Phase structure of the CP(1) model in the presence of a topological $\theta$-term

TL;DR

This study uses tensor renormalization group methods to analyze the phase structure of the CP(1) model with a topological theta-term, revealing a first-order transition at theta=pi and challenging previous claims of a critical coupling below 1.1.

Contribution

First numerical tensor network analysis of the CP(1) model with a theta-term, providing new insights into its phase transitions and challenging prior conjectures.

Findings

Identifies a first-order phase transition at θ=π.

Finds no evidence of a second-order transition below β=1.1.

Suggests any critical coupling, if it exists, exceeds β=1.1.

Abstract

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 $\beta_c>1.1$.