We consider two partially frustrated S = 1/2 antiferromagnetic spin systems
on the triangular and pentagonal lattices. In an elementary plaquette of the
two lattices, one bond has exchange interaction strength $\alpha$ ($\alpha \leq
1$) whereas all other bonds have exchange interaction strength unity. We show
that for $\alpha$ less than a critical value $\alpha_{c}$, collinear
N\'eel-type ordering is possible in the ground state. The ground state energy
and the excitation spectrum have been determined using linear spin wave theory
based on the Holstein-Primakoff transformation.Comment: Four pages, LaTeX, Four postscripts figures, Phys. Rev. B58, 73
  (1998

D. A. Huse

D. Yoshika

Indrani Bose

M. E. Zhitomirsky

M. H. Waldor

P. W. Anderson

R. Deutscher

R. Kubo

Th. Jolicoeur

Uma Bhaumik

W. F. Wolff

Y. R. Wang

English

arXiv

Crossref

Collinear Néel-type ordering in partially frustrated lattices

We consider two partially frustrated S=1/2 antiferromagnetic spin systems on the triangular and pentagonal lattices. In an elementary plaquette of the two lattices, one bond has exchange interaction strength a (a&#8804;1) whereas all other bonds have exchange interaction strength unity. We show that for a less than a critical value ac, collinear Néel-type ordering is possible in the ground state. The ground-state energy and the excitation spectrum have been determined using linear spin-wave theory based on the Holstein-Primakoff transformation

