Full counting statistics for interacting trapped fermions

Abstract

We study N spinless fermions in their ground state confined by an external potential in one dimension with long range interactions of the general Calogero-Sutherland type. For some choices of the potential this system maps to standard random matrix ensembles for general values of the Dyson index β. In the fermion model β controls the strength of the interaction, β = 2 corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions N D in a domain D of macroscopic size in the bulk of the Fermi gas. We predict that for general β the variance of N D grows as A β log N + B β for N 1 and we obtain a formula for A β and B β . This is based on an explicit calculation for β ∈ {1, 2, 4} and on a conjecture that we formulate for general β. This conjecture further allows us to obtain a formula for the higher cumulants of N D . Our results for the variance in the microscopic regime are found to be consistent with the predictions of the Luttinger liquid theory with parameter K = 2/β, and allow to go beyond. In addition we present families of interacting fermion models in one dimension which, in their ground states, can be mapped onto random matrix models. We obtain the mean fermion density for these models for general interaction parameter β. In some cases the fermion density exhibits interesting transitions, for example we obtain a noninteracting fermion formulation of the Gross-Witten-Wadia model.

On the generating function of the Pearcey process

“…The asymptotics for the variance of the counting functions of other classical point processes such as the sine, Airy and Bessel point processes are also of the same form [42,15,17,14] (with different values for c 1 and c 2 ). This phenomena is expected to be universal, in the sense that it is expected to hold for many point processes in random matrix theory and other related fields, see the very general predictions [40,41]. We emphasize that the proof of (1.17…”

Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators

“…There is also a correspondance between free fermions on the unit circle and the circular unitary ensembles [22]. One can realize Gaussian/circular β-ensembles for any β > 0 by considering the ground state of certain interacting Fermi gases known as Calogero-Sutherland systems [36,80] (these are the only cases where such exact mappings exist). This suggests that more generally, β-ensembles fall in the universality class of certain interacting 1-dimensional Fermi gases.…”

“…Finally, there are also explicit results on number variance fluctuations, cf. [80] and the perspectives below.…”

“…This problem has been extensively studied recently in the context of random matrix theory based on so-called Fisher-Hartwig asymptotics and it relates optimal particles' rigidity and Gaussian multiplicative chaos, e.g. [80,21,39] In a gapped system (Integer Quantum Hall states linked with Berezin-Toeplitz quantization), the socalled area law holds and macroscopic CLTs have been rigorously established in [18,34].…”

Quantum Generalized Hydrodynamics of the Tonks-Girardeau gas: density fluctuations and entanglement entropy

“…The operator φ(s) satisfies the U(1) current algebra [∂ φ(s), ∂ φ(s )] = −2πiδ (s − s ) as a consequence of (25). The two-point function in the ground state of the quadratic Hamiltonian (32) is…”