Momenta spacing distributions in anharmonic oscillators and the higher order finite temperature Airy kernel

TL;DR

This paper rigorously derives the limiting distribution of the extreme momentum of noninteracting fermions in anharmonic traps at positive temperature, revealing a hierarchy governed by a Painlevé-II integro-differential system.

Contribution

It introduces a novel higher order finite temperature Airy kernel and connects it to a Painlevé-II hierarchy using Riemann-Hilbert techniques.

Findings

Edge momentum distribution follows a Painlevé-II hierarchy.

Established a connection between Painlevé-II hierarchy and an integro-differential mKdV hierarchy.

Extended known results from harmonic to anharmonic traps at finite temperature.

Abstract

We rigorously compute the integrable system for the limiting $(N\rightarrow\infty)$ distribution function of the extreme momentum of $N$ noninteracting fermions when confined to an anharmonic trap $V(q)=q^{2n}$ for $n\in\mathbb{Z}_{\geq 1}$ at positive temperature. More precisely, the edge momentum statistics in the harmonic trap $n=1$ are known to obey the weak asymmetric KPZ crossover law which is realized via the finite temperature Airy kernel determinant or equivalently via a Painlev\'e-II integro-differential transcendent, cf. \cite{LW,ACQ}. For general $n\geq 2$, a novel higher order finite temperature Airy kernel has recently emerged in physics literature \cite{DMS} and we show that the corresponding edge law in momentum space is now governed by a distinguished Painlev\'e-II integro-differential hierarchy. Our analysis is based on operator-valued Riemann-Hilbert techniques which produce a Lax pair for an operator-valued Painlev\'e-II ODE system that naturally encodes the aforementioned hierarchy. As byproduct, we establish a connection of the integro-differential Painlev\'e-II hierarchy to a novel integro-differential mKdV hierarchy.