Tensor renormalization group approach to the $O(2)$ models via symmetry-twisted partition functions

TL;DR

This paper develops a tensor renormalization group method using symmetry-twisted partition functions to analyze phase transitions and symmetry breaking in $O(2)$ models in two and three dimensions, including BKT transitions.

Contribution

It introduces a novel approach combining symmetry-twisted partition functions with tensor renormalization group to detect phase transitions and symmetry breaking in $O(2)$ models.

Findings

Symmetry-twisted partition functions detect spontaneous symmetry breaking.

The method accurately determines the BKT transition point.

It successfully identifies phase transitions in generalized $O(2)$ models.

Abstract

We investigate critical phenomena in the $O(2)$ models using symmetry-twisted partition functions that can be efficiently computed within the tensor renormalization group framework. We first demonstrate, taking the three-dimensional model as an example, that symmetry-twisted partition functions detect the spontaneous breaking of global continuous symmetry. We then consider the same model in two dimensions, where the Berezinskii--Kosterlitz--Thouless (BKT) transition occurs. Since symmetry-twisted partition functions directly provide the helicity modulus at a finite twist angle, we determine the BKT transition point. These results are presented based on Ref.~\cite{Akiyama:2026dzg}. Finally, in addition to the original paper~\cite{Akiyama:2026dzg}, we apply this approach to the two-dimensional generalized $O(2)$ model and confirm that it successfully identifies the phase transitions between the ferromagnetic and nematic phases, as well as between the nematic and paramagnetic phases.