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Casimir effect on the : U(1) gauge in two spatial

M. N. Chernodub,1, 2, 3 V. A. Goy,4, 2 and A. V. Molochkov2 1Laboratoire de Math´ematiques et Physique Th´eoriqueUMR 7350, Universit´ede Tours, 37200 France 2Soft Laboratory, Far Eastern Federal University, Sukhanova 8, Vladivostok, 690950, Russia 3Department of Physics and Astronomy, University of Gent, Krijgslaan 281, S9, B-9000 Gent, Belgium 4School of Natural Sciences, Far Eastern Federal University, Sukhanova 8, Vladivostok, 690950, Russia (Dated: September 8, 2016) We propose a general numerical method to study the Casimir effect in lattice gauge . We illustrate the method by calculating the density of zero- fluctuations around two parallel wires of finite static permittivity in Abelian in two spatial dimensions. We discuss various subtle issues related to the lattice formulation of the problem and show how they can successfully be resolved. Finally, we calculate the Casimir potential between the wires of a fixed permittivity, extrapolate our results to the limit of ideally conducting wires and demonstrate excellent agreement with a known theoretical result.

I. INTRODUCTION lattice discretization) described by spatially-anisotropic and -dependent static permittivities ε(x) and per- meabilities µ(x) at zero and finite temperature in theories The influence of the physical objects on zero-point with various gauge groups. (vacuum) fluctuations is generally known as the Casimir The structure of the paper is as follows. In Sect. II effect [1–3]. The simplest example of the Casimir effect is we review the implementation of the Casimir boundary a modification of the vacuum energy of electromagnetic conditions for ideal conductors and propose its natural field by closely-spaced and perfectly-conducting parallel counterpart in the . We also discuss plates which leads to attraction of the plates to each a simple lattice implementation of materials character- other. If the objects are made of real materials, then their ized by static permittivity ε and permeability µ. Sec- intrinsic physical properties affect the energy of zero- tion III is devoted to discussion of the Casimir problem point fluctuations in accordance with the Lifshitz the- in (2+1) dimensional Abelian gauge theory. We discuss ory [4]. Generally, the Casimir forces depend on permit- geometrical setup, describe relevant , derive tivity, permeability, conductivity of the material which, the effect analytically in the continuum and in general, are complex functions of the electromagnetic rederive it again in the Euclidean lattice spacetime. The wave frequency. A calculation of the Casimir energy, in latter formula turns out to be crucial for precise compari- general, is quite involved. son – done in IV – of the results of our numerical Even in the case of perfect conductors the analytical simulations with the theoretical prediction in perfectly calculation of the Casimir forces may be a challenge if the conducting limit. In Sect. IV we calculate zero-point elec- objects are not flat. These forces may be either attractive tromagnetic fields around dielectric wires, study Casimir or repulsive depending on geometrical shape of the con- energy, make various scaling checks and discuss subtleties ductors. There are various numerical tools to compute of determination of Casimir energy in continuum based Casimir [5] including worldline Monte-Carlo on the lattice results. Finally, we calculate numerically methods [6, 7]. the dependence of the Casimir energy on static permit- In our paper we would like to join investigation of tivity ε of the wires. The last section is devoted to our the Casimir physics using Monte-Carlo methods of lat- conclusions. tice gauge theories that are usually applied to nonpertur- bative studies in . A similar philosophy arXiv:1609.02323v1 [hep-lat] 8 Sep 2016 albeit with different technical implementations has al- II. THE METHOD ready been successfully used to study the Casimir forces between ideal conductors in Refs. [8–10]. A. Casimir boundary conditions in continuum In order to illustrate our method we compute the Casimir between two parallel wires in an 1. Abelian gauge theory in (3+1) dimensions Abelian gauge model in two spatial dimensions in ther- modynamic equilibrium. We calculate the zero-point en- The Casimir effect is best probed by perfectly con- ergy for the wires described by a finite static permittivity ducting metallic surfaces of fixed geometrical shapes. In (dielectric constant). We extrapolate our numerical re- (3+1) dimensions a perfectly conducting surface im- sults to the limit of ideal conductors and demonstrating poses the following (Casimir) boundary conditionsS on the feasibility of the method by confronting our findings electromagnetic field Aµ: with known analytical results. We notice that our ap-

proach can be easily generalized to study the Casimir en- B (x) = 0 , E (x) = 0 , (1) ⊥ ergy for materials of various shapes (consistent with the x k x ∈S ∈S 2 where B (x) is the magnetic field normal to the surface that the Casimir condition (8) is reduced to the single ⊥ at the point x and E (x) is electric field which is tan- constraint E2(x) = 0 or to gential to the surface atk the point x. The ideal metallic F 02(x) = 0 , (9) surface imposes three conditions on electromagnetic field as it restricts one magnetic component B and two elec- where x belongs to the world surface of the wire. Equa- ⊥ tric components E . tion (9) implies that the component of the electric field k For a flat static surface perpendicular to the third axis tangential to the wire should vanish at each point of the x3, conditions (1) take the simple form: wire. E (x) = E (x) = B (x) = 0 , x , (2) 1 2 3 ∈ S B. Casimir boundaries in lattice gauge theory which can identically be rewritten in terms of the field- strength 1. Lattice Abelian gauge theory: F = ∂ A ∂ A , (3) µν µ ν − ν µ as follows: In order to set up a lattice reformulation of the Casimir boundary conditions (5) and (8) let us first consider a F 01(x) = F 02(x) = F 12(x) = 0 , x . (4) simplest possible case given by a compact U(1) gauge ∈ S model (compact electrodynamics) on the lattice. In general, the Casimir conditions (1) can also be The action of the compact electrodynamics is formu- rewritten in the Lorentz- form, lated in D-dimensional Euclidean spacetime: µν X nν Fe (x) = 0 (in D=3+1) , (5) S[θ] = β (1 cos θ ) , (10) − P 1 P where nµ is a vector tangential to the world-volume of the metallic surface at the point x and the tilde means where β is the lattice (to be defined the duality operation: below). The sum in Eq. (10) goes over all elementary plaquettes P of the lattice. Each plaquette P is defined µν 1 µναβ by the position of one of the corners of the plaquette Fe =  Fαβ (in D=3+1) . (6) 2 x = (x1, . . . , xD) and by the directions of two orthogonal vectors µ < ν in the plaquette plane, P = x, µν with For a flat static (x1, x2) surface the normal vector is di- µ, ν = 1,...D. The plaquette angle θ is{ constructed} rected along the third direction, nµ = δµ3 and the bound- P from the elementary link angles θ ( π, +π) which ary conditions for the gauge field (5) are indeed reduced x,µ ∈ − to constraints (2) and (4). belong to the perimeter ∂P of the plaquette P : θ = θ + θ θ θ . (11) Px,µν x,µ x+ˆµ,ν − x+ˆν,µ − x,ν 2. Abelian gauge theory in (2+1) dimensions The partition function of the model is as follows: Z S[θ] = θ e− , (12) In 2+1 dimensions the Casimir effect is formulated as Z D an interaction between metallic wires in two dimensional spatial volume. The world-surface of a wire is a two- where dimensional surface as one comes from the spa- Z Y Z π θ dθl , (13) tial dimension of the wire and another dimension comes D ≡ π from the time coordinate. Moreover, in 2+1 dimensions l − the dual object to the field strength tensor (6) is a vector: is the integration measure over the lattice gauge field θl. The action (10) is invariant under the 2π shifts of the µ 1 µαβ Fe =  Fαβ (in 2+1) . (7) plaquette variable, 2 θP θP + 2πn , n Z . (14) Therefore boundary condition (5) reduces in (2 + 1) di- → ∈ mensions to the following constraint: Therefore the 2π-shifted values of the plaquette strength tensor are physically equivalent to each other indicating µ nµFe (x) = 0 (in 2+1) . (8) that the Abelian is a compact . The angular variable θxµ has a sense of a “latticized” For a straight static wire parallel to the x2 axis, the Abelian gauge field Aµ, θxµ = aAµ(x), where a is the normal vector nµ is directed along the x1 coordinate, so lattice spacing (i.e., the length of the elementary link of the lattice). In continuum limit a 0 the plaquette variable (11) reduces, for finite values→ of the gauge field Aµ, to the field strength tensor (3) 1 A two-dimensional surface has a three-dimensional world-volume which include two spatial coordinates and one time coordinate. 2 θPx,µν = a Fµν (x) , (15) 3

