Critical Exponents of the Classical 3D Heisenberg Model: A Single-Cluster Monte Carlo Study

TL;DR

This study uses single-cluster Monte Carlo simulations to accurately determine critical exponents of the 3D Heisenberg model, leveraging larger lattices and improved estimators to reduce systematic errors.

Contribution

It introduces an efficient simulation approach enabling precise measurement of critical exponents for the 3D Heisenberg model with larger lattices and advanced analysis techniques.

Findings

Verification of reduced critical slowing down at phase transition.

Accurate estimates of critical exponents $ u,eta, ho, ext{etc.}$.

Improved estimators for correlation length and susceptibility.

Abstract

We have simulated the three-dimensional Heisenberg model on simple cubic lattices, using the single-cluster Monte Carlo update algorithm. The expected pronounced reduction of critical slowing down at the phase transition is verified. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors. In one set of simulations we employ the usual finite-size scaling methods to compute the critical exponents $\nu,\alpha,\beta,\gamma, \eta$ from a few measurements in the vicinity of the critical point, making extensive use of histogram reweighting and optimization techniques. In another set of simulations we report measurements of improved estimators for the spatial correlation length and the susceptibility in the high-temperature phase, obtained on lattices with up to $100^3$ spins. This enables us to compute independent estimates of $\nu$ and $\gamma$ from power-law fits of their critical divergencies.