Full counting statistics for interacting trapped fermions

Smith, Naftali R.; Le Doussal, Pierre; Majumdar, Satya; Schehr, Grégory

doi:10.21468/SciPostPhys.11.6.110

SciPost Physics

Full counting statistics for interacting trapped fermions

Naftali R. Smith, Pierre Le Doussal, Satya N. Majumdar, Grégory Schehr

SciPost Phys. 11, 110 (2021) · published 22 December 2021

doi: 10.21468/SciPostPhys.11.6.110
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Abstract

We study $N$ spinless fermions in their ground state confined by an external potential in one dimension with long range interactions of the general Calogero-Sutherland type. For some choices of the potential this system maps to standard random matrix ensembles for general values of the Dyson index $\beta$. In the fermion model $\beta$ controls the strength of the interaction, $\beta=2$ corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions ${\cal N}_{\cal D}$ in a domain $\cal{D}$ of macroscopic size in the bulk of the Fermi gas. We predict that for general $\beta$ the variance of ${\cal N}_{\cal D}$ grows as $A_{\beta} \log N + B_{\beta}$ for $N \gg 1$ and we obtain a formula for $A_\beta$ and $B_\beta$. This is based on an explicit calculation for $\beta\in\left\{ 1,2,4\right\} $ and on a conjecture that we formulate for general $\beta$. This conjecture further allows us to obtain a universal formula for the higher cumulants of ${\cal N}_{\cal D}$. Our results for the variance in the microscopic regime are found to be consistent with the predictions of the Luttinger liquid theory with parameter $K = 2/\beta$, and allow to go beyond. In addition we present families of interacting fermion models in one dimension which, in their ground states, can be mapped onto random matrix models. We obtain the mean fermion density for these models for general interaction parameter $\beta$. In some cases the fermion density exhibits interesting transitions, for example we obtain a noninteracting fermion formulation of the Gross-Witten-Wadia model.

TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.11.6.110
TI  - Full counting statistics for interacting trapped fermions
PY  - 2021/12/22
UR  - https://scipost.org/SciPostPhys.11.6.110
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 11
IS  - 6
SP  - 110
A1  - Smith, Naftali R.
AU  - Le Doussal, Pierre
AU  - Majumdar, Satya
AU  - Schehr, Grégory
AB  - We study $N$ spinless fermions in their ground state confined by an external potential in one dimension with long range interactions of the general Calogero-Sutherland type. For some choices of the potential this system maps to standard random matrix ensembles for general values of the Dyson index $\beta$. In the fermion model $\beta$ controls the strength of the interaction, $\beta=2$ corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions ${\cal N}_{\cal D}$ in a domain $\cal{D}$ of macroscopic size in the bulk of the Fermi gas. We predict that for general $\beta$ the variance of ${\cal N}_{\cal D}$ grows as $A_{\beta} \log N + B_{\beta}$ for $N \gg 1$ and we obtain a formula for $A_\beta$ and $B_\beta$. This is based on an explicit calculation for $\beta\in\left\{ 1,2,4\right\} $ and on a conjecture that we formulate for general $\beta$. This conjecture further allows us to obtain a universal formula for the higher cumulants of ${\cal N}_{\cal D}$. Our results for the variance in the microscopic regime are found to be consistent with the predictions of the Luttinger liquid theory with parameter $K = 2/\beta$, and allow to go beyond. In addition we present families of interacting fermion models in one dimension which, in their ground states, can be mapped onto random matrix models. We obtain the mean fermion density for these models for general interaction parameter $\beta$. In some cases the fermion density exhibits interesting transitions, for example we obtain a noninteracting fermion formulation of the Gross-Witten-Wadia model.
ER  -

@Article{10.21468/SciPostPhys.11.6.110,
	title={{Full counting statistics for interacting trapped fermions}},
	author={Naftali R. Smith and Pierre Le Doussal and Satya N. Majumdar and Grégory Schehr},
	journal={SciPost Phys.},
	volume={11},
	pages={110},
	year={2021},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.11.6.110},
	url={https://scipost.org/10.21468/SciPostPhys.11.6.110},
}

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Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.

¹ ² Naftali R. Smith,
³ Pierre Le Doussal,
³ Satya Majumdar,
⁴ Grégory Schehr

Funder for the research work leading to this publication

Agence Nationale de la Recherche [ANR]