Random antiferromagnetic quantum spin chains: Exact results from scaling of rare regions

TL;DR

This paper develops a scaling theory for the critical and Griffiths phases of random antiferromagnetic XY and XX spin chains, providing exact critical exponents and confirming them with numerical simulations based on free fermion mappings.

Contribution

It introduces a phenomenological scaling approach based on rare regions and random walk distributions, extending understanding of dynamical and surface properties in disordered quantum spin chains.

Findings

Exact critical decay exponents derived for volume and surface

Numerical results confirm conformal decay profiles

Non-universal power-law autocorrelation decay in Griffiths phase

Abstract

We study XY and dimerized XX spin-1/2 chains with random exchange couplings by analytical and numerical methods and scaling considerations. We extend previous investigations to dynamical properties, to surface quantities and operator profiles, and give a detailed analysis of the Griffiths phase. We present a phenomenological scaling theory of average quantities based on the scaling properties of rare regions, in which the distribution of the couplings follows a surviving random walk character. Using this theory we have obtained the complete set of critical decay exponents of the random XY and XX models, both in the volume and at the surface. The scaling results are confronted with numerical calculations based on a mapping to free fermions, which then lead to an exact correspondence with directed walks. The numerically calculated critical operator profiles on large finite systems (L<=512) are found to follow conformal predictions with the decay exponents of the phenomenological scaling theory. Dynamical correlations in the critical state are in average logarithmically slow and their distribution show multi-scaling character. In the Griffiths phase, which is an extended part of the off-critical region average autocorrelations have a power-law form with a non-universal decay exponent, which is analytically calculated. We note on extensions of our work to the random antiferromagnetic XXZ chain and to higher dimensions.