Full text of DOI:10.1007/s100510050004

Eur. Phys. J. B 13, 15–19 (2000)
THE EUROPEAN
PHYSICAL JOURNAL B
EDP Sciences
Societa` Italiana di Fisica
Springer-Verlag 2000
c
ꢀ
Ground state phase diagram of a spin-2 antiferromagnet
on the square lattice?
H. Niggemanna, A. Klu¨mper, and J. Zittartz
Institut fu¨r Theoretische Physik, Zu¨lpicher Strasse 77, 50937 K¨oln, Germany
Received 8 April 1999
Abstract. We use the vertex state model approach to construct optimum ground states for a large class
of quantum spin-2 antiferromagnets on the square lattice. Optimum ground states are exact ground states
of the model which minimize all local interaction operators. The ground state contains two continuous pa-
rameters and exhibits a second order phase transition from a disordered phase with exponentially decaying
correlation functions to a N´eel ordered phase. The behaviour is very similar to that of the corresponding
ground state of a quantum spin-3/2 model on the hexagonal lattice, which has been investigated in an
earlier paper.
PACS. 75.10.Jm Quantized spin models – 05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
1 Introduction
zero yields the equivalence
H|Ψ0i = 0 ⇐⇒ hij|Ψ0i = 0
for all nearest neighbours i and j.
The matrix product technique has found a large number
of applications in many particle physics. They are nat-
ural realizations of so-called optimum ground states for
one-dimensional quantum spin models [1–3] and of sta-
tionary states for one-dimensional stochastic models [4,5].
The vertex state model approach used in this work is a
generalization of the matrix product technique to more
complicated lattice types [6,7].
(2)
If such a global state |Ψ0i exists we call it an optimum
ground state since the global ground state energy E0 takes
the lowest possible value. The r.h.s. of (2) can be used
to find local conditions for the existence of an optimum
ground state and its realization in terms of a vertex state
model.
In the following we shall follow closely the proce-
dure presented in [6], where the construction of a vertex
state model for an antiferromagnetic spin-23 model on the
hexagonal lattice is explained in detail. Section 2 contains
a parametrization of all spin-2 nearest neighbour interac-
tions which obey a certain set of symmetries. In Section 3
we present the vertices which are used to construct the
global ground state. The corresponding ground state con-
ditions are given in terms of the parameters of the general
local interaction operator. Properties of the global Hamil-
tonian are investigated in Section 4. In particular we re-
port on a second order quantum phase transition, which
has also been found in the case of the hexagonal lattice [6].
Section 5 contains a short conclusion.
As explained in [8–10] there is a connection between
the matrix product technique and the density matrix
renormalization group (DMRG) method [11]. The ground
state corresponding to the fixed point of the DMRG pro-
cedure has a matrix product structure. The significance
of the vertex state model approach with regard to a gen-
eralization of the DMRG method to higher dimensions is
discussed in [12]. Further information about the DMRG
method and its relations to the matrix product approach
and other methods can be found in [13].
In this paper we consider a quantum spin-2 model on
the square lattice with nearest neighbour interaction
X
H =
hij .
(1)
hi,ji
2 Parametrization of nearest-neighbour
interactions
The operator hij is the same on all bonds. Normalizing
the energy such that the lowest eigenvalue of each hij is
The local interaction operator hij is supposed to have at
least the following symmetries:
1) parity invariance, i.e. [hij, Pij ] = 0, where Pij is the
operator which exchanges the spins at lattice sites i
and j;
?
Work performed within the research program of the Son-
derforschungsbereich 341, K¨oln-Aachen-Ju¨lich, Germany.
a
e-mail: hn@thp.uni-koeln.de

16
The European Physical Journal B
2) rotational invariance in the xy-plane of spin space, i.e. necessary to generate the hij eigenstates from the orthog-
[hij , Siz + Sjz] = 0;
onal basis
3) spin-flip invariance, i.e. hij is invariant under the trans-
|00i, |11i + |11i, |22i + |22i.
