Free fermions and the classical compact groups

Abstract

There is a close connection between the ground state of non-interacting fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: i) we consider the full family of admissible quantum boundary conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded interval, and the corresponding projection correlation kernels; ii) we construct the grand canonical extensions at finite temperature of the projection kernels, interpolating from Poisson to random matrix eigenvalue statistics. The scaling limits in the bulk and at the edges are studied in a unified framework, and the question of universality is addressed. Whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is, a priori, not obvious. We complete the picture by constructing a finite temperature extension of the Haar measure on the classical compact groups. The eigenvalue statistics of the resulting grand canonical matrix models (of random size) corresponds exactly to the grand canonical measure of non-interacting free fermions with classical boundary conditions.

35 pages, 5 figures. Final version

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• Non-interacting fermions in hard-edge potentials
• Interplay between transport and quantum coherences in free fermionic systems
• Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble
• Matrix models for classical groups and Toeplitz$\pm$Hankel minors with applications to Chern-Simons theory and fermionic models
• Exact equivalences and phase discrepancies between random matrix ensembles
• Free fermions and $\alpha$-determinantal processes
• Noninteracting trapped Fermions in double-well potentials: inverted parabola kernel
• Quench dynamics of noninteracting fermions with a delta impurity
• Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation
• The semiclassical limit of a quantum Zeno dynamics
• Truncated quantum observables and their semiclassical limit
• Quantum Systems with jump-discontinuous mass. I