Classical Spin Liquid: Exact Solution for the Infinite-Component Antiferromagnetic Model on the Kagomé Lattice

15 Pages·0.64 MB·English

by  D. A. Garanin

PHYSICALREVIEW B VOLUME59,NUMBER 1 1 JANUARY1999-I,443-456
Classical spin liquid: Exact solution for the infinite-component antiferromagnetic
9 model on the kagom´e lattice
9
9
1 D. A. Garanin∗ and Benjamin Canals†
Max-Planck-Institut fu¨r Physik komplexer Systeme, N¨othnitzer Strasse 38, D-01187 Dresden, Germany
n
a (Received 22 May 1998)
J
5 Thermodynamic quantities and correlation functions (CF’s) of the classical antiferromagnet on the
1 kagom´e lattice are studied for the exactly solvable infinite-component spin-vector model, D .
→ ∞
In this limit, the critical coupling of fluctuations dies out and the critical behavior simplifies, but

] theeffect of would be Goldstone modes preventingordering at any nonzero temperature is properly
h
accountedfor. Incontrasttoconventionaltwo-dimensionalmagnetswithcontinuoussymmetryshow-
c
e ingextendedshort-rangeorderatdistancessmallerthanthecorrelation length,r<ξc exp(T∗/T),
m correlations in the kagom´e-lattice model decay already at the scale of the lattice∼spaci∝ng due to the
strong degeneracy of the ground state characterized by a macroscopic number of strongly fluctuat-
t- ing local degrees of freedom. At low temperatures, spin CF’s decay as S0Sr 1/r2 in the range
ta a0 ≪ r ≪ ξc ∝ T−1/2, where a0 is the lattice spacing. Analytical reshults fior∝the principal ther-
s modynamicquantities in ourmodel are in fairly good quantitativeagreement with theMonte Carlo
.
t simulations for the classical Heisenberg model, D = 3. The neutron-scattering cross section has its
a
maxima beyond the first Brillouin zone; at T 0 it becomes nonanalytic but does not diverge at
m →
any q.
-
d
PACS number(s): 75.50.Ee, 75.40.Cx
n
o
c
[
I. INTRODUCTION field transition temperature, TMFA, which in fact shows
3 c
that there is no phase transition at this temperature be-
v
2 Classical antiferromagnets on kagom´e and pyrochlore cause of fluctuations. This results in the smooth tem-
6 lattices built of corner-sharing triangles and tetrahedra, peraturedependence ofthethermodynamicquantities2,3
3 respectively, are examples of frustrated systems which and in the diffuse magnetic neutron scattering.6
5 cannot order because of the high degeneracy of their Thedegeneracyofthegroundstateofthesemodelscan
0 ground state1 and ensuing large fluctuations. Monte be lifted by small perturbations, such as dipole-dipole
8
Carlo(MC)simulationsforkagom´e2 andpyrochlore3lat- interactions, lattice distortions, next-nearest-neighbor
9
/ tices with the nearest-neighbor (nn) interaction J show (nnn) or long-range interactions, and quantum effects.
at a smooth temperature dependence of the heat capacity, This may be a reason why pyrochlore antiferromag-
m C(T), in the entire temperature range. The spin corre- nets usually freeze into a spin-glass state with lowering
- lation functions (CF’s) of both models show only weak temperature.7,8 Theoretically, the most transparent way
d short-rangeorder at T<J and decay at distances of the to lift the degeneracy is to include nnn interactions in
n order of several lattice∼spacings a . This is in a strik- the Hamiltonian.1 Experiment and MC simulations on
o ing contrast to the long-range ord0er in common three- pyrochlores9 show ordering with an unusual critical be-
c
: dimensionalmagnetsandtheextendedshort-rangeorder havior (β ≈ 0.18) in this case. According to the spin-
v [strong correlation at the distances r<ξ exp(T /T) wave results of Ref. 10 for the kagom´e lattice, at low
c ∗
Xi with T∗ J] in common two-dimensio∼nal m∝agnets. temperatures dipole-dipole interactions favor the planar
Spin-w∼ave calculations starting from one of the or- q=0 phase which is characterizedby the same ordering
r
a dered states of the kagom´e lattice4 (see, also, Ref. 5 for pattern in each of the elementary triangles.
the quantum case) yield a twofold degenerate Goldstone A more subtle mechanism for lifting the degeneracy
mode, as well as a zero-energymode for all values of the and selection of definite ordering patterns is the nonlin-
wave vector q in the Brillouin zone, the latter reflecting earinteractionofspinwavesforclassicalsystemsatvery
the instability of the ground states. Mean-field approxi- low temperatures, typically T <0.01J. For the kagom´e
mation(MFA)atelevatedtemperaturesforbothkagom´e lattice, nonlinear effects (therm∼al fluctuations) favor the
and pyrochlore lattices1 reflects the same behavior. The coplanarspinconfigurationwiththe√3 √3short-range
×
maximaleigenvaluesoftheFourier-transformedexchange order in the case of the Heisenberg model, D = 3, as
interaction matrix are q independent for both models was suggested by the results of MC simulations2,11,12
(and twofold degenerate for pyrochlore). Thus the sys- and high-temperature series expansions.4 Extension of
temcannotchoosetheorderingwavevectoratthemean- the √3 √3 short-range order into the true long-range
×
order in the limit T 0 is, however, hampered by for-
→
1
mation of chiral domain walls which cost no energy but The D = modelproperlyaccountsforthe profound
provide a gain in entropy at low concentrations.12 The role played,∞especially in low dimensions, by the Gold-
configuration selection at low temperatures only occurs stone or would be Goldstone modes. At the same time,
if the number of spin components D is low enough. So, the less significant effects of the critical fluctuation cou-
the early MC simulations of Ref. 11 for the kagom´e an- plingleading,e.g.,tothequantitativelydifferentnonclas-
tiferromagnet showed selection of a coplanar state for sical critical indices, die out in the limit D . Thus
→ ∞
D = 3, but no such selection for D 4. For the py- this model is a relatively simple yet a powerful tool for
≥
rochlorelattice,earlysimulationsshowedtheselectionof classicalspinsystems. Itshouldnotbemixedupwiththe
the collinear spin ordering for the Heisenberg model at N-flavorgeneralizationofthe quantumS =1/2model35
low temperatures,9 although according to the recent re- in the limit N , including its 1/N expansion.15,36
→ ∞
sults of Ref. 13 this happens only for the plane rotator TheN-componentnonlinearσ-model(see,e.g.,Refs.37,
model, D = 2, and not for higher spin dimensionalities. as well as Ref. 38, and 39 for the 1/N expansion) is a
Theaboveresultsareinaccordwithgeneralcriterionfor quantum extension of Eq. (1.1) in the long-wavelength
selectionoforderedstatesasafunctionofspinandspace region at low temperatures. Effective free energies for
dimensions for corner sharing objects, which was formu- the n-componentorderparameterappear,insteadofEq.
