present approach we have a rather unique opportunity to construct stressfree networks without a huge computational overhead. The idea is to start, as before, from an “empty” lattice and add one bond at a time to it, applying the pebble game at each stage. If adding a trial bond would result in that bond being redundant and hence create a stressed region, then that move is abandoned. Thus the network self-organizes in such a way that there is no stress in it at all. Note that the pebble game now serves not only as a tool to analyze the network, as before, but also as a decision-making mechanism when building the network. It is not possible to keep adding bonds beyond a certain point, without introducing stress (this is considered in more detail below). How should we proceed then? While going on with some sort of self-organization would be reasonable (as some bonds would create less stress than others), it is impossible to analyze this within our model, so we start inserting bonds completely at random, once avoiding stress becomes impossible. General Properties. First of all, how long is it possible to keep adding bonds to a network without introducing stress? It is certainly impossible to have more independent constraints then there are degrees of freedom in the network. Now recall that in the Maxwell counting approximation, the rigidity transition occurs when the numbers of constraints and of degrees of freedom balance. Thus it is certainly not possible to have an unstressed network with the mean coordination above where Maxwell counting predicts the transition (that is, above for central-force networks in 2d and for glassy networks in 3d). This provides an upper limit (still not always reachable, as we will see) for the unstressed networks. Note, though, that since the Maxwell counting percolation limit is not exact, this does not mean that rigid networks are necessarily stressed! The actual rigidity transition may occur below the point where Maxwell counting puts it. This is a very important point that leads to possibility of an intermediate phase, as described below. Secondly, we know that the Maxwell counting result for the number of floppy modes would be exact if all constraints in the network were independent. But this is exactly what we have in our case! Thus the number of floppy modes in Maxwell counting is exact for as

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long as we are able to keep the network unstressed. Hence we follow the Maxwell result for the number of floppy modes in the floppy and intermediate phases. We now analyze some specific cases in more detail. Intermediate Phase in 2D Central-Force Networks. Let us first prove that it is indeed possible to reach the Maxwell counting limit without any stress in this case (and for any CF networks), provided that the fully coordinated (undiluted) network has no floppy modes (which is the case for triangular networks). As we have seen before, generally speaking, we should distinguish carefully between constraints and bonds. A constraint can be thought of as one algebraic relation for the coordinates of atoms; stress appears whenever one or more of such relations are not satisfied. A bond can have several associated constraints, as in bond-bending networks. In the case of CF networks, though, each bond has only one associated constraint (the distance between the sites it connects), so “bonds” and “constraints” are identical. Recall once again that every single constraint can be either independent (in which case it reduces the number of floppy modes of the network

by 1), or redundant (so it does not change the number of floppy modes). At the point where stress becomes inevitable any trial bond would cause stress (be redundant). So all the bonds, which will be subsequently inserted, are redundant. Thus f will remain constant up to the very end (which is the full lattice), therefore f = 0 at this point. Since Maxwell counting is still exact there, the proof is complete. We would like to emphasize that equivalence of “bonds” and “constraints” was essential for this proof (we used these terms

interchangeably). See the next subsection for comparison. Secondly, it is possible to establish a relation between the self-organized networks and those obtained by usual completely random insertion (to which we for simplicity refer as “random” in contrast to “self-organized” in what follows). Indeed, assume we are using the same random list of M bonds to build a random network and a self-organized one, trying to insert bonds as they are listed. For the random network, all the M bonds will get in; for the self-organized network, some of them will be, generally speaking, rejected, so that will be inserted. The bonds rejected in the self-organized network will be redundant in the random one; they do not influence the number of floppy modes, the configuration of rigid clusters (and thus whether or not rigidity percolation occurs) and the redundancy or independence of all the subsequently inserted bonds. Thus all these characteristics will be identical for the two networks. The consequence is that there is a

correspondence between self-organized and random networks having the same number of floppy modes; in particular, rigidity percolation occurs at the same number of floppy modes. This analysis allows us to make a very important conclusion. Since in random networks rigidity percolates at a non-zero f and the same has to be true for self-organized networks (because of the just mentioned consequence), yet stress appears exactly at f = 0, we conclude that there exists an intermediate phase, which is rigid (i.e. the infinite rigid cluster exists), but unstressed (so, evidently, there is no stress percolation). This is different from the situation with random insertion, where the rigidity and stress percolation thresholds always coincide (see Figure 3). It could be possible that stress does not percolate immediately after it is introduced; we will see from simulation results that this is not the case, so the upper boundary of the intermediate phase (the stress transition) may be defined as either the point where stress first appears, or equivalently, the point where it percolates. As is seen from our consideration, it lies at As we have mentioned, the fractions of bonds in percolating rigid and stressed clusters (denoted and respectively) can serve as order parameters. Now, since there is an intermediate phase where rigidity percolates, while stress does not, these two parameters 52

turn zero at different points, between which the intermediate phase lies. Besides, since the number of floppy modes is zero above the stress transition, the whole network is rigid, and thus is identically 1. These facts are illustrated in Figure 3.

