Table Of ContentCasimir effect due to fluctuations of geometry in a quantum antiferromagnet
Anuradha Jagannathana and Attila Szallasb
aLaboratoire de Physique des Solides, CNRS-UMR 8502,
Universit´e Paris-Sud, 91405 Orsay, France and
b Wigner Research Centre for Physics, Hungarian Academy of Sciences H-1525 Budapest, P.O.Box 49, Hungary
(Dated: October17, 2012)
We show that fluctuations of geometrical connectivity can give rise to a Casimir type attractive
or repulsive force in a two dimensional Heisenberg antiferromagnet. The type of fluctuation that
we consider, called a phason flip, is well known in quasicrystals, but less so in periodic structures.
2 Astheclassical groundstateenergyoftheantiferromagnet isunaffectedbythistypeoffluctuation,
1 energy changes are purely of quantum origin. We calculate the effective interaction between two
0 parallel domain walls, defining a slab of thickness d, in such an antiferromagnet within linear spin
2 wavetheory. Theinteractionisanisotropic, andforaparticularorientation oftheslab wefindthat
it decays as 1/d, thus,more slowly than the electromagnetic Casimir effect in thesame geometry.
t
c
O PACSnumbers: 75.10.Jm,71.23.Ft,71.27.+a
6
1
determinedbygeometricaldefectstermedphasonflips,in
whichthe bonds ofa singlespinflipbetweentwoequiva-
]
l lentlocalconfigurations. We assumethatthetransitions
e
- betweenthetwoconfigurationscanoccureasilyandthat
r
the magnitude of the nearest neighbor Heisenberg spin-
t
s
spin couplings, J, is unaffected by the change. Such a
.
t situationcouldarisein structures inwhichthere aretwo
a
m equivalent chemical pathways to generate antiferromag-
neticHeisenbergsuperexchange,aswewilldiscussatthe
-
d end of this paper. This type of disorder is reminiscent
n
of the ”two level systems“ that were proposed for amor-
o
phous systems [7], in order to explain the anomalously
c
FIG. 1. The staggered dice lattice. The sites are colored
[ highscatteringrateofphononsseenatlowtemperatures.
red or blue according to whether they belong in the A or B
While phason flip disorder is well-known in the field of
1 sublattice.
quasicrystals [8], its effects have only recently been con-
v
2 sideredinaperiodicstructure,namelythestaggereddice
4 antiferromagnet [9]. Compared to the disordered quan-
4 tum spin models studied in the literature such as bond-
4 The Casimir effect [1] refers to the net force between
disorderedmodels or site-dilution models [10–14]), pha-
. two electrically neutral objects that arises due to vac-
0 son flip disorder leads to some novel properties. In con-
uum fluctuations of the electromagnetic field when they
1 trast to bond-fluctuation or site-dilution, a phason flip
2 are placed suffiently close together. Initially considered
does not modify the classical ground state energy of the
1 as a somewhat mysterious apparition of “something out
antiferromagnet. It does however modify the quantum
v: ofnothing”,the phenomenondescribedbyCasimirisfar correctionsto the groundstate energy and the spectrum
i more widespread, and arises quite generally due to fluc-
X of excitations around the ground state. As a result, an-
tuations in media where long ranged correlations [2] are
tiferromagnetism was found to be enhanced rather than
r
present. Thermal fluctuations in superfluids [3] and su-
a diminishedasinothertypesofdisorderedantiferromag-
perconductors[4]giverisetolongrangedforces. Casimir
nets[9]. Thisphenomenoncanbeconsideredasavariant
forces have been shown to exist in nematic liquid crys-
ofthe order-by-disordereffectoriginallyfoundbyVillain
tals [5], or even granular systems [6]. In this paper we
for a frustrated [15] spin model [16].
