Noninteracting fermions at finite temperature in a d -dimensional trap: Universal correlations

Abstract

We study a system of $N$ noninteracting spinless fermions trapped in a confining potential, in arbitrary dimensions $d$ and arbitrary temperature $T$. The presence of the confining trap breaks the translational invariance and introduces an edge where the average density of fermions vanishes. Far from the edge, near the center of the trap (the so-called ``bulk regime''), where the fermions do not feel the curvature of the trap, physical properties of the fermions have traditionally been understood using the local density (or Thomas-Fermi) approximation. However, these approximations drastically fail near the edge where the density vanishes and thermal and quantum fluctuations are thus enhanced. The main goal of this paper is to show that, even near the edge, novel universal properties emerge, independently of the details of the shape of the confining potential. We present a unified framework to investigate both the bulk and the edge properties of the fermions. We show that for large $N$, these fermions in a confining trap, in arbitrary dimensions and at finite temperature, form a determinantal point process. As a result, any $n$-point correlation function, including the average density profile, can be expressed as an $n\ifmmode\times\else\texttimes\fi{}n$ determinant whose entry is called the kernel, a central object for such processes. Near the edge, we derive the large-$N$ scaling form of the kernels, parametrized by $d$ and $T$. In $d=1$ and $T=0$, this reduces to the so-called Airy kernel, that appears in the Gaussian unitary ensemble (GUE) of random matrix theory. In $d=1$ and $T>0$ we show a remarkable connection between our kernel and the one appearing in the ($1+1$)-dimensional Kardar-Parisi-Zhang equation at finite time. Consequently, our result provides a finite-$T$ generalization of the Tracy-Widom distribution, that describes the fluctuations of the position of the rightmost fermion at $T=0$, or those of the largest single-fermion momentum. In $d>1$ and $T\ensuremath{\ge}0$, while the connection to GUE no longer holds, the process is still determinantal whose analysis provides a new class of kernels, generalizing the $1d$ Airy kernel at $T=0$ obtained in random matrix theory. Some of our finite-temperature results should be testable in present-day cold-atom experiments, most notably our detailed predictions for the temperature dependence of the fluctuations near the edge.