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arXiv:2103.15855 · Physical Review Research

Learning algorithm reflecting universal scaling behavior near phase transitions

Nishad Maskara, Michael Buchhold, Manuel Endres, Evert van Nieuwenburg

Published May 9, 2022• 8 citations

Abstract

Machine-learning-inspired techniques have emerged as a new paradigm for analysis of phase transitions in quantum matter.In this work, we introduce a supervised learning algorithm for studying critical phenomena from measurement data, "neural network scaling," which is based on iteratively training convolutional networks of increasing (spatial) complexity and test it on the transverse field Ising chain and q = 6 Potts model.At the continuous Ising transition, our scaling procedure directly reflects the hallmark of a continuous (second-order) phase transition, divergence of a characteristic length scale.Specifically, we extract a classification length scale by measuring the response of the classification accuracy while varying the largest convolution size (architecture of the network).We observe empirically the scaling exponent of the classification length is consistent with a power law with the correlation length exponent ν = 1.Furthermore, we demonstrate the versatility of our algorithm by showing the universal scaling behaviors persist across a variety of measurement bases, including when the order parameter is nonlocal.Finally, we show that the classification length scale remains finite for the q = 6 Potts model, which has a first-order transition and lacks a divergent correlation length.

DOI: 10.1103/physrevresearch.4.l022032

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The main question raised in the article is whether a neural network trained on a spin lattice model in one universality class can be used to test a model in another universality class. The quantities of interest are the critical phase transition temperature and the correlation length exponent. In other words, the question of transfer learning is how "universal" the trained network is and under what conditions. For this purpose, we applied a supervised learning procedure to three two-dimensional models for which critical properties are precisely known: The Ising model, the four-state Potts model, and the Baxter-Wu model. We consider two datasets: One with spin configurations and one with binding energy configurations. We find that estimates of the critical temperature agree well with the known results for both datasets, but not with the results of cross-testing using the energy datasets of the two models: The four-state Potts model and the Ising model. Estimates of the critical length exponent are less regular, and appear to be more accurate for energy datasets. A good example is the cross-testing using the energy dataset between the Ising model and the Baxter-Wu model in both training and testing directions.

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 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.