considerthe interactionenergyinathinslabofthe spin-
The Heisenberg model considered is
1 antiferromagnetic Heisenberg model on the staggered
2
dice lattice illustrated in Fig.1. The interfaces corre- H =J S~ S~ (1)
i j
spondtoarowofphasonflipsofthestructure,andhence ·
X
hi,ji
to boundary conditions of a novel type compared to the
usual Dirichlet or Neumann boundary conditions. with J >0, where S~ are the spin operators for the sites
i
In our model, periodic boundary conditions are as- i of the staggered dice lattice and the sum runs over all
sumed so that there are no external boundaries. The pairs of spins i,j joined by an edge of length a. On
parallel slab geometry in this antiferromagnet is instead the dice lattice, one can distinguish hexagonal “cages”
2
formedbysixspins,ofwhichthreearecoupledtothespin
at the cage center. This can occur in one of two ways. 5 5
The staggered dice lattice is formed by assembling such
6 6
hexagonal cages such that the bonds alternate between r2 2 2
the ”up” and”down”configurations. The lattice vectors
4 4
of the SDL shown in Fig.1 are given by ~r = √3axˆ and
1 3 3
~r = 3ayˆ, and the symmetry of the structure is rectan-
2
1 1
gular. The SDL structureis unfrustrated,ascanbe seen
in Fig.1, where spins are colored red or blue, according
to the sub-lattice and one expects rotational symmetry r1
to be broken in the ground state, which is N´eel ordered,
FIG. 2. Bond configurations before (left) and after (right)
with spins on sublattice A aligned along, say, the +z- two phason flips. Thesix sites of the unit cell are numbered.
direction, and spins on sublattice B aligned along z.
−
We recall that in the ordinary dice (also known as T3)
lattice, the central sites are linked to three neighbors in
a uniformway insteadofthis alternatingstaggeredfash- limit that ~k tends to zero one finds the linear relation-
ion, and that the result is a ferrimagnet [17]. In spin ship for the lowestenergyGoldstone modes ¯hω(~k)=v k
s
wave theory, one represents the spin operators in terms
ofHolstein-Primakoffbosonoperatorsai(a†i)forsitesion (for k ≪a−1)where vs =6q421J is the spin waveveloc-
the A-sublattice andb (b†)for sites i onthe B-sublattice ity. The ground state energy per site was calculated in
i i linear spin wave theory to be E = 0.6517(7)J in the
[18]. The linear spin wave Hamiltonian for the SDL lat- 0 −
thermodynamic limit, which compares well to the result
tice ofNsitesobtainedfromEq.1afterneglectinghigher
E = 0.6639(1)J obtained from quantum Monte Carlo
order terms is 0 −
(QMC) simulations [9].
Fig.2 illustrates the effect of a pair of phason flips
HLSW = 2JS(S+1)N +H(2) withinasingleunitofthe dicelattice. Theinteriorspins
−
H(2) =JS (a†a +b b†+a†b†+a b ) (2) S~3 and S~6 are coupled with three cage spins in different
i i j j i j i j
ways in the left hand and the right-hand figures. As the
X
hi,ji
total number of bonds is constant, the total classicalen-
One can introduce the Fourier transformed set of opera- ergy is unchanged by phason flips. As can be seen from
tors Fig.2, the transition between the two configurations im-
a = 1 e−i~k.R~na (3) plies a change of the sublattice of the center spin. In
µk µn order to preserve the antiferromagnetic condition on the
√N
c Xn total spin, S =0, the number of flips on sublattices A
tot
1
b = ei~k.R~nb (4) and B respectively, are therefore taken to be equal in all
µk µn
√Nc Xn the (finite) systems that we consider. Since phason flips
do not introduce any frustration, N´eelantiferromagnetic
where the sum runs over the N = N/6 unit cells sit-
c orderisalwayspreservedandlinearspinwavetheorycan
uated at positions R~ and µ = 1,2,3 labels the three
n thus be usedto study the systemwithin the entire range
sites belonging to eachsublattice within a unit cell. The
of values of the number of flips, 0 N N/3 where N
f
Hamiltonian now reads ≤ ≤
is the total number of spins. Periodic boundary condi-
H(2) =JS zµδµµ′(a†µkaµk+bµkb†µk)+ tions are assumed along x and y directions.
Xk µX,µ′ The energy cost per phason flip can be calculated in
(γµµ′(k)aµkbµ′k+h.c.) (5) spin wave theory by a numerical diagonalization proce-
dure. Itwasfoundtobeapproximately0.06J,farsmaller
where z = 4,5 and 3 are the coordination numbers for
µ than, for example, the energy of a spin vacancy which is
µ=1,2,3 and γ(k) is the 3 3 matrix
about 0.6J [19]. The energy of interaction as a function
×
f∗(1+f∗) 1 f∗f∗ of the distance has a short distance component – of the
2 1 1 2
1 1+f1 1+f1∗ (6) orderofone unit cellspacing–andananisotropicpower
f∗ 1+f 0 law decay at long distance. The sign of the interaction
2 1 energyoftwophasonsdependsonthesublattice: forpha-
with f = exp(i~k.~r ). We consider henceforth the solu- sonsonoppositesublattices,theinteractionisattractive,
µ µ
tions obtained for S = 1, the case for which the quan- and for those on the same sublattice it is repulsive. The
2
tum fluctuations are strongest. The energies of the spin rapiddecayofperturbationsaroundaphasonflipcanbe
excitations in the model are determined upon diagonal- understood in terms of a continuum version of the dis-
izationby means of a Bogoliubovtransformation. In the creteHamiltonianofEq.1bythefactthataflipproduces
3
EHdL
0.0622
0.0620
0.0618
-7
-8 BN
FIG. 3. Illustration of domain wall configurations used in 0.0616 -9 AN
-10
thestudyofCasimireffect. a)Slabalignedalong~r1 direction
(shaded light blue) b) Slab aligned along ~r1 +~r2 direction. 8 9 10 11
Sites on the interfaces belong in either the A-sublattice (red 0.0614
points) or B-sublattice (bluepoints).
