Simulating Correlations with Computers: Lecture Notes of the Autumn School on Correlated Electrons 2021 3958065295, 9783958065291

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Eva Pavarini and Erik Koch (Eds.)

Simulating Correlations with Computers

Modeling and Simulation

Simulating Correlations with Computers Eva Pavarini and Erik Koch (Eds.)

Modeling and Simulation Band / Volume 11 ISBN 978-3-95806-529-1

ISBN 978-3-95806-529-1

9 783958 065291

11

Schriften des Forschungszentrums Jülich Reihe Modeling and Simulation

Band / Volume 11

Forschungszentrum Jülich GmbH Institute for Advanced Simulation

Lecture Notes of the Autumn School on Correlated Electrons 2021 Eva Pavarini and Erik Koch (Eds.)

Simulating Correlations with Computers Autumn School organized by the Institute for Advanced Simulation at Forschungszentrum Jülich 20 – 24 September 2021

Schriften des Forschungszentrums Jülich Modeling and Simulation ISSN 2192-8525

Band / Volume 11 ISBN 978-3-95806-529-1

Bibliographic information published by the Deutsche Nationalbibliothek. Die Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.

Publisher: Cover Design:

Forschungszentrum Jülich GmbH Institute for Advanced Simulation

Printer:

Schloemer & Partner GmbH, Düren

Copyright:

Forschungszentrum Jülich 2021

Grafische Medien, Forschungszentrum Jülich GmbH

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This is an Open Access publication distributed under the terms of the Creative Commons Attribution License 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Contents Preface 1. Second Quantization and Jordan-Wigner Representations Erik Koch 2. Fundamentals of Quantum Chemistry Klaus Doll 3. Lies My Teacher Told Me About Density Functional Theory: Seeing Through Them with the Hubbard Dimer Kieron Burke 4. Hubbard Dimer in GW and Beyond Pina Romaniello 5. Dynamical Mean-Field Theory for Materials Eva Pavarini 6. Green Functions and Self-Energy Functionals Robert Eder 7. Green Functions in the Renormalized Many-Body Perturbation Theory V´aclav Janiˇs 8. An Essential Introduction to NEGF Methods for Real-Time Simulations Gianluca Stefanucci 9. Orbital Entanglement and Correlation Christian Schilling 10. Analog Quantum Simulations of the Hubbard Model Walter Hofstetter 11. Programming Quantum Computers Kristel Michielsen 12. Quantum Chemistry on Quantum Computers Libor Veis 13. Quantum Computing – Quo Vadis? David DiVincenzo Index

Preface The combinatorial growth of the Hilbert space makes the many-electron problem one of the grand challenges of theoretical physics. Progress relies on the development of non-perturbative methods, based on either wavefunctions or self energies. This made, in recent years, calculations for strongly correlated materials a reality. These simulations draw their power from three sources: theoretical advances, algorithmic developments, and the raw power of massively parallel supercomputers. Turning to quantum hardware could give quantum materials science the ultimate boost. Before quantum parallelism can be exploited, however, many questions, algorithmic and engineering, need to be addressed. This year’s school will provide students with an overview of the state-of-the-art of manybody simulations and the promises of quantum computers. After introducing the basic modeling techniques and the concept of entanglement in correlated states, lectures will turn to methods that do not rely on wavefunctions, comparing density-functional theory, the GW method and dynamical mean-field approaches. Advanced lectures will broaden the discussion, addressing topics from the Luttinger-Ward functional to non-equilibrium Green functions. As a glimpse of future possibilities, the basics of quantum computing and its possible uses in materials simulations will be outlined. A school of this size and scope requires backing from many sources. This is even more true during the Corona pandemics, which provided scores of new challenges. We are very grateful for all the practical and financial support we have received. The Institute for Advanced Simulation at the Forschungszentrum J¨ulich and the J¨ulich Supercomputer Centre provided the major part of the funding and were vital for the organization of the school as well as for the production of this book. The Institute for Complex Adaptive Matter (ICAM) supplied additional funds and ideas for successful online formats. The nature of a school makes it desirable to have the lecture notes available when the lectures are given. This way students get the chance to work through the lectures thoroughly while their memory is still fresh. We are therefore extremely grateful to the lecturers that, despite tight deadlines, provided their manuscripts in time for the production of this book. We are confident that the lecture notes collected here will not only serve the participants of the school but will also be useful for other students entering the exciting field of strongly correlated materials. We are grateful to Mrs. H. Lexis of the Verlag des Forschungszentrum J¨ulich and to Mrs. D. Mans of the Grafische Betriebe for providing their expert support in producing the present volume on a tight schedule. We heartily thank our students and postdocs who helped with proofreading the manuscripts, often on quite short notice: Elaheh Adibi, Julian Mußhoff, Neda Samani, and Xue-Jing Zhang. Finally, our special thanks go to Dipl.-Ing. R. H¨olzle for his invaluable advice on the innumerable questions concerning the organization of such an endeavor, and to Mrs. L. Snyders for expertly handling all practical issues.

