Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature

Abstract

We consider $N$ non-interacting fermions in an isotropic $d$-dimensional harmonic trap. We compute analytically the cumulative distribution of the maximal radial distance of the fermions from the trap center at zero temperature. While in $d=1$ the limiting distribution (in the large $N$ limit), properly centered and scaled, converges to the squared Tracy-Widom distribution of the Gaussian Unitary Ensemble in Random Matrix Theory, we show that for all $d>1$, the limiting distribution converges to the Gumbel law. These limiting forms turn out to be universal, i.e., independent of the details of the trapping potential for a large class of isotropic trapping potentials. We also study the position of the right-most fermion in a given direction in $d$ dimensions and, in the case of a harmonic trap, the maximum momentum, and show that they obey similar Gumbel statistics. Finally, we generalize these results to low but finite temperature.

28 pages, 4 figures

References in corpus (9)

• Many-Body Physics with Ultracold Gases
• Theory of ultracold Fermi gases
• Single-Atom Resolved Fluorescence Imaging of an Atomic Mott Insulator
• A Quantum Gas Microscope for Fermionic Atoms
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory
• Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Phase transitions and edge scaling of number variance in Gaussian random matrices

Cited by in corpus (18)

• Extreme value statistics of correlated random variables: a pedagogical review
• Multicritical edge statistics for the momenta of fermions in non-harmonic traps
• Wigner function of noninteracting trapped fermions
• Universality of the third-order phase transition in the constrained Coulomb gas
• Counting statistics for non-interacting fermions in a $d$-dimensional potential
• Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble
• Anomalous 1D fluctuations of a simple 2D random walk in a large deviation regime
• Third-order phase transition: random matrices and screened Coulomb gas with hard walls
• Last-passage time for linear diffusions and application to the emptying time of a box
• Brownian flights over a circle
• Noninteracting trapped Fermions in double-well potentials: inverted parabola kernel
• Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature
• Kernels for noninteracting fermions via a Green's function approach with applications to step potentials
• Hole probability for noninteracting fermions in a $d$-dimensional trap
• Edge fluctuations for random normal matrix ensembles
• Exact first-order effect of interactions on the ground-state energy of harmonically-confined fermions
• Universality of Top Rank Statistics for Brownian Reshuffling
• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"