Unconventional quantum ordered and disordered states in the highly frustrated spin-1/2 Ising-Heisenberg model on triangles-in-triangles lattices

0.78 MB·English

by  Jana Cisarova

APS/123-QED
Unconventional quantum ordered and disordered states in the highly frustrated
spin-1/2 Ising-Heisenberg model on triangles-in-triangles lattices
Jana Cˇisa´rov´a1,2, and Jozef Streˇcka1,
∗ †
1Department of Theoretical Physics and Astrophysics, Faculty of Science,
P. J. Sˇafa´rik University, Park Angelinum 9, 040 01, Koˇsice, Slovak republic
2Institute of Theoretical Physics, Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland
(Dated: December 21, 2012)
2
1 Ground state and thermodynamics of geometrically frustrated spin-1/2 Ising-Heisenberg model
0 on two differentbut topologically related triangles-in-triangles lattices isinvestigated in particular.
2 A rigorous mapping based on generalized star-triangle transformations enables one to derive exact
c analytical results for the free and internal energy, spontaneous magnetization, entropy and specific
e heat from the known exact results for the corresponding spin-1/2 Ising model on a triangular lat-
D tice. It is demonstrated that several unconventional quantum ordered and disordered phases may
become the ground state in addition to the classical ferromagnetic phase with a perfect alignment

0 of all constituent spins. A mutual competition between the ferromagnetic Ising and XXZ Heisen-
2 berg interactions may lead to an appearance of a peculiar quantum ferromagnetic phase, in which

the magnetic moment of the Heisenberg spins undergoes a quantum reduction to one third of the
]
h saturation value in contrast with the fully saturated magnetic moment of the classical Ising spins.
c In the highly frustrated regime, we have found either the disordered quantum paramagnetic phase
e with the substantial residual entropy or a similar but strikingly spontaneously long-range ordered
m phase,inwhichaquantumreductionofthemagneticmomentoftheHeisenbergspinscausesasmall
quantumreductionofthemagneticmomentofotherwisepurelyclassicalIsingspins. Itisevidenced
-
t thatthespecificheatexhibitsdiversethermaldependenceswithorwithoutalogarithmicdivergence
a
superimposed either on one or two round Schottky-typemaxima.
t
s
t. PACSnumbers: 05.50.+q,64.60.F-,75.10.Jm,75.30.Kz,75.40.Cx
a Keywords: Ising-Heisenbergmodel,geometricspinfrustration,triangles-in-triangleslattices
m
-
d I. INTRODUCTION reentrant phase transitions,10,11 the quantum reduction
n of the magnetization,13,14 the presence of magnetization
o jumps and intermediate plateaus in the magnetization
c Frustrated two-dimensional (2D) quantum spin mod- process,3,15,16 ”order-from-disorder” effect,17,18 the ap-
[
els belong to the most challenging issues to deal with in pearanceofseveralunusualspin-liquidgroundstates,1,2,4

2 the modern statistical and condensed matter physics.1–4 the enhanced magnetocaloric effect,19–21, the localized
v A considerable interest in investigating 2D geometri- magnons in a close vicinity of saturation field,22–25 etc.
7
cally frustrated spin models is closely connected with Despite a great effort in this research area, the classi-
3
3 an existence of numerous layered insulating magnetic cal and quantum frustrated spin models defined on 2D
0 materials,4–7 which can be characterized by incapabil- ”triangles-in-triangles”(TIT)latticeshavereceivedmuch
. ity of spins to simultaneously satisfy all pair spin-spin less attention so far. TIT lattices generally consists of
0
interactions. In general, the magnetism of the vast ma- smaller triangular entities embedded in either some or
1
2 jority of insulating materials is well captured by the all triangular cells of 2D triangle-based lattices such as
1 quantum Heisenberg model and its various extensions.8 triangular or kagom´e lattice. The sites of original lat-
: It should be mentioned, however, that a mutual inter- tice might be thus called as nodal lattice sites, while
v
play between the geometric frustration and quantum the sites of smaller triangular entities may be viewed as
i
X fluctuations is not only responsible for an extraordi- decorating lattice sites. In this way, one may consider
r nary diverse magnetic behavior, but is also the main various geometrically frustrated spin models on differ-
a
cause of a mathematical intractability of geometrically ent but geometrically related TIT lattices. If all lattice
frustrated Heisenberg models with exception of a few sites are occupied by the Ising spins, one considers the
prominent exactly solved cases.9 On the other hand, classical Ising model whose magnetic behavior is not af-
the classical Ising model is still exactly solvable on sev- fected by quantum fluctuations at all. If all lattice sites
eral frustrated 2D lattices and it may thus provide a are occupied by quantum Heisenberg spins, one turns to
valuable insight into cooperative phenomena originating the quantum Heisenberg model that is basically affected
from the spin-frustration effect.10,11 The most limiting through quantum effects. If the nodal lattice sites are
drawback of the Ising model unfortunately lies in the occupied by the Ising spins and the decorating sites by
lack of appropriate experimental realizations which are the quantum Heisenberg ones, one considers the hybrid
rather scarce.8,12 Among the most striking phenomena Ising-Heisenberg model affected through local quantum
emerging in geometrically frustrated classical and quan- fluctuations only.
tum spin systems one could mention the existence of Up to now, there exist only a few theoretical studies
2
ofclassicalandquantumspinmodelsononeveryspecial to the partially ordered and partially disordered ground
exampleofTITlatticecalledasthetriangulatedkagom´e state of the second model, which is characterized by the
lattice.26–35 A particular interest in studying the role of obvious quantum reduction of the spontaneous magneti-
geometric frustration in the triangulated kagom´e lattice zation.
has been stimulated by the family of polymeric coordi- The organization of this paper is as follows. Both in-
nation compounds Cu9X2(cpa)6 nH2O (X=F,Cl,Br and vestigatedmodelsareintroducedinSectionIIalongwith
·
cpa=carboxypentonic acid), which provides an experi- basicstepsoftheirexactanalyticaltreatment. Themost
mental realization of highly frustrated magnetic materi- interestingresultsforthegroundstate,orderparameters
als with a remarkablyhigh frustrationratio (>130) and and overall thermodynamics are then discussed in Sec-
possibly spin-liquid ground state.36–41 In addition, there tion III. Finally, several concluding remarks are drawn
are also experimental indications of one-third plateau in Section IV.