0.0612
gradient terms of high (third and above) order.
We now present results for the energy dependence as
0.0610
a function of thickness for a horizontal (i.e. having its
orientation parallel to the ~r direction) slab of thickness
1
d. Fig.ashowstheconfigurationstudied: thelowerrowof
0.0608 d
A-sublatticephasonsisshowninred,whiletheupperrow 0 10 20 30 40
correspondstoB-sublatticephasons,showninblue. The
resultingsystemisanalogoustoaparallelplatecapacitor: FIG. 4. Change in ground state energy per unit length e(d)
outside“theparallellinesonehastheoriginaldicelattice, plotted as a function of the distance d between two rows of
defects for three system sizes (N = 4800,5808,6912). The
while“inside”theplatesonehasthefullyflippedversion
linesrepresentafittoEq.7. Theinsetshowsinalog-logplot
of the originallattice. For each value of d, we calculated
thescaling with 1/N of theAN and BN coefficients.
the difference of ground state energy E (d) E where
N 0
−
E is the ground state energy of the defect-free SDL.
0
Dividing this by the length of the domain wall L , we
x
obtainthe energyper unitlength, ǫ. As Fig.4shows,the Whenthehorizontalslabcalculationsarerepeatedbut
interaction between the lines is attractive. We fitted the for domain walls of phason flips on the same sublattice,
energy change to the expression the resulting force is repulsive, with the same d depen-
dence. This behavior is consistent with the earlier stud-
ies of the interaction between two phason flips: repul-
(E (d) E ) 1 1
N 0
− =eN(d)=AN +BN + (7) sive for same sublattice, attractive for different sublat-
L (cid:18)d L d(cid:19)
x y− tices [9]. This mechanismfor repulsiveCasimirforce can
where the secondterm ofthe termin bracketsarisesdue becontrastedwiththesystemdescribedin[20]whereina
to the periodic boundary conditions (L is the periodic topological insulator is used as a dielectric medium, and
y
lengthinthey-direction). Fitting tothedatausingEq.7 in which the sign of the interaction is varied by tuning
fordistancesthataresufficientlylarge,weobtaintheco- the magnetoelectric polarizability of the medium. The
efficients A and B for each sample size. The energy Casimir effect is strongest for the horizontal configura-
N N
versus distance plot for three different sample sizes are tion, where the domain walls are rows of closely spaced
showninFig.4,alongwiththecurvesgivenbyA +B /d same-sublattice flips. All other orientations of the do-
N N
for N =4800,5888and 6912. Both A and B coefficients main walls lead to faster decay. In particular, a domain
followafinitesizescalingin1/N,inaccordancewiththe wallalongthe~r +~r directionforexample(Fig.),which
1 2
scaling expected for the energy per length of a 1D anti- alternates flips on the A- and B-sublattices, leads to a
ferromagnet. Extrapolation to the infinite system yields short range contact-type interaction.
theinterfaceenergyperlength: e(d)=A+B/d,withthe In sum, the interaction energy E(d) between domain
values A = 0.062J/a and B = 0.001J. If one takes the wallswhichareparalleltoxˆdependsonthe separationd
exchange coupling J to have a value comparable to that as L /d. This dependence on the inverse of the thick-
x
−
of typical Heisenberg antiferromagnets, namely a frac- ness can be guessed from a dimensional analysis. On
tion ofaneV, andthat the edge length is ofthe orderof the one hand, the interaction energy is a dimensionless
anAngstrom,oneobtainsaninteractionenergyofabout quantity when expressed in units of J (the only energy
1pJ/m. scale in the problem), and on the other hand, this en-
4
ergy is expected to be proportional to the length of the
interface, which the interaction energy is proportional
to L /d. This distance dependence of the energy can
x
be compared with that of the electromagnetic field in
a similar geometry. In that case, a dimensional analy-
sis predicts that the interaction energy due to quantum
fluctuations, which is proportional to ¯hc, should drop
off as L /d2. Our model also differs in several essen-
x
tialrespectsfromthe one consideredin[21]betweentwo
rows of holes in a square lattice antiferromagnet, where
a Casimir effect is observed. For that problem, the in-
teraction, always attractive was found to decay as 1/d2.
