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Solving Sign Problem: Phase-Structure Analysis of a Lattice Model with a Topological θ Term Using Tensor Renormalization Group Approach

Published Jan 1, 2026• 0 citations
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Abstract

This study clarifies the phase structure of the two-dimensional lattice CP(1) model with a θ term by employing the tensor renormalization group method and conformal field theory, thus enabling the circumvention of the sign problem and logarithmic corrections.
DOI
10.7566/jpsht.6.013

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Katsumasa Nakayama, Lena Funcke, Karl Jansen +2 more

We numerically study the phase structure of the CP(1) model in the presence of a topological $\theta$-term, a regime afflicted by the sign problem for conventional lattice Monte Carlo simulations. Using a bond-weighted Tensor Renormalization Group method, we compute the free energy for inverse couplings ranging from $0\leq \beta \leq 1.1$ and find a CP-violating, first-order phase transition at $\theta=\pi$. In contrast to previous findings, our numerical results provide no evidence for a critical coupling $\beta_c<1.1$ above which a second-order phase transition emerges at $\theta=\pi$ and/or the first-order transition line bifurcates at $\theta\neq\pi$. If such a critical coupling exists, as suggested by Haldane's conjecture, our study indicates that is larger than $\beta_c>1.1$.

•Jan 1, 2022•1 citation
Abstract We investigate the phase structure of a 2D lattice CP(1) model with a θ\thetaθ term. In particular, we aim to identify a critical region expected to exist along a θ=π\theta =\piθ=π line. To explore the phase structure nonperturbatively and avoid the sign problem, we employ the tensor renormalization group method. We make two improvements compared to previous tensor network studies. The first improvement involves refining the initial tensor. Specifically, we construct it using a quadrature method, which achieves higher accuracy compared to the conventional approach. The second improvement consists of analyzing the phase structure using the information on the conformal field theory, namely the central charge and the scaling dimensions, which can be accessed relatively easily via the tensor renormalization group method. Thanks to these improvements, we identify both the onset of the critical region, βc=0.5952(8)\beta _{\rm c}=0.5952(8)βc​=0.5952(8), and its universality class as the SU(2)k=1{}_{k=1}k=1​ Wess–Zumino–Witten model.
Phase structure analysis of CP(1) model with θθθ term by tensor renormalization group
We analyze the phase structure of 2d lattice CP(1) model with θθθ term by using the bond-weighted tensor renormalization group method. We propose a new tensor network representation for the model using the quadrature scheme and confirm that its accuracy is better than that of the conventional character-like expansion. As a probe to study the phase structure, we adopt the central charge and the scaling dimensions. The numerical results indicate an existence of critical point at θ=πθ=πθ=π, which is consistent with the Haldane's conjecture.
Phase structure analysis of 2d CP(1) model with θ\thetaθ term by tensor renormalization group
We analyze the phase structure of 2d lattice CP(1) model with θ\thetaθ term by using the bond-weighted tensor renormalization group method. We propose a new tensor network representation for the model using the quadrature scheme and confirm that its accuracy is better than that of the conventional character-like expansion. As a probe to study the phase structure, we adopt the central charge and the scaling dimensions. The numerical results indicate an existence of critical point at θ=π\theta=\piθ=π, which is consistent with the Haldane's conjecture.
Phase structure of the CP(1) model in the presence of a topological θ\thetaθ-term
We numerically study the phase structure of the CP(1) model in the presence of a topological θ\thetaθ-term, a regime afflicted by the sign problem for conventional lattice Monte Carlo simulations. Using a bond-weighted Tensor Renormalization Group method, we compute the free energy for inverse couplings ranging from 0≤β≤1.10\leq \beta \leq 1.10≤β≤1.1 and find a CP-violating, first-order phase transition at θ=π\theta=\piθ=π. In contrast to previous findings, our numerical results provide no evidence for a critical coupling βc<1.1\beta_c<1.1βc​<1.1 above which a second-order phase transition emerges at θ=π\theta=\piθ=π and/or the first-order transition line bifurcates at θ≠π\theta\neq\piθ=π. If such a critical coupling exists, as suggested by Haldane's conjecture, our study indicates that is larger than βc>1.1\beta_c>1.1βc​>1.1.