wykresx.eps


Acta Polytechnica Vol. 51 No. 1/2011

Thermal Entanglement and Critical Behavior of Magnetic Properties on
a Triangulated Kagomé Lattice

N. Ananikian, L. Ananikyan, L. Chakhmakhchyan, A. Kocharian

Abstract

The equilibrium magnetic and entanglement properties in a spin-1/2 Ising-Heisenberg model on a triangulated Kagomé
lattice are analyzed by means of the effective field for the Gibbs-Bogoliubov inequality. The calculation is reduced to
decoupled individual (clusters) trimers due to the separable character of the Ising-type exchange interactions between
the Heisenberg trimers. The concurrence in terms of the three qubit isotropic Heisenberg model in the effective Ising
field in the absence of amagnetic field is non-zero. Themagnetic and entanglement properties exhibit common (plateau,
peak) features driven by a magnetic field and (antiferromagnetic) exchange interaction. The (quantum) entangled and
non-entangled phases can be exploited as a useful tool for signalling the quantum phase transitions and crossovers at
finite temperatures. The critical temperature of order-disorder coincides with the threshold temperature of thermal
entanglement.

Keywords: Triangulated Kagomé lattice, Ising-Heisenberg model, Gibbs-Bogoliubov inequality, entanglement, concur-
rence.

1 Introduction
Geometrically frustrated spin systems exhibit fasci-
nating new phases of matter, a rich variety of un-
usual ground states and thermal properties as a re-
sult of zero and finite temperature phase transitions
driven by quantum and thermal fluctuations, respec-
tively [1]. The efforts aimed at a better understand-
ing of these phenomena have stimulated an inten-
sive search for transition-metal magnetic and molec-
ularmaterialswhose paramagneticmetal centers can
be strongly frustrated by local geometric structures.
One of the most interesting geometrically frustrated
magnetic two-dimensional structures is the triangu-
lated Kagomé (triangles-in-triangles) lattice, which
can be applied to the magnet Cu9X2(cpa)6 ·nH2O
(X=F, Cl, Br and cpa=carboxypentonic acid) [2].
The magnetic architecture of these series of com-
pounds, which can be regarded as a triangulated
Kagomé lattice (Fig. 1), is currently under ac-

tive theoretical investigation [3]. The spin-
1
2
Ising-

Heisenberg model on this lattice, which takes into
account quantum interactions between Cu2+ ions in
a-sites, in the limit when monomeric b-spins having
anexchangeof Isingcharacter, providesa richphysics
anddisplays the essential featuresof the copperbased
coordination compounds [4, 5].
Entanglement is a generic feature of quantumcor-

relations in systems,which cannot be quantified clas-
sically [6]. It provides a new perspective for under-
standing quantumphase transitions (QPTs) and col-
lective many-body phenomena in condensed matter

physics. A key novel observation is that quantum
entanglement can play an important role in proxim-
ity to QPTs controlled by quantum fluctuations in
the vicinity of quantum critical points. A new line of
research points to a connection between the entan-
glement of a many-particle system and the existence
ofQPTs and scaling [7]. The basic features of entan-

glement in spin-
1
2
finite systems are fairlywell under-

stood by now,while the role of local cluster topology
and spin correlations in the thermodynamic limit still
remains unanswered. Effective field theories can be
offered by using the Gibbs-Bogoliubov inequality for
studying the thermodynamic and thermal entangle-
ment properties ofmany-body systems [8]. Although
the method is not exact, it is still possible to see re-
gions of criticality [9].
Unlike a classical transition, controlled by tem-

perature, a quantum phase transition (QPT) is
driven solely by (quantum) interactions. In the
case of the triangulated Kagomé lattice, each a-
type trimer interacts with its neighboring trimer
through the (ferromagnetic) Ising-type exchange, i.e.
a classical interaction. Therefore, the states of two
neighboring a-clusters become separable (unentan-
gled) [6]. Thus, the concurrence (a measure of en-
tanglement [10]), which characterizes quantum (non
classical) features for each trimer, in the effective
field, can be calculated separately. The key result of
the currentwork is a comparative analysis of specific
(peakandplateau) features in themagnetic and ther-
mal entanglement properties of the spin-1/2 Ising-
Heisenberg model on a triangulated Kagomé lattice.