Bhaumik, Uma

Bose, Indrani

Indian Academy of Sciences

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Collinear Néel-type ordering in partially frustrated lattices
Uma Bhaumik and Indrani Bose
Department of Physics
Bose Institute
93/1, Acharya Prafulla Chandra Road
Calcutta-700 009, India.
Abstract
We consider two partially frustrated S = 1
2
antiferromagnetic spin
systems on the triangular and pentagonal lattices. In an elemen-
tary plaquette of the two lattices, one bond has exchange interaction
strength α (α ≤ 1) whereas all other bonds have exchange interac-
tion strength unity. We show that for α less than a critical value
αc, collinear Néel-type ordering is possible in the ground state. The
ground state energy and the excitation spectrum have been deter-
mined using linear spin wave theory based on the Holstein-Primakoff
transformation.
P.A.C.S. Numbers: 75.10.Jm, 75.30.Ds
Frustrated spin sytems show a tendency to be magnetically disordered.
Frustration may occur due to the presence of further neighbour interactions,
besides the nearest-neighbour(NN) ones, as well as due to the topology of
the underlying lattice. A well-known example of the latter is the triangular
lattice. The ground state of the Ising Antiferromagnet(AFM), defined on the
triangular lattice, is highly degenerate with the entropy per site being a finite
quantity. The Ising AFM thus does not order at any temperature. In this
context, the question of interest is whether for a quantum spin Hamiltonian
long range magnetic order can exist in the ground state. The quantum
Heisenberg AFM has been widely studied on the triangular lattice [1, 2, 3, 4,
5] and there is now more or less a cosensus that AFM long range order(LRO)
exists in the ground state. The ground state of the classical spin systems
exhibits non-collinear 1200 ordering. The ordering is partially destroyed but
still exists once the quantum nature of the spins is taken into account .
Frustrated lattices, in general, exhibit non-collinear Néel-type magnetic
order whereas non-frustrated lattices like the square lattice exhibit collinear
1
Néel-type order in the classical ground state. A partially frustrated lattice
is obtained when not all the exchange interaction strengths along NN bonds
have equal values. In this case, the classical ground state, in a particular
parameter regime, exhibits collinear Néel-type ordering similar to that in the
case of a non-frustrated lattice. The ordering is not destroyed when quantum
fluctuations are taken into account. Two examples of this will be given in
this Brief Report, namely, those of the triangular and the pentagonal lattices.
The result for the classical ground state of the triangular lattice is already
known [6]. In the so called Row model, the strength of the unequel exchange
interaction is α (α ≤ 1) whereas all other exchange interaction strengths
have value unity (Fig.1). For (α ≤ αc), the ground state has collinear order
whereas for (α > αc), non-collinear spiral ordering occurs. This ordering
becomes the 120o ordering when α = 1. For α ≤ αc, we determine the exci-
tation spectrum as well as the quantal correction to the ground state energy
using linear spin wave (LSW) theory based on the Holstein-Primakoff(HP)
transformation. We next consider the case of the partially frustrated pentag-
onal lattice. The Ising AFM has been previously studied on the pentagonal
lattice [7, 8] and the ground state has been found to be disordered. In this
Brief Report, we study the spin S = 1
2
HAFM on the pentagonal lattice for
the first time. The elementary plaquette of a pentagonal lattice is a pentagon
and hence the lattice is topologically frustrated. We study the model in a
limited parameter regime, namely, the one in which collinear Néel-type order
exists in the ground state.
Consider a single triangular plaquette with three NN bonds. One of the
bonds has exchange interaction strength α, the other two have exchange
interaction strengths unity. The three spins sit at the three vertices and
interact through NN interaction. If the spins are treated as classical vectors,
it is easy to show that for α ≤ 1
2
, collinear Néel-type ordering is obtained in
