Lifshitz tails at spectral edge and holography with a finite cutoff

TL;DR

This paper develops a holographic model using 2D dilaton gravity to describe Lifshitz tails in spectral densities of disordered systems, linking geometric path fluctuations to disorder effects.

Contribution

It introduces a novel holographic approach employing polymer representation of JT gravity to model Lifshitz tails in disordered quantum systems.

Findings

Identifies small loop regimes responsible for Lifshitz tail formation.

Relates disorder strength to boundary dilaton value.

Shows path fluctuations follow KPZ scaling, enabling dual description.

Abstract

We propose the holographic description of the Lifshitz tail typical for one-particle spectral density of bounded disordered system in $D=1$ space. To this aim the "polymer representation" of the Jackiw-Teitelboim (JT) 2D dilaton gravity at a finite cutoff is used and the corresponding partition function is considered as the weighted sum over paths of fixed length in an external magnetic field. We identify the regime of small loops, responsible for emergence of a Lifshitz tail in the Gaussian disorder, and relate the strength of disorder to the boundary value of the dilaton. The geometry corresponding to the Poisson disorder in the boundary theory involves random paths fluctuating in the vicinity of the hard impenetrable cut-off disc in a 2D plane. It is shown that the ensemble of "stretched" paths evading the disc possesses the Kardar-Parisi-Zhang (KPZ) scaling for fluctuations, which is the key property that ensures the dual description of the Lifshitz tail in the spectral density for the Poisson disorder.