4 where higher O(a ) corrections are not shown. equivalent to the requirement cos θPx,µν = 1 at the cor- Substituting Eq. (15) into Eq. (10), expanding over a responding plaquettes hence Eq. (18). and keeping the leading term only, we get the action of The relations from Eq. (18) are implemented on the the continuum U(1) gauge theory: set of plaquettes which includes, at all timeslices PS 1 Z (i) all spatial plaquettes P belonging to the plane; S[A] = dDx F 2 (x) , (16) x,12 4g2 µν (ii) all spatiotemporal plaquettes Px,14 and Px,24 which where we have identified the sum on the lattice with the are formed by one link in the Euclidean time direc- D P R D integral in a continuum limit, a x d x, and also tion and another link which belongs to the plane. imposed the following relation between→ the lattice spac- ing a, the lattice coupling constant β and the continuum In D = 3 Euclidean spacetime we consider a straight electric g: wire in x2 direction positioned at x1 = 0. In the case of an ideal metal, the corresponding boundary condition is 1 given by the lattice version of Eq. (9): β = 2 4 D . (17) g a − cos θx,23 = 1 , (in 3D) , (19) Notice that the above formula indicates that the dimen- sionality of the continuum coupling g in D spacetime where the x3 axis is now associated with the Euclidean dimensions is [g] = mass4 D. “time” direction. − One of the ways to implement the boundary conditions In addition to regular configurations, the ac- of the type (18) and (19) is to add a set of Lagrange tion (16) describes also singular topological configura- factors that affect the plaquettes in . To this end the tions, Abelian monopoles. We are not going to discuss standard U(1) action (10) has to beP changedS as follows: the monopoles in the present paper. To this end we work in a weak-coupling regime where the monopole density X Sλ[θ; ] = βP (λ) cos θP , (20) is negligibly small. We leave a detailed discussion of PS P the monopole effects on Casimir forces for a sequel ar- ticle [11]. where ( β , P / , S βP (λ) = ∈ P (21) 2. Lattice Abelian gauge theory: Boundary conditions β + λ , P , ∈ PS where β is given in Eq. (17) for appropriate dimension D. The Casimir boundary conditions force certain compo- The Lagrange multiplier λ is then sent to infinity in nents of the field strength tensor to vanish at the metallic order to enforce the condition (19). Consequently, the surfaces (5) and (8) in D = 3 + 1 and D = 2 + 1, respec- partition function (12) of the model in the presence of tively. Due to the linear character of these equations, the Casimir plates becomes as follows: the same conditions hold in the Euclidean spacetime in

D = 4 and D = 3 equations, correspondingly. [ ] = lim λ[ ] , (22) In the lattice gauge theory the boundary conditions (5) Z PS λ + Z PS Z→ ∞ and (8) correspond to the vanishing of the field strength Sλ[θ; S ] λ[ ] = θ e− P . (23) tensor (11) – up to the discrete transformations (14) – Z PS D at a certain set of the plaquettes P that are ei- S A different type of the boundary conditions based on ther touching at or belonging to the world-volume∈ P of the the lattice Chern-Simons action was proposed both in metallic surfaces. compact [8] and non-compact [9] U(1) gauge theories. For the sake of simplicity, below we consider examples Another approach – which was used in a non-compact of a static metallic surface in D = 4 and a straight wire in version of the theory – is to put a certain set of link D = 3. The generalization to more complicated surfaces variables θ to zero in a fixed gauge [10]. In our paper is straightforward. l we work with explicitly gauge-invariant approach keeping In D = 4 Euclidean spacetime we consider a straight the values of the Lagrange multiplier λ finite thus allow- plate in (x , x ) plane positioned at x = 0. The corre- 1 2 3 ing us to simulate a system with a finite permittivity ε. sponding boundary conditions are derived from Eq. (4): We will also take the limit λ in order to compare our numerical results with the→ expression ∞ for the ideal cos θx,14 = cos θx,24 = cos θx,12 = 1 , (in 4D) , (18) metals [Eq. (37) below]. where x = (x1, . . . , x4) and the x4 axis is the Euclidean The advantage of our approach is that we use different “time” direction. Condition (18) is valid for all x1, x2, parameters λij different for different orientations Pij of x4 at fixed x3 = 0. In order to derive Eq. (18) we no- the plaquettes in , and keep them finite. In (3 + 1) PS ticed that the condition Fµν (x) = 0 implies, because of dimensions the spatial-temporal components of λ may equivalence (14), the validity of the lattice constraint be associated with generally anisotropic components of

θPx,µν = 2πn where n Z. The latter constraint is the dielectric permittivity εi via the relation λi4 βεi. ∈ ∼ 4

The spatial-spatial components may be related to the all calculated R-dependent quantities (potentials, energy component of the diamagnetic permeability µi as follows: densities, fields, etc) should be invariant under the spatial 1 λij βijkµk− . flip R Ls R. Our∼ approach gives us certain freedom in simulations → − of Casimir systems with electromagnetic properties close to reality. Indeed, instead of enforcing ideal metal con- ditions (18) at infinite λ, a finite value of λ may allow us to simulate finite values of relative permittivity (di- electric constant) ε and relative permeability (encoded in diamagnetic constant µ) of the material. In order to illustrate our approach, below we study zero-point inter- actions of thin wires with a finite permittivity ε.

