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arXiv:2501.17815

A machine-learning study of phase transitions in Ising, Blume-Capel, and Ising-metamagnet models

Vasanth Kumar Babu, Rahul Anil Pandit

Published Jan 29, 2025• 0 citations
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Abstract

We combine machine-learning (ML) techniques with Monte Carlo (MC) simulations and finite-size scaling (FSS) to study continuous and first-order phase transitions in Ising, Blume-Capel, and Ising-metamagnet spin models. We go beyond earlier studies that had concentrated on obtaining the correlation-length exponent $ν$. In particular, we show (a) how to combine neural networks (NNs), trained with data from MC simulations of Ising-type spin models on finite lattices, with FSS to obtain both thermal magnetic exponents $y_t = 1/ν$ and $y_h$, respectively, at both critical and tricritical points, (b) how to obtain the NN counterpart of two-scale-factor universality at an Ising-type critical point, and (c) FSS at a first-order transition. We also obtain the FSS forms for the output of our trained NNs as functions of both the temperature and the magnetic field.

DOI
10.48550/arxiv.2501.17815

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•Oct 14, 2019•0 citations
We combine machine-learning (ML) techniques with Monte Carlo (MC) simulations and finite-size scaling (FSS) to study continuous and first-order phase transitions in Ising, Blume-Capel, and Ising-metamagnet spin models. We go beyond earlier studies that had concentrated on obtaining the correlation-length exponent ννν. In particular, we show (a) how to combine neural networks (NNs), trained with data from MC simulations of Ising-type spin models on finite lattices, with FSS to obtain both thermal magnetic exponents yt=1/νy_t = 1/νyt​=1/ν and yhy_hyh​, respectively, at both critical and tricritical points, (b) how to obtain the NN counterpart of two-scale-factor universality at an Ising-type critical point, and (c) FSS at a first-order transition. We also obtain the FSS forms for the output of our trained NNs as functions of both the temperature and the magnetic field.
We combine machine-learning techniques with Monte Carlo (MC) simulations and finite-size scaling (FSS) to study continuous and first-order phase transitions in Ising, Blume-Capel, and Ising-metamagnet spin models. We go beyond earlier studies that had concentrated on obtaining the correlation-length exponent ν. In particular, we show (a) how to combine neural networks (NNs), trained with data from MC simulations of Ising-type spin models on finite lattices, with FSS to obtain both thermal magnetic exponents y_{t}=1/ν and y_{h}, respectively, at both critical and tricritical points, (b) how to obtain the NN counterpart of two-scale-factor universality at an Ising-type critical point, and (c) how to get FSS at a first-order transition. We also obtain the FSS forms for the output of our trained NNs as functions of both the temperature and the magnetic field.
We investigate phase transitions in the two-dimensional Ising model with competing nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions, where the NNN coupling generates a degenerate superantiferromagnetic (SAF) ground state, by combining machine-learning methods with Monte Carlo simulations and finite-size scaling (FSS). Going beyond earlier studies, we show that unsupervised learning can recover order-parameter and Binder cumulant analogs that obey FSS. Principal component analysis (PCA) reveals the two-component structure of the SAF order parameter, and k-means clustering is used to visualize qualitative differences between ordered and disordered configurations in the principal-component space. A fully connected neural network (FCNN), trained without prior knowledge of the order parameter or the critical temperature, learns the strength of SAF stripe ordering and uses it as the internal feature for phase classification. FSS of thermodynamic observables, the PCA-derived quantity, and the FCNN output yields consistent estimates of the critical temperature and correlation-length exponent. These results demonstrate that machine learning can decode ordering patterns, extract meaningful order-parameter analogs, and reproduce critical behavior in frustrated Ising systems.
Abstract Studying percolation phase transitions offers valuable insights into the characteristics of phase transitions, shedding light on the underlying mechanisms that govern the formation of global connectivity within a system. We explore the percolation phase transition in the 3D cubic Ising model by employing two machine learning techniques. Our results demonstrate that machine learning methods can distinguish different phases during the percolation transition. Through the finite-size scaling analysis on the output of the neural networks, the percolation temperature and a correlation length exponent in the geometrical percolation transition are extracted and compared to those in the thermal magnetization phase transition within the 3D Ising model. These findings provide a valuable method for enhancing our understanding of the properties of the QCD critical point, which belongs to the same universality class as the 3D Ising model.
Abstract We investigate the connection between the supervised learning of the binary phase classification in the ferromagnetic Ising model and the standard finite-size-scaling theory of the second-order phase transition. Proposing a minimal one-free-parameter neural network model, we analytically formulate the supervised learning problem for the canonical ensemble being used as a training data set. We show that just one free parameter is capable enough to describe the data-driven emergence of the universal finite-size-scaling function in the network output that is observed in a large neural network, theoretically validating its critical point prediction for unseen test data from different underlying lattices yet in the same universality class of the Ising criticality. We also numerically demonstrate the interpretation with the proposed one-parameter model by providing an example of finding a critical point with the learning of the Landau mean-field free energy being applied to the real data set from the uncorrelated random scale-free graph with a large degree exponent.
We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.
In recent years Machine Learning has proved to be successful in many technological applications and scientific tasks, such as image and speech recognition, natural language understanding and more. In this work we apply both supervised and unsupervised Machine Learning to three key problems in statistical physics: design of Hamiltonian from data, phase recognition and study of critical properties of a system undergoing phase transition. As an example of model design, we use a dataset with spin configurations and corresponding energies randomly sampled from a one dimensional Ising model. We then make two guesses for the correct Hamiltonian: the former involving interactions of spins with an external local field, the latter involving two body interactions among spins. The coupling constants are determined with a linear penalized regression, comparing the effects of L1 and L2 penalization terms of the cost function. We pay specific attention to the problem of overfitting and to the validation process, which is critical for accepting or rejecting the proposed model. The phase recognition problem is faced with the two dimensional Ising and XY models as examples. After showing the limits of a simple softmax regression for this task, we build suitable neural networks to overcome these limits. In particular, a feed forward network is built and the learning process is investigated for the Ising model; while a more sophisticated convolutional network is proposed for the XY model in order to detect local topological structures. The last part of the work is dedicated to the unsupervised study of phase transitions, and the determination of critical properties (order parameter, critical temperature, critical exponents). The discussed techniques are Principal Component Analysis (PCA) for dimensional reduction and K-means clustering for organizing data into subsets with specific properties. PCA is applied to the two dimensional Ising, Potts and XY models, and it is used to find suitable order parameters. The study of the proposed order parameter as a function of temperature provides evidence of phase transition, and a finite size scaling allows to extrapolate both the critical temperature in the thermodynamic limit, and the ν\nuν critical exponent for the correlation length. K-means clustering is applied to equilibrium configurations of the two dimensional Ising model, before and after the dimensional reduction, and the critical temperature is estimated. Moreover, a clustering of relaxation curves of magnetization in a Monte Carlo dynamics is used to build a phase diagram on the parameter space of temperature and magnetic field. A synergy of C++, Python and Wolfram Mathematica 12.0 is used throughout this work in order to sample input datasets and to build and control customized neural networks and learning tools. The most relevant codes are provided in the appendix.