Ground state modulations in the CPN−1 model

Abstract

In this work we examine a system consisting of a confined one-dimensional arrangement of atoms that we describe by using the 2-dimensional CP N −1 model, restricted to an interval and at finite temperature. We develop a method to obtain the bulk and boundary parts of the one-loop effective action as a function of the effective mass of the fluctuations. The formalism has the advantage of allowing for a systematic analysis of a large class of boundary conditions and to model the (adiabatic) response of the ground state to changes in the boundary conditions. In the case of periodic boundary conditions, we find that inhomogeneous phases are disfavored for intervals of large size. Away from periodic boundary conditions, our numerical results show that the ground state has a generic crystal-like structure that can be modulated by variations of the boundary conditions. The results presented here could be relevant for experimental implementations of nonlinear sigma models and could be tested by lattice numerical simulations.

Remarks on the large-NCPN−1model

“…Here, we follow Ref. [31] and perform a coordinate transformation, x →x ¼ x=l and τ →τ ¼ τ=l (τ is the Wick-rotated Euclidean time and β ¼ 1=T in the expression above represents the inverse temperature), in order to rescale the interval to one of unit length. These rescaled coordinates are dimensionless and we use the symbol ∇ð¼ l∇Þ to indicate differentiation with respect to the rescaled coordinatex.…”

“…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.…”

“…Taking the limits of μ → 0 and M → 0 in the previous formulas recuperate, respectively, known formal expressions (See, for example, Refs. [31,[49][50][51]).…”

“…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.…”

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.…”