Phase Transition of the <i>q</i> -State Clock Model: Duality and Tensor Renormalization

Abstract

We investigate the critical behavior and the duality property of the\nferromagnetic $q$-state clock model on the square lattice based on the\ntensor-network formalism. From the entanglement spectra of local tensors\ndefined in the original and dual lattices, we obtain the exact self-dual points\nfor the model with $q \\leq 5 $ and approximate self-dual points for $q \\geq 6$.\nWe calculate accurately the lower and upper critical temperatures for the\nsix-state clock model from the fixed-point tensors determined using the\nhigher-order tensor renormalization group method and compare with other\nnumerical results.\n