Figure 3. Order parameters and for self-organized and random triangular networks. It is seen that the intermediate phase (shaded) is formed in the self-organized case, extending from 3.905 to 4, while in the random case the two thresholds coincide and there is no intermediate phase. All results are averages over two realizations on 400×400 networks.

Given the discussion of the floppy modes in the random and self-organized networks, it is tempting to suggest that the same relation holds for the just defined rigidity order

parameter. The subtlety is that the relation is defined in terms of sites (i.e., same sites are in the percolating cluster and same sites are pivot joints on its border), while the order parameter is defined in terms of bonds. Of course, there is no direct correspondence between bonds, as there are different numbers of bonds in related random and selforganized networks. Still it might be safely assumed that the rigid cluster size critical exponents are the same for rigidity percolation in random and self-organized networks. Other critical exponents may be different, though. It is interesting to note that since f given by Maxwell counting is exact in the whole unstressed region, in both the floppy and the intermediate phase f is a perfect straight line and the rigidity transition does not show up in f .

Results of our simulations of this model are shown in Figures 3 and 4. The simulations were done for networks with periodic boundary conditions in both directions. There are several facts to be inferred (besides confirming all the results we have obtained

so far). We see that stress percolates immediately after it appears at (this fact was mentioned above). Second, the cluster size critical exponent for the stressed cluster is quite small (smaller than the one for the rigid cluster). In random networks, the stressed cluster exponent is larger than the rigid cluster exponent, which is because the stressed percolating cluster is smaller than the rigid cluster (the former being a subset of the latter) and the two thresholds coincide.

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Figure 4. Number of floppy modes per degree of freedom for self-organized and random triangular networks. Thresholds are shown with different symbols. The intermediate phase in the self-organized case is shaded. Note that rigidity percolation occurs at the same f in the random and self-organized cases. The self-

organized plot is strictly linear up to

and coincides with Maxwell counting.

Intermediate Phase in 3D Bond-Bending Networks. In case of glassy networks there is a slight problem with implementing our general algorithm of self-organization. In the CF case we were starting from an empty lattice to ensure that it had no stress initially. In the present case the initial dilution can only go as far as to the point where any further dilution would create a 1-coordinated site. At this limit there are no bonds with both ends being sites of coordination 3 and higher, so that further dilution is impossible. It is

generally not true that this final network is unstressed. For smaller networks (~104 sites and less), it is possible to pick those that are unstressed; for larger ones such cases are rare, and it is reasonable to assume that the fraction of constraints that are redundant is a constant in the thermodynamic limit. This constant seems to be very low, though (in our simulations, typically about 0.05% of constraints were redundant). Besides, the number of redundant constraints does not grow when new bonds are inserted according to our algorithm (up to the stress transition), so this problem is largely irrelevant. Unlike the case of CF networks, BB networks have more than one constraint associated with each bond. When a new bond is added, not only the distance between the sites it connects is fixed, but the angles between the new bond and those stemming out of the two sites at either end of that bond are fixed as well. Any bond that has at least one redundant constraint associated with it would cause stress. Some of the stress-causing bonds have only part of the associated constraints redundant and the rest independent, and such a bond will change the number of floppy modes. This makes some of our conclusions made for CF networks invalid in this case. Firstly, this invalidates the proof of the reachability of the Maxwell counting limit ( in this case). This is because even when at the upper reachable limit all the as yet uninserted bonds would cause stress, some of these bonds may further decrease the number of floppy modes and thus this number is not necessarily zero at this point.

