Casimir effect for Dirac lattices

1.9 MB·

by  M. Bordag

Casimir effect for Dirac lattices
M. Bordag
Institut fu¨r Theoretische Physik, Universit¨at Leipzig, Germany∗
I.G. Pirozhenko
Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia
Dubna State University, Dubna, Russia†
We consider polarizable sheets, which recently received some attention, especially in context of
the dispersion interaction of thin sheets like graphene. These sheets are modeled by a collection
of delta function potentials and resemble zero range potentials, known in quantum mechanics. We
developatheoreticaldescriptionandapplytheso-calledTGTG-formulatocalculatetheinteraction
of two such lattices. Thereby we make use of the formulation of the scattering of waves off such
sheets provided earlier. We consider all limiting cases, providing link to earlier results. Also, we
7
discuss the relation to the pairwise summation method.
1
0
PACSnumbers: 03.65.Nk,03.65.-w
2

n
a I. INTRODUCTION [10]. In electrodynamics, the self energy of a delta func-
J tion potential is singular and one needs a renormaliza-

1 This paper is a contribution to the discussion of van tion. In terms of quantum field theory this setup was
der Waals and Casimir forces between surfaces. Last considered in [11].
]
h years, much attention was paid to the interaction be-
p tweenslabsoffinite,andespeciallysmall,thickness. This Recently, a sheet of delta function potentials was used
- triggeredbyagrowinginteresttotwodimensionalstruc- in[7]tomodelapolarizablesheet. Forinstance, scatter-
t
n tures (sheets) like 2d electron gas, monoatomically lay- ing off such sheet was investigated and subsequently the
a ers and graphene, as well as to the interaction between transition to a continuous sheet as well.
u
them. As discussed in [1], the situation with sheets hav-
q
[ ing only in-plane polarizability is relatively clear. Here In the present note we consider a setup of two such
onecanformulateahydrodynamicplasmamodelandcal- sheets, hold in parallel at some separation. This is a
1 culate the quantities of interest [2]. The same holds for typical situation for the Casimir effect. We use the well
v graphene described by the Dirac equation model for the known scattering approach, also called ’TGTG’-formula,
1
π-electrons [3], which are responsible for the interaction which can be considered also as a generalization of the
6
2 with the electromagnetic field. Lifshitz formula. Making use of the translational invari-
0 Thesituationwithperpendicularpolarizabilityismore ance of the lattice, the T-matrices can be calculated in
0 complicated. Whilein[4]noresponsetotheelectromag- momentum representation; for a one dimensional lattice
1. netic field was found, in [5] it was shown that such re- even explicitly. Further, we allow for a displacement of
0 sponsetakesplace. Inthatpaperthesheetwasmodelled one lattice with respect to the other. Also we study the
7 as a lattice of harmonic oscillators (dipoles), vibrating limitingcasesofsmallandlargeseparation,resp. oflarge
1 in direction perpendicular to the sheet, and the limit of and small lattice spacing. For large separation, the limi-
v: zero spacing of this lattice was investigated. This was tingcasecorrespondstotheinteractionoftwosheetscar-
i extended in [6] to a sheet, continuous from the very be- rying delta function potential, and for small separation
X ginning,havingparallelorperpendicularpolarizabilities. the limiting case corresponds to the interaction of two
r Later, a similar setup was considered using point dipoles points carrying delta function potential. In this paper
a
[7]. These can be represented by Dirac delta functions we consider only a scalar field. Extensions to the elec-
forming a so-called ’Dirac lattice’. In fact, there repre- tromagnetic field should be quite straight forward. It is
sent point scatterers. alsoanintentionofthispapertoprovideaframeworkfor
Such lattices, taken alone, have a well known inter- calculating dispersion forces in realistic, experimentally
naldynamics,thesimplestcasebeingtheKronig-Penney relevant situations, where the atomistic structure must
model (’Dirac comb’) [8]. In quantum mechanics their be accounted for.
use is known as ’method of zero-range potential’ [9]. As
well known, in more than one dimension a Hamilton op- The paper is organized as follows. In the next section
erator with a delta function potential is not self adjoint we provide the necessary formulas to calculate the vac-
uum energy for two and one dimensional lattices. In the
thirdsectionwederivealllimitingcasesanddiscusstheir
interrelations. In the fourth section we compare the re-
∗ [email protected] sultswithpairwisesummation. Someconclusionsandan
† [email protected] appendix complete the paper.
