CCasimir effect for Dirac lattices
M. Bordag
Institut f¨ur Theoretische Physik, Universit¨at Leipzig, Germany ∗ I.G. Pirozhenko
Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, RussiaDubna State University, Dubna, Russia † We consider polarizable sheets, which recently received some attention, especially in context ofthe dispersion interaction of thin sheets like graphene. These sheets are modeled by a collectionof delta function potentials and resemble zero range potentials, known in quantum mechanics. Wedevelop a theoretical description and apply the so-called TGTG-formula to calculate the interactionof two such lattices. Thereby we make use of the formulation of the scattering of waves off suchsheets provided earlier. We consider all limiting cases, providing link to earlier results. Also, wediscuss the relation to the pairwise summation method.
PACS numbers: 03.65.Nk,03.65.-w
I. INTRODUCTION
This paper is a contribution to the discussion of vander Waals and Casimir forces between surfaces. Lastyears, much attention was paid to the interaction be-tween slabs of finite, and especially small, thickness. Thistriggered by a growing interest to two dimensional struc-tures (sheets) like 2d electron gas, monoatomically lay-ers and graphene, as well as to the interaction betweenthem. As discussed in [1], the situation with sheets hav-ing only in-plane polarizability is relatively clear. Hereone can formulate a hydrodynamic plasma model and cal-culate the quantities of interest [2]. The same holds forgraphene described by the Dirac equation model for the π -electrons [3], which are responsible for the interactionwith the electromagnetic field.The situation with perpendicular polarizability is morecomplicated. While in [4] no response to the electromag-netic field was found, in [5] it was shown that such re-sponse takes place. In that paper the sheet was modelledas a lattice of harmonic oscillators (dipoles), vibratingin direction perpendicular to the sheet, and the limit ofzero spacing of this lattice was investigated. This wasextended in [6] to a sheet, continuous from the very be-ginning, having parallel or perpendicular polarizabilities.Later, a similar setup was considered using point dipoles[7]. These can be represented by Dirac delta functionsforming a so-called ’Dirac lattice’. In fact, there repre-sent point scatterers.Such lattices, taken alone, have a well known inter-nal dynamics, the simplest case being the Kronig-Penneymodel (’Dirac comb’) [8]. In quantum mechanics theiruse is known as ’method of zero-range potential’ [9]. Aswell known, in more than one dimension a Hamilton op-erator with a delta function potential is not self adjoint ∗ [email protected] † [email protected] [10]. In electrodynamics, the self energy of a delta func-tion potential is singular and one needs a renormaliza-tion. In terms of quantum field theory this setup wasconsidered in [11].Recently, a sheet of delta function potentials was usedin [7] to model a polarizable sheet. For instance, scatter-ing off such sheet was investigated and subsequently thetransition to a continuous sheet as well.In the present note we consider a setup of two suchsheets, hold in parallel at some separation. This is atypical situation for the Casimir effect. We use the wellknown scattering approach, also called ’TGTG’-formula,which can be considered also as a generalization of theLifshitz formula. Making use of the translational invari-ance of the lattice, the T -matrices can be calculated inmomentum representation; for a one dimensional latticeeven explicitly. Further, we allow for a displacement ofone lattice with respect to the other. Also we study thelimiting cases of small and large separation, resp. of largeand small lattice spacing. For large separation, the limi-ting case corresponds to the interaction of two sheets car-rying delta function potential, and for small separationthe limiting case corresponds to the interaction of twopoints carrying delta function potential. In this paperwe consider only a scalar field. Extensions to the elec-tromagnetic field should be quite straight forward. It isalso an intention of this paper to provide a framework forcalculating dispersion forces in realistic, experimentallyrelevant situations, where the atomistic structure mustbe accounted for.The paper is organized as follows. In the next sectionwe provide the necessary formulas to calculate the vac-uum energy for two and one dimensional lattices. In thethird section we derive all limiting cases and discuss theirinterrelations. In the fourth section we compare the re-sults with pairwise summation. Some conclusions and anappendix complete the paper. a r X i v : . [ qu a n t - ph ] J a n II. VACUUM ENERGY FOR POINTSCATTERERS
