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arXiv:2506.18561 · Physical review. E

Stability of universal properties against perturbations of the Markov chain Monte Carlo algorithm

Matteo Bacci, Claudio Bonati

Published Aug 26, 2025• 0 citations
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Abstract

We numerically investigate the stability of universal properties at continuous phase transitions against perturbations of the Markov chain Monte Carlo algorithm used to simulate the system. We consider the three-dimensional XY model as a test bed, and both local (single-site Metropolis) and global (single-cluster) updates, introducing deterministic truncation-like perturbations and stochastic perturbations in the acceptance probabilities. In (almost) all the cases, we find a remarkable stability of the universal properties, even against large perturbations of the Markov chain Monte Carlo algorithm, with critical exponents and scaling curves consistent with those of the standard XY model within statistical uncertainties. Only for the single-cluster update with very large truncation error does something different happen, but large scaling corrections prevent us from precisely assessing the critical properties of the transition, and, in particular, to understand whether the critical behavior observed corresponds to a known universality class.

DOI
10.1103/vdxg-mmkl

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We numerically investigate the stability of universal properties at continuous phase transitions against perturbations of the Markov chain Monte Carlo algorithm used to simulate the system. We consider the three-dimensional XY model as a test bed, and both local (single-site Metropolis) and global (single-cluster) updates, introducing deterministic truncation-like perturbations and stochastic perturbations in the acceptance probabilities. In (almost) all the cases, we find a remarkable stability of the universal properties, even against large perturbations of the Markov chain Monte Carlo algorithm, with critical exponents and scaling curves consistent with those of the standard XY model within statistical uncertainties. Only for the single-cluster update with very large truncation error does something different happen, but large scaling corrections prevent us from precisely assessing the critical properties of the transition, and, in particular, to understand whether the critical behavior observed corresponds to a known universality class.
We perform a comprehensive numerical study of how variations in microscopic coupling strength affect critical behavior in the two-dimensional Ising model. Using extensive Monte Carlo simulations based on the Wolff cluster algorithm, we investigate the clean Ising model on a square lattice for interaction strengths in the range 0.80 ≤ J ≤ 1.20. While the exact solution at J = 1 has long established the model’s critical properties, a systematic numerical assessment of universality under continuous coupling rescaling has not been performed. Here, we address this gap by analyzing systems of linear size L = 20 − 500, allowing for high-precision finite-size scaling of thermodynamic observables. For each coupling, the critical temperature T_C(J) is determined via finite-size extrapolation of the susceptibility maxima, yielding results in quantitative agreement with Onsager’s exact prediction. At the extrapolated critical points, we extract the critical exponents ν, β, γ, and α through detailed scaling analyses. These exponents remain consistent with the exact Ising benchmarks across the full range of interaction strengths, confirming that the universal aspects of the phase transition are preserved. Our findings demonstrate that modifying the microscopic coupling strength rescales the critical temperature without altering the asymptotic critical behavior, thereby providing a clear and quantitative confirmation of the robustness of the two-dimensional Ising universality class.
The dipolar universality class describes the phase transition in 3D ferromagnets with strong dipolar interactions, as first discussed by Aharony and Fisher in the 1970s. While this universality class has been studied theoretically using renormalization group methods, as well as experimentally, little is known about it from Monte Carlo simulations. In this paper, we aim to bridge this gap. We introduce a lattice model that faithfully implements the transverse constraint on the order parameter. We introduce a Markov Chain Monte Carlo algorithm which involves a combination of local Metropolis updates preserving the constraint, and a global update of the zero mode. We perform simulations on cubic lattices up to volume 48×48×4848\times 48 \times 4848×48×48. We observe a continuous phase transition between the disordered and ordered phases. We obtain estimates of the critical exponent ννν and the critical Binder ratio, and compare them with results from other techniques. We also study whether rotational invariance emerges at the critical point, and discuss the influence of the toroidal geometry on this question.
We use the single-cluster Monte Carlo update algorithm to simulate the Ising model on two-dimensional Poissonian random lattices with up to 80 000 sites which are linked together according to the Voronoi-Delaunay prescription. In one set of simulations we use reweighting techniques and finite-size scaling analysis to investigate the critical properties of the model in the very vicinity of the phase transition. In the other set of simulations we study the approach to criticality in the disordered phase, making use of improved estimators for measurements. From both sets of simulations we obtain clear evidence that the critical exponents agree with the exactly known exponents for regular lattices, i.e., that (lattice) universality holds for the two-dimensional Ising model.
We perform large-scale Monte Carlo simulations of the classical XY model on a three-dimensional L×L×LL\times L \times LL×L×L cubic lattice using the graphics processing unit (GPU). By the combination of Metropolis single-spin flip, over-relaxation and parallel-tempering methods, we simulate systems up to L=160. Performing the finite-size scaling analysis, we obtain estimates of the critical exponents for the three-dimensional XY universality class: α=−0.01293(48)α=-0.01293(48)α=−0.01293(48) and ν=0.67098(16)ν=0.67098(16)ν=0.67098(16). Our estimate for the correlation-length exponent ννν, in contrast to previous theoretical estimates, agrees with the most recent experimental estimate νexp=0.6709(1)ν_{\rm exp}=0.6709(1)νexp​=0.6709(1) at the superfluid transition of 4^44He in a microgravity environment.
We present a Monte Carlo study of the two-component 4 model on the simple cubic lattice in three dimensions. By suitable tuning of the coupling constant, , we eliminate leading-order corrections to scaling. High-statistics simulations using finite-size scaling techniques yield = 0.6723(3)[8] and = 0.0381(2)[2], where the statistical and systematical errors are given in the first and second bracket, respectively. These results are more precise than any previous theoretical estimate of the critical exponents for the 3D XY universality class.