3. Non-Abelian gauge theory: Boundary conditions

For the sake of completeness here we also formulate the Casimir boundary conditions for SU(N) gauge theory. FIG. 1. The Casimir problem in two spatial dimensions. The non-Abelian analogues of the conditions (18) and (19) are given by a substitution of the cosines cos θP by The lattice Casimir condition (19) for ideally conduct- ing wires implies vanishing of the “23”-component of the 1 SP [U] = 1 Re Tr UP , (24) lattice field strength tensor at the plaquettes which be- − N long to the world-surfaces of the Casimir wires. These plaquettes are visualized in Fig. 2. In order to impose where U = U U U † U † is the SU(N) pla- Px,µν x,µ x+ˆµ,ν x+ˆν,µ x,ν the Casimir condition we simulate the compact gauge quette constructed from the Ux,µ link fields. The partition function in the presence of the plates is given model with the action (20). The coordinate-dependent by Eqs. (22) and (23), where the non-Abelian action is: gauge coupling βP , reflects the boundary conditions at the plates (21): X  Sλ[U; ] = βP SP [U] , (25) βP (ε) = β 1 + (ε 1) (δµ,2δν,3 δµ,3δν,2) PS x,µν − − P (δ + δ ) . (27) · x,l1 x,l2 and the plaquette-dependent non-Abelian coupling con- At the plaquettes belonging to the world surfaces of the stant β is given by Eq. (21). The SU(N) lattice coupling P wires the lattice coupling constant is equal to βP = εβ β is related to the continuum non-Abelian charge g sim- while outside of the wires, at the bulk majority of the ilarly to the Abelian case (17): β = 2N/(g2a4 D). − lattice plaquettes, the coupling constant is βP = β. In terms of our notations of Eq. (21), λ = (ε 1)β. The quantity ε is the static permittivity of the material.− Be- III. CASIMIR PROBLEM IN 3D ABELIAN low we study the dependence of the Casimir forces on the GAUGE THEORY value of the static permittivity ε. In our simulations we realize the case of perfectly con- A. Geometrical setup ducting wires by taking eventually the limit of large per- mittivity ε . This is possible because in two spatial In order to test our approach we study the compact dimensions→ the ∞ magnetic permeability does not exist and U(1) lattice gauge theory in D = 3 Euclidean dimen- a wire with infinite static dielectric permittivity affects 3 sions. We consider a symmetric Ls cubic lattice which the electromagnetic field in the same way as an ideal corresponds to a zero-temperature theory. We impose the metal (cf. Section 5.1 of Ref. [2]). Mathematically, in periodic boundary conditions at the opposite sides of the the limit of large dielectric permittivity ε a compo- lattice along all three directions. According to Eq. (17) nent of the electric field parallel to the wire→ ∞vanishes (9) the lattice gauge coupling is given by thus mimicking an ideal metal. 1 β = . (26) g2a B. Observables Our physical setup is shown in Fig. 1. We consider the simplest case of two parallel static straight wires along The most important associated with the the direction x separated by the distance R = l l . Casimir effect is the energy-momentum tensor of the 2 | 2 − 1| The wires are located at positions x1 = l1 and x1 = l2. gauge field (16), Due to the periodic boundary conditions the wires divide µν 1 µα ν 1 µν αβ T = F Fα + η FαβF . (28) the x1 axis into two intervals, R and Ls R. Therefore, 4g2 − − 5

The normalized energy density (32) is a local quan- tity which is equal to a change in the energy density of vacuum fluctuations that appears due to the presence of the wires. In the geometry of our problem the energy density (32) depends only on the coordinate x1, which is transverse to the wires themselves. Therefore it is nat- ural to introduce the (Casimir ) energy density per unit length of the wires:

+ Z∞ 1 V (R) = dx (x ) E2 , (33) Cas 1 ER 1 ≡ −2g2 y −∞ which is a finite quantity both in ultraviolet and infrared FIG. 2. The geometry of the Casimir problem in two spatial limits. In Eq. (33) the quantity dimensions. The shadowed planes indicate the plaquettes S where the boundary condition (19) is implemented. P Z (x) = dx [ (x) ] , (34) hhO ii 1 hO iR − hOi0 In Minkowski spacetime the energy density of the gauge corresponds to the excess of the expectation value of the field is: operator evaluated per unit length of the wires. In the latticeO 00 1 2 2 2 T = 2 Bz + Ex + Ey , (29) 2g Ls 1 X− where we took into account the fact that in the spacetime (x) = [ (x1) ] , (35) hhO iilat hO iR − hOi0 with the metric (+, , ) one has F01 = Ex, F02 = Ey x1=0 and F = B . − − 12 z and the lattice Casimir energy density per unit length of From Eq.− (29) we conclude that the Euclidean energy- the wires (33) takes the following compact form: momentum tensor has the following form: lat 1 VCas (R) = β cos θ23 , (36) T 00 = B2 E2 E2 , (30) hh ii E 2g2 z − x − y where the lattice coupling β is given in Eq. (26). since as we pass from the Minkowski space to the Eu- clidean space the terms with electric field in Eq. (29) change their signs, E2 E2 and E2 E2, while the C. Casimir energy: continuum vs. lattice x → − x y → − y one with the magnetic field remains intact, B2 B2. z → z Therefore the 1. General remarks The rotational of the problem – evident from Fig. 2 – imposes the following constraint on the expecta- A photon in two spatial dimensions has only one phys- tions values: ical degree of freedom. A field, corresponding to this de- 2 2 gree of freedom should vanish at the wire. Therefore, it is Bz = Ex . (31) natural to expect that the Casimir energy in a monopole- Indeed, the problem is invariant under a π/2 rotation free U(1) gauge theory coincides with the one of the free ± about the x x1 axis. The rotation interchanges the real-valued field with a Dirichlet boundary condi- ≡ axes x2 x3, and leads to the transformation of the tion imposed on the field at the wire. The latter energy ↔ ± field strengths: Bz Ex thus enforcing Eq. (31). is known to be as follows [12] From Eqs. (30) and↔ ± (31) we get the normalized energy density: ζ(3) 1 VCas(R) = 2 , (37) 00 00 − 16π R R(x) = TE (x) R TE (x) 0 E − where ζ(x) is the zeta-function with ζ(3) 1.20206. 1  2 2  = E E (x) , (32) The aim of this section is to derive a lattice≈ version of 2g2 y 0 − y R zero-point (Casimir) energy density between two static where the subscript “0” indicates that the expectation straight wires (37) in the non-compact, monopole-free value is taken in the absence of the wires while the sub- U(1) gauge theory. To this end we rederive Eq. (37) in script “R” means that the corresponding average is taken continuum spacetime and then briefly repeat the deriva- in the presence of the wires separated by the distance R. tion on the lattice in a weak-coupling regime where the By construction, the energy density (32) is free from ul- monopole density is negligibly small. The case with traviolet divergencies that emerges in the limit a 0. monopoles will be treated in details in Ref. [11]. → 6

