Phase transitions of the five-state clock model on the square lattice

Using the tensor renormalization group method based on the higher-order singular value decomposition,we have studied the phase transitions of the five-state clock model on the square lattice.The temperature dependence of the specific heat indicates the system has two phase transitions, as verified clearly in the correlation function. By investigating the magnetic susceptibility, we can only obtain the upper critical temperature as Tc2 = 0.9565(7). From the fixed-point tensor, we locate the transition temperatures at Tc1 = 0.9029(1) and Tc2 = 0.9520(1), consistent with the MC and the DMRG results.