### Abstract

We investigate the lattice ℂP^{N−1} sigma model on Ss1(large) ×Sτ1(small) with the ℤ_{N} symmetric twisted boundary condition, where a sufficiently large ratio of the circumferences (L_{s} ≫ L_{τ}) is taken to approximate ℝ × S^{1}. We find that the expectation value of the Polyakov loop, which is an order parameter of the ℤ_{N} symmetry, remains consistent with zero (|〈P〉| ∼ 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 |〈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 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 |〈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 ∝ 〈T_{xx} − T_{ττ}〉. The result is consistent with the absence of a phase transition between large and small β regions.

Original language | English |
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Article number | 11 |

Journal | Journal of High Energy Physics |

Volume | 2020 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2020 Aug 1 |

### Keywords

- Lattice Quantum Field Theory
- Sigma Models
- Solitons Monopoles and Instantons
- Wilson
- ’t Hooft and Polyakov loops

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

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## Cite this

^{N−1}model with ℤ

_{ N}twisted boundary condition: bions, adiabatic continuity and pseudo-entropy.

*Journal of High Energy Physics*,

*2020*(8), [11]. https://doi.org/10.1007/JHEP08(2020)011