In that case, each row of holes is characterized by the
microscopiclengthscalea,andthe regionbetweenis the
unperturbed antiferromagnet. In contrast, the slab ge-
ometry that we consider separatestwo regionswhichare
distinct, although related by phason flips.
A back-of-the-envelopecalculationfor this inversedis-
tancedependenceofEcanbegiven. Wenotethatatlong
distance,thetwophasoninteractionenergydecaysasr−3
[9]. Theddependenceoftheenergyoftheslabofphasons
canbe estimatedby computing the energyofinteraction
of a single A-sublattice phasononthe edge due to its in-
teractions with successive rows of B-sublattice phasons,
which is approximately ddy Lx/2dx(x2 + y2)−3/2
0 0 ∼
1/d for Lx d a. AdRded toR this is a contribution of
≫ ≫
the oppositesign,due tothe interactionsofsamesublat-
FIG. 5. (upper) Two examples of rows of cubes showing the
ticephasons,butthedistancesaredifferentandtherefore vertices before projection. Red sites project onto hexagonal
the contributions will not completely cancel. This rough cagevertices,whilethebluesitecanlieeitheraboveorbelow
calculation for the energy per unit length makes use of the 3-fold plane bisecting the cubes. The upper row shows a
the simplified asymptotic form of the 2-phason interac- uniform choice of blue sites, and the lower row a staggered
choice. (lower)The sites after projection on the three-fold
tion,anddoesnottakeintoaccountthedetailedangular
plane. The upper row has the same decoration in all cells,
dependence of the anisotropic decay.
while the lower row alternates between two configurations.
We briefly addressthe questionofpossible experimen-
tal realizations of the ordinary dice and staggered dice
lattices. On the one hand, the ordinary dice lattice can
beenrealizedusingopticaltrappingofcoldatoms[22],as the hexagonal cell of the dice lattice correspond to the
arealizationoftightbindingmodelswithso-calledDirac- six vertices lying closest to this plane (shown in red in
Weylfermions. Itwouldbeinterestingtoinvestigatepos- Fig.5 for a row of cubes). The two remaining apical ver-
sibilities for using suchsystemsto realizethe Heisenberg tices shown in blue, both project onto the central site
model. On the other hand, in conventional condensed in the cage but with different connectivities. If the one
matter systems, the staggereddice lattice which is a rel- oftheseconfigurationsisselectedrandomly,forexample,
atively complex structure, could perhaps be realized in one obtains the phason-disordered case. One could in
layered compounds. As an example of how this may be principle impose a given configuration by suitably engi-
done, we can consider how the Heisenberg model on the neeringthesubstrateonwhichthespinsystemisbuilt,so
twodimensionalKagomelattice(dualofthedicelattice) as to slightly bias one orother ofthe two configurations.
has been experimentally realized in a number of com- Transitions could then be induced by small changes of
pounds. One of these, called volborthite, has spins asso- the distances of the apical atoms.
ciatedwithCuatoms,whichinteractviasuper-exchange Inconclusion,we havecomputedthe Casimirforcesin
processes. When each of these magnetically active lay- a two dimensional antiferromagnet based on the honey-
ers is viewed from above, the spins and the bonds are comblattice. Thefluctuationshaveageometricalorigin,
seen to project to the Kagome structure. In an analo- involving changes of local topology, and may be realized
gousway,our model onthe dice lattice couldbe realized in quasi-two-dimensionalstructures in which the nearest
byconsideringspinsonverticesofcubes,andconsidering neighborenvironmentsallowforlocaltransitionsbetween
their projection on a plane perpendicular to one of the two differentconformationswith little or no energy cost.
body diagonals that bisects the cube. The six sites of The interactionenergydepends onthe orientationofthe
5
Casimir slab, and can be attractive or repulsive depend- 2339 (2008)).
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inverseoftheslabthickness. Themagnitudeoftheeffect
(2000).
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We wouldlike tothank B.Douc¸otandThorstenEmig
Rev. B 69 214424 (2004).
for useful discussions.
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spins on vertices of a triangle.
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