Multicritical edge statistics for the momenta of fermions in non-harmonic traps

TL;DR

This paper investigates the universal edge statistics of fermion momenta in non-harmonic traps, revealing new distributions governed by Painlevé equations, especially near flat trap minima, with implications for cold atom experiments.

Contribution

It introduces novel momentum edge statistics for flat traps with potentials $V(x) \\sim x^{2n}$, generalizing Airy kernel results and connecting to Painlevé hierarchies.

Findings

New universal distributions for momentum maxima derived from Painlevé equations

Explicit kernels based on generalizations of the Airy kernel

Potential experimental signatures in cold atom systems

Abstract

We compute the joint statistics of the momenta $p_i$ of $N$ non-interacting fermions in a trap, near the Fermi edge, with a particular focus on the largest one $p_{\max}$. For a $1d$ harmonic trap, momenta and positions play a symmetric role and hence, the joint statistics of momenta is identical to that of the positions. In particular, $p_{\max}$, as $x_{\max}$, is distributed according to the Tracy-Widom distribution. Here we show that novel "momentum edge statistics" emerge when the curvature of the potential vanishes, i.e. for "flat traps" near their minimum, with $V(x) \sim x^{2n}$ and $n>1$. These are based on generalisations of the Airy kernel that we obtain explicitly. The fluctuations of $p_{\max}$ are governed by new universal distributions determined from the $n$-th member of the second Painlev\'e hierarchy of non-linear differential equations, with connections to multicritical random matrix models. Finite temperature extensions and possible experimental signatures in cold atoms are discussed.