2
II. VACUUM ENERGY FOR POINT in terms of the T-operators of the corresponding lattice.
SCATTERERS Further, the free Green’s function,
(cid:90) d3k ei(cid:126)k((cid:126)x−(cid:126)x(cid:48))
We consider two types of lattices. First, two dimen- G ((cid:126)x−(cid:126)x(cid:48))= , (7)
sional lattices, A and B, given by the lattice vectors 0 (2π)3 ω2−k2−k32+i0
(cid:18) (cid:19) (cid:18) (cid:19) enters this formula. We will use below also the following
a +c a
(cid:126)aA = n , (cid:126)aB = n , (1) two representations,
n b n 0
eiω|(cid:126)x| (cid:90) d2k eik(x−x(cid:48))+iΓ|x3−x(cid:48)3|
For these, we use the following notations. A three di- G ((cid:126)x)= = , (8)
0 4π|(cid:126)x| (2π)2 2iΓ(k)
mensional vector is denoted by an arrow, a two dimen-
sional vector parallel to the (x,y)-plane is denoted in √
(cid:18) (cid:19) with Γ(k)= ω2−k2+i0 and k =|k|.
x
bold,forinstance(cid:126)x= . Thelatticesitesaregiven The T-operators can be defined from
x
3
(cid:18)n (cid:19) G((cid:126)x,(cid:126)x(cid:48)) (9)
by a =a 1 , where n and n are integers, a is the
n n 1 2 (cid:90) (cid:90)
2 =G ((cid:126)x−(cid:126)x(cid:48))− dz 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)),
lattice spacing; the lattice is rectangular. The lattice B 0 0 0
is in the (x,y)-plane and the lattice A is parallel to B
whereG((cid:126)x,(cid:126)x(cid:48))istheGreen’sfunctionofequation(3)and
on a separation b and shifted within that plane by the
can be related to the ansatz
displacement vector c.
Second we consider one dimensional lattices (chains), (cid:88)
G((cid:126)x,(cid:126)x(cid:48))=G ((cid:126)x−(cid:126)x(cid:48))− G ((cid:126)x−(cid:126)a )Φ−1 G ((cid:126)a −(cid:126)x(cid:48)).
A and B, given by the lattice vectors 0 0 n n,n(cid:48) 0 n(cid:48)
n,n(cid:48)
    (10)
an+c an
Inserting this ansatz into the equation
(cid:126)anA = 0 , (cid:126)anB = 0 , (2)
b 0 (cid:32) (cid:33)
−ω2−∆+g(cid:88)(cid:0)δ(x−(cid:126)aA)+δ(x−(cid:126)aB)(cid:1) G((cid:126)x,(cid:126)x(cid:48))
n n
where n is a (single) integer. These chains are on the n
x-axis (B) resp. parallel to it (A). Both are in the plane =δ((cid:126)x−(cid:126)x(cid:48)) (11)
y = 0. Their separation is b as above, the displacement
and using the equation
c is a number.
With these notations, we consider a scalar field φ((cid:126)x), (cid:0)−ω2−∆(cid:1)G ((cid:126)x,(cid:126)x(cid:48))=δ((cid:126)x−(cid:126)x(cid:48)) (12)
whose wave equation after Fourier transform in time is 0
(cid:32) (cid:33) for the free Green’s function, one comes to the equation
−ω2−∆+g(cid:88)(cid:0)δ((cid:126)x−(cid:126)aA)+δ((cid:126)x−(cid:126)aB)(cid:1) φ((cid:126)x)=0, (cid:88)
n n −Φ−1 +gδ −g G ((cid:126)a −(cid:126)a )Φ−1 =0. (13)
n n,n(cid:48) n,n(cid:48) 0 n m m,n(cid:48)
(3) m
where(cid:126)aA and(cid:126)aB are given by (1). In this equation, the
n n With the definition
deltafunctionsarethreedimensional,hencethecoupling
1
ghasthedimensionoflength. AsmentionedintheIntro-
Φ = δ −G ((cid:126)a −(cid:126)a ) (14)
duction,oneshouldrememberthattheaboveequationis n,n(cid:48) g n,n(cid:48) 0 n n(cid:48)
notwelldefinedandg mustbeviewedasabarecoupling
Eq. (13) becomes
which needs to undergo a renormalization.
Thevacuumenergyoftheinteractionoftwolatticesis (cid:88)
Φ Φ−1 =δ (15)
given by the mentioned ’TGTG’-formula, n,m m,n(cid:48) n,n(cid:48)
m
1 (cid:90) ∞
E0 = 2π dξTrln(1−M), (4) and Φ−n,1n(cid:48) is the inverse matrix to Φn,n(cid:48).
0 The diagonal elements of Φ contain G (0) which
n,n(cid:48) 0
is not well defined. Referring to [7] for a discussion, we
and ξ is the imaginary frequency, ω =iξ. The kernel M
renormalize the coupling,
is the product of two other kernels,
(cid:90) 1 1
−G (0)→ , (16)
M((cid:126)x,(cid:126)x(cid:48))= dx(cid:48)(cid:48)NA((cid:126)x,(cid:126)x(cid:48)(cid:48))NB((cid:126)x(cid:48)(cid:48),(cid:126)x(cid:48)), (5) g 0 g
and after that we get
each of which is given by
(cid:40)
(cid:90) 1, (n=n(cid:48))
N ((cid:126)x,(cid:126)x(cid:48))= dx(cid:48)(cid:48)G ((cid:126)x,(cid:126)x(cid:48)(cid:48))T ((cid:126)x(cid:48)(cid:48),(cid:126)x(cid:48)), (6) Φn,n(cid:48) = g (17)
A,B 0 A,B −G ((cid:126)a −(cid:126)a ), (n(cid:54)=n(cid:48))
0 n n(cid:48)
3
which is well defined. inverse. As a consequence, the inversion of this matrix
Now, comparing Eqs. (10) and (10), we identify the can be calculated by Fourier transform,
T-operator to be
f =a2(cid:90) dk eikanf˜(k), f˜(k)=(cid:88)e−ikanf .
