Griffiths-McCoy Singularities in the Random Transverse-Field Ising Spin Chain

TL;DR

This paper investigates Griffiths singularities in the paramagnetic phase of the random transverse-field Ising chain, revealing power-law behaviors in dynamical quantities and relating critical exponents to the dynamical exponent z.

Contribution

The study extends previous work by analyzing additional quantities like non-linear susceptibility and energy-density autocorrelation, establishing their power-law singularities and their relation to the dynamical exponent z.

Findings

Power-law singularities in dynamical quantities in the Griffiths phase.

The critical exponents depend on the distance from the critical point.

The decay of autocorrelation functions follows specific power laws related to z.

Abstract

We consider the paramagnetic phase of the random transverse-field Ising spin chain and study the dynamical properties by numerical methods and scaling considerations. We extend our previous work [Phys. Rev. B 57, 11404 (1998)] to new quantities, such as the non-linear susceptibility, higher excitations and the energy-density autocorrelation function. We show that in the Griffiths phase all the above quantities exhibit power-law singularities and the corresponding critical exponents, which vary with the distance from the critical point, can be related to the dynamical exponent z, the latter being the positive root of [(J/h)^{1/z}]_av=1. Particularly, whereas the average spin autocorrelation function in imaginary time decays as [G]_av(t)~t^{-1/z}, the average energy-density autocorrelations decay with another exponent as [G^e]_av(t)~t^{-2-1/z}.