Comparative study of the critical behavior in one-dimensional random and aperiodic environments

TL;DR

This paper compares the critical behavior of quantum spin chains and random walks in one-dimensional random and aperiodic environments, revealing universal scaling laws and differences in singularities.

Contribution

It provides a detailed analysis of the scaling and critical exponents in fluctuating environments, highlighting universal behaviors and distinctions between random and aperiodic systems.

Findings

Scaling t ~ log^{1/omega} at criticality

Universal functions of omega for some critical exponents

Absence of Griffiths singularities in aperiodic systems

Abstract

We consider cooperative processes (quantum spin chains and random walks) in one-dimensional fluctuating random and aperiodic environments characterized by fluctuating exponents omega>0. At the critical point the random and aperiodic systems scale essentially anisotropically in a similar fashion: length (L) and time (t) scales are related as t ~ log^{1/omega}. Also some critical exponents, characterizing the singularities of average quantities, are found to be universal functions of omega, whereas some others do depend on details of the distribution of the disorder. In the off-critical region there is an important difference between the two types of environments: in aperiodic systems there are no extra (Griffiths)-singularities.