(5)
formation Sz → −Sz.
In order to find a parametrization of all local inter-
action operators which fulfill the above set of symmetries
we write down the general form of the simultaneous eigen-
states of Siz + Sjz (eigenvalue m) and Pij (eigenvalue p)
This requires 3 continuous parameters. The explicit
parametrization within this subspace is not needed in this
paper.
Any local interaction operator hij which obeys sym-
metries 1 and 2 can be written in terms of projectors onto
the local eigenstates (3)
m = 4, p = 1, |v4i = |22i
m = −4, p = 1, |v−4i = |22i
X
m = 3, p = 1, |v3+i = |12i + |21i
p = −1, |v3−i = |12i − |21i
hij
=
λk|vkihvk|.
(6)
k
m = −3, p = 1, |v−+3i = |12i + |21i
p = −1, |v−−3i = |12i − |21i
The λk are real parameters. Spin-flip invariance (symme-
try 3) is achieved by choosing the same λ-coefficients for
Siz + Sjz eigenstates corresponding to eigenvalues m and
−m. This leads to the following general representation of
the local interaction operator:
m = 2, p = 1, |v2+1i = |11i + A2 (|02i + |20i)
|v2+2i = A|11i − (|02i + |20i)
p = −1, |v2−i = |02i − |20i
m = −2, p = 1, |v−+21i = |11i + A2 (|02i + |20i)
|v−+22i = A|11i − (|02i + |20i)
hij = λ4(|v4ihv4| + |v−4ihv−4|)
+ λ+3 (|v3+ihv3+| + |v−+3ihv−+3|)
+ λ−3 (|v3−ihv3−| + |v−−3ihv−−3|)
+ λ2+1(|v2+1ihv2+1| + |v−+21ihv−+21|)
+ λ2+2(|v2+2ihv2+2| + |v−+22ihv−+22|)
p = −1, |v−−2i = |02i − |20i
m = 1, p = 1, |v1+1i = (|01i + |10i) + B(|12i + |21i)
|v1+2i = B(|01i + |10i) − (|12i + |21i)
p = −1, |v1−1i = (|01i − |10i) + C(|12i − |21i)
|v1−2i = C(|01i − |10i) − (|12i − |21i)
m = −1, p = 1, |v−+11i = (|01i + |10i) + B(|12i + |21i)
|v−+12i = B(|01i + |10i) − (|12i + |21i)
p = −1, |v−−11i = (|01i − |10i) + C(|12i − |21i)
|v−−12i = C(|01i − |10i) − (|12i − |21i)
m = 0, p = 1, |v0+1i, |v0+2i, |v0+3i : see below
p = −1, |v0−1i = (|11i − |11i) + D(|22i − |22i)
|v0−2i = D(|11i − |11i) − (|22i − |22i).
+ λ−2 (|v2−ihv2−| + |v−−2ihv−−2|)
(7)
+ λ1+1(|v1+1ihv1+1| + |v−+11ihv−+11|)
+ λ1+2(|v1+2ihv1+2| + |v−+12ihv−+12|)
+ λ1−1(|v1−1ihv1−1| + |v−−11ihv−−11|)
+ λ1−2(|v1−2ihv1−2| + |v−−12ihv−−12|)
+ λ0+1|v0+1ihv0+1| + λ0+2|v0+2ihv0+2| + λ0+3|v0+3ihv0+3|
+ λ−01|v0−1ihv0−1| + λ−02|v0−2ihv0−2|.
The total number of parameters is 22: there are 15 λ-
parameters, the superposition parameters A, B, C, D, and
the 3 rotation parameters in the subspace m = 0, p = 1.
This includes two trivial parameters, namely an offset and
a scale.
Equation (7) is the most general interaction opera-
tor between adjacent spin-2 sites which has symmetries
1–3. For most values of the 22 parameters the correspond-
ing global Hamiltonian (1) has a very complicated ground
state. However, in the next section we shall construct
optimum ground states for a submanifold of interaction
operators.
