In this note we consider a non-linear, large-N <math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math> sigma model on a finite size interval with periodic boundary conditions, at finite temperature and chemical potential in the regime of βμ large. Our goal is to extend previous calculations and obtain the coefficients of the derivative expansion of the one-loop effective action in the region of βμ large by carrying out the appropriate analytical continuation. This calculation complements previous results and allows us to conclude that the ground state remains homogeneous in this regime as long as it is assumed to be a slowly varying function of the spatial coordinates. While this is reasonable at the two extremes of small or large chemical potential, for intermediate values of the chemical potential and small enough temperature, one might expect (by analogy with other models) that lower energy crystalline solutions may exist. In this case a simple derivative expansion, like the one discussed here, would need to be modified in order to capture these features.

Japan

Antonino Flachi

United States

Guglielmo Fucci

One-loop effective action of the Model at Large

In this work we examine a system consisting of a confined one-dimensional arrangement of atoms that we describe by using the 2-dimensional <math><mi>C</mi><msup><mi>P</mi><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math> 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.

Muneto Nitta

Satoshi Takada

Ryosuke Yoshii

Ground state modulations in the <math><mi>C</mi><msup><mi>P</mi><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math> model

In this work we analyze the Casimir energy and force for a scalar field endowed with general self-adjoint boundary conditions propagating in a higher dimensional piston configuration. The piston is constructed as a direct product I×N , with I=[0,L]⊂R and N a smooth, compact Riemannian manifold with or without boundary. The study of the Casimir energy and force for this configuration is performed by employing the spectral zeta function regularization technique. The obtained analytic results depend explicitly on the spectral zeta function associated with the manifold N and the parameters describing the general boundary conditions imposed. These results are then specialized to the case in which the manifold N is a d -dimensional sphere.