2. Zero-point energy in continuum limit The integration under the exponent is taking place along the two-dimensional world surface, and the integrations Boundary conditions in integral form. A perfect infinitely over the Lagrange multipliers h+ and h enforce the Casimir conditions (42) at x = +R/2 and− at x = R/2 thin metallic wire forces the electric field along the wire 1 1 − to vanish (9) at any point of its world trajectory. In 3 plates, respectively. spacetime dimensions the world-trajectory of a wire is a two-dimensional surface S which can be parametrized Zero-point energy in continuum. In the presence of the by a vector x¯ = x¯(τ, ξ). Here τ and ξ are time-like Casimir surface the partition function of the and space-like parameters. For example, one can use the can be written asS follows following parametrization for a pair of two static wires Z Z V (R) S[A] placed at x± = R/2: Z e−A· = A e− δ [F ] = h Z [h], (48) 1 ± S ≡ D S D S  R  x¯ (τ, ξ) (x1, x2, x3) = , ξ, τ , (38) where h = h+ h and the photon action S[A] is ± ≡ ± 2 given inD Eq. (16)D withD − D = 3. For the static parallel where the subscript “ ” parameterizes the left/right seg- wires (50) Eq. (48) can be used to define the Casimir ments of the wires. ± energy per unit length of the wires: A surface S can be described by the antisymmetric 1 tensor function V (R) = log Z , (49) Z Z − S ∂x¯[µ, ∂x¯ν] (3)  A sµν (x) = dτ dξ δ x x¯(τ, ξ) , (39) where is the area of each wire ( = TL for ∂τ ∂ξ − a longA wire of length L which exists time T ).A where a b = a b a b . For the parallel static [µ, ν] µ ν − ν µ The h-dependent partition function (48) can be rewrit- straight wires (38) the world surface (39) has the fol- ten as follows: lowing form: Z Z   3 1 2 µ s± (x) = (δ δ δ δ ) δ(x R/2) . (40) Z [h] = A exp d x Fµν + iAµJ µν µ,2 ν,3 − ν,3 µ,2 1 ∓ S D −4 The Casimir condition (9) can conveniently be rewrit-  1 Z  = C exp d3x d3y J (x; h)D(x y)J (y; h) , (50) ten in a covariant form with the help of the quantity (39): −2 µ − µ µν F (x)sµν (x) = 0 . (41) where we have performed the Gaussian integration over For the static straight wires one automatically gets from the photon field Aµ. Notice that the source tensor (46) Eqs. (40) and (41): enters Eq. (50) via the conserved vector ν µ F ( R/2, x , x ) = 0 . (42) Jµ(x; h) = ∂ Jµν (x; h) , ∂ Jµ(x; h) = 0 . (51) 23 ± 2 3 In the path-integral formalism the Casimir condi- In Eq. (50) the quantity C stands for an inessential con- tion (41) can be implemented with the help of a δ func- stant. From now on we omit this and similar constant tional which can formally be written as follows: factors to simplify our notations.

Y  µν  Substituting Eq. (50) into Eq. (48) one gets δ [F ] = δ F (x)sµν (x) . (43) S Z 1 R 2 2 T x d x d y Λ (~x)Kb (~x ~y)Λ(~y) Z = Λ e− 2 − , (52) The infinite product of the δ functions (43) can be imple- S D mented with the help of the over where we introduced the two-dimensional vector ~x = the Lagrange multiplier h: (x2, x3) on the worldsheet of the wires. We also intro- Z  Z  duced the vector field: i 3 µν δ [F ] = h exp d x h(x)F (x)sµν (x) ,(44)   S D 2 h+(~x) Z  Z  Λ(~x) = , (53) i 3 µν h (~x) h exp d x F (x)Jµν (x; h) , (45) − ≡ D 2 and the matrix where we have introduced the source tensor:  ∂2 ∂2  K(~x) = + (54) x x x b 2 2 Jµν ( ; h) = h( )sµν ( ) . (46) ∂x2 ∂x3   In the case of two parallel Casimir plates (38) one gets D(0, x2, x3) D( R, x2, x3) − , Z Z  Z Z D(+R, x2, x3) D(0, x2, x3) δ [F ] = h+ h exp i dx2 dx3 S D D − with the scalar   X R Z d3k eikx 1 ha(x2, x3)F23 a , x2, x3 . (47) x 2 D( ) = 3 2 = , (55) a= 1 (2π) k 4π x ± | | 7 which obeys the differential equation This equation has the following solutions:

∆D(x) = δ(x) , (56) ~q  ~q R − κ (~q) = | | 1 e−| | . (67) 2 ± − 2 ± where ∆ ∂µ is the three-dimensional Laplacian. The Gaussian≡ integral (52) is given (up to an inessen- The solutions are characterized by the discrete index tial multiplicative constant) by the following determi- and continuous parameter ~q. The phase space associated± nant: with these variables is 1/2 Z 2 Z = det− K.b (57) d q X S Tr , (68) ~q ≡ A (2π)2 The Casimir energy per unit wire length can be deduced ± from Eqs. (49) and (57): where is the area in the transverse ~x (x , x ) plane. A ≡ 2 3 1 1 Thus, we get for Eq. (59): V (R) = log det− 2 Kb Tr log K.b (58) A· − ≡ 2 X Z d2q Tr log Kb log κi = log det Qb(R, ~q) In order to evaluate the potential in Eq. (58) we notice ≡ A (2π)2 first that formally this expression can be written as a sum i Z 2  2  over all eigenvalues κi d q ~q  2 ~q R = log 1 e− | | . (69) X A (2π)2 4 − Tr log Kb log κi , (59) ≡ i Then Eq. (58) leads us to the following expression for the potential: of the operator Kb: V (R) = V0 + VCas(R) , (70) KLb = κL m (60) where where L = L(~x) is an eigenvector of the operator Kb. In 1 Z d2q ~q 2 the explicit form the eigenvalue equation (60) reads as V = log (71) follows: 0 2 (2π)2 4 Z d2y Kb(~x ~y)L(~y) = κL(~x) . (61) is the divergent contribution to the potential V (R). Since − this contribution does not depend on the distance R be- tween the plates we can safely omit it. The second term Using the integral representation (55), the operator K b in Eq. (70) is the finite Casimir energy in Eq. (54) can be rewritten as follows Z 2   Z 3 2 2  ip1R  1 d q 2 ~q R d k p + p 1 e− 2 3 VCas(R) = 2 log 1 e− | | , (72) Kb(~x) = 3 2 2 2 ip1R .(62) 2 (2π) − − (2π) p1 + p2 + p3 e 1 Next, we represent the eigenvector L in terms of its The integral can be evaluated explicitly, the final result Fourier transform, for the Casimir energy is shown in Eq. (37). Z d2q L(~x) = L(~q)ei~q~x , (63) (2π)2 3. Zero-point energy on the lattice substitute Eqs. (62) and (63) into Eq. (61) and integrate The lattice derivation of the zero point energy follows over ~y. We obtain for Eq. (61): closely the derivation in the continuum limit. On the lat- 2 3 Z d q h i tice of the volume Ls the lattice momenta are quantized: Qb(R, ~q) κ L(~q)ei~q~x = 0 , (64) (2π)2 − 2πni pi = , ni = 0, 1,...,Ls 1 . (73) where Ls − Z 2  ip R  dp1 ~q 1 e 1 The lattice counterpart of Eq. (65) is as follows: Qb(R, ~q) = − − 2π p2 + ~q 2 eip1R 1 1 3    ~q R  P 2πni ~q 1 e−| | Ls 1 1 cos − Ls = | | ~q R . (65) 1 X i=2 − − 2 e−| | 1 Qblat (R; n2, n3) = L 3   − s P 2πni n1=0 1 cos Ls Since Eq. (64) should be valid for all vectors ~q, we j=1 − arrive to the following equation for the eigenmodes L: 2πiRn1 ! 1 e− Ls h i 2πiRn1 , (74) Qb(R, ~q) κ L(~q) = 0 . (66) · Ls − e 1 8 where R = 0,...Ls 1. The zero-point energy is given allows us to equilibrate the integration errors accumu- by the lattice versions− of Eqs. (58) and (69): lated at the Casimir planes and outside the planes. This equilibration is particularly important in a limit of high Ls 1 Ls 1 − − th 1 X X permittivities ε . V (R) = log det Qb(R; n2, n3) . (75) → ∞ lat 2 We eliminate long autocorrelation lengths in Markov n2=0 n3=0 chains of configurations using overrelaxation steps which The zero-point energy given by Eqs. (74) and (75) can separate gauge field configurations sufficiently far from easily be computed on the lattice. It contains an inessen- each other [13]. We also use self-tuning adaptive algo- tial constant and the physical R-dependent term. rithm in order to control acceptance rate in the HMC in range [0.70, 0.85]. The basic parameters of our simulation are presented in Table I. 0 ● ● ● ● ● ● ●●●●●●●●●●●● ● ● ● ● ● Trajectories per one value of ε 2 ... 4 105 × 0.00 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4 -1 ● ● Trajectories for thermalization 2 10 × -0.02 Overrelaxation steps between trajectories 5

lat -2 ● ● 3

V Lattice size 24

2 -0.04 a Range of gauge coupling β = 3 7 -3 ∼ -0.06 Values of permittivity ε per single value of β 20 ≈ -4 -0.08 ● ● 0 5 10 15 20 TABLE I. Basic simulation parameters. ● ● 0 5 10 15 20 We use Nvidia graphics processing units (GPU) R/a GTX980 with CUDA architecture as main coprocessors. Since our involves only nearest-neighbor in- FIG. 3. Theoretically calculated zero-point energy (75) be- teractions, the simulations can be parallelized by assign- tween two parallel wires separated by the distance R on ing one GPU thread to perform calculations at each site L = 24 symmetric lattice. The inset shows the energy with s of the lattice. In order to increase efficiency of the cal- the R = 0 and R = Lsa points excluded. culations we perform the simulations at the GPUs only thus decreasing data transfer between CPU and GPU. As an example, we show the theoretical lattice zero- point energy in Fig. 3 for Ls = 24 symmetric lat- tice. Obviously, the energy is invariant under the flips B. Electromagnetic fields around wires R aLs R. Notice that the depend of the energy on→ the distance− between the plates is very steep at small In order to characterize the effect of the wires on the distances R and it is almost flat at R 4a. Therefore, & local behavior of electromagnetic fields we calculate the numerically, the basics Casimir physics can be detected lattice quantities (no sum over ij indices is imposed): only at small separations between the wires where the lat ultraviolet lattice artifacts are particularly strong. We a4 F 2 (x) = 2 cos θ cos θ  . (76) will show that this problem is indeed rather serious but ij R − h iji0 − h ijiR it can nevertheless be successfully circumvented. The quantity (76) is equivalent, up to higher-order O(a6) terms, to a corresponding component of the electromag- netic field strength squared. IV. CASIMIR ENERGY IN SIMULATIONS In Fig. 4 we show the components of the field strength (76) as the function of the x x1 coordinate A. Numerical setup which is normal to the wires. We show≡ the data for the lattice coupling β = 3 at fixed permittivity ε = 6 and for We first generate configurations with a trivial relative three different distances between the wires, R/a = 1, 2, 8. permittivity ε = 1 of the wires, so that at the very fist All dimensionful quantities are shown in units of lattice step the wires are not visible. Then we gradually increase spacing. As expected, each wire strongly suppresses the the permittivity of the wires keeping the Wilson gauge electric field component parallel to the wire Ey. In accor- coupling β fixed. The configurations with higher ε are dance with the geometry of the problem, the wires affect generated starting from configurations with lower ε. also the components Ex and Bz by suppressing their fluc- We generate configurations of the gauge fields using a tuations Ex and Bz in an equivalent way (the latter is Hybrid Monte Carlo (HMC) which combines well seen for small separations between the wires). At 2 2 2 advantages of a molecular approach and stan- each wire Ex R = Bz R Ey while outside the 2 2 2 dard Monte-Carlo methods [13]. In the molecular dy- wires Ex R = Bz R Ey . The general features of namics component we use a second-order minimum norm the fields around the wires,∼ shown in Fig. 4, are rather integrator [14] with multiple time scales [15]. The latter universal: the increase of permittivity ε leads to further 9