We consider two types of lattices. First, two dimen-sional lattices, A and B, given by the lattice vectors (cid:126)a A n = (cid:18) a n + c b (cid:19) , (cid:126)a B n = (cid:18) a n (cid:19) , (1)For these, we use the following notations. A three di-mensional vector is denoted by an arrow, a two dimen-sional vector parallel to the ( x, y )-plane is denoted inbold, for instance (cid:126)x = (cid:18) x x (cid:19) . The lattice sites are givenby a n = a (cid:18) n n (cid:19) , where n and n are integers, a is thelattice spacing; the lattice is rectangular. The lattice Bis in the ( x, y )-plane and the lattice A is parallel to Bon a separation b and shifted within that plane by thedisplacement vector c .Second we consider one dimensional lattices (chains),A and B, given by the lattice vectors (cid:126)a A n = an + c b , (cid:126)a B n = an , (2)where n is a (single) integer. These chains are on the x -axis (B) resp. parallel to it (A). Both are in the plane y = 0. Their separation is b as above, the displacement c is a number.With these notations, we consider a scalar field φ ( (cid:126)x ),whose wave equation after Fourier transform in time is (cid:32) − ω − ∆ + g (cid:88) n (cid:0) δ ( (cid:126)x − (cid:126)a A n ) + δ ( (cid:126)x − (cid:126)a B n ) (cid:1)(cid:33) φ ( (cid:126)x ) = 0 , (3)where (cid:126)a A n and (cid:126)a B n are given by (1). In this equation, thedelta functions are three dimensional, hence the coupling g has the dimension of length. As mentioned in the Intro-duction, one should remember that the above equation isnot well defined and g must be viewed as a bare couplingwhich needs to undergo a renormalization.The vacuum energy of the interaction of two lattices isgiven by the mentioned ’TGTG’-formula, E = 12 π (cid:90) ∞ dξ Tr ln(1 − M ) , (4)and ξ is the imaginary frequency, ω = iξ . The kernel M is the product of two other kernels, M ( (cid:126)x, (cid:126)x (cid:48) ) = (cid:90) dx (cid:48)(cid:48) N A ( (cid:126)x, (cid:126)x (cid:48)(cid:48) ) N B ( (cid:126)x (cid:48)(cid:48) , (cid:126)x (cid:48) ) , (5)each of which is given by N A,B ( (cid:126)x, (cid:126)x (cid:48) ) = (cid:90) dx (cid:48)(cid:48) G ( (cid:126)x, (cid:126)x (cid:48)(cid:48) ) T A,B ( (cid:126)x (cid:48)(cid:48) , (cid:126)x (cid:48) ) , (6) in terms of the T -operators of the corresponding lattice.Further, the free Green’s function, G ( (cid:126)x − (cid:126)x (cid:48) ) = (cid:90) d k (2 π ) e i(cid:126)k ( (cid:126)x − (cid:126)x (cid:48) ) ω − k − k + i , (7)enters this formula. We will use below also the followingtwo representations, G ( (cid:126)x ) = e iω | (cid:126)x | π | (cid:126)x | = (cid:90) d k (2 π ) e i k ( x − x (cid:48) )+ i Γ | x − x (cid:48) | i Γ( k ) , (8)with Γ( k ) = √ ω − k + i k = | k | .The T -operators can be defined from G ( (cid:126)x, (cid:126)x (cid:48) ) (9)= G ( (cid:126)x − (cid:126)x (cid:48) ) − (cid:90) dz (cid:90) dz (cid:48) G ( (cid:126)x − (cid:126)z ) T ( (cid:126)z, (cid:126)z (cid:48) ) G ( (cid:126)z (cid:48) − (cid:126)x (cid:48) ) , where G ( (cid:126)x, (cid:126)x (cid:48) ) is the Green’s function of equation (3) andcan be related to the ansatz G ( (cid:126)x, (cid:126)x (cid:48) ) = G ( (cid:126)x − (cid:126)x (cid:48) ) − (cid:88) n , n (cid:48) G ( (cid:126)x − (cid:126)a n )Φ − n , n (cid:48) G ( (cid:126)a n (cid:48) − (cid:126)x (cid:48) ) . (10)Inserting this ansatz into the equation (cid:32) − ω − ∆ + g (cid:88) n (cid:0) δ ( x − (cid:126)a A n ) + δ ( x − (cid:126)a B n ) (cid:1)(cid:33) G ( (cid:126)x, (cid:126)x (cid:48) )= δ ( (cid:126)x − (cid:126)x (cid:48) ) (11)and using the equation (cid:0) − ω − ∆ (cid:1) G ( (cid:126)x, (cid:126)x (cid:48) ) = δ ( (cid:126)x − (cid:126)x (cid:48) ) (12)for the free Green’s function, one comes to the equation − Φ − n , n (cid:48) + gδ n , n (cid:48) − g (cid:88) m G ( (cid:126)a n − (cid:126)a m )Φ − m , n (cid:48) = 0 . (13)With the definitionΦ n , n (cid:48) = 1 g δ n , n (cid:48) − G ( (cid:126)a n − (cid:126)a n (cid:48) ) (14)Eq. (13) becomes (cid:88) m Φ n , m Φ − m , n (cid:48) = δ n , n (cid:48) (15)and Φ − n , n (cid:48) is the inverse matrix to Φ n , n (cid:48) .The diagonal elements of Φ n , n (cid:48) contain G (0) whichis not well defined. Referring to [7] for a discussion, werenormalize the coupling,1 g − G (0) → g , (16)and after that we getΦ n , n (cid:48) = (cid:40) g , ( n = n (cid:48) ) − G ( (cid:126)a n − (cid:126)a n (cid:48) ) , ( n (cid:54) = n (cid:48) ) (17)which is well defined.Now, comparing Eqs. (10) and (10), we identify the T -operator to be T A ( (cid:126)z, (cid:126)z (cid:48) ) = (cid:88) n , n (cid:48) δ ( (cid:126)z − (cid:126)a A n )Φ − n , n (cid:48) δ ( (cid:126)a A n (cid:48) − (cid:126)z (cid:48) ) (18)for one lattice and by the same formula with B in placeof A for the other. Inserting this expression into (6) weget N A ( (cid:126)x, (cid:126)x (cid:48) ) = (cid:88) n , n (cid:48) G ( (cid:126)x − (cid:126)a A n )Φ − n , n (cid:48) δ ( (cid:126)a A n (cid:48) − (cid:126)x (cid:48) ) (19)and, again, the same with A → B . Further, insertingthese into (5), we get M ( (cid:126)x, (cid:126)x (cid:48) ) = (cid:90) dx (cid:48)(cid:48) N A ( (cid:126)x, (cid:126)x (cid:48)(cid:48) ) N B ( (cid:126)x (cid:48)(cid:48) , (cid:126)x (cid:48) ) , (20)for the kernel M . Eqs. (19) and (20) represent thegeneral expressions for the kernel entering the ’TGTG’-formula (4) for a generic lattice of delta functions. Inthe following we consider first two dimensional latticesas given by (1), and, subsequently one dimensional lat-tices (chains) as given by Eq. (2). A. Vacuum energy for two dimensional lattices
In this subsection we consider lattices as given by Eq.(1). These have translational invariance with respect toa lattice step. Equivalently, we have a n − a n (cid:48) = a n − n (cid:48) ,and the quantities entering Eq. (14) and (15), dependonly on differences, Φ n , n (cid:48) = Φ n − n (cid:48) and the same for its inverse. As a consequence, the inversion of this matrixcan be calculated by Fourier transform, f n = a (cid:90) d k (2 π ) e i ka n ˜ f ( k ) , ˜ f ( k ) = (cid:88) n e − i ka n f n . (21)This way we getΦ − n , n (cid:48) = a (cid:90) d k (2 π ) e i k ( a n − a n (cid:48) ) φ ( k ) (22)with ˜ φ ( k ) = (cid:88) n e − i ka n Φ n . (23)Here we insert (17) and come to˜ φ ( k ) = 1 g − (cid:88) n (cid:48) e − i ka n G ( a n ) , = 1 g − π J ( ω, k ) , (24)where Eq. (8) was used and we introduced J ( ω, k ) = (cid:88) n (cid:48) | a n | e iω | a n | + i ka n . (25)As usual, in the primed sum the term with n = 0 isdropped. Finally, we mention the relation (cid:88) m e i ka m Φ − m , m (cid:48) = 1˜ φ ( k ) e i ka m (cid:48) , (26)which follows with the Fourier transform (21).Now we return to the kernel N , (19), and insert (8) forthe Green’s function, N A ( (cid:126)x, (cid:126)x (cid:48) ) = (cid:88) n , m (cid:90) d k (2 π ) e i k ( x − a n − c )+ i Γ | x − b | i Γ( k ) Φ − n − m δ ( a m + c − x (cid:48) ) δ ( b − x (cid:48) ) . (27)Next we use (26) and get N A ( (cid:126)x, (cid:126)x (cid:48) ) = (cid:90) d k (2 π ) e i k ( x − c )+ i Γ | x − b | i Γ( k ) ˜ φ ( k ) (cid:88) m e − i ka m δ ( a m + c − x (cid:48) ) δ ( b − x (cid:48) ) . (28)The corresponding expression for N B ( (cid:126)x, (cid:126)x (cid:48) ) follows from here with c = 0 and b = 0. Using these, we can writedown the kernel M ( (cid:126)x, (cid:126)x (cid:48) ), Eq. (5), M ( (cid:126)x, (cid:126)x (cid:48) ) = (cid:90) dx (cid:48)(cid:48) (cid:90) d k (2 π ) e i k ( x − c )+ i Γ | x − b | i Γ( k ) ˜ φ ( k ) (cid:88) m e − i ka m δ ( a m + c − x (cid:48)(cid:48) ) δ ( b − x (cid:48)(cid:48) ) · (cid:90) d k (cid:48) (2 π ) e i k (cid:48) x (cid:48)(cid:48) + i Γ (cid:48) | x (cid:48)(cid:48) | i Γ( k (cid:48) ) ˜ φ ( k (cid:48) ) (cid:88) m (cid:48) e − i k (cid:48) a m (cid:48) δ ( a m (cid:48) − x (cid:48) ) δ ( x (cid:48) ) . (29)Carrying out the integration over (cid:126)x (cid:48)(cid:48) using the delta func-tions, we come to a sum over m . Before doing this sum,we split the momenta k and k (cid:48) into quasi momentum andinteger part, k = q + 2 πa N , k (cid:48) = q (cid:48) + 2 πa M , (30)where N and M are integer vectors like n in a n . Theintegration becomes (cid:82) d k = (cid:82) d q (cid:80) N and the compo- nents of q = ( q , q ) are restricted to − πa ≤ q , < πa .Now the sum over m appearing in (29) gives (cid:88) m e − i ( k − k (cid:48) ) a m = (cid:18) πa (cid:19) δ (2) ( q − q (cid:48) ) , (31)where the dependence on N and M dropped out. Then(29) turns into M ( (cid:126)x, (cid:126)x (cid:48) ) = 1 a (cid:90) d q (2 π ) (cid:88) N e i x ( q + πa N ) − i πa Nc + i Γ | x − b | i Γ( k ) ˜ φ ( k ) (cid:88) M e i πa Mc + i Γ (cid:48) | b | i Γ( k (cid:48) ) ˜ φ ( k (cid:48) ) (cid:88) m (cid:48) e − i qa m (cid:48) δ ( a m (cid:48) − x (cid:48) ) δ ( x (cid:48) ) . (32)This expression for M ( (cid:126)x, (cid:126)x (cid:48) ) must be inserted into thegeneralized Lifshitz formula (4) under the sign of thetrace. In doing so, the delta function in one factor M ( (cid:126)x, (cid:126)x (cid:48) ) turns the x in the next factor into a m (cid:48) . Again,the summation over m (cid:48) gives a delta function for thequasi momenta. In this way, the products of the factors M ( (cid:126)x, (cid:126)x (cid:48) ) is diagonal in q and we can define h ( ω, q ) = 1 a (cid:88) N e i Γ( k ) b + i πa Nc i Γ( k ) ˜ φ ( k ) (33)with k given by (30), and we rewrite (32) in the form M ( (cid:126)x, (cid:126)x (cid:48) ) = a (cid:90) d q (2 π ) e i qx h ( ω, q ) h ( ω, q ) ∗ (cid:88) m e − i qa m δ ( a m − x (cid:48) ) δ ( x (cid:48) ) . (34)In this way, the factors M ( (cid:126)x, (cid:126)x (cid:48) ) entering (10) becomediagonal. The x -integration in the trace is to be takenover one cell and it turns the sum into unity, (cid:90) d(cid:126)x e i qx (cid:88) m e − i qa m δ ( a m − x ) δ ( x ) = 1 . (35)As a consequence, for the vacuum energy per lattice cellwe get the expression E = 12 (cid:90) ∞ dξπ a (cid:90) d q (2 π ) ln (cid:0) − | h ( iξ, q ) | (cid:1) . (36)This is the final general formula for the vacuum energy oftwo lattices as given by Eq. (1). The numerical result forthe vacuum energy for one cell of the lattices is presentedat Fig. 1. B. Vacuum energy for one dimensional lattices(chains)
In this subsection we consider lattices as given by Eq.(2). The calculations are mostly in parallel to the pre-ceeding subsection. The Fourier transform is now one dimensional, f n = a (cid:90) dk π e ik an ˜ f ( k ) , ˜ f ( k ) = (cid:88) n e − ik an f n , (37)where we used for the momentum the notation k sincethe lattice is parallel to the x -axis. Following Eqs. (22)and (23), we get ˜ φ ( k ) = (cid:88) n e − ik an Φ n , (38)which after renormalization gives˜ φ ( k ) = 1 g − π (cid:88) n (cid:54) =0 e iω | n | + ik an a | n | . (39)The sum is one dimensional and can be done explicitly,˜ φ ( k ) = 1 g − πa ln (cid:0) e iωa − k a ) e iωa (cid:1) . (40)By the way, the zeros of this function determine the zonesin the Kronig-Penney model.Now we consider the kernel N A , Eq. (6), and insertthe Green’s function from (8) and the relation (cid:88) n e ik an Φ − mm (cid:48) = 1˜ φ ( k ) e − k am (cid:48) (41)to get N A ( (cid:126)x, (cid:126)x (cid:48) ) = (cid:90) d k (2 π ) e ik ( x − an − c )+ i Γ | x − b | i Γ( k ) ˜ φ ( k ) (cid:88) m e − ik am δ ( am + c − x (cid:48) ) δ ( x (cid:48) ) δ ( b − x (cid:48) ) . (42)In parallel to (29), but carrying out the integration over x (cid:48)(cid:48) , we get further M ( (cid:126)x, (cid:126)x (cid:48) ) = 1 a (cid:90) d k (2 π ) e ik ( x − c )+ ik x + i Γ | x − b | i Γ( k ) ˜ φ ( k ) (cid:88) m e − ik am · (cid:90) d k (cid:48) (2 π ) e ik (cid:48) ( am + c )+ i Γ (cid:48) | x (cid:48)(cid:48) | i Γ( k (cid:48) ) ˜ φ ( k (cid:48) ) (cid:88) m (cid:48) e − ik (cid:48) am (cid:48) δ ( am − x (cid:48) ) δ ( x (cid:48) ) δ ( x (cid:48) ) . (43)We introduce a splitting of the momenta like (30), but now only for k , k = q + 2 πa N, k (cid:48) = q + 2 πa M, (44)where N and M are integers, and come to M ( (cid:126)x, (cid:126)x (cid:48) ) = 1 a (cid:90) dq π (cid:90) dk π (cid:88) N e iq ( x − c )+ i πa Nc + ik x + i Γ | x − b | i Γ( k ) ˜ φ ( k ) · (cid:90) dk (cid:48) π (cid:88) M e iqc + i πa Mc + i Γ (cid:48) | b | i Γ( k (cid:48) ) ˜ φ ( k (cid:48) ) (cid:88) m e − iqam δ ( am − x (cid:48) ) δ ( x (cid:48) ) δ ( x (cid:48) ) . (45)Following the same discussion concerning the trace as inthe preceding subsection, we are lead to introduce thenotation h ( ω, q ) = 1 a (cid:90) dk π (cid:88) N e i πa Nc + i Γ | b | i Γ( k ) ˜ φ ( k ) (46)with Γ = (cid:113) ω − k − k + i , k = q + 2 πa N. (47)Finally, we get the vacuum energy for one cell of thechains, E = 12 (cid:90) ∞ dξπ a π/a (cid:90) − π/a dq π ln (cid:0) − | h ( iξ, q ) | (cid:1) . (48)This is the final general formula for the vacuum energyof two lattices as given by Eq. (2).It should be mentioned that the formula for h ( iξ, q )can be simplified since the integration over k in (46)can be carried out. First, we perform the Wick rotationand define Γ = iγ, γ = (cid:113) ξ + k + k . (49) We get h ( iξ, q ) = 1 a (cid:90) dk π (cid:88) N e i πa Nc − γb − γ ˜ φ ( k ) . (50)Changing the integration from k for γ and using theintegral representation K ( z ) = (cid:90) ∞ e z cosh( θ ) dθ (51)one comes to h ( iξ, q ) = 12 πa (cid:88) N K ( (cid:112) ξ + k b ) − φ ( k ) e i πa Nc (52)with ˜ φ ( k ) given by Eq. (40). The numerical result forthe vacuum energy for one cell of the chains is presentedat Fig. 1 (inset).The influence of the displacement c on the interactionbetween the chains is shown in Fig. 2. Similar profilesmay be found in [12] for the Casimir-Polder potentialbetween an atom and a one-dimensional grating. FIG. 1. The vacuum energy per unit as a function of distancebetween lattices for different values of the coupling. Herelattices are exactly opposite to one another, c = 0. From leftto right g/a = 0 . , . , .
