Self-consistent analytic solutions in twisted ℂPN−1 model in the large-N limit

In this paper we examine analytically the large-N gap equation and its solution for the 2D CP N −1 sigma model defined on a Euclidean spacetime torus of arbitrary shape and size (L, β), β being the inverse temperature. It is shown that the system has a unique homogeneous phase, with the CP N −1 fields n i acquiring a dynamically generated mass λ ≥ Λ 2 (analogous to the mass gap of SU (N ) Yang-Mills theory in 4D), for any β and L. We comment on several related topics discussed in the recent literature. One concerns the possibility, which we will exclude, of a "Higgs-like" -or deconfinement -phase at small L and at zero temperature. Another topic involves "soliton-like" (inhomogeneous) solutions of the generalized gap equation. In both cases, a proper treatment of the zero modes is essential. A related question concerns a possible instability of the standard CP N −1 vacuum on R 2 , which is shown not to occur. The CP N −1 model with twisted boundary conditions is also analyzed. The θ dependence, and different limits involving N , β and L are briefly discussed.

Section: Discussionmentioning

confidence: 99%

Large-N ℂℙN −1 sigma model on a Euclidean torus: uniqueness and stability of the vacuum

Bolognesi

Gudnason

Konishi

et al. 2019

“…Self-consistent analytic solutions for this case was obtained in refs. [61,62]. A similar problem in a two-dimensional disk was discussed in refs.…”

Section: Jhep08(2018)007mentioning

confidence: 62%

“…In the case of a finite interval with the Dirichlet boundary condition, it is inevitable that the Higgs field and mass gap function are inhomogeneous [59][60][61][62][63]. Self-consistent analytic solutions for this case was obtained in refs.…”

Section: Jhep08(2018)007mentioning

confidence: 99%

“…Multiple Higgs solitons with arbitrary separation have also been obtained. The twisted boundary condition results in the flavor rotating solutions in the CP N −1 model [62] (see also [63]). …”

Section: Jhep08(2018)007mentioning

confidence: 99%

“…In this case the homogeneous Higgs solution with the flavor rotation [62] can also be the solution of the same boundary condition.…”

Section: Confining Soliton Lattice Solutionmentioning

confidence: 99%

See 2 more Smart Citations

Confining solitons in the Higgs phase of ℂP N −1 model: self-consistent exact solutions in large-N limit

Nitta¹,

Yoshii²

2018

Self Cite

Abstract:The quantum CP N −1 model is in the confining (or unbroken) phase with a full mass gap in an infinite space, while it is in the Higgs (broken or deconfinement) phase accompanied with Nambu-Goldstone modes in a finite space such as a ring or finite interval smaller than a certain critical size. We find a new self-consistent exact solution describing a soliton in the Higgs phase of the CP N −1 model in the large-N limit on a ring. We call it a confining soliton. We show that all eigenmodes have real and positive energy and thus it is stable.

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

Fujimori

Itou

Misumi

et al. 2020

Self Cite

We investigate the lattice CP N −1 sigma model on S 1 s (large) × S 1 τ (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 1 s 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 τ τ. The result is consistent with the absence of a phase transition between large and small β regions.