lated in Ref. 13. Quantum fluctuations were shown to (1.1),inconventionaltheoriesofcriticalphenomena. Us-
stabilizethe√3 √3phaseforS 1,14 buttheyshould ing them for the 1/n expansion (see, e.g., Ref. 40) is a
destroyordering×forlowspinvalue≫sS.5,15,16,17Oneofthe matter of taste. While yielding the same results for the
possiblemechanismsforthatistunnelingoftheweather- criticalindicesasthelattice-based1/Dexpansion,24,25,26
wane (hard hexagon) mode in the √3 √3 structure.18 it misses the absolute values of the nonuniversal quanti-
×
It should be stressed, however, that the subtle effects ties. The same comment also applies to the spatially
quoted above can be easily overwhelmed by more triv- inhomogeneoussystemsin the limitD =n= ,suchas
∞
ial and robustones, and they are much easier to observe semi-infinite ferromagnets (cf. Refs. 41 and 42).
in simulations than in experiment. The first task of the In this article the solution for the isotropic antiferro-
theory is thus to describe the principalfeaturesof classi- magnetic infinite-component spin-vector model on the
calspin models onfrustrating lattices, as, e.g.,a smooth kagom´e lattice will be given. The qualitatively simi-
variation of the thermodynamic quantities in the whole lar results for the pyrochlore lattice will be presented
temperature range. The simplest approach, the MFA, is in a subsequent communication. As long as the system
clearlyinapplicableinthiscase,whereasthemorepower- studied is homogeneous, isotropic, and in zero magnetic
ful tools of the theory of criticalphenomena, such as the field, the standard spherical model22,23 can be applied,
renormalizationgroup,seemtohavenotbeenyetapplied too. Such an approach for nonordering frustrated three-
to these lattices. dimensional systems has been advocated in Ref. 43. We
The “next simplest” approximation for classical spin prefer, however,to use the more general framework.
systems, which follows the MFA, consists in generalizing The rest of this article is organized as follows. In Sec.
the Heisenberg Hamiltonian for the D-component spin II the structure of the kagom´e lattice and its collective
vectors:19,20 spin variables are described. In Sec. III the formalism
of the D = model is tailored for the kagom´e lattice.
= H· sr 1 Jrr′sr·sr′, sr =1 (1.1) The diagram∞s of the classical spin diagram technique
H − − 2 | |
r rr′ that do not disappear in the limit D are summed
X X → ∞
up. Thegeneralanalyticalexpressionsforthethermody-
and taking the limit D . In this limit the problem
→ ∞ namic functions and spin CF’s for all temperatures are
becomes exactly solvable for all lattice dimensionalities,
obtained. In Sec. IV the thermodynamic quantities of
d, and the partition function of the system coincides21
the kagom´e antiferromagnet (AFM) are calculated and
with thatofthe sphericalmodel.22,23 The D = model
∞ compared with MC simulation results in the whole tem-
possesses, however, a number of important advantages
perature range. In Sec. V the real space correlation
withrespecttothesphericalone. (i)The1/Dexpansion
functions are computed. In Sec. VI the neutron scat-
is possible,24,25,26 including the case of low-dimensional
tering cross section is worked out. In Sec. VII possible
systems.27,28 The calculations can be done conveniently
improvements of the present approach, such as the 1/D
in the framework of the diagram technique for classical
expansion, are discussed.
spinsystems.29,27,30 (ii) Inclusionofanisotropicterms in
Eq. (1.1)is possible, too,which allowsus to describe or-
dering in low dimensions, including thin films31 and do-
II. LATTICE STRUCTURE AND THE
main walls.32 (iii) In spatially inhomogeneous cases the
HAMILTONIAN
D = model yields physically correct results, in con-
∞
trast to the spherical model failing on the global spin
constraint.33 (iv) Below the Curie temperature T or in The kagom´e lattice shown in Fig. 1 consists of corner-
c
sharing triangles. Each node of the corresponding Bra-
amagneticfield,theD = modeldescribesbothtrans-
∞ vais lattice (i.e., each elementary triangle in Fig. 1) is
verse and longitudinal CF’s (Ref. 34) that differ from
numbered by i,j = 1,...,N. Each site of the elemen-
each other, in contrast to the single CF in the spherical
tary triangle is labeled by the index l = 1,2,3. It is
model.
2
1
slq = slie−iq·rli, sli = N slqeiq·rli, (2.4)
0,1 1,1 2,1 Xi Xq
* where the wave vector q belongs to the hexagonal Bril-
louin zone specified by the corners ( π/3, π/√3) and
± ±
1 2 ( 2π/3,0)(seeFig. 2). TheFourier-transformedHamil-
±
tonian reads
0,0 1,0 2,0
= 1 Vll′sl·sl′ , (2.5)
3 H 2N ll′q q q −q
X
where the interaction matrix is given by
0,-1 1,-1
0 a b a cos(u q)
Vˆ =2J a 0 c , b ≡cos(v ·q) (2.6)
q
  ≡ ·
b c 0 c cos(w q).
≡ ·
 
At the second stage, the Hamiltonian (2.5) is finally
FIG. 1. Structure of the kagom´e lattice. The elementary
diagonalized to the form
trianglesarelabeledbythepairsofnumbersn ,n according
u v
to Eq. (2.1), and the sites on triangles (the sublattices) are 1
= V˜nσn·σn , (2.7)
labeledbyl=1,2,3. Theconfigurationshowncorrespondsto H −2N q q −q
thecoplanar√3 √3structurecharacterizedbytheordering nq
X
×
wave vector q given byEq. (2.3).