Eva Pavarini and Erik Koch August 2021

1

Second Quantization and Jordan-Wigner Representations Erik Koch Julich Supercomputer Centre ¨ Forschungszentrum Julich ¨

Contents 1

Many-electron states

2

Second quantization 2.1 Creation and annihilation operators . . 2.2 Representation of Slater determinants 2.3 Representation of n-body operators . 2.4 Transforming the orbital basis . . . .

3

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5 5 7 8 12

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13 13 15 16 18

Jordan-Wigner representation 4.1 Spins, hard-core bosons, and fermions . . . . . . . . . . . . . . . . . . . . . .

18 22

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Exact diagonalization 3.1 Variational principles . . . . . . . . . . . . . . 3.2 Matrix eigenvalue problem . . . . . . . . . . . 3.3 Dimension of the Hilbert space and sparseness . 3.4 Non-interacting electrons . . . . . . . . . . . .

A Appendix A.1 Atomic units . . . . . . . . . . . . . A.2 Non-orthonormal basis . . . . . . . A.3 Pauli matrices . . . . . . . . . . . . A.4 Some useful commutation relations .

E. Pavarini and E. Koch (eds.) Simulating Correlations with Computers Modeling and Simulation Vol. 11 Forschungszentrum Julich, 2021, ISBN 978-3-95806-529-1 ¨ http://www.cond-mat.de/events/correl21

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24 24 25 26 28

1.2

1

Erik Koch

Many-electron states

One of the great mysteries of quantum mechanics is the existence of indistinguishable objects. Classically this is not possible: objects can always be distinguished, at least by their position in space, meaning that indistinguishable objects must be identical. This is Leibniz’ Principle of the Identity of Indiscernibles [1]. For quantum objects, however, the uncertainty principle makes the distinction of particles by their position impossible. This allows for the existence of elementary particles. They form the basic units of all matter. So, quite remarkably, all the different objects we know are made of indistinguishable building blocks. In the formalism of quantum mechanics, indistinguishability means that no observable lets us distinguish one of these particles from the other. Consequently, every observable for, e.g., electrons, must treat each electron in the same way. Thus, in principle, observables must act on all the electrons in the universe. In practice we can, of course, distinguish electrons localized on the moon from those in our lab to an excellent approximation. Thus, for all practical purposes, we can restrict ourselves to the electrons in the system under consideration, assuming that the differential overlap with all other electrons vanishes. Any observable M (x1 , . . . , xN ) for the N electrons in our system must then be symmetric under permutations of the coordinates xi . The consequences are straightforward: An observable M (x) acting on a single-particle degree P of freedom x must act on all indistinguishable particles in the same way, i.e., i M (xi ). LikeP wise, a two-body observable M (x, x0 ) must act on all pairs in the same way, i,j M (xi , xj ) with M (x, x0 ) = M (x0 , x). We can thus write any observable in the form 1 X 1 X (2) M (xi , xj ) + M (3) (xi , xj , xk ) + · · · (1) 2! 3! i i6=j i6=j6=k X X X (0) (1) (2) =M + M (xi ) + M (xi , xj ) + M (3) (xi , xj , xk ) + · · · , (2)

M (x) = M (0) +

X

i

M (1) (xi ) +

i