the ground state. Let 0, Q, Q′ be the respective orientations of the spins with
respect to the z-axis. The first and third spins interact through exchange
interaction of strength α. In the collinear state, Q = π and Q′ = 0. For
α > 0.5, Q′ = 2Q in the ground state with Q = cos−1(− 1
2α
), i.e., a spiral
ordering is obtained. We first consider the Row model (Fig.1) with α ≤ 1
2
.
The sites belonging to alternate rows (shown by dashed lines) belong to one
2
of the sublattices A or B. The interaction Hamiltonian is
H =
∑
i∈A,B,δ=â2 â3
Si · Si+δ + α
∑
i∈A,B,δ=â1
Si · Si+δ (1)
â1, â2, â3 are the NN vectors in the horizontal and oblique directions respec-
tively. The spins on the A(B) sublattice are pointing up(down). This is the
classical ground state. The foundation of spin wave theory is the assumption
that AFM LRO exists in the ground state and the amplitude of zero-point
motion produced by quantum fluctuation about the classical ordered state is
small. This assumption fails if quantal corrections diverge. The first step in
the LSW theory is to transform the operators to the bosonic operators. The
HP transformations connecting the spin operators to the bosonic operators
aj ,bj ’s are given in the lowest order by
S
†
Aj
=
√
2saj
S
†
Bj
=
√
2s b†j
SzAj = s− a
†
jaj
−SzBj = s− b
†
jbj (2)
where j denotes the lattice site and S is the magnitude of the spins. We
will ultimately consider the case of S = 1
2
. S†, Sz are the spin raising
operator and z component of the spin respectively. One can similarly define
the spin lowering operator S−j . We will not exhibit the different steps of LSW
theory as these are standard [9, 10]. The Hamiltonian(1) is expressed in
terms of bosonic operators and then Fourier transformed. The Hamiltonian
contains only quadratic operators and so can be diagonalized by the well-
known Bogolyubov transformation. The diagonalized Hamiltonian is given
by
H = N S(S + 1) (2α− 4) + S
∑
k
ωk + S
∑
k
ωk(c
†
kck + d
†
kdk) (3)
where ck,dk are the new transformed operators and N is the total number of
sites. The excitation spectrum ωk is given by
ωk =
√
√
√
√
(
1 − α sin2
k · a1
2
)2
− γ2k (4)
3
where
γk =
1
z
∑
δ=â2,â3
eik·δ (5)
and z, the number of NNs along the oblique directions, is 4. Fig.2 shows the
excitation spectrum for α = 0.3 and S = 1
2
. The ground state energy Eg (the
sum of the first two terms in (3)) is given by Eg = -0.6025. One can verify
that for α = 0, the excitation spectrum for the square lattice is recovered.
As long as α < 1
2
, ωk is positive for all momentum wave vectors. This shows
that the choice of the starting ground state is correct.
We next turn to the case of the pentagonal lattice. A cross-section of
the lattice is shown in Fig 3. The lattice is a non-Bravais lattice and has
two types of sites with coordination numbers 3 and 4 respectively. The NN
exchange interaction strengths along the solid lines are given by α (α ≤ 1).
Consider a single pentagonal plaquette . The classical ground state for dif-
ferent values of α can be determined by minimizing the exchange interaction
energy with respect to the spin orientation angles. For α < 0.4, one finds
that the collinear Néel state is the ground state. The question is whether this
order survives once the quantal fluctuations are taken into account. Again,
we perform LSW theory to get an answer. Fig.3 shows the six sublattices
(AiBi, i = 1, 2, 3) corresponding to the six inequivalent sites. The A(B)
sublattices contain up(down) spins. The interaction Hamiltonian is given by
H =
∑
<i j>
Ji jSi · Sj (6)
where ¡i j¿ denotes NNs and Ji j=1 for NNs shown by solid lines in Fig.1.
The NN vectors δi’s are given by
δ1 = x̂,
δ2 = ŷ,
δ3 =
1
2
x̂+
√
3
2
ŷ,
δ4 = −
1
2
x̂+
√
3
2
ŷ (7)
and similarly for the vectors in the reverse directions. The HP transforma-
tions connecting the spin operators to bosonic operators ai and bi are given
4
by (2). Each of the six kinds of HP bosons forms a rectangular sublattice
with dimension (1, 4+
√
3). After Fourier transformation, the Hamiltonian(6)
becomes
H = H0 +H1
where
(8)
H0 = NS(S + 1)(−8 + 2α)
H1 = S
∑
k
ψ
†
k Mkψ (9)
ψ
†
k = (a
†
1k a
†
2k a
†
3k b1k b2k b3k) (10)
Mk =
[
Z1k Z2k
Z
†
2k Z1k
]
Z1k =