√3 where V˜n = 2Jν (q) are the eigenvalues of the matrix
q n
Vll′ taken with the negative sign,
q
convenient to use the dimensionless units in which the
interatomic distance equals 1 and hence the lattice pe- ν =1, ν =( √1+8abc 1)/2. (2.8)
1 2,3
riod equals 2. The triangles numbered byi,j =1,...,N ± −
can be obtained from each other by the translations The diagonalizing transformation has the explicit form
rlj =rli+nu2u+nv2v, (2.1) Un−l1(q)Vqll′Ul′n′(q)=V˜qnδnn′, (2.9)
where rli is the position of a site on the lattice, nu and where the summation over the repeated indices is im-
nv areintegers,2uand2varetheelementarytranslation plied and Uˆ is the real unitary matrix, Uˆ−1 = UˆT, i.e.,
vectors (lattice periods), and Un−l1 = Uln. The columns of the matrix Uˆ are the three
normalized eigenvectors U = (U ,U ,U ) of the in-
u=(1,0), v=( 1/2,√3/2). (2.2) n 1n 2n 3n
− teraction matrix Vˆ:
Oneofthemostsymmetricphasesofthekagom´eAFM
U =(ac bν , ab cν , ν2 a2)/ Q ,
is the so-called√3 √3 phase which is shown in Fig. 1. n − n − n n− n
This coplanar phas×e can be described by the complex Qn =(νn2 −a2)2+(ab−cνn)2+(acp−bνn)2. (2.10)
“spin” s˜ sx+isy = exp(iq ·r+iφ ), where the or-
r ≡ r r √3 0 The eigenvector U corresponding to the dispersionless
deringwavevectorq canbewritteninthreeequivalent 1
√3 eigenvalueν =1canberepresentedintheunnormalized
forms: 1
form as
u
2π
q√3 =− 3 × v (2.3) U1 =[sin(w·q), sin(v·q), sin(u·q)]. (2.11)
w,

The normalized eigenvectors satisfy the requirements of
wherew=( 1/2, √3/2). Inthisphasespinsrotateby orthogonality and completeness, respectively,
− −
240 =120 asrchangesbythelatticeperiod2ineach
o−fthe◦directio◦nsu,v, andw makingtheangle120 with Uln(q)Uln′(q)=δnn′, Uln(q)Ul′n(q)=δll′. (2.12)
◦
each other. Another realization of the √3 √3 phase,
inwhichspinsrotateby 120 ,isdescribed×byq with TheFouriercomponentsofthespinsslqandthecollective
− ◦ √3 spin variables σnq are related by
positive sign. In addition, the √3 √3 phase can be
×
describedby appropriatecombinationsofdifferentforms sl =U (q)σn, σn =slU (q). (2.13)
q ln q q q ln
of q given above.
√3
To facilitate the diagram summation in the next sec- The largestdispersionlesseigenvalue ν of the interac-
1
tion, it is convenient to put the Hamiltonian (1.1) into a tion matrix [see Eq. (2.8)] manifests frustration in the
diagonalform. First, one goes to the Fourier representa- system which precludes an extended short-range order
tion according to even in the limit T 0. Independence of ν of q signals
1
→
3
n √3n n +√3n √2
Uˆq ∼= √16 −−nxx+−√3nyy −−nyy−√3nxx √2, (2.15)
2n 2n √2
x y
1  
where n q/q.
≡
Inthenextsectiontheequationsdescribingspincorre-
lation functions of the classical kagom´e antiferromagnet
0 in the large-D limit will be obtained with the help of
the classical spin diagram technique. The readers who
are not interested in details can skip to Eq. (3.16) or
directly to Sec. IV.
-1 Π
€€€€€€€€€€€€€€€
(cid:143)!3!!
III. CLASSICAL SPIN DIAGRAM TECHNIQUE
0 q
-2 y AND THE LARGE-D LIMIT
22 ΠΠ
--€€€€€€€€€€€€€€€€€€€€€€€€
Π
33 00 -€€€€€€€€€€€€€€€ The exact equations for spin correlation functions in
q 2 Π (cid:143)!3!! the limit D , as well as the 1/D corrections, can
x €€€€€€€€€€€€ → ∞
3 be the most conveniently obtained with the help of the
classical spin diagram technique.29,27,30 A perturbative
FIG. 2. Reduced eigenvalues of the interaction matrix,
expansionofthethermalaverageofanyquantity char-
νn(q) = V˜qn/(2J) of Eq. (2.7), plotted over the Brillouin acterizing a classical spin system (e.g., = s —Athe z
zone. A zi
spin component on the lattice site i) can be obtained by
rewriting Eq. (1.1) as = + , where is, e.g.,
0 int 0
H H H H
that 1/3 of all spin degrees of freedom are local and can the mean-field Hamiltonian, and expanding the expres-
rotate freely. The other two eigenvalues satisfy sion
ν (q)=1 q2/2, ν (q)= 2+q2/2 (2.14) 1 N
2 ∼ − 3 ∼− = dsj exp( β ), sj =1, (3.1)
hAi A − H | |
at small wave vectors, q2 ≡qx2+qy2 ≪1. The eigenvalue Z Z jY=1
ν which becomes degenerate with ν in the limit q 0
2 1 → whereβ =1/T,inpowersof int. TheintegrationinEq.
is related, as we shall see below, to the usual would be H
(3.1)iscarriedoutwithrespecttotheorientationsofthe
Goldstone mode destroying the long-range order in low-
D-dimensionalunitvectorss oneachofthelatticesites.
j
dimensional magnets with a continuous symmetry. The
Averages of various spin-vector components on various
eigenvalueν ispositioned,inthelong-wavelengthregion,
3 lattice sites with the Hamiltonian can be expressed
0
much lower than the first two, and it is tempting to call H
throughspincumulants,orsemi-invariants,whichwillbe
it the “optical” eigenvalue. In fact, however, the eigen-
considered below, in the following way:
values of the interaction matrix are not the same as the
normalmodesofthesystemthatappearinthedynamics. s =Λ ,
αi 0 α
Whereasν givesrisetothezero-energyspin-wavebranch h i
1 s s =Λ δ +Λ Λ , (3.2)
αi βj 0 αβ ij α β
correspondingtotheabsenceofarestoringforceforsmall h i
s s s =Λ δ +Λ Λ δ
deviations from one of degenerate ground states, ν and αi βj γk 0 αβγ ijk αβ γ ij
2 h i
ν3 hybridize to the double-degenerate Goldstone mode +ΛβγΛαδjk +ΛγαΛβδki+ΛαΛβΛγ,
withenergy q atsmallwavevectors,asinconventional
antiferromag∝nets.5,4 With increasing q the eigenvalue ν etc., where δij, δijk, etc., are the site Kroneckersymbols
2
equal to 1 for all site indices coinciding with each other
decreaseswhereasν increases;atthecornersoftheBril-
3
and to zero in all other cases. For the one-site averages
louin zone they become degenerate: ν = ν = 1/2.