inthe low-temperaturemagnetizationprocess,40,41 while
the temperaturedependence ofthe susceptibility implies
two basically different exchange interactions in this in-
II. MODEL AND SOLUTION
triguing series of magnetic compounds.37–41 A stronger
antiferromagnetic coupling has been assigned to the ex-
change interaction inside the smaller triangles and a Let us introduce the spin-1/2 Ising-Heisenberg model
possibly much weaker ferromagnetic coupling to the ex- on two different TIT lattices, which are schematically
change interaction within the larger triangles of the tri- depicted on the left-hand-side of Fig. 1(a)-(b). As il-
angulated kagom´e lattice.40,41 ESR measurements gave lustrated, the Ising spins σ = 1/2 placed at nodal lat-
the almost isotropic g-value and this result is taken to tice sites of a simple triangular lattice (full circles) are
mean that Cu X (cpa) nH O compounds afford a suit- mutually inter-connected through smaller triangles of
9 2 6 2
able experimental realiza·tionof the isotropic Heisenberg the quantum Heisenberg spins S = 1/2 (empty circles),
model.39,41 Of course, the quantum Heisenberg model which are either placed into each up-pointing triangle
on the triangulated kagom´e lattice cannot be rigorously (Fig.1(a))orintoeachtriangle(Fig.1(b))oftheunderly-
solvedandthus, this model has been dealt with the help ing triangularlattice. Fromthis perspective, the smaller
of different approximate perturbation approaches31,32 trianglesof three quantumHeisenberg spins (Heisenberg
and numerical exact diagonalization.33–35 However, the trimers)areembedded in the largertrianglesofthe clas-
substantial difference between the exchange interactions sical Ising spins and hence, both investigated magnetic
within the smaller and larger triangles of the triangu- structures fall into the class of TIT lattices. The total
latedkagom´elatticeprovidesaplausiblegroundforintro- Hamiltonian of the spin-1/2 Ising-Heisenberg model on
ducing the simplified Ising-Heisenbergmodel, which cor- both aforedescribed TIT lattices can be defined as
rectlyreproducesthestrongerHeisenbergcouplinginside
the smaller triangles and merely approximatesthe much = J ◦−◦ ∆ SˆxSˆx+SˆySˆy +SˆzSˆz J ◦−•σˆzSˆz, (1)
weakerexchangeinteractioninsidethelargertrianglesby H − H i j i j i j − I k l
the Ising coupling. A comparison between the exact so- <Xi,j>h (cid:16) (cid:17) i <Xk,l>
lutionsoftheIsing26,27 andIsing-Heisenberg28–30 models
where σˆz and Sˆα (α = x,y,z) denote spatial compo-
ontriangulatedkagom´elattice revealsthat a presenceof k i
nents of the spin-1/2 operator of the Ising and Heisen-
localquantumfluctuationsleadstoamarkeddecreasein
berg spins, respectively. The parameter J stands for
the residual entropy of the quantum paramagnetic state H
the XXZ Heisenberg interaction between the nearest-
of the latter classical–quantummodel.
neighborHeisenbergspinsfromthesmallinnertriangles,
Ontheotherhand,thereisalackoftheoreticalstudies ∆ is a spatial anisotropy in this interaction and the pa-
concerning with the classical and quantum spin models rameter J represents the Ising-type interaction between
I
defined on other geometrically frustrated TIT lattices, the nearest-neighbor Ising and Heisenberg spins.
which could be derived for instance from a simple trian- Forfurtherconvenience,letusrewritethetotalHamil-
gular lattice. The main goal of the present work is to tonian as a sum over the six-spin cluster Hamiltonians
fill in this gap, whereas the main emphasis will be laid schematically illustrated in Fig. 1(c)-(d)
on a comparative study of the Ising-Heisenberg models
defined on two different but geometrically related TIT γN
lattices each of them descended from a simple triangu- ˆ = ˆ . (2)
k
lar lattice. It will be demonstrated that two different H H
k=1
X
magnetic architectures, even though they are geometri-
cally closely related, will cause a different level of local Here, N stands for the total number of the Ising spins
quantumfluctuationsandthusfundamentallychangethe andγN denotesthetotalnumberofsix-spinclusters,i.e.
magnetic behavior of the considered spin models espe- γ = 1 for the TIT lattice drawn in Fig. 1(a) and γ = 2
cially in the highly frustrated regime. It actually turns for the TIT lattice shown in Fig. 1(b). It is of principal
out that the ground state of the first considered model importance that all the interaction terms including the
in the highly frustrated regime is disordered in contrast threeHeisenbergspinsfromthekthinnertrianglebelong
3
J
eff
Y-Ñ
(a)
2J
eff
Y-Ñ
Y-Ñ
(b)
(c) s s s S s s J s
k3 k3 k1 k1 k2 k1 eff k2
S JIJ JIS Y-Ñ J J S J S Y-Ñ J J
k2 H k3 eff eff kJ2 HJ k3 eff eff
I I
S (d)
s s s J s s s
k1 k1 k2 k1 eff k2 k3 k3
FIG. 1: (a)-(b) The spin-1/2 Ising-Heisenberg model on two different but geometrically related TIT lattices and its rigorous
mappingtothespin-1/2 Isingmodel on a triangular lattice. The full circles denotelattice positions oftheIsing spinsσ=1/2
and the empty ones lattice positions of the Heisenberg spins S = 1/2; (c)-(d) Schematic representation of two equivalent
star-triangle transformations used for establishing a precise mapping correspondence between both themodels.
