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Antiferromagnetic Long-Range Order in the Anisotropic Quantum Antiferromagnetic Long-Range Order in the Anisotropic Quantum
Spin Chain Spin Chain
Josef Kurmann
Harry Thomas
Gerhard Müller
University of Rhode Island
, gmuller@uri.edu
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Citation/Publisher Attribution Citation/Publisher Attribution
Kurmann, J., Thomas, H., & Müller, G. (1982). Antiferromagnetic long-range order in the anisotropic
quantum spin chain
.
Physica A: Statistical Mechanics and its Applications
,
112
(1-2), 235-255. doi:
10.1016/0378-4371(82)90217-5
Available at: http://dx.doi.org/10.1016/0378-4371(82)90217-5
This Article is brought to you for free and open access by the Physics at DigitalCommons@URI. It has been
accepted for inclusion in Physics Faculty Publications by an authorized administrator of DigitalCommons@URI. For
more information, please contact digitalcommons@etal.uri.edu.
Antiferromagnetic Long-Range Order in the Anisotropic
Quantum Spin Chain
Josef Kurmann
1
, Harry Thomas
1
, Gerhard Müller
2
1
Institut für Physik, Universität Basel, CH-4056 Basel, Switzerland
2
Department of Physics, University of Rhode Island, Kingston RI 02881, USA
This is a study of the ground-state properties of the one-dimensional spin-s (1/2 < s < ∞) anisotropic XY Z
antiferromagnet in a magnetic field of arbitrary direction. It provides the first rigorous results for the general
case of this model in non-zero field. By exact calculations we find the existence of an ellipsoidal surface
h = h
N
in field space where the ground state is of the classical two-sublattice Néel type with non-zero
antiferromagnetic long-range order. At h
N
there are no correlated quantum fluctuations. We give upper and
lower bounds for the critical field h
c
where antiferromagnetic long-range order is suppressed by the field. The
zero-temperature phase diagrams are discussed for a few representative cases.
1. Introduction
As fundamental results of the theory of critical phenomena it has been found that the impor-
tance of fluctuations in a system of interacting spins depends crucially on
(i) the dimensionality of the system,
(ii) the range of interaction,
(iii) the symmetry of the Hamiltonian in spin space,
(iv) at T → 0, the question whether the spins are treated quantum-mechanically or as classical
n-vectors.
Accordingly, among the most strongly fluctuating systems of interacting spins are one-dimensional
(1D) spin-
1
2
chains with nearest-neighbour exchange coupling. Here the thermal fluctuations pre-
vent ordering at any non-zero temperature, and in cases where the Hamiltonian has a continuous
rotational symmetry in spin space, the zero-point fluctuations may destroy long-range order (LRO)
even at T = 0 [1]. In the light of these findings it is of particular interest to study the ground-state
(GS) properties of the 1D quantum antiferromagnet (AFM) in a magnetic field. The Hamiltonian
reads
H =
N
X
l=1
J
x
S
x
l
S
x
l+1
+ J
y
S
y
l
S
y
l+1
+ J
z
S
z
l
S
z
l+1
− h · S
l
. (1.1)
The exchange parameters J
µ
are considered to be non-negative. We shall see that the GS properties
depend strongly on the commutability of the exchangeand Zeeman parts H
EX
and H
ZE
of the
Hamiltonian. For the s =
1
2
XYZ model (1.1) at h = 0, a number of exact results such as the
energies of the GS [2] and of the low-lying excited states [3] are already known. The present work
contains the first rigorous results for the general XYZ model in non-zero field. In order to emphasize
quantum effects we shall compare our results for the quantum chain (1.1) with those of the classical
counterpart (1.1) where the spin operators are replaced by three-dimensional vectors of length s.
It is evident from all known results on T = 0 properties of the 1D AFM (1.1) in zero and non-
zero magnetic field that the GS is usually very complicated [4-13]. Below a certain critical value
h
c
of the field, the GS either displays true AFM LRO or only "incipient" LRO, depending on the
symmetry of the Hamiltonian. In the latter case the asymptotic behavior R → ∞ of the correlation
functions hS
µ
l
S
µ
l+R
i is governed by a power-law decay. In the case of true LRO the appropriate
1
Antiferromagnetic long-range order in the anisotropic quantum spin chain
correlation function approaches a constant hS
µ
l
i
2
, and the correlations of the fluctuations S
µ
l
−hS
µ
l
i
decay exponentially.
