Anderson's orthogonality catastrophe in one dimension induced by a magnetic field

TL;DR

This paper investigates how a magnetic field affects the orthogonality catastrophe in a one-dimensional Fermi gas, revealing significant phase effects on ground state overlaps despite gauge invariance.

Contribution

It provides the first analysis of Anderson's orthogonality catastrophe induced by a magnetic potential in one dimension, including asymptotics for ground state overlaps under different boundary conditions.

Findings

Overlap asymptotics for periodic boundary conditions derived from Toeplitz determinant theory.

Upper bounds on overlap for Dirichlet boundary conditions established.

Two-term asymptotics of ground-state energy differences analyzed.

Abstract

According to Anderson's orthogonality catastrophe, the overlap of the $N$-particle ground states of a free Fermi gas with and without an (electric) potential decays in the thermodynamic limit. For the finite one-dimensional system various boundary conditions are employed. Unlike the usual setup the perturbation is introduced by a magnetic (vector) potential. Although such a magnetic field can be gauged away in one spatial dimension there is a significant and interesting effect on the overlap caused by the phases. We study the leading asymptotics of the overlap of the two ground states and the two-term asymptotics of the difference of the ground-state energies. In the case of periodic boundary conditions our main result on the overlap is based upon a well-known asymptotic expansion by Fisher and Hartwig on Toeplitz determinants with a discontinuous symbol. In the case of Dirichlet boundary conditions no such result is known to us and we only provide an upper bound on the overlap, presumably of the right asymptotic order.