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

Abstract

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.

Remarks on the large-NCPN−1model

“…This conclusion seems to be perfectly in tune with that of Ref. [28] and the additional term is the missing ingredient that brings to an agreement the results of Refs. [28,31].…”

“…The above expression for the one-loop effective action at large-N at finite temperature and chemical potential is readily obtained after path-integration over the fields n k and n à k and, for μ ¼ 0 coincides with those of Refs. [16,25,28,31]. As explained at the beginning of this section, the constraint (2) has been incorporated by means of a Lagrange multiplier M 2 (as δS=δM 2 ¼ 0) that operates as an effective mass.…”

“…In the case of periodic boundary conditions, both solutions (constant or inhomogeneous) are possible and the usual expectation is that the constant solution is the lower energy one. For the model at hand, this point has been debated recently [27,28].…”

“…1 The second reason is to inspect whether a transition from a massive to a massless phase may or may not occur and whether there is any clear mechanism to exclude the existence of a massless phase (both possibilities have been entertained in the literature with differing conclusions; see Ref. [28] and references given there). Extending the calculation to finite density gives us an excuse to reconsider this debated matter.…”

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

“…Such similarities can be explained by several physical setups, in which the two-dimensional CP N −1 sigma model effectively describes various physical properties of four-dimensional gauge theories; non-Abelian vortices in the non-Abelian gauge-Higgs models [5-10, 10, 11] and dense QCD [12][13][14][15], long strings in Yang-Mills theories [16], and an appropriately compactified Yang-Mills theory [17]. Non-perturbative properties of the CP N −1 model have long been studied analytically by the gap equations with the large-N approximation [2][3][4][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] and by lattice simulations [36][37][38][39][40][41][42][43][44][45][46][47][48][49]. In the previous work [47,48] of the present authors, they have studied the CP N −1 model on S 1 s (large) × S 1 τ (small) by lattice Monte Carlo simulations.…”