Tensor renormalization group calculations of partition-function ratios

TL;DR

This work investigates dimensionless partition-function ratios $X_1$ and $X_2$ as probes of phase transitions in two-dimensional lattice models. Using the bond-weighted tensor renormalization group (BWTRG), the authors compute these ratios for the Ising, three-state Potts, and four-state Potts models and connect their critical values to modular-invariant torus partition functions from conformal field theory (CFT). They demonstrate that $X_1$ and $X_2$ obey the same finite-size scaling as the Binder parameter and confirm universal CFT predictions at criticality, while the four-state Potts model exhibits logarithmic corrections, in line with known CFT results. The study of anisotropic systems shows that universal values depend on the correlation-length ratio $\xi_x/\xi_y$, with tilted tensor-network realizations capturing this dependence. Overall, the results validate BWTRG as an effective tool for extracting CFT data and guiding finite-size analyses in two-dimensional critical systems.

Abstract

The behavior of dimensionless quantities defined as ratios of partition functions is analyzed to investigate phase transitions and critical phenomena. At criticality, the universal values of these ratios can be predicted from conformal field theory (CFT) through the modular-invariant partition functions on a torus. We perform numerical calculations using the bond-weighted tensor renormalization group for three two-dimensional models belonging to different universality classes: the Ising model, the three-state Potts model, and the four-state Potts model. The partition-function ratios obey the same finite-size scaling form as the Binder parameter, and their critical values agree well with the universal values predicted by CFT. In the four-state Potts model, we observe logarithmic corrections in the system-size dependence of these ratios.

Figures (7)

• Figure 1: (a) The size dependence of $X_1$ near the critical temperature on the Ising model and (b) its FSS plot. The horizontal dashed line indicates the universal value predicted by CFT. The horizontal axis of the FSS plot is scaled by the deviation from the estimated critical temperature. The critical exponent related to the correlation length is taken as the exact value $\nu=1$. The reduced temperature is denoted by $t \equiv (T - T_\text{c})/T_\text{c}$. Simulations were performed using BWTRG with the bond dimension $\chi=128$.
• Figure 2: The bond-dimension dependence of $X_1$ (circle) and $X_2$ (square) at criticality on the Ising model. The dashed horizontal lines indicate the universal values predicted by CFT.
• Figure 3: The size dependence of the partition-function ratios defined from larger cluster sizes at criticality on the Ising model. Circle symbols represent $X$'s without skew, while square symbols represent those with skew. Simulations were performed using BWTRG with the bond dimension $\chi=128$.
• Figure 4: The size dependence of $X_1$ (circle) and $X_2$ (square) near the critical temperature on the three-state Potts model. The red symbols represent the data at criticality. The horizontal dashed (dotted) line indicates the universal value of $X_1$ ($X_2$) predicted by CFT. Simulations were performed using BWTRG with the bond dimension $\chi=144$.
• Figure 5: The size dependence of $X_1$ near the critical temperature on the four-state Potts model. The red symbols represent the data at criticality. The horizontal dashed line indicates the universal value of $X_1$ predicted by CFT. The inset shows the deviation from the universal value as a function of $\log_{10} L$. The dotted line is proportional to $(\log L)^{-1}$. Simulations were performed using BWTRG with the bond dimension $\chi=144$.