Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions
We study the isotropic Heisenberg chain with nearest and next-nearest neighbor interactions. The ground state phase diagram is constructed in dependence on the additional interactions and an external magnetic field. The thermodynamics is studied by use of finite sets of nonlinear integral equations...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2007
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| Цитувати: | Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions / Ch. Trippe, A. Klumper // Физика низких температур. — 2007. — Т. 33, № 11. — С. 1213-1221. — Бібліогр.: 14 назв. — англ. |
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| author | Trippe, Ch. Klumper, A. |
| author_facet | Trippe, Ch. Klumper, A. |
| citation_txt | Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions / Ch. Trippe, A. Klumper // Физика низких температур. — 2007. — Т. 33, № 11. — С. 1213-1221. — Бібліогр.: 14 назв. — англ. |
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| description | We study the isotropic Heisenberg chain with nearest and next-nearest neighbor interactions. The ground state phase diagram is constructed in dependence on the additional interactions and an external magnetic field. The thermodynamics is studied by use of finite sets of nonlinear integral equations resulting from integrability. The equations are solved numerically and analytically in suitable limiting cases. We find second and first order transition lines. The exponents of the low-temperature asymptotics at the phase transitions are determined.
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| first_indexed | 2025-12-07T17:49:41Z |
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| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, No. 11, p. 1213–1221
Quantum phase transitions and thermodynamics of
quantum antiferromagnets with competing interactions
Christian Trippe and Andreas Kl�mper
Fachbereich C – Physik, Bergische Universität Wuppertal, 42097 Wuppertal, Germany
E-mail: trippe@physik.uni-wuppertal.de
E-mail: kluemper@physik.uni-wuppertal.de
Received March 26, 2007
We study the isotropic Heisenberg chain with nearest and next-nearest neighbor interactions. The ground
state phase diagram is constructed in dependence on the additional interactions and an external magnetic
field. The thermodynamics is studied by use of finite sets of nonlinear integral equations resulting from
integrability. The equations are solved numerically and analytically in suitable limiting cases. We find se-
cond and first order transition lines. The exponents of the low-temperature asymptotics at the phase transi-
tions are determined.
PACS: 75.10.Jm Quantized spin models;
75.10.Pq Spin chain models;
75.40.–s Critical-point effects, specific heats, short-range order.
Keywords: integrable systems, thermodynamics, critical behavior, Bethe-ansatz.
1. Introduction
Low-dimensional quantum systems are of considerable
current interest. On one hand they can be studied in exper-
iments, where they are realized as quasi 1D or 2D subsys-
tems. On the other hand, some of the 1D systems, like the
spin-1/2 Heisenberg chain, can be solved exactly or other-
wise non-perturbativly. The method of exactly solving
quantum spin systems via Bethe ansatz is essentially re-
stricted to 1D models, but allows for solving models of
coupled chains, see [1] and references therein. Depending
on the topology, coupled chains (or spin ladders) may be
considered as interpolations between 1D and 2D systems,
or in the case of spin ladders with zigzag interactions the
system may be viewed as a single chain with longer range
interactions.
In many cases these models are only studied in the
ground state and without an external magnetic field, see
e.g. [2]. Considering nonzero temperature and an external
magnetic field is of interest for two reasons. First, the
magnetic field can lead to (several) quantum phase transi-
tions in the ground state, see e.g. [3]. Second, nonzero
temperature and magnetic field are required for compari-
son with experimental work.
The main goal of this paper is to study the thermo-
dynamical properties of two quantum spin chains with
competing interactions in an external magnetic field.
Both are generalisations of the standard spin-1/2 Hei-
senberg chain and can be solved exactly via Bethe ansatz
[4]. Here we investigate in more detail the thermody-
namics which also leads to new information on the ground
state.
The paper is organized in the following way. First we
introduce in Sec. 2 the Hamiltonians of the two models in-
vestigated in this paper. Then in Sec. 3 we show the rela-
tion of these Hamiltonians to a row-to-row transfer matrix
and present the definition of a quantum transfer matrix
which allow for exactly solving the models via Bethe
ansatz. In Sec. 4 we derive nonlinear integral equations
determining the thermodynamical properties of the mod-
els and consider their zero temperature limit. Using these
equations we discuss the ground state phase diagrams in
Sec. 5 and present several results for the magnetic suscep-
tibility, magnetisation and specific heat in Sec. 6. Finally,
we summarise our results.
