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arXiv:2212.14655 · Progress of Theoretical and Experimental Physics

Machine learning phases of an Abelian gauge theory

Jhao-Hong Peng, Yuan-Heng Tseng, Fu-Jiun Jiang

Published Jul 1, 2023• 6 citations
arXivHTMLPDF

Abstract

Abstract The phase transition of the two-dimensional U(1) quantum link model on the triangular lattice is investigated by employing a supervised neural network (NN) consisting of only one input layer, one hidden layer of two neurons, and one output layer. No information on the studied model is used when the NN training is conducted. Instead, two artificially made configurations are considered as the training set. Interestingly, the obtained NN not only estimates the critical point accurately but also uncovers the physics correctly. The results presented here imply that a supervised NN, which has a very simple architecture and is trained without any input from the investigated model, can identify the targeted phase structure with high precision.

DOI
10.1093/ptep/ptad096

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Abstract The phase transition of the two-dimensional U(1) quantum link model on the triangular lattice is investigated by employing a supervised neural network (NN) consisting of only one input layer, one hidden layer of two neurons, and one output layer. No information on the studied model is used when the NN training is conducted. Instead, two artificially made configurations are considered as the training set. Interestingly, the obtained NN not only estimates the critical point accurately but also uncovers the physics correctly. The results presented here imply that a supervised NN, which has a very simple architecture and is trained without any input from the investigated model, can identify the targeted phase structure with high precision.
We study the phase transitions of the two-dimensional antiferromagnetic Ising model with nearest J1J_1J1​ and next-to-nearest J2J_2J2​ interactions on the triangular lattice for J2/J1=0.1,0.5J_2/J_1 = 0.1, 0.5J2​/J1​=0.1,0.5 and 1.0. The method of supervised neural networks (NN) is employed for the investigation. While supervised NN is used, no real spin configurations are needed for the training. In addition, two kinds of configurations having their spins be arranged in a staggered pattern are considered as the training set. Remarkably, with this unconventional training strategy, not only the critical temperatures of the studied J2/J1J_2/J_1J2​/J1​ are computed accurately by the resulting NN, but also the nature of the investigated phase transitions are determined correctly. Specifically, the phase transitions associated with J2/J1=0.1,0.5J_2/J_1 = 0.1, 0.5J2​/J1​=0.1,0.5 and 1.0 are first order. These conclusions are consistent with the known results obtained by other methods. Since the training strategy is simple, the NN calculations is highly efficient. It remains to examine whether the unconventional training approach considered in this study can be used to investigate other models with untypical phase transitions or with nontrivial ground state configurations.
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.
Abstract We study the phase transitions of the 2D antiferromagnetic Ising model with nearest J1J_1J1​ and next-to-nearest J2J_2J2​ interactions on the triangular lattice for J2/J1=0.1,0.5J_2/J_1 = 0.1, 0.5J2​/J1​=0.1,0.5, and 1.0. The method of supervised neural networks (NNs) is employed for the investigation. Although supervised NN is used, no real spin configurations are needed for the training. In addition, two kinds of configurations which have their spins arranged in a staggered pattern are considered as the training set. Remarkably, with this unconventional training strategy, not only are the critical temperatures of the studied J2/J1J_2/J_1J2​/J1​ computed accurately by the resulting NN, but also the nature of the investigated phase transitions is determined correctly. Specifically, the phase transitions associated with J2/J1=0.1,0.5J_2/J_1 = 0.1, 0.5J2​/J1​=0.1,0.5, and 1.0 are first order. These conclusions are consistent with the known results obtained by other methods. Since the training strategy is simple, the NN calculations are highly efficient. It remains to be examined whether the unconventional training approach considered in this study can be used to investigate other models with untypical phase transitions or with nontrivial ground-state configurations.
We study the phase transitions of three-dimensional (3D) classical O(3)O(3)O(3) model and two-dimensional (2D) classical XY model, as well as both the quantum phase transitions of 2D and 3D dimerized spin-1/2 antiferromagnets, using the technique of supervised neural network (NN). Moreover, unlike the conventional approaches commonly used in the literature, the training sets employed in our investigation are neither the theoretical nor the real configurations of the considered systems. Remarkably, with such an unconventional set up of the training stage in conjunction with some semiexperimental finite-size scaling formulas, the associated critical points determined by the NN method agree well with the established results in the literature. The outcomes obtained here imply that certain unconventional training strategies, like the one used in this study, are not only cost-effective in computation but are also applicable for a wide range of physical systems.
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.
A universal (supervised) neural network (NN), which is only trained once on a one-dimensional lattice of 200 sites, is employed to study the phase transition of the two-dimensional (2D) 5-state ferromagnetic Potts model on the square lattice. In particular, the NN is obtained by using merely two artificially made configurations as the training set. Due to the elegant features of the considered NN, results associated with systems consisting of over 4000000 spins can be obtained with ease, and convincing evidence showing the investigated phase transition is weakly first order is reached. The outcomes demonstrated here can hardly be achieved with the standard NNs that are commonly used in the literature.