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Acta Polytechnica Vol. 51 No. 1/2011

The rest of the paper is organized as follows: in
Sec. 2 we introduce the Ising-Heisenberg model on
the triangulated Kagomé lattice and provide a vari-
ational solution based on the Gibbs-Bogoliubov in-
equality. The basic principles for calculating entan-
glement measure and some of the results on intrin-
sic properties are introduced in Sec. 3. In Sec. 4
we present a comparison of magnetic properties and
thermal entanglement. Concluding remarksaregiven
in Sec. 5.

2 Basic formalism

We consider the spin-
1
2
Ising-Heisenberg model on a

triangulated Kagomé lattice (TKL) (Fig. 1) consist-
ing of two types of sites (a and b). Since the exchange
coupling between Cu2+ ions is almost isotropic, it
is more appropriate to apply isotropic Heisenberg
model. There is a strong Heisenberg Jaa (antiferro)
exchange coupling between trimeric sites of type a
and a weaker Ising-type (ferro) exchange (Jab) be-
tween trimeric types a and monomeric b. Thus, the
Kagomé lattice of the Ising spins (monomers) con-
tains inside each triangle unit a smaller triangle of
Heisenberg spins (trimer).

Fig. 1: A cross-section of TKL. The solid lines represent
intra-trimerHeisenberg interactions Jaa, while thebroken
lines labelmonomer-trimer Ising interations Jab. The cir-
cle marks the k-th cluster. Saki presents the Heisenberg
spins and Sbki the Ising spins

The Hamiltonian can be written as follows:

H = Jaa
∑
(i,j)

Sai S
a
j − Jab

∑
(k,l)

(Sz)ak · (S
z)bl −

H

2N
3∑

i=1

3[(Sz)aj +
1
2
(Sz)bj], (1)

whereSa = {Sax, S
a
y , S

a
z } is theHeisenberg spin-

1
2
op-

erator, and Sb is the Ising spin. Jaa > 0 corresponds
to antiferro-coupling and Jab > 0 ferro-couplings.
Here, the total number of sites is 3N, where the first
twosummations runover a−a and a−b nearestneigh-
bors, respectively, and the last sum incorporates the
effect of a uniform magnetic field.
The variational Gibbs-Bogoliubov inequality is

adopted to solve the Hamiltonian (1)

F ≤ F0 + 〈H − H0〉0, (2)
where H is the real Hamiltonian which describes the
system, and H0 is the trial one. F and F0 are free en-
ergies corresponding to H and H0, respectively, and
〈. . .〉0 denotes the thermal average over the ensemble
defined by H0. Following [4], the trial Hamiltonian
is reduced to

H0 =
∑

k∈trimers
Hc0, (3a)

Hc0 = λaa
(
Sak1S

a
k2
+Sak2S

a
k3
+Sak1S

a
k3

)
−

3∑
i=1

[
γa(S

z)aki +
γb
2
(Sz)bki

]
. (3b)

In this Hamiltonian, the stronger quantum Heisen-
berg antiferromagnetic interactions between a-sites
are treated exactly, while the weaker Ising-type ones
between the a- and b-sites (|Jab/Jaa| ≈ 0.025 [11])
are replacedby self-consistent (effective) fields of two
types: γa and γb.
The variational parameters γa, γb and λaa can

be found by minimizing the RHS of (2). Using the
fact that in terms of (3b) Sa and Sb are statisti-
cally independent, and taking into account 〈Sx〉0 =
〈Sy〉0 =0asingle sitemagnetization, 〈(Sz)a〉0 = ma,
〈(Sz)b〉0 = mb on a and b-sites, we obtain λaa = Jaa,
γa =2Jabmb+H, γb =4Jabma+H. The eigenvalues
of Hac0 are:

E1 =
3
4
(λaa +2γa);E2 = E3 =

1
4
(−3λaa +2γa)

E4 =
1
4
(3λaa +2γa);E5 = E6 =

1
4
(−3λaa −2γa)