2. Hamiltonians
In this paper we investigate the properties of two sys-
tems. First, the Hamiltonian of the Bethe ansatz solvable
isotropic spin-1/2 chain with nearest neighbour interac-
tions and three spin interactions between successive
spins, referred to as next-nearest neighbor model or «mo-
© Christian Trippe and Andreas Kl�mper, 2007
del with NN interaction» [4–6], can be written in the form
� � � �NN hJ J� � �1 2
2
2� , where
�1 1
1
1
4
� ��
�
�
��
�
� S Si i
i
L
(1)
is the Hamiltonian of the standard spin-1/2 Heisenberg
chain, and
��2
1
1 1� � �
�
� ��S S Si
i
L
i i (2)
contains the three spin interactions. The Zeeman term
�h i
z
i
L
h S� �
�
�
1
(3)
takes account of an homogeneous external magnetic field
h. The coupling � 2 determines the relative strength of the
three-spin interactions.
The second system considered in this paper contains
nearest and next-nearest neighbour interactions as well as
four-spin interactions (in the following referred to as
«model with NNN interactions» [4,7]) which has the form
� � � �NNN hJ J� � �1 3
3
3� where
�3 1 2 1 1 2 1 2
1
2
2 2� � � � �� � � � � � �S S S S S S S S S Si i i i i i i i i i( )( ) ( )( ) .S Si i
i
L
�
�
��
��
�
��
� 1
1
1
8
(4)
The operators �i , (i �1 2 3, , ) and �h commute mutually as
well as with a transfer matrix t( )� constructed in the next
Section. These properties allow for exactly solving the
models via Bethe ansatz.
3. Transfer matrices
The R-matrix belonging to the Heisenberg chain is
given by
R��
��
�
�
�
�
�
�
�
�� � � � � � � �( , ) ( )� � � (5)
which is a solution of the Yang-Baxter equation. Here the
indices in the first column denote states in the auxiliary
space and the indices in the second column denote states
in the quantum space. For a more detailed description of
the notation see [8]. With ( ) ( , ) ( , )L R ej j�
�
��
��
�
�� � � �� the
operator defined on a chain of L sites
t L LL( ) [ ( , ) ( , )]� � �� tr 0 01� (6)
yields a family of commuting transfer matr ices
[ ( ), ( )]t t� � � 0 for arbitrary � �, � �. The operators (�i ,
i �1 2 3, , ) are given as logarithmic derivatives of this
transfer matrix at the shift point � � 0
�1
11
2
0
2
� �� ( )( ) ;
L
(7)
�2
2
4
0
4
� �
i iL
� ( )( ) ; (8)
�3
31
8
0
4
� � �� ( )( )
L
, (9)
where � � �( ) ln ( )� t .
Next, we introduce R matrices R R��
��
��
��� � � �( , ) ( , )�
by a clockwise and R R��
��
��
��� � � �( , ) ( , )� by an anticlock-
wise rotation and define a transfer matrix t( )� in the same
way as t above. The partion function of the models with-
out magnetic field can be expressed as
Z t u t
N
i
i
N
�
�
�
�
�
�
�
�
�� �
�
!lim ( ) ( )tr
1
2
0 (10)
with appropriate spectral parameters u i [9], depending
on the model*. The column-to-column transfer matrix of
the corresponding two dimensional L N� lattice is called
quantum transfer matrix (QTM). It is defined by
tQTM DL
N
QTM
L
N
QTM
uN L
QTM
L
Q
( ) ( , ) ( , ) ( , )� � � �� � tr 0
1 2 2
0
1
�
TM
u( , )� 1
�
�
��
��
(11)
where the magnetic field h is included by means of
twisted boundary conditions via the diagonal matrix
D h h� � diag (exp ( ),exp ( ))� �2 2 and
L
R e j
R e
j
QTM
j
j
�
� ��
��
�
�
��
��
�
�
� �
� �
� �
( , )
( , ) ,
~
( , )
�
�
even ,
R e jj��
��
�
�� �( , ) , odd .