TA((cid:126)z,(cid:126)z(cid:48))= (cid:88)δ((cid:126)z−(cid:126)aA)Φ−1 δ((cid:126)aA −(cid:126)z(cid:48)) (18) n (2π)2 n
n n,n(cid:48) n(cid:48) n
n,n(cid:48) (21)
This way we get
for one lattice and by the same formula with B in place
of A for the other. Inserting this expression into (6) we Φ−1 =a2(cid:90) dk eik(an−an(cid:48)) 1 (22)
get n,n(cid:48) (2π)2 φ˜(k)
N ((cid:126)x,(cid:126)x(cid:48))= (cid:88)G ((cid:126)x−(cid:126)aA)Φ−1 δ((cid:126)aA −(cid:126)x(cid:48)) (19) with
A 0 n n,n(cid:48) n(cid:48)
n,n(cid:48) φ˜(k)=(cid:88)e−ikanΦ . (23)
n
and, again, the same with A → B. Further, inserting n
these into (5), we get
Here we insert (17) and come to
(cid:90)
M((cid:126)x,(cid:126)x(cid:48))= dx(cid:48)(cid:48)NA((cid:126)x,(cid:126)x(cid:48)(cid:48))NB((cid:126)x(cid:48)(cid:48),(cid:126)x(cid:48)), (20) φ˜(k)= g1 −(cid:88)(cid:48)e−ikanG0(an),
n
1 1
for the kernel M. Eqs. (19) and (20) represent the = − J (ω,k), (24)
general expressions for the kernel entering the ’TGTG’- g 4π 1
formula (4) for a generic lattice of delta functions. In
where Eq. (8) was used and we introduced
the following we consider first two dimensional lattices
as given by (1), and, subsequently one dimensional lat- (cid:88)(cid:48) 1
tices (chains) as given by Eq. (2). J1(ω,k)= |a |eiω|an|+ikan. (25)
n
n
As usual, in the primed sum the term with n = 0 is
dropped. Finally, we mention the relation
A. Vacuum energy for two dimensional lattices
(cid:88) 1
eikamΦ−1 = eikam(cid:48), (26)
In this subsection we consider lattices as given by Eq. m,m(cid:48) φ˜(k)
m
(1). These have translational invariance with respect to
a lattice step. Equivalently, we have a −a = a , which follows with the Fourier transform (21).
n n(cid:48) n−n(cid:48)
and the quantities entering Eq. (14) and (15), depend NowwereturntothekernelN,(19),andinsert(8)for
only on differences, Φ = Φ and the same for its the Green’s function,
n,n(cid:48) n−n(cid:48)
(cid:88)(cid:90) d2k eik(x−an−c)+iΓ|x3−b|
N ((cid:126)x,(cid:126)x(cid:48))= Φ−1 δ(a +c−x(cid:48))δ(b−x(cid:48)). (27)
A (2π)2 2iΓ(k) n−m m 3
n,m
Next we use (26) and get
(cid:90) d2k eik(x−c)+iΓ|x3−b| (cid:88)
N ((cid:126)x,(cid:126)x(cid:48))= e−ikamδ(a +c−x(cid:48))δ(b−x(cid:48)). (28)
A (2π)2 2iΓ(k)φ˜(k) m 3
m
The corresponding expression for N ((cid:126)x,(cid:126)x(cid:48)) follows from here with c = 0 and b = 0. Using these, we can write
B
down the kernel M((cid:126)x,(cid:126)x(cid:48)), Eq. (5),
(cid:90) (cid:90) d2k eik(x−c)+iΓ|x3−b| (cid:88)
M((cid:126)x,(cid:126)x(cid:48))= dx(cid:48)(cid:48) e−ikamδ(a +c−x(cid:48)(cid:48))δ(b−x(cid:48)(cid:48))
(2π)2 2iΓ(k)φ˜(k) m 3
m
·(cid:90) (d22πk)(cid:48)2 e2iikΓ(cid:48)x(k(cid:48)(cid:48)+(cid:48))iφΓ˜((cid:48)|kx(cid:48)3(cid:48)(cid:48))| (cid:88)e−ik(cid:48)am(cid:48)δ(am(cid:48) −x(cid:48))δ(x(cid:48)3). (29)
m(cid:48)
4
Carryingouttheintegrationover(cid:126)x(cid:48)(cid:48)usingthedeltafunc- nents of q = (q ,q ) are restricted to −π ≤ q < π.