In a recent study on the isotropic s =
1
2
XY AFM (J
x
= J
y
= J, J
z
= 0) with an in-plane field
h = h
x
[14] we discovered that at the special value h
N
=
√
2J of the field the GS becomes identical
to the GS of the corresponding classical chain, i.e. it factorises into single-site states exhibiting the
same expectation values hS
µ
l
i as in the classical two-sublattice Néel-type state with spins of the two
sublattices being in a spin-flop configuration within the XY plane. The quantum fluctuations which
are considerable both below and above h
N
become completely uncorrelated at this particular value
h
N
. The existence of a quasi-classical GS is actually a general feature of the anisotropic quantum
AFM chain in a magnetic field. In section 2 we present a general solution for the conditions under
which a Néel-type GS exists for the Hamiltonian (1.1) with arbitrary s. We give explicit results for
the GS configurations for fields varying in either of the coordinate planes in spin space. In section
3 we elucidate the role of the particular field h
N
in the classical limit. For this purpose we discuss
briefly the effect of a magnetic field on the GS and the spin-wave normal modes of the classical
chain. The aim of section 4 is to determine the critical field h
c
where (true or incipient) AFM LRO
is suppressed by the magnetic field. In section 5 we use these results for a discussion of the phase
diagrams of a few representative classes of system (1.1).
2. Néel ground state of the quantum antiferromagnet
In this section we investigate the existence of a Néel-type GS of Hamiltonian (1.1). It is conve-
nient to treat first the case s =
1
2
, and then generalize the results to arbitrary s. In a first step we
look for an eigenstate of (1.1) which factorises into alternating single-site states as follows:
|Ni =
N
Y
l=1
|ψ
l
i, (2.1a)
|ψ
2l−1
i = a
1
|2l − 1, ↑i + b
1
|2l − 1, ↓i, (2.1b)
|ψ
2l
i = a
2
|2l, ↑i + b
2
|2l, ↓i. (2.1c)
We use z as quantization axis. It is useful to represent the complex coefficients a
i
and b
j
in terms
of angular coordinates, θ
j
, φ
j
, j = 1, 2:
a
1
= cos
θ
1
2
exp
−ı
φ
1
2
, b
1
= sin
θ
1
2
exp
ı
φ
1
2
,
a
2
= cos
θ
2
2
exp
−ı
φ
2
2
, b
2
= sin
θ
2
2
exp
ı
φ
2
2
, (2.2)
The resulting expectation values of the single-site spin components are
hψ
2l−1
|S
x
2l−1
|ψ
2l−1
i =
1
2
sin θ
1
cos φ
1
,
hψ
2l−1
|S
y
2l−1
|ψ
2l−1
i =
1
2
sin θ
1
sin φ
1
,
hψ
2l−1
|S
z
2l−1
|ψ
2l−1
i =
1
2
cos θ
1
, (2.3)
and analogous expressions for sites 2l. This is equivalent to saying that the spins of one sublattice
(2l −1) are pointing in direction (θ
1
, φ
1
) and those of the other sublattice (2l) in direction (θ
2
, φ
2
)
of a polar coordinate system. Thus, for the particular Néel state (2.1), the original nontrivial
many-body problem reduces to a simple eigenvalue problem involving only two lattice sites:
H
l,l+1
|ψ
l
i|ψ
l+1
i =
N
|ψ
l
i|ψ
l+1
i, (2.4)
with
H
l,l+1
= J
x
S
x
l
S
x
l+1
+J
y
S
y
l
S
y
l+1
+J
z
S
z
l
S
z
l+1
−
1
2
h
x
(S
x
l
+S
x
l+1
)+h
y
(S
y
l
+S
y