"
#
$
%
$
(12)
The monodromy matrix corresponding to the QTM is a
representation of the Yang-Baxter algebra with intertwi-
ner R. The partion function is given by
Z tN
QTM L� tr ( ( )) .0 (13)
1214 Fizika Nizkikh Temperatur, 2007, v. 33, No. 11
Christian Trippe and Andreas Klümper
* Note that we ignored in (10) the additive constants of equations (7)–(9).
Hence the free energy in the thermodynamic limit is de-
termined by
f T
N
QTM� �
� �
lim ln ( )& 0 (14)
where &QTM is the largest eigenvalue of the QTM.
4. Nonlinear integral equations
For the standard spin-1 2 Heisenberg chain one can
derive different sets of nonlinear integral equations
(NLIE) determining the thermodynamical properties.
Historically first, an infinite set of equations via TBA [10]
was obtained. Then, as a second possibility, a set of only
two equations [11,12] was derived. In fact, it is also possi-
ble to find an arbitrary number of equations interpolating
between these extreme schemes [13].
For the models investigated in this paper the set of two
coupled nonlinear integral equations was derived in [4]
where also certain parameter ranges were treated numeri-
cally. Here, we performed further numerical studies of
these equations and found that they are not valid for low
temperatures in the vicinity of the phase coexistence if
straight integration contours are used (see Sec. 5). This is
due to the fact that the imaginary parts of some Bethe
ansatz numbers grow strongly, leading to a crossing of the
integration contours by singularities of the integrands.
To determine the free energy also for the cases where
the two NLIE with standard contours are not valid, it is
useful to utilize the fusion hierarchy of this model. One
obtains in the usual way an infinite set of NLIE. The first
equation is
ln ( )
cos ( )
( ) ( ln ( ))y x
x
Z x s Y x1 2� � � �
v�
'h
* (15)
with * denoting convolution* and s x x( ) [ cos ( )]� �2 1h '
being the integration kernel. Z x( ) depends on the model
and is given by
Z x
x
x
( )
sin ( )
( )
,
�
� �
'
'
2
2
2
v
h
cosh
for the model with NN interactions ,
tanh
cosh
for the model with N� �
'
'3
3
22 1
v
( )
( )
,
x
x
�
NN interactions .
"
#
$
$
%
$
$
(16)
The other equations are independent of the model and
read
ln ( ) ( ln ( ))( ) .y x s Y Y xj j j� � �* 1 1 (17)
The magnetic field does not enter explicitly in these equa-
tions, it only fixes the asymptotic behavior of the y-func-
tions. For zero magnetic field it reads
lim ( ) ( )
| |x
jy x j j
� �
� � 2
and for h ( 0
lim ( )
| |
( )
x
j
j j
y x
z z
z z� �
� � �
�
�
�
�
�
�
�
�
�
�
�
1 1
1
2
1 (18)
with z h� exp ( )� 2 . Note that the asymptotic behavior
and therefore the system of equations is invariant under a
change of sign of the magnetic field. The free energy is
given by
f e T dx
Y x
x
� �
��
�
)0
1
2
ln ( )
( )cosh '
(19)
with e J0 2� � ln for the model with NN interactions and
e J
J
0 3
3
2
3
8
3� � �ln ( )� *
for the model with NNN interactions, * denoting the
Riemann *-function.
It is possible to close the set of infinitely many integral
equations after the ( )k �1 th equation. The minimal param-
eter k �1gives the set of two NLIE presented in [4]. Next
we show the results for k + 2 following [13].