0.00 ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■0.00◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■0.00◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆●■ ◆● ◆●■ ◆● ◆● ◆● -0.01 ◆● ◆● -0.01 -0.01 2 2 2 ●● Ex R=1a ●● Ex R=2a ●● Ex R=8a ��� ��� ��� 〉 -0.02 〉 〉 � 2 � -0.02 2 � -0.02 2 ■■ ■■ ■■ � Ey β� =3 Ey β� =3 Ey β=3 〈 〈 〈 � � � � -0.03 2 � -0.03 2 � -0.03 2 ◆◆ Bz ε=6 ◆◆ Bz ε=6 ◆◆ Bz ε=6 -0.04 -0.04 -0.04 ■ ■ ■ ■ ■ ■ 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

x1/a x1/a x1/a

FIG. 4. Excess in expectation values of the components of the field strength tensor squared (76) vs the coordinate x1 normal to the wires at the = lattice 3 and twocoupling di↵erentβ = values 3 andat of fixedpermittivity permittivity". Theε = positions 6. The positionsof the wires ofthe is marked wires are bymarked the vertical by the pink vertical lines, thepink distance lines, the between distances the between wires is theR =4 wiresa. are R/a = 1, 2, 8 for left, middle and right plots, respectively.

0.0 ◆●■ suppression of the field fluctuations at the positions of forfields each are nonzero. Notice and thatare independent at mentioned on the values distance of theR ●■ ●■ ●■ lattice coupling the density of the topological lattice the wires-0.1 without changing●■ ●■ ●■ ●■ the●■ ●■ qualitative●■ ●■ ●■ ●■ ●■ ●■ shape●■ ●■ ●■ of the between the wires. profiles. 2 2 configurations, Abelian monopoles, is extremely small, Ex, Bz 4 6 -0.2 ⇢mon 10 g [16]. Therefore we neglect the monopoles

Evidently,� the larger permittivity ε, the stronger ef- 2 ij in our⇠ analysis. fects ofF the wires◆ on electromagnetic fields are. In order ●■ � -0.3 0.056 to characterize the dependence of a quantity on the permittivity ε we◆ fitted the corresponding numericalO data 1 -0.4 ◆ 2 ▲ ▲ ▼▲ ■ ▲ 2 0.0 ε→+∞ ◆ ▲ ▼ ■ ◆ ▼ ◆●■▼ ◆ ● ◆ ■ ◆ Ey 0.054 ■▲ ▼ ● E

◆ ∞ ▼ y by the function ◆ ◆ ◆

◆ � ▲ ◆ ◆ ◆ ◆ 5

◆ 2 -0.5 ◆ ◆ ◆ ◆ ◆ ij ▼ ● C -0.5 ◆ ◆ ◆ ◆ ● β=◆3. ◆ � F 5(λ) =10 +15 O20 , 25 30 (77) 4 ◆●■■ 0.052g ∞ ● ■ β=4. O O ε + ε )/ ●■ ● ◆ ●■ ● ● ε O R ■ ■ ■

( ■ where , C and ε > 0 are the fitting parameters. -1.0 ◆ β=5.

Cas 2 2 O∞ O O 0.050 ◆ ▲ E B FIG. 5. The integral excess in the expectation values of elec- V , Examples of the normalized mean expectation values of 0 excl x ▲ β=x6. tromagnetic fields (34) and (35) vs permittivity " for L =24, -1.5 the electromagnetic fields integrated along the coordinate 1 excl ▼ β=7. =3andR = 4. The statistical errors are smaller than the 1 ▼ 2 3 4 5 6 7 8 x normal to the wires, Eqs. (34) and (35), are shown in size1 of the symbols. The lines are the fits by the function (77). 2 excl Fig. 5. The fits by the function (77) describe the data -2.0 R/a 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 very well with χ2/d.o.f. 1. ≈ FIG. 6. Integral expectationRg values2 of electromagnetic fields (34) and (35) vs the distance between the wires R in the ◆●■ ● 0.00.056 ■ FIG.perfect-metal 7. The zero-point limit (ε potential) for theVCas latticebetweenLs = perfectly 24 at β con-= 3. ●■ → ∞ ●■ ●■ ducting wires separated by the distance R in units of the con- -0.1 ●■ ●■ ●■ ●■ ●■ ●■ ●■ ●■ ●■ ●■1●■ ●■ ●■ ●■ ●■ 0.054 E2 tinuum coupling constant g at various lattice coupling con- ∞ 2 2 y � 5 stants . The lines represent the fits by Eq. (78) with zero,

2 E ,B ij x z

-F 0.2 ◆ ◆ ◆ ◆ ◆

� ◆ one and two first points removed at each .