Secondly, the nice relation between random and self-organized networks no longer holds, because out of the redundant bonds by which the two differ, some (namely, the partially redundant ones) change f , rigidifying the network and changing the 54

configuration of rigid clusters. Still the equality of critical exponents for rigid cluster sizes in random and self-organized cases probably holds. At the same time, some facts are unchanged. In particular, f given by Maxwell counting is still exact in the unstressed region. Most importantly, the intermediate phase still exists. The results of simulations done for the diluted diamond lattice are given in Figures 5 and 6. As in the previous subsection, we use periodic boundary conditions in all directions. We note in addition to the graphs that, as in the CF case, stress percolates immediately

after it appears. The intermediate phase extends from to 2.392 (not reaching 2.4). Again, the stress transition is sharper than the rigidity transition. Our results are consistent with the second order transition with the very small critical exponent or a first order transition is more likely. Another feature of the plot in Fig. 5 is that the rigidity order parameter is not exactly unity in the stressed phase (which is expected, as some floppy modes remain in the stressed phase) and the second transition shows up as a kink in the rigidity order parameter. In conclusion to this section, we would like to mention that it is possible within our approach to establish a hierarchy of stress-causing bonds (by the number of associated

redundant constraints) and when stress becomes inevitable, first put those having one redundant constraint, then those having two, and so on. Exactly at only those bonds having no associated independent constraints will remain uninserted. It is unlikely, though, that there is a good correlation between the number of redundant constraints and the actual increase in stress energy, as the distribution of stresses caused by different bonds is quite wide, so this complication seems unreasonable.

Figure 5. The order parameters and for the self-organized diluted diamond lattice. The intermediate phase is shaded. Circles are average over 4 networks with 64,000 sites, triangles are averages over 5 networks with 125,000 sites. The dashed lines are the power law fit below the stress transition and for guidance of the eye above. Note the break in the slope at the stress transition.

Elastic Properties of Self-Organized Networks. So far our study of self-organized networks was limited to their geometrical properties. Of course, this work becomes really meaningful when we turn to what the physical consequences of self-organization are. The simplest quantity to look at is the elasticity of the networks of springs. Unfortunately, the 55

pebble game, being concerned with the geometric properties only, is unable to help us find the numerical values of elastic constants, so we have to do a usual relaxation using, for

example, the conjugate gradient method [36] and consider particular configurations, and not just the connectivity. So far in this preliminary study, we have only considered the 2d case. The first and quite surprising fact is that in case of periodic boundary conditions in all directions the elastic constants are exactly zero in the intermediate phase, regardless of the

size of the supercell and despite the existence of the percolating rigid cluster. Indeed, periodic boundary conditions mean that positions of images of same site in different supercells are fixed with respect to each other. The network is built stressless with these additional constraints taken into account. The exact specification of these constraints beyond stating what sites are involved is determined by the particular size and shape of the supercell, but is never taken into account (just as particular bond lengths never matter in determination of stressed regions). So straining the network by changing this size and shape leaves it stressless. The important thing here is that straining does not add any new

constraints. We confirmed this result numerically by doing exact diagonalization of the dynamical matrix (similar to [37]), rather than by relaxation, which ensures better precision.

Figure 6. The fractions of floppy modes per degree of freedom for the diluted diamond lattice (both selforganized and random cases). Different thresholds and the Maxwell prediction for the rigidity threshold are shown with different symbols. The intermediate phase in the self-organized case is shaded. The Maxwell

counting line is seen only above the stress transition point in self-organized networks, as below this point it coincides with the self-organized line.Note that the rigidity transition in the two cases no more occurs at the same f . Instead, the values of are close, which is probably coincidental.

Of course, for different boundary conditions the elastic constants may be non-zero for finite samples, but are expected to vanish in the thermodynamic limit. We consider the busbar geometry, in which busbars are applied to two opposite sides of the network and it is strained perpendicular to the busbars. The network is built assuming open boundaries at the busbars and periodic boundary conditions parallel to the busbars. The first and the last rows of sites are assumed belonging to the respective busbar (i.e., attached rigidly to it). In addition, when building the network, we consider the sites belonging to each busbar as being fixed with respect to each other, connecting them with fictitious bonds and considering these bonds as belonging to the network. This makes the open boundaries “less open” and eliminates certain boundary effects, as will be clear from an analogy in the next 56

section with connectivity percolation. The arguments of the previous paragraph do not apply here, as the network is built not assuming a fixed distance between the busbars (as if it is allowed to relax) and straining changes and fixes it thus imposing an additional constraint.

Figure 7. An example of the triangular self-organized network 150×150 in the intermediate phase (at ). The thickest bonds belong to the applied-stress backbone, those of medium thickness are in the percolating rigid cluster (but not in the backbone), the thinnest ones are not in the percolating cluster. The busbars are shown schematically.

