Extreme statistics of the excitations in the random transverse Ising chain

István A Kovács,Tamás Pető,Ferenc Iglói

doi:10.1103/physrevresearch.3.033140

Abstract

In random quantum magnets, like the random transverse Ising chain, low energy excitations are localized in rare regions and there are only weak correlations between them. It is an open question whether these correlations are relevant in the sense of the renormalization group. To answer this question, we calculate the distribution of the excitation energy of the random transverse Ising chain in the disordered Griffiths phase with high numerical precision by the strong disorder renormalization group method and---for shorter chains---by free-fermion techniques. Asymptotically, the two methods give identical results, which are well fitted by the Fr\'echet limit law of the extremes of independent and identically distributed random numbers. Considering the finite size corrections, the two numerical methods give very similar results, but these differ from the correction term for uncorrelated random variables, indicating that the weak correlations between low-energy excitations in random quantum magnets are relevant.

Highlights

Many-body systems in the presence of quenched disorder have unusual dynamical properties due to rare-region effects
It is an intriguing problem to what extent the localized excitations in random quantum magnets are independent? Are weak correlations between the rare regions manifested in some effects, such as in the form of finite-size corrections? In this paper, we address this question and consider the Published by the American Physical Society
We have considered a paradigmatic model of random quantum magnets, the random transverse Ising model in 1D, and studied the distribution of low-energy excitations in the paramagnetic Griffiths phase, with extensive numerical methods

Summary

Introduction

Many-body systems in the presence of quenched disorder have unusual dynamical properties due to rare-region effects In these systems—due to extreme fluctuations of strong couplings—domains are formed, which can remain locally ordered even in the paramagnetic phase. The relaxation time τ , associated with turning the spins in such domains, can be extremely large and it has no upper limit in the thermodynamic limit. These types of Griffiths singularities are responsible for nonanalytical behavior of several physical quantities (susceptibility, specific heat, autocorrelation function) in the so called Griffiths phase, which is an extended part of the paramagnetic phase [1].

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