Confusion scheme in machine learning detects double phase transitions and quasi-long-range order

Thanks to the development of machine learning techniques, it has been shown that the supervised learning can be useful to study critical phenomena of various systems. However, the supervised learning cannot be done without labels which require knowledge about critical behavior of the system. To overcome this barrier, unsupervised machine learning methods have been considered and the confusion scheme has been proposed. In this study, we use the confusion scheme of the unsupervised learning and investigate critical behavior of various systems which exhibit single (double) phase transitions with (without) quasi-long-range order. In detail, we choose the two-color Ashkin-Teller model, the $XY$ model, and the eight-state clock model as such systems and snapshots of the spin configurations at various temperatures are collected via Monte Carlo simulations to be used as input data for the unsupervised machine learning. We also put focus on the size dependence of results and validate the availability of the confusion scheme in thermodynamic limit. Our results indicate that the confusion scheme of the unsupervised learning successfully locates the approximate transition points for all models and becomes more accurate as the system size is increased. We also find a characteristic feature of the result which reflects the presence of a quasi-long-range order. We conclude that regardless of the presence of a quasi-long-range order, single and double phase transitions can be detected via the confusion scheme.

Figure 1

Specific heat $c$ from MC simulations (a),(b) and the test accuracy $a$ in the confusion scheme (c),(d) for the two-color AT model for $K_4/K_2=1$ (a),(c) and for $K_4/K_2=7$ (b),(d) for various system sizes $L=12,16,20,24,28$, and 40. The piecewise linear grey curves in (c) and (d) are the expected ideal confusion scheme curves with the critical temperature picked at the maximum of the specific heat for the largest system size $L=40$. The peak positions of the accuracy measurements agree with the corresponding peak positions of the ideal confusion scheme results.

Figure 2

Specific heats $c$ (a),(b) and the accuracy $a$ (c),(d) for the $XY$ model (a),(c), and the eight-state clock model (b)(d). Confusion scheme successfully detects (c) the single and (d) the double transitions, and the Kosterlitz-Thouless transitions around $T\approx 1.0$ are reflected in the form of the sharper peaks for the larger system sizes. The grey curves in (c) and (d) show expected ideal confusion scheme curves of the largest system size $L=40$.

Figure 3

Accuracy $a$ for the two-color AT model with (a) $K_4/K_2=1$, (b) $K_4/K_2=7$, (c) for the $XY$ model, and (d) for the eight-state clock model, all for the system size $L=20$. We use different sizes $H=200,400,800,1600$ of the hidden layer for the feedforward network, and the result from the convolutional neural network (CNN) is also presented.

Figure 4

The total prediction error $E_{\mathrm{tot}}\equiv {\sum }_T^{\prime }[1-a\left(T^{\prime }\right)]$ with the test accuracy $a\left(T^{\prime }\right)$ at the reference temperature $T^{\prime }$ is displayed as a function of the size $H$ of the hidden layer network, for the AT model at (a) $K_4/K_2=1$ and (b) $K_4/K_2=7$. Insets are for $a\left(T^{\prime }\right)$.