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

We investigate the lattice ${\mathbb C} P^{N-1}$ sigma model on $S_{s}^{1}$(large) $\times$ $S_τ^{1}$(small) with the ${\mathbb Z}_{N}$ symmetric twisted boundary condition, where a sufficiently large ratio of the circumferences ($L_{s}\gg L_τ$) 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 ($|\langle P\rangle|\sim 0$) from small to relatively large inverse coupling $β$ (from large to small $L_τ$). As $β$ increases, the distribution of the Polyakov loop on the complex plane, which concentrates around the origin for small $β$, isotropically spreads and forms a regular $N$-sided-polygon shape (e.g. pentagon for $N=5$), leading to $|\langle P\rangle| \sim 0$. 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 $β$, we find that a regular-polygon shape of the Polyakov-loop distribution, even if it is broken, tends to be restored and $|\langle P\rangle|$ gets smaller as the number of samples increases. To discuss the adiabatic continuity of the vacuum structure from another viewpoint, we calculate the $β$ dependence of ``pseudo-entropy" density $\propto\langle T_{xx}-T_{ττ}\rangle$. The result is consistent with the absence of a phase transition between large and small $β$ regions.