2 3
The q dependences of the eigenvalues ν over the−whole (i = j = k = ...) Eq. (3.2) reduces to the well-known
n
representationof moments throughsemi-invariants,gen-
Brillouin zone is shown in Fig. 2.
eralizedforamulticomponentcase. Inthe graphicallan-
Incontrasttothesmoothbehaviorofν ,thediagonal-
n
izing matrix Uˆ composedof the eigenvectorsU [see Eq. guage(seeFig. 3)thedecomposition(3.2)correspondsto
n
all possible groupings of small circles (spin components)
(2.10)]has muchmore complicatedstructure as function
of q. This results in the intricate behavior of the spin intoovalblocks(cumulantaverages). The circlescoming
from (the “inner”circles)areconnectedpairwiseby
CF’s and neutron-scattering cross sections at low tem- int
H
the wavy interaction lines representing βJ . In diagram
peratures, which will be considered in Sec. V. Here we ij
only show the nonanalytic limiting form of Uˆ at small expressions, summations over site indices i and compo-
nentindicesαofinnercirclesarecarriedout. Oneshould
wave vectors,
not take into account disconnected (unlinked) diagrams
[i.e.,thosecontainingdisconnectedpartswithno“outer”
circlesbelongingto inEq. (3.1)],sincethese diagrams
A
4
are compensated for by the expansion of the partition
function in the denominator of Eq. (3.1). Considera-
Z
tion of combinatorial numbers shows that each diagram
containsthe factor1/ns,wherens isthe numberofsym- n1 n1'
metry group elements of a diagram [see, e.g., the factor l'
1/2! in Eq. (3.12) below]. The symmetry operations do
l l
not concern outer circles, which serve as a distinguish-
able “root” to build up more complicated (e.g., renor-
n n' n n' n n'
malized) diagrams. For spatially homogeneous systems, 1 1
it is more convenient to use the Fourier representation
and to calculate integrals over the wave vectors in the FIG. 3. Self-consistent Gaussian approximation (SCGA)
for classical spin systems in the nonordered state. (a) the
Brillouin zone rather than lattice sums. As due to the
Dysonequationforthespincorrelationfunction;(b)theblock
Kronecker symbols in Eq. (3.2) lattice sums are subject
summationfortherenormalizedpaircumulantspinaverages.
to the constraint that the coordinates of the circles be-
longing to the same block coincide with each other, the
sum of wavevectors coming to or going out of any block
given spin from its neighbors, which implies a Gaussian
alonginteractionlines is zero. So,forour modelthe pair
statisticsofmolecularfieldfluctuations. Theappropriate
cumulant average of the Fourier components defined by
diagram sequence for the nonordered state, s = 0, is
z
Eq. (2.4) reads h i
represented in Fig. 3. Its analytical form for the square
hslαqslβ′q′i0,cum =ΛαβNδq′,−qδll′, (3.3) loarttbiecleowmoTdceilnisthgeivoerndeirningRemf.od2e7l.s tIhneaavmeraaggneestpicinfipelod-
larization s = 0 appears. The additional diagrams
where δll′ is the sublattice Kronecker symbol. The cu- h zi 6
and corresponding analytical expressions can be found
mulant spin averages in Eq. (3.2) can be obtained by
inRefs. 29,30,and28. The SCGA equationsin the spa-
differentiating the generating function Λ(ξ) over appro-
priate components of the dimensionless field ξ βH:29 tially inhomogeneous case and their large-D limit have
≡ been derived (and applied to domain walls) in Ref. 32.
∂pΛ(ξ) In all cases above, the SCGA equations have been
Λ (ξ)= ,
α1α2...αp ∂ξ ∂ξ ...∂ξ written for diagonal Hamiltonians describing the sim-
α1 α2 αp
plest one-sublattice magnets. For nondiagonal Hamilto-
Λ(ξ)=ln (ξ), (3.4)
Z0 nians,suchasEq. (2.5),thematrixinteractionlines,here
where ξ ξ , βVll′, complicate the formalism. Simplification can be
≡| | − q
achieved by using the diagonalized Hamiltonian, for our
0(ξ)=const ξ−(D/2−1)ID/2 1(ξ) (3.5) model, Eq. (2.7). For the latter, however, the σ coun-
Z × −
terparts of the one-site cumulant averagesdo not have a
isthepartitionfunctionofaD-componentclassicalspin,
transparent meaning anymore since σ is a combination
and I (ξ) is the modified Bessel function. For the two
ν ofspinsondifferentsitesandsublattices. Thustheσ cu-
lowest-order cumulants the differentiation in Eq. (3.4)
mulants should be specially worked out as follows. The
leads to the following expressions:
pair σ cumulant (which is in our model explicitly diago-
Λ (ξ)=B(ξ)ξ /ξ, nal in the spin-component indices α,β) can be rewritten
α α
in terms of the initial spin variables as
B(ξ) ξ ξ ξ ξ
Λ (ξ)= δ α β +B (ξ) α β. (3.6)
αβ ξ (cid:18) αβ − ξ2 (cid:19) ′ ξ2 hσαnqσαnq′′i0,cum =Uln(q)Ul′n′(q′)hslαqslα′q′i0,cum. (3.9)
where δ is the Kroneckersymbol for spin components,
αβ With the use of Eq. (3.3) and the first of the relations
(2.12), this can be simplified to the final form
B(ξ)=dΛ(ξ)/dξ =I (ξ)/I (ξ) (3.7)
D/2 D/2 1
−
is the Langevinfunction of D-component classicalspins, hσαnqσαnq′′i0,cum =ΛααNδq′,−qδnn′. (3.10)
andB (ξ)=dB/dξ(seethedetailsinRef.30). If =0,
′ 0
H Now, with the help of the results just obtained, the sec-
as is the case for our model in the absence of a magnetic
ond diagramin the sum in Fig. 3b can be written in the
field,thepairspincumulantinEq. (3.6)simplifiestothe
analytical form
obvious form
Λαβ(0)=Λαα(0)δαβ, Λαα(0)=1/D. (3.8) A2 =Uln(q)Uln′(q)ΛααββLβl, (3.11)
As was shown in Ref. 29 (see also Ref. 27) the limit where the summation over the spin-component index β
D forthespin-vectormodeliscompletelydescribed isimpliedandthequantityLβl inthelowest(thesecond)
→∞
by the self-consistent Gaussian approximation (SCGA), order of the perturbation theory is given by
since all diagrams not accounted for by the SCGA van-
1 1
ish in this limit. The SCGA consists in taking into ac- L(β2l) = 2!ΛββN Uln1(p)Ul′n1(p)βV˜pn1
count pair correlations of the molecular field acting on a Xl′p Xn1
5
× Uln′1(p)Ul′n′1(p)βV˜pn′1. (3.12) The analytical expression for the σ CF in the SCGA,
Xn′1 whichsatisfiesthe DysonequationshowninFig. 3a,has
the Ornstein-Zernike form
Here β = 1/T in front of V˜ cannot be confused with DΛ˜
the spin component index β. This expression can be σn(q)= αα . (3.18)
simplified in two ways. First, one can perform the sum 1−Λ˜ααβV˜qn
over the index l and use the first of Eqs. (2.12), which
′
This expression differs from that obtained by Reimers
leads to
on the mean-field basis6 by the replacement of the bare
L(2) = 1Λ 1 (βV˜n1)2U2 (p). (3.13) cumulant Λαα = 1/D by its renormalized value Λ˜αα de-
βl 2! ββN p ln1 termined by the diagram series Fig. 3b. The summation
Xn1p ofthesediagramsisdocumentedinthemostdetailedway
by Eqs. (3.16)–(3.19) of Ref. 30. The result for Λ˜ is
Second, inverting the transformation(2.9) one can write αα
given by the second line of Eq. (3.6) averaged over the
L(2) = 1Λ 1 (βVll′)2. (3.14) Gaussianfluctuationsofallcomponentsofthemolecular
βl 2! ββN p fieldξ withthedispersiondefinedbythequantityLα. In
Xl′p our model, fluctuations of different components of ξ are
independent fromeachother andofthe samedispersion,
Taking into account the explicit form of Vpll′ given by L = L. Thus the quantity Λ˜ is diagonal and inde-
α αβ
Eq. (2.6), one can see that after the integrationoverthe
pendentofα. Inthelarge-D limitthemultiple Gaussian
wave vector p expression (3.14) becomes independent of integraldetermining Λ˜ is dominatedby the stationary
αα
the sublattice index l. After this observation one can point and the result simplifies to27
symmetrize Eq. (3.13) with respect to l. This leads to
the vanishing of the diagonalization matrix by virtue of 2 1
Λ˜ = . (3.19)
thefirstofEqs. (2.12)andtotheappearanceofthefactor αα D1+ 1+8L/D
1/3. Now the summation over l in Eq. (3.11) simplifies,
and the diagonalizationmatrices convert, again, to δnn′. HerethedispersionLcorrepspondingtothediagramseries
Theresultinthesecondorderoftheperturbationtheory in Fig. 3 is given by the formula
has the form
Λ˜ dq (βV˜n)2
A2 =ΛααββL(β2), L(β2) = 3Λ·β2β!N1 Xn1p(βV˜pn1)2 (3.15) L= 3·α2α!Xn v0Z(2π)d1−Λ˜αqαβV˜qn (3.20)
generalizing Eq. (3.15). Here, the summation
(δnn′ has been omitted) and it is independent of the (1/N) ...isreplacedbytheintegrationovertheBril-
eigenvalue index n and of the wave vector q. q
louinzone,v istheunit-cellvolume,anddisthespatial
0
The mechanism of the simplification of diagram ex- dimensPionality. Forthekagom´elatticewehavev =2√3
0
pressions demonstrated above can be shown to work for
and d=2. The expression for L can be simplified to
whatever complicated diagrams. In all cases oval blocks
roerpigriensaeln,tnocnudmiuaglaonntalsipzeind,avveerrsaiognesofΛtαh1eαc2.l.a.αsspi,caalsspinintdhie- L= P2¯Λ˜−1, P¯ ≡ 31 Pn, (3.21)
agramtechnique. Inalltheelementsconnectedtoagiven αα n
X
block summation over the eigenvalue indices n is carried
where P is the lattice Green function associated with
out. The diagonalizing matrix Uˆ disappears completely n
the eigenvalue n:
if correlation functions for the σ variables,
dq 1
D 1 P =v . (3.22)
σn(q)= σn σn = σnσn , (3.16) n 0 (2π)d1 Λ˜ βV˜n
Nh αq α,−qi Nh q −qi Z − αα q
are considered. After the calculation of the latter the Now one can eliminate L from Eqs. (3.19) and (3.21),
true spin CF’s can be found from the formula which yields the basic equation of the large-D model,
sll′(q)=Uln(q)Ul′n(q)σn(q) (3.17) DΛ˜ααP¯ =1. (3.23)
following from Eq. (2.13). Note that σn(q) are eigen- This nonlinear equation determining Λ˜αα as a func-
values ofthe correlationmatrixsll′(q) andthey describe tion of temperature differs from those considered
earlier29,27,28,30 byamorecomplicatedformofP¯ reflect-
independentlinearresponsestoappropriatewave-vector-
ingthelatticestructure. Theformofthisequationissim-
dependent fields. As can be seen from Eq. (3.17), the
ilartothatappearinginthetheoryoftheusualspherical
eigenvectors describing the “normal modes” of the sus-
ceptibility are those of the interactionmatrix Vll′ in Eq. model.22,23Themeaningsofbothequationsare,however,
q different. Whereas in the standard spherical model a
(2.5).
similarequationaccountforthe prettyunphysicalglobal
6
spin constraint, Eq. (3.23) here is, in fact, the normal- The susceptibility per spin symmetrized over sublattices
ization condition s2 =1 for the spin vectors on eachof can be expressed through the spin CF’s as
h ri
the lattice sites r [see Eq. (3.1)]. Indeed, calculating the
spin autocorrelation function in the form symmetrized χ = 1 sll′(q). (3.27)
q
over sublattices with the help of Eqs. (3.17), (2.12), and 3DT
ll′
(3.18), one obtains X
With the use of Eq. (3.17) this can be rewritten in the
dq 1 form
s2 =v sll(q)
h ri 0 (2π)d3
Z Xl χ = 1 W2(q)σn(q), W (q) U (q),
=v0 (2dπq)d31 σn(q)=DΛ˜ααP¯. (3.24) q 3DT Xn n n ≡Xl ln
Z Xn (3.28)
That is, the spin-normalization condition is automati-
where the diagonalized CF’s are given by Eq. (3.18).
callysatisfiedinourtheorybyvirtueofEq. (3.23). After
From Eq. (2.15) it follows that in the limit q 0 one
Λ˜ hasbeenfoundfromthisequation,thespinCF’sare →
αα hasW =W =0andW =√3. Thusthehomogeneous
readily given by Eqs. (3.18) and (3.17). 1 2 3
susceptibility χ χ simplifies to
Toavoidpossibleconfusion,weshouldmentionthatin ≡ 0
the paper of Reimers, Ref. 6, where Eq. (3.18) with the 1
bare cumulant Λ = 1/D has been obtained, the theo- χ= σ3(0). (3.29)
αα DT
retical approach has been called the “Gaussian approx-
imation (GA)”. This term taken from the conventional As we shall see in the next section, disappearance of the
theory of phase transitions based on the Landau free- terms with n = 1 and 2 from this formula ensures the
energy functional implies that the Gaussian fluctuations nondivergence of the homogeneous susceptibility of the
oftheorderparameterareconsidered. Inthemicroscopic kagom´e antiferromagnet in the limit T 0. The situa-
→
language,thismerelymeanscalculatingcorrelationfunc- tion for q=0 is much more intricate and it will be con-
6
tions of fluctuating spins after applying the MFA. Such sidered below in relation to the neutron-scattering cross
an approach is known to be inconsistent, since corre- section.
lations are taken into account after they had been ne-
glected. As a result, forthe kagom´e lattice one obtains a
phase transition at the temperature T =TMFA =2J/D IV. THERMODYNAMICS OF THE KAGOME´
c c
but immediately finds that the approach breaks down ANTIFERROMAGNET
below T because of the infinitely strong fluctuations. In
c
contrastto this MFA-basedapproach,the self-consistent Toputtheresultsobtainedaboveintotheformexplic-
Gaussian approximation used here allows, additionally, itly well behaved in the large-D limit and allowing a di-
totheGaussianfluctuationsofthemolecular field,which rectcomparisonwiththeresultsobtainedbyothermeth-
renormalizeΛ andleadtotheabsenceofaphasetran- odsforsystemswithfinitevaluesofD,itisconvenientto
αα
sitionforthisclassofsystems. TheSCGAis,inasense,a usethemean-fieldtransitiontemperatureTMFA =2J/D
c
“double-Gaussian”approximation: Thediagramseriesin as the energy scale. With this choice, one can introduce
Fig. 3a allows for the Gaussian fluctuations of the order the reduced temperature θ and the so-calledgap param-
parameter, whereas that in Fig. 3b describes Gaussian eter G according to
fluctuations of the molecular field.
T D
To close this section, let us work out the expressions θ , G Λ˜ . (4.1)
for the energy and the susceptibility of the kagom´e anti- ≡ TcMFA ≡ θ αα
ferromagnet. For the energy of the whole system, using
In these terms, Eq. (3.23) rewrites as
Eqs. (2.7) and (3.16), as well as the equivalence of all
spin components, one obtains θGP¯(G)=1 (4.2)
N dq
U = = v V˜nσn(q). (3.25) and determines G as function of θ. Here P¯(G) is defined
tot hHi − 2 0 (2π)d q
n Z by Eq. (3.21), where
X
To obtain the energy per spin U, one should divide this dq 1 1
P =v , P = , (4.3)
expressionby 3N. With the use of Eq. (3.18), the latter n 0 (2π)d1 Gν (q) 1 1 G
can be expressed through the lattice Green’s function P¯ Z − n −
of Eq. (3.21); then with the help of Eq. (3.23) it can be andthereducedeigenvaluesν (q)aregivenbyEq. (2.8).
n
put into the final form The σ CF’s of Eq. (3.18), which are proportional to the
integrands of P , can be rewritten in the form
n
T 1
U = D . (3.26)
2 (cid:18) − Λ˜αα(cid:19) σn(q)= θG . (4.4)
1 Gν (q)
n
−
7
Further, it is convenient to consider the reduced energy G
per spin defined by 1.0
U˜ U/U , U = J, (4.5)
0 0
≡ | | − 0.8
where U is the energy per spin at zero temperature.
0
With the help of Eq. (3.26) U˜ can be written as
0.6 1/θ, MFA
U˜ =θ 1/G. (4.6)
−
The homogeneous susceptibility χ of Eq. (3.29) can be 0.4
rewrittenwiththehelpofEq. (2.14)inthereducedform
G 0.2 1 − θ/3
χ˜ 2Jχ= . (4.7)
≡ 1+2G
The sense of calling G the “gap parameter” is clear 0.0
from Eq. (4.4). If G = 1, then the gap in correla- 0.0 0.5 1.0 1.5 2.0 2.5 3.0
tion functions closes: σ1 turns to infinity, and σ2 di- θ ≡ T/TMc FA
FIG.4. Temperature dependence of the gap parameter G
verges at q 0. For nonordering models, it happens
→ for the kagom´e antiferromagnet.
only in the limit θ 0, however. The solution of Eq.
→
(4.2) satisfies G 1 and goes to zero at high tem- ~
peratures. If θ ≤1, the function P¯ is dominated by U
0.0
≪
P =1/(1 G),whereasP remainsoforderunityandP
1 3 2
−
divergesonlylogarithmically,asinusualtwo-dimensional
systems: P2 ∼=(√3/π)ln[c/(1−G)], c∼1. The ensuing -0.2
asymptotic form of the gap parameter at low tempera-
tures reads
-0.4
θ θ 2 √3 3c
G=1 ln , θ 1. (4.8)
∼ − 3 − 3 π θ ≪
(cid:18) (cid:19) -0.6
At high temperatures, Eq. (4.2) requires small values
of G. Here, the limiting form of P¯ can be shown to be
P¯ =1+G2. The corresponding asymptote of G has the -0.8
∼
form
1 1 -1.0
G∼= θ (cid:18)1− θ2(cid:19), θ ≫1. (4.9) 0 1 2 3 θ 4≡T/TMc FA5
FIG.5. Temperature dependenceof the reduced energy of
The numerically calculated temperature dependence of
thekagom´e antiferromagnet.
G is shown in Fig. 4. Note that in the MFA one has
G=1/θ which attains the value 1 at θ =1.
The temperature dependence of the reduced energy of
The reduced variables introduced at the beginning of
Eq. (4.6) is shown in Fig. 5. Its asymptotic forms fol-
this section are very convenient for the comparison of
lowing from Eqs. (4.8) and (4.9) are given by
the results for D = with those for finite values of D,
∞
whichareobtainedbyothermethods. Theexpected dis-
1/θ, θ 1
U˜ = − ≫ (4.10) crepancies are of order 1/D which is not too much for
∼ 1+(2/3)θ, θ 1.
(cid:26) − ≪ D = 3. (Note that the D = approximation can be
∞
improved by the 1/D expansion.27,28) To compare with
This implies the reduced heat capacity C˜ = dU˜/dθ is
the MC simulation data of Ref. 2 for the heat capacity
equal to 2/3 at low temperatures, in contrast to C˜ = 1 of the Heisenberg model we will use, instead of C˜, the
for usual two-dimensional lattices in the same approxi- true heatcapacityC =dU/dT =(D/2)C˜ [seeEqs. (4.1)
mation. The latter result is solely due to the term linear
and (4.5)], which in our approach tends to D/3 1
in θ in Eq. (4.6), whereas G only exponentially deviates ⇒
at low temperatures. The fairly good agreement on the
from1atlowtemperatures. Forthekagom´elattice,there
high-temperature side of Fig. 6 is not surprising,since a
is a linear in θ contribution to the gap parameter G of
nontrivial dependence on D appears only at order 1/T3
Eq. (4.8), which leads to C˜ =2/3. This reflects the fact
for the nn correlationfunction and hence for the energy,
that one of three modes in the kagom´e lattice [see Eq.
and thus at order 1/T4 for the heat capacity [see the
(2.7)] is dispersionless, and hence 1/3 of all spin degrees
combination n+2 D +2 in Eq. (3.10a) of Ref. 4].
of freedom in the system are local and free, making no ≡
The reasonable agreement with the MC results at low
contribution to the heat capacity.
8
C = dU/dT ~χ
1.0 0.4
0.8
0.3
0.6
0.2
0.4
0.1
0.2
0.0 0.0
0.0 0.5 1.0 1.5 0 1 2 3 4 5
θ ≡T/TMc FA θ ≡ T/TMFA
c
FIG. 6. Temperature dependence of the heat capacity of FIG. 7. Temperature dependence of the reduced uniform
the kagom´e antiferromagnet. The MC results of Ref. 2 for susceptibilityofthekagom´eantiferromagnet. TheMCresults
theHeisenberg model (D=3) are represented bycircles. of Ref. 12 for the Heisenberg model (D =3) are represented
bycircles.
temperatures is better than expected and can be inter-
preted as a compensation of errors. Indeed, for finite transversetothelocalorientationofspins,χ =1/(2J0),
values of D one should take into account the 1/D cor- and one longitudinal susceptibility χ which⊥vanishes in
rections to the present D = results. For conventional the zero-temperature limit. After kthe averaging over
magnets, this leads in the fi∞rst order in 1/D to the re- all orientations of spins one obtains the exact result
placement of C = D/2 by C = (D 1)/2 in the limit χ=(1 1/D)/(2J0)atT =0, whichdiffers significantly
T 0.27,28 This result is exact and p−hysically transpar- from th−at for the D = model. Taking into account
ent→as following from the constraint s = 1 in counting the first-order 1/D corre∞ction leads to a rather accurate
r
ofthespindegreesoffreedom;itdoes|no|tchangefurther result for χ(T) in the whole temperature range,27 which
at higher orders of the 1/D expansion. For the kagom´e has a well-known flat maximum at T <J. Returning to
lattice, the same counting argument suggests to replace the kagom´e antiferromagnet,one can s∼tate that the 1/D
D by D 1, which would yield C = (D 1)/3 2/3 corrections to the susceptibility are smaller than in the
for T −0. On the other hand, inclusio−n of th⇒e 1/D conventional case. The small maximum of χ˜ in the data
correct→ions reduces the degeneracy of the ground state, of Ref. 12 is probably a 1/D effect arising due to the
and the heat capacity should increase again. This de- increase of the longitudinal susceptibility of spins with
generacy reduction manifests itself by the appearance of temperature at low temperatures, similarly to that in
the q dependence of the correlation function σ1 of Eq. conventional low-dimensional antiferromagnets (see Ref.
αα
(4.4). On the high-temperature side, the degeneracy of 27 for details).
the largest eigenvalue of the susceptibility matrix is re-
moved at order 1/T8 [see Eqs. (3.29) and (3.31) of Ref.
