Tunable Casimir repulsion with three dimensional topological insulators

0.39 MB·English

by  Adolfo G. Grushin

Tunable Casimir repulsion with three dimensional topological insulators
Adolfo G. Grushin1 and Alberto Cortijo2
1Instituto de Ciencia de Materiales de Madrid (CSIC),
Sor Juana In´es de la Cruz 3, Madrid 28049, Spain.
2Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom
In this Letter, we show that switching between repulsiveand attractive Casimir forces by means
ofexternaltunableparameterscould berealized withtwotopological insulatorplates. Wefindtwo
regimes where a repulsive (attractive) force is found at small (large) distances between the plates,
1 cancelingoutatacriticaldistance. Forafrequencyrangewheretheeffectiveelectromagneticaction
1 is valid, this distance appears at length scales corresponding to 1−ǫ(ω)∼ 2αθ.
π
0
2

n The full experimental accessibility to micrometer and
a sub-micrometer size physics and the possibility of devel-
J
oping applications has turned the understanding of phe-
1
nomenaatthesescalestobeoffundamentalimportance.
1
Within this scenario, the Casimir force [1] arises when

] two objects are placed near each other at distances of a
i
c few micrometers. In the general case of two dielectrics
s thesituationiswelldescribedbythetheorydevelopedby
-
l Dzyaloshinskiiet. al [2]wheretheopticalresponseofthe
r
t materialdetermines the magnitude andbehaviourofthe
m
force. In the simplest case of a mirror symmetric situa-
. tionatheoremensuresthattheforceisalwaysattractive
t
a [3,4],resultinginaproblemfornano-mechanicaldevices.
m
To revert the sign, one must search non symmetric sit-
- uations, usually adding complexity to the problem. The
d FIG. 1: Different configurations of the Casimir effect with
n firstCasimirrepulsionproposal,knownasDzyaloshinskii identicalTI coveredwith athinmagnetic layer.(a) Themag-
o repulsion, was recently confirmed experimentally [5] and netization is of the same sign on each surface resulting in a
c it involves a third dielectric medium between the plates, case where θ1 = θ2 giving Casimir attraction. In (b), (c)
[ excludingthepossibilityoffrictionlessdevicesandquan- and (d) the magnetizations have opposite signs on the sur-
2 tum levitation. In turn, vacuum mediated proposals in- face(θ1 =−θ2)leadingtoattraction whend>dm,repulsion
v clude magnetic versus non-magnetic situations [6] and when d < dm and to a quantum levitation configuration at
1 dm where thenet force is zero.
the use of metamaterials [7–9]. In this Letter, we report
8
a new method for obtaining a twofold tunable Casimir
4
3 repulsion. By use of the optical properties of topologi- energy density (CED) stored by the plates is given by
. cal insulators (TI) it is feasible to achieve all situations [16]:
2
between repulsion to attraction by using two control-
0
0 lable parameters: the distance between dielectrics, and Ec(d) = ∞dξ d2kk logdet 1−R ·R e−2k3d ,
1 the sign of the topologicalmagnetoelectric polarizability A~ 2π (2π)2 1 2
Z0 Z
: (TMEP) θ, where the latter allows to tune the optical (cid:2) (cid:3)(1)
v
i properties of the mentioned materials. where A is the plate area, k = k2+ξ2/c2 is the wave
X 3 k
r TmIestaalrleiccbhoaurancdtaerryizesdtabteysapbroutlkecitnesdublaytitnimgbeerheavveriosualrswyimth- vectorperpendiculartotheplateqs,kk isthevectorparal-
a leltothe platesandξ istheimaginaryfrequencydefined
metry [10, 11]. The topologicalprotection of edge states
as ω = iξ. The matrices R are 2×2 reflection matri-
1,2
ensures their stability against non-magnetic perturba-
ces of media 1 and 2 containning the Fresnel coefficients
tions. The 3Dcounterpartof this noveltopologicalstate
defined as:
wasshowntoexistinaBi Sb alloy[12]andinthesto-
x 1−x
ichiometric crystals Bi2Se3, Bi2Te3,TlBiSe2 and Sb2Te3 R= Rs,s(iξ,kk) Rs,p(iξ,kk) , (2)
[13–15]. Rp,s(iξ,kk) Rp,p(iξ,kk)
(cid:20) (cid:21)
The Casimir force is intimately related to the optical
where R describes the reflection amplitude of an in-
properties of the two dielectric bodies [2]. For instance, i,j
cident wave with polarization i which is reflected with
consider the situation where two dielectric parallel semi-
polarization j. The polarizations R (p) describe paral-
infinite bodies (labeled 1 and 2) are placed at a distance s
lel(perpendicular)polarizationwithrespecttotheplane
d from each other in vacuum. In this case, the Casimir
of incidence. The Casimir force per unit area on the
2
plates is obtained by differentiating expression (1). A face) which can be safely described with the TMEP and
positive (negative) force, or equivalently a positive (neg- a dielectric function, as shown in earlier works [11, 18].
ative) slope of E (d), corresponds to attraction (repul- Hence,tonumericallycomputetheCEDbymeansof(1)
c
sion) of the plates. amodelforthedielectricfunctionisnecessary(wehence-
The electromagnetic response of a dielectric, which de- forthassumeµ=1). Duetothelowconcentrationoffree
fines the reflection matrices, is governed by Maxwell’s carriersin insulators the most general phenomenological
equations derived from the ordinary electromagnetic ac- model to describe the optical response of a dielectric is
tion S = dx3dt ǫE2−(1/µ)B2 , being E and B the a sum of oscillators to account for particular absorption
0
electric and magnetic fields respectively. TI in three di- resonances [21]. When only one oscillator is considered
R (cid:0) (cid:1)
mensionsarewelldescribedbyaddingatermoftheform (see however [22]) the dielectric function evaluated can
S = (α/4π2) dx3dtθ E ·B, where α = 1/137 is the be written as:
θ
fine structure constant and θ is the TMEP (axion field)
[11, 17]. BecauRse of time reversalsymmetry, this term is ǫ(iξ)=1+ ωe2 , (4)
ξ2+ω2 +γ ξ
a good description of the bulk of a trivial insulator (e.g. R R
vacuum)whenθ =0andofthebulkofaTIwhenθ =π. In this model, ω is the resonant frequency of the os-
R
However, the axion coupling is only a good description cillator while ω accounts for the oscillator strength.
e
of both the bulk and the boundary of a TI when a time The damping parameter γ satisfies γ << ω play-
R R R
reversal breaking perturbation is induced on the surface ing therefore a secondary role. In what follows, we have
and the system becomes fully gapped. In this situation, rescaled all quantities in units of ω leaving the quan-
R
θ can be shown to be quantized in odd integer values of tity ǫ(0) ≡ 1+(ω /ω )2 as the only parameter of the
e R
π such that θ = (2n+1)π where n ∈ Z. The value of n model. A good candidate to be described by this model
isdeterminedbythenatureofthetimereversalbreaking is the TI TlBiSe [15]. This material has experimentally
2
perturbation,whichcouldbecontrollableexperimentally [9] (neglecting free carrier contributions and assuming
bycoveringtheTIwithathinmagneticlayer. Inparticu- high frequency transparency) ǫ(0) ∼ 4 and has a single
lar,positiveornegativevaluesofθarerelatedtodifferent resonant frequency near 56cm−1. Other TI could need
signsofthemagnetizationonthesurface[18]. Aswewill more oscillators to be added in (4).
demonstrate in what follows, the Casimir force is very We have computed the CED between two TI plates de-
sensitive to the value of θ and the tunability of its sign scribed by the TMEP θ and θ and the value of the
1 2
will allow us to describe a mechanism for switching be- dielectric constant at zero frequency ǫ(0) by numerical
tween repulsive and attractive forces, evaluationofexpression(1). Theresultsaresummarized
The electromagnetic response of a system in the pres- in figures 2 and 3 where the CED is plotted against the
enceofaθtermisstilldescribedbytheordinaryMaxwell dimensionless distance d¯≡ dω /c. From Fig. 2 (a) it
R
equations but the constituent relations which define the is clear that opposite signs of θ lead to the existence
1,2
electricdisplacementDandthemagneticfieldH acquire of a minimum (d¯ ) where the net force is zero. The
m
an extra term proportionalto θ [19] D =ǫE+α(θ/π)B behaviour is attractive when both signs become equal
and H = B/µ−α(θ/π)E. We note that eq.(1) can be suggesting that it is possible to tune the Casimir force
easilymodifiedtotakeintoaccountmagnetoelectriccou- by tuning the relative signs of θ, i.e switching the mag-
plings(thishappensalsoinchiralmetamaterials[9]). The netizations of the coverings. The existence of a mini-
resultisthesameequationwiththeproperreflectionma- mum is analytically shown below in terms of the rela-
trices. It is then possible to derive by means of ordinary tive importance of the off-diagonalterms (3) againstthe
electromagnetictheorythe reflectioncoefficientsofa TI- diagonal terms and in terms of the relative sign of the
vacuum interface. For a TI characterizedby a frequency TMEPs. Atlargedistancesthe diagonaltermsdominate
dependentdielectricfunctionǫ(ω)andaTMEPθ there- and the usual Casimir attraction is recovered. At small
flection coefficients will take a symmetric form [1] where distances,theoff-diagonaltermsdominateandtheirsign
the off diagonal coefficient can be expressed as: determineswhethertheCEDapproaches±∞(i.e. repul-
sive orattractive)leading to a minimum at intermediate
R (iξ,k ,θ)=sgn(θ)r (iξ,k ,θ), (3)
s,p k sp k distances if the signs of θ are opposite.
1,2
wherer isanevenfunctionofθ. Whenθ =0,R =0, To prove the existence of the minimum we consider the
sp s,p
leading to the usual attractive Casimir force due to the Fresnel equations for TI obtained earlier in [1] which
non-mixing of polarizations [21]. When θ 6=0 the reflec- lead to eq. (3) added to the following properties of the
tioncoefficientsmixpolarizationsandthe signofθ plays dielectric function: finite dielectric permittivity at zero
acrucialroleonthesignoftheCasimirforce(seebelow). frequency (ǫ(0) < ∞) and high frequency transparency,
In what follows we will consider that the surface of the ǫ(ω) → 1 when ω → ∞. For analytical traceability we
TIs of the Casimir system are covered by a thin mag- assume θ = −θ ≡ θ and that the dielectric function
1 2
netic layer as shown in Fig.1. This effectively turns the (4) describes the TI, although the derivation does not
TI into a full insulator (both in the bulk and on the sur- depend on the explicit form of the dielectric function as
3
(a) 2
20
(b)
0.4
15
1 0.2
10
0
0
5 −0.2
E /Ec0−1 θθ1 == 53ππ E /Ec0 0 0.02 0.04 0.06 0.08 0.1
1
θ = π −5 θ = π
−2 1 θ = 3π
θ1 = −π −10 θ = 5π
−3 θ1 = −3π −15 θ = 7π

0 0.02 0.04 0.06 0.08 0.1 −20
ωRd/c 0 0.02 0.04ω d/c0.06 0.08 0.1
R
FIG.2: Casimirenergydensity(inunitsofE0 =A~c/(2π)2(ωR/c)3)asafunctionofthedimensionlessdistanced¯forωe/ωR =
0.45. In (a) θ2 = −π is fixed. Whenever sgn(θ1) = −sgn(θ2) a minimum d¯m appears leading to a vanishing net force on the
plates. Increasing θ1 within positive values suppresses the minimum shifting d¯m towards lower values (if both signs are equal
thenonly attractive behaviouroccurs). Complete repulsion isachieved when oneof theTMEP ismuch biggerthan theother.
(b)Theoptimalsituationθ1 =−θ2 =θ. Differentvaluesofθ showthattheminimumisenhancedwhenthedifferencebetween
thetwo values is as small as possible (θ=π).
long as it fulfils the mentioned conditions. r ) and r =2|α¯|/D where D =1+ǫ(0)+α¯2+ ǫ(0)χ
p sp
In(1)wecanrescaleξandk tocontaind¯whichgivesan , χ is a frequencyandmomentumdependentfunction ir-
overallfactor1/d¯3andforceskthereflectionmatricestobe relevantforthepresentdiscussionandα¯ =αθ/π.pInthis
evaluatedattherescaledfrequencyandmomentaξ/d¯and long distance limit, depending on the values of ǫ(0) dif-
k /d¯. Hence, E (d¯→ 0) → ±∞ and E (d¯→ ∞) → 0 ferent behaviors emerge. Since ǫ(0)≥1 we now consider
k c c
since the behaviour of ǫ(iξ) ensures that the reflection thetwoextremelimits,onewhereǫ(0)=1andtheother
matrices are not singular when evaluated at iξ/d¯ → 0 with ǫ(0)>>1.