to the same cluster Hamiltonian ˆ spindegreesoffreedomofthekthHeisenbergtrimer,the
k
H
cluster partition function will solely depend on the
3 Zk
ˆ = J [∆(Sˆx Sˆx +Sˆy Sˆy )+Sˆz Sˆz ] three nodal Ising spins σk1, σk2, andσk3 attachedto the
Hk − H k,i k,i+1 k,i k,i+1 k,i k,i+1 kth Heisenberg trimer. Moreover, one may take advan-
i=1
X tage of the generalized star-triangle transformation42–45
3
−JIXi=1Sˆkz,i(σˆkz,i+σˆkz,i+1), (Sˆkα,4 ≡Sˆkα,1,σˆkα,4 ≡σˆkα,1) Zk=(Aσkze1x,pσkz[β2,Jσkz3()σ=z σTzrk+exσp(z−σβzHˆ+k)σz σz )], (5)
(3) eff k1 k2 k2 k3 k3 k1
whichcanbeusedinordertoreplacethiseffectiveBoltz-
which in turn ensures a validity of the standard com-
mutation relation [ˆ , ˆ ] = 0 between different cluster mann’s factor by the equivalent expression depending
i j
H H on the three nodal Ising spins only. The above map-
Hamiltonians i = j. Owing to this fact, the total parti-
6 ping transformation generally represents a set of eight
tion function can be partially factorized into a product
algebraic equations, which can be obtained from Eq. (5)
ofcluster partitionfunctions andone may traceoutspin
by substituting all available spin configurations of three
degrees of freedom of different Heisenberg trimers inde-
Ising spins involved therein. In an absence of the exter-
pendently of each other according to the relation
nalmagneticfield,oneactuallygetsjusttwoindependent
γN γN equations; the first equation for two uniform configura-
= Trkexp( β ˆk)= k, (4) tions with the three equally aligned Ising spins
Z − H Z
{Xσi}kY=1 {Xσi}kY=1
V ( 1/2, 1/2, 1/2)
1 k
where β = 1/(k T), k is Boltzmann’s constant, T is ≡Z ± ± ±
B B
3 3
tshuemmabastoilounteovteermsppeinrasttuartee,stohfeasllytmhbeoIlsing{σspi}indseannodtetshae =2exp(cid:18)4βJH(cid:19)cosh(cid:18)2βJI(cid:19)
P
symbol Tr stands for a trace over spin degrees of free- 1 1
k +2exp βJ (1 4∆) cosh βJ
dom of the kth Heisenberg trimer. After tracing out the −4 H − 2 I
(cid:20) (cid:21) (cid:18) (cid:19)
4
1 1 on a simple triangular lattice, since the generalizedstar-
+4exp βJ (1+2∆) cosh βJ
H I
−4 2 triangle transformation (5) is performed locally for the
(cid:20) (cid:21) (cid:18) (cid:19)
3 same smallest commuting spin cluster in both the mod-
=Aexp βJeff , (6) els (see Fig. 1(c)-(d)). As a matter of fact, the most
4
(cid:18) (cid:19) crucial difference between both the mappings lies just
and the second equation for six non-uniform configura- in a strength of the effective nearest-neighbor coupling
tionswithoneIsingspinpointinginoppositewithrespect γJeff,whichistwiceaslargefortheTITlatticeshownin
to the other two Fig. 1(b) compared to that displayed in Fig. 1(a). The
other mapping parameter A represents just less impor-
V ( 1/2, 1/2, 1/2)= ( 1/2, 1/2, 1/2) tant multiplicative factor in the mapping relation (10)
2 k k
≡Z ± ± ∓ Z ± ∓ ±
between both partition functions. It is worthwhile to re-
= ( 1/2, 1/2, 1/2)
k
Z ∓ ± ± mark, moreover, that the exact result for the partition
3 1
=2exp βJ cosh βJ function of the spin-1/2 Ising model on the triangular
H I
4 2 lattice is well known46–49
(cid:18) (cid:19) (cid:18) (cid:19)
1 1
+2exp −4βJH(1+2∆) cosh 2βJI IM 1 2π 2π
(cid:20) (cid:21) (cid:18) (cid:19) ln Z2 = 8π2 ln C1−D1cosθ
+2exp 1βJ (1 ∆) cosh 1βQ+ (cid:18) (cid:19) Z0 Z0 (cid:20)
H
−4 − 2
(cid:20) (cid:21) (cid:18) (cid:19) D1cosφ D1cos(θ+φ) dθdφ,
− −
1 1 (cid:21)
+2exp βJ (1 ∆) cosh βQ
(cid:20)−4 H − (cid:21) (cid:18)2 −(cid:19) C1 = cosh3 βγJeff +sinh3 βγJeff
2 2
1 (cid:18) (cid:19) (cid:18) (cid:19)
=Aexp βJ . (7)
−4 eff βγJeff
(cid:18) (cid:19) D1 = sinh , (12)
2
(cid:18) (cid:19)
For abbreviation, we have introduced here two auxiliary
functions Q± defined as follows and hence, the partition function of the spin-1/2 Ising-
HeisenbergmodelontheTITlatticescanalsobeexactly
J ∆ 2 calculated by inserting the exact result (12) for the par-
Q = H J +2(J ∆)2. (8)
± s(cid:18) 2 ± I(cid:19) H ttirtiiaonngufulanrctlaiotnticZeIwMitohftthheeesffpeinct-i1v/e2nIesainregstm-noedigelhboonrtihne-
Itis noteworthythattwoalgebraicequations(6)–(7)un- teraction γJeff into the rigorous mapping relation (10).
ambiguously determine yet unspecified mapping param- At this stage, other basic thermodynamic quantities
eters A and J can also be calculated in a rather straightforward way
eff
within the framework of this rigorous mapping method.