(3)
In this table the canonical spin-2 basis states are de-
noted as
Siz|2i = 2|2i
Siz|2i = −2|2i
Siz|1i = 1|1i
Siz|1i = −1|1i .
Siz|0i = 0
(4)
As can be seen from (3), the local eigenstates are com-
pletely fixed in the subspaces m = ꢀ4, ꢀ3 and m =
ꢀ2, p = −1. Except for m = 0, p = 1 the remain-
ing subspaces are two-dimensional. Within such a two-
dimensional subspace the eigenstates of hij can be rotated
(but have to be orthogonal), so a superposition parameter
has to be introduced. In (3) these superposition parame-
ters are denoted as A, B, C, and D.
3 The vertex state model
The subspace m = 0, p = 1 is 3-dimensional, hence
a single superposition parameter is not sufficient to cover Following the procedure explained in [6] we define a set of
all possible orientations of the hij eigenstates within this vertices with binary arrow variables on the bonds. On the
subspace. Instead, an arbitrary 3-dimensional rotation is square lattice there are 24 = 16 different vertices:

H. Niggemann et al.: Ground state phase diagram of a spin-2 antiferromagnet on the square lattice
17
product of all generated local spin states, and summing
out all bond variables:
¡A
¡A
A¡
A¡
µ2
¨
H
H
¨
¨
H
H
¨
⊗
X Y
µ3
:
b|2i
:
b|2i
H
H
¨
¨
¨
¨
H
H
|Ψ0i =
·
(10)
µ1
A¡
A¡
¡A
¡A
i
i
{µ}
µ4
¡A
¡A
A¡
A¡
H
H
H
H
¨
¨
¨
¨
A global state of this form is called a vertex state model.
The local ground state condition (2) requires that |Ψ0i
is annihilated by every local interaction operator. In case
of a vertex state model this is certainly fulfilled if hij an-
nihilates all two-spin states which are generated by all
possible concatenations of neighbouring vertices. The list
of all these two-spin states can be easily obtained from the
vertices (8):
: σa|1i
: σa|1i
¨
¨
¨
¨
H
H
H
H
A¡
A¡
¡A
¡A
¡A
¡A
A¡
A¡
¨
¨
H
H
H
¨
¨
H
:
:
a|1i
a|1i
:
:
a|1i
a|1i
H
H
¨
¨
¨
H
H
¨
A¡
A¡
¡A
¡A
|12i + σ|21i
a2|11i + b|02i
a2|11i + b|20i
|01i + σ|10i
|01i + b|12i
|12i + σ|21i
A¡
A¡
¡A
¡A
a2|11i + b|02i
a2|11i + b|20i
|01i + σ|10i
¨
H
H
¨
¨
¨
H
H
H
¨
¨
H
H
H
¨
¨
A¡
A¡
¡A
¡A
(8)
(11)
|01i + b|12i
¡A
¡A
A¡
A¡
¨
H
H
¨
¨
¨
H
H
: σa|1i
: σa|1i
|10i + b|21i
|10i + b|21i
H
¨
¨
H
H
H
¨
¨
|00i + σa2|11i
a2|11i + σb2|22i
|00i + σa2|11i
a2|11i + σb2|22i.
¡A
¡A
A¡
A¡
¡A
¡A
A¡
A¡
If these 16 pair states are local ground states of hij
then the global state |Ψ0i is an optimum ground state
of H. Comparing (11) with the general form of the eigen-
states (3) yields the following conditions for the model
parameters:
H
¨
¨
H
H
H
¨
¨
:
:
|0i
|0i
:
:
|0i
|0i
¨
H
H
¨
¨
¨
H
H
A¡
A¡
¡A
¡A
A¡
A¡
¡A
¡A
H
H
¨
¨
¨
H
H
¨
λσ3 = λ+21 = λ−2 = λ+11 = λ1−1 = λ1σ2 = 0
λ+01 = λ−01 = λ+02 = λ0−2 = 0
¨
¨
H
H
H
¨
¨
H
A¡
A¡
¡A
¡A
(12)
b
A = , B = C = b, D = 0
a2
λ4, λ−3 σ, λ+22, λ1−2σ, λ0+3 > 0 .
¡A
¡A
A¡
A¡
H
H
¨
¨
¨
H
H
¨
: σ|0i
: σ|0i .
¨
¨
H
H
H
¨
¨
H
Note that in the 3-dimensional subspace m = 0, p = 1,
only one two-spin state is not annihilated by the local
interaction (λ+03 > 0). The corresponding eigenstate is
¡A
¡A
A¡
A¡
Unlike classical vertices, the “weights” of these vertices
are single-spin states α|mi, where
σ
a2
1
b2
|v0+3i = |00i − (|11i + |11i) + (|22i + |22i). (13)
1
2
m = (number of outgoing arrows
Conditions (12) reduce the number of free continuous pa-
rameters in the Hamiltonian from 22 to 7 (5 λ-parameters
and a, b). This includes an overall scale, so the number of
non-trivial interaction parameters is 6.
(9)
− number of incoming arrows).
The prefactor α is not fixed by this scheme. It must be
adjusted to satisfy the local ground state condition.
In (8) we have introduced 3 parameters, a and b are
real, σ = ꢀ1. Note that b is the coefficient of the |2i and
|2i states, and the prefactors of the |1i and |1i states are
either a or σa.
Of course the general local interaction operator (7) can
also be written in terms of the canonical single-spin op-
erators Sx, Sy, Sz. In this representation hij is a poly-
nomial in the three operators SixSjx + SiySjy, SizSjz, and
(Siz)2 + (Sjz)2, which are all compatible with symmetries
The set of vertices (8) is the two-dimensional analogue 1, 2, and 3. In the present case of spin-2 the polynomial
to the matrices used in the matrix product technique for consists of 22 terms. As functions of the parameters in-
spin chains. The global ground state is obtained by at- troduced above the coefficients of these 22 terms can be
0
taching a vertex to each lattice site, taking the tensorial obtained by writing the expectation values hvk|hij|vk i in

18
The European Physical Journal B
35
both representations. This yields a set of linear equations
for the coefficients.
30
25
20
15
10
5
An important special case with a rather simple Hamil-
tonian is the isotropic point. In this case the ground state
√
√
1
2
Neel
parameters are a =
6, b = − 6, σ = −1, where |Ψ0i
coincides with the VBS ground state on the square lattice
discussed in [14]. The corresponding local interaction is
the SO(3) invariant operator
b2
7
7
1
hij = SiSj + (SiSj)2 + (SiSj)3 + (SiSj)4, (14)
disordered
10 45 90
VBS
which projects onto the subspace with maximum total
spin (Si + Sj)2 = 4(4 + 1). It is shown in [14] that on a fi-
nite square lattice with periodic boundary conditions, the
VBS state has exponentially decaying two-point correla-
tion functions and no N´eel order. In addition, the existence
of an energy gap above the ground state is conjectured.
0
0
1
2
3
4
a2
5
6
7
8
9
Fig. 1. Phase diagram as a function of a2 and b2.
4 Properties of the global ground state
No exact solution of this classical vertex model is avail-
able, so we have applied a Monte-Carlo algorithm to this
As can be seen from the vertices (8) the constructed system. As on the hexagonal lattice [6] the model exhibits
ground state |Ψ0i is invariant under a global spin flip a phase transition from a disordered phase to a N´eel or-
Sz → −Sz for all values of the parameters a, b, and σ. dered phase. Figure 1 shows the phase diagram, which has
Therefore the single-spin magnetization and the global been obtained from the numerics.
magnetization hStzotali vanish, which indicates an antifer-
– For large values of a2 and small values of b2 longitu-
romagnet.
dinal correlation functions decay exponentially to zero
As on the hexagonal lattice there are two noteworthy
as a function of distance. There is no long-range order.
special cases. In the limit b → ∞ the vertex state model is
The global ground state is disordered.
dominated by the |2i- and |2i-vertices, which can only be
arranged in a “checkerboard” configuration. Thus in the
limit b → ∞ the ground state is simply the sum of both
possible fully polarized N´eel states. The other important
– In the opposite case of small a2 and large b2, longitu-
dinal correlation functions show an alternating long-
range behaviour. The ground state has N´eel order.
√
√
1
Interestingly, as a function of a2 and b2 the phase tran-
sition line is a straight line. The slope is 3.0 ꢀ 0.1 and its
intercept is given by 3.7 ꢀ 0.3. The analytical explanation
of this simple geometrical shape is an open problem.
According to the simulations, the phase transition is
of second order. Figure 2 shows the critical correlation
function for a system of 30 × 30 lattice sites at a2 = 2.
Agreement with the algebraic function
special case is the isotropic point, a =
6, b = − 6, σ =
−1, which has already been discussed2above. Most impor-
tantly, it has no N´eel order. It will be seen in this section
that there are two phases with and without N´eel order
which are separated by a critical transition line.
In order to calculate properties of the global ground
state for general values of the parameters we investigate
the inner product hΨ0|Ψ0i. As explained in [6] it is given by
the partition function of the classical vertex model with
vertices defined as
ꢁ
ꢂ
1
r
1
√
√
fl(r) = cl
+
(16)
31 − r
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
µ2 ν2
µ3
ν3
µ4 ν4
µ2
i
ν2
i
*
+
is excellent if cl is used as a fitting parameter. Therefore,
as in the hexagonal lattice case, the critical exponent η is
equal to 1/2.
µ1
ν1
µ3
ν3
=
·
(15)
µ1
ν1
µ4
ν4
For the classical vertex model (15) we define p as
↓↑
the probability to find a pair of antiparallel arrows on
These vertices carry two independent sets of bond vari- a bond. In contrast to the hexagonal lattice p decays
↓↑
ables. Their weights are ordinary numbers. Due to the very slowly as a function of the parameters. For instance
assignment scheme (9) only those classical vertices have a
p
≈ 3% at a2 = 4.0, b2 = 15.7 (this is in the regime
↓↑
non-vanishing weight, where the number of outgoing ar- of the phase transition line). Thus vertices with unequal
rows on the µ-bonds and on the ν-bonds are equal. Only arrow pairs on their bonds (off-diagonal vertices) cannot
70 out of the 256 vertices fulfil this condition. Quantum be neglected. Although no reduction to a simpler, exactly
mechanical expectation values in the vertex state model solvable model is available, we conjecture that the phase
correspond to statistical expectation values in the classical transition corresponds to two simultaneous Ising transi-
vertex model (15).
tions, as on the hexagonal lattice.

H. Niggemann et al.: Ground state phase diagram of a spin-2 antiferromagnet on the square lattice
19
3.2
the transition corresponds to two simultaneous Ising tran-
sitions. Interestingly the phase transition line is a straight
line as a function of a2 and b2.
For the standard spin-2 Heisenberg model on the
square lattice with just bilinear exchange the ground state
is known to have N´eel order. Our results show that this
order is destroyed upon introducing higher isotropic spin-
exchange operators, however N´eel order is restored in the
presence of anisotropic terms.
3
2.8
2.6
2.4
2.2
2
hS1zSrziΨ
0
1.8
1.6
1.4
1.2
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2
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The global ground state is unique for finite system size
and 0 < a2 < ∞, 0 < b2 < ∞. The proof is a slightly
modified version of the one presented in [6].
5 Conclusion
We have applied the vertex state model approach to the
ground state problem of a class of spin-2 antiferromagnets
on the square lattice. The global ground state contains two
continuous parameters a and b which parametrize z-axis
anisotropy. Complete SO(3) invariance and extreme Ising
anisotropy are contained as special cases.
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exhibits a second order phase transition from a disordered
phase to a N´eel ordered phase. For the critical correlation
functions we have found η = 1/2 and conjecture that