and iξ/d¯→∞. In the limit where ǫ(0)>>1, the condition |r |,|r | >>
s p
The way the integralapproaches these limits determines |r | is always satisfied. When ǫ(0) is strictly infinity we
sp
the sign of E (d¯). For instance, if the integrand is pos- recovertheidealcaseofanordinarymetalwithr =±1
c s,p
itive at small distances and negative at large distances, and r =0. The integrand at large distances is a nega-
sp
necessarilyaminimumexistsatanintermediatedistance tivequantityandsoE (d¯)approacheszerofromnegative
c
d¯ . In what follows it will be shown that this is exactly values. From the previous discussion at small distances
m
whathappensunlessǫ(0)=1,wherebothlimitsarepos- E (d¯) → +∞, therefore there must be a minimum at
c
itive and hence long range repulsion is obtained. Under an intermediate distance 0 < d¯ < ∞ since the func-
m
these conditions the diagonal terms in (2) are equal for tion must cross the x axis. In the unrealistic case where
both TI (which we label r and r ) and the off-diagonal ǫ(0) = 1 one can check that |r | >> |r |,|r | making
s p sp s p
terms givenby (3) have opposite overallsigns, but equal E (d¯) alwayspositive for alldistances. In this casethere
c
absolute value given by the function r . Introducing is nominimum andthe forceisalwaysrepulsive. By this
sp
these inside (1) the integrand reads: analytical analysis and when θ = −θ we expect that
1 2
as we increase ǫ(0) from one, a minimum develops at an
I = log[1+e−2k3(r)(2rs2p−rp2−rs2)+ (5) intermediate distance d¯m. This distance shifts to lower
values as we increase ǫ(0) until, at ǫ(0) = ∞ we recover
+e−4k3(r)(rs2p−rprs)2], the metallic case where complete attraction occurs. In
the case where θ = θ the signs inside (5) change and
1 2
wherek(r) isnowevaluatedattherescaledfrequencyand make the logarithm to be negative recovering attraction
3
momenta just as the reflection matrices. In the limit of at all intermediate distances (for more details see [22]).
small distances (d¯ → 0) and using the high frequency The consistency of these analytical expectations is con-
transparency of the dielectric function it can be shown firmed numerically with the results shown in figures 2
that |r |,|r | <<|r | since the first are of order α2 and and 3.
s p sp
thesecondareoforderα. Hencetheintegrandispositive From these we infer that in order to enhance as much
and so Ec(d¯→0)→+∞. as possible the minimum, it is necessary to search for
Now we consider the limit of large distances (d¯→ ∞). a situation where θ ≡ θ = −θ . The CED for differ-
1 2
In this limit, the reflection coefficients take the form ent values of θ satisfying this condition are depicted in
r = 1−ǫ(0)−α¯2 /D (a similar expression holds for
s
(cid:0) (cid:1)
4
coating is to gap the surface states. We have estimated
a) 4 the parasitic magnetic forces between the magnetic lay-
ers following [24]. The dipole-dipole interaction is of the
2 order of attoN at distances of 50 nm and the magnetic
Casimir force[24] is ∼1fN, much smaller than the force
0 describedherewhichisoftheorderof5pN.Theproposed
0
E
/c effect could also be explored in other magnetodielectric
E −2 ωe/ωR = 0.2 materials such as Cr2O3 which can be described by a
ω /ω = 0.3 higher axion coupling[25]. However, these materials in-
e R
−4 ωe/ωR = 0.4 duce more general magnetoelectric couplings [26] which
ω /ω = 0.5 we will consider in a future work.
e R
−6 ω /ω = 0.6 We acknowledge F. de Juan, M.A.H. Vozmediano, F.
e R
Guinea, J. Sabio, and B. Valenzuela for very useful dis-

0 0.02 0.04 0.06 0.08 0.1 cussions and suggestions. A.G.G acknowledges MICINN
ω d/c
R (FIS2008-00124)andA.C.acknowledgesEPRSCScience
and Innovation Award(EP/G035954)for funding.
FIG. 3: Effect of parameter ǫ(0) with fixed θ1 = −θ2 = π.
The effect of increasing ǫ(0) is to develop a minimum, which
shifts tosmaller d¯m as ǫ(0) is increased.
[1] H. B. G. Casimir, Proc. Kon. Neder. Akad. Wet. 51, 793
figure Fig. 2 (b). Under these circumstances the mini-
(1948).
mum is more prominent when θ = π, i.e. when θ takes
[2] I.Dzyaloshinskii,E.M.Lifshitz,andL.P.Pitaevskii,Adv.
its smallest possible value. The general analytical anal- Phys. 10, 165 (1961).
ysis, supported by the numerical evaluation for different [3] O. Kenneth and I. Klich, Phys. Rev. Lett 97, 160401
parameters, suggest that a simple experimental set-up (2006).
(Fig. 1) could switch from complete Casimir attraction [4] C. P. Bachas, arXiv p.0611082 (2006).
[5] J.Munday,F.Capasso,andV.A.Parsegian,Nature457,
to a stable quantum levitation regime by reversing the
170 (2009).
magnetization of one of the layers covering one of the
[6] T. H. Boyer, Phys. Rev.A 9, 2078 (1974).
TI. In this process the system will turn from a symmet-
[7] F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni, Phys.
ric situation where θ1 = θ2 resulting in attraction to a Rev.Lett. 100, 183602 (2008).
non-symmetricsituationwheretheoptimumconditionis [8] U. Leonhardt and T. G. Philbin, New J.Phys. 9, 254
satisfied (θ =−θ ) and a stable minimum appears. (2007).