A= V V3 14, βJ =ln V1 , (9) Usingthestandardrelationsofthermodynamicsandsta-
1 2 eff V tistical physics, the free energy F of the spin-1/2 Ising-
(cid:18) 2(cid:19)
(cid:0) (cid:1) Heisenberg model on the TIT lattices can be expressed
whichensureageneralvalidityofthe star-triangletrans- through the free energy F of the corresponding spin-
IM
formation (5) on assumption that the mapping parame- 1/2 Ising model on the triangular lattice
ters A and J are given by Eqs. (6)–(9).
eff
Substituting the generalized star-triangle transforma-
F =F (β,γJ ) γNk T lnA. (13)
IM eff B
tion (5) into the relation (4) one easily gets a rigorous −
mapping correspondence between the partition function
Similarly, the internal energy U of the spin-1/2 Ising-
ofthespin-1/2Ising-HeisenbergmodelontheTITlat-
Z Heisenberg model on the TIT lattices can be connected
tice and respectively, the partition function of the
ZIM to the internal energy UIM of the spin-1/2 Ising model
spin-1/2 Ising model on a simple triangular lattice
onthetriangularlatticewiththetemperature-dependent
nearest-neighbor interaction γJ
(β,J ,J ,∆)=AγN (β,γJ ), (10) eff
H I IM eff
Z Z
γN W 3W U (β,γJ ) W W
which is unambiguously given by the Hamiltonian with U= 1+ 2 + IM eff 1 2 .(14)
the effective nearest-neighbor interaction γJeff − 4 (cid:18)V1 V2 (cid:19) Jeff (cid:18)V1 − V2 (cid:19)
= γJ σzσz. (11) For completeness, let us merely quote that the explicit
HIM − eff i j form of the internal energy U of the spin-1/2 Ising
<i,j> IM
X
modelonthe triangularlatticecanbe forinstance found
Despite the different spatial symmetries, the Ising- inRef.[49],whiletwonewfunctionsW andW entering
1 2
Heisenberg model on the two distinct TIT lattices can theestablishedmappingrelationship(14)fortheinternal
be rigorously mapped to the same effective Ising model energy are explicitly given by the following formulas
5
∂V 3 3 3 3
1
W = = exp βJ J cosh βJ +2J sinh βJ
1 H H I I I
∂β 2 4 2 2
(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
1 1 1 1
exp βJ (1 4∆) J (1 4∆)cosh βJ 2J sinh βJ
H H I I I
− 2 −4 − − 2 − 2
(cid:20) (cid:21)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
1 1 1
exp βJ (1+2∆) J (1+2∆)cosh βJ 2J sinh βJ ,
H H I I I
− −4 2 − 2
(cid:20) (cid:21)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
∂V 3 3 1 2 1
2
W = = exp βJ J cosh βJ + J sinh βJ
2 H H I I I
∂β 2 4 2 3 2
(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
1 1 1 1
exp βJ (1+2∆) J (1+2∆)cosh βJ 2J sinh βJ
H H I I I
− 2 −4 2 − 2
(cid:20) (cid:21)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
1 1 1 1
JH(1 ∆)exp βJH(1 ∆) cosh βQ+ +cosh βQ−
− 2 − −4 − 2 2
(cid:20) (cid:21)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
1 1 1
+ exp βJ (1 ∆) Q+sinh βQ+ +Q sinh βQ . (15)
H − −
−4 − 2 2
(cid:20) (cid:21)(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
Furthermore, one may also readily obtain exact results resultant exact expression for the single-site magnetiza-
for the entropy S and specific heat C by making use of tion of the Ising spins in the spin-1/2 Ising-Heisenberg
the standard thermodynamic relations model on the TIT lattice
1
S =kBlnZ+ UT, C = ∂∂UT, (16) mI = 12"1− (Vγ +31V6Vγ)1γ(VV23γγ Vγ)3#8 , (19)
1 2 1 − 2
aftersubstitutingthepreviouslyderivedexactresultsfor
the partition function (10) and the internal energy (14) whereasγ =1fortheTITlatticeshowninFig. 1(a)and
into the relations (16). γ =2 for the TIT lattice drawn in Fig. 1(b).
Next, let us calculate the spontaneous magnetization Owing to the commuting character of different cluster
oftheIsingandHeisenbergspins,respectively. Thespon- Hamiltonians, the ensemble average of any function f2
taneous magnetization of the Ising spins can be simply of the Ising and Heisenberg spins belonging to the kth
calculatedby exploitingthe exactmapping theoremsde- cluster Hamiltonian (3) can be calculated from the gen-
veloped by Barry and co-workers50–53 eralized form of Callen-Suzuki spin identity56–58
hf1(σˆiz,σˆjz,...,σˆkz)i=hf1(σˆiz,σˆjz,...,σˆkz)iIM, (17) hf2(Sˆkα1,Sˆkα2,Sˆkα3,σˆkz1,σˆkz2,σˆkz3)i=
Tr f (Sˆα ,Sˆα,Sˆα ,σˆz ,σˆz ,σˆz )exp( β ˆ )
according to which the canonical ensemble average k 2 k1 k2 k3 k1 k2 k3 − Hk .(20)
ofanyfunctionf1 oftheIsingspinsinthespin-1/2Ishi·n·g·i- * Trkexp(−βHˆk) +
Heisenberg model on the TIT lattice directly equals the
The exact spin identity (20) allows us to express, after
canonical ensemble average of the same function
h···iIM straightforward but a little bit tedious calculation, the
f of the Ising spins in the equivalent spin-1/2 Ising
1 spontaneous magnetization of the Heisenberg spins m
model on the triangular lattice. This result is taken to H
in terms of the triplet correlation t σˆz σˆz σˆz
mean that the spontaneous magnetization of the Ising IM ≡ h k1 k2 k3iIM
between three Ising spins from the kth cluster Hamilto-
spins m in the spin-1/2 Ising-Heisenberg model on the
I nian and the formerly derived the single-site magnetiza-
TIT lattice caninturnbe calculatedfromthe single-site
tion m of the Ising spins
magnetization m of the spin-1/2 Ising model on the IM
IM
triangular lattice54
m Q Q 2t Q Q
m Sˆz = IM 1+ 2 + IM 1 3 2 ,(21)
mI≡hσˆizi=hσˆiziIM ≡mIM(β,γJeff), H≡h ki 2 (cid:18)V1 V2(cid:19) 3 (cid:18)V1 − V2(cid:19)
1
1 16z6 8 and two newly defined functions Q1 and Q2
m = 1 , (18)
IM 2 − (1+3z2)(1 z2)3
(cid:20) − (cid:21) 3 3
Q =3exp βJ sinh βJ
1 H I
4 2
with the parameter z = exp( γβJeff/2). Substituting (cid:18) (cid:19) (cid:18) (cid:19)
−
the explicit form (9) of the effective nearest-neighborin- 1 1
+exp βJ (1 4∆) sinh βJ
teraction γJeff into the formula (18) one finally gets the −4 H − 2 I
(cid:20) (cid:21) (cid:18) (cid:19)
6
1 1 given by Eqs. (6) and (7) instead of β. It is quite
+2exp βJ (1+2∆) sinh βJ ,
H I
−4 2 evident from the critical condition (25) that the spin-
(cid:20) (cid:21) (cid:18) (cid:19)
3 1 1/2 Ising-Heisenberg model on the TIT lattice is spon-
Q2=3exp βJH sinh βJI taneouslylong-rangeorderedwheneverthe effective cou-
4 2
(cid:18) (cid:19) (cid:18) (cid:19) pling is greaterthan the criticalvalue γβJ >ln3, oth-
eff
1 1
erwise it becomes disordered for γβJ <ln3. Thus, an
+exp βJ (1+2∆) sinh βJ eff
H I
−4 2 intersection of the effective coupling γβJ with its crit-
(cid:20) (cid:21) (cid:18) (cid:19) eff
1 1 ical value provides a feasible criterion for obtaining the
+exp βJ (1 ∆) cosh βQ+
H critical points of the spin-1/2 Ising-Heisenberg model on
−4 − 2
(cid:20) (cid:21) (cid:18) (cid:19) both TIT lattices, which will be thoroughly discussed in
1 1
exp βJ (1 ∆) cosh βQ . (22) Section IIIB.
