Casimir force for the maxwell-chern-simons-proca model

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Casimir Force for the Maxwell-Chern-Simons-Proca Model
Carlos Rafael M. S., J. F. de Medeiros Neto - Universidade Federal do Par´ a Rudnei O. Ramos -Departamento de F´sica Te´ rica, Universidade do Estado do Rio de Janeiro, 20550-013, Rio de Janeiro, RJ, Brazil ı o

Junho de 2011, Bel´ m, Pa, Brasil e

Introducao ¸˜
In a recent work, two of us considered the vortex condensation in the CSHmodel [1], specialized to the case of the self-dual potential for the scalar field [2, 3], in which case vortices can be considered as noninteracting. The strategy used to study the vortex condensation problem in the CSH model was to make explicit the vortex excitations in the functional action, by making use of a series of dual transformations for the original Lagrangian fields, obtaining anequivalent action, in which it became clear the vortex contributions, represented by a complex scalar vortex field ψ coupled to a gauge field. In a (2+1)-dimensional Euclidean space-time, the dual action can be written as S= +
2 d3x c1α2Hµν + ic2α2ǫµνγ hµ∂ν hγ |∂µψ + ic3αhµψ|2 (1) (∂µhµ)2 2 2 , M |ψ| + 2α


The Boundary Conditions

NWe must investigate if the boundary conditions considered forgauge field can be mapped properly in well defined boundary conditions for the scalar fields {φ, ϕ} (and viceversa), in order to obtain the Casimir force for the MPCS associated to the initial vortex model. For this purpose, we need to invert the equations (given in [8]), in order to obtain Ai in terms of {φ, ϕ}. Initially, we have 1 √ 1 √ 2φ − 2ϕ A1 = 2θµ1 2θν1 (5)

which implies that A1 must alsoobey Neumann boundary conditions: ∂1A1(0) = ∂1A1(L) = 0. (19) In order to find the behavior of A2 (or a function of of A2), we proceed inverting the equations of Ref. [8] that lead from A1, A2, π 1, π 2 to {φ, ϕ, πϕ, πφ}. We find √ √ 1 1 A2 = √ µ1πφ − M 2µ1∂2∂1φ − θν1πϕ + θ 2N ν1∂2∂1ϕ. 2 2 (20) with 1 1 µ µ µ µ M= , N =− . 1+ + 1+ + 2 2 2 2 2θ 4m 4m 2θ 4m 4m (21) For the geometry that we areconsidering here, it is useful to consider the fields φ and ϕ as written as transverse spatial Fourier transforms together with Fourier transforms in time [7]. For φ, we write φ(x, y, t) = dω −iω t e 2π dk ik ˜ e φ(ω, k, x). 2π (22)

where µ1 and ν1 are differential operators defined by 1 2 2 µ1 = − θ K11 − θQ12 + S 22 2 and 1 2 2 ν1 = − θ K11 + θQ12 + S 22 2 where (i = 1, 2)
−1 −1


where α is anarbitrary mass parameter, c1, c2 and c3 are parameters related to the original CSH model and M is a dynamical induced mass term [1]. The last term in Eq. (??) is an usual gauge fixing parameter. In the present work, we find an expression for the Casimir force for the Maxwell-Proca-Chern-Simons model, obtained from Eq. (??) by considering the vortex field in its nontrivial vacuum expectation value.This is a new result concerning vortices in planar systems, and deserves attention, since it is known that “the properties of superconductors can be affected by their shape. This effect is increasengly noticeable as the size of the superconductor decreases” (as indicated in Ref. [4]) and the Casimir force may have great importance at nanoscale materials.



µ 1 ˜ ∂i ∂j ǫij + 2 ∂i∂j Kij≡ ηij + 2 , Qij ≡ m 2 m µ2 1 ˜ 1+ ∂i ≡ ǫij ∂ j , S ij ≡ 2 4m2


∇2 − m2 η ij + ∂ i∂ j .

Taking analogous equations for ϕ and remembering that we are considering Neumann boundary conditions for φ, we see that Eq. (22), together with the definitions of πφ and πϕ, implies that A2, given in Eq. (20), must also obey Neumann boundary conditions: ∂1A2(0) = ∂1A2(L) = 0. (23)

(9) Since thephysical boundary conditions are not taken in the momentum space, Eq. (5) implies that we need to invert, in the real space-time, the differential operators µ1 and ν1, given in Eq‘s. (6) e (7) . In this aim, we consider the following operation: µ−2φ 1 1 2 2 = − θ K11 − θQ12 + S 22 φ = A − B∂1 φ, (10) 2 1 µ 1 µ2 A= θ θ− + m2 + , 2 2 2 4 and µ2 1 θ2 + θµ + + 1 B= 2m2 4 (12)

3 The Casimir Force
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