1 2
To conclude, for this appealing situation to be experi- [9] R. Zhaoet al., Phys. Rev.Lett. 103, 103602 (2009).
[10] C. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802
mentally accessible one has to search for realizations of
(2005).
ω where the typical distances between the plates are
R [11] X. Qi, T. L. Hughes, and S. Zhang, Phys. Rev. B 78,
at least of order 0.1µm - 1µm. For a frequency range
195424 (2008).
where the axion lagrangian is valid [17], the minimum [12] D. Hsieh et al., Nature 452, 970 (2008).
is expected to appear at a position where diagonal and [13] Y. L. Chen et al., Science 325, 178 (2009).
off-diagonal terms are similar in magnitude, i.e. length [14] Y. Xia et al., Nature Phys.5, 398 (2009).
scales corresponding to 1−ǫ(ω) ∼ 2αθ. Low values of [15] T. Sato et al., arXiv: arXiv:1006.2437 (2010).
π [16] M. T. Jaekel and S. Reynaud, Journal de Physique I 1,
ǫ(0)(typically less than10)favorthis situationsince the
1395 (1991).
minimum is realized at larger distances. While the elec-
[17] A.M.Essin,J.E.Moore,andD.Vanderbilt,Phys.Rev.
tromagnetic parameters of TI are still not well charac- Lett 102, 146805 (2009).
terized, a low ǫ(0) could be achieved by using thin films [18] X.-L. Qi et al., Science 323, 1184 (2009).
or by air injection which will lower the bulk dielectric [19] F. Wilczek, Phys. Rev.Lett 58, 1799 (1987).
response. For TlBiSe we estimate from numerical in- [1] M.-C. Chang and M.-F. Yang, Phys. Rev. B. 80, 113304
2
tegration that the minimum of the CED appears at a (2009).
[21] G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepa-
distanceofd=0.1µmandwithaCEDofthesameorder
nenko,Rev.Mod. Phys.81, 1827 (2009).
than for the usual metal-vacuum-metal system at 1µm,
[22] See the Auxiliary information document for details:
hencebeingstillexperimentallyaccessible. Wemustnote
EPAPS No. (????).
howeverthatthisestimationrequiresahighTMEPvalue [9] C.L.MitsasandD.I.Siapkas,SolidStateComm.83,857
(θ ∼ 10π) in order to shift the minimum to observable (1992).
distances. Therefore the proposed effect is on the verge [24] P. Bruno, Phys. Rev.Lett 88, 240401 (2002).
of experimental accessibility and should encourage ex- [25] F. W.Hehl et al., Phys. Lett. A 372, 1141 (2008).
[26] A. M. Essin et al., arXiv: 1002.0290 (2010).
perimental efforts to attain full optical characterization
ofTI.We stressherethatthe onlyeffectofthe magnetic
5
Auxiliary Material:
Tunable Casimir repulsion with three dimensional topological insulators
Attraction versus repulsion from Fresnel coefficients:
1.Fresnel coefficients for topological insulators: In order to compute the Casimir energy E (d), the Fresnel coeffi-
c
cients which define the reflection matrices of a topological insulator have to be calculated. These coefficients relate
the amplitude of the incident and reflected electric fields for different polarizations. Since the normal components of
D andB andtangentialcomponentsofE andH mustbecontinuousalongtheinterfaceonecanobtainthefollowing
reflection amplitudes by defining H and D as in the main text [1, 2]:
E(r) 1 n2−n2−α¯2+n n χ 2sgn(θ −θ )|α¯|n E(i)
s = 1 2 1 2 − 2 1 1 s , (6)
Ep(r) ! ∆(cid:18) 2sgn(θ2−θ1)|α¯|n1 −n21+n22+α¯2+n1n2χ− (cid:19) Ep(i) !
e2
where α¯ = α(θ −θ )/π, α is the fine structure constant (α = ), n is the refractive index of material i, ∆ =
2 1 ~c i
n2+n2+α¯2+n n χ and:
1 2 1 2 +
k2 k2
ξ2+ k ± ξ2+ k
n2 n2
1 2
χ = (cid:18) (cid:19) . (7)
±
k2 k2
ξ2+ k ξ2+ k
n2 n2
s 1 2
(cid:18) (cid:19)(cid:18) (cid:19)
When α¯ =0 and after a little algebra, these coefficients reduce to the ordinary Fresnel coefficients for a dielectric-
dielectric interface [3]. Since we are considering a topological insulator-vacuum interface, n =1, θ =0, θ =θ and
1 1 2
n2(ω)=ǫ(ω), therefore α¯ =αθ/π. By its definition, ∆is alwayspositive andthe off diagonalterms havea anoverall
2
sign governed by the sign of θ.
