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

Katsumasa Nakayama, Lena Funcke, Karl Jansen, Ying-Jer Kao, Kuehn, Stefan

Published Jan 1, 2022• 1 citation
Publisher

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 $\beta_c>1.1$.

DOI
10.3204/pubdb-2021-03419

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Phase structure analysis of CP(1) model with θθθ term by tensor renormalization group
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Phase structure analysis of 2d CP(1) model with θ\thetaθ term by tensor renormalization group
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We apply the higher-order tensor renormalization group to the lattice $\mathrm{CP}(N\ensuremath{-}1)$ model in two dimensions. A tensor network representation of the $\mathrm{CP}(N\ensuremath{-}1)$ model in the presence of the $\ensuremath{\theta}$ term is derived. We confirm that the numerical results of the CP(1) model without the $\ensuremath{\theta}$ term using this method are consistent with that of the O(3) model which is analyzed by the same method in the region $\ensuremath{\beta}\ensuremath{\gg}1$ and that obtained by the Monte Carlo simulation in a wider range of $\ensuremath{\beta}$. The numerical computation including the $\ensuremath{\theta}$ term is left for future challenges.
We numerically study the phase structure of the two-flavor Schwinger model with matrix product states, focusing on the (1+11+11+1)-dimensional analog of the CPCPCP-violating Dashen phase in QCD. We simulate the two-flavor Schwinger model around the point where the positive mass of one fermion flavor corresponds to the negative mass of the other fermion flavor, which is a sign-problem afflicted regime for conventional Monte Carlo techniques. Our results indicate that the model undergoes a CPCPCP-violating Dashen phase transition at this point, which manifests itself in abrupt changes of the average electric field and the analog of the pion condensate in the model. Studying the scaling of the bipartite entanglement entropy as a function of the volume, we find clear indications that this transition is not of first order.