1. Inset: the same for two parallelDirac chains.FIG. 2. The factor η = E ( c (cid:54) = 0) /E ( c = 0) is plotted as afunction of c/a for g/a = 0 . β = b/a of the chains. From bottom to top β = 0 . , . , . , . III. LIMITING CASES
In this section we discuss the limiting cases of the inter-action lattices of delta functions. There are two limitingcases for each, two and one dimensional lattices. Onelimiting case is for large separation, or equivalently fordimensional reasons, for small lattice step. The other isfor small separation, or equivalently, large lattice step.For large separation, the two dimensional lattices turninto planes, carrying a delta function potential. For thesethe Casimir effect was first investigated in [13]. The onedimensional lattices (chains) turn into lines, carrying a(two dimensional) delta function potential. This case wasnot considered so far. However, it turns out that that itis equivalent to the finite size cylinders with Dirichletboundary conditions for turning their radius to zero, asconsidered in [14] and [15].For small separation, in the limit, only one delta func-tion for the one lattice interacts with the closest one inthe other lattice. This is equivalent to the Casimir-Polderinteraction of two points carrying a delta function poten-tial each. This is the scalar version of the well knowninteraction of two dipoles, which was considered withinthe scattering approach in [16]. Also, it can be consid-ered as limiting case of the interaction of two spherescarrying delta function potential for vanishing radius ofthe spheres. The corresponding calculation is shown inthe Appendix.The considered cases and their interrelations are shown in Fig. 3.
A. Two dimensional lattices at large separation
We start from Eq. (36) for the vacuum energy and con-sider the limit of lattice spacing a →
0, or equivalently,separation b → ∞ . At the end, we will get tow parallelcontinuous sheets for which we need the energy densityper unit area which is a E . The integration region in q , given by − πa ≤ q , < πa , turns into a whole R . Thefunction h ( ω, k ), (33), reads with imaginary frequency, h ( iξ, q ) = 1 a (cid:88) N e − γ ( k ) b + i πa Nc − γ ( k ) ˜ φ ( k ) (53)with γ = (cid:112) ξ + k , k i = q i + πa N i (i=1,2). For a → N = 0 vanish and weget k i = q i . Also we need the function ˜ φ , given by (24) and the function J , (25), which reads now J ( iξ, k ) = (cid:88) n (cid:48) | a n | e − ξ | a n | + i ka n . (54)For a →
0, the sum turns over into an integral, whichcan be calculated easily, J ( iξ, k ) → a → πa γ , (55)see Eq. (134) in [7]. This way we get h ( iξ, k ) → a → r e − ξb , (56)where r = 11 − a g γ . (57) FIG. 3. Possible transitions from one dimensional lattices (chains) and two dimensional lattices of delta functions for largeand small separation. For convenience, the two dimensional lattices are shown for c = 0, i.e., without displacement. For largeseparation, the lattices turn into continuous plates and the chains into wires. For small separation both turn into the closespints. Also shown are the transitions from spheres and cylinders to plates and wires. The vacuum energy (36) becomes1 a E → a → (cid:90) ∞ dξπ (cid:90) d q (2 π ) ln (cid:16) − (cid:0) r e − γb (cid:1) (cid:17) . (58)In fact, r , Eq. (57), is the well known reflection coeffi-cient for a plane carrying a delta function potential and(58) gives the Casimir effect for two such sheets. Theratio g/a , having the meaning of coupling per unit area,is the strength of the delta function potential in the cor-responding equation, where the delta function is one di-mensional and well defined such that this strength carriesover from the equation to the reflection coefficient with-out change. This is in opposite to the strength g in Eq.(36) which after renormalization (16) has little to do withthe g in equation (3). The other way round, one can usethe established above relation to the continuous sheets as normalization of the coupling g after renormalization. B. One dimensional lattices (chains) at largeseparation