H −
− −4 − 2
(cid:20) (cid:21) (cid:18) (cid:19)
To complete our exactcalculationofthe single-sitemag-
III. RESULTS AND DISCUSSION
netizationofthe Heisenbergspins m itisnowsufficient
H
to findthe tripletcorrelationt σˆz σˆz σˆz ofthe
spin-1/2Ising model on the triIaMng≡ulhark1latkt2icek,3wiIhMich has In this section, let us proceed to a discussion of the
been rigorously calculated by Baxter and Choy55 most interesting results obtained for the ground state
and finite-temperature properties of the spin-1/2 Ising-
m y 2z2+1 (y+3)(y 1) Heisenberg model on two considered TIT lattices. First,
IM
tIM = 4 1+2 − y+−z2 2 − ,(23) it is worth mentioning that all the results derived in
" p− # the foregoing section are valid regardless of whether the
interaction constants are assumed to be ferromagnetic
where y = exp(γβJ ) and z = exp( γβJ /2) as be-
eff − eff or antiferromagnetic. While the change in character of
fore. Substituting the effective nearest-neighbor interac-
the Heisenberg interaction has a profound effect upon
tion γJ into the parameters y and z entering the for-
eff the magnetic behavior of the spin-1/2 Ising-Heisenberg
mula(23)oneobtainstherequiredclosed-formexpression
model on the TIT lattices, the respective change in the
for the triplet correlation of the Ising spins
Ising interactionJ J merely causes a rather trivial
I I
tIM = m4IM 1+2V12γ −(V2γV22γV+γ)V21γV2γ fsoprien,wreevewrislallhσenizce→fo→r−thσ−irzesotfriacltlotuhreaItstienngtisopninosn.lyTtohetrhee-
" 1 − 2 particularcaseofthemodelwiththe ferromagneticIsing
(Vγ +3Vγ)(Vγ Vγ) interaction JI > 0, while the respective behavior of the
−2V1γp 1 (V1γ −2 V2γ1)2− 2 #.(24) mcaondseilmwpitlyhbtheedaendtuifceerdrofmroamgntehteicrIessinugltsinttoerbaectpiornesJenIt<ed0.
Let us finally mention that the Ising interaction will be
Last but not least, let us make a few comments on a
subsequently used as the energy unit when defining two
critical behavior of the spin-1/2 Ising-Heisenberg model
dimensionless parameters: the dimensionless tempera-
on the TIT lattices. It should be noted here that the
ture k T/J and a relative strength of the Heisenberg
B I
critical behavior is always accompanied with a non-
interaction with respect to the Ising one J /J .
H I
analytic behavior of the partition function or some of its
higher-order temperature derivative. However, it may
be easily understood from Eqs. (6)-(9) that the map-
A. Ground state
ping parameter A as well as any temperature derivative
of it is a smooth continuous function and therefore, it
Let us examine first the ground state of the spin-
must not cause a critical behavior of the spin-1/2 Ising-
1/2Ising-Heisenbergmodel ontwo geometricallyrelated
Heisenberg model. Bearing this in mind, the spin-1/2
TIT lattices under consideration. Because of the com-
Ising-HeisenbergmodelontheTITlatticesbecomescrit-
muting character of different cluster Hamiltonians, the
icalif andonly if the partitionfunction ofcorresponding
overallground-statespinarrangementis simply givenby
spin-1/2 Ising model on the triangular lattice becomes
a tensor product over the lowest-energy eigenstates of
critical as well. Accordingly, the critical condition al-
the cluster Hamiltonian (3). If the lowest-energy eigen-
locating critical points of the spin-1/2 Ising-Heisenberg
value of the cluster Hamiltonian (3) is degenerate, how-
model on the TIT lattices can readily be obtained from
ever, there does not exist a unique ground state but the
a comparisonofthe effective coupling ofthe correspond-
highly degenerate ground-state manifold whose degener-
ingspin-1/2Isingmodelonthetriangularlatticewithits
acy might represent rather complex counting problem.