l+1
)+h
z
(S
z
l
+S
z
l+1
)
. (2.5)
2
Antiferromagnetic long-range order in the anisotropic quantum spin chain
Apart from the solution of (2.4) for |h| → ∞ which is not interesting for our purposes, we find
that there exists a solution with energy (per site)
N
= E
N
/N = −
1
4
(J
x
+ J
y
+ J
z
) (2.6)
for finite fields, provided the field vector points to the surface of an ellipsoid characterized by
h
2
x
(J
x
+ J
y
)(J
x
+ J
z
)
+
h
2
y
(J
x
+ J
y
)(J
y
+ J
z
)
+
h
2
z
(J
x
+ J
z
)(J
y
+ J
z
)
= 1. (2.7)
In other words, we can freely choose a field direction (Θ, P hi) in spin space such that h =
h(sin Θ cos Φ, sin Θ sin Φ, cos Θ), and shall always find a Néel-type eigenstate if the magnitude h of
the field takes the value
h
N
=
(J
x
+ J
y
)(J
x
+ J
z
)(J
y
+ J
z
)
(J
y
+ J
z
) sin
2
Θ cos
2
Φ + (J
x
+ J
z
) sin
2
Θ sin
2
Φ + (J
x
+ J
y
) cos
2
Θ
1/2
. (2.8)
We now show that the existence of a Néel eigenstate |Ni for s =
1
2
implies the existence of a
Néel eigenstate for any s. For this purpose, we note that the unitary transformation U
l
to a new
basis consisting of the state
|ψ
l
i = a
l
|l, ↑i + b
l
|l, ↓i (2.9a)
given by (2.1b,c), and an orthogonal state
|ϕ
l
i = −b
∗
l
|l, ↑i + a
∗
l
|l, ↓i (2.9b)
with a
l
, b
l
given by (2.2), defines a rotation R
l
of the spin vector S
l
U
−1
l
S
µ
l
U
l
=
X
µ
0
R
µµ
0
l
S
µ
0
l
. (2.10)
We call the new spin components S
ξ,η,ζ
l
. Then the local ζ-axis represents the spin direction at site
l in the Néel state. (There exists actually a whole family of such transformations, because (2.9b)
may still be multiplied by an arbitrary phase factor, corresponding to a rotation about the ζ-axis).
If the Hamiltonian H has a Néel eigenstate (2.1), then the transformed Hamiltonian
H
0
= U
−1
HU, (2.11)
where U = Π
l
U
l
, has a ferromagnetic eigenstate. Expressing H
0
in terms of the operators S
ζ
l
and
S
±
l
= S
ξ
l
± ıS
η
l
,
H
0
=
1
2
X
nn
AS
ζ
l
S
ζ
l
0
+ (BS
−
l
+ B
∗
S
+
l
)S
ζ
l
0
+ (CS
−
l
S
−
l
0
+ C
∗
S
+
l
S
+
l
0
) + DS
+
l
S
−
l
0
−
X
l
h
ζ
S
ζ
l
+ h
+
S
−
l
+ h
−
S
+
l
, (2.12)
where the sum nn is over nearest neighbours, the A, B, C are linear combinations of the exchange
constants J
µ
and h
ζ
and h
±
= h
ξ
± ıh
η
represent the components of h along the new axes, then
we find as conditions for a ferromagnetic eigenstate with spin direction along ζ
C = 0, Bs + h
+
= 0. (2.13)
Now, the rotations R
l
defined by (2.10) may be applied to the spin operators for any spin quantum
number s. In other words, the SU(2) unitary transformation (2.9) induces an SU(2s + 1) unitary
transformation by the requirement that both define the same rotation for spin vectors. One can
therefore transform the Hamiltonian for arbitrary spin s to the same form (2.12) as above. This
3
Antiferromagnetic long-range order in the anisotropic quantum spin chain
proves the following theorem : If H has a Néel eigenstate for exchange constants J
µ
and field h
for s =
1
2
, then H has also a Néel eigenstate for the same exchange constants and field 2sh for
arbitrary s.