This case is to a large extend in parallel to the pre-ceding subsection. We start from (48) and consider theenergy per unit length, a E . Further, we substitute ξ → ξ/b and q → q/b . This way we get a factor 1 /b in front and all further dependence is on a/b .Again in the function h ( iξ, k ), Eq. (40), accountingfor (44), only N = 0 gives a non vanishing contribution.A somehow different feature appears from ˜ φ , ( ?? ), whichhas a logarithmic behaviour now,˜ φ ( k ) = 1 g − b πa (cid:18) − ab ξ + ln (cid:18) (cid:18) cosh (cid:16) ab ξ (cid:17) − cos (cid:16) ab k (cid:17) (cid:19)(cid:19)(cid:19) , = 1 g − b πa (cid:16) (cid:16) ab (cid:17) − ab ξ + ln (cid:0) ξ + k (cid:1) + . . . (cid:17) , → a → − ln( ab )2 πa/b . (59)The last line is the leading order and one observes thatthe dependence on the coupling g is lost. Inserting into(52) we get h ( ξ, k ) → a → b ln( a/b ) K (cid:18)(cid:113) ξ + k (cid:19) . (60) Finally, inserting into (48), we get1 a E → a → b (cid:90) ∞ dξπ (cid:90) dk π ln (cid:32) − (cid:18) a/b ) K (cid:18)(cid:113) ξ + k (cid:19)(cid:19) (cid:33) , (61)and further expanding the logarithm,1 a E → a → − πb a/b ) , (62)where (cid:82) ∞ dp p K ( p ) = was used.It is to be mentioned, that this result coincides withthe large separation limit of the interaction between twoparallel cylindrical shells carrying δ -potentials (see Ap-pendix B), considered also in [17]. The same limit maybe reproduced from [15], Eq. (88), for the interaction ofa conducting cylinder with a conducting plane after tak-ing into account the different geometries and a factor of2 for the polarizations. C. Two dimensional lattices at short separation
Here we consider the vacuum energy (36) for the inter-action of two dimensional lattices for small separation, orequivalently for large lattice steps. In this case the inte-gration region for the integration over q in (36) shrinksto zero and this integral gives a (cid:90) d q (2 π ) f ( q ) → a →∞ f (0) , (63)where f ( q ) is some arbitrary function. In the function h ( ω, q ), (24), we first consider the function J ( ω, k ), (25).For a → ∞ , it simply vanishes. As a consequence, ˜ φ ( k ),(24), turns into ˜ φ ( k ) → a →∞ g . (64) Now in the function h ( ω, q ), the sum over N turns intoan integration according to2 πa N → a →∞ k, a (cid:88) N → a →∞ (cid:90) d k (2 π ) . (65)This way, we get h ( iξ, q ) → a →∞ (cid:90) d k (2 π ) g − γ e − ξb + i kc . (66)The integration can be carried out and after some calcu-lation we arrive at h ( iξ, q ) → a →∞ g πd e − ξd , (67)where d = √ b + c is the separation between the closesdelta functions from the two lattices. With these, thevacuum energy turns into E → a →∞ (cid:90) ∞ dξπ ln (cid:18) − (cid:16) g πd e − ξd (cid:17) (cid:19) , (68)which is the Casimir-Polder interaction of two points atseparation d carrying a delta function potential each.It coincides with the interaction of two spheres carry-ing delta function potential at large separation (or smallradius). The corresponding formula is displayed in theAppendix. Again, we remark that the coupling g in (68)is after renormalization and, therefore, is not fixed. D. One dimensional lattices (chains) at shortseparation
We consider the vacuum interaction of two chains asgiven by Eq. (48) for short separation. Similar to thepreceding subsection the integration over q shrinks to apoint, a (cid:90) dq π f ( q ) → a →∞ f (0) , (69)Next we consider the function ˜ φ ( k ), (40), and have˜ φ ( k ) → a →∞ g . (70)in the function h ( ω, q ). The sum over N turns into anintegration according to2 πa N → a →∞ k , a (cid:88) N → a →∞ (cid:90) dk π . (71)and we get from (52) h ( iξ, q ) → a →∞ g (cid:90) ∞−∞ dk π K ( (cid:112) ξ + k b ) e ikc , = g πd e − ξd , (72)where again the integration over k was carried out and d = √ b + c is the separation between the closest deltafunctions from the two chains.Now, inserting (69) and (72) into (48), we get E → a →∞ (cid:90) ∞ dξπ ln (cid:18) − (cid:16) g πd e − ξd (cid:17) (cid:19) , (73)which is the same expression as (68) and represents theCasimir-Polder interaction of two points carrying deltafunction potential as discussed at the end of the precedingsubsection. IV. COMPARISON WITH PAIRWISESUMMATION
