Moving from continuous to discrete symmetry in the 2D XY model

TL;DR

This paper investigates how discretizing the 2D XY model into clock models affects its phase structure, using tensor network methods to analyze the impact of perturbations and the dominant role of core tensors.

Contribution

It introduces a tensor network approach to compare discretizations of the XY model and reveals the influence of core tensors and perturbations on phase transitions.

Findings

Discretization into clock models alters phase structure at low temperatures.

Core tensors have a dominant influence on phase behavior.

Small perturbations can induce a non-zero critical temperature.

Abstract

We study the effects of discretization on the U(1) symmetric XY model in two dimensions using the Higher Order Tensor Renormalization Group (HOTRG) approach. Regarding the $Z_N$ symmetric clock models as specific discretizations of the XY model, we compare those discretizations to ones from truncations of the tensor network formulation of the XY model based on a character expansion, and focus on the differences in their phase structure at low temperatures. We also divide the tensor network formulations into core and interaction tensors and show that the core tensor has the dominant influence on the phase structure. Lastly, we examine a perturbed form of the XY model that continuously interpolates between the XY and clock models. We examine the behavior of the additional phase transition caused by the perturbation as the magnitude of perturbation is taken to zero. We find that this additional transition has a non-zero critical temperature as the perturbation vanishes, suggesting that even small perturbations can have a significant effect on the phase structure of the theory.