� ● 2 ij ◆■ 0.052 ● C. Zero-point energy ◆ ●■ ●

� F ● -0.3 ■ ●■ ■ ●■ ■ The zero-point energy shows a very good scaling in the ◆ 2 2 1. Matching to continuum potential: strong lattice features 0.050 ◆ Ex , Bx ultraviolet limit. Indeed, according to Eq. (26) the phys- -0.4 ◆ 2 ◆ E ◆ ◆ y ical lattice spacings for = 3 and =7di↵er more than 1 2 3 ◆ 4◆ ◆ 5 6 7 8 ◆ ◆ ◆ ◆ ◆ ◆ ◆ byThe a factor integral of two, excess and in yet the theEy datacomponent for V determineexpressed -0.5 R/a ◆ ◆ Cas 5 10 15 20 25 30 inthe physical zero-point units energy almost induced coincide by with the wires each other, via Eq. Fig. (36). 7. However,The components the scalingEx and is notBz perfectdo not contribute because the to data the en- at FIG. 6. Integral expectation valuesε of electromagnetic fields (34) and (35) vs the distance between the wires R in the smallergy density – in lattice as they units exactly – wire cancel separations each other.R/a Independs Fig. 7 perfect-metalFIG. 5. The integral limit (" excess) in for the the expectation lattice Ls =24at values of elec-=3. verywe show strongly the zero-point on the distance. potential Indeed,VCas asas we a already function no- of !1 tromagnetic fields (34) and (35) vs permittivity ε for L = 24, ticedthe distance form Fig.R, 3, expressed the lattice in physical Casimir units potential of the is verycou- β = 3 and R = 4. The statistical errors are smaller than the steeppling constantat small R/ag2. We(where plotted the theultraviolet data for lattice a set of artifacts lattice size of the symbols. The lines are the fits by the function (77). components Ex and Bz do not contribute to the energy arecoupling strong) constants and veryβ flat= 3,..., even7, at and moderate convertedR/a the(where lat- density as they exactly cancel each other. In Fig. 7 we thetice statistical data from noise the is lattice strong). units Therefore, (in term we of expect the lattice that showThe the fits zero-point by Eq. (77) potential allowV usCas toas determine the function the of inte- the thespacing latticea) particularities to the physical should units be (in veryg) using for this Eq. quan- (26) distancegral excessR, in expressed the expectation in physical values units of electromagnetic of the coupling tity,for each and,β consequently,. Notice that this at quantity mentioned cannot values be reliably of the constantfields in theg2. limit We of plotted ideal conductivity. the data for In a Fig. set 6 of we lattice show comparedlattice coupling to (fittedβ the by) density the continuum of the topological formula for lattice the couplingthese quantities constants as functions =3,..., of7, the and distance converted between the lat- the zero-pointconfigurations, potential Abelian (37). monopoles, is extremely small, 4 6 ticewires dataR. Noticefrom the that lattice even units in the (in limit term of of infinite the lattice sepa- ρmonIn order10− tog check[16]. the Therefore importance we neglect of lattice the particular-monopoles ∼ spacingration, Ra) to the, the physical integral units excesses (in ing) electromagnetic using Eq. (26) ities,in our we analysis. have fitted the lattice zero-point potential V → ∞ Cas 10

respectively [according to Eq. (79) the theoretical value ▲▼ ▲ ▼▲◆ ■ ●▲ ◆ ■ 0.0 ε→+∞ ▲ ◆ ●▲ ▼ ■ ◆ ◆●■▼ for the strength is C 0.0239]. Thus, we observe that ▼ ■ ▼ th ▲ ◆ ≈ ▼ ● the fit of the lattice data by the theory-inspired continu- -0.5 ● β=3. ous function (78) is strongly dependent on the region of 4 ■ ■ β=4. the fit. We conclude that the fitting of the lattice data ◆

( R )/ g for the zero-point energy in (2 + 1)d lattice gauge theory -1.0 ◆ β=5.

Cas ▲ by the continuum function is ambiguous and, therefore, V 0 excl ▲ β=6. meaningless. -1.5 1 excl ▼ β=7. ▼ 2 excl -2.0 2. Matching zero-point energy to the lattice formula 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 R g2 There are two subtle points which have to be taken into account in order to recover the continuum version FIG. 7. The zero-point potential VCas between perfectly con- ducting wires separated by the distance R in units of the con- of the Casimir potential (the zero-point energy) from the tinuum coupling constant g at various lattice coupling con- lattice data: stants β. The lines represent the fits by Eq. (78) with zero, 1. As we mentioned, we should fit the data for the one and two first points removed at each β. lattice potential by the “latticisized” version (74) and (75) of the 1/R2 potential of continuum the- The zero-point energy shows a very good scaling in the ory (37). This procedure allows us to take into ultraviolet limit. Indeed, according to Eq. (26) the phys- account the lattice features of the potential both ical lattice spacings for β = 3 and β = 7 differ more than at short and long distances. The latter takes into account periodicity and mirror (R Lsa R) in- by a factor of two, and yet the data for VCas expressed → − in physical units almost coincide with each other, Fig. 7. variance of the potential. However, the scaling is not perfect because the data at 2. Since our model is compact, the relation between small – in lattice units – wire separations R/a depends continuum and lattice field strength squared (76) is very strongly on the distance. Indeed, as we already no- valid up to O(a6) terms. These corrections are not ticed form Fig. 3, the lattice Casimir potential is very essential in β limit where the lattice spac- steep at small R/a (where the ultraviolet lattice artifacts → ∞ 2 ing (26) is small, a g− . At finite values of the are strong) and very flat even at moderate R/a (where lattice coupling β the next-to-the-leading correc- the statistical noise is strong). Therefore, we expect that tions in lattice spacing a can be taken into account the lattice particularities should be very essential for the by replacing in Eq. (36) the Wilson lattice coupling Casimir potential, and, consequently, this quantity can- β by its Villain counterpart [17]: not be reliably compared to (fitted by) the continuum formula for the zero-point potential (37).    I0(β) In order to check the importance of lattice particular- βV (β) = 2 log , (80) I1(β) ities, we have fitted the lattice zero-point potential VCas by the following continuum function: where I0 and I1 are the modified Bessel functions. C For reference, the Villain coupling βV is 20% (7%) V (R) = lat , (78) fit − R2 smaller at than the Wilson coupling β at β = 3 (β = 7). where the Clat is the sole fitting parameter. In the perfect-metal limit of the theoretical value of Clat is (37): We calculate numerically the Casimir potential using Eq. (36) in which the Wilson coupling β is substituted ζ(3) Cth C lim C(ε) = 0.02391 .... (79) by its Villain counterpart (80), β βV . Then we fit the ∞ ε → ≡ ≡ →∞ 16π ≈ lattice data for the Casimir potential by the theoretical First, we have fitted by the continuum function (78) formula given by Eqs. (74) and (75): the whole range of the lattice data for VCas . Then we Clat have excluded the data at the smallest separation, R/a, V fit (R) = V th (R) , (81) lat C lat at each value of β and fitted the data again. Next, we th removed yet another point 2R/a for each value of β and where, for the sake of convenience, we introduced Cth, made the fit once again. These three fits are shown in Eq. (79). In Eq. (81) the prefactor Clat plays a role of a Fig. 7 by the solid, dashed and dot-dashed curves and single fit parameter. marked, respectively, by “0, 1, 2 excl”. For these fits The examples of the lattice fits in the ideal-metal we get different values of the would-be continuum en- limit ε are shown in Fig. 8. The lattice func- → ∞ ergy Clat = 0.0358(6), 0.0264(10), 0.0184(14) with the tion (81) describes the numerical data almost perfectly 2 2 quality of the fit defined by χ /d.o.f. = 1.45, 0.39, 0.17, with χ /d.o.f . 1. The results are essentially robust 11 against removal one or two points in the ultraviolet re- 3. Casimir effect at finite permittivity gion as the corresponding fitting parameters Clat coin- cide with each other within error bars. The latter prop- Our approach allows us to study dependence of the erty highlights the correctness of the chosen fit method. Casimir force on the permittivity ε of the wires. Since in our setup the wires are infinitely thin, the system can be characterized, in a continuum limit, by the following space-dependent permittivity: 0.00 ε→+∞ ε(x) = 1 + εδ(x l ) + δ(x l ) (82) -0.01 − 1 − 2 -0.02 where x x1 and l1 l2 = R according to Fig. 1. We have≡ repeated| the− analysis| of the previous section -0.03 V ( R )