A special feature of van der Waals and Casimir forcesis their multi particle character. While in certain cases apairwise summation may give a good approximation, ingeneral it will not. In this section we compare the resultsof the exact calculation for point scatterers with the re-sult of pairwise summation, i.e., without multi particleforces.To this end we perform a pairwise summation of allindividual Casimir-Polder interactions F ( z ) = (cid:88) n F n cos( ϕ n ) , F n ( z ) = − dE CP ( r ) dr (cid:12)(cid:12)(cid:12)(cid:12) r = r n , (74)where E CP is given by (68). For two chains n = n , r n = √ z + n a and cos( ϕ n ) = 1 / (cid:112) an/z ) ;for two 2D-lattices a double summation is required n = { n , n } , r n = (cid:112) z + n a + n a and cos( ϕ n ) = FIG. 4. The difference between exact result (48) (dashedline) and pairwise summation of individual Casimir-Polderinteractions for chains (73), g/a = 0 . / (cid:112) an /z ) + ( an /z ) . And finally, the interac-tion energy per one δ -function is E pw ( b ) = (cid:90) ∞ b F ( z ) dz = ∞ (cid:88) n = −∞ E CP ( r n ) (75)As follows from Section III, Eqs. (68) and (73), thepairwise summation is a good approximation at shortdistances, where the exact formulas tend to a two-pointCasimir-Polder interaction. At medium and large dis-tances the pairwise summation overestimates the vac-uum energy. The corresponding curves for Dirac chainsat medium distances are presented at Fig. (4). Herethe pairwise summation of Casimir-Polder interactionsis performed for N = 1000 Dirac δ -s. The evaluationaccording to the exact formula, Eq. (48), was also trun-cated at N = 1000 in Eq. (40), though the result onlyweakly depends on N at medium and large distances.The result of pairwise summation depends consider-able on the coupling g at large separations, while theasymptote of the exact result, Eq. (62), is coupling in-dependent. V. CONCLUSIONS
We considered T -operators for a two-dimensional anda one dimensional lattices of δ -functions. These involvelattice sums, which can be expressed in terms Hurwitzzeta function. Further we used this T -operator to formu-late the kernel in the TGTG formula for the dispersioninteraction of two such lattices. This can be viewed asa kind of generalized Lifshitz formula and a represents afinite (converging) expression for the interaction energy.We considered the cases of the interaction of two paral-lel two-dimensional lattices and, of 1-dimensional parallellattices (chains). The generalization to rotations is leftfor future work.We consider in detail to limiting cases and show thetransition from lattices to plans for large separation andto the Casimir-Polder interaction of two lattice sites at0small separation. These limiting cases are in agreementwith earlier results. Our formulas appear to interpolatebetween these and establish the link between these.. ACKNOWLEDGEMENT
We acknowledge partial support from the Heisenberg-Landau Programme.
Appendix A: The Casimir effect for two spherescarrying delta function potentials
In this appendix we display basic formulas for the in-teraction of two spheres carrying delta function potential (’semitransparent’ spheres) using the by now well knownscattering approach (’TGTG’-formula). Although suchkind of calculations are, in much more general form, con-tained in a number of papers, for example in [18], specificformulas for one of the most simple special cases maybe of use. The first calculation involving delta functionpotential is [19], which was focused on the correctionsbeyond PFA.The basic setup is given by the equation (cid:16) − ω − ∆ + g πR (cid:16) δ ( | (cid:126)x | − R ) + δ ( | (cid:126)x − (cid:126)d | − R ) (cid:17)(cid:17) φ ( (cid:126)x ) = 0 , (A1)where R is the radius of the spheres, one at the origin,the other at separation d = | (cid:126)d | . These delta functionsare one dimensional and the coupling g does not undergoany renormalization.Within the chosen approach, we need for the T -operator for a single sphere. In this case the equationis (cid:16) − ω − ∆ + g πR δ ( | (cid:126)x | − R ) (cid:17) φ ( (cid:126)x ) = 0 . (A2)The scattering problem for a single sphere was consid-ered, for example, in [20] and we use some notations fromthere. The Green’s function for Eq. (A2), in a sphericalbasis, is G ( (cid:126)x, (cid:126)x (cid:48) ) = (cid:88) lm Y lm (Ω) d l ( r, r (cid:48) ) Y lm (Ω (cid:48) ) ∗ (A3)and the free radial Green’s function is d (0) ( r, r (cid:48) ) = 1 rr (cid:48) j l ( ωr < ) h (1) l ( ωr > ) , (A4)where j l ( z ) = (cid:112) π/ (2 z ) J l +1 / ( x ) and h (1 , l ( z ) = (cid:112) π/ (2 z ) H (1 , l +1 / ( x ) are spherical Bessel functions. TheirWronskian is j (cid:48) l ( z ) h l ( z ) − j l ( z ) h (cid:48) l ( z ) = 1 /iz . The deltafunction in (A2) results in matching conditions on theradial function d l ( r, r (cid:48) ), which must be continuous andobey ∂ r d l ( r, r (cid:48) ) | r = R +0 − ∂ r d l ( r, r (cid:48) ) | r = R − = − g πR d l ( R, r (cid:48) ) . (A5)With the ansatz d l ( r, r (cid:48) ) = d (0) ( r, r (cid:48) ) − d (0) ( r, R )Φ − ( ω ) d (0) ( R, r (cid:48) ) (A6) we get Φ − ( ω ) = iωg/ π g πR iωj l ( ωR ) h l ( ωR ) . (A7)After Wick rotation, this expression becomesΦ − ( iξ ) = − gR/ π g πR I l + ( ξR ) K l + ( ξR ) (A8)with the modified Bessel functions I ν ( z ) and K ν ( z ). Eq.