To calculate the Casimir energy stored between the plates one computes:
E (d) ∞dξ d2k
c = k logdet 1−R ·R e−2k3d , (8)
A~ 2π (2π)2 1 2
Z0 Z
(cid:2) (cid:3)
equation (1) in the main paper, where R is the 2x2 matrix defined in (6).
i
2. Analytical existenceofaminimumandtheroleof differentparameters: Toprovetheexistenceoftheminimumit
isenoughtoassumethatthereflectioncoefficientsaregivenbytheexpression(6)addedtotheanalyticalpropertiesof
thedielectricfunction,i.e. finitedielectricpermittivityatzerofrequency(ǫ(0)<∞)andhighfrequencytransparency:
ǫ(ω)→1 when ω →∞. For an insulator one can assume the dielectric function:
ω2
ǫ(iξ)=1+ e , (9)
ξ2+ω2 +γ ξ
R R
although the derivation does not depend on the analytical form of the dielectric function as long as it fulfils the
mentioned conditions. In particular, it holds for more than one oscillator. In addition, and for analytical traceabil-
itywemightfurtherassumetheparticularsituationwhereθ =−θ (wherelabels1and2identifytheCasimirplates).
1 2
It is straightforward to check that if in the expression for E (d) we rescale ξ and k to contain d this gives an
c k
overallfactor 1/d3 and forces the reflection matrices to be evaluated at the rescaledfrequency and momenta ξ/d and
k /d. Thus all of the d dependence can be transferred to the reflection matrices and the overall1/d3 prefactor. The
k
expression for the energy turns to be:
E (d) 1 ∞ ξ k ξ k
c = dξ d2k logdet 1−R , k ·R , k e−2k3 , (10)
E d3 k 1 d d 2 d d
0 Z0 Z (cid:20) (cid:20) (cid:21) (cid:20) (cid:21) (cid:21)
6
wherek2(r) =ξ2+k2 isdefinedthroughtherescaledvariables(whichincludecthespeedoflightinvacuumandω
3 k R
defined in the main text) and E = A~c/(2π)2(w /c)3 where A is the area of the plates and the reflection matrices
0 R
are evaluated at these rescaled variables with a rescaled dielectric function:
2
ω
e
ω
ǫ(iξ/d)=1+ (cid:18) R(cid:19) . (11)
2
ξ γ ξ
R
+1+
d ω d
(cid:18) (cid:19) R
Thebehaviourofǫ(iξ)ensuresthatthereflectionmatricesarenotsingularwhenevaluatedatξ/d→0andξ/d→∞.
Hence we can conclude that E (d→0)→±∞ and E (d→∞)→0.
c c
The waythe integrandapproachesthese limits determines the sign of E (d). For instance, if the integrandis positive
c
at small distances and negative at large distances, necessarily a minimum exists at an intermediate distance d . In
c
what follows it will be shown that this is exactly what happens when θ = −θ (unless ǫ(0) = 1, where both limits
1 2
are positive and hence long range repulsion is obtainned). The first step is to evaluate the integrand in (8). For the
situation in which θ =−θ =θ the reflection matrices describing both topological insulators are:
1 2
r (iξ,k ) ±r (iξ,k )
R = s k sp k . (12)
± ±r (iξ,k ) r (iξ,k )
sp k p k
(cid:20) (cid:21)
Introducing this inside (8) we obtain the following integrand:
I =logdet 1−R+·R−e−2k3(r) =log 1+e−2k3(r) 2rs2p−rp2−rs2 +e−4k3(r) rs2p−rprs 2 ,
h i h (cid:0) (cid:1) (cid:0) (cid:1) i
Notice that the last term although always positive, will play no role in what follows since it is always suppressed
over the first term (note that r ,r ,r <1).
s p sp
In the limit of small distances (d → 0) and using the high frequency transparency of the dielectric function it
easy to show that |r | <<|r | (i=s,p): Notice first that the denominator ∆ defined in (6) is common to all terms
i sp
so it cannot play a role on the relative magnitude of the coefficients. We need however to study the behaviour of χ
−
at small distances, given by:
k2
ξ2+k2− ξ2+ k
ξ k k n2
χ , k = (cid:18) 2(cid:19) . (13)
−
d d
(cid:18) (cid:19) ξ2+k2 ξ2+ k2k
k n2
s 2
(cid:16) (cid:17)(cid:18) (cid:19)
Remembering that at small distances (large frequencies) we have transparency, n2(ξ/d) = ǫ(ξ/d) → 1 then it is
2
straightforwardto see that χ →0 and hence:
−
−α¯2
r = =−r ,
s 2+α¯2+χ p
+
and
2α¯
r = ,
sp 2+α¯2+χ
+
since the first are of order O(α2) and the second are of order O(α) (remember that α¯ is proportional to the fine
structure constant α) we have that |r | << |r | (i = s,p). Hence the integrand is positive (since the integrand has
i sp
the form I =ln(1+A) where A>0) and so E (d→0)→+∞.