2 − 4α sin2 k·δ1
2
0 0
0 3 0
0 0 3



Z2k =



0 x⋆1 x2
x1 0 x
⋆
x⋆2 x 0



x1 = e
ik·δ1 , x2 = e
ik·δ2 , x = eik·δ3 + eik·δ4 (11)
N is the total number of sites. Following Jolicoeur and Le Guillou , we
consider the generalised Bogolyubov transformation matrix T as follows
[
α(k)
β(k)
]
= T
[
a(k)
b†(k)
]
(12)
where α(k) is the column vector with three components αn(k) (n=1,2,3) and
similarly for β†(k). In order to satisfy the bosonic commutation relations, T
5
has to satisfy
T−1 = η T η (13)
where
η =
[
I 0
0 −I
]
(14)
and I is the 3 × 3 identity matrix. Finally we have
H1 = S
∑
k
[α†(k) β(k)]η T [ηMk]T
−1
[
α(k)
β†(k)
]
(15)
To diagonalize H1 a suitable choice for T is
T−1 = (v1 v2 v3w1w2w3) (16)
where vn and wn are the eigenvectors of ηMk, with corresponding eigen-
values (ωn ,−ωn). The eigenvalues of ηMk occur in pairs with ωn¿0. The
matrix ηMk can be diagonalized numerically. We obtain three excitation
spectra each of which is doubly degenerate corresponding to the two sublat-
tices Ai, Bi. The diagonalized Hamiltonian has the form
H = H0 + S
3
∑
n=1
ωn(α
†
nαn + β
†
nβn) (17)
H0 = NS(S + 1)(−8 + 2α) + S
3
∑
n=1
ωn (18)
Fig.4 shows the three excitation spectra for α =0.27 and S = 1
2
. The posi-
tivity of the spectra shows that the collinear Néel state with quantum cor-
rections is still the ground state. For α beyond the critical value αc ≈0.32
this is no longer so. The ground state energy Eg is given by Eq.(18) and has
the value Eg =-0.4835 for α=0.27.
For both the triangular and pentagonal lattices, there is a critical value
αc of α below which the classical ground state has collinear Néel-type order.
As shown by LSW theory, the order is maintained, albeit with quantum cor-
rections, when the quantum nature of the spins is taken into account. Let
6
us now consider the case α > αc. For the triangular lattice, as mentioned
before, the ground state shows spiral ordering. For the pentagonal lattice,
however, the problem is more difficult. Consider the case α=1. Calculations
for an elementary plaquette show that in the classical ground state the dif-
ference in the successive spin orientation angles is 1440. Unlike the case of
the triangular lattice, the spin arrangement of a single pentagon cannot be
repeated for the whole lattice. The determination of the classical ground
state structure thus becomes computationally more difficult. The parameter
region α > αc for the pentagonal lattice has not been studied as yet. The
value of αc can be taken as a measure of frustration in a spin system. The
values for the square, triangular and pentagonal lattices are αc =1, 0.5 and
∼0.32 respectively. Thus the pentagonal lattice appears to be more frus-
trated than the triangular lattice. Several studies have been undertaken in
the recent past to understand the effect of topological frustration on ground
state properties. In two dimensions, the most studied cases are those of the
triangular and Kagomé lattices. Frustration, in general, leads to more disor-
der in the ground state. The interest in spin-disordered states has arisen in
connection with high temperature superconductivity. The pentagonal lattice
is another topologically frustrated lattice which has not been studied so far in
the context of quantum spin systems. The present study constitutes a small
begining which will hopefully lead to more exhaustive studies in future.
Acknowledgement
The Authors thank Sitabhra Sinha and Asimkumar Ghosh for computational
help.
7
Figure Captions
Fig.1 The partially frustrated Row model defined on the tringular lattice. A
and B denote the two sublattices. The dashed bonds have exchange in-
teraction strength α(α ≤ 1), all other bonds (solid lines) have exchange
interaction strength unity.
Fig.2 Excitation spectrum of the Row Model for α=0.3.
Fig.3 The partially frustrated pentagonal lattice. The six sublattices are
Ai and Bi (i=1,2,3). The dashed bonds have exchange interaction
strength α(α < 1), all other bonds (solid lines) have exchange interac-
tion strength unity.
Fig.4 Excitation spectra for the pentagonal lattice for α=0.27.
8
References
[1] D.A. Huse and V.Elser, Phys. Rev. Lett. 60, 2531 (1988)
[2] D. Yoshika and J.Miyazaki, J.Phys.Soc.Jpn.60, 614 (1991)
[3] R.Deutscher, H.U.Everts, S.Miyashita and J.Wintel, J.Phys.A:
Math.Gen. 23, L 1043 (1993)
[4] Y.R.Wang, Phys.Rev.B 45, 12604 (1992)
[5] Th. Jolicoeur and J.C. Guillou, Phys. Rev. B 40, 2727 (1989)
[6] M.E.Zhitomirsky, Phys. Rev. B 54, 353 (1996)
[7] W.F.Wolff and J.Zittartz, Z.Phys. B 45, 139 (1982)
[8] M.H.Waldor, W.F.Wolff and J.Zittartz, Phys. Lett. 104A, 355 (1984)
[9] P.W.Anderson, Phys.Rev. 86, 694 (1952)
[10] R.Kubo, Phys.Rev. B 86, 568 (1952)
9
0
1
2
3
4
0
2
4
6
8
0
1
2
3
4
Kxky
w
Figure 2
0
1
2
3
4
0
0.2
0.4
0.6
0
1
2
3
4
KxKy
w
(k
)
Figure 4
Figure 3
A1 A1 A1 A1
A1 A1 A1 A1
B1 B1 B1
A3 A3 A3
A2 A2 A2
B2B2B2B2
B3B3B3 B3
Figure 1
A A A A
B B B B
A A A A
A A A A
B B B B


