Lattice $\mathbb{C}{P}^{N-1}$ model with ${\mathbb{Z}}_N$ twisted boundary condition: bions, adiabatic continuity and pseudo-entropy

Abstract

We investigate the lattice $\mathbb{C}{P}^{N-1}$ sigma model on $S_s^1$(large) $\times$ $S_{\tau }^1$(small) with the ${\mathbb{Z}}_N$ symmetric twisted boundary condition, where a sufficiently large ratio of the circumferences ($L_s\gg L_{\tau }$) is taken to approximate $\mathbb{R}\times S^1$. We find that the expectation value of the Polyakov loop, which is an order parameter of the ${\mathbb{Z}}_N$ symmetry, remains consistent with zero ($|⟨P⟩|\sim 0$) from small to relatively large inverse coupling $\beta$ (from large to small $L_{\tau }$). As $\beta$ increases, the distribution of the Polyakov loop on the complex plane, which concentrates around the origin for small $\beta$, isotropically spreads and forms a regular $N$-sided-polygon shape (e.g. pentagon for $N=5$), leading to $|⟨P⟩|\sim 0$. By investigating the dependence of the Polyakov loop on $S_s^1$ direction, we also verify the existence of fractional instantons and bions, which cause tunneling transition between the classical $N$ vacua and stabilize the ${\mathbb{Z}}_N$ symmetry. Even for quite high $\beta$, we find that a regular-polygon shape of the Polyakov-loop distribution, even if it is broken, tends to be restored and $|⟨P⟩|$ gets smaller as the number of samples increases. To discuss the adiabatic continuity of the vacuum structure from another viewpoint, we calculate the $\beta$ dependence of ``pseudo-entropy" density $\propto ⟨T_{xx}-T_{\tau \tau }⟩$. The result is consistent with the absence of a phase transition between large and small $\beta$ regions.

Comments: 31 pages, 13 figures; (v2) minor corrections, to appear in JHEP