E7 =
1
4
(3λaa −2γa);E8 =

3
4
(λaa −2γa) (4)

and the corresponding eigenvectors given by

|ψ1〉 = |000〉

|ψ2〉 =
1
√
3

(
q|001〉+ q2|010〉+ |100〉

)
|ψ3〉 =

1
√
3

(
q2|001〉+ q|010〉+ |100〉

)
|ψ4〉 =

1
√
3
(|001〉+ |010〉+ |100〉)

|ψ5〉 =
1
√
3

(
q|110〉+ q2|101〉+ |011〉

)
(5)

|ψ6〉 =
1
√
3

(
q2|110〉+ q|101〉+ |011〉

)
|ψ7〉 =

1√
3
(|110〉+ |101〉+ |011〉)

|ψ8〉 = |111〉,

where q = ei2π/3 (these eigenvectors should be also
the eigenstates of cyclic (rotation) operator P with
eigenvalues 1, q and q2, satisfying q2 + q +1=0).

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Acta Polytechnica Vol. 51 No. 1/2011

For the above-defined a- and b-single site magne-
tizations we obtain (here and further the Boltzman’s
constant is set to be kB =1):

ma = (6a)

1
6

3sinh
(
3γa
2T

)
+2e

3λaa
2T sinh

(
γa
2T

)
+sinh

(
γa
2T

)
cosh

(
3γa
2T

)
+2e

3λaa
2T cosh

(
γa
2T

)
+cosh

(
γa
2T

) ,
mb =

1
2
tanh

( γb
2T

)
. (6b)

For the Gibbs-Bogoliubov free energy (FGB) of the
system we obtain the following expression:

FGB
N
=

λaa
2
+4Jabmamb −2T

[
1
3
ln
{
4e

3Jab
2T

cosh
( γa
2T

)
+2cosh

( γa
2T

)
+2cosh

(
3γa
2T

)}
+

1
2
ln
{
2cosh

( γb
2T

)}]
. (7)

3 Concurrence and thermal
entanglement

The effective field treatment of (1) transformsmany-
body systemtoa reduced“single”cluster study. This
allows us to study, in particular, the thermal (local)
entanglement properties of the a-sublattice in terms
of a three-qubit (isotropic) Heisenberg model in a
self-consistent field γa, which carries the properties
of the whole system. As a measure of the pairwise
entanglement, we use concurrence C(ρ) [10]. The
corresponding density matrix ρ is defined as

C(ρ)= max{λ1 − λ2 − λ3 − λ4,0}, (8)

where λi are the square roots of the eigenvalues of the
operator ρ̃ = ρ12(σ

y
1 ⊗ σ

y
2)ρ

∗
12(σ

y
1 ⊗ σ

y
2) in descend-

ing order. Since we consider pairwise entanglement,
we use the reduced density matrix ρ12 = Tr3ρ. In
the effective field, due to the classical character of
the Ising interaction (Sec. 1) between trimers, the
concurrence for each decoupled Heisenberg cluster
can be calculated individually. In our case, the den-

sity matrix has the form ρ =
1
Z

8∑
k=1

exp(−Ek/T)|ψk〉

〈ψk|, Ek and |ψk〉 taken from (4) and (5) and
Z is the partition function [Z = Trρ =

e−
3(2γa+λaa)

4T

(
1+ e

γa
T

)(
1+ e

2γa
T +2e

2γa+3λaa
2T

)
].

While the construction of ρ̃ does not depend on
whether γa is an effective parameter or a real mag-
netic field, the self-consistent field solution for γa is
crucial in obtaining the final results. In this paper
we skip specific derivations and rather focus only on
the final results. Concurrence C(ρ) is given by [12]:

C(ρ)=
2
Z

max(|y| −
√

uv,0), (9)

where

u =
1
3
e
2γa−3λaa

4T

(
1+3e

γa
T +2e

3λaa
2T

)
v =

1
3
e−

3(2γa+λaa)
4T

(
3+ e

γa
T +2e

2γa+3λaa
2T

)
(10)

w =
1
3
e−

2γa+3λaa
4T

(
1+ e

γa
T

)(
1+2e

3λaa
2T

)
y = −

1
3
e−

2γa+3λaa
4T

(
1+ e

γa
T

)(
−1+ e

3λaa
2T

)
.