4; the effect vanishes, however, for D ]. At low V. REAL-SPACE CORRELATION FUNCTIONS
→ ∞
temperatures,theresultingheatcapacitybecomes11/12
(Ref. 2), which is not far away from our result C 1. Thelong-wavelength,low-temperaturebehaviorofthe
→
The reduces uniform susceptibility χ˜ calculated from σ correlationfunctionsofEq. (4.4)isgiven,accordingto
Eq. (4.7) is shown in Fig. 7. Again, our results Eqs. (2.8), (2.14), and (4.8), by
are in a fairly good agreement with the MC data of
Ref. 12, which are, in turn, in accord with the high- 3κ2 θ
σ1 =3, σ2 = , σ3 = , (5.1)
temperature series expansion (HTSE) results of Ref. 4 ∼ ∼ κ2+q2 ∼ 3
(not shown). Here, in contrast to the heat capacity, our
result χ˜ = 1/3, or χ = 1/(6J), at T = 0 is exact. This where the quantity κ2 = 2θ/3 in σ2 defines the correla-
followsfromthe factthat the zero-temperaturesuscepti- tion length
bilities of the classical kagom´e antiferromagnet have the
1/2
same value χ=1/(6J)for all directions of the field with 1 3
ξ = = . (5.2)
c
respect to the three spins on a triangle being mutually κ 2θ
(cid:18) (cid:19)
oriented at 120 . On the contrary, for conventional low-
◦
Theappearanceofthislengthparameterimpliesthatthe
dimensionalantiferromagnets,whichshowtwo-sublattice
real-spacespinCF’sdefined,accordingtoEqs. (3.17)and
short-range correlations, there are D 1 susceptibilities
− (2.4), by
9
slilj′ =v0 (2dπq)deiq·(rli−rlj′)Uln(q)Ul′n(q)σn(q) (5.3) ofTthoesoturddeyrroefalt-hsepalacettcicoerrseplaatciionngfaunndcttioonlisstatthdeisptaanrtciecs-
Z
ular cases of the general formula (5.6), it is convenient
decayexponentiallyatlargedistancesatnonzerotemper-
to enumerateCF’s bythe numbersn andn definedby
u v
atures. Incontrasttoconventionallattices,divergenceof Eq. (2.1), as is shown in Fig. 1. Thus sll′ is the cor-
ξ atθ 0doesnotleadheretoanextendedshort-range nu,nv
c relation function of the l sublattice spin of the “central”
order, i→.e., to strong correlation at distances r<ξ . The
c triangle (0,0) with the l sublattice spin of the triangle
zero-temperature CF’s are purely geometrical q∼uantities translated by (n ,n ). ′(Note that sl′l = sll′ , in
which are dominated by σ1 and have the form u v nu,nv 6 nu,nv
general.) There is a number of several useful relations
slilj′ =3v0 (2dπq)deiq·(rli−rlj′)Ul1(q)Ul′1(q) bslnel,t0woevenerclorartelTati=on0fuisnczteiroonsb.yFvirirsttu,ethoefsEumqs.of(t5h.4e)CaFn’ds
Z (2.12):
=3v dq cos[q·(rl rl′)]
0 (2π)d i− j s11 +s22 +s33 =0, T =0. (5.9)
n,0 n,0 n,0
Z
sin(ul·q)sin(ul′·q) , (5.4) Takingintoaccountthesymmetryofthelattice,onecan
× sin2(u·q)+sin2(v·q)+sin2(w·q) put this relation into the form of the “star rule”
where, according to Eq. (2.11), sll +sll +sll =0, l=1,2,3 (5.10)
n,0 0,n n,n
u w, u v, u u. (5.5) for the sum of the correlation functions along the direc-
1 2 3
≡ ≡ ≡ tions u, v, and w [see Eqs. (2.2) and (2.3)]. The star
Atlargedistances r rl rl′,the smallvaluesofq are rule does not hold at nonzero temperatures, which can
importantinEq. (5.i4j)≡. Tih−usojnecanexpandthesinesto be easily seen from the HTSE for the spin CF’s starting
the lowestorderanduse (u·q)2+(v·q)2+(w·q)2 =−32q2. from 1/T2n for s1n1,0 and s2n2,0 and from 1/T2n+1 for s3n3,0.
Afterthatintegrationcanbedoneanalyticallyandyields Moredetailedanalysisshowsthatinthelow-temperature
the asymptotic result region the sum in Eqs. (5.9) and (5.10) is smaller than
s33 byafactoroforder(κn)2ln[1/(κn)]atthe distances
n,0
sll′ = 2√3 (ul·ul′)ri2j −2(ul·rij)(ul′·rij) (5.6) κn≪1. Thus the star rule can be used with a good ac-
ij ∼ π r4 curacy in the whole range θ 1. An additional relation
ij ≪
canbefoundfromtheconditionthatatzerotemperature
for r 1. That is, at zero temperature spin CF’s the sum of spins in each of the triangles is zero. Thus
ij
≫
decrease at the scale of the lattice spacing and decay ac- one obtains, e.g., the “triangle rule”
cordingtoapowerlaw1/r2 atlargedistances. Theform
sl1 +sl2 +sl3 =0, T =0 (5.11)
of Eq. (5.6) is that of the dipole-dipole interaction in nu,nv nu,nv nu,nv
a two-dimensional world. Here the elementary transla- for all l, n , and n , as well as similar relations.
u u
tion vectors ul associated with each of three sublattices ThemostnontrivialoftherelationsbetweenspinCF’s
[see Eqs. (5.5), (2.2), and (2.3)] play the role of dipole is the “hexagon rule”
moments.
At nonzero temperatures, an additional exponential shex ( 1)ζsrr′ =σ1δr hex (5.12)
decay of the correlationfunctions appears, which is gov- ≡r′ hex − ∈
X∈
erned by the correlation length ξ of Eq. (5.2). For
c for the correlators between a site r and all the sites r
θ 1,the third-eigenvalueterm,n=3,inEq. (5.3)can ′
≪ belonging to hexagons,which are taken with alternating
still be neglected, and one can use the first and second
signs. Ifthesiteritselfbelongstothehexagon,theright-
columnsofthelong-wavelengthformofthediagonalizing
matrix U (q), Eq. (2.15). The resulting CF sll′(q) of handsideofEq. (5.12)isnonzeroandtheautocorrelation
ln
function s in the sum is taken with the positive sign.
Eq. (3.17), which enters Eq. (5.3), has the form rr
As follows from Eq. (4.4) and the temperature depen-
sll′(q)∼= κ2(−1+3δlκl′2)++2q(2ul·q)(ul′·q). (5.7) dinenthceisocfatsheefrgoampp1aartamhiegtherteGm,ptehreatquuraensttitoy3sahetxlocwhatnegmes-
peratures. This very deep relation has been derived in
Whereas the κ2 term in the numerator yields only small Ref. 4 from the condition that the largest eigenvalue of
contributions θ in the real-space CF’s, that in the de- the correlation matrix σ1 is independent of q. For mod-
nominator resu∝lts in the additional exponentially decay- els with finite D this condition and hence the hexagon
ing factor rule (5.12) is violated only at order 1/T8 of the HTSE.4
For our D = model, σ1 given by Eq. (3.18) remains
∞
dispersionless at all temperatures, and the hexagon rule
1, κn 1
κr K (κr )= ≪ (5.8) is always satisfied.
ij 1 ij ∼ πκrij/2e−κrij, κn≫1, At long distances, the zero-temperature sublattice-
q diagonal CF’s in the horizontal direction, which follow
in Eq. (5.6).  from Eq. (5.6), have the form
10