Tensor renormalization group approach to critical phenomena via symmetry-twisted partition functions

TL;DR

This work develops a tensor renormalization group (TRG) framework to efficiently compute symmetry-twisted partition functions $Z[A]$ and uses their ratios to diagnose spontaneous symmetry breaking (SSB) and critical behavior. By enforcing symmetry constraints in the tensor network, the authors evaluate twisted partition functions and perform finite-size scaling to extract critical temperatures and exponents, as well as the helicity modulus for Berezinskii-Kosterlitz-Thouless (BKT) transitions. They demonstrate the approach on the 2D Ising model (discrete $\mathbb{Z}_2$ symmetry) and the 3D $O(2)$ model (continuous $U(1)$ symmetry), obtaining $T_c$ and $\nu$ for the latter and confirming CFT predictions at criticality for the former. They also directly compute the helicity modulus in the 2D $O(2)$ model to determine the BKT transition temperature $T_{\mathrm{BKT}}$, finding results in agreement with established literature. Overall, the work shows that twisted partition functions, computed via TRG, provide a powerful tool for extracting universal low-energy data from lattice models with global symmetries, with potential extensions to generalized $O(2)$ models and gauge theories.

Abstract

The locality of field theories strongly constrains the possible behaviors of symmetry-twisted partition functions, and thus they serve as order parameters to detect low-energy realizations of global symmetries, such as spontaneous symmetry breaking (SSB). We demonstrate that the tensor renormalization group (TRG) offers an efficient framework to compute the symmetry-twisted partition functions, which enables us to detect the symmetry-breaking transition and also to study associated critical phenomena. As concrete examples of SSB, we investigate the two-dimensional (2D) classical Ising model and the three-dimensional (3D) classical $O(2)$ nonlinear sigma model, and we identify their critical points solely from the twisted partition function. By employing the finite-size scaling argument, we find the critical temperature $T_c=2.2017(2)$ with the critical exponent $ν= 0.663(33)$ for the 3D $O(2)$ model. In addition, we also study the Berezinskii-Kosterlitz-Thouless (BKT) criticality of the 2D classical $O(2)$ model by extracting the helicity modulus from the twisted partition functions, and we obtain the BKT transition temperature, $T_{\mathrm{BKT}}=0.8928(2)$.

Figures (10)

• Figure 1: $Z_{-1}/Z_{1}$ in the 2D Ising model. The vertical dashed line indicates the exact critical point $T_{c}=2/(\log(1+\sqrt{2}))$. The horizontal dashed line denotes the exact value of $Z_{-1}/Z_{1}$ for the 2D Ising CFT, which is given by Eq. \ref{['eq:exact_zpz_2DIsing']}. The computation is done by the BTRG with the bond dimension $D_{\rm BTRG}=64$.
• Figure 2: $Z_{-1}/Z_{1}$ as a function of $\log_{2}L$ in the 2D Ising model, with the temperature deviation $\Delta T=1.0\times10^{-6}$ from the critical point. The horizontal dashed line denotes the exact value of $Z_{-1}/Z_{1}$ for 2D Ising CFT. The computation is done by the BTRG with the bond dimension $D_{\rm BTRG}=64$.
• Figure 3: Finite-size scaling analysis for $Z_{-1}/Z_{1}$ as a function of $tL$ in the 2D Ising model. The results of different volumes follow the universal curve as a function of $tL$ with $t=(T-T_c)/T_c$. The computation is done by the BTRG with the bond dimension $D_{\rm BTRG}=64$.
• Figure 4: $Z_{\alpha=\pi}/Z_{0}$ in the 3D $O(2)$ model. The vertical dashed line indicates the critical point $T_c^{\mathrm{(MC)}} = 2.2018441(5)$ obtained by the Monte Carlo simulation in Ref. Xu:2019mvy. The gray band denotes the value of $Z_{\alpha=\pi}/Z_{\alpha=0}$ at the critical point from the Monte Carlo simulation with $L=2^{4}$Gottlob_1994. The twisted partition function gives a clear signal for the $U(1)$ symmetry breaking. The computation is done by the ATRG with the bond dimension $D_{\rm ATRG}=96$.
• Figure 5: Finite-size scaling analysis for $Z_{\alpha=\pi}/Z_{0}$ in the 3D $O(2)$ model. $Z_{\alpha=\pi}/Z_0$ with different volumes follow the universal curve when they are plotted as a function of $tL^{1/\nu}$. The computation is done by the ATRG with the bond dimension $D_{\rm ATRG}=96$.