1 2 a 1,2 a
tions, we come to a sum over m. Before doing this sum, Now the sum over m appearing in (29) gives
wesplitthemomentakandk(cid:48) intoquasimomentumand
integer part,
(cid:88)e−i(k−k(cid:48))am =(cid:18)2π(cid:19)2δ(2)(q−q(cid:48)), (31)
2π 2π a
k=q+ N, k(cid:48) =q(cid:48)+ M, (30) m
a a
where N and M are integer vectors like n in a . The where the dependence on N and M dropped out. Then
n
integration becomes (cid:82) d2k = (cid:82) d2q(cid:80) and the compo- (29) turns into
N
M((cid:126)x,(cid:126)x(cid:48))= a12 (cid:90) (d2π2q)2 (cid:88)eix(q+2aπ2NiΓ)−(ki2)aπφ˜N(kc+)iΓ|x3−b| (cid:88)e2ii2Γaπ(Mk(cid:48)c)+φ˜i(Γk(cid:48)|(cid:48)b)| (cid:88)e−iqam(cid:48)δ(am(cid:48) −x(cid:48))δ(x(cid:48)3). (32)
N M m(cid:48)
This expression for M((cid:126)x,(cid:126)x(cid:48)) must be inserted into the M((cid:126)x,(cid:126)x(cid:48)) is diagonal in q and we can define
generalized Lifshitz formula (4) under the sign of the
trace. In doing so, the delta function in one factor 1 (cid:88) eiΓ(k)b+i2aπNc
h(ω,q)= (33)
M((cid:126)x,(cid:126)x(cid:48)) turns the x in the next factor into am(cid:48). Again, a2 2iΓ(k)φ˜(k)
the summation over m(cid:48) gives a delta function for the N
quasi momenta. In this way, the products of the factors
with k given by (30), and we rewrite (32) in the form
(cid:90) d2q (cid:88)
M((cid:126)x,(cid:126)x(cid:48))=a2 eiqxh(ω,q)h(ω,q)∗ e−iqamδ(a −x(cid:48))δ(x(cid:48)). (34)
(2π)2 m 3
m
In this way, the factors M((cid:126)x,(cid:126)x(cid:48)) entering (10) become dimensional,
doviaegroonnael.ceTllhaenxd-iinttteugrrnastitohneisnumtheinttroacueniitsy,to be taken f =a(cid:90) dk1eik1anf˜(k ), f˜(k )=(cid:88)e−ik1anf ,
n 2π 1 1 n
n
(cid:90) (cid:88) (37)
d(cid:126)xeiqx e−iqamδ(am−x)δ(x3)=1. (35) where we used for the momentum the notation k since
1
m the lattice is parallel to the x-axis. Following Eqs. (22)
and (23), we get
As a consequence, for the vacuum energy per lattice cell
we get the expression φ˜(k )=(cid:88)e−ik1anΦ , (38)
1 n
n
E = 1(cid:90) ∞ dξ a2(cid:90) d2q ln(cid:0)1−|h(iξ,q)|2(cid:1). (36) which after renormalization gives
0 2 π (2π)2
0
φ˜(k )= 1 − 1 (cid:88) eiω|n|+ik1an. (39)
Thisisthefinalgeneralformulaforthevacuumenergyof 1 g 4π a|n|
n(cid:54)=0
twolatticesasgivenbyEq. (1). Thenumericalresultfor
thevacuumenergyforonecellofthelatticesispresented The sum is one dimensional and can be done explicitly,
at Fig. 1.
φ˜(k )= 1 − 1 ln(cid:0)1+e2iωa−2cos(k a)eiωa(cid:1). (40)
1 g 4πa 1
Bytheway,thezerosofthisfunctiondeterminethezones
B. Vacuum energy for one dimensional lattices
in the Kronig-Penney model.
(chains)
Now we consider the kernel N , Eq. (6), and insert
A
the Green’s function from (8) and the relation
In this subsection we consider lattices as given by Eq.