Collinear néel-type ordering in partially frustrated lattices

Publications of the IAS Fellows

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98 Collinear Ne´el-type ordering in partially frustrated lattices
Uma Bhaumik and Indrani Bose
Department of Physics
Bose Institute
93/1, Acharya Prafulla Chandra Road
Calcutta-700 009, India.
Abstract
We consider two partially frustrated S = 1
2
antiferromagnetic spin
systems on the triangular and pentagonal lattices. In an elemen-
tary plaquette of the two lattices, one bond has exchange interaction
strength α (α ≤ 1) whereas all other bonds have exchange interac-
tion strength unity. We show that for α less than a critical value
αc, collinear Ne´el-type ordering is possible in the ground state. The
ground state energy and the excitation spectrum have been deter-
mined using linear spin wave theory based on the Holstein-Primakoff
transformation.
P.A.C.S. Numbers: 75.10.Jm, 75.30.Ds
Frustrated spin sytems show a tendency to be magnetically disordered.
Frustration may occur due to the presence of further neighbour interactions,
besides the nearest-neighbour(NN) ones, as well as due to the topology of
the underlying lattice. A well-known example of the latter is the triangular
lattice. The ground state of the Ising Antiferromagnet(AFM), defined on the
triangular lattice, is highly degenerate with the entropy per site being a finite
quantity. The Ising AFM thus does not order at any temperature. In this
context, the question of interest is whether for a quantum spin Hamiltonian
long range magnetic order can exist in the ground state. The quantum
Heisenberg AFM has been widely studied on the triangular lattice [1, 2, 3, 4,
5] and there is now more or less a cosensus that AFM long range order(LRO)
exists in the ground state. The ground state of the classical spin systems
exhibits non-collinear 1200 ordering. The ordering is partially destroyed but
still exists once the quantum nature of the spins is taken into account .
Frustrated lattices, in general, exhibit non-collinear Ne´el-type magnetic
order whereas non-frustrated lattices like the square lattice exhibit collinear
1
Ne´el-type order in the classical ground state. A partially frustrated lattice
is obtained when not all the exchange interaction strengths along NN bonds
have equal values. In this case, the classical ground state, in a particular
parameter regime, exhibits collinear Ne´el-type ordering similar to that in the
case of a non-frustrated lattice. The ordering is not destroyed when quantum
fluctuations are taken into account. Two examples of this will be given in
this Brief Report, namely, those of the triangular and the pentagonal lattices.
The result for the classical ground state of the triangular lattice is already
known [6]. In the so called Row model, the strength of the unequel exchange
interaction is α (α ≤ 1) whereas all other exchange interaction strengths
have value unity (Fig.1). For (α ≤ αc), the ground state has collinear order
whereas for (α > αc), non-collinear spiral ordering occurs. This ordering
becomes the 120o ordering when α = 1. For α ≤ αc, we determine the exci-
tation spectrum as well as the quantal correction to the ground state energy
using linear spin wave (LSW) theory based on the Holstein-Primakoff(HP)
transformation. We next consider the case of the partially frustrated pentag-
onal lattice. The Ising AFM has been previously studied on the pentagonal
lattice [7, 8] and the ground state has been found to be disordered. In this
Brief Report, we study the spin S = 1
2
HAFM on the pentagonal lattice for
the first time. The elementary plaquette of a pentagonal lattice is a pentagon
and hence the lattice is topologically frustrated. We study the model in a
limited parameter regime, namely, the one in which collinear Ne´el-type order
exists in the ground state.
Consider a single triangular plaquette with three NN bonds. One of the
bonds has exchange interaction strength α, the other two have exchange
interaction strengths unity. The three spins sit at the three vertices and
interact through NN interaction. If the spins are treated as classical vectors,
it is easy to show that for α ≤ 1
2
, collinear Ne´el-type ordering is obtained in
