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

### Aurélien Grabsch 1 Satya N. Majumdar 1 Gregory Schehr 1 Christophe Texier 1

#### SciPost Physics Journals, the SciPost Foundation, 2018

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.

• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques