Machine-Learning Studies on Spin Models

Abstract

With the recent developments in machine learning, Carrasquilla and Melko have proposed a paradigm that is complementary to the conventional approach for the study of spin models. As an alternative to investigating the thermal average of macroscopic physical quantities, they have used the spin configurations for the classification of the disordered and ordered phases of a phase transition through machine learning. We extend and generalize this method. We focus on the configuration of the long-range correlation function instead of the spin configuration itself, which enables us to provide the same treatment to multi-component systems and the systems with a vector order parameter. We analyze the Berezinskii-Kosterlitz-Thouless (BKT) transition with the same technique to classify three phases: the disordered, the BKT, and the ordered phases. We also present the classification of a model using the training data of a different model.

Figure 1

A schematic diagram of the fully connected neural network in the present simulation.

Figure 2

(a) The output layer averaged over a test set as a function of T for the 2D 3-state Potts model. The system sizes are L = 24, 32, and 48. The samples of T within the ranges $0.85\le T\le 0.94$ and $1.06\le T\le 1.15$ are used for the training data. In the inset, the finite-size scaling plot is given, where the horizontal axis is chosen as $t{L}^{\mathrm{1/}\nu }$ with $t=(T-T_c)/J$. The values of T_c and v are $T_c=1/\ln (1+\sqrt{3})=0.995$ and v = 5/6, respectively. (b) The same plot for the 2D 5-state Potts model. The system sizes are the same. The samples of T within the ranges $0.7\le T\le 0.79$ and $0.91\le T\le 1.0$ are used for the training data.

Figure 3

(a) The output layer for the 5-state Potts model using the training data of the 3-state Potts model. (b) The output layer for the 3-state Potts model using the training data of the 5-state Potts model.

Figure 4

(a) The output layer averaged over a test set as a function of T for the 2D 6-state clock model. The system sizes are L = 24, 32, 48, and 64. The samples of T within the ranges $0.4\le T\le 0.64$, $0.77\le T\le 0.83$, and $0.96\le T\le 1.2$ are used for the training data. (b) The same plot for the 2D 4-state clock model. The samples of T within the ranges $0.9\le T\le 1.06$ and $1.2\le T\le 1.4$ are used for the training data.

Figure 5

The output layer for the 4-state clock model using the training data of the 6-state clock model.