Dynamic critical exponents of Swendsen-Wang and Wolff algorithms by nonequilibrium relaxation

TL;DR

This study uses nonequilibrium relaxation to determine the dynamic critical exponents of the Swendsen-Wang and Wolff algorithms in the 2D Ising model, revealing size-dependent relaxation behaviors and deriving an exact spectrum for a 1D case.

Contribution

It introduces a nonequilibrium relaxation method to accurately measure dynamic exponents and provides new insights into the size dependence of relaxation times for these algorithms.

Findings

Relaxation times follow a logarithmic size dependence up to L=8192.

Effective dynamic exponent z_exp=1.19(2) for Wolff with disordered initial state.

Derived an exact dynamic spectrum for the 1D Ising chain.

Abstract

With a nonequilibrium relaxation method, we calculate the dynamic critical exponent z of the two-dimensional Ising model for the Swendsen-Wang and Wolff algorithms. We examine dynamic relaxation processes following a quench from a disordered or an ordered initial state to the critical temperature T_c, and measure the exponential relaxation time of the system energy. For the Swendsen-Wang algorithm with an ordered or a disordered initial state, and for the Wolff algorithm with an ordered initial state, the exponential relaxation time fits well to a logarithmic size dependence up to a lattice size L=8192. For the Wolff algorithm with a disordered initial state, we obtain an effective dynamic exponent z_exp=1.19(2) up to L=2048. For comparison, we also compute the effective dynamic exponents through the integrated correlation times. In addition, an exact result of the Swendsen-Wang dynamic spectrum of a one-dimension Ising chain is derived.