Machine learning of frustrated classical spin models. I. Principal component analysis

This work aims at determining whether artificial intelligence can recognize a phase transition without prior human knowledge. If this were successful, it could be applied to, for instance, analyzing data from the quantum simulation of unsolved physical models. Toward this goal, we first need to apply the machine learning algorithm to well-understood models and see whether the outputs are consistent with our prior knowledge, which serves as the benchmark for this approach. In this work, we feed the computer data generated by the classical Monte Carlo simulation for the $XY$ model in frustrated triangular and union jack lattices, which has two order parameters and exhibits two phase transitions. We show that the outputs of the principal component analysis agree very well with our understanding of different orders in different phases, and the temperature dependences of the major components detect the nature and the locations of the phase transitions. Our work offers promise for using machine learning techniques to study sophisticated statistical models, and our results can be further improved by using principal component analysis with kernel tricks and the neural network method.

Figure 1

Schematic of (a) square, (b) triangular, and (c) union jack lattices.

Figure 2

(a) The eigenvalues of the $\mathcal{S}$ matrix for the $XY$ model in a two-dimensional $18\times 18$ square lattice. Two eigenvalues are significantly larger than the others. (b) The projection of all $N=9000$ data sets onto the two-dimensional principal subspace $l_n=(l_n^1,l_n^2)$ ($n=1,\cdots ,N$). The temperature of the data set ranges from $0.2J$ to $1.8J$, with $\mathrm{\Delta }T=0.2J$, and at each temperature 1000 data sets are taken from the classical Monte Carlo simulation. The color bar indicates the temperature at which the data are generated.

Figure 3

Temperature-resolved PCA for the square-lattice $XY$ model in $L=18\times 18,L=24\times 24$, and $L=36\times 36$ lattices, respectively. We consider a temperature range from $0.2J$ to $1.8J$, with $\mathrm{\Delta }T=0.1J$, and at each fixed temperature, $N_0=1000$ data sets are input. The dashed line indicates the expected temperature for the KT transition in the square-lattice $XY$ model.

Figure 4

Schematics of two types of chiral orders for the $XY$ model in a triangular lattice.

Figure 5

The eigenvalues of the $\mathcal{S}$ matrix for the $XY$ model in a two-dimensional $18\times 18$ triangular lattice. Four eigenvalues are significantly larger than the others. The temperature of the data set ranges from $0.3J$ to $0.7J$, with $\mathrm{\Delta }T=0.05J$, and at each temperature $N_0=1000$ data sets are taken from the classical Monte Carlo simulation.

Figure 6

The projection of data for the $XY$ model in a triangular lattice to the principal subspace. The principal subspace is generated by data from all temperature. (a)–(f) For $(l_n^1,l_n^2),(l_n^1,l_n^3),(l_n^1,l_n^4),(l_n^2,l_n^3),(l_n^2,l_n^4)$, and $(l_n^3,l_n^4)$, respectively. The color bar indicates the temperature at which the data are generated.

Figure 7

Temperature-resolved PCA for the triangular-lattice $XY$ model in $L=18\times 18,L=24\times 24$, and $L=36\times 36$ lattices. We consider a temperature range from $0.3J$ to $0.7J$, with $\mathrm{\Delta }T=0.025J$, and at each fixed temperature, $N_0=1000$ data sets are input. The dashed line indicates the expected temperature for the KT transition in the triangular-lattice $XY$ model.

Figure 8

Schematics of two types of chiral orders for the $XY$ model in the union jack lattice at low temperature and an intermediate antiferromagnetic order at intermediate temperature.

Figure 9

The eigenvalues of the $\mathcal{S}$ matrix for the $XY$ model in a two-dimensional $18\times 18$ union jack lattice. Four eigenvalues are significantly larger than the others. The temperature of the data set ranges from $0.2J$ to $1.0J$, with $\mathrm{\Delta }T=0.1J$, and at each temperature $N_0=1000$ data sets are taken from the classical Monte Carlo simulation.

Figure 10

The projection of data for the $XY$ model in the union jack lattice to the principal subspace. (a)–(f) correspond to $(l_n^1,l_n^2),(l_n^1,l_n^3),(l_n^1,l_n^4),(l_n^2,l_n^3),(l_n^2,l_n^4)$, and $(l_n^3,l_n^4)$, respectively. The color bar indicates the temperatures at which the data are generated.

Figure 11

Temperature-resolved PCA for the union-jack-lattice $XY$ model in $L=18\times 36,L=24\times 48$, and $L=36\times 72$ lattices, respectively. We consider a temperature range from $0.2J$ to $1.0J$, with $\mathrm{\Delta }T=0.025J$, and at each fixed temperature, $N_0=1000$ data sets are input. The dashed line indicates the expected temperature for the Ising transition and the KT transition in the union-jack-lattice $XY$ model. The inset shows a comparison of the critical behavior between the union jack lattice and square lattice; the eigenvalues are rescaled so that the data coincide at $T/T_c=1$, and here we consider $L=36\times 72$ for the union jack lattice and $L=36\times 36$ for the square lattice.