Table Of ContentSurprise(s) in magnets without net moments
Kedar Damle1
1Tata Institute of Fundamental Research, 1, Homi Bhabha Road, Mumbai 400005, India
(Dated: January 12, 2009)
We are all familiar with ferromagnetic and antiferromagnetic materials, in which the localized
ionicmoments(incaseofionicinsulators)ortheelectronicspins(incaseofmetals)gointoalong-
range ordered state witha net macroscopic moment (in case of ferromagnets) or a net macroscopic
sublatticemagnetization(incaseofantiferromagnets). Howeverthisbehaviourisfarfromubiquitous
eveninionicinsulatorswithwell-developedlocalmoments. Indeed,therearemanyionicinsulators
inwhichthedominantinteractionsbetweenthelocalmomentscompetewitheachother,leadingto
acooperativeparamagneticstatewithnoorderingofthemomentsdowntothelowesttemperatures
accessible to experiments. The physics of such magnets without net moments has some interesting
4 and surprising aspects, which are touched upon in this brief review.
1
0
PACSnumbers: 75.10.Jm05.30.Jp71.27.+a
2
n
a Overview: In many ionic insulators, some of the ions case, for instance, in the compound MnO which has a
J have non-zero ground-state angular momemtum, and an Neel ordering temperature of approximately 116 K.
2 associated magnetic moment. These localized magnetic However, there are many other examples in which the
moments usually interact with each other in two ways: magneticionsarecoupledbyexchangecouplingsthatdo
]
l On the one hand, by virtue of a being magnetic dipoles, not obey this bipartite constraint—in other words, there
e
- each moment reacts to the dipole field of all other mag- aretrianglesinthenearestneighbourconnectivityofthe
r neticdipoles,givingrisetoalong-rangeanisotropicmag- lattice. Such magnetic lattices with triangular motifs in
t
s netic dipolar interaction. On the other hand, the virtual them are good examples of geometric frustration. To see
.
t hopping of charge carriers between neighbouring mag- the significance of such triangular motifs, it is enough
a
m netic ions gives rise to a short-ranged (typically nearest tonotethattheNeel(antiferromagnetic)statealongany
neighbour) interaction generally known as the exchange axisnisfrustratedinthepresenceofsuchtriangles,since
-
d interaction [1]. there is no unique way of satisfying all the exchange in-
n This exchange energy can be written as E = teractions (Fig ) fully.
o (cid:80)
c J (cid:104)ij(cid:105)Si·Sj ;J >0, where J is the exchange constant In many situations [2], this results in a macroscopic
[ andthesubscriptsrefertopairsofnearest-neighbourmo- degeneracy of classical minimum energy configurations.
ments. In many commonly occuring cases, J is positive At intermediate temperatures T that are less than the
1
and therefore antiferromagnetic in nature [1], in that it exchange J, but are not small enough for the quantum
v
5 encourages antiparallel alignment of pairs of neighbour- mechanical nature of spins to matter, the spin correla-
7 ing moments. In many situations, this exchange con- tions(measured,say,byneutronscatteringexperiments)
3 stantismuchlargerthanthelong-rangedipolarcoupling, inthesystemsimplyreflectthismacroscopicdegeneracy,
0
which can then be left out of the analysis to a very good and can be modeled in a universal way in terms of aver-
.
1 approximation, while in other cases, the dipolar cou- agesoveranensemblethatgivesacertainweighttoeach
0 plingcandominateoverthenearest-neighbourexchange. of these minimum energy configurations [3]. To a first
4
[Here, the spins S are of course quantum-mechanical op- approximation, this weight is of course uniform and the
1
: erators; however, for many purposes at not too low tem- same for each minimum energy microstate, and the sub-
v peratures,theycanbeusefullyapproximatedbyclassical leadingeffectsofcanonicalfluctuations(thatincreasethe
i
X vectors of fixed length, particularly if the spin quantum energy from this minimal value) can also be included in
r number S is 3/2 or higher.] a more sophisticated treatment. Many examples of such
a When the magnetic ions form a bipartite lattice (in frustratedmagnetsareknown. Onthepyrochlorelattice,
which the lattice can be broken up into two sublattices theseincludetheCu2+ basedS =1/2magnetparamela-
in such a way that the nearest neighbour exchange cou- conite[4]andtheCr3+basedS =3/2magnetsCdCr2O4
pling connects only pairs of spins belonging to different and HgCr2O4 [5]. Several interesting examples have also
sublattices), this antiferromagnetic exchange energy is been studied on the kagome lattice—these include Cu2+
minimized by the so-called Neel state in which all spins based S = 1/2 volborthite and other systems[6], Ni2+
lie along a spontaneously chosen axis n and every spin based S = 1 magnets Ni3V2O8[7], Cr3+ based S = 3/2
points anti-parallel to its nearest neighbours [In two and systems[8],andFe3+ basedS =5/2magnetsFejarosite
higher dimensions, this picture also gives an essentially [9].
correctcaricatureofthegroundstateofthefullquantum Much of the interest in frustrated magnetism arises
problem on a square or hypercubic lattice.] This is the fromthenon-trivialnatureoftheresultingstate. Inpar-
2
+n
by the competing antiferromagnetic interactions by en-
suringthateachtrianglehasexactlyonefrustratedbond
(pair of parallel Ising spins).
Furthermore, these minimally frustrated configura-
tionshavearelatively‘clean’characterizationintermsof
−n ?
dimercoversofthedualhoneycomblattice. Moreexplic-
itly,considerthehoneycombnetformedbyforminglinks
FIG. 1: Three spins interacting antiferromagnetically with between the centers of triangles across the shared side of
each other cannot satisfy the demands of all the exchange the triangle. If we place hard-rods on each honeycomb
interactions
link that crosses a frustrated bond in a minimally frus-
trated configuration, then each honeycomb lattice site
will have exactly one hard-rod covering it. Such configu-
ticular, as we will see explicitly in examples below, the rationsofhard-rodsdefineso-called‘dimercovers’ofthe
resulting intermediate state is not a simple paramagnet honeycomblattice,andclearly,thereisaone-to-onemap-
witheachmomentfluctuatingmoreorlessindependently ping between dimer covers of the honeycomb lattice and
of the others. Instead, it is typically a non-trivial coop- minimum frustration states of the classical Ising model
erative paramagnet, with a complicated pattern of cor- on the triangular lattice.
relationsbetweenfarawaymoments. Suchacooperative As the temperature T falls well below the exchange
paramagnetic state can have some very unusual proper- energy scale J, most triangles of the lattice will satisfy
ties, and that is the really interesting thing about these the minimum frustration condition and have exactly one
systems. Finally, it is also useful to keep in mind that frustratedbond. Indeed,oneexpectsthatatypicallylow-
the ultimate fate of such magnets at very low temper- temperature configuration will differ from a minimally
atures is less universal, and depends sensitively on the frustrated T = 0 configuration only by an exponentially
effectsofquantumfluctuationsandother(subdominant) small (O(e−J/kBT)) density of triangles with three frus-
interactions acting in this subspace. tratedbonds. Ifweignoretheeffectsofsuchdefects, the
Toy model—The triangular lattice antiferromagnet: T (cid:28)J properties of this system can thus be modeled by
Fortunately for us, the triangular lattice Ising antifer- calculating the properties of the ensemble of minimum
romagnet provides a simple example where a great deal frustration states, with equal weight to each such state.
of the foregoing can be illustrated quite explicitly. Here Thisismostconvenientlydoneindimerlanguage,and
‘Ising’ refers to the fact that we consider a simplified one learns two important things upon translating back
situation in which ‘spins’ that are not unit vectors, but to the language of Ising spins: The first is that the Ising
instead discrete variables that can take two values ±1. spins are not ordered even at T = 0, i.e there will be
Such models are generally referred to as Ising models, no magnetic Bragg peaks even in a hypothetical neutron
after Ising, who studied them first in his Ph.D thesis. scattering experiment performed on our Ising magnet.
Although this model looks very over-simplified at first The second is that the correlations between spins do not
sight, it does have the potential to describe real mag- decay away to zero on the scale of a few lattice spacings,
nets at least in some cases. This is because real mag- as would be expected for a simple paramagnet in which
nets have local moments that often correspond to ionic the moments fluctuate independently of each other. In-
ground state multiplets with a non-zero value of orbtital stead, the correlations decay to zero very slowly, as the
angular-momentum. Insuchcases,spin-orbitcouplingin inverse square-root of the separation between the spins.
the presence of strong crystal field effects can induce a TheT =0stateisthusacooperativeparamagnetinthe
single-ion anisotropy term −D(cid:80) (Sz)2 in the magnetic momentsarecorrelatedwitheachotherovermacroscopic
i i
Hamiltonian. If D is large (compared to J) and posi- distances although there is no long-range ordering. In
tive, then the spins predominantly prefer to be in one of this simple toy model, it is also possible to put back the
two states Sz =±S (for spin S moments), which can be exponentially small density of defect triangles into our
thoughtofasthetwoIsingstatesinourforegoingdescrip- description,anditcanbeshownthatthesedonotchange
tion [One example of this is the Kagome lattice antifer- theT =0pictureinanystrikingway—allthathappensis
romagnet Nd-Langasite, where a description in terms of thatthedefecttrianglesdisrupttheslowpower-lawdecay
a Ising magnet on the Kagome lattice apparently works of correlations beyond a length-scale ξ that corresponds
quite well] to the typical inter-defect distance, and the correlations
With that background, we now ask: What configu- decay exponentially rapidly to zero for r (cid:29)ξ.
rations minimize the nearest neighbour Ising exchange Therearethustwodifferentbutrelatedquestionsthat
(cid:80)
energy E =J σ σ ? Clearly, the answer is all con- one needs to keep in mind when thinking about the low
(cid:104)ij(cid:105) i j
figurations in which each triangle has either two ‘up’ temperature properties of such magnets: The first is the
(+1) spins and one ‘down’ (−1) spin, or vice versa. nature of the degenerate minimum exchange energy con-
These configurations minimize the ‘frustration’ induced figurations, and the ensemble they define. In particular,
3
allothersfreecorrespondstoaheightconfigurationwith
(x,y) maximum tilt (gradient) in one of the principal direc-
(0,0)
tions of the triangular lattice. By inspection, we also see
that such a staggered configuration cannot be changed
T1 into any other dimer configuration by any local moves
that do not involve a macroscopically large number of
h_
R links,andthushasvery few nearbydimerconfigurations.
A
Conversely,dimerconfigurationsthatcanbetransformed
e T0 into other valid dimer configurations in a large number
B
ofwaystendtohavezeroaveragetiltinheightlanguage.
h_
L
(x,0) With this motivation, one postulates a coarse-grained
‘free energy’ that captures the entropic weight of differ-
ent height configurations and writes the T =0 partition
function as
FIG.2: Thehoneycomblatticedualtothetriangularlattice,
the definition of the electric field and heights, and the path Z =(cid:90) Dh(x)exp(cid:18)−K (cid:90) d2x(∇h)2(cid:19) , (2)
followedtoreachpointx,y)onthetriangularlatticestarting 2
from the origin (0,0)
where K is a phenomenological ‘stiffness’ parameter. Of
course,inordertousethiseffectivefieldtheory,oneneeds
in the limiting case of zero temperature, physical quan- a prescription for writing the local spin density σ(r) in
titiescanbemodeledbyaveragesoverthisrestricteden- terms of the height fields. To understand this correspon-
sembleofminimumenergyconfigurations. Thesecondis dence, one may start by fixing one spin, say the spin on
the nature of the thermally induced defects that allow a the site at the origin, to be up σ(r = 0) = +1, and the
systemtolocallydeviatefromaminimumenergyconfig- corresponding height to be zero h(r = 0) = 0. Now,
uration,andthepropertiesofadilute,extremelycoldgas we note that 3h jumps by an odd number whenever one
of these defects—the low temperature properties of the crosses an unfrustrated bond (across which the spin flips
magnet depend both on the nature of the ground state sign), while the height jump is even across a frustrated
ensemble,andthestatisticalmechanicsofthedefectgas. bond (across which the spin remains unchanged).
Effective field theory for the T → 0 limit: In order to Thisimmediatelyimpliesthataspinatsomeothersite
prepare the ground for our later discussion, it is useful r will be up if and only if 3h(r) is even [10]. This pro-
to spend a little time understanding these results from videsonepieceofthecorrespondencebetweentheheight
the perspective of a coarse-grained effective field theory. field and the spin field. The second, and in many con-
Theideaisthinkofthedimeroccupationonalinkasthe textsmorecrucial,pieceofthecorrespondenceisslightly
value of an electric field e on the corresponding link of trickier to understand, and is best appreciated by trying
thehoneycomblattice,withthesignconvenitonthatthe topredictthespinvalueatsiter=(x,y)bygoingacross
e on each link always points from the A sublattice site xlinksofthedualhoneycomblatticeindirectionT ,fol-
0
to the B sublattice site of that link (see Fig 2). Clearly, lowed by y links of the dual lattice in direction T [11].
1
thedimerconstraintnowtranslatestothestatementthat Now, each dimer crossed in the process guarantees that
thereisastatic+chargeoneachAsublatticesiteofthe the spin state has not changed, while each empty link
honeycomb net, and a static − charge on each B sublat- corresponds to a flip in the spin state. We may there-
(cid:80)
tice site. Now, we solve for this Gauss’s law divergence fore write σ(r) = exp(iπ (1−n ))σ(0), where n is
l l l
constraint by writing e in terms of a height field (which the dimer number on all the links l thus encountered.
is the two dimensional analog of the vector potential of Rewriting this in terms of the height field allows us to
ordinary electrodynamics). argue that σ(r)∼exp(2πi(x+y)/3+iπh(x,y))+h.c.
Thisheightfieldhisdefinedontheoriginaltriangular Thus,boththezeromomentumandmomentum±Q≡
lattice sites, but is quite distinct from the original spins, ±(2π/3,2π/3)componentsofthespinfieldhaveasimple
and in terms of h, we may write the electric field on link localrepresentationintermsofexponentialsoftheheight
l as field, and the long-distance properties of the spin corre-
lations may be captured by the correspondence: σ(r) =
1
e − =h −h (1) c cos(πh(r)+2π(x+y)/3)+c cos(3πh(r)), where c
l 3 L(l) R(l) Q 0 Q
and c are non-universal scale factors. Using this and
0
where R(l) and L(l) are triangluar lattice sites to the the known value of K [10], we can calculate the T = 0
right and left of this link (as defined when looking down correlators of the Ising spins, and find that the leading
√
the link from its A sublattice end (see Fig 2). termatlargeseparationr goesascos(2πr/3)/ r. Thus,
Wenownotethatthestaggereddimerconfigurationin the triangular Ising magnet is anything but a simple un-
which we occupy all links of one orientation and leave correlated paramagnet, although it does not order at all
4
discussion of the various technical aspects and subtleties
involved.]
Inthesematerials,Dy3+ andHo3+ occupythevertices
ofthepyrochlorelatticetetrahedra(showninFig3)and
carry a ground state magnetic dipole moment µ=10µ
B
(where µ is the Bohr magneton) that has its origins in
B
thespin-orbitcoupledgroundstatemultipletforthisva-
lence state. Crystal field effects result in a strong ‘easy-
axis’ energy that forces eacn moment to lie along the
tetrahedralbodydiagonalthatpassesthroughthecorre-
sponding pyrochlore lattice site. Thus, if one considers a
FIG. 3: The pyrochlore structure made up of corner sharing
tetrahedra whose centers form the diamond lattice. In the single tetrahedron, each moment has two choices: It can
spin-ice compounds, Ho3+ or Dy3+ moments occupy vertices eitherpointinwardtowardsthecenterofthetetrahedron,
of the tetrahedra shown forming spin-ice. or outward away from the center of the tetrahedron—in
both cases, it must lie precisely along the corresponding
body-diagonal.
This degree of freedom can be thought of as an Ising
spinσ,andthemagneticpropertiesofthesematerialscan
againbemodeledassomesortofIsingmodel. Sinceeach
dipole is shared by one up-pointing tetrahedron and one
down-pointingtetrahedron,aconvenientsign-convention
f or this mapping is that σ = +1 if the corresponding
dipolepointsoutwardswhenviewedfromtheup-pointing
!"#$%&’(’
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from the down-pointing tetrahedron to which it belongs.
WhatistheHamiltonianorenergyfunctionalthatde-
scribes the energetics of theseIsing spins? The answer is
a little complicated: It turns out that the nearest neigh-
bourexchangecouplinginthesesystemsisweak(oforder
1K in temperature units), and the long range magnetic
dipole interactions between the magnetic moments is ac-
FIG. 4: Tetrahedra violating the two-in two-out ice-rule are tually more important.
actuallymagneticmonopoles. Iftheywereboundinnearest-
The Hamiltonian is thus the well-known classical ex-
neighbour pairs (as on left), then the lowest lying excita-
pressionfortheinteractionenergyofabunchofmagnetic
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f a ‘dumbell-model’ [14] for the interaction energy. The
+’-(&%++!*?+’!"3(-"
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"(*!#*’,(/"
!"#%"%;@+"4,,/0*’(*+"&#!#1#4("*’
&$#*"#6!"4’,("=6#*"%-’*+*+"1’
!,%
,&’
*6((s"p$$’i(n4-*i’1ce"9c1o#m8’(p4o*+u"n0*d#s*[$T,-h*e"1i&n"!t#e*6r!e"s#te%*d!’(r4e"(a*d*"e%*r.s,!h7o,u*+ld*+"alFs!o!+"r(e’6-%&!,
"p%%yrochlore lattice.
#*+("/f.*e!+#r"
*t’3,#o($6#$"t’Oh,#.*e’,*+(o"r,7i.g#1!i!n#’"4!a("l+*"’a
’4r+
t*+;i#c!@4l+"e’#%s#*(R"%/*e**f++s""!1."!.",2"!–"/1#’2$4!6"%#f’/,o(5r&,a!.,63m’(/
",o%(r03(e"!"5/d%e&*#!t,!a*(’4
i$l"e"%3;d’/P",(-
""3.",!!9 The blue end represents a fictitious positive magnetic
*+"F!!+"(’6%$#-#(/+"(
"*+"("#!"%*("’4+7,6!1,/"$6$*’1#*"$5.#’$%96(/"!"%*’1#*’(4*+"*’1"
%
#$"#*$,-*"1&"!#*6!";@+"!,$",.*+"1’%%’(4Q,6$,17’(*"!#
*’,(’%*+"!".,!"
$"#!K#$*+,64+
(,(>
,(0(’(4’*-’$$$"#/*,$,
#$$57,6(/&#’!%,.&#!*’
$"%9
,(%’/"!#7$5%$,-’(4/,-(/’26%’,(#(/
5
charge +q /2, and the red end represents a fictitious urations as before: One simply considers all up-pointing
m
negative magnetic charge −q /2 (the reason for the fac- tetrahedra, and agrees to put a dimer through each out-
m
tor of two in this definition will be clear below). The ward pointing magnetic moment. To ensure consistency,
magnitudesofthesechargesareadjustedtogivethecor- theruleisreversedforalldown-pointingtetrahedra,that
rect magnetic dipole strength of µ = 10µ by requiring is, eachinwardpointingspinofadown-pointingtetrahe-
B
that q /2 = µ/a , where a is twice the distance from dron corresponds to a dimer on the diamond lattice link
m d d
the vertex of a tetrahedron to its body-center along the passing through that spin. This gives a dimer model on
body diagonal (equivalently, a is the nearest-neighbour the diamond lattice, with two dimers touching each dia-
d
distance of the dual diamond lattice). The Ising degree mond lattice vertex, and one can then use efficient loop
of freedom σ at each pyrochlore site now corresponds to algorithms to sample all the minimum energy configura-
the orientation of this dumbell, and the Ising spin σ at tions in this dimer representation [16, 17].
the vertex of an up-pointing tetrahedron is +1 if the red What about excited states? Naively, one might imag-
end of the dumbbell is located at its body-center, and ine that the lowest lying excited states may be con-
−1iftheblueendofthedumbbellislocatedatitsbody- structed by starting with an arbitrary minimum energy
center. configuration, andflippingoneIsingspin. Suchaflipped
Theoriginalenergyfunctionalcannowbesimply(but spin would give rise to two nearest neighbour tetrahe-
approximately) reproduced by postulating a fictitious drathatviolatetheicerule—oneofthemwillhavethree
Coulomb interaction between these fictitious magnetic out-pointing spins, and one of them will have three in-
charges: pointing spins.
However, the language of fictitious magnetic charges
µ Q Q
V (r ) = 0 α β ;α(cid:54)=β allows us to think a little bit more deeply. For consider
m αβ 4π r
αβ flippingasinglespinσ from+1to−1. Thiscreatestwo
αβ
1
= v Q2 ;α=β equal and opposite fictitious magnetic charges at near-
2 0 α est neighbour sites α and β: Q = +q , Q = −q .
α m β m
Hereµ isthevacuumpermeability,Q isthetotalmag- Thought of in this way, there is nothing special about
0 α
netic charge on diamond lattice site α (corresponding to having these two charges ±qm at nearest neighbour lo-
the body-center of tetrahedron α) and the ‘self-energy’ cations. By flipping a string of Ising spins, we can pull
constant v is adjusted to correctly reproduce the inter- these charges further and further apart from each other
0
action energy between nearest neighbour dipoles. until they are separated by distance r. Now, these two
Naturally, this statement is only approximate, but the charges interact with an attractive Coulombic potential
approximation involved is such that the difference be- that falls off as 1/r. As we know, a 1/r attraction is not
tween the real interaction energy and the approximate confining, in the sense that it only takes a finite amount
formobtainedbyourdeviceofintroducingfictitiousmag- ofworktoseparatethetwochargestoinfinity. Thus,the
neticchargesfallsoffrapidlywithdistancerbetweenthe dominant excitations will not correspond to two charges
magnetic dipoles, and is very small everywhere. ±qm bound tightly at nearest-neighbour distance, but
This way of thinking immediately yields dividends rather two independent and ‘free’ charges that can be at
when we ask for the nature of the minimum energy con- arbitrary separations from each other.
figurations of the Ising spins σ. Using the electrostatic We may now translate back to the language of the
analogy, it becomes clear that the minimum energy con- original magnetic moments and Ising spins: The dom-
figurations are precisely those configurations for which inant low energy excitations consist of arbitrarily long
the total (fictitious) magnetic charge on the body-center stringsofflippedIsingspins. Somemoreanalysisreveals
ofeachtetrahedroniszeroQ =0forallα. Translatedto that the end points of these strings are genuine mag-
α
Ising variables, this is the ‘two-in two-out’ ‘ice rule’ that netic monopoles of strength ±q , in the sense that the
m
saysthattwoverticesofeachtetrahedronmusthaveIsing associated magnetic field configuration would induce a
spinspointinginwardstowardsitsbody-center,whiletwo currentinasuperconductingringifoneendofthestring
must have Ising spins pointing outwards away from its passed through the ring. This current would be identi-
body-center. [Here ‘ice-rule’ refers to the analogy to fied as a monopole signal in a standard monopole search
Pauling’sideasabouttheentropyofice,andNagle’sunit experiment such as the Stanford experiment to detect
model for ice [15]]. fundamental magnetic monopoles in cosmic radiation!
How many such minimum energy configurations are So far, all of this is strictly a T = 0 argument about
there? Theansweristhatthesetofminimumenergycon- ground states and low-lying excited states. One may
figurations has macroscopic entropy, i.e it scales as the worry that entropic effects associated with thermal fluc-
exponentialofthenumberofsystemsites. Thisisclosely tuations at non-zero temperature would somehow spoil
analogous to the triangular lattice example we discussed all this. Perhaps fortuitously, the answer is no! It turns
earlier,andindeed,thespin-iceminimumenergyconfigu- out that the most important consequence of entropic ef-
rationscanalsobecharacterizedintermsofdimerconfig- fects is a modification of the prefactor of the Coulombic
6
isotropic nearest neighbour antiferromagnetic exchange
coupling J ≈ 100K (in temperature units), forming two
Kagome layers coupled through apical sites in the mid-
dle (Fig 5). The spins in the dimer layer only interact
withinapairviaanisotropicantiferromagneticexchange
J(cid:48) ≈ 200K (in temperature units), thus forming a sys-
temofdecoupledspin-dimers. TheexcessGaintroduced
by having p< 1, substitute for the Cr3+ ions and intro-
ducenon-magneticimpuritieswithS =0. Fromdetailed
studies [21], it is known that the Ga have a slight prefer-
enceforgoingintotheKagomeandisolateddimerlayers,
rather than substitute for the apical Cr3+.
With this background, let us now think classically
(S =3/2 is large enough that we expect a classical anal-
ysistobeaccurateexceptatextremelylowtemperatures
atwhichquantumfluctuationsstarttoplayanimportant
role) and ask for the classical minimum energy configu-
rations of the exchange Hamiltonian
H = J (cid:88)((cid:88)S(cid:126) − h )2+ J (cid:88)((cid:88)S(cid:126) − h )2
2 i 2J 2 i 2J
i∈ (cid:52) i∈(cid:52)
(cid:52) (cid:52)
where referstothetetrahedrathathaveasoneoftheir
facesth(cid:52)eup-pointing(down-pointing)trianglesintheup-
per (lower) Kagome layer, and (cid:52) refers to the down-
pointing (up-pointing) triangles in the upper (lower)
Kagomelayer,andhistheexternalmagneticfield,which
we now proceed to set to zero.
When written in this form, it is clear that this energy
functional has enormously many ground states, which
correspond to all possible configurations in which each
FIG. 5: The kagome bilayer structure of SCGO, with each
Kagomelayermadeupofcornersharingtriangles,andjoined simplex (a tetrahedron or a triangle) has zero net spin.
to the other by apical sites. Cr3+ ions with S =3/2 occupy Let us now dope the system with non-magnetic impuri-
thecornersofthesetrianglesaswellastheapicalsites. Other ties. If a simplex has a single non-magnetic site, it can
Cr3+ ions form the spacer layers consisting of isolated spin stillarrangethespinsontheothersitestoadduptozero,
dimers (Figure is taken from the arXiv version of Ref [21])
andthusasinglevacancyonasimplexhasnosignificant
effect on the properties of the system.
What about a correlated defect consisting of two va-
attractionbetweenouremergentmagneticmonopoles,by
cancies on a single simplex? If this simplex is a tetra-
an amount proportional to T that can be calculated pre-
hedron, again, nothing much happens. However, if this
cisely by computer simulations [17, 18, 20]. This is a
simplex is a triangle, then it becomes impossible to sat-
fairlyinnocuouseffect,andisnotexpectedtochangethe
isfy the zero spin constraint on this simplex. However,
basic picture outlined above.
all neighbouring simplices can still satisfy the zero net
Impurity induced half-orphan S =3/4 spins in SCGO:
spin constraint. One therefore expects an infinitesimal
The second of our promised surprises is Henley’s [22]
magnetic field, say h = (cid:15)zˆ to immediately polarize the
identification of the unusual defect induced local mo-
netspinofthetrianglewithtwodefects,whilehavingno
ments that may exist in the pyrochlore slab mag-
effect on any other simplex.
net SCGO. SCGO is an abbreviation for the oxide
What is the total spin of the resulting state? Since
SrCr Ga O , where the listed formula is only notional
9 3 19 each physical spin is shared by two simplices, and only
since this ideal stoichiometric composition can never be
one simplex has non-zero net spin, we may write Sz =
in the laboratory. What is instead commonly prepared (cid:16) (cid:17) tot
is SrCr9pGa12−9pO19, with p ranging all the way from 12 (cid:80) Sz +(cid:80)(cid:52)S(cid:52)z = 3/4! Thus, such a correlated
roughly 0.5 to 0.98 [21]. defect(cid:52)giv(cid:52)es rise to a spin of S = 3/4—these have been
The ideal stoichiometry corresponds to S = 3/2 Cr3+ referedtointheliteratureas‘half-orphans’[22],andcon-
ionsoccupyingthesitesofa‘pyrochloreslab’,inaddition situte the second of our promised surprises.