c
7
Now we consider the limit of large distances (d→∞). In this limit, the reflection coefficients take the form
1−ǫ(0)−α¯2+ ǫ(0)χ
−
r = ,
s
1+ǫ(0)+α¯2+ ǫ(0)χ
p +
for the diagonal part (with a similar expression for r ) and p
p
2α¯
r = ,
sp
1+ǫ(0)+α¯2+ ǫ(0)χ
+
p 2
for the off diagonal. We have defined the quantity ǫ(0)≡1+ we .
wR
In this case, depending on the values of ǫ(0) different behaviour(cid:16)s em(cid:17)erge. Notice, that from its definition, ǫ(0) ≥ 1
and so we can distinguish to extreme limits, one where ǫ(0)=1 and the other with ǫ(0)>>1.
When ǫ(0) = 1 one can see that we return to the previous case since the quantity χ in this limit also goes
−
to zero. Hence |r | >> |r | (i = s,p) is satisfied for all distances and E (d) is always positive .Therefore, using
sp i c
the fact that that E (d) → 0 when d → ∞ and that E (d) → +∞ when d → 0 we deduce that there is no mini-
c c
mumandthattheforceisalwaysrepulsive. Thisisconfirmedbythenumericalcalculationspresentedinthemaintext.
When ǫ(0)>> 1, we see that the opposite condition, |r | >> |r |, is satisfied. Even in the worst case when χ
s,p sp −
is smallest, ǫ(ξ) in r is always larger than 2α in r . The integrand at large distances is a negative quantity (since
s sp
the integrand now has the form I = ln(1−B) with B > 0) and so E (d) approaches to zero from negative values
c
(E (d)→−∞ for d→∞). Since at small distances E (d)→+∞ we conclude that there must be a minimum at an
c c
intermediate distance 0<d <∞, since the function must cross the d axis.
m
Notice that when ǫ(0) is strictly infinity we recover the case of an ideal metal where r = ∓1 and r = 0, with
s,p sp
attraction at all distances.
To sum up, by this analytical analysis we expect the following situation to occur (when θ =−θ ). As we increase
1 2
ǫ(0)fromone,aminimumdevelopsatanintermediatedistanced . Thisdistanceshiftstolowervaluesasweincrease
m
ǫ(0) until, at ǫ=∞ we recover the ideal metal case where complete attraction occurs.
In the case where θ =θ the signs inside I change to give:
1 2
I =log 1−e−2k3(r) 2rs2p+rp2+rs2 +e−4k3(r) rs2p−rprs 2 .
h (cid:0) (cid:1) (cid:0) (cid:1) i
Thepredominantterminsidethelogarithmisalwaysnegativeandhencetheintegrandisalwaysnegative. Therefore
at small distances E (d) → −∞ and at large distances E (d) → 0 approaching this limit from negative values,
c c
recovering attraction at all intermediate distances.
Finally, itcanbe checkedbyanalogousmethods, thatthe casewhere oneθ iszeroandthe other oneis not(dielectric
- topological insulator case) results in Casimir attraction for all distances.
Comparison with chiral metamaterials:
It is instructive to compare our results with the results obtainned by Zhao et al. in [4]. In their work,the relevant
parameter to obtain repulsion is the chirality which mixes transverse electric and transverse magnetic polarizations
giving in turn the off-diagonal r components. When calculating the Casimir force, by differentiation of the Lifshitz
sp
formula (Eq. (1) in the main text) they obtain the following integrand (in our notation):
(r2+r2−2r2 )e−2kd−(r2 +r r )2e−4kd
s p sp sp s p
J = . (14)
1−(r2+r2−2r2 )e−2kd+(r2 +r r )2e−4kd
s p sp sp s p
Notethatwhenr becomesdominant,J turnsnegativeasdiscussedin[4],enablingthusthepossibilityofrepulsion
sp
whenr issufficientlylarge. Inchiralmetamaterialsthisoccursforlargechiralityathighfrequencies(shortdistances)
sp
8
100
50
/EC0 0
E
−50 ω2=0.5, ω2 = 0
a b
ω2 = ω2 = 0.5, β = 103
a b
−100 ω2=0 , ω2 = 0.5, β = 103
a b
−150
0 0.02 ωd/c 0.04
1
FIG. 4: Effect of the addition of a high frequency oscillator to the dielectric permittivity (4). The units are the same as in
the figures of the main text. Green squares represent the high frequency oscillator (eq (16) with ωa =0), blue circles the low
frequency oscillator (eq (16) with ωb =0) and thedashed red line represents themodel in (16)
hence giving repulsion at short distances. The comparison with the topological insulator case is straightforward.