critical value
Typical ground-state phase diagrams in the ∆ J /J
H I
−
γβ J =ln3 Vγ(β )=3Vγ(β ). (25) plane including all possible ground states are depicted
c eff ⇔ 1 c 2 c in Fig. 2(a)-(b) for both investigated TIT lattices. Al-
Here, it is emphasized that the inverse critical temper- thoughtheground-stateboundariesareidenticalforboth
ature β = 1/(k T ) enters the parameters V and V theinvestigatedmodels,thereisafundamentaldifference
c B c 1 2
7
in the character of spin arrangements of some ground-
state phases. As one can see from Fig. 2(a), the Ising- (a) 3
Heisenberg model on the first TIT lattice from Fig. 1(a)
displays two spontaneously long-range ordered and one
2
disordered ground state, more specifically, the classical QFP
ferromagnetic phase (CFP), the quantum ferromagnetic
|
phase(QFP),andthedisorderedquantumparamagnetic JI1
|
phase(QPP).Allthreephasesaremutuallyseparatedby / H CFP
tgwlgaowirlhwviogoieuecnnslnhmitndb-CCeeewysnsnFFtettearoPPhretofge--efQQydsoca.ibPFoslAltcenPPaoitdsg:(ihnineOctetneoiiDondsIunstuPliadsobn)tuyg:ebsseac(sJJeoJJficxHHmwIIorpsnepte==sl-acltorti−a∆retindsud2ge1t−,Hri+Cn)2et1ghiF∆ps,ePeh(,ntafehwbosneeerietr∆rthgrgreaays>snpppsio1eneif)tcrs((itfot22eihnv67ceest)) (b) --| J021230DP 1 2 3QQPFPP4 5
N Nγ |JI1
|CFPi= |↑iσiz |↑↑↑iSkz1,Skz2,Skz3, (28) J / H CFP
i=1 k=1
Y Y 0
dominatesintheprevailingregionoftheparameterspace
with the ferromagnetic Heisenberg interaction J > 0.
H -1
If the Heisenberg intra-trimer interaction has a strong ODP
DP
easy-plane anisotropy, however, one may also encounter
-2
a more striking spontaneous long-range order
0 1 2 3 4 5

N Nγ FIG. 2: (Color online) Ground-state phase-diagrams of the
1
QFP = ( + + ) (,29) spin-1/2 Ising-Heisenberg model on the two TIT lattices
| i |↑iσiz √3 |↑↑↓i |↑↓↑i |↓↑↑iSkz1,Skz2,Skz3 shown in Fig. 1(a) and (b). For a detailed description of
i=1 k=1
Y Y
thephases see the text.
in which a perfect alignment of all the Ising spins
is accompanied with a symmetric quantum superposi-
tion of three up-up-down spin states of the Heisenberg
trimers. Accordingly, the spontaneous magnetization of
the Heisenberg spins undergoes a quantum reduction of
the magnetization to one third of the saturation magne-
tization and hence, we will refer to QFP as to the quan-
tum ferromagnetic phase in view of a parallel orienta-
tion of both sublattice magnetizations of the Ising and to twoHeisenbergspins creatingsingletpairwillbe con-
Heisenberg spins, respectively. An origin of this uncon- sequently decoupled from the rest of the respective spin
ventionalspontaneouslong-rangeorderingcanberelated clusterandinthisway,thespinfrustrationoftheHeisen-
toacompetitionbetweentwodifferentbutferromagnetic berg trimers is also mediated to the Ising spins. Despite
interactions, namely, the easy-axis Ising and easy-plane the spin frustration, the Ising spins still have a tendency
Heisenberg (∆>1) interaction. of showing ferromagnetic short-rangeorder as evidenced
Last but not least, the parameter region with the an- by the zero-temperature limit of the effective coupling
tiferromagnetic Heisenberg interaction is dominated by lim γβJ = ln2 (γ = 1) in the corresponding spin-
T 0 eff
QPP without any spontaneous long-range order. The 1/2 I→sing model on the triangular lattice implying that
disordered phase occurs on behalf of a spin frustration σzσz 0.06564. The ferromagnetic short-rangeor-
h i jin.n. ≃
of the Heisenberg trimers, which are incapable to simul- der of the Ising spins is related to tenfold degeneracy of
taneously satisfy specific requirements of the antiferro- the lowest-energy eigenstate of the cluster Hamiltonian
magnetic intra-trimer interaction between the Heisen- (3), whereas four out of ten lowest-energy eigenstates of
berg spins. If the antiferromagnetic intra-trimer inter- theHeisenbergtrimeroccuraslongasthethreeenclosing
action is stronger than the threshold value (27), then, Ising spins are equally aligned and another six provided
the Heisenberg trimers have a tendency to form one sin- that one out of three Ising spins points in opposite with
gletdimerandoneunboundspin. TheIsingspincoupled respect to the other two (see Fig. 3 for pictorial repre-
8
s s
sentation of all lowest-energy eigenstates) k3 k3
|↑↑↑iσkz1,σkz2,σkz3 √112((|↑↓↑i−|↓↑↑i))Skz1,Skz2,Skz3 Sk2 Sk3 Sk2 Sk3
|↑↑↑iσkz1,σkz2,σkz3 √2 |↑↑↓i−|↓↑↑i Skz1,Skz2,Skz3
1 ( ) s S s s S s
|↓↓↓iσkz1,σkz2,σkz3 √2 |↑↓↓i−|↓↑↓i Skz1,Skz2,Skz3 k1 k1 k2 k1 k1 k2
|↓↓↓iσkz1,σkz2,σkz3 √12(|↑↓↓i−|↓↓↑i)Skz1,Skz2,Skz3 sk3 sk3
||QOPDPPii=( ||↓↑↑↓↑↑iiσσkkzz11,,σσkkzz22,,σσkkzz33 √√1122((||↑↑↓↑↑↓ii−−||↓↓↑↑↑↑ii))SSkkzz11,,SSkkzz22,,SSkkzz33 Sk2 Sk3 Sk2 Sk3
1 ( )
||↑↑↑↓↓↓iiσσkkzz11,,σσkkzz22,,σσkkzz33 √√122(||↑↑↑↓↓↓ii−−||↑↓↓↑↑↓ii)SSkkzz11,,SSkkzz22,,SSkkzz33 sk1 Sk1 sk2 sk1 Sk1 sk2
1 ( )
|↓↑↓iσkz1,σkz2,σkz3 √2 |↑↓↓i−|↓↓↑i Skz1,Skz2,Skz3 s s s
1 ( ) . k3 k3 k3
|↓↓↑iσkz1,σkz2,σkz3 √2 |↓↑↓i−|↓↓↑i Skz1,Skz2,Skz3
(30) Sk2 Sk3 Sk2 Sk3 Sk2 Sk3
If the three enclosingIsing spins are equally aligned(say
all three Ising spins are pointing up), one actually finds sk1 Sk1 sk2 sk1 Sk1 sk2 sk1 Sk1 sk2
just the two linearly independent lowest-energy eigen- s s s
k3 k3 k3
statescomposedofonesingletdimerandoneHeisenberg
spin polarized by the two surrounding Ising spins into S S S S S S
their direction (see the first four eigenstates in Eq. (30) k2 k3 k2 k3 k2 k3
schematically drawn in the first two rows of Fig. 3). On
theotherhand,thereexistsjustonelowest-energyeigen- sk1 Sk1 sk2 sk1 Sk1 sk2 sk1 Sk1 sk2
state of the Heisenberg trimer being composed of the
FIG.3: Thelowest-energyeigenstatesoftheclusterHamilto-
singlet pair and one polarized Heisenberg spin on as-
nian (3), which constitute the ground-state manifold for the
sumptionthatoneIsingspinispointinginoppositewith
QPP and ODP phases. The spin polarization is given by an
respect to the other two. Under this condition, the po-
arrow and an ovaldenotes a singlet-dimer state.
