We investigate the lattice ℂPN−1 sigma model on Ss1(large) ×Sτ1(small) with the ℤN symmetric twisted boundary condition, where a sufficiently large ratio of the circumferences (Ls ≫ Lτ) is taken to approximate ℝ × S1. We find that the expectation value of the Polyakov loop, which is an order parameter of the ℤN symmetry, remains consistent with zero (|〈P〉| ∼ 0) from small to relatively large inverse coupling β (from large to small Lτ). 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 |〈P〉| ∼ 0. By investigating the dependence of the Polyakov loop on Ss1 direction, we also verify the existence of fractional instantons and bions, which cause tunneling transition between the classical N vacua and stabilize the ℤN 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 |〈P〉| 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 ∝ 〈Txx − Tττ〉. The result is consistent with the absence of a phase transition between large and small β regions.
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