Lattice ℂPN−1 model with ℤN twisted boundary condition: bions, adiabatic continuity and pseudo-entropy

Abstract

A bstract We investigate the lattice ℂ P N− 1 sigma model on $$ {S}_s^1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>s</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> (large) × $$ {S}_{\tau}^1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>τ</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> (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 $$ {S}_s^1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>s</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> 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.

Topics & Concepts