2 β=3β=5β=7 for a wide range of ε. We have found that at finite ε the a -0.04 data match very well the lattice version (75) of the 1/R2 Simulation -0.05 potential. The corresponding continuum counterpart of Theory fit the zero-point potential is as follows: -0.06 C(ε) 2 4 6 8 V (R, ε) = . (83) Cas − R2 R/a In Fig. 10 we show the dependence of the strength C(ε) of the Casimir interaction (83) on the permittivity of the FIG. 8. Fits of the numerical data for the zero-point energy by wires as compared to the strength of the effect in the the lattice potential (81) on the lattice Ls = 24 at various β. The wires are perfectly conducting. ideal-metal limit at infinite permittivity. The numerical data show very good scaling as the data points corre- sponding to different values of the lattice coupling β lie The dependence of the best fit parameter Clat on the on the same curve. The dependence of the strength fac- value of the lattice coupling β in the ideal-metal limit is tor C(ε) on permittivity can be very well described by shown in Fig. 9. First of all, we notice that the numeri- the following function2: cal result matches perfectly the theoretical prediction be- ε 1 cause Clat = Cth within small error bars for all studied C(ε) = − C , (84) values of the lattice coupling β. We may attribute this ε + 2 ∞ property to inevitable finite-size corrections which may where the factor C for an ideal conductor is given in appear due to Wilsonian cos-type of the chosen action. Eq. (79). The function∞ in Eq. (84), shown in Fig. 10 We also notice from Fig. 9 that the Casimir effect is inde- by the solid line, describes the numerical data almost pendent on the lattice coupling which implies a very good perfectly. scaling towards the continuum limit. The latter matches

● ● well with the theoretical fact that the Casimir energy be- ● ■ ● ● ● ● ■ ●■ ●■ ■ ●■▲ ◆●▲■ ◆■◆ ◆ ●▲ ▲■▼◆▲◆ tween ideally conducting plates should not depend on the ●▲■ ▼▲◆■▼◆▼ 0.8 ●▲■◆▲▼◆▼ ● ▲■▼◆▼ coupling constant. ■▲◆▼ ● β=3 ● ▲◆▼ ◆▲■ ▼ ▲ ▼ 0.6 ■◆▼ ●▲▼ ■ β=4 C(ε) ◆ 1.4 ▲■▼ C∞ 0.4 ◆▼ ◆ β=5 ●▲ 1.2 ■ ▼ ▲ β=6 0.2 ▲◆ ε-1 1.0 ▼ C(ε)= C∞ ε+2 ▼ β=7 Clat 0.8 0.0 ◆●▼▲■ Cth 0.6 0 5 10 15 20 25 30 0.4 ε

0.2 ε→+∞ FIG. 10. The strength factor C of the zero-point energy (83) 0.0 as the function of permittivity ε of the wires. The solid line 3 4 5 6 7 corresponds to the function (84). β

FIG. 9. The ratio of numerically simulated vs. theoretically calculated prefactors of the Casimir energy at various lattice coupling constants β for perfectly conducting wires. 2 A fit of the lattice data by the function C(ε) = ε+ε1 gives the C∞ ε+ε2 following best fit parameters: ε1 = −1.01(1) and ε1 = 1.99(7). 12

V. CONCLUSIONS In the limit of an ideally conducting wire, ε , our result for the zero-point energy agrees very well→ with ∞ the analytical formula (37), Fig. 9. At finite values of the rel- We proposed a simple method to calculate the zero- ative permittivity of the wires, ε > 1 the vacuum energy point (Casimir) vacuum energy using Monte-Carlo meth- has the Casimir form (83) with a modified prefactor (84) ods in the framework of lattice gauge theory. shown in Fig. 10. Generally, our lattice data exhibit very accurate ultraviolet scaling. The Casimir energy is associated with modification of Our method is suitable for calculations of the zero- the zero-point vacuum fluctuations in the presence of point energy in thermodynamic equilibrium for materials physical objects which impose certain boundary condi- described by (generally, space-dependent) static permit- tions on (electromagnetic) fields and/or affect the fields tivity ε(x) and permeability µ(x). It can be generalized via relative permittivity ε and/or permeability µ of the to calculate the vacuum energy for spatially anisotropic material. In our numerical method the materials are de- materials of various shapes (with an appropriate dis- scribed by space- and orientation-dependent gauge cou- cretization), at zero and finite temperatures, and for any, pling (21) in the lattice action (20). including non-Abelian, gauge groups. In order to illustrate our approach we calculated the zero-point energy between two parallel thin wires char- acterized by a finite static permittivity ε. We carried out ACKNOWLEDGMENTS our simulations in a weak-coupling regime of an Abelian gauge theory in two spatial dimensions. The appropri- The work was supported by the Federal Target Pro- ate modification of the lattice gauge coupling is given by gramme for Research and Development in Priority Areas Eq. (27). In the continuum limit the wires are described of Development of the Russian Scientific and Technolog- by the spatially dependent permittivity (82). ical Complex for 2014-2020 (Contract 14.584.21.0017).

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