Critical exponents of random XX and XY chains: Exact results via random walks

TL;DR

This paper derives exact critical exponents for random XX and XY spin chains at phase transitions, linking quantum critical behavior to properties of random walks, and providing precise decay and dynamical exponents.

Contribution

It provides exact analytical results for critical exponents in disordered quantum spin chains using a novel connection to random walk persistence.

Findings

Exact decay exponents for correlations are determined.

Critical and Griffiths singularities are characterized precisely.

Dynamical and correlation length exponents are explicitly calculated.

Abstract

We study random XY and (dimerized) XX spin-1/2 quantum spin chains at their quantum phase transition driven by the anisotropy and dimerization, respectively. Using exact expressions for magnetization, correlation functions and energy gap, obtained by the free fermion technique, the critical and off-critical (Griffiths-McCoy) singularities are related to persistence properties of random walks. In this way we determine exactly the decay exponents for surface and bulk transverse and longitudinal correlations, correlation length exponent and dynamical exponent.