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

David S. Dean 1 Pierre Le Doussal 2 Satya. N. Majumdar 3 Gregory Schehr 3, *

Journal of Statistical Mechanics: Theory and Experiment, IOP Science, 2017, 2017 (6), pp.063301 (1-39). 〈10.1088/1742-5468/aa6dda〉

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.

  • 1. LOMA - Laboratoire Ondes et Matière d'Aquitaine
  • 2. LPTENS - Laboratoire de Physique Théorique de l'ENS
  • 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques