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# A complete quasiclassical map for the dynamics of interacting fermions

## Abstract

We present a strategy for mapping the dynamics of a fermionic quantum system to a set of classical dynamical variables. The approach is based on imposing the correspondence relation between the commutator and the Poisson bracket, preserving Heisenberg's equation of motion for one-body operators. In order to accommodate the effect of two-body terms, we further impose quantization on the spin-dependent occupation numbers in the classical equations of motion, with a parameter that is determined self-consistently. Expectation values for observables are taken with respect to an initial quasiclassical distribution that respects the original quantization of the occupation numbers. The proposed classical map becomes complete under the evolution of quadratic Hamiltonians and is extended for all even order observables. We show that the map provides an accurate description of the dynamics for an interacting quantum impurity model in the coulomb blockade regime, at both low and high temperatures. In conclusion, the numerical results are aided by a novel importance sampling scheme that employs a reference system to reduce significantly the sampling effort required to converge the classical calculations.

- Authors:

- Univ. of California, Berkeley, CA (United States); Tel Aviv Univ., Tel Aviv (Israel)
- Univ. of California, Berkeley, CA (United States)
- Univ. of California, Berkeley, CA (United States); Tel Aviv Univ., Tel Aviv (Israel); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Univ. of California, Berkeley, CA (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Kavli Energy NanoScience Inst., Berkeley, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences & Engineering Division; USDOE

- OSTI Identifier:
- 1564042

- Alternate Identifier(s):
- OSTI ID: 1527065

- Grant/Contract Number:
- AC02-05CH11231

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 150; Journal Issue: 23; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Levy, Amikam, Dou, Wenjie, Rabani, Eran, and Limmer, David T. A complete quasiclassical map for the dynamics of interacting fermions. United States: N. p., 2019.
Web. doi:10.1063/1.5099987.
```

```
Levy, Amikam, Dou, Wenjie, Rabani, Eran, & Limmer, David T. A complete quasiclassical map for the dynamics of interacting fermions. United States. doi:10.1063/1.5099987.
```

```
Levy, Amikam, Dou, Wenjie, Rabani, Eran, and Limmer, David T. Wed .
"A complete quasiclassical map for the dynamics of interacting fermions". United States. doi:10.1063/1.5099987.
```

```
@article{osti_1564042,
```

title = {A complete quasiclassical map for the dynamics of interacting fermions},

author = {Levy, Amikam and Dou, Wenjie and Rabani, Eran and Limmer, David T.},

abstractNote = {We present a strategy for mapping the dynamics of a fermionic quantum system to a set of classical dynamical variables. The approach is based on imposing the correspondence relation between the commutator and the Poisson bracket, preserving Heisenberg's equation of motion for one-body operators. In order to accommodate the effect of two-body terms, we further impose quantization on the spin-dependent occupation numbers in the classical equations of motion, with a parameter that is determined self-consistently. Expectation values for observables are taken with respect to an initial quasiclassical distribution that respects the original quantization of the occupation numbers. The proposed classical map becomes complete under the evolution of quadratic Hamiltonians and is extended for all even order observables. We show that the map provides an accurate description of the dynamics for an interacting quantum impurity model in the coulomb blockade regime, at both low and high temperatures. In conclusion, the numerical results are aided by a novel importance sampling scheme that employs a reference system to reduce significantly the sampling effort required to converge the classical calculations.},

doi = {10.1063/1.5099987},

journal = {Journal of Chemical Physics},

number = 23,

volume = 150,

place = {United States},

year = {2019},

month = {6}

}

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