(A8) is in agreement with Eq. (23) in [20].The ’TGTG’-formula can be written in a spherical ba-sis, E = 12 (cid:90) ∞ dξπ Tr ln(1 − M ) (A9)where the kernel is M l,l (cid:48) ; m = l + l (cid:48) (cid:88) l (cid:48)(cid:48) = | l − l (cid:48) | N l,l (cid:48)(cid:48) ; m N l (cid:48)(cid:48) ,l (cid:48) ; m (A10)and the trace is over the orbital momenta. All expres-sions are diagonal in the magnetic quantum number m .The kernels in (A10) are N l,l (cid:48) ; m (A11)= 1 R (cid:114) π ξR K l (cid:48)(cid:48) + ( ξd ) H l (cid:48)(cid:48) l,l (cid:48) I l + ( ξR ) I l (cid:48) + ( ξR )Φ − ( iξ )with the factors H l (cid:48)(cid:48) l,l (cid:48) resulting from the matrix elementsfor the transition between the two centers. These aregiven, e.g., by Eq. (10.129) in [21] or Eq. (5) in [22], andcorrespond to the U in [18], Eq. (2.25). For g → ∞ , this1formula turns into that for Dirichlet boundary conditionson the spheres.For us, the important property is that for d → ∞ ,the leading order comes from l = l (cid:48) = l (cid:48)(cid:48) = 0 and with H , = 1 we get N → d →∞ g πd E − ξd . (A12)The dependence on the radius R of the spheres drops outin this limit. By inserting this into (A9) we get E → d →∞ (cid:90) ∞ dξπ ln (cid:18) − (cid:16) g πd e − ξd (cid:17) (cid:19) (A13)for the interaction of two spheres carrying delta functionpotential in the limit of large separation. Appendix B: The Casimir effect for two cylinderscarrying delta function potentials
The derivation of the vacuum energy for two paral-lel cylinders carrying delta function potentials may befound, for example in [17] or [23], EL = 14 π ∞ (cid:90) dξ ξ Tr ln(1 − M ) , (B1)with M given by (A10) and N l,l (cid:48) ; m = gRK l + l (cid:48) ( ξd ) I l (cid:48) ( ξR )1 + gRI l (cid:48) ( ξR ) K l (cid:48) ( ξR ) . (B2) Here we compute the large distance limit of this vacuumenergy. The leading contribution for large separation d comes from the s-wave. After the substitution ξ → ξ/d in (B1), the relevant matrix elements entering the tracemay be rewritten in the form M ll (cid:48) = δ l δ l g R K ( ξ ) I ( ξR/d )(1 + gRI ( ξR/d ) K ( ξR/d )) . (B3)With allowance for the behaviour of the Bessel functionsat small arguments, I ( x ) = 1 + O ( z ) and K ( x ) = − γ − ln( x/
2) + O ( z ), one arrives at the expression M ll (cid:48) = (B4) δ l δ l (cid:48) K ( ξ )ln ( R/d ) (cid:20) γ + ln( ξ/ − ( gR ) − )ln( R/d ) (cid:21) − + O (( R/d ) ) , which can be easily expanded in powers of small1 / ln( R/d ). This expansion we substitute into (B1) andin the leading order arrive at EL = − πd ∞ (cid:90) dξ ξ K ( ξ )ln ( R/d ) = − πd (ln( R/d )) , (B5)which coincides with (73) which was obtained from thechains. [1] G. Barton, “Casimir effects in monatomically thin insu-lators polarizable perpendicularly: nonretarded approxi-mation,” New J. Phys. , 063028 (2013).[2] G. Barton, “Casimir effects for a flat plasma sheet: I.Energies,” J. Phys. A: Math. Gen. , 2997–3019 (2005).[3] M. Bordag, I. V. Fialkovsky, D. M. Gitman, and D. V.Vassilevich, “Casimir interaction between a perfect con-ductor and graphene described by the Dirac model,”Phys. Rev. B , 245406 (2009).[4] Prachi Parashar, Kimball A. Milton, K. V. Shajesh, andM. Schaden, “Casimir interaction energy for magneto-electric δ -function plates,” Phys. Rev. D , 085021(2012).[5] G. Barton, “Monolayers polarizable perpendicularly:The Maxwellian response functions,” Ann. Phys. ,534 (2015).[6] M. Bordag, “Monoatomically thin polarizable sheets,”Phys. Rev. D , 125015 (2014).[7] M. Bordag and J.M. Munoz-Castaneda, “Dirac lat-tices, zero-range potentials and self adjoint extension,”Phys. Rev. D , 065027 (2015).[8] R. de L. Kronig and W. G. Penney, “Quantum Mechan-ics of Electrons in Crystal Lattices,” Proc.Roy.Soc.Lond.,Ser. A , 499 (1931). [9] Yu.N. Demkov and V.N. Ostrovskii, Zero-Range Poten-tials and Their Applications in Atomic Physics (PlenumPress, New York and London, 1988).[10] F. A. Berezin and L. D. Faddeev, “A remark onSchr¨odinger’s equation with a singular potential,” SovietMath. Dokl. , 372–375 (1961).[11] R.W. Jackiw, Diverse Topics in Theoretical and Math-ematical Physics, section I.3: