Remarks on the large-NCPN−1model

Abstract

In this paper we consider the CP N−1 model confined to an interval of finite size at finite temperature and chemical potential. We obtain, in the large-N approximation, a mixed-gradient expansion of the one-loop effective action of the order parameter associated with the effective mass of the quantum fluctuations. This expansion gives an expression for the thermodynamic potential density as a functional of the order parameter, generalizing previous calculations to arbitrarily large order and to the case of finite chemical potential and allows one to discuss some generic features of the ground state of the model. The technique used here relies on analytic regularization and provides an efficient scheme to extract the coefficients of the expansion. Once a solution for the ground state is known these coefficients can be used to deduce some generic properties of the ground state as a function of external conditions. We also show that there can be no transition to a massless phase for any value of the external conditions considered and clarify a seemingly important point regarding the regularization of the effective action connected to the appearance of logarithmic divergences and to the Mermin-Wagner-Hoenberg-Coleman theorem.

One-loop effective action of the ${\mathbb C}P^{N-1}$ model at large $μβ$

“…and it is identical to that of 31 in the present case. In the expression (5), we have chosen the background-field configuration to lie along the k = 1 direction, i.e., n 1 = σ , n k = 0 with k > 1.…”

“…Obviously the opposite choice is also allowed. This same method was used in 31 in the range β µ ≪ 1. With this understanding, the large-z asymptotic expansion of ϖ (a) can be found to be ϖ…”

“…In this section, we briefly summarize how to obtain the derivative expansion of the bulk oneloop effective action obtained in Refs. 30,31 . For brevity, we simply outline the main steps and refer the reader to the original references for additional details.…”

“…To this end, we need to assume a spatially-dependent ground state from the beginning of our analysis. A general approach to deal with this situation in the large-N limit has been discussed, for example, in references 30,31 where a derivative expansion for the one-loop effective action Γ M 2 as a functional of the mass gap M 2 has been obtained in the form,…”

“…The quantity α 0 represents the vacuum energy contribution (see reference 25 ). The explicit form of the coefficients α p , with p ≥ 1 has been obtained in 30 at finite temperature (T = 1/β ), and in 31 in the presence of a finite chemical potential µ associated with the first component of the complex parameter n i . In those papers the calculations were performed in the range β µ < 2π which led the authors to conclude that the inhomogeneous phases are energetically unfavorable.…”