The Random Transverse Ising Spin Chain and Random Walks

TL;DR

This paper investigates the critical and off-critical behavior of the random transverse-field Ising chain, extending analysis to surface properties and ferromagnetic phases, with exact calculations and numerical simulations revealing scaling laws and Griffiths-McCoy singularities.

Contribution

It introduces a phenomenological theory for scaling and Griffiths-McCoy singularities, and relates surface magnetization to random walk survival probabilities, extending previous studies.

Findings

Surface magnetization linked to adsorbing walk survival probability

Exact critical exponents calculated for the model

Griffiths-McCoy singularities characterized by a single varying exponent

Abstract

We study the critical and off-critical (Griffiths-McCoy) regions of the random transverse-field Ising spin chain by analytical and numerical methods and by phenomenological scaling considerations. Here we extend previous investigations to surface quantities and to the ferromagnetic phase. The surface magnetization of the model is shown to be related to the surviving probability of an adsorbing walk and several critical exponents are exactly calculated. Analyzing the structure of low energy excitations we present a phenomenological theory which explains both the scaling behavior at the critical point and the nature of Griffiths-McCoy singularities in the off-critical regions. In the numerical part of the work we used the free-fermion representation of the model and calculated the critical magnetization profiles, which are found to follow very accurately the conformal predictions for different boundary conditions. In the off-critical regions we demonstrated that the Griffiths-McCoy singularities are characterized by a single, varying exponent, the value of which is related through duality in the paramagnetic and ferromagnetic phases.