Single-ion and exchange anisotropy effects and multiferroic behavior in high-symmetry tetramer single molecule magnets

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by  Richard A. Klemm

Single-ion and exchange anisotropy effects and multiferroic behavior in
high-symmetry tetramer single molecule magnets
Richard A. Klemm1,∗ and Dmitri V. Efremov2,†
1Department of Physics, University of Central Florida, Orlando, FL 32816 USA
2Institut fu¨r Theoretische Physik, Technische Universita¨t Dresden, 01062 Dresden, Germany
(Dated: February 2, 2008)
We study single-ion and exchange anisotropy effects in equal-spin s tetramer single molecule
1
magnets exhibiting T , D , D , C , C , or S ionic point group symmetry. We first write the
d 4h 2d 4h 4v 4
group-invariant quadratic single-ion and symmetric anisotropic exchange Hamiltonians in the ap-
8
propriatelocalcoordinates. WethenrewritetheselocalHamiltoniansinthemolecularorlaboratory
0
representation, along with the group-invariant Dzyaloshinskii-Moriya (DM), and isotropic Heisen-
0
berg, biquadratic, and three-center quartic Hamiltonians. Using our exact, compact forms for the
2
single-ionspinmatrixelements,weevaluatetheeigenstateenergiesanalytically tofirstorderinthe

n microscopic anisotropyinteractions, correspondingtothestrongexchangelimit, andprovidetables
a of simple formulas for the energies of the lowest four eigenstate manifolds of ferromagnetic (FM)
J and anitiferromagnetic (AFM) tetramers with arbitrary s . For AFM tetramers, we illustrate the
1
9 first-order level-crossing inductions for s = 1/2,1,3/2, and obtain a preliminary estimate of the
1
1 microscopicparametersinaNi fromafittomagnetization data. Accurateanalyticexpressionsfor
4
the thermodynamics, electron paramagnetic resonance absorption and inelastic neutron scattering

] cross-section are given, allowing for a determination of threeof themicroscopic anisotropy interac-
l
e tionsfrom thesecond excitedstatemanifold ofFMtetramers. Wealso predict thattetramerswith
- symmetries S and D should exhibit both DM interactions and multiferroic states, and illustrate
r 4 2d
t ourpredictions for s1 =1/2,1.
s
.
t PACSnumbers: 75.75.+a,75.50.Xx,73.22.Lp,75.30.Gw,75.10.Jm
a
m
- I. INTRODUCTION The Cu tetramer Cu OCl (TPPO) , where TPPO is
d 4 4 6 4
n triphenylphosphine oxide, has four spin 1/2 ions on
o Single molecule magnets (SMM’s) have been a topic the corners of a regular tetrahedron, with an s = 2
c ground state and approximate T symmetry.[12, 13, 14]
of great interest for more than a decade,[1] because of d
[
their potential uses in quantum computing and/or mag- In this case, there are no single-ion anisotropy effects,

2 neticstorage,[2]whicharepossibleduetomagneticquan- but anisotropic symmetric exchange interactions were
v thought to be responsible for the zero-field energy
tum tunneling (MQT) and entangled states. In fits to a
8
splittings.[12, 15] The Co , Co (hmp) (MeOH) Cl ,
wealth of data, the Hamiltonian within an SMM cluster 4 4 4 4 4
1
where hmp is hydroxymethylpyridyl, and Cr ,
5 was assumed to be the Heisenberg exchange interaction 4
2 plusweakertotal(global,orgiant)spinanisotropyinter- [Cr4S(O2CCH3)8(H2O)4](NO3)2 H2O, compounds
·
2. actions, with a fixed overall total spin quantum number have s = 6 ground states with spin 3/2 ions on the cor-
ners of tetrahedrons.[16, 17] Those compounds have S
1 s.[1] MQT and entanglement were only studied in this 4
7 simple model. and approximate D2d symmetry, respectively.[16, 17] A
0 number of high symmetry s =4 ground state Ni struc-
The simplest SMM clusters are dimers.[3, 4, 5] 4
: tures with spin 1 ions were reported.[18, 19, 20, 21, 22]
v Surprisingly, two antiferromagnetic dimers,
Xi an Fe2, [Fe(salen)Cl]2, where salen is N,N′- Two of these, [Ni(hmp)(ROH)Cl]4, where R is an alkyl
group, such as methyl, ethyl, or 3,3-dimethyl-1-butyl
ethylenebis(salicylideneiminato), and a Ni ,
r 2 and hmp is 2-hydroxymethylpyridyl, form tetramers
a Na Ni (C O ) (H O) , appear to have substantial
2 2 2 4 3 2 2
with precise S group symmetry.[21, 22] Two others,
single-ion anisotropy without any appreciable total 4
Ni (ROH)L , where R is methyl or ethyl and H L
spin anisotropy.[5, 6, 7] The presence of single-ion or 4 4 2
is salicylidene-2-ethanolamine, had approximate S
exchange anisotropy actually precludes the total spin 4
symmetry, although the precise symmetry was only
s from being a good quantum number.[4, 5] Although
C .[18] Several planar Mn compounds with the Mn+3
the most common SMM clusters have ferromagnetic 1 4
spin 2 ions on the corners of squares were made, with
(FM) intramolecular interactions and contain n 8
≥ overall s = 8 tetramer ground states.[23] Although two
magnetic ions,[8, 9] a number of intermediate-sized FM
of these complexes had only approximate S symmetry,
SMM clusters with n = 4 and rather simple molecular 4
one of these complexes, Mn Cl (L’) , where H L’
structures were recently studied. Fits to electron 4 4 4 2
is 4-t-butyl-salicylidene-2-ethanolamine, had perfect
paramagnetic resonance (EPR) Ni data assuming a
4
S symmetry.[23] Inelastic neutron scattering (INS)
fixed s were also problematic, suggesting single-ion or 4
experiments provided strong evidence for single-ion
exchange anisotropy in that tetramer, as well.[10, 11]
2
anisotropy in a Co and a Ni with approximate S
4 4 4
symmetry.[17, 18]
We note that ab initio calculations of the intramolec- 4 1
ular spin-spin interactions in SMM clusters have not yet 1
been always successful in calculating even the strongest, 3
y
intramolecular isotropic Heisenberg interactions accu- a
y a
rately, and have been incapable of calculating any of the z
x
local anisotropic spin-spin interactions within an SMM x
cluster.[24, 25, 26] Even to obtain the Heisenberg in- 4 a
teractions accurately, it seems one needs to extend the a 2 3 a 2
local spin-density approximation (LSDA) to include on-
site repulsions with strength U (the LSDA+U model), FIG. 1: Td (left) and D4h (right) ion sites (filled). Circle:
which would have to be introduced phenomenologically origin. Arrows: local axial zˆnTd (left), azimuthal xˆDn4h (right)
tofitthelowesttwoenergylevelmanifoldsinzeroapplied single-ionvectors. TheaxialvectorszˆnD4h =zˆ,normaltothe
ionic plane.
magnetic field.[26, 27, 28, 29, 30] We therefore define a
microscopicmodelto be amodelconstructedintermsof
the individual spins and from the local interactions be-
tween them, with parameters describing the strengths of
the various types of local spin-spin interactions and in- 1
teractionsbetweenthelocalspinsandthemagneticfield.
Thisisdistinctfromamodelconstructedsolelyfromthe 1 3 1 1 3 1 1
anisotropiesofthetotalspinofanSMMcluster,whichwe y y
z c z c
denote asaphenomenologicalmodel. Ourdefinitionofa
4
microscopicmodelisanalogoustothe standardmodelof x 1 x
a 4 a
the interactions of quarks and gluons within a hadron.
a 2 1 a 2
Recently there have been microscopic treatments of
dimers,[4,5]trimers,andtetramers,includingZeemang-
FIG.2: D (left)andS (right)ionsites(filled). Circle: ori-
tensor anisotropy, single-ion anisotropy, and anisotropic 2d 4
gin. Arrows: local axial single-ion vectors. The g = D ,S
2d 4
exchange interactions.[31] Most of those treatments and axial vectors zˆg make the angles θg with the z axis, and the
1 1
theirrecentextensionstomoregeneralsystemsexpressed S axial vector zˆS4 also makes the angle α with the x axis,
4 1 1
the single-spin matrix elements only in terms of Wigner where cosα =sinθS4cosφS4.
1 1 1
3j, 6j, and 9j symbols.[31, 32] While such treatments
are very helpful in fitting experimental data, more com-
pact analytic forms are desirable to study microscopic discuss the six structures and the generalquadratic spin
models of FM SMM clusters in which the MQT and Hamiltonian. In Sec. III, we write the single-ion and
entanglement issues crucial for quantum computing can symmetric anisotropic exchange Hamiltonians in terms
be understood. We constructed the quadratic single-ion ofthelocalcoordinates,andtheantisymmetricexchange
and anisotropic near-neighbor (NN) and next-nearest- Hamiltonian in the molecular coordinates. In Sec. IV,
neighbor (NNN) exchange SMM cluster Hamiltonians we impose the operations of the six group symmetries,
from the respective local axial and azimuthal vector and discuss the effects of antisymmetric anisotropic ex-
groups for equal-spin tetramer SMM clusters with point changeinteractionsandthe relatedelectric polarizations
group symmetries g = T , D , D , C , C , and S , in lower symmetry systems. In Sec. V, the resulting
d 4h 2d 4h 4v 4
and found compact analytic expressions for the single- group-symmetricHamiltonians arewritten in the molec-
spin matrix elements of four general spins. Each local ular representation, and the isotropic biquadratic ex-
vector group generates site-dependent molecular single- change interactions are introduced. Section VI contains
ionandexchangeanisotropy. WethenshowthatforD2d theeigenstatesofthefullHamiltoniantofirstorderinthe
and S symmetries, the antisymmetric exchange inter- anisotropy and NN biquadratic exchange interactions.
4
actions lead to non-vanishing spin currents that may be These eiqenstates are used to obtain the level-crossing
accompanied by electric polarizations, leading to multi- inductions for AFM tetramers, and particular examples
ferroic effects. We evaluate the magnetization, specific with s = 1/2,1,3/2 are presented. In Sec. VI, we
1
heat, EPR and INS transitions in the Hartree approxi- also evaluate quantitatively some effects of antisymmet-
mation, and provide a procedure for extracting three of ric anisotropic exchange and provide our related predic-
theeffectivesite-independentmicroscopicparametersus- tions for multiferroic behavior. In Sec. VII, the self-
ing EPR. We also show analytically how to include the consistent Hartree approximation (or strong-exchange
effects of weak biquadratic exchange. limit) is used to provide simple but accurate results for
An outline of the paper is as follows. In Sec. II, we the thermodynamics, EPRresonantinductions, andINS
3
g = D or S , c/a > 1, approximately as in a Co ,[17]
2d 4 4
or c/a < 1, as in some Mn and a Ni .[19, 23] For com-
1 1 1 1’ 1 1’ parison with the planar sy4mmetries g4 = C4h,D4h, and
4 D , we assume for g = D ,S that c/a < 1, so that
4 4v 2d 4
there are four NN sites and two NNN sites. For each g,
y y xˆ,yˆ,zˆ are the molecular (or laboratory)unit coordinate
a a
axis vectors.
x x
The most general Hamiltonian quadratic in the four
3 2 3 2 spin operators Sn may be written for group g as
1 a 1 1’ a 1’
4 4
FIG. 3: C (left) and C (right) ion sites (filled). Circle: Hg =−µB B·↔ggn·Sn+ Sn·D↔gn,n′ ·Sn′, (2)
origin. Arr4ohws: local azim4uvthalxˆC4h (left)axialzˆC4v (right) nX=1 n,Xn′=1
n n
single-ionvectors. TheaxialvectorszˆnC4h =zˆ. TheC4v axial where µB is the Bohr magneton and B =
vectors each make the angle θ1 = π/2−θ1′ with the z axis. B(sinθcosφ,sinθsinφ,cosθ) is the magnetic induc-
Thedottedarrows(equivalenttotheD azimuthalsingle-ion
4h tion at an arbitrary direction (θ,φ) relative to the
vectors xˆD4h) are their projections in the xy plane.
n molecular (or cluster) coordinates (xˆ,yˆ,zˆ).[31, 37]
For simplicity, we take ↔gg to be diagonal, isotropic,
n
cross-sections,anddescribehowEPRexperimentsinthe and site-independent, so that the Zeeman interaction
excited states of FM tetramers can provide a measure of may be written in terms of a single gyromagnetic ra-
someofthemicroscopicanisotropyinterationsstrengths. tio γ 2µB. Thus in the following, g only refers to
Finally,inSec. VIII,wediscussthesignificanceofourre- the mo≈lecular group. We separate D↔gn,n′ into its sym-
sults,andprovideapreliminaryfittomagnetizationdata metric andantisymmetric parts,D↔g =D↔g,s +D↔g,a ,
n,n′ n,n′ n,n′
onanAFMNi4 tetramer,andinSec. IX,wepresentour respectively. For n′ = n, the single-ion D↔g is neces-
conclusions. n,n
↔
sarily symmetric, so Dg,a = 0. For each g, the four
n,n
↔
Dg,s contain the local single-ion structural information,
II. STRUCTURES AND BARE HAMILTONIAN n,n ↔
and the six distinct symmetric Dg,s contain the local
n,n′
symmetric exchange structural information, which lead
For SMM clusters with ionic site point groups g =
to the isotropic, or Heisenberg, exchange interactions,
T ,D , we assume the four equal-spin s ions sit on
d 4h 1 and the remaining symmetric anisotropic exchange in-
opposite corners of a cube or square of side a centered ↔
teractions. The six distinct antisymmetric Dg,a con-
at the origin, as pictured in Fig. 1. For clusters with n,n′
tain additional local structural information which lead
g = D ,S , we take the ions to sit on opposite corners
2d 4
to the Dzyaloshinskii-Moriya (DM) interactions.[38, 39]
of a tetragonal prism with sides (a,a,c) centered at the
Physically, the symmetric anisotropic exchange interac-
origin,as in Fig. 2. The ions for g =C ,C also sit on
4h 4v
tions also contain the intramolecular dipole-dipole inter-
the corners of a square of side a centered at the origin,
actions, which can be even larger in magnitude than the
as pictured in Fig. 3, but the ligand groups have differ-
terms originating from actual anisotropic exchange.[37,
ent symmetries than for the simpler D case pictured
4h
40]
in Fig. 1.[33] In each case, we take the origin to be at
the geometric center, so that 4 r = 0, where the As is well known, each of the symmetric rank-three
n=1 n tensors(or matrices)D↔g canbe diagonalizedby three
relative ion site vectors are P n,n′
rotations: arotationbythe angleφg aboutthe molec-
a (2n 1)π (2n 1)π n,n′
rn = √2hsin(cid:16) −4 (cid:17)xˆ+cos(cid:16) −4 (cid:17)yˆi urolatartzedaxx˜isa,xtihs,enfoallorwoetadtiboynabyrotthaetioanngblye θtnhg,en′anabgloeuψtgthe
c( 1)nzˆ. (1) about the rotated z˜ axis.[41] This necessarily leadsn,tno′
−2 −
Tetrahedrons with g = T , c/a = 1, approximately as the three principal axes xˆ˜gn,n′, yˆ˜gn,n′, and zˆ˜gn,n′. For the
d single-ion axes with n′ = n, we denote these principal
in Cu4,[12] are a four-spin example of the equivalent- axes to be xˆ˜g, yˆ˜g, and zˆ˜g, respectively, which are writ-
neighbor model.[34] In squares with g = D , C , or n n n
4h 4h ten explicitly in Sec. III. The non-vanishing matrix ele-
C ,c=0. ThehighD symmetryisapproximatelyex-
4v 4h mentsintheselocally-diagonalizedsymmetricmatrixco-
hibitedbythesquareNd4 compound,Nd4(OR)12,where ordinates are D˜g,s , D˜g,s and D˜g,s . Since the
R is 2,2-dimethyl-1-propyl,in which the Nd+3 ions have n,n′,xx n,n′,yy n,n′,zz
↔
equal total angular momentum j =9/2.[35, 36] We note structuralinformationineachoftheD˜g,s dependsupon
n,n′
thattheMn clusterswithapproximateorexactS sym- the localenvironment,inthe absenceofmoleculargroup
4 4
metry also have c = 0.[23] In tetragonal prisms with g symmetry, each of these angles would in principle be
4
different from one another. basiselements arethe vectorsthat diagonalizethe single
AlthoughanantisymmetricexchangematrixD↔gn,,an′ can ionmatrixfromD↔g toD↔˜g .[41]Sinceweemploythese
n,n n,n
generally be diagonalized by a unitary transformation,
vectors repeatedly, we write them here for simplicity of
it contains at most three independent, real parameters,
presentation. The diagonalized vector set elements may
which can be incorporated into the components of a
be written in the molecular (xˆ,yˆ,zˆ) representation as
three-vector, dg , with an effective spin-spin interac-
n,n′
ttoionwroiftethienftohremmdognl,enc′u·l(aSrnr×epSrens′e),n[t3a8t,io3n9].whichiseasiest xˆ˜g =  scionsφφggnccoossψψgng+−ccoossθθgngcsoinsφφggnssiinnψψngg , (7)
n n n n n n
For the six high-symmetrygroupsunder study, we an- sinθgsinψg
alyze the effects of molecular group symmetry upon the  n n 
single-ion and anisotropic exchange parts of Hg. The yˆ˜g =  −csionsφφggnssiinnψψgng+−ccoossθθgngcsoinsφφgng ccoossψψngg ,(8)
group symmetries further restrict the number of inde- n − n n n n n
sinθgcosψg
pendent parameters.  n n 
Inthe absenceofanyanisotropyinteractions,the bare sinθgsinφg
Hamiltonian g is given by the Zeeman and Heisenberg zˆ˜g =  sinθngcosφng , (9)
H0 n − n n
interactions, cosθg
 n 
H0g = −JγgB(S·1S−S2J+g′(SS12·SS33++SS23·SS44)+S4 S1),(3) wgehniecrhalsaqtuisafdyraxˆ˜tgnic×sinyˆ˜ggnle=-ionzˆ˜gna.nisWoterotphyeninwterriatectitohneamsost
− · · · ·
which can be rewritten as
4
g,ℓ = Jg (S zˆ˜g)2
g = JgS2 γB S (Jg′ −Jg)(S2 +S2 ),(4) Hsi −nX=1(cid:16) a,n n· n
H0 − 2 − · − 2 13 24
+Jg [(S xˆ˜g)2 (S yˆ˜g)2] , (10)
whereS =S +S ,S =S +S ,andS =S +S is e,n n· n − n· n (cid:17)
13 1 3 24 2 4 13 24
thetotalspinoperator,[36]andwedroppedanirrelevant,
in terms of the site-dependent axial and azimuthal in-
overallconstant. In Eq. (4),
teractions Jg ,Jg , analogous in notation to that for
a,n e,n
J′ = J , (5) homoionic dimers.[4, 5] In terms of the diagonalizedma-
Td Td trix elements, Jg = D˜g,s (D˜g,s +D˜g,s )/2
J′ = J (6) − a,n n,n,zz − n,n,xx n,n,yy
g 6 g and Jg =(D˜g,s D˜g,s )/2.
− e,n n,n,xx− n,n,yy
for g =D ,S ,D ,C , and C . In terms ofthe diag-
2d 4 4h 4h 4v
onalized matrix elements, 2J = D˜g,s +D˜g,s and
− g 1,2,xx 1,2,yy
2J′ = D˜g,s +D˜g,s , for instance. For our c/a < 1 B. Local symmetric anisotropic exchange
−convgention1,,3,xJx and1,3,yJy′ are the NN and NNN Heisen- Hamiltonian
− g − g
berg interactions for g =D ,S , C ,C and D .
2d 4 4v 4h 4h
In addition to the single-ion interactions, the other
microscopic anisotropic interactions are the anisotropic
III. THE SINGLE-ION AND ANISOTROPIC
exchange interactions, which include the intracluster
EXCHANGE HAMILTONIANS
dipole-dipole interactions.[40] The intercluster dipole-
dipole interactions can lead to low-T hysteresis in the
To take account of the molecular group g symme- phenomenologicaltotalspinmodel,[42]butinthe micro-
tries, it is useful to write the single-ion and symmetric scopic individual spin model, are generally much weaker
anisotropic exchange interactions in terms of the local than the intracluster ones due to the largerdistances in-
coordinates. In this section, we write the local Hamil- volved. Hence,weneglectthoseandallotherintercluster
tonian for these interactions, and the molecular Hamil- interactions, such as those mediated by phonons. As for
tonian for the antisymmetric exchange interactions. In the single-ion interactions, we first construct the sym-
Sec. IV, we then impose the group symmetries on these metric anisotropic exchange Hamiltonian g in the lo-
interactions for C4h,D4h,C4v,S4,D2d, and Td molecular cal group coordinates. In this case, therHeaaere distinct
group symmetries, respectively. local vector sets for the NN and NNN exchange inter-
actions. Diagonalization of the symmetric anisotropic
↔ ↔
A. Local single-ion Hamiltonian exchangematrixDgn,,sn′ leadstoD˜gn,,sn′ andthevectorba-
sis {xˆ˜gn,n′,yˆ˜gn,n′,zˆ˜gn,n′}, given by Eqs. (7)-(9) with the
Forthesingle-ionanisotropy,wedefinethelocalvector subscript n replaced by n,n′.
basisforthenthsitetobe xˆ˜g,yˆ˜g,zˆ˜g foreachg. These ThelocalsymmetricanisotropicexchangeHamiltonian
{ n n n}
5
Hag,eℓ is then generally given by g θ1g φg1 ψ1g
g,ℓ = 2 6−2q Jf,g (S zˆ˜g )(S zˆ˜g ) C4h 0 φg1 0
Hae −Xq=1 nX=1h n,n+q n· n,n+q n+q · n,n+q D4h 0 π4 0
+Jc,g (S xˆ˜g )(S xˆ˜g ) C4v,D2d θ1g 34π −π2
n,n+q(cid:16) n· n,n+q n+q· n,n+q S θS4 φS4 ψS4
(S yˆ˜g )(S yˆ˜g ) , (11) 4 1 1 1
− n· n,n+q n+q· n,n+q (cid:17)i Td tan−1√2 32π 0
wherewedefineS5 S1,asifthefourNNspinswereon TABLE I: Lists of the single-ion parameter sets µg.
≡ 1
a ring. In Eq. (11), the axial and azimuthal interaction
strengths Jf,g =D˜g,s (D˜g,s +D˜g,s )/2and
Jc,g =(−D˜gn,s,n′ Dn˜,ng,′s,zz−)/2,n,ans′,fxoxrthens,nin′,gylye-ionin- These symmetry operations are represented by the ma-
− n,n′ n,n′,xx− n,n′,yy trices for λ = 1,...,26 listed in Subsection A of the
teraction strengths. The subscripts a,e and superscripts Oλ
Appendix. For each g, we require g to be invariant
f,c correspond to our dimer notation.[5] under each symmetry operation g fHor each allowed λ.
For the six g cases under studyO,λthe set g of group
{Oλ}
operations greatly reduces the number of single-ion and
C. Antisymmetric anisotropic exchange Hamiltonian
symmetricanisotropicexchangeparameters. Aswe shall
see,ineachgroupg,thesereducethesingle-ionandsym-
As noted above, we write the antisymmetric
metric anisotropic exchange interaction strength set to
anisotropic exchange, or Dzyaloshinskii-Moriya
(DM),[38, 39] Hamiltonian g in the molecular Jg Jg,Jg,Jg ,Jg , (13)
HDM { j} ≡ { a e f,q c,q}
representation,[37]
for q = 1,2, which are independent of the site index n.
That is, for each g, there are at most two single-ion,
2 6−2q
g = dg S S . (12) two NN and two NNN symmetric anisotropic exchange
HDM Xq=1 nX=1 n,n+q·(cid:16) n× n+q(cid:17) interaction strengths. In addition, for these six g cases,
the group operations further limit the number of vector
We note that in these molecular coordinates, the DM set parameters to
interaction three-vectors dg depend explicitly upon
n,n+q µg = θg,φg,ψg , (14)
the exchange bond indices n,n+q for each group g. We 1 { 1 1 1}
then employ the local group symmetries to relate them µg1q = {θ1gp,φg1p,ψ1gp}, (15)
to one another. where p = q+1 = 2,3, and we used the notation θg =
The rules for the directions of the dgn,n+q were given θg , θg = θg , etc. Some of these parameters may1be
1,1 1p 1,p
by Moriya,[38] and were employed for a dimer example
further restricted. In addition, however, the molecular
by Bencini and Gatteschi.[37] The Moriya rules are: (1)
single-ionandanisotropicexchangeHamiltonianscontain
dg vanishes if a center of inversion connects r and
n,n′ n both site-independent and site-dependent terms.
risnn′.o(r2m)aWl thoetnhae mmiirrrroorr ppllaannee.co(3n)taWinhsernnaanmdirrrno′r,pdlgna,nne′ anFisoortrHopDgicMe,xwcheafinrgsetpimairp,o[3s7e,t3h8e]Manodriytahernuliemspoonseeathche
is the perpendicular bisectorofrn−rn′, dgn,n′ lies inthe required group symmetries on the six pairs. For the
mirror plane. (4) When a two-fold rotation axis is the six groups under study, the group symmetries place re-
perpendicular bisector of rn−rn′, then dgn,n′ is orthog- strictionsuponthedgn,n+q,leadingtothe anisotropicex-
onal to the rotation axis. (5) When rn rn′ is an r-fold change parameter set
rAostantoiotnedaxaibsowviet,hwre>sh2a,ltlhienncodrgnp,onr′aitsepta−hraeslleelrtuolersni−nrtnh′e. dg = {dgz,dgx1,dgy1,dgx2,dgy2}. (16)
molecular representation. For example, in NaV O , the
2 5 For each of the six g cases, the NNN DM parameter set
lackofinversionsymmetrybetweeninteractingspinshas
has at least one more restriction than does the NN DM
been shown to lead to a DM interaction.[43]
parameterset. Somegsymmetriesleadtosite-dependent
signs of the components of dg.
IV. GROUP SYMMETRY INVARIANCE
C. Imposing the group symmetries
A. General considerations
In Subsection A ofthe Appendix, we describe the ma-
In this section, we impose the set of allowed group trices for λ = 1,...,26 representing the group sym-
λ
O
g symmetry operations upon the full Hamiltonian g. metry operations for g =C ,D ,C ,S ,D , and T .
4h 4h 4v 4 2d d
H
6
g θg φg ψg Then, imposing symmetry, we have S =S , and
1p 1p 1p either θC4h = πO/26or θC4h = 0, both oOf6whnich lenad to
1 1
C4h 0 φC1p4h 0 invariance of this part of the Hamiltonian under 6 We
D 0 π 0 therefore take the easy-axis case, θC4h = 0. CaOrrying
4h 4 1
C4v 0 (p−42)π 0 out similar transformations on the azimuthal single-ion
Hamiltonian leads to
S θS4 φS4 ψS4
4 1p 1p 1p
D2d θ1D22dδp,2+ π2δp,3 (2−(p1−)p1π) (p−22)π JeC,n4h = JeC4h, π (23)
χC4h = φC4h +ψC4h =χC4h + . (24)
TABLEII:ListsoftherelevantNN(p=2)andNNN(p=3) n n n n+1 2
parameter sets µg .
1p We could then choose ψC4h = 0, leaving one free angle
1
parameter φC4h, as listed in Table I, plus the two inter-
g dg dg dg dg dg 1
z x1 y1 x2 y2 action strengths JC4h,JC4h.
a e
C ,D dg 0 0 0 0 The symmetric anisotropic exchange Hamiltonian,
4hS4 4h dSzz4 dSx14 dSy14 dxS24 dSy24 HsimaCe4ihla,ℓrcfaasnhbioenm. TadheeionpvaerraiatniotnusnfdoerrtOhe1,oOth2e,rOfi6viengasvyemry-
D dD2d 0 dD2d dD2d 0 metriesarelistedinSubsectionAoftheAppendix,along
2d z y1 x2
with the associated matrices. Our results for the single-
Td,C4v 0 0 0 0 0 ion,symmetricanisotropicexchange,andDMinteraction
parameters are compiled in Tables I-III.
TABLE III:Lists of the DMparameter sets dg.
D. Induced electric polarizations
For eachmolecular groupg,the allowedsymmetry oper-
ations commute with the Hamiltonian. For a partic-
λ
ular λ,O λrn = rn′(λ). We therefore take Sn = S(rn), As shown by Katsura et al.,[44] the spin-orbit inter-
so that OλSn =S( λrn)=S(rn′(λ))=Sn′(λ). actions between spins at sites n and n′ can induce an
For CO symmetOry, besides the trivial identity oper- electric polarization
4h
ation, the allowed group operations are clockwise and
counterclockwise rotations by π/2 about the z axis, and Pn,n′ ∼rˆn,n′ ×(Sn×Sn′), (25)
reflections in the xy plane.[33] These operations are rep- where ˆrn,n′ is a unit vector directed from site n to site
resentedrespectivelyby the matrices 1,2,6. We use this n′. In our model, the thermodynamic averages of such
O
simplecasetoillustratehowthesymmetriesareimposed.
polarizations vanish in the absence of DM interactions,
We first consider the axial part of HsCi4h,ℓ, and set but non-vanishing in-plane vector components dgq of the
dg DM interaction parameter sets allow them to become
4 4
JC4h(S zˆ˜C4h)2 = JC4h (S ˆz˜C4h)2 T. finite. Tetramers with the rather low molecular group
nX=1 a,n n· n nX=1 a,n O1 n· n O1 symmetriesS4andD2dhavenooverallcenterofinversion
(17) symmetry,andcontainacomplexsetofDMinteractions.
Dependinguponthepolarizabilityoftheattachedligand
Weinterpretzˆ˜C4h asitsvectortranspose,(zˆ˜C4h)T inEq. groups,thismayleadtoacombinedspin-inducedelectric
n n
(17), and obtain polarization
1Sn = Sn+1, (18) 1 4
(zˆ˜Cn4hO)TO1T = (cid:16)−sinθnC4hcosφCn4h, Ps ∝ 2n,Xn′=1rˆn,n′ ×hSn×Sn′i, (26)
sinθC4hsinφC4h,cosθC4h . where ... representsthethermodynamicaverageinthe
− n n n (cid:17) h i
presence of the full Hamiltonian, including the relevant
(19)
DM interactions.
Substituting these into the right-hand side of Eq. (17), BesidesthedirectDMinteractions,wepredictthepos-
setting n n + 1 in the left-hand side, and equating sibility of dual, or induced, DM interactions. Although
→
coefficients of S S for α,β =x,y,z leads to dg DM interactions between individual spin pairs are al-
n+1,α n+1,β q
lowed in tetramers with the lowest C and D sym-
4h 4h
JaC,4nh = JaC,4nh+1 =JaC4h, (20) metries studied, the group symmetry causes the dipole
θC4h = θC4h =θC4h, (21) moments on opposite sides of their square geometries to
n n+1 1
π cancel one another. Although we have not studied this
φCn4h = φCn+4h1+ 2. (22) point in detail, tetramers with these symmetries can in
7
principle be made to exhibit additional effective DM in- where
tTehraucst,ioinnstbetyraamppelriscawtiiothnoSfa,nDelec,troirclfioewlderEsy6=m0m.[e4t4r,y4,5a] J˜g = Jg+δJg, (30)
4 2d
multiferroic effect can occur,[45] in which both DM in- J˜g′ = Jg′ +δJg′, (31)
teractions and Ps 6= 0. More generally, multiferroic ef- where Jg,Jg′ are given by Eqs. (5) and (6), and the δJg
fects arise in systems such as some dimers, trimers, and andδJ′ aregiveninintermsoftheparameterssetsµg in
tetramersthatgenerallydonothaveacenterofinversion g 1p
SubsectionCoftheAppendix. Forthethreeplanarsym-
at the midpoints of the rn,n′.[5] metries, g = C ,D , and C , δJ = δJ′ = 0. For T
4h 4h 4v g g d
symmetry, there are no group-satisfying azimuthal sym-
metric exchange vectors, so JTd = 0 for q = 1,2. How-
V. THE HAMILTONIAN IN THE MOLECULAR c,q
REPRESENTATION ever, the axial vectors parallel to rn,n′ satisfy all of the
group symmetries, so that the JTd could exist, provided
f,q
A. The molecular single-ion Hamiltonian that JfT,d1 = JfT,d2. However, the requirement J˜Td = J˜T′d
topreservetheT symmetryofthe renormalizedHeisen-
d
berg interactions forces JTd = JTd/2. Hence, we must
To make contact with experiment, we use the group f,2 f,1
symmetriestorewriteHsgi,ℓ inthemolecular(xˆ,yˆ,zˆ)rep- conclude that JfT,dq =0 for q =1,2 and δJTd =δJT′d =0.
resentation, g,ℓ also leads to additional interactions δ g in the
Hae Hae
molecular frame,
g = JgS2 +( 1)nJg (S2 S2 )
Hsi −Xn (cid:16) z n,z − xy n,x− n,y δ g = 2 6−2q Jg S S
Hae (cid:20) q,z n,z n+q,z
+ Kg (n)S S , (27) Xq=1 nX=1
αβ n,α n,β
αX6=β (cid:17)
+( 1)n+1 Jg
− (cid:18) q,xy
where α,β = x,y,z, and we subtracted an irrelevant
constant. g contains the site-independent interactions S S S S
Jg and thHesisite-dependent interactions ( 1)nJg and ×h n,x n+q,x− n,y n+q,yi
Kzg (n),whicharewrittenintermsofthep−aramexteyrsets + Kg (n)S S , (32)
αβ q,αβ n,α n+q,β(cid:19)(cid:21)
µg in Subsection B of the Appendix. Most important is αX6=β
1
the result that for Td symmetry, where α,β =x,y,z. As for the single-ion interactions in
the molecular representation, the site-independent sym-
JzTd = 0. (28) metricexchangeinteractionsJg contributetotheeigen-
q,z
state energies to first order, but the first-order contribu-
The first-order contributions to the eigenstate ener-
tions to the eigenstate energies from the site-dependent
gies from the site-dependent interactions ( 1)nJg and
Kg (n)vanish. Hence,theseinteractionson−lycontxryibute interactions vanish. Both the site-independent and site-
αβ dependent symmetric exchange interactions are given in
to the eigenstate energies to second and higher orders in
terms of the parameter sets µg in Subsection C of the
the interactions Jg and Jg. For C ,D and S , the ef- 1p
a e 4v 2d 4 Appendix.
fective axial site-independent interactions Jg arise from
z
a combination of the local axial and azimuthal interac-
tions Jag andJeg. For g 6=Td, Jzg canbe large,evenif the C. Antisymmetric exchange Hamiltonian
molecular structure is nearly T .
d
In Secs. IV and V, we already evaluated the anti-
symmetric exchange Hamiltonians in the molecular rep-
B. Symmetric anisotropic exchange in the molecular
resentation, and the parameter sets are listed in Table
representation
III. These six g may be combined as
HDM
We then construct the group-invariant symmetric 2 6−2q
g = S S dg(n)δ zˆ
aonrdisiontartoepsicfoerxtchheansgixe Hgasmymiltmoneitarniesi.n tFhoer mgo=lecDular,Sco-, HDM Xq=1 nX=1(cid:0) n× n+q(cid:1)·(cid:16) z q,1
2d 4
there are renormalizations of the isotropic exchange in- +dgsin(nπ/2)+(zˆ dg)cos(nπ/2) ,
teractions, modifying g to q × q (cid:17)
H0 (33)
g,r = J˜gS2 γB S (J˜g′ −J˜g)(S2 +S2 ), where the scalar dgz(n) and the two two-vectors dgq all
H0 − 2 − · − 2 13 24 vanishforg =C4v,Td,butforthe other foursymmetries
(29) are given in Subsection C of the Appendix.
8
Wenotethatbothsite-dependentandsite-independent the six g symmetries may be written as
DM interactions give rise to second-order eigenstate en-
2
ergy corrections, and can only be neglected in fits to
g = g (36)
experiment for tetramers with symmetries very close to Ht Ht,q
Xq=1
T ,C , or higher. In Subsection D of the Appendix, we
d 4v
give g for the lower symmetry C13 tetramers. 4 4
HDM 2v Htg,1 = −Jtg,1 Sn·Sn+1 Sn′ ·Sn′+1 ,(37)
Xn=1 nX′=1(cid:0) (cid:1)(cid:0) (cid:1)
odd even
4 2
D. Biquadratic and three-center quartic isotropic Htg,2 = −Jtg,2 Sn·Sn+1 Sn′ ·Sn′+2 .(38)
exchange interactions nX=1nX′=1(cid:0) (cid:1)(cid:0) (cid:1)
In the previous subsections, we listed the quadratic VI. EIGENSTATES OF THE FULL
single-ion and anisotropic exchange interactions for the HAMILTONIAN
six high-symmetry tetramer groups under study. How-
ever,in the lower-symmetryAFM tetramers Ni4Mo12 , A. Induction representation
{ }
with C symmetry,[46] and Ni and Co [2 2] grids (or
1v 4 4
×
rhombuses),withapproximateC13 symmetry,[47,48,49]
2v We assume a molecular Hamiltonian
fits to magnetization data were facilitated by the inclu-
sion of biquadratic interactions.[46, 47, 48] In the for- g = g,r+ g +δ g + g + g. (39)
H H0 Hsi Hae HDM Hb
mer case, the powder fits assumed field-dependent inter-
action parameters, but the single-ion interactions were To take proper account of B in g,r, we construct our
H0
assumed to have T symmetry, which vanish to first or- SMM eigenstates in the induction representation by
d
derintheirstrength,andtheanisotropicexchangeinter-
actions were neglected. In the latter C13 case, the au- xˆ cosθcosφ sinφ sinθcosφ xˆ′
2v −
thorsneglectedtheNNDMinteractionsgivenintheAp-  yˆ = cosθsinφ cosφ sinθsinφ  yˆ′ ,
pendix. Subsequently, Kostyuchenko showed that three-  zˆ   sinθ 0 cosθ  zˆ′ 
   −  
center isotropic quartic interactions should be compa-
(40)
rable in magnitude to the biquadratic interactions, and
providedafitto the midpointsofthe level-crossingmag- sothatB =Bzˆ′. Asubsequentarbitraryrotationabout
netization behavior on Ni4Mo12 , without making the zˆ′ does not affect the eigenstates.[5] We then set ¯h = 1
{ }
assumption of strong field dependence to the Heisen- and write
berg interactions.[50] Here we provide preliminary fits
to the AFM Ni Mo magnetization data, extending S2 ψs13,s24 = s(s+1)ψs13,s24 , (41)
{ 4 12} | s,m i | s,m i
the treatment of Kostyuchenko to include the first or- S2 ψs13,s24 = s (s +1)ψs13,s24 , (42)
der anisotropy interactions, which can fit the widths of 13| s,m i 13 13 | s,m i
the transitions, as well as the midpoints. More com- S224|ψss,1m3,s24i = s24(s24+1)|ψss,1m3,s24i, (43)
pletefitstothoseexperimentsandtoexperimentsonthe S ψs13,s24 = mψs13,s24 , (44)
z˜| s,m i | s,m i
gridSMM’s willbe presentedelsewhere.[51]Suchfitsare S ψs13,s24 = Aσ˜m ψs13,s24 , (45)
greatlyaidedbyananalytictreatmentofbiquadraticand σ˜| s,m i s | s,m+σ˜i
three-center isotropic quartic exchange. Am = (s m)(s+m+1), (46)
s −
p
For tetramers with the six g symmetries under study,
whereS =S +iσ˜S withσ˜ = . Forbrevity,wedefine
the biquadratic interactions may be written as σ˜ x˜ y˜ ±
ν s,m,s ,s ,s , (47)
13 24 1
≡ { }
2
ν s,s ,s ,s , (48)
g = g (34) ≡ { 13 24 1}
Hb Hb,q
Xq=1 where ν excludes m, and write ν ψs13,s24 . g
| i ≡ | s,m i Hb,2
6−2q and g,r are both diagonal in this representation, but
Hbg,q = −Jbg,q nX=1(cid:0)Sn·Sn+q(cid:1)2. (35) tHhsgeii,rδHHin0agtee,raacntidonHsbgt,1renargethnsott,oabnedswmealtlh,erreelfaotrieveastsoumJ˜e
g
and J˜′. From Eqs. (41) to (44), ν′ g,r + g ν =
For g = T , we take JTd = JTd, but otherwise Jg = g h |H0 Hb,2| i
Jg . For tdhe six g symmb,1etries,b,2g is invariant undbe,r2a6ll Eνg,0δν′,ν, where
b,1 Hb
of the appropriate symmetries. J˜
The three-center quartic interactions for systems with Eνg,0 = − 2gs(s+1)−γBm+δEνg,0 (49)
9
2
1
δEg = (J˜′ J˜)s (s +1)
ν,0 −2nX=1h g − g n,n+2 n,n+2 2.0
Jg AFM DM
+ b2,2[−2s1(s1+1)+sn,n+2(sn,n+2+1)]2 s1 = 1/2
Jg 1.5 S4
+ t,2[ 2s (s +1)+s (s +1)]
2 − 1 1 n,n+2 n,n+2 B || (111)
2 M/1.0 d/|Jg|=0.1
s(s+1) sn′,n′+2(sn′,n′+2+1) ,
×(cid:16) −nX′=1 (cid:17)i
(50) 0.5 J’g/Jg=0.5 J’g/Jg=1.5
Type I
where the J˜g and J˜g′ are given by Eqs. (5), (6), and Type II
(129)-(132). Since g,r and g are invariant under all 0.0
rotations, Eg is inHde0pendenHt bo,f2θ,φ. 0 1 2
ν,0 B/|J |
g
B. First-order eigenstates FIG.4: PlotsofthemagnetizationM/γ versusγB/J˜g ofan
s = 1/2 tetramer at T = 0 with c= a, g = S , dg|=|dg =
1 4 z 1y
gInthaesind′,gu+ction′,gr+epre′,sgen.tatigon,iwsaewscrailtaerHinsgdi+epδeHndageen+t dang2xd=dadsg2hyed=cdu,rdveg1xsa=re0,fodr/t|Jh˜ge|T=yp0e.1I,(aJn˜g′d/JB˜g||=(1111.5)).aTnhdeTsoylpide
oHfDtMhe diHrescitionHoafe BH. DWMe tHhbe,n1 make a standard per- II (J˜g′/J˜g =0.5) tetramers, respectively.
turbationexpansionfor the sevenremaining microscopic
anisotropy energies Jg Jg,Jg,Jg ,Jg ,Jg for
{ j} ≡ { a e f,q c,q b,1} For all six g symmetries, Eg has a form analogous
q = 1,2 small relative to J˜ , J˜′ .[5] To do so, it is nec- ν,1
| g| | g| to that of the equal-spin dimer in the absence of az-
essaryto evaluate the single-ionmatrix elements analyt-
imuthal single-ion and symmetric anisotropic exchange
ically, as they contain much of the interesting physics.
interactions.[5] For these high-symmetry tetramers, to
Compact expressions for these matrix elements for gen-
first order in the anisotropy interactions, the azimuthal
eral (s ,s ,s ,s ) are given in Subsection E of the Ap-
1 2 3 4 single-ion and anisotropic exchange interactions merely
pendix.
renormalize the respective effective site-independent ax-
AtarbitraryB angles(θ,φ),thefirstordercorrections
ial interactions. Thus, to first order, we only have two
Eνg,1 = hν|Hs′,ig + Ha′,eg +HD′,gM + Hbg,1 +Htg,1|νi to the effectiveisotropicexchange,twobiquadraticisotropicex-
eigenstate energies for g = C4h, D4h, C4v, S4, D2d, and change, and three effective anisotropy interactions, J˜g,
Td symmetries are J˜′, Jg , Jg , Jg, Jg , and Jg , which are fixed for a
g b,1 b,2 z 1,z 2,z
particular SMM. Nevertheless, the first-order eigenstate
Eνg,1 = J˜zg2,ν[m2−s(s+1)]−δJ˜zg,ν ecnonertgaiienstEheνgse=seEvνeg,n0e+ffeEcνgt,i1v,egiinvteenrabcytioEnqsst.re(n49g)thasnidnw(5a1y)s,
[3m2−s(s+1)]J˜g,νcos2θ, (51) that depend strongly upon the quantum number set ν
− 2 z and upon θ. These different ν,θ dependencies can be
J˜g,ν = Jga+ Jg c− 1Jg a−, (52) employedto provide definitive measures of at least some
z z ν − 1,z ν − 2 2,z ν of the seven ν-independent effective isotropic exchange
δJ˜g,ν = Jgb+ 1Jg (b++b−) 1Jg b− and anisotropy interactions.
z z ν − 4 1,z ν ν − 4 2,z ν
+Jg +Jg , (53)
b,1Bν t,1Tν
C. Type I and Type II tetramers
where analyticexpressionsfor the a±, b±, c±, and
ν ν ν Bν Tν
for general ν are given in Subsection F of the Appendix, There are at least two types of FM and AFM
along with Tables IV-VII and VIII-XI of their simple tetramers. To the extent that single-ion, symmetric
analytic forms for the lowest four eigenstate manifolds anisotropic exchange, and biquadratic exchange interac-
of FM and AFM tetramers, respectively. We note that tions are small relative to the Heisenberg interactions,
all of these interaction coefficients are invariant under there are just two types of tetramers. The criterion is
s s , as expected. The DM and all site-dependent simply based upon J˜′ J˜ in δEg , which we assume
13 ↔ 24 g − g ν,0
interactions vanish in this first-order perturbation. Sec- to be larger in magnitude than all anisotropy and bi-
ond order corrections to the eigenstate energies will be quadratic interaction strengths. Type I tetramers have
presented elsewhere.[51] J˜′ J˜ > 0, which can occur for either sign of J˜, pro-
g − g g
10
)
s
4
t 0.6
i
n AFM DM AFM DM
u
y s1 = 1/2 3 s1 = 1 Jg’/Jg=0.5
ar 0.4 S4 S4 Type I
tr B || (111) B||(111) Jg’/Jg=1.5
i
b d/|Jg|=0.1 M/2 Type II
r
a J’g/Jg=0.5
( 0.2
| J’g/Jg=1.5
Ps Type I 1 d/|Jg|=0.01
Type II
|
0.0
0 1 2 0
B/|J |
g 0 1 2 3 4 5
B/|J |
g
FIG. 5: Plots of the magnitude of the spin-derived polariza-
tion P in arbitrary units versus γB/J˜ at T = 0 of an
s g
dsd1g2ays=h=e|d1d/,c2|udrtg1vexetrs=aam0re,erdfo/wr|Ji˜ttghh|e=cT=0y.p1ae,,aIgn(dJ=˜g′B/SJ|˜4|g(,|1=d11gz|1)..=5)Tdhag1neyds=oTliyddpg2xaen=IdI Fsd1g2IxG=.=71:dtg2Peytlor=atsmdoe,frtdahg1texTm=a=g0n,0edt/wiz|Jia˜tgthi|ocn==M0./a0γ1,,gvae=nrsduSsB4γ,|B|d(gz1/1|J=1˜g)|.dog1Tyfah=ne
(J˜g′/J˜g =0.5) tetramers, respectively. solid anddashedcurvesarefortheTypeI(J˜g′/J˜g =1.5) and
TypeII (J˜g′/J˜g =0.5) tetramers, respectively.
)
s
nit 1s).0
itrary u0.4 SAs14F =M 1 D/2M B=|Jg| 00ary unit..68 Jg’/Jg=0.5 d/|Jg|=0.01 AsS14F =M 1 DM
rb tr B||(111)
a0.2 bi
| (s B=1.5|Jg| 0ar.4 Jg’/Jg=1.5
P (

| |
0.s2
P
0.0 |
0.00 0.25 0.50
0.0
0 1 2 3 4 5
B/|J |
g
FIG. 6: Plots of the magnitude of the spin-derived polar-
ization P in arbitrary units versus θ/π at T = 0 of an
s
Ttdsi1gzvhe=el=ysd.og1l1iy|d/=2an|dtdg2extdr=aamshdeeg2ryd=wcuidtrhv,edscg1xar==e f0oa,r,dγ/gBJg/=|=J˜g0S|.4=1,,1aJ.˜n5g′d,/1Jφ˜,gr=e=sπp/ec14-,. Ftsdi1g2IoyGn==.|8P1d:s,t|Pdeilg1tnoxrtas=amrobe0fri,ttrdhwa/eir|tyJmh˜gua|cng=ni=ti0st.u0avd1,ee,rgsaounf=sdtγhBSBe4|/s,|p(|J1di˜ng1gz|-1d)=a.etrTidvTg1heyed==spo0olilddaog2rfxaiznaa=dn-
dashed curves are for the Type I (J˜g′/J˜g = 1.5) and Type II
(J˜g′/J˜g =0.5) tetramers, respectively.
videdg =T . ForTypeI,thelowestenergystateineach
d
6
smanifoldoccursforthemaximumvaluess ,s =2s .
13 24 1
Thus, atlowT,Type I tetramersbehaveas pairsofspin las for first-order eigenstate energy parameters a±, b±,
ν ν
2frsu1stdraimteedr,s.wiTthyptheeIlIowteetsrtameneerrsgywistthateJ˜g′in−eaJ˜cgh<s m0aanrie- c(sν±1,3,asn2d4)B=ν w(2itsh1,a2rsb1i)t,ra(sry13s,si2n4)th=e (tsh/r2ee,ss/p2e)cifaolrceavseens osf,
fold occurring for the minimal s13,s24 values. For even and(s13,s24)=[(s±1)/2,(s∓1)/2)]forodds aregiven
s, these minima occur for s ,s = s/2, but for odd s, inSubsectionsGandHoftheAppendix. Forsufficiently
13 24
the energy minimum is doubly degenerate, occurring at strong J˜g′ −J˜g, these formulas can apply to the lowest
s ,s = (s 1)/2,(s 1)/2. Hence, explicit formu- energy eigenstate in each s manifold. When J˜′ J˜ is
13 24 ± ∓ g − g