Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization

TL;DR

This paper explores the localization and delocalization transitions of elastic lines in disordered media, revealing connections to random matrix theory, KPZ universality, and extending the Baik-Ben Arous-Peche transition to complex defect distributions.

Contribution

It establishes a novel link between elastic line pinning transitions and BBP phase transitions in random matrices, and analyzes fluctuations using advanced probabilistic methods.

Findings

Delocalization transition coincides with BBP transition in random matrix spectrum.

Ground state energy fluctuations follow Tracy-Widom distribution in delocalized phase.

Localized phase fluctuations are described by KPZ universality class distributions.

Abstract

We revisit the problem of an elastic line (e.g. a vortex line in a superconductor) subject to both columnar disorder and point disorder in dimension $d=1+1$. Upon applying a transverse field, a delocalization transition is expected, beyond which the line is tilted macroscopically. We investigate this transition in the fixed tilt angle ensemble and within a one-way model where backward jumps are neglected. From recent results about directed polymers and their connections to random matrix theory, we find that for a single line and a single strong defect this transition in presence of point disorder coincides with the Baik-Ben Arous-Peche (BBP) transition for the appearance of outliers in the spectrum of a perturbed random matrix in the GUE. This transition is conveniently described in the polymer picture by a variational calculation. In the delocalized phase, the ground state energy exhibits Tracy-Widom fluctuations. In the localized phase we show, using the variational calculation, that the fluctuations of the occupation length along the columnar defect are described by $f_{KPZ}$, a distribution which appears ubiquitously in the Kardar-Parisi-Zhang universality class. We then consider a smooth density of columnar defect energies. Depending on how this density vanishes at its lower edge we find either (i) a delocalized phase only (ii) a localized phase with a delocalization transition. We analyze this transition which is an infinite-rank extension of the BBP transition. The fluctuations of the ground state energy of a single elastic line in the localized phase (for fixed columnar defect energies) are described by a Fredholm determinant based on a new kernel. The case of many columns and many non-intersecting lines, relevant for the study of the Bose glass phase, is also analyzed. The ground state energy is obtained using free probability and the Burgers equation.