When introducing the boundary conditions as described above, we will have non-zero stress when an external strain is applied, and some of the bonds will be stressed. These bonds are said to belong to the applied stress backbone [21] (which we refer to as simply backbone in what follows). It can be found easily by the pebble game using a method proposed by Moukarzel [38], which in our case consists in putting an additional bond across the network emulating the external strain, and finding those bonds in which stress is induced. A typical result is shown in Figure 7. It is seen that the backbone has filamentary structure. We note that stress in this backbone was created by putting just one extra bond and thus it is enough to take any one bond out of the backbone for it to be destroyed, so it is extremely fragile. Also, since the backbone always has only one redundant bond (when the bond across is added), it does not grow throughout the intermediate phase after it appears at the rigidity transition, because growth can only occur by adding new redundant bonds. This means that for any given sample the elastic constants are the same throughout the intermediate phase (here we mean finite samples, of course, as in the infinite limit the elastic constants are zero).

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Figure 8. The elastic modulus c11 for self-organized triangular networks. Each point corresponds to one sample (their linear sizes are specified by different symbols). The intermediate phase is shaded. The dashed line is the mean-field linear dependence, reaching 1 at the full coordination.

We found the elastic modulus c11 numerically in both the intermediate and stressed

phases. The triangular lattice was distorted by random displacement of atoms. For displacements along each axis uniform distribution on an interval (–0.1; 0.1) in units of the lattice constant was chosen, but the results are only slightly sensitive to the width of the distribution. Equilibrium lengths of springs were chosen equal to the distance between the atoms they connect, so the initial network is unstressed. Thus subtraction of two large energies when finding elastic constants is avoided. The results are shown in Fig. 8. Predetermining the applied stress backbone speeds up the relaxation greatly, as was first pointed out in Ref. [21]. Still, we were unable to reach full relaxation in the intermediate phase in all but the smallest samples (up to 30×30). The values in the intermediate phase are very low and are assumed to go to zero in the limit of large samples. We are currently doing finite size scaling to test this. Above the stress transition, the modulus seems to grow linearly, but, of course, it is hopeless to try and determine the critical exponent with reasonable precision from our data. Self-Organization in Connectivity Percolation

The model. It is interesting and useful to see if similar phenomena are possible in the more familiar case of connectivity percolation, especially as connectivity percolation is easier to study and understand. The essence of our algorithm of building self-organized networks in the rigidity case is rejecting stress-causing bonds (or those having redundant constraints). As we have seen, in the CF case, when “bonds” and “constraints” are the same, we may equivalently formulate this as rejecting redundant or irrelevant bonds. In bond connectivity percolation we also can build the networks by inserting bonds one by one; most importantly, there is a clear analog to redundant bonds. The relevant property now is connectivity, by which we mean the presence or absence of paths connecting any two sites of the network. Redundant bonds are those which connect sites already connected, that is would close a loop in the network. Thus the analog of self-organization is building loopless networks. There are other equivalent ways to draw this parallel. The first is based on the fact that connectivity percolation can be considered as rigidity percolation with the sites having one

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degree of freedom regardless of the lattice dimensionality. Each site thus has one coordinate and each bond is a relation between the coordinates of the sites it connects. Then the concepts of rigid clusters and clusters in the usual connectivity sense coincide. The number of floppy modes f is now the number of clusters. A redundant bond in the rigidity sense is the one that does not change f, it is also stress-causing, as it would introduce a relation between coordinates that cannot generally be satisfied. On the other hand, viewed from the connectivity perspective, such a bond connects the sites belonging

to the same cluster and closing a loop, and our model is again recovered. Yet another way is to recall that rigidity percolation with angular constraints in 2D (or with angular and dihedral constraints in 3D) is equivalent to connectivity percolation. Then stresslessness is equivalent to looplessness. Connectivity percolation and related phenomena were studied so extensively in all imaginable flavors that it would be strange if this and similar models were not studied before. Indeed exactly this model was proposed as far back as 1979 [39] and rediscovered in 1996 [40]. Besides, there was an extensive study of loopless graphs (trees) in relation to various phenomena ranging from resistance of a network between two point contacts (considered by Kirchhoff in mid nineteenth century [41]) to river networks [42] to certain optimization problems [43,44]. In many of these and other papers the algorithm for building trees was equivalent to ours. Still, we consider this model from a different perspective. Given that connectivity percolation can be considered as rigidity percolation with one degree of freedom per site, we can apply the usual two-dimensional pebble game with the simple modifications. Of course, the essence of our self-organization algorithm is still the rejection of bonds that are not independent. The pebble game allows the determination of all analogs of the quantities considered for rigidity. The Intermediate Phase. In this section we carry out the same kind of analysis as was done for rigidity percolation.