Analytical derivation shows that the integrand for opposing signs of the topological magnetoelectric response (θ =
1
−θ ) has the exact same form as the integrand given by (14) (with properly modified reflection coefficients). In the
2
last section we showed that, for high frequencies (short distances) r becomes dominant and hence, the topological
sp
part becomes dominant analogous to the case of large chirality in chiral metamaterials. When both topological
magnetoelectricterms have equalsigns (θ =θ ) one obtains the sameintegrandwiththe importantdifference ofthe
1 2
signs in front of the off-diagonal terms:
(r2+r2+2r2 )e−2kd−(r2 −r r )2e−4kd
s p sp sp s p
J = . (15)
1−(r2+r2+2r2 )e−2kd+(r2 −r r )2e−4kd
s p sp sp s p
As shownin the last section, evenwith dominantr repulsionis not possible at any frequency. Therefore the case
sp
with opposing θ signs is analogous to a high chirality metamaterial.
1,2
Note as well, thatthe chiralityfunction is frequency dependent while the topologicalmagnetoelectric responseis not,
since itisa topologicalcontributionandit doesnotgetrenormalized. Thereisalwaysanintrinsichighenergycut-off
of the order of the lattice spacing but since we are interested in the µm scale these effects are neglected. Hence, with
opposing signs for θ there is always a region of repulsion for low enough distances.
1,2
Related to the previously discussed works there are other studies where repulsion is realised in the context of
dielectric and magnetic anisotropy [6–8] in metamaterials (leaving chirality aside). Anisotropy can mix polarizations
and thus, under the right circumstances induce repulsion as shown by [6–8]. The problem of finding anisotropic
correctionsto the Casimir force in topologicalinsulators is not at all trivial. In the present work,we have focused on
the effect of the topological magneotelectric response on the Casimir force and so we have not included the effect of
anisotropy which we leave for future work.
For a full comparison of the cited works with the results here presented it would be necessary to include anisotropy
in our study in order to draw clear conclusions. Our work is thus closer to that of chiral metamaterials, and so we
leave comparison with anisotropic metamaterials for a detailed study of anisotropy. Nevertheless one could expect
that since anisotropy is responsible in metamaterials for repulsion at certain distances (for some specific range of
parameters), anisotropy can compete with the axionic response to develop richer behaviours.
9
Effect of including two oscillators:
Inthe maintextwehavemodelledTlBiSe withasingleoscillatorgivenexperimentallyby[9]. Howeveringeneral
2
otheroscillatorsatdifferentfrequenciescanbepresentwhenconsideringothertopologicalinsulators. Nevertheless,as
stated above,the proof for the existence of the minimum still holds even when more oscillators are consideredin (4),
as long as high frequency transparency and finite zero frequency dielectric response is respected, which is in general
true for insulators at low temperatures. The presence of other resonance frequencies can modify the position of the
minimum just as discussed in the main text. To explore this issue a little further we have studied the case when the
dielectric function is given by:
ω2 ω2
ǫ(iξ)=1+ a + b , (16)
ξ2+1+γ ξ ξ2+β2+γ ξ
R R
in units of ω the frequency of the firstoscillator. The parameterβ ≡ ω2 measures the position of the resonanceof
1 ω1
thesecondoscillatorwithrespecttothefirst. Toillustratetheeffectofaddingahighfrequencyoscillatorwestudythe
particularcasewhereβ ∼103 andω2 =ω2 =0.5showninFigure4. We haveplottedthe caseswherebothoscillators
a b
are included red dashed curve) together with the two isolated oscillators (green squares for ω = 0 ω 6= 0 and blue
a b
circlesfor ω 6=0 ω =0). Figure 4showsthat the effect ofahighfrequency oscillatoris suppressedwheneverω 6=0
a b a
and it is only when ω = 0 that it plays a role shifting the repulsive zone to higher distances, as expected from the
a
analysisinthe maintext. Althoughthe minimum stillexists inthis lastcase,the dielectric functionis soclose to one
that nearly complete repulsion is obtainned, as discussed in the first section. These analysis can be summarized up
as follows: 1) The minimum still exists when more oscillators are considered, in agreement with the analytical proof
and2)Thelowerfrequencywilldominatethepositionoftheminimumsinceitscontributiontothedielectricfunction
will be less suppressed.
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