larized Heisenberg spin is always connected to the two
equally aligned Ising spins and the singlet pair decou-
ples the third unequally oriented Ising spin from the re-
maining spins of the relevant spin cluster (see the last there emerge either two or one lowest-energy eigenstate
six eigenstates in Eq. (30) schematically illustrated in depending onwhetherthe threeenclosingIsingspins are
the last two rows of Fig. 3). Altogether, the ground- equally aligned or not.
statemanifoldinherenttothe QPPwillbe characterized As far as the ground state of the spin-1/2 Ising-
byaratherhugebutnon-trivialmacroscopicdegeneracy, Heisenberg model on the other TIT lattice (Fig. 1(b))
which will be explored in a more detail in Section IIID is concerned, the ground-state boundaries and the re-
by the analysis of entropy. spective spin arrangements of two previously reported
TheotherdisorderedstateDPemergingforthepartic- spontaneously long-range ordered phases CFP and QFP
ular limiting case of ∆=0 (i.e. the Ising limit) deserves remain unchanged. However, it surprisingly turns out
aspecialmention,sincethereappearsadramaticchange that the Ising-Heisenberg model on the second TIT lat-
inthecharacterofthelowest-energyeigenstatesthatcon- tice displays even in the highly frustrated region a quite
stitute the ground-statemanifold due to a complete lack peculiarspontaneouslong-rangeorderinsteadofthe dis-
of local quantum fluctuations. It can be easily proved ordered QPP for arbitrary but non-zero anisotropy pa-
that there exist just three different lowest-energy spin rameter∆=0(notethatthecharacteroftheotherdisor-
6
configurationsonthesmallerinnertrianglesforanygiven deredstateDPtoemergefor∆=0remainsunchanged).
configurationofthethreeenclosingIsingspins(seeFig.4 The existence of this unconventional spontaneous order-
for the pictorial representation of the lowest-energycon- ingcouldbeascribedtothequantumorder-from-disorder
figurations). Accordingly, the overall degeneracy of the effect, which partially lifts the ground-state degeneracy
ground-statemanifoldbecomes for the DP phase a quite of QPP due to a strengthening of local quantum fluc-
simple counting problem Ω = 2N3γN = (2 3γ)N, be- tuations invoked by a higher number of the Heisenberg
×
causetheIsingspinsresidingonthesitesofthelargertri- trimersinthesecondTITlatticeshowninFig.1(b). The
angles contribute to the overalldegeneracy by the factor quantumorder-from-disordereffectgenerallyactsagainst
2N and the additional factor 3γN is related to the three- thespinfrustrationanditmaythuscauseanappearance
fold degeneracy of the lowest-energy spin configurations of the unusual partially orderedand partially disordered
on the smaller inner triangles. Obviously, the similar phase (ODP) with a spontaneously broken Z symme-
2
procedure cannot be used for finding out the degener- try. Note furthermore that typical spin arrangements to
acyoftheground-statemanifoldfortheotherdisordered emerge in ODP are completely the same as specified by
QPP of the more general Ising-Heisenberg model, since Eq. (30)forQPPandschematicallyillustratedinFig.3.
9
s s s
k3 k3 k3
(a)
Sk2 Sk3 Sk2 Sk3 Sk2 Sk3 1.6 0.0 = 0
-0.2
sk1 s Sk1 sk2 sk1 s Sk1 sk2 sk1 s Sk1 sk2 1eff .2 -0.5 ln 3
k3 k3 k3 J
-0.8
sskk11 SSSkk2ss2kk33 SSkk11SSSkk33 sskk22 ssk1k1 SSSkk22sskk33 SSkk11SSSkk33 sskk22 sskk11 SSSkk22sskk33 SSkk11SSSkk33 sskk22 (b) J 000eff 12.....048060.-01-.1-00.01..91-1-.003-..802.k5B-T00 ../23 JJIH0 0/. .0J4I =0 -.25.=0 00.6
k2 k3 k2 k3 k2 k3 1.2-0.9 ln 3
S S S
sk1 k1 sk2 sk1 k1 sk2 sk1 k1 sk2 0.8
-1.0
FIG. 4: The lowest-energy spin configurations, which consti- 0.4 -1.1
tute the ground-state manifold for the DP phase to emerge
-1.3
in the Ising limit ∆ = 0. Another twelve lowest-energy spin 0.0
configurations can be obtained from the spin configurations JH / JI = -2.0
drawninthelasttwolinesbyrotatingthemaroundthethree- 0.0 0.2 0.4 0.6 0.8
fold C3 symmetry axis. kBT / JI
FIG.5: (Coloronline)Temperaturedependencesoftheeffec-
tive nearest-neighbor interaction γβJeff of the corresponding
spin-1/2 Ising model on thetriangular lattice for the specific
choice of the anisotropy parameter ∆=0 and various values
of the interaction ratio JH/JI. Broken lines show the critical
The remarkable spontaneous order of the ODP will be value of the effective coupling above (below) which the spin
convincingly evidenced in Sections IIIB and IIIC by a system becomes spontaneously ordered (disordered).
