Abstract
In this paper, we consider the ${\mathbb C}P^{N-1}$ 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 the Mermin-Wagner-Hoenberg-Coleman (MWHC) theorem.
Highlights
The CPN−1 model is 1 þ 1 dimensional (d 1⁄4 1) field theory, consisting of N complex scalar fields ni (i 1⁄4 1; 2; ...; N) with an action of the form ZS 1⁄4 dxdtjDμnij2; ð1Þ with Dμ 1⁄4 ∂μ − iAμ, with the U(1) gauge field Aμ lacks at classical level a kinetic term
Issues being currently debated have to do with whether the model can develop a massless ground state for small enough interval size, how the properties of the ground state depend on the external conditions, and how everything fits under the umbrella of the large-N approximation
To inspect whether a spatially modulated solution may become energetically disfavored, as external conditions are varied, we need to compute the dependence of the coefficients π0ð1Þ, π0ð3Þ and π0ð5Þ on the temperature and on the chemical potential
Summary
The CPN−1 model is 1 þ 1 dimensional (d 1⁄4 1) field theory, consisting of N complex scalar fields ni (i 1⁄4 1; 2; ...; N) with an action of the form. II, we introduce the main setup and notation, and illustrate the calculation of the effective action at finite temperature, density, and size using zeta-regularization This calculation is essentially a repetition of that of Ref. [31] with two major differences: the first being the inclusion of a chemical potential, and the second being a different regularization that allows to capture the infrared behavior of the model and leads to the appearance of a logarithmic contribution This is an issue of some importance, since it is this term that eventually prevents a massless ground state to be realized and locks the system into a massive phase. Some formulas involving polylogarithmic functions used in the computation are given in Appendix
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