Universal ground state properties of free fermions in a $d$-dimensional trap

TL;DR

This paper analyzes the ground state properties of free fermions in a $d$-dimensional trap, revealing universal behaviors at the edges and providing exact calculations of correlation functions and kernels for large particle numbers.

Contribution

It introduces a universal scaling framework for the edge properties of free fermions in arbitrary dimensions, generalizing known results and deriving exact correlation kernels.

Findings

Density has finite support with a universal edge scaling behavior.

Edge kernel generalizes the Airy kernel to higher dimensions.

Correlation functions have a determinantal structure enabling exact large-$N$ calculations.

Abstract

The ground state properties of $N$ spinless free fermions in a $d$-dimensional confining potential are studied. We find that any $n$-point correlation function has a simple determinantal structure that allows us to compute several properties exactly for large $N$. We show that the average density has a finite support with an edge, and near this edge the density exhibits a universal (valid for a wide class of potentials) scaling behavior for large $N$. The associated edge scaling function is computed exactly and generalizes to any $d$ the edge electron gas result of Kohn and Mattsson in $d=3$ [Phys. Rev. Lett. 81, 3487 (1998)]. In addition, we calculate the kernel (that characterizes any $n$-point correlation function) for large $N$ and show that, when appropriately scaled, it depends only on dimension $d$, but has otherwise universal scaling forms, at the edges. The edge kernel, for higher $d$, generalizes the Airy kernel in one dimension, well known from random matrix theory.