morecomprehensiveanalysisofthe criticalbehaviorand
the order parameter. However, the spontaneous order
of ODP could also be evidenced by the effective cou-
pling of the corresponding spin-1/2 Ising model on the B. Critical behavior
triangular lattice tending towards the low-temperature
asymptotic limit lim γβJ = ln4 for γ = 2, which
T 0 eff Before proceeding to a detailed discussion of the crit-
is evidently above the→critical value γβ J = ln3 and
c eff ical behavior, let us explore in detail temperature de-
is thus consistent with the spontaneously long-range or-
pendences ofthe effective interactionγβJ ofthe corre-
eff
dered ground state. It should be pointed out, moreover,
spondingspin-1/2Isingmodelonthetriangularlatticeas
thatthefinitezero-temperaturelimitoftheeffectivecou-
depicted in Figs. 5 and 6 for two TIT lattices at various
plingalsoservesinevidenceofanimperfectspontaneous
values of the interaction ratio J /J and two different
H I
long-range ordering to emerge in ODP, which is also
values of the exchange anisotropy ∆ = 0 and 1, respec-
confirmed by non-zero albeit not fully saturated zero-
tively. It can be clearly seen from Fig. 5 that the effec-
temperaturevaluesofthespontaneousmagnetizationsof
tive coupling in the Ising limit ∆=0 either diverges for
the Ising and Heisenberg spins mI = 21(112859)18 ≃ 0.47482 JH/JI > 1 or it tends towards zero for JH/JI < 1 as
and mH = (2√2721 − 61)(112859)18 ≃ 0.16408 obtained from temperatu−regoestozero. Inthe lattercase,theeff−ective
Eqs. (19) and (21), respectively. Altogether, it could coupling is always below its critical value γβ J = ln3
c eff
be concluded thatthe local quantumfluctuations arere- andthis resultprovesanexistenceofthe DPinthe Ising
sponsible in ODP not only for the quantum reductionof limit ∆ = 0 of both the investigated TIT models for
the magnetic moment of the Heisenberg spins, but more any temperature if J /J 1. Contrary to this, the
H I
≤ −
strikingly,theyalsoindirectlycauseasmallbutnon-zero effective coupling γβJ either diverges or asymptoti-
eff
(cca. 5%) quantum reduction of the magnetic moment callyreachessome finite value for arbitrarybut non-zero
of the classical Ising spins as well. anisotropy parameter ∆ = 0 as shown in Fig. 6 for one
6
10
(a) (a) 0.8
= 1 0.0
0.0
1.6 0.5
-0.5 0.9
JI0.6
J 1eff .2-2-/03.6 -0.2 ln 3 k T / Bc 1.2 1.0
(b)J 000eff 22.....048005.0--2-00/3..067.1JH0 /.- 20Jk.IB5 =T0 .-/3-2 J1.I0.00.40.00.5= 01.6 (b) / J cI00001.....02492 -1 0= 41.0 2JH.0 2/1 J.5I 0 3 . 00.491.00.55
1.5 TB 1.2
k
-0.2 ln 3 0.6 1.5
1.0
-1.0 2.0
0.5 -0.7 0.3 = 4.0
JH / JI = -2.0
0.0
0.0
0.0 0.2 0.4 0.6 0.8
-3 -2 -1 0 1 2 3 4 5
FIG.6: (Coloronline)TempekrBaTtu /r eJIdependencesoftheeffec- JH / JI
FIG. 7: (Color online) Critical temperature kBTc/JI of the
tive nearest-neighbor interaction γβJeff of the corresponding spin-1/2Ising-HeisenbergmodelontwoTITlatticesfromFig.
spin-1/2 Ising model on thetriangular lattice for the specific
1(a)-(b)asafunctionoftheinteractionratioJH/JIforseveral
choice of theanisotropy parameter ∆=1 and various values
valuesof theexchange anisotropy ∆.
of the interaction ratio JH/JI. Broken lines show the critical
value of the effective coupling above (below) which the spin
system becomes spontaneously ordered (disordered).
the anisotropyparameter ∆. If one considers the partic-
ularcaseoftheeasy-axisexchangeanisotropy∆<1,the
critical temperature of the first lattice model (Fig. 7(a))
particularchoiceof∆. While theeffectivecouplingβJ
eff monotonically decreases with decreasing the ratio J /J
H I
of the first TIT lattice always remains below its critical
untilitcompletelyvanishesattheground-stateboundary
valueinthehighlyfrustratedregimeJ /J < 2/(2+∆)
H I − (27)betweenCFPandQPP.Thehighestcriticaltemper-
(Fig. 6(a)), the twice as large effective coupling 2βJ
eff ature of CFP
of the second TIT lattice is strong enough in order to
induce the spontaneous long-range order at sufficiently lim kBTc = ln 2+√3 −1 0.75933 (31)
low temperatures even in the highly frustrated region JH JI ≃
J /J < 2/(2+∆) (Fig. 6(b)). This result thus pro- JI→∞ h (cid:16) (cid:17)i
H I
−
videsanindependentconfirmationofthequantumorder- can be accordingly acquired in the asymptotic limit
from-disorder effect, which arises from the local quan- J /J . On the other hand, the critical temper-
H I
→ ∞
tum fluctuations governingthe magnetic behavior of the ature exhibits a more striking non-monotonous depen-
Heisenberg trimers. Finally, it is worthy to recall that dencewhenassumingtheeasy-planeexchangeanisotropy
an intersection of the effective coupling γβJ with the ∆ > 1, because the sufficiently strong ferromagnetic
eff
relevant critical point of the spin-1/2 Ising model on the Heisenberg interaction then favors a presence of uncon-
triangular lattice can be straightforwardly used in order ventional quantum ferromagnetic ordering QFP before
to locate critical points of the spin-1/2 Ising-Heisenberg the classical CFP one. The critical temperature there-
model on the TIT lattices, which in fact represents the fore tends just to one third of the previously reported
numerical solution of the critical condition (25). asymptotic value
ThecriticaltemperatureoftwoinvestigatedTITmod-
k T 1
elsisplottedinFig.7againstarelativestrengthbetween lim B c = 3ln 2+√3 − 0.25311, (32)
the Heisenbergand Isinginteractionfor severalvalues of JH JI ≃
JI→∞ h (cid:16) (cid:17)i