Fluctuations of observables for free fermions in a harmonic trap at finite temperature

TL;DR

This paper analyzes fluctuations of linear statistics for noninteracting fermions in a harmonic trap at finite temperature, deriving general relations and demonstrating ensemble dependence of fluctuations.

Contribution

It introduces a general relation for occupation number fluctuations applicable to arbitrary traps and ensembles, and provides explicit formulas for harmonic traps.

Findings

Fluctuations depend on the statistical ensemble in the thermodynamic limit.

Derived compact expressions for variance of linear statistics in harmonic traps.

Validated analytical results with numerical simulations.

Abstract

We study a system of 1D noninteracting spinless fermions in a confining trap at finite temperature. We first derive a useful and general relation for the fluctuations of the occupation numbers valid for arbitrary confining trap, as well as for both canonical and grand canonical ensembles. Using this relation, we obtain compact expressions, in the case of the harmonic trap, for the variance of certain observables of the form of sums of a function of the fermions' positions, $\mathcal{L}=\sum_n h(x_n)$. Such observables are also called linear statistics of the positions. As anticipated, we demonstrate explicitly that these fluctuations do depend on the ensemble in the thermodynamic limit, as opposed to averaged quantities, which are ensemble independent. We have applied our general formalism to compute the fluctuations of the number of fermions $\mathcal{N}_+$ on the positive axis at finite temperature. Our analytical results are compared to numerical simulations. We discuss the universality of the results with respect to the nature of the confinement.