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

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.

Figures (5)

• Figure 1: ATRG results for the three-dimensional $O(2)$ model with bond dimension $D_{\rm ATRG}=96$. (a) $Z_{\pi}/Z_{0}$ against temperature, varying $L$. The gray band indicates the MC estimate of $Z_{\pi}/Z_{0}$ at criticality for $L=2^4$ from Ref. Gottlob_1994. The red vertical dashed line denotes the estimate $T_c=2.2018441(5)$ obtained in a recent MC study Xu:2019mvy. (b) Finite-size scaling for $Z_{\pi}/Z_{0}$.
• Figure 2: BTRG results for the two-dimensional $O(2)$ model with bond dimension $D_{\rm BTRG}=200$. (a) $\Upsilon_{\pi/6}(2L,L)$ against temperature, varying $L$. The solid line shows the NK criterion, $\Upsilon_{\alpha}=2T/\pi$. (b) $T^{*}(L)$ as a function of $1/(\ln bL)^{2}$. Squares represent $T^{*}(L)$, and the circle denotes the extrapolated $T_{\rm BKT}$ at $L\to\infty$.
• Figure 3: $Z_{\alpha}/Z_{0}$ at $L=2^{7}$ for the two-dimensional generalized $O(2)$ model ($q=2$, $\Delta=0.16$) with bond dimension $D_{\rm BTRG}=100$. Different symbols represent different twist angles $\alpha$.
• Figure 4: BTRG results for the two-dimensional generalized $O(2)$ model ($q=2$, $\Delta=0.16$) with bond dimension $D_{\rm BTRG}=200$. (a) $Z_{\pi}/Z_{0}(L,L)$ against temperature, varying $L$. The red horizontal line denotes the exact value of $Z_{\pi}/Z_{0}$ for the two-dimensional Ising universality class. (b) Finite-size scaling of $Z_{\pi}/Z_{0}$.
• Figure 5: BTRG results for the two-dimensional generalized $O(2)$ model ($q=2$, $\Delta=0.16$) with bond dimension $D_{\rm BTRG}=200$. (a) $\Upsilon_{\pi/6}(2L,L)$ against temperature, varying $L$. The solid line shows the NK criterion, $\Upsilon_{\alpha}=2q^{2}T/\pi$. (b) $T^{*}(L)$ as a function of $1/(\ln bL)^{2}$. Squares represent $T^{*}(L)$, and the circle denotes the extrapolated $T_{\rm BKT}$ at $L\to\infty$.