the ground state. Let 0, Q, Q′ be the respective orientations of the spins with
respect to the z-axis. The first and third spins interact through exchange
interaction of strength α. In the collinear state, Q = pi and Q′ = 0. For
α > 0.5, Q′ = 2Q in the ground state with Q = cos−1(− 1
2α
), i.e., a spiral
ordering is obtained. We first consider the Row model (Fig.1) with α ≤ 1
2
.
The sites belonging to alternate rows (shown by dashed lines) belong to one
2
of the sublattices A or B. The interaction Hamiltonian is
H =
∑
i∈A,B,δ=aˆ2 aˆ3
Si · Si+δ + α
∑
i∈A,B,δ=aˆ1
Si · Si+δ (1)
aˆ1, aˆ2, aˆ3 are the NN vectors in the horizontal and oblique directions respec-
tively. The spins on the A(B) sublattice are pointing up(down). This is the
classical ground state. The foundation of spin wave theory is the assumption
that AFM LRO exists in the ground state and the amplitude of zero-point
motion produced by quantum fluctuation about the classical ordered state is
small. This assumption fails if quantal corrections diverge. The first step in
the LSW theory is to transform the operators to the bosonic operators. The
HP transformations connecting the spin operators to the bosonic operators
aj ,bj ’s are given in the lowest order by
S
†
Aj
=
√
2saj
S
†
Bj
=
√
2s b†j
SzAj = s− a
†
jaj
−SzBj = s− b
†
jbj (2)
where j denotes the lattice site and S is the magnitude of the spins. We
will ultimately consider the case of S = 1
2
. S†, Sz are the spin raising
operator and z component of the spin respectively. One can similarly define
the spin lowering operator S−j . We will not exhibit the different steps of LSW
theory as these are standard [9, 10]. The Hamiltonian(1) is expressed in
terms of bosonic operators and then Fourier transformed. The Hamiltonian
contains only quadratic operators and so can be diagonalized by the well-
known Bogolyubov transformation. The diagonalized Hamiltonian is given
by
H = N S(S + 1) (2α− 4) + S
∑
k
ωk + S
∑
k
ωk(c
†
kck + d
†
kdk) (3)
where ck,dk are the new transformed operators and N is the total number of
sites. The excitation spectrum ωk is given by
ωk =
√√√√(1− α sin2 k · a1
2
)2
− γ2k (4)
3
where
γk =
1
z
∑
δ=aˆ2,aˆ3
eik·δ (5)
and z, the number of NNs along the oblique directions, is 4. Fig.2 shows the
excitation spectrum for α = 0.3 and S = 1
2
. The ground state energy Eg (the
sum of the first two terms in (3)) is given by Eg = -0.6025. One can verify
that for α = 0, the excitation spectrum for the square lattice is recovered.
As long as α < 1
2
, ωk is positive for all momentum wave vectors. This shows
that the choice of the starting ground state is correct.
We next turn to the case of the pentagonal lattice. A cross-section of
the lattice is shown in Fig 3. The lattice is a non-Bravais lattice and has
two types of sites with coordination numbers 3 and 4 respectively. The NN
exchange interaction strengths along the solid lines are given by α (α ≤ 1).
Consider a single pentagonal plaquette . The classical ground state for dif-
ferent values of α can be determined by minimizing the exchange interaction
energy with respect to the spin orientation angles. For α < 0.4, one finds
that the collinear Ne´el state is the ground state. The question is whether this
order survives once the quantal fluctuations are taken into account. Again,
we perform LSW theory to get an answer. Fig.3 shows the six sublattices
(AiBi, i = 1, 2, 3) corresponding to the six inequivalent sites. The A(B)
sublattices contain up(down) spins. The interaction Hamiltonian is given by
H =
∑
<i j>
Ji jSi · Sj (6)
where ¡i j¿ denotes NNs and Ji j=1 for NNs shown by solid lines in Fig.1.
The NN vectors δi’s are given by
δ1 = xˆ,
δ2 = yˆ,
δ3 =
1
2
xˆ+
√
3
2
yˆ,
δ4 = −
1
2
xˆ+
√
3
2
yˆ (7)
and similarly for the vectors in the reverse directions. The HP transforma-
tions connecting the spin operators to bosonic operators ai and bi are given
4
by (2). Each of the six kinds of HP bosons forms a rectangular sublattice
with dimension (1, 4+
√
3). After Fourier transformation, the Hamiltonian(6)
becomes
H = H0 +H1
where
(8)
H0 = NS(S + 1)(−8 + 2α)
H1 = S
∑
k
ψ
†
kMkψ (9)
ψ
†
k = (a
†
1k a
†
2k a
†
3k b1k b2k b3k) (10)
Mk =
[
Z1k Z2k
Z
†
2k Z1k
]
Z1k =

