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arXiv:1704.01628 · Physical Review Letters

Kinetic Energy of a Trapped Fermi Gas at Finite Temperature

Jacek Grela, Satya N. Majumdar, Grégory Schehr

Published Sep 26, 2017• 16 citations
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Abstract

We study the statistics of the kinetic (or, equivalently, potential) energy for N noninteracting fermions in a 1d harmonic trap of frequency ω at finite temperature T. Remarkably, we find an exact solution for the full distribution of the kinetic energy, at any temperature T and for any N, using a nontrivial mapping to an integrable Calogero-Moser-Sutherland model. As a function of temperature T and for large N, we identify (i) a quantum regime, for T∼ℏω, where quantum fluctuations dominate and (ii) a thermal regime, for T∼Nℏω, governed by thermal fluctuations. We show how the mean and the variance as well as the large deviation function associated with the distribution of the kinetic energy cross over from the quantum to the thermal regime as T increases.

DOI
10.1103/physrevlett.119.130601

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We study the statistics of the kinetic (or, equivalently, potential) energy for N noninteracting fermions in a 1d harmonic trap of frequency ω at finite temperature T. Remarkably, we find an exact solution for the full distribution of the kinetic energy, at any temperature T and for any N, using a nontrivial mapping to an integrable Calogero-Moser-Sutherland model. As a function of temperature T and for large N, we identify (i) a quantum regime, for T∼ℏω, where quantum fluctuations dominate and (ii) a thermal regime, for T∼Nℏω, governed by thermal fluctuations. We show how the mean and the variance as well as the large deviation function associated with the distribution of the kinetic energy cross over from the quantum to the thermal regime as T increases.
We present closed analytical expressions for the particle and kinetic-energy spatial densities at finite temperatures for a system of noninteracting fermions (bosons) trapped in a d-dimensional harmonic-oscillator potential. For d=2d=2d=2 and 3,3,3, exact expressions for the N-particle densities are used to calculate perturbatively the temperature dependence of the splittings of the energy levels in a given shell due to a very weak interparticle interaction in a dilute Fermi gas. In two dimensions, we obtain analytically the surprising result that the l degeneracy in a harmonic-oscillator shell is not lifted in the lowest order even when the exact, rather than the Thomas-Fermi expression for the particle density is used. We also demonstrate rigorously (in two dimensions) the reduction of the exact zero-temperature fermionic expressions to the Thomas-Fermi form in the large-N limit.
We study N N spinless fermions in their ground state confined by an external potential in one dimension with long range interactions of the general Calogero-Sutherland type. For some choices of the potential this system maps to standard random matrix ensembles for general values of the Dyson index \beta β . In the fermion model \beta β controls the strength of the interaction, \beta=2 β = 2 corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions N_D N D in a domain D D of macroscopic size in the bulk of the Fermi gas. We predict that for general \beta β the variance of N_D N D grows as A_{\beta} \log N + B_{\beta} A β log N + B β for N \gg 1 N ≫ 1 and we obtain a formula for A_\beta A β and B_\beta B β . This is based on an explicit calculation for \beta\in\left\{ 1,2,4\right\} β ∈ { 1 , 2 , 4 } and on a conjecture that we formulate for general \beta β . This conjecture further allows us to obtain a universal formula for the higher cumulants of N_D N D . Our results for the variance in the microscopic regime are found to be consistent with the predictions of the Luttinger liquid theory with parameter K = 2/\beta K =
We compute the full probability distribution of the moment of inertia I \propto \sum_{i=1}^N \vec{r}_i^{\,2} I ∝ ∑ i = 1 N r → i 2 of a gas of N N noninteracting bosons trapped in a harmonic potential V(r) = (1/2)\, m\, \omega^2 r^2 V ( r ) = ( 1 / 2 ) m ω 2 r 2 , in all dimensions and at all temperature. The appropriate thermodynamic limit in a trapped Bose gas consists in taking the limit N\to ∞ N → ∞ and \omega\to 0 ω → 0 with their product \rho = N \omega^d ρ = N ω d fixed, where \rho ρ plays the role analogous to the density in a translationally invariant system. In this thermodynamic limit and in dimensions d;gt 1 d > 1 , the harmonically trapped Bose gas undergoes a Bose-Einstein condensation (BEC) transition as the density \rho ρ crosses a critical value \rho_c(\beta) ρ c ( β ) , where \beta β denotes the inverse temperature. We show that the nature of the condensation is different for 1<d≤ 2 1 < d ≤ 2 and d;gt 2
One-particle properties of noninteracting fermions in a one-dimensional harmonic trap and at zero temperature are studied. Exact expressions and asymptotic results for a large fermion number N are given for the particle density distribution n0(z,N).{n}_{0}(z,N).n0​(z,N). For large N and near the classical boundary at the Fermi energy, the density displays increasing fluctuations. A simple scaling of these tails of the density distribution with respect to N is established. The Fourier transform of the density distribution is calculated exactly. It displays a small but characteristic hump near 2kF,{2k}_{F},2kF​, with kF{k}_{F}kF​ being a properly defined Fermi wave number. This is due to Friedel oscillations, which are identified and discussed. These quantum effects are missing in the semiclassical approximation. Momentum distributions are also evaluated and discussed. As an example of a time-dependent one-particle problem, we calculate exactly the evolution of the particle density when the trap is suddenly switched off, and find a simple scaling behavior, in agreement with recent general results.
We derive simple analytical expressions for the particle density $\ensuremath{\rho}(r)$ and the kinetic energy density $\ensuremath{\tau}(r)$ for a system of noninteracting fermions in a ddd-dimensional isotropic harmonic oscillator potential. We test the Thomas-Fermi (TF, or local-density) approximation for the functional relation $\ensuremath{\tau}[\ensuremath{\rho}]$ using the exact $\ensuremath{\rho}(r)$ and show that it locally reproduces the exact kinetic energy density $\ensuremath{\tau}(r)$, including the shell oscillations, surprisingly well everywhere except near the classical turning point. For the special case of two dimensions (2D), we obtain the unexpected analytical result that the integral of ${\ensuremath{\tau}}_{\mathrm{TF}}[\ensuremath{\rho}(r)]$ yields the exact total kinetic energy.
We extend to finite temperature a Green's-function method that was previously proposed to evaluate ground-state properties of mesoscopic clouds of noninteracting fermions moving under harmonic confinement in one dimension. By calculations of the particle and kinetic-energy density profiles, we illustrate the role of thermal excitations in smoothing out the quantum shell structure of the cloud and in spreading the particle spill out from quantum tunnel at the edges. We also discuss the approach of the exact density profiles to the predictions of a semiclassical model often used in the theory of confined atomic gases at finite temperature.