Transverse-field Ising spin chain with inhomogeneous disorder

TL;DR

This paper investigates how inhomogeneous disorder near the boundary affects the critical and off-critical properties of the transverse-field Ising spin chain, revealing new relevance criteria and critical behaviors through exact and numerical methods.

Contribution

It provides exact analytical results and numerical verification for the impact of boundary inhomogeneity on surface magnetization and critical phenomena in the disordered quantum spin chain.

Findings

Inhomogeneity with slow decay ($\,\kappa<1/2$) is relevant, altering surface order and magnetization.

At the marginal case ($\kappa=1/2$), the surface magnetization exhibits a power-law decay with a variable critical exponent.

Griffiths phase properties remain unaffected by boundary inhomogeneity.

Abstract

We consider the critical and off-critical properties at the boundary of the random transverse-field Ising spin chain when the distribution of the couplings and/or transverse fields, at a distance $l$ from the surface, deviates from its uniform bulk value by terms of order $l^{-\kappa}$ with an amplitude $A$. Exact results are obtained using a correspondence between the surface magnetization of the model and the surviving probability of a random walk with time-dependent absorbing boundary conditions. For slow enough decay, $\kappa<1/2$, the inhomogeneity is relevant: Either the surface stays ordered at the bulk critical point or the average surface magnetization displays an essential singularity, depending on the sign of $A$. In the marginal situation, $\kappa=1/2$, the average surface magnetization decays as a power law with a continuously varying, $A$-dependent, critical exponent which is obtained analytically. The behavior of the critical and off-critical autocorrelation functions as well as the scaling form of the probability distributions for the surface magnetization and the first gaps are determined through a phenomenological scaling theory. In the Griffiths phase, the properties of the Griffiths-McCoy singularities are not affected by the inhomogeneity. The various results are checked using numerical methods based on a mapping to free fermions.