Lattice ℂP N−1 model with ℤ N twisted boundary condition: bions, adiabatic continuity and pseudo-entropy

Abstract

Abstract We investigate the lattice ℂP N−1 sigma model on S s 1 $$ {S}_s^1 $$ (large) × S τ 1 $$ {S}_{\tau}^1 $$ (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 S s 1 $$ {S}_s^1 $$ direction, we also verify the existence of fractional instantons and bions, which cause tunneling transition between the classical N vacua and stabilize the 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.

Keywords

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