In order to prove that the Néel state |Ni is the GS of H, it is sufficient to show that
N
is the
GS energy of the pair Hamiltonian (2.5). The proof is especially simple if the field is along one of
the axes. For h = h
x
, we find that we can write the pair Hamiltonian in the form
H
l,l+1
−
N
=
1
2
(J
x
+ J
y
)
(s −S
ζ
l
)(s −S
ζ
0
l+1
) + (s − S
ζ
0
l
)(s −S
ζ
l+1
)
+ J
z
s
2
− (S
ξ
l
S
ξ
l+1
+ S
η
l
S
η
l+1
+ S
ζ
l
S
ζ
l+1
)
− (h − h
x
N
)S
x
l
, (2.14)
where S
ζ
0
l
refers to the spin configuration shifted by a lattice constant, and
N
= −s
2
(J
x
+J
y
+J
z
).
For h = h
x
N
, we have thus expressed H
l,l+1
−
N
as a sum of positive operators, which proves that
|ψ
l
i|ψ
l+1
i is the GS of H
l,l+1
and therefore |Ni is the GS of H. For s =
1
2
, we have checked the
GS property of |ψ
l
i|ψ
l+1
i numerically for various sets of exchange constants and fields in arbitrary
directions on the ellipsoid (2.7). But for s >
1
2
and general field direction we have as yet found no
formal proof.
The spin configuration of the Néel state is determined by the eigenvalue equation (2.4) which
leads in terms of the new variables
θ =
1
2
(θ
1
+ θ
2
), φ =
1
2
(φ
1
+ φ
2
); α =
1
2
(θ
1
− θ
2
), β =
1
2
(φ
1
− φ
2
) (2.15)
to the following equations
J
y
+ J
z
0 0 0 −h
z
−h
y
0 J
x
+ J
z
0 −h
z
0 h
x
0 0 J
x
+ J
y
−h
y
−h
x
0
0 −h
z
−h
y
J
y
+ J
z
0 0
−h
z
0 −h
x
0 J
x
+ J
z
0
−h
y
h
x
0 0 0 J
x
+ J
y
cos θ cos φ
cos θ sin φ
sin θ cos β
cos α sin φ
cos α cos φ
sin α sin β
= 0. (2.16)
In order to survey the diversity of solutions, it is sufficient to study the special cases with the field
in the xy plane, i.e. Θ = π/2, for (i) J
x
≥ J
y
≥ J
z
, (ii) J
z
≥ J
x
≥ J
z
, and (iii) J
y
≥ J
z
≥ J
x
. The
results for a definite ordering J
x
≥ J
y
≥ J
z
with the field in either of the three coordinate planes
may then be obtained by a suitable transformation of axes.
In case (i) (J
x
≥ J
y
≥ J
z
) the Néel state configuration obviously lies in the xy plane, i.e.
θ = π/2, α = 0. The angle φ characterizes the direction of the net magnetization, and 2β is the
angle between the directions of the sublattice magnetizations. The solution of (2.16) yields
tan φ = tan Φ
J
x
+ J
z
J
y
+ J
z
, (2.17)
cos β =
"
(J
x
+ J
z
)(J
y
+ J
z
)
(J
x
+ J
z
) sin
2
Φ + (J
y
+ J
z
) cos
2
Φ
(J
x
+ J
y
)
(J
x
+ J
z
)
2
sin
2
Φ + (J
y
+ J
z
)
2
cos
2
Φ
#
1/2
. (2.18)
In case (ii) (J
z
≥ J
x
≥ J
z
) the sublattice spins lie in a plane perpendicular to the xy plane. Here,
we have θ = π/2, β = 0, and the angle between the directions of the two sublattice magnetizations
is 2α. Eqs. (2.16) are then solved by (2.17) and
cos α =
"
(J
x
+ J
y
)
(J
x
+ J
z
)
2
sin
2
Φ + (J
y
+ J
z
)
2
cos
2
Φ
(J
x
+ J
z
)(J
y
+ J
z
)
(J
x
+ J
z
) sin
2
Φ + (J
y
+ J
z
) cos
2
Φ
#
1/2
. (2.19)
The case (iii) (J
y
≥ J
z
≥ J
x
) shows more complex behavior since we know from (i) and (ii)
that for Φ = 0 the spins of the two sublattices lie in the xy-plane, whereas for Φ = π/2 they are
pointing out of the xy-plane. In fact there is a critical field direction Φ
F
given by
tan Φ
F
=
J
y
+ J
z
J
x
+ J
z
J
y
− J
z
J
z
− J
x
1/2
, (2.20)
4
