# Semiclassical approach to dynamics of interacting fermions

## Abstract

Understanding the behaviour of interacting fermions is of fundamental interest in many fields ranging from condensed matter to high energy physics. Developing numerically efficient and accurate simulation methods is an indispensable part of this. Already in equilibrium, fermions are notoriously hard to handle due to the sign problem. Out of equilibrium, an important outstanding problem is the efficient numerical simulation of the dynamics of these systems. In this work we develop a new semiclassical phase-space approach (a.k.a. the truncated Wigner approximation) for simulating the dynamics of interacting fermions in arbitrary dimensions. As fermions are essentially non-classical objects, a phase-space is constructed out of all fermionic bilinears. Classical phase-space is thus comprised of highly non-local (hidden) variables representing these bilinears, and the cost of the method is that it scales quadratic rather than linear with system size. We demonstrate the strength of the method by comparing the results to the exact quantum dynamics of fermion expansion in the Hubbard model and quantum thermalization in the Sachdev–Ye–Kitaev (SYK) model for small systems, where the semiclassics nearly perfectly reproduces correct results. We furthermore analyse fermion expansion in a larger, intractable by exact methods, 2D Hubbard model, which is directly relevant to recent coldmore »

- Authors:

- Department of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215 (United States)
- (Belgium)

- Publication Date:

- OSTI Identifier:
- 22701531

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics

- Additional Journal Information:
- Journal Volume: 384; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMPUTERIZED SIMULATION; FERMIONS; HIDDEN VARIABLES; HIGH ENERGY PHYSICS; HUBBARD MODEL; INTERACTIONS; PHASE SPACE; QUANTUM FIELD THEORY; SEMICLASSICAL APPROXIMATION

### Citation Formats

```
Davidson, Shainen M., E-mail: shainen@bu.edu, Sels, Dries, Theory of Quantum and Complex Systems, Universiteit Antwerpen, Universitetisplein 1, B-2610 Antwerpen, and Polkovnikov, Anatoli.
```*Semiclassical approach to dynamics of interacting fermions*. United States: N. p., 2017.
Web. doi:10.1016/J.AOP.2017.07.003.

```
Davidson, Shainen M., E-mail: shainen@bu.edu, Sels, Dries, Theory of Quantum and Complex Systems, Universiteit Antwerpen, Universitetisplein 1, B-2610 Antwerpen, & Polkovnikov, Anatoli.
```*Semiclassical approach to dynamics of interacting fermions*. United States. doi:10.1016/J.AOP.2017.07.003.

```
Davidson, Shainen M., E-mail: shainen@bu.edu, Sels, Dries, Theory of Quantum and Complex Systems, Universiteit Antwerpen, Universitetisplein 1, B-2610 Antwerpen, and Polkovnikov, Anatoli. Fri .
"Semiclassical approach to dynamics of interacting fermions". United States. doi:10.1016/J.AOP.2017.07.003.
```

```
@article{osti_22701531,
```

title = {Semiclassical approach to dynamics of interacting fermions},

author = {Davidson, Shainen M., E-mail: shainen@bu.edu and Sels, Dries and Theory of Quantum and Complex Systems, Universiteit Antwerpen, Universitetisplein 1, B-2610 Antwerpen and Polkovnikov, Anatoli},

abstractNote = {Understanding the behaviour of interacting fermions is of fundamental interest in many fields ranging from condensed matter to high energy physics. Developing numerically efficient and accurate simulation methods is an indispensable part of this. Already in equilibrium, fermions are notoriously hard to handle due to the sign problem. Out of equilibrium, an important outstanding problem is the efficient numerical simulation of the dynamics of these systems. In this work we develop a new semiclassical phase-space approach (a.k.a. the truncated Wigner approximation) for simulating the dynamics of interacting fermions in arbitrary dimensions. As fermions are essentially non-classical objects, a phase-space is constructed out of all fermionic bilinears. Classical phase-space is thus comprised of highly non-local (hidden) variables representing these bilinears, and the cost of the method is that it scales quadratic rather than linear with system size. We demonstrate the strength of the method by comparing the results to the exact quantum dynamics of fermion expansion in the Hubbard model and quantum thermalization in the Sachdev–Ye–Kitaev (SYK) model for small systems, where the semiclassics nearly perfectly reproduces correct results. We furthermore analyse fermion expansion in a larger, intractable by exact methods, 2D Hubbard model, which is directly relevant to recent cold atom experiments. - Highlights: • We propose a semiclassical method to simulate dynamics of interacting fermions. • A phase-space is constructed out of all fermionic bilinears. • We study quantum thermalization in the Sachdev–Ye–Kitaev model.},

doi = {10.1016/J.AOP.2017.07.003},

journal = {Annals of Physics},

issn = {0003-4916},

number = ,

volume = 384,

place = {United States},

year = {2017},

month = {9}

}