Noninteracting fermions in a trap and random matrix theory

Abstract

We review recent advances in the theory of trapped fermions using techniques borrowed from random matrix theory (RMT) and, more generally, from the theory of determinantal point processes. In the presence of a trap, and in the limit of a large number of fermions N 1, the spatial density exhibits an edge, beyond which it vanishes. While the spatial correlations far from the edge, i. e. close to the center of the trap, are well described by standard manybody techniques, such as the local density approximation (LDA), these methods fail to describe the fluctuations close to the edge of the Fermi gas, where the density is very small and the fluctuations are thus enhanced. It turns out that RMT and determinantal point processes offer a powerful toolbox to study these edge properties in great detail. Here we discuss the principal edge universality classes, that have been recently identified using these modern tools. In dimension d = 1 and at zero temperature T = 0, these universality classes are in one-toone correspondence with the standard universality classes found in the classical unitary random matrix ensembles: soft edge (described by the "Airy kernel") and hard edge (described by the "Bessel kernel") universality classes. We further discuss extensions of these results to higher dimensions d ≥ 2 and to finite temperature. Finally, we discuss correlations in the phase space, i.e., in the space of positions and momenta, characterized by the so called Wigner function. arXiv:1810.12583v1 [cond-mat.stat-mech]

Extreme value statistics of correlated random variables: A pedagogical review

“…The importance of these fluctuations means that traditional theories such as LDA break down in this edge region [230]. Recently, there has been some progress, using techniques from RMT and determinantal processes, to study analytically this edge region (for a short review see [231]). In particular, a natural observable to probe the fluctuations at the edge is the position of the fermion which is the farthest from the center of the trap [206,232], hence the connection to EVS.…”

Quantum information scrambling after a quantum quench

“…If x and x are in different intervals A i , then x − x ∝ d, and the integral is dominated by the stationary point at k = −π/2. Crucially, the saddle point contribution vanishes at k = π/2, and one has to consider the order k 3 , which gives (8) is ubiquitous in freefermion models and it is related to the Airy processes [72][73][74][75][76][77][78]. At this point, the mutual information is dominated by the elements of C x,x coming from x, x in different intervals and, since is finite, the same d −2/3 behavior holds in general.…”

Free fermions at the edge of interacting systems

“…Refs. [20][21][22][23]), we do not need it here, and rely instead on standard quantum mechanics techniques to solve our quantum mechanics problems. From this perspective, Airy and T-W scaling follow at a fundamental level [10] from the free fermions Hamiltonian (2) and its Dirac sea ground state.…”