 2− 4α sin
2 k·δ1
2
0 0
0 3 0
0 0 3


Z2k =

 0 x
⋆
1 x2
x1 0 x
⋆
x⋆2 x 0


x1 = e
ik·δ1 , x2 = e
ik·δ2 , x = eik·δ3 + eik·δ4 (11)
N is the total number of sites. Following Jolicoeur and Le Guillou , we
consider the generalised Bogolyubov transformation matrix T as follows[
α(k)
β(k)
]
= T
[
a(k)
b†(k)
]
(12)
where α(k) is the column vector with three components αn(k) (n=1,2,3) and
similarly for β†(k). In order to satisfy the bosonic commutation relations, T
5
has to satisfy
T−1 = η T η (13)
where
η =
[
I 0
0 −I
]
(14)
and I is the 3× 3 identity matrix. Finally we have
H1 = S
∑
k
[α†(k) β(k)]η T [ηMk]T
−1
[
α(k)
β†(k)
]
(15)
To diagonalize H1 a suitable choice for T is
T−1 = (v1 v2 v3w1w2w3) (16)
where vn and wn are the eigenvectors of ηMk, with corresponding eigen-
values (ωn ,−ωn). The eigenvalues of ηMk occur in pairs with ωn¿0. The
matrix ηMk can be diagonalized numerically. We obtain three excitation
spectra each of which is doubly degenerate corresponding to the two sublat-
tices Ai, Bi. The diagonalized Hamiltonian has the form
H = H0 + S
3∑
n=1
ωn(α
†
nαn + β
†
nβn) (17)
H0 = NS(S + 1)(−8 + 2α) + S
3∑
n=1
ωn (18)
Fig.4 shows the three excitation spectra for α =0.27 and S = 1
2
. The posi-
tivity of the spectra shows that the collinear Ne´el state with quantum cor-
rections is still the ground state. For α beyond the critical value αc ≈0.32
this is no longer so. The ground state energy Eg is given by Eq.(18) and has
the value Eg =-0.4835 for α=0.27.
For both the triangular and pentagonal lattices, there is a critical value
αc of α below which the classical ground state has collinear Ne´el-type order.
As shown by LSW theory, the order is maintained, albeit with quantum cor-
rections, when the quantum nature of the spins is taken into account. Let
6
us now consider the case α > αc. For the triangular lattice, as mentioned
before, the ground state shows spiral ordering. For the pentagonal lattice,
however, the problem is more difficult. Consider the case α=1. Calculations
for an elementary plaquette show that in the classical ground state the dif-
ference in the successive spin orientation angles is 1440. Unlike the case of
the triangular lattice, the spin arrangement of a single pentagon cannot be
repeated for the whole lattice. The determination of the classical ground
state structure thus becomes computationally more difficult. The parameter
region α > αc for the pentagonal lattice has not been studied as yet. The
value of αc can be taken as a measure of frustration in a spin system. The
values for the square, triangular and pentagonal lattices are αc =1, 0.5 and
∼0.32 respectively. Thus the pentagonal lattice appears to be more frus-
trated than the triangular lattice. Several studies have been undertaken in
the recent past to understand the effect of topological frustration on ground
state properties. In two dimensions, the most studied cases are those of the
triangular and Kagome´ lattices. Frustration, in general, leads to more disor-
der in the ground state. The interest in spin-disordered states has arisen in
connection with high temperature superconductivity. The pentagonal lattice
is another topologically frustrated lattice which has not been studied so far in
the context of quantum spin systems. The present study constitutes a small
begining which will hopefully lead to more exhaustive studies in future.
Acknowledgement
The Authors thank Sitabhra Sinha and Asimkumar Ghosh for computational
help.
7
Figure Captions
Fig.1 The partially frustrated Row model defined on the tringular lattice. A
and B denote the two sublattices. The dashed bonds have exchange in-
teraction strength α(α ≤ 1), all other bonds (solid lines) have exchange
interaction strength unity.
Fig.2 Excitation spectrum of the Row Model for α=0.3.
Fig.3 The partially frustrated pentagonal lattice. The six sublattices are
Ai and Bi (i=1,2,3). The dashed bonds have exchange interaction
strength α(α < 1), all other bonds (solid lines) have exchange interac-
tion strength unity.
Fig.4 Excitation spectra for the pentagonal lattice for α=0.27.
8
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9
0
1
2
3
4
0
2
4
6
8
0
1
2
3
4
Kxky
w
Figure 2
0
1
2
3
4
0
0.2
0.4
0.6
0
1
2
3
4
KxKy
w
(k)
Figure 4
Figure 3
A1 A1 A1 A1
A1 A1 A1 A1
B1 B1 B1
A3 A3 A3
A2 A2 A2
B2B2B2B2
B3B3B3 B3
Figure 1
A A A A
B B B B
A A A A
A A A A
B B B B


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