Dynamical Critical Properties of the Random Transverse-Field Ising Spin Chain

TL;DR

This paper investigates the dynamical critical behavior of the random transverse-field Ising chain at criticality, revealing a multiscaling phenomenon in correlation functions through numerical analysis of free fermion mappings.

Contribution

It provides a detailed numerical study of the distribution of local correlation functions at criticality, highlighting a new type of multiscaling behavior distinct from multifractality.

Findings

Correlation functions decay algebraically with time.

Distribution of the log of correlation functions follows a specific scaling form.

Identifies a new multiscaling behavior in critical dynamics.

Abstract

We study the dynamical properties of the random transverse-field Ising chain at criticality using a mapping to free fermions, with which we can obtain numerically exact results for system sizes, L, as large as 256. The probability distribution of the local imaginary time correlation function S(tau) is investigated and found to be simply a function of alpha = -log S(tau) / log(tau). This scaling behavior implies that the typical correlation function decays algebraically where the exponent is determined from the distribution of alpha. The precise value for the exponent depends on exactly how the ``typical'' correlation function is defined. The form of P(alpha) for small alpha gives a contribution to the average correlation function, which varies as a power of the logarithm of time, which was obtained recently in Europhys. Lett. 39, 135 (1997). These results represent a type of ``multiscaling'' different from the well-known ``multifractal'' behavior.