(2). The calculations are mostly in parallel to the pre- (cid:88)eik1anΦ−1 = 1 e−k1am(cid:48) (41)
ceeding subsection. The Fourier transform is now one mm(cid:48) φ˜(k )
n 1
5
to get
(cid:90) d2k eik1(x1−an−c)+iΓ|x3−b| (cid:88)
N ((cid:126)x,(cid:126)x(cid:48))= e−ik1amδ(am+c−x(cid:48))δ(x(cid:48))δ(b−x(cid:48)). (42)
A (2π)2 2iΓ(k)φ˜(k ) 1 2 3
1 m
In parallel to (29), but carrying out the integration over x(cid:48)(cid:48), we get further
1(cid:90) d2k eik1(x1−c)+ik2x2+iΓ|x3−b| (cid:88)
M((cid:126)x,(cid:126)x(cid:48))= e−ik1am
a (2π)2 2iΓ(k)φ˜(k )
1 m
·(cid:90) (d22πk)(cid:48)2 eik21(cid:48)i(Γam(k+(cid:48)c))φ˜+(ikΓ(cid:48)(cid:48)|)x(cid:48)3(cid:48)| (cid:88)e−ik1(cid:48)am(cid:48)δ(am−x(cid:48)1)δ(x(cid:48)2)δ(x(cid:48)3). (43)
1 m(cid:48)
We introduce a splitting of the momenta like (30), but now only for k ,
1
2π 2π
k =q+ N, k(cid:48) =q+ M, (44)
1 a 1 a
where N and M are integers, and come to
M((cid:126)x,(cid:126)x(cid:48))= 1(cid:90) dq (cid:90) dk2 (cid:88)eiq(x1−c)+i2aπNc+ik2x2+iΓ|x3−b|
a 2π 2π 2iΓ(k)φ˜(k )
N 1
·(cid:90) dk2(cid:48) (cid:88)eiqc+i2aπMc+iΓ(cid:48)|b| (cid:88)e−iqamδ(am−x(cid:48))δ(x(cid:48))δ(x(cid:48)). (45)
2π 2iΓ(k(cid:48))φ˜(k(cid:48)) 1 2 3
M 1 m
Following the same discussion concerning the trace as in We get
the preceding subsection, we are lead to introduce the
notation h(iξ,q)= 1(cid:90) dk2 (cid:88)ei2aπNc−γb. (50)
h(ω,q)= 1(cid:90) dk2 (cid:88) ei2aπNc+iΓ|b| (46) a 2π N −2γφ˜(k1)
a 2π 2iΓ(k)φ˜(k )
N 1
Changing the integration from k for γ and using the
2
with
integral representation
(cid:113) 2π
Γ= ω2−k12−k22+i0, k1 =q+ a N. (47) (cid:90) ∞
K (z)= ezcosh(θ)dθ (51)
0
Finally, we get the vacuum energy for one cell of the 0
chains,
one comes to
π/a
E0 = 12(cid:90)0∞ dπξ a (cid:90) 2dπq ln(cid:0)1−|h(iξ,q)|2(cid:1). (48) h(iξ,q)= 2π1a(cid:88)K0(−(cid:112)2ξφ˜2(k+)k12b)ei2aπNc (52)
−π/a N 1
This is the final general formula for the vacuum energy
of two lattices as given by Eq. (2). with φ˜(k1) given by Eq. (40). The numerical result for
It should be mentioned that the formula for h(iξ,q) the vacuum energy for one cell of the chains is presented
can be simplified since the integration over k in (46) at Fig. 1 (inset).
2
can be carried out. First, we perform the Wick rotation The influence of the displacement c on the interaction
and define between the chains is shown in Fig. 2. Similar profiles
(cid:113) may be found in [12] for the Casimir-Polder potential
Γ=iγ, γ = ξ2+k2+k2. (49) between an atom and a one-dimensional grating.
1 2
6
III. LIMITING CASES
Inthissectionwediscussthelimitingcasesoftheinter-
action lattices of delta functions. There are two limiting
cases for each, two and one dimensional lattices. One
limiting case is for large separation, or equivalently for
dimensional reasons, for small lattice step. The other is
for small separation, or equivalently, large lattice step.
FIG.1. Thevacuumenergyperunitasafunctionofdistance Forlargeseparation, thetwodimensionallatticesturn
between lattices for different values of the coupling. Here intoplanes,carryingadeltafunctionpotential. Forthese
lattices are exactly opposite to one another, c=0. From left
the Casimir effect was first investigated in [13]. The one
torightg/a=0.01,0.05,0.1. Inset: thesamefortwoparallel
dimensional lattices (chains) turn into lines, carrying a
Dirac chains.
(twodimensional)deltafunctionpotential. Thiscasewas
not considered so far. However, it turns out that that it
is equivalent to the finite size cylinders with Dirichlet
boundary conditions for turning their radius to zero, as
considered in [14] and [15].
For small separation, in the limit, only one delta func-
tion for the one lattice interacts with the closest one in
theotherlattice. ThisisequivalenttotheCasimir-Polder
interactionoftwopointscarryingadeltafunctionpoten-
tial each. This is the scalar version of the well known
interaction of two dipoles, which was considered within
the scattering approach in [16]. Also, it can be consid-
ered as limiting case of the interaction of two spheres
FIG. 2. The factor η = E(c (cid:54)= 0)/E(c = 0) is plotted as a carrying delta function potential for vanishing radius of
functionofc/aforg/a=0.1anddifferentseparationsβ =b/a the spheres. The corresponding calculation is shown in
of the chains. From bottom to top β =0.1,0.2,0.4,0.6. the Appendix.
The considered cases and their interrelations are shown in Fig. 3.
A. Two dimensional lattices at large separation and the function J , (25), which reads now
1
(cid:88)(cid:48) 1
WestartfromEq. (36)forthevacuumenergyandcon- J1(iξ,k)= |a |e−ξ|an|+ikan. (54)
n
sider the limit of lattice spacing a → 0, or equivalently, n
separation b → ∞. At the end, we will get tow parallel
For a → 0, the sum turns over into an integral, which
continuous sheets for which we need the energy density
can be calculated easily,
per unit area which is 1 E . The integration region in
a2 0
q, given by −π ≤ q < π, turns into a whole R2. The 2π
function h(ω,ka), (331),2, reaads with imaginary frequency, J1(iξ,k) → a2γ, (55)
a→0
see Eq. (134) in [7]. This way we get
1 (cid:88) e−γ(k)b+i2aπNc
h(iξ,q)= a2 −2γ(k)φ˜(k) (53) h(iξ,k) → re−ξb, (56)
N a→0
where
(cid:112)
with γ = ξ2+k2, k = q + 2πN (i=1,2). For a → 0
i i a i 1
all contributions except that for N = 0 vanish and we r = . (57)
get k = q . Also we need the function φ˜, given by (24) 1− a22γ
i i g
7
FIG. 3. Possible transitions from one dimensional lattices (chains) and two dimensional lattices of delta functions for large
and small separation. For convenience, the two dimensional lattices are shown for c=0, i.e., without displacement. For large
separation, the lattices turn into continuous plates and the chains into wires. For small separation both turn into the closes
pints. Also shown are the transitions from spheres and cylinders to plates and wires.
The vacuum energy (36) becomes as normalization of the coupling g after renormalization.
a12E0 → 12(cid:90) ∞ dπξ (cid:90) (2dπq)2 ln(cid:16)1−(cid:0)re−γb(cid:1)2(cid:17). (58)
a→0 0
B. One dimensional lattices (chains) at large
In fact, r, Eq. (57), is the well known reflection coeffi- separation
cient for a plane carrying a delta function potential and
(58) gives the Casimir effect for two such sheets. The
This case is to a large extend in parallel to the pre-
ratiog/a2,havingthemeaningofcouplingperunitarea,
ceding subsection. We start from (48) and consider the
is the strength of the delta function potential in the cor-
energy per unit length, 1E . Further, we substitute
responding equation, where the delta function is one di- a 0
ξ → ξ/b and q → q/b. This way we get a factor 1/b2
mensionalandwelldefinedsuchthatthisstrengthcarries
in front and all further dependence is on a/b.
over from the equation to the reflection coefficient with-
out change. This is in opposite to the strength g in Eq. Again in the function h(iξ,k1), Eq. (40), accounting
(36)whichafterrenormalization(16)haslittletodowith for (44), only N =0 gives a non vanishing contribution.
the g in equation (3). The other way round, one can use Asomehowdifferentfeatureappearsfromφ˜,(??),which
the established above relation to the continuous sheets has a logarithmic behaviour now,
8
1 b (cid:18) a (cid:18) (cid:18) (cid:16)a (cid:17)2 (cid:16)a (cid:17)2(cid:19)(cid:19)(cid:19)
φ˜(k ) = − − ξ+ln 2 cosh ξ −cos k ,
1 g 4πa b b b 1
= 1 − b (cid:16)2ln(cid:16)a(cid:17)− aξ+ln(cid:0)ξ2+k2(cid:1)+...(cid:17),
g 4πa b b 1
ln(a)
→ − b . (59)
2πa/b
a→0
The last line is the leading order and one observes that Finally, inserting into (48), we get
the dependence on the coupling g is lost. Inserting into
(52) we get
b (cid:18)(cid:113) (cid:19)
h(ξ,k1) → ln(a/b)K0 ξ2+k12 . (60)
a→0
1 1 (cid:90) ∞ dξ (cid:90) dk (cid:32) (cid:18) 1 (cid:18)(cid:113) (cid:19)(cid:19)2(cid:33)
aE0 → 2b2 π 2π1 ln 1− ln(a/b)K0 ξ2+k12 , (61)
a→0 0
and further expanding the logarithm, Now in the function h(ω,q), the sum over N turns into
an integration according to
1 1 1
aE0a→→0 − 8πb2ln(a/b)2, (62) 2πN → k, 1 (cid:88) → (cid:90) dk . (65)
a a2 (2π)2
a→∞ a→∞
N
where (cid:82)∞dppK (p)2 = 1 was used.
0 0 2 This way, we get
It is to be mentioned, that this result coincides with
the large separation limit of the interaction between two (cid:90) dk g
parallel cylindrical shells carrying δ-potentials (see Ap- h(iξ,q) → (2π)2−2γe−ξb+ikc. (66)
a→∞
pendix B), considered also in [17]. The same limit may
The integration can be carried out and after some calcu-
be reproduced from [15], Eq. (88), for the interaction of
lation we arrive at
a conducting cylinder with a conducting plane after tak-
ing into account the different geometries and a factor of h(iξ,q) → g e−ξd, (67)
2 for the polarizations. 4πd
a→∞
√
where d= b2+c2 is the separation between the closes
delta functions from the two lattices. With these, the
C. Two dimensional lattices at short separation vacuum energy turns into
1(cid:90) ∞ dξ (cid:18) (cid:16) g (cid:17)2(cid:19)
Hereweconsiderthevacuumenergy(36)fortheinter- E0 → 2 π ln 1− 4πde−ξd , (68)
actionoftwodimensionallatticesforsmallseparation,or a→∞ 0
equivalently for large lattice steps. In this case the inte-
which is the Casimir-Polder interaction of two points at
gration region for the integration over q in (36) shrinks
separation d carrying a delta function potential each.
to zero and this integral gives
It coincides with the interaction of two spheres carry-
ing delta function potential at large separation (or small
(cid:90) d2q
a2 f(q) → f(0), (63) radius). The corresponding formula is displayed in the
(2π)2 a→∞ Appendix. Again, we remark that the coupling g in (68)
is after renormalization and, therefore, is not fixed.
where f(q) is some arbitrary function. In the function
h(ω,q),(24),wefirstconsiderthefunctionJ (ω,k),(25).
1
For a→∞, it simply vanishes. As a consequence, φ˜(k), D. One dimensional lattices (chains) at short
(24), turns into separation
1 We consider the vacuum interaction of two chains as
φ˜(k) → g. (64) given by Eq. (48) for short separation. Similar to the
a→∞
9
preceding subsection the integration over q shrinks to a
point,
(cid:90) dq
a f(q) → f(0), (69)
2π
a→∞
Next we consider the function φ˜(k ), (40), and have
1
1
φ˜(k1) → g. (70)
a→∞
in the function h(ω,q). The sum over N turns into an
integration according to FIG. 4. The difference between exact result (48) (dashed
line) and pairwise summation of individual Casimir-Polder
2π 1(cid:88) (cid:90) dk interactions for chains (73), g/a=0.1.
N → k, → . (71)
a a 2π
a→∞ a→∞
N
(cid:112)
1/ 1+(an /z)2+(an /z)2. And finally, the interac-
and we get from (52) 1 2
tion energy per one δ-function is
(cid:90) ∞ dk (cid:112)
h(iξ,q) → g 2πK0( ξ2+k2b)eikc, (cid:90) ∞ (cid:88)∞
a→∞ −∞ Epw(b)= F(z)dz = ECP(rn) (75)
g
= e−ξd, (72) b n=−∞
4πd
As follows from Section III, Eqs. (68) and (73), the
where again the integration over k was carried out and
√ pairwise summation is a good approximation at short
d= b2+c2 is the separation between the closest delta
distances, where the exact formulas tend to a two-point
functions from the two chains.
Casimir-Polder interaction. At medium and large dis-
Now, inserting (69) and (72) into (48), we get
tances the pairwise summation overestimates the vac-
1(cid:90) ∞ dξ (cid:18) (cid:16) g (cid:17)2(cid:19) uum energy. The corresponding curves for Dirac chains
E0 → 2 π ln 1− 4πde−ξd , (73) at medium distances are presented at Fig. (4). Here
a→∞ 0 the pairwise summation of Casimir-Polder interactions
is performed for N = 1000 Dirac δ-s. The evaluation
which is the same expression as (68) and represents the
according to the exact formula, Eq. (48), was also trun-
Casimir-Polder interaction of two points carrying delta
cated at N = 1000 in Eq. (40), though the result only
functionpotentialasdiscussedattheendofthepreceding
weakly depends on N at medium and large distances.
subsection.
The result of pairwise summation depends consider-
able on the coupling g at large separations, while the
asymptote of the exact result, Eq. (62), is coupling in-
IV. COMPARISON WITH PAIRWISE dependent.
SUMMATION
A special feature of van der Waals and Casimir forces V. CONCLUSIONS
is their multi particle character. While in certain cases a
pairwise summation may give a good approximation, in
We considered T-operators for a two-dimensional and
generalitwillnot. Inthissectionwecomparetheresults
a one dimensional lattices of δ-functions. These involve
of the exact calculation for point scatterers with the re-
lattice sums, which can be expressed in terms Hurwitz
sult of pairwise summation, i.e., without multi particle
zetafunction. FurtherweusedthisT-operatortoformu-
forces.
late the kernel in the TGTG formula for the dispersion
To this end we perform a pairwise summation of all interaction of two such lattices. This can be viewed as
individual Casimir-Polder interactions a kind of generalized Lifshitz formula and a represents a
(cid:12) finite (converging) expression for the interaction energy.
F(z)=(cid:88)Fncos(ϕn), Fn(z)=− dECdPr(r)(cid:12)(cid:12)(cid:12) , We considered the cases of the interaction of two paral-
n r=rn leltwo-dimensionallatticesand,of1-dimensionalparallel
(74) lattices (chains). The generalization to rotations is left
where EC√P is given by (68). For two chains n = for future work.