First of all we describe Maxwell counting, as this, although simple, is rarely discussed in relation to connectivity percolation. For a network with N sites the number of degrees of freedom is now simply N, the number of constraints is, as before, so the number of floppy modes per site is and this becomes zero at Since, as we have seen, connectivity percolation is nothing but a kind of rigidity

percolation on a CF network with 1 degree of freedom per site, all of the general analysis for CF networks in the previous section is valid. Specifically, Maxwell counting is exact in the “unstressed” (this now means loopless) phase; the limit is reachable without creating loops; the relation between random and self-organized networks also holds. The order parameters are defined analogously to the rigidity case. The first parameter is (by analogy) the size of the percolating (connectivity) cluster. However, the difference is that now the clusters (including the percolating one) can be defined in terms of either bonds or sites (there are no “pivot joints” that would be shared between several clusters). Therefore, there is a possibility to define this order parameter as the fraction of sites (instead of bonds) in the percolating cluster. This makes the relation between the order parameters of self-organized and random networks with the same number of “floppy modes” (clusters) exact. Yet, to be consistent, we ignore this possibility and define the order parameters as fractions of bonds, not sites. The second order parameter is, logically, the fraction of “stressed” bonds (bonds in loops). We do not show the results of simulations (which were done for the square lattice) as they are very similar to those in rigidity case, except that the “stressed” cluster critical exponent is larger, not smaller than the connected cluster exponent. Existence of the

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intermediate phase is confirmed in the range from to 2 for the square net. The lower transition coincides with the result obtained in Ref. [40]. Conductivity. Similarly to the elasticity case, we consider the busbar geometry here in two variants, with and without fictitious bonds making sites at the busbars rigid with respect to each other (we refer to these two cases as boundary conditions A and B respectively). As in rigidity, it is possible to find the conductivity backbone by Moukarzel’s method [38] (for B all the fictitious busbar bonds have to be put prior to placing the bond across, while they are already in the network in case A). Two examples corresponding to A and B are shown in Fig. 9. For A the backbone consists of just one path, while for B it is tree-like with branching near the busbars. Analog of these “boundary effects” in B is what was eliminated in study of elasticity, when the boundary conditions

analogous to A were chosen. The simulations for conductivity in 2D can be done very efficiently with the FrankLobb algorithm [45], whose only limitation is that it is applicable for the open boundary conditions only.

Figure 9. Examples of self-organized square networks in the intermediate phase with boundary conditions A (left panel) and B (right panel), as described in the text. The thickest bonds are in the conducting backbone,

those of medium thickness are in the percolating cluster (but not in the backbone), the thinnest are not in the percolating cluster. The busbars are shown schematically.

It is known from work on a river network model built in the same way as our network [42] that the backbone branches (in fact, all network branches) are fractal and the fractal dimension is

The only essential difference between the river network model and

our one is that they consider spanning trees (i.e., all sites are in the connecting cluster), which in our case corresponds only to This should not matter, though, since it is the dimensionality of the network that the cluster actually spans (i.e., of the connecting

cluster) that is important and this dimensionality is 2 everywhere in the intermediate phase. Thus we come to a conclusion that the fractal dimension of backbone branches is the same throughout the intermediate phase and equals 1.22. We confirm this fact in our simulations. We note that this differs from both the random walk result (d = 2) and that for selfavoiding random walks (d = 4/3) – in our case branches are more “straight” than both of these walks. Then for boundary conditions A it is obvious that the fraction of bonds in the backbone is and the conductance is Our simulations confirm this result, for both variants of boundary conditions. The effective conductivity in 2D is

equal to the conductance. Thus we come to the conclusion that the conductivity does indeed go to zero in the thermodynamic limit for the intermediate phase.