First, we find that concurrence C(ρ) as an entan-
glement measure exhibits critical behavior upon the
temperature variation shown inFig. 2 in the absence
of a field.

Fig. 2: Concurrence C(ρ) versus temperature field for
Jaa =1, α =0.025 and H =0

The system is entangled at a relatively low tem-
perature, below the threshold, Tth. This effect ap-
parently occurs because of the Ising-type interaction
replacedby the effective field γa =2Jabmb+H acting
upon the a-spins, which isnon-zero at H = 0. Thus,
effective field provides a solution for an entanglement
resource in the absence of a magnetic field.
Another important observation: the threshold

temperature at which C(ρ) becomes zero coincides
with the critical temperature Tc at which spon-
taneous magnetization m vanishes for the smooth
(second order) phase transition between ordered-
disordered phases. Expanding ma into series near
the phase transition point:

m = am + bm3 + cm5 + . . . (11)

one finds Tc from the condition a = 1, b < 0 (for
the case Jaa = 1 and α = 0.025, Tc = 0.0102062).
The coincidence of the critical and threshold temper-
atures for magnetization and concurrence is a con-
sequence of the fact that at Tc the system under-
goes the order-disorder phase transition and the sec-
ond term in γa also vanishes (mb = 0, when H = 0
and T ≥ Tc). In general, we find a number of other
similarities between the magnetic properties and the
entanglement of the system. Under variation of H,
the entanglement andmagnetic properties showvery

9



Acta Polytechnica Vol. 51 No. 1/2011

rich behavior in the low-temperature region. Fig. 3
presents the three-dimensional plot of the concur-
rence as a function of the temperature and external
magnetic field.
Wewill study some other features in the behavior

of C(ρ) by returning to the magnetic and entangle-
ment ground state properties in Sec. 4.

Fig. 3: Concurrence C(ρ) versus temperature T and ex-
ternal magnetic field H for Jaa =1, α =0.025

4 Quantum critical points and
phase diagrams

Nowwe consider themany-body quantumeffects rel-
evant to entanglement properties and discuss some
similarities between magnetic (statistical) properties
and (quantum) entanglement. As statistical char-
acteristics, the density distribution of susceptibility

χ =
∂ma
∂H

reduced per one a-site is shown in Fig.

4(a) as a function of the coupling constant Jaa and
the external field H, at a relatively high temperature
T = 0.1, higher than Tc. The white stripes, which
have a certain a finite width due to nonzero temper-
ature, correspond to the peaks of the susceptibility.
A similar density plot is shown for entanglement in
Fig. 4 (b) for the same range of Jaa − H parame-
ters. A comparison of these two graphs shows that
the general behavior of the statistical properties in χ
resembles the features of the quantum concurrence.

(a) (b)

Fig. 4: Density plot of (a) susceptibility χ, (b) concur-
rence C(ρ) versus H and Jaa for α =0.025 and T =0.1

Our calculations show that the peaks inmagnetic
susceptibility correspond to the phase boundaries on
the Jaa − H density diagram in concurrence C(ρ) at
which the quantum coherence vanishes. As can be
seen, this is true only for the Ising-Heisenbergmodel
on theTKLlatticewith Jaa > 0coupling; for Jaa < 0
coupling the system is non-entangled, C(ρ)=0.
Thus, the extremal behavior of χ is not repro-

duced by the concurrence density. Hence, although
the critical behavior of the two characteristics coin-
cides for the antiferromagnetic region, only (quan-
tum)concurrence canbeusedasa reference forquan-
titative analysis of QPTs.
In addition,we studyhere thequantumcriticality

in the ground state phase diagram resulting from the
magnetic field variation in themagnetization and en-
tanglement properties of the a-sublattice. Fig. 5(a)
shows a phase diagramof constantmagnetization for
the a-sublattice. This diagram differentiates the fol-
lowingphases for Jaa > 0: Phase I corresponds to the
spontaneous magnetization ma = 1/6, when spins
in the a-sublattice are in one of the available (↑↑↓)
configurations; Phase II corresponds to one of the
possible configurations (↓↓↑) with ma = −1/6.

