# Phase transition of the q-state clock model: duality and tensor renormalization

## TL;DR

This paper studies the critical behavior and duality of the q-state clock model on a square lattice using tensor-network methods, identifying self-dual points and critical temperatures for various q values.

## Contribution

It introduces a tensor-network approach to analyze the duality and critical points of the q-state clock model, providing exact and approximate self-dual points and precise critical temperatures.

## Key findings

- Exact self-dual points for q ≤ 5
- Approximate self-dual points for q ≥ 6
- Accurate critical temperature estimates for q=6

## Abstract

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