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The results in both the intermediate and the stressed phase are shown in Fig. 10. Just as for elastic constants, the dependence in the stressed phase is linear, but now much larger sizes are available, so this linearity may be exact, but we know of no reason for this to be so. Note the finite value of the conductivity in the intermediate phase, which is a finite size effect. This value would be constant for boundary conditions A, as the conducting backbone consists of just one stem not changing across the intermediate phase. Here this value changes slightly across the intermediate phase. We mention here briefly that our preliminary results in 3 dimensions show that the conductivity is also zero in the thermodynamic limit in the intermediate phase, and it goes to zero with increasing size even faster than in two dimensions.

Figure 10. Conductivity for resistor networks with present bonds having resistance R1 = 1 and missing having resistance

(diamonds); superconducting networks (R 1 = 0, R2 = 200, circles); mixed

networks (R1 = 1, R2 = 200, triangles). All results are averages over 10 square networks 100×100 with open boundary conditions parallel to the busbars, the busbar sites are treated as in case B (see text).

Superconducting Networks. We have seen that in the thermodynamic limit the

conductivity is zero in both the disconnected and intermediate phases (just as elastic constants were zero in both the floppy and intermediate phases in the rigidity case). These results make us wonder if the lower transition shows up in any physical quantities for infinite networks. One possibility is to consider superconductor networks instead of resistor networks. In this model all the existing bonds are replaced with conductors of zero resistance (“superconductors”), while all the absent bonds are equal resistors with finite resistance. It turns out that the same kind of correspondence between random and self-organized networks with the same f we had for clusters is valid for the conductance in this case. Indeed, these networks differ by redundant bonds that connect sites already connected. All the connected sites have zero potential difference (as they are connected with superconductors), so putting redundant bonds does not change the distribution of the potential and thus does not influence the conductance. It is known [46] that in the random case the resistivity is zero above the threshold and non-zero below it, with the critical exponent the same as for the conductivity of resistor networks (1.30). Thus in the self-organized case the resistivity will turn zero in the point related to the percolation threshold of random networks by the above relation, i.e., at the 61

lower transition. The critical exponent will be the same as in the random case (1.30), but this is now different from the value for of resistor networks Mixture of Two Sorts of Resistors. We can now “combine” the resistor and superconductor models by introducing two sorts of resistors, with resistances R1 and R2, R1 < R2, and putting R1 resistors in place of present bonds and R2 resistors in place of missing bonds. Assume now that R1 n c, n=nc, and nEc, but their number will be very small and will be neglected here. One of the important ideas in the MIT field was the notion of a minimum metallic conductivity σmin developed by Mott [16 ]. This idea is based on the Ioffe-Regel (IR) criterion that where kF is the Fermi wavevector and is the mean free path and that the Boltzmann conductivity was only meaningful for where d is the atomic spacing (the donor spacing in n-type Si). Mott employed the Boltzmann result to obtain

Invoking the IR criterion and

Mott obtained (4)

where the coefficient C depends on the number of valley v of a multivalley semiconductor. The Mott notion was that would drop discontinuously to zero for nnc) for finite T is the e-e interaction theory of Altshuler and Aronov (A-A) [36] for diffusing electrons in strongly disordered metals. A very large variety

of disordered metals exhibit the correction This correction, plus a second T1/2 correction from ionized impurity scattering, will provide an excellent description of the doped Si and Ge data for n>nc. The A-A correction for the d=3 case takes the form

(9a)

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Figure 3. Extrapolated T=0 values of versus the uniaxial stress S from Ref.[42]. The solid line is the region which is reproducible in 3 samples and yields s= 0.49. The inset shows data from Ref. [41] from individual samples. (Copyright by the American Physical Society)

where (9b)

where x = (2k F/κ)2 where this κ is the screening wave vector. The factor is known as the exchange-Hartree factor, or the singlet and triplet (spin=1) scattering amplitudes. The form of F= x-1 ln(1+x) is for a Fermi liquid. Because the A-A theory is only first order in the disorder many believe the theory is not valid very close to nc. An early perturbative form of the theory used (4/3-2F) for the exchange-Hartree factor and this was used by Rosenbaum et al. [37] to explain their data using x = 0.2x(n/1018)1/3 based on free electron-like expressions for kF and k. Had they used for x in the range 0.3 to 1.0 they would not have been able to explain their negative m(n) values for n>1.05nc unless they reversed the sign of the exchange-Hartree factor. An examination of the A-A integral and final result in Eqs 5.3 and 5.4 of their review indicates the sign is negative for d=3 and opposite that for d=1. However, the integral diverges for all and analytical continuation cannot be used from d=1 to d=3. Choosing an appropriate cutoff frequency for the integral of (10