toformingalayerofdecoupledpairs(Fig5). Thespinsin To complete our discussion, we must also note that
the pyrochlore slab interact with each other through an these half-oprhans come with a statutory warning: As
7
in our earlier example, this is again a purely T = 0 [6] F. Bert et. al., Phys. Rev. Lett. 95, 087203 (2005);
statement relying on minimizing the interaction energy. Y. S. Lee (unpublished).
Again, it is not at all obvious that any of this survives [7] G. Lawes et. al., Phys. Rev. Lett. 93, 247201 (2004);
M. Kenzelmann et. al., Phys. Rev. B 74, 014429 (2006).
the entropic effects of thermal fluctuations at non-zero
[8] I. Hagemann et. al., Phys. Rev. Lett. 86, 894 (2001);
temperature. Indeed, unlike in the previous example in
D.Bono,P.Mendels,G.Collin,andN.Blanchard,Phys.
which entropic effects have been analyzed and are now
Rev. Lett. 92, 217202 (2004).
well-understood, not much is known about the effects [9] K. Matan et. al., Phys. Rev. Lett. 96, 247201 (2006);
of thermal fluctuations on these half-orphans, although T. Yildrim and A. B. Harris, Phys. Rev. B 73, 214446
diluted SCGO has been studied using computer simula- (2006).
tions and phenomenological approaches [23, 24]. It thus [10] B.Nienhuis,H.J.Hilhorst,andH.W.J.Blote,J.Phys.
A: Math. Gen. 17, 3559 (1984).
remains an open question whether these S = 3/4 spins
[11] D. Heidarian and K. Damle, in preparation.
survive the effects of non-zero temperature and can be
[12] C. Castelnovo, R. Moessner, and S. L. Sondhi, Nature
‘seen’ in experiments, and we are working on providing
451, 42 (2008).
some definite answers soon [25]. [13] L. D. C. Jaubert and P. C. W. Holdsworth, Nature
Acknowledgements: The author acknowledges use- Physics (to appear).
ful discussions with D. Dhar, and with his collaborators [14] S. Isakov, R. Moessner, and S. L. Sondhi, Phys. Rev.
A. Banerjee, D. Heidarian, R. Moessner, and A. Sen. Lett. 95, 217201 (2005)
[15] J. F. Nagle, Chem. Phys. 43, 317 (1979).
Funding from DST SR/S2/RJN-25/2006, and the hospi-
[16] A.W.SandvikandR.Moessner,Phys.Rev.B73,144504
tality of School of Physical Sciences, JNU and Physics
(2006).
Department, IIT Bombay during the completion of this
[17] A. Banerjee and K. Damle, in preparation.
contribution is also gratefully acknowledged. [18] D. A. Huse, W. Krauth, R. Moessner, and S. L. Sondhi,
Phys. Rev. Lett. 91, 167004 (2003)
[19] M. Hermele, M. P. A. Fisher, and L. Balents, Phys Rev
B 69, 064404 (2004).
[20] C. L. Henley, Phys. Rev. B 71, 014424 (2005)
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InterScience-Wiley, New York (1963). [22] C. L. Henley, Can. J. Phys. 79, 1307 (2001).
[2] A. P. Ramirez, Annu. Rev. Mater. Sci. 24, 453-480 [23] P.SchifferandI.Daruka,Phys.Rev.B56,13712(1997)
(1994). [24] R. Moessner and A. J. Berlinsky, Phys. Rev. Lett. 83,
[3] R. Moessner, Can. J. Phys. 79, 1283 (2001). 3293 (1999).
[4] L.Pinsard-Gaudard,J.Rodriguez-Carvajal,A.Gukasov, [25] A. Sen, K. Damle, and R. Moessner, in preparation
and P. Monod, Phys. Rev. B 69, 104408 (2004).
[5] H. Ueda et. al., Phys. Rev. Lett. 94, 047202 (2005).
