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

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.

36 pages, 6 pdf figures

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• Counting statistics for non-interacting fermions in a $d$-dimensional potential
• Theory of Non-Interacting Fermions and Bosons in the Canonical Ensemble
• Non-interacting fermions in hard-edge potentials
• Free fermions and $\alpha$-determinantal processes
• Semi-classical limit of large fermionic systems at positive temperature
• Correlations of occupation numbers in the canonical ensemble and application to BEC in a 1D harmonic trap
• Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature
• General truncated linear statistics for the top eigenvalues of random matrices