We demonstrate that a machine learning technique with a simple feedforward neural network can sensitively detect two successive phase transitions associated with the Berezinskii-Kosterlitz-Thouless (BKT) phase in q-state clock models simultaneously by analyzing the weight matrix components connecting the hidden and output layers. We find that the method requires only a data set of the raw spatial spin configurations for the learning procedure. This data set is generated by Monte Carlo thermalizations at selected temperatures. Neither prior knowledge of, for example, the transition temperatures, number of phases, and order parameters nor processed data sets of, for example, the vortex configurations, histograms of spin orientations, and correlation functions produced from the original spin-configuration data are needed, in contrast with most of previously proposed machine learning methods based on supervised learning. Our neural network evaluates the transition temperatures as T2/J=0.921 and T1/J=0.410 for the paramagnetic-to-BKT transition and BKT-to-ferromagnetic transition in the eight-state clock model on a square lattice. Both critical temperatures agree well with those evaluated in the previous numerical studies.
Berezinskii--Kosterlitz--Thouless transition of the two-dimensional XY model on the honeycomb lattice
The Berezinskii--Kosterlitz--Thouless (BKT) transition of the two-dimensional XY model on the honeycomb lattice is investigated using both the techniques of Neural Network (NN) and Monte Carlo simulations. It is demonstrated in the literature that with certain plausible assumptions, the associated critical temperature TBKT,H is found to be 21 exactly. Surprisingly, the value of TBKT,H obtained from our NN calculations is 0.560(9) which deviates significantly from 21. In addition, based on the helicity modulus, the TBKT,H determined is 0.571(8) agreeing well with that resulting from the NN estimation. The outcomes presented in this study indicate that a detailed analytic calculation is desirable to solve the found discrepancy.
Abstract The Berezinskii–Kosterlitz–Thouless (BKT) transition of the two-dimensional XY model on the honeycomb lattice is investigated using both the techniques of Neural network (NN) and Monte Carlo simulations. It is demonstrated in the literature that, with certain plausible assumptions, the associated critical temperature TBKT,H is found to be 1/2 exactly. Surprisingly, the value of TBKT,H obtained from our NN calculations is 0.572(3), which deviates significantly from 1/2. In addition, based on the helicity modulus, the TBKT,H determined is 0.576(4), agreeing well with that resulting from the NN estimation. It will be interesting to carry out a more detailed analytic calculation to obtain a theoretical value consistent with the numerical result reached here.