Abstract
We study a system of 1D non-interacting 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)ℒ=∑nh(xn). 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}_+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.
Highlights
A remarkable recent achievement is the development of Fermi quantum microscopes [16,17,18], allowing the direct measurement of the fermions’ positions in a confining trap
When quantum correlations are dominant, like for the problem we aim to study, it is well known that the most efficient approach is supplied by the grand canonical ensemble in which the temperature T and the chemical potential μ are fixed, while the energy and the number of fermions fluctuate
It is straightforward to show that the canonical occupation numbers exhibit the particle-hole symmetry around the Fermi level: nN +k c which is exactly the same as the grand canonical relation (60). These symmetries (60,63), along with relations (53,57) on the occupation numbers will be essential for our study of linear statistics for fermions in a harmonic trap
Summary
The recent experimental progresses in cold atoms [1,2,3], which have made accessible new types of observables, have stimulated a renewed interest in the study of fermionic systems over the past few years. A remarkable recent achievement is the development of Fermi quantum microscopes [16,17,18], allowing the direct measurement of the fermions’ positions in a confining trap Motivated by this context, several theoretical studies have focused on different observables counting the fermions in a given spatial domain When quantum correlations are dominant, like for the problem we aim to study, it is well known that the most efficient approach is supplied by the grand canonical ensemble in which the temperature T and the chemical potential μ are fixed, while the energy and the number of fermions fluctuate. Our aim here is to introduce a general framework to analyse the fluctuations of arbitrary linear statistics within both the grand canonical and canonical ensembles
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