(a) (b)

Fig. 5: (a) Phase diagram of the a-sublattice for |α| =
0.025; (b)Density plot of concurrence C(ρ) versus H and
Jaa for |α| =0.025 at zero temperature

In the ferromagnetic case (Jaa < 0) we have full
spin saturation in regions III and IV, with the value
of the maximum magnetization per atom ma = 1/2
[configuration (↓↓↓)] and ma = −1/2 [configuration
(↑↑↑)], respectively. Phase I contains the two-fold
degenerate states |ψ5〉 and |ψ6〉, while Phase II con-
tains the two-fold degenerate states |ψ2〉 and |ψ3〉
with C(ρ) = 1/3. Phases III and IV correspond
to states |ψ1〉 and |ψ8〉, respectively. These phases
are non-entangled, (C(ρ) = 0) [Fig. 5(b)]. The area
of non-zero entanglement coincideswith phase I+II,
where |ma| = 1/6, while the one with zero entangle-
ment (C(ρ)=0) corresponds to phase III+IV with
|ma| =1/2.

5 Conclusion
In this work we have demonstrated correlations be-
tween magnetic properties and quantum entangle-

ment in the spin-
1
2
Ising-Heisenberg model on a tri-

10



Acta Polytechnica Vol. 51 No. 1/2011

angulated Kagomé lattice. We adopted the varia-
tionalmean-field like treatment (based on theGibbs-
Bogoliubov inequality) to decouple clusters in effec-
tive (interconnected) fields of two types (consisting
of Heisenberg a trimers and Ising-type b monomers).
Each of these fields taken separately describes not
only the corresponding (a- or b- type) spins, but the
system as a whole.
We used concurrence as a computable measure

of bipartite entanglement for the trimeric units in
terms of the isotropic Heisenberg model in the effec-
tive magnetic field γa. Using the fact that “a subdi-
visions” are separable, we studied the entanglement
for each of them individually in an effective Ising-
type field, (γb). The model exhibits quantum criti-
cality, which can be identified and characterized by
studying the behavior of the magnetic and entangle-
ment properties with respect to the interaction, the
magnetic field and the temperature that control the
transition. It turned out that entanglement does not
vanish in the zero external field, as happens for the
common three qubit (isotropic) Heisenberg model.
We find that the temperature at which entangle-

mentbecomes zerocoincideswith the critical temper-
ature of the second order phase transition at which
spontaneous magnetization disappears. In addition,
we show that in the antiferromagnetic region (the in-
teractions between trimeric a sites are exactly of this
type) themagnetic susceptibility peaks coincidewith
the boundary lines at which entanglement vanishes.
However, this does not take place in the ferromag-
netic case. Therefore, one can detect a quite visible
correlation for the lineboundariesbetween thephases
on the density diagrams for entanglement and mag-
netization as a signature of the corresponding quan-
tum phase transition. Note that the disordered spin
liquid state can also exist in the ground state of the
frustrated spin system, on the assumption that there
is a sufficiently strong antiferromagnetic intra-trimer
interaction.
Finally, themagnetization,magnetic susceptibili-

ties and (quantum) entanglement features can be ex-
ploited to signal andunderstand thequantumcritical
points and phase transitions.

Acknowledgement

This work was supported by the PS-1981, PS-
2033ANSEF andECSP-09-08-saspNFSAT research
grants.

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N. Ananikian
Yerevan Physics Institute
Alikhanian Br. 2
0036 Yerevan, Armenia

L. Ananikyan
E-mail: lev.ananikyan@gmail.com
Yerevan Physics Institute
Alikhanian Br. 2
0036 Yerevan, Armenia

L. Chakhmakhchyan
Yerevan Physics Institute
Alikhanian Br. 2
0036 Yerevan, Armenia
Yerevan State University
A. Manoogian 1
0025 Yerevan, Armenia

A. Kocharian
Yerevan Physics Institute
Alikhanian Br. 2
0036 Yerevan, Armenia
Department of Physics
California State University
Los Angeles, CA 90032, USA

12