Machine-learning study using improved correlation configuration and application to quantum Monte Carlo simulation

We use the Fortuin-Kasteleyn representation-based improved estimator of the correlation configuration as an alternative to the ordinary correlation configuration in the machine-learning study of the phase classification of spin models. The phases of classical spin models are classified using the improved estimators, and the method is also applied to the quantum Monte Carlo simulation using the loop algorithm. We analyze the Berezinskii-Kosterlitz-Thouless (BKT) transition of the spin-1/2 quantum $XY$ model on the square lattice. We classify the BKT phase and the paramagnetic phase of the quantum $XY$ model using the machine-learning approach. We show that the classification of the quantum $XY$ model can be performed by using the training data of the classical $XY$ model.

Figure 1

Examples of the spin configuration $\left\{s_i\right\}$ [(a), (d)], correlation configuration $\left\{g_i(L/2)\right\}$ [(b), (e)], and improved correlation configuration $\left\{{\tilde{g}}_i(L/2)\right\}$ [(c), (f)] of the 2D three-state Potts model. The upper figures [(a)–(c)] are snapshots at the low temperature, $T=0.9$, and the lower figures [(d)–(f)] are those at the high temperature, $T=1.06$. The borders of FK clusters for the spin configuration are drawn by lines. They are copied in an improved correlation configuration.

Figure 2

(a) The output layer averaged over a test set as a function of $T$ for the 2D three-state Potts model. The system sizes are $L=32$, 48, and 64. 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 comparison is made between the results of improved correlation and those of ordinary correlation in the case of $L=48$. The exact $T_c$, $1/ln(1+\sqrt{3})\approx 0.995$, is shown as a dashed line for convenience. (b) The same plot for the 2D six-state clock model. The system sizes are the same. 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.

Figure 3

Comparison of snapshots of $\left\{{\tilde{g}}_i(L/2)\right\}$ of the 2D quantum [(a), (b)] and classical [(c), (d)] $XY$ models. Examples of snapshots below $T_{\mathrm{BKT}}$ [(a), (c)] and those above $T_{\mathrm{BKT}}$ [(b), (d)] are displayed.

Figure 4

(a) The output layer averaged over a test set as a function of $T$ for the 2D spin-1/2 $XY$ model. The system sizes are $L=32$, 48, and 64. The samples of $T$ within the ranges $0.16\le T\le 0.32$ and $0.38\le T\le 0.54$ are used for the training data. (b) The classification of the quantum $XY$ model using the training data of the classical $XY$ model (plane rotator). The samples of $T$ within the ranges $0.50\le T\le 0.84$ and $0.96\le T\le 1.30$ for the classical $XY$ model are used for the training data.