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arXiv:2007.00355 · Physical review. E

Mapping distinct phase transitions to a neural network

Dimitrios Bachtis, Gert Aarts A, Biagio Lucini A

Published Nov 16, 2020• 41 citations
arXivHTMLPDF

Abstract

We demonstrate, by means of a convolutional neural network, that the features learned in the two-dimensional Ising model are sufficiently universal to predict the structure of symmetry-breaking phase transitions in considered systems irrespective of the universality class, order, and the presence of discrete or continuous degrees of freedom. No prior knowledge about the existence of a phase transition is required in the target system and its entire parameter space can be scanned with multiple histogram reweighting to discover one. We establish our approach in q-state Potts models and perform a calculation for the critical coupling and the critical exponents of the ϕ^{4} scalar field theory using quantities derived from the neural network implementation. We view the machine learning algorithm as a mapping that associates each configuration across different systems to its corresponding phase and elaborate on implications for the discovery of unknown phase transitions.

DOI
10.1103/physreve.102.053306

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•Jan 1, 2020•7 citations
We demonstrate, by means of a convolutional neural network, that the features learned in the two-dimensional Ising model are sufficiently universal to predict the structure of symmetry-breaking phase transitions in considered systems irrespective of the universality class, order, and the presence of discrete or continuous degrees of freedom. No prior knowledge about the existence of a phase transition is required in the target system and its entire parameter space can be scanned with multiple histogram reweighting to discover one. We establish our approach in q-state Potts models and perform a calculation for the critical coupling and the critical exponents of the ϕ^{4} scalar field theory using quantities derived from the neural network implementation. We view the machine learning algorithm as a mapping that associates each configuration across different systems to its corresponding phase and elaborate on implications for the discovery of unknown phase transitions.
Over the past several years, there have been many studies demonstrating the ability of deep neural networks to identify phase transitions in many physical systems, notably in classical statistical physics systems. One often finds that the prediction of deep learning methods trained on many ensembles below and above the critical temperature T_{c} behaves similarly to an order parameter, and this analogy has been successfully used to locate T_{c} and estimate universal critical exponents. In this work, we pay particular attention to the ability of a convolutional neural network to capture these critical parameters for the two-dimensional Ising model when the network is trained on configurations at T=0 and T=∞ only. We directly compare its output to the same network trained at multiple temperatures below and above T_{c} to gain understanding of how this extreme restriction of training data can impact a neural network's ability to classify phases. We find that the network trained on two temperatures is still able to identify T_{c} and ν, while the extraction of γ becomes more challenging.
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.
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.
Conventionally, the training of a neural network for learning phases of matter uses real physical quantities as the training set. However, it has been demonstrated in several studies that this may not be required. Here we investigate the phase transitions of the two-dimensional (2D) q-state Potts models on the square lattice using a pre-trained neural network (NN). The employed NN was trained previously using two artifically made configurations as the training set. Hence no training is conducted for the present study. Remarkably, the used NN not only calculates the critical points of the considered phase transitions precisely, but also determines the nature of these phase transition definitely without ambiguity. Our results as well as the outcomes found previously suggest that training a NN using real physical quantities as the training set is not required to obtain a working NN. These unconventional studies also imply that if the configuration space of a system can be classified into two categories, then the NN considered here is likely applicable to study the phase transition of that system. Comparison between our approach and other known NN methods for studying the 2D q-state Potts models is briefly discussed as well.
A universal supervised neural network (NN) relevant to compute the associated criticalities of real experiments studying phase transitions is constructed. The validity of the built NN is examined by applying it to calculate the criticalities of several three-dimensional (3D) models on the cubic lattice, including the classical O(3)O(3)O(3) model, the 5-state ferromagnetic Potts model, and a dimerized quantum antiferromagnetic Heisenberg model. Particularly, although the considered NN is only trained one time on a one-dimensional (1D) lattice with 120 sites, yet it has successfully determined the related critical points of the studied 3D systems. Moreover, real configurations of states are not used in the testing stage. Instead, the employed configurations for the prediction are constructed on a 1D lattice of 120 sites and are based on the bulk quantities or the microscopic states of the considered models. As a result, our calculations are ultimately efficient in computation and the applications of the built NN is extremely broaden. Considering the fact that the investigated systems vary dramatically from each other, it is amazing that the combination of these two strategies in the training and the testing stages lead to a highly universal supervised neural network for learning phases and criticalities of 3D models. Based on the outcomes presented in this study, it is favorably probable that much simpler but yet elegant machine learning techniques can be constructed for fields of many-body systems other than the critical phenomena.
The numerical emulation of quantum systems often requires an exponential number of degrees of freedom which translates to a computational bottleneck. Methods of machine learning have been used in adjacent fields for effective feature extraction and dimensionality reduction of high-dimensional datasets. Recent studies have revealed that neural networks are further suitable for the determination of macroscopic phases of matter and associated phase transitions as well as efficient quantum state representation. In this work, we address quantum phase transitions in quantum spin chains, namely the transverse field Ising chain and the anisotropic XY chain, and show that even neural networks with no hidden layers can be effectively trained to distinguish between magnetically ordered and disordered phases. Our neural network acts to predict the corresponding crossovers finite-size systems undergo. Our results extend to a wide class of interacting quantum many-body systems and illustrate the wide applicability of neural networks to many-body quantum physics.