(cid:112)
n, r = z2+n2a2 and cos(ϕ ) = 1/ 1+(an/z)2; We consider in detail to limiting cases and show the
n n
for two 2D-lattices a double summation is required transition from lattices to plans for large separation and
(cid:112)
n = {n ,n }, r = z2+n2a2+n2a2 and cos(ϕ ) = to the Casimir-Polder interaction of two lattice sites at
1 2 n 1 2 n
10
small separation. These limiting cases are in agreement (’semitransparent’spheres)usingthebynowwellknown
with earlier results. Our formulas appear to interpolate scattering approach (’TGTG’-formula). Although such
between these and establish the link between these.. kindofcalculationsare,inmuchmoregeneralform,con-
tainedinanumberofpapers,forexamplein[18],specific
formulas for one of the most simple special cases may
ACKNOWLEDGEMENT be of use. The first calculation involving delta function
potential is [19], which was focused on the corrections
We acknowledge partial support from the Heisenberg- beyond PFA.
Landau Programme.
Appendix A: The Casimir effect for two spheres
carrying delta function potentials
In this appendix we display basic formulas for the in-
teractionoftwospherescarryingdeltafunctionpotential The basic setup is given by the equation
(cid:16) g (cid:16) (cid:17)(cid:17)
−ω2−∆+ δ(|(cid:126)x|−R)+δ(|(cid:126)x−d(cid:126)|−R) φ((cid:126)x)=0, (A1)
4πR2
where R is the radius of the spheres, one at the origin, we get
the other at separation d = |d(cid:126)|. These delta functions
iωg/4π
areonedimensionalandthecouplingg doesnotundergo Φ−1(ω)= . (A7)
any renormalization. 1+ 4πgRiωjl(ωR)hl(ωR)
Within the chosen approach, we need for the T-
operator for a single sphere. In this case the equation After Wick rotation, this expression becomes
is
−gR/4π
(cid:16)−ω2−∆+ 4πgR2δ(|(cid:126)x|−R)(cid:17)φ((cid:126)x)=0. (A2) Φ−1(iξ)= 1+ 4πgR2Il+21(ξR)Kl+21(ξR) (A8)
with the modified Bessel functions I (z) and K (z). Eq.
The scattering problem for a single sphere was consid- ν ν
(A8) is in agreement with Eq. (23) in [20].
ered,forexample,in[20]andweusesomenotationsfrom
The’TGTG’-formulacanbewritteninasphericalba-
there. The Green’s function for Eq. (A2), in a spherical
sis,
basis, is
(cid:88) 1(cid:90) ∞ dξ
G((cid:126)x,(cid:126)x(cid:48))= Y (Ω)d (r,r(cid:48))Y (Ω(cid:48))∗ (A3) E = Trln(1−M) (A9)
lm l lm 0 2 π
lm 0
and the free radial Green’s function is where the kernel is
1 l+l(cid:48)
d(0)(r,r(cid:48))= j (ωr )h(1)(ωr ), (A4) (cid:88)
rr(cid:48) l < l > Ml,l(cid:48);m = Nl,l(cid:48)(cid:48);mNl(cid:48)(cid:48),l(cid:48);m (A10)
l(cid:48)(cid:48)=|l−l(cid:48)|
where j (z) = (cid:112)π/(2z)J (x) and h(1,2)(z) =
l l+1/2 l
(cid:112)π/(2z)H(1,2) (x) are spherical Bessel functions. Their and the trace is over the orbital momenta. All expres-
l+1/2 sions are diagonal in the magnetic quantum number m.
Wronskian is jl(cid:48)(z)hl(z)−jl(z)h(cid:48)l(z) = 1/iz2. The delta The kernels in (A10) are
function in (A2) results in matching conditions on the
radial function dl(r,r(cid:48)), which must be continuous and Nl,l(cid:48);m (A11)
obey 1 (cid:114) π
= K (ξd)Hl(cid:48)(cid:48)I (ξR)I (ξR)Φ−1(iξ)
g R3 2ξR l(cid:48)(cid:48)+21 l,l(cid:48) l+12 l(cid:48)+21
∂ d (r,r(cid:48)) −∂ d (r,r(cid:48)) =− d (R,r(cid:48)).
r l |r=R+0 r l |r=R−0 4πR2 l
(A5) with the factors Hl(cid:48)(cid:48) resulting from the matrix elements
l,l(cid:48)
With the ansatz for the transition between the two centers. These are
given,e.g.,byEq. (10.129)in[21] orEq. (5)in[22],and
d (r,r(cid:48))=d(0)(r,r(cid:48))−d(0)(r,R)Φ−1(ω)d(0)(R,r(cid:48)) (A6) correspond to the U in [18], Eq. (2.25). For g →∞, this
l