SciPost Physics

SciPost Phys. 4, 014 (2018) · published 24 March 2018

• doi: 10.21468/SciPostPhys.4.3.014
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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.

• Kulkarni et al., Density profile of noninteracting fermions in a rotating two-dimensional trap at finite temperature
  Phys. Rev. A 107, 023302 (2023) [Crossref]
• Lacroix-A-Chez-Toine et al., Non-interacting fermions in hard-edge potentials
  J. Stat. Mech. 2018, 123103 (2018) [Crossref]
• Giraud et al., Correlations of occupation numbers in the canonical ensemble and application to a Bose-Einstein condensate in a one-dimensional harmonic trap
  Phys. Rev. A 97, 053615 (2018) [Crossref]
• Mohanta et al., Full statistics of nonequilibrium heat and work for many-body quantum Otto engines and universal bounds: A nonequilibrium Green's function approach
  Phys. Rev. E 108, 064127 (2023) [Crossref]
• Lewin et al., Semi-classical limit of large fermionic systems at positive temperature
  60, 091901 (2019) [Crossref]
• Dean et al., Noninteracting fermions in a trap and random matrix theory
  J. Phys. A: Math. Theor. 52, 144006 (2019) [Crossref]
• Barghathi et al., Theory of noninteracting fermions and bosons in the canonical ensemble
  Phys. Rev. Research 2, 043206 (2020) [Crossref]
• Krajenbrink et al., Systematic time expansion for the Kardar–Parisi–Zhang equation, linear statistics of the GUE at the edge and trapped fermions
  Nuclear Physics B 936, 239 (2018) [Crossref]
• Smith et al., Counting statistics for noninteracting fermions in a d -dimensional potential
  Phys. Rev. E 103, L030105 (2021) [Crossref]
• Cunden et al., Free fermions andα-determinantal processes
  J. Phys. A: Math. Theor. 52, 165202 (2019) [Crossref]
• Grabsch, General truncated linear statistics for the top eigenvalues of random matrices
  J. Phys. A: Math. Theor. 55, 124001 (2022) [Crossref]