One can find suitable functions b, b , B b( ): ( )x x� �1
and B b( ): ( )x x� �1 satisfying B B( ) ( ) ( )x x Y xk� and
therefore yielding the functional relation
y x
i
y x
i
Y x x xk k k� � �� � �1 1 2
2 2
( ) ( ) ( ) ( ) ( ) .B B (20)
For (20) the cases k �1and k � 2 are exceptional. For k �1
the equation is not used, for k � 2 we useY0 1, . This leads
to the following NLIE
ln ( ) ln ( )( ) ln ( )( ) .y x s Y x s xk k� �� �1 2* * BB (21)
Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions
Fizika Nizkikh Temperatur, 2007, v. 33, No. 11 1215
* ( )( ) ( ) ( )g h t g t h d* � �
��
�
) � � �.
Finally, two NLIE equations for b, b close the equation
system exactly
ln ( ) ( ln )( ) ( ln )( ) ( ln )( ) ;b B Bx
h
s Y x x x ik� � � � ��
�
- -
2
1* * *
(22)
ln ( ) ( ln )( ) ( ln )( ) ( ln )( ) .b B Bx
h
s Y x x x ik� � � � � ��
�
- -
2
1* * *
(23)
In (21) for k � 2 and in (22), (23) for k �1, instead of
s Y* ln 0 the inhomogeneity of (15) has to be used. The ker-
nel function - is given by
-
'
( )
( )
.
| |
x dk
k
k
ikx�
��
� �
)
1
2 2 2
2e
cosh
e (24)
Here, the magnetic field enters the equations explicitly.
The different sets of equations for different k are all
equivalent. The equations for larger k allow for the inves-
tigation of lower temperatures even with straight con-
tours.
Calculating the zero temperature limit of the fusion hi-
erarchy with . i iT y� ln in the usual way [10] one obtains
only one dressed energy . .: � 1 which can have negative
values, all other dressed energies are strictly positive and
hence do not contribute. The dressed energy . is deter-
mined by
. �
'
�
� �
. � . �( )
( )
( ) ( )�
� �
�)
1 1
1 2 0d
�
(25)
with
. �
�
�
'
�
�
�
'
0 2
2 2
2
2
1 4
8 2
1 4
16 1
( )
( )
| | , ,
� �
�
�
�
�
�
�
J
J
h
J
NNfor �
2 1
1 4
2
2 3
�
�
�
�
�
"
#
$
$
%
$
$ ( )
| | ,h NNNfor �
(26)
and � � � /{ | ( ) }� . �� 0 . Note that only the absolute va-
lue of the magnetic field enters the equation (25) via the
bare energy (26). This is due to the fact that only the abso-
lute value of h enters the fusion hierarchy via the asymp-
totic behavior (18). This statement becomes even more
obvious if one derives (25) from the system of only two
NLIE. Here one of the steps of the derivation uses the ob-
servation that the function b drops out for h 0 0 and b
drops out for h / 0 and in both cases (25) is obtained. Fur-
thermore a rescaling of the couplings � 2, � 3 was applied
by � �� 2v for the model with NN interactions and
� �� 3
2
v for the model with NNN interactions. This will
be useful in the discussion of the ground state phase dia-
gram in the next Section as in the new variable certain
critical points occur at � � 1 1.
The dressed energy can be solved in two limiting ca-
ses. The first one is at h � 0 and | |� 2 1 by Fourier trans-
form
. � '�
� '�
'�
�
( )
( )
[ ( )] ,
cos ( )
[
�
� �
� � �
v
v
cosh
tanh for
h
1
1
�NN
2 2� '�tanh for( )] .�NNN
"
#
$$
%
$
$
The second case is for h h f+ where the integral van-
ishes and . � . �( ) ( )� 0 . h f is the saturation field, i.e. the
value of the magnetic field corresponding to the phase
transition into the ferromagnetic phase.
5. Phase diagrams of the ground state
In this section we first give our results on the ground
state phase diagram for both models (Fig. 1). Our results
differ a little from those of [4]. For all couplings � there is
a phase transition into a ferromagnetic phase and phase
coexistence for the lines | |� + 1, h � 0. But only in the
model with NNN interactions and positive coupling � we
find a phase transition between a commensurate and an
incommensurate phase.
Looking at the dressed energy for the models with zero
magnetic field one can understand the phase diagrams.
The phase transitions in dependence on the magnetic field
h correspond to the opening and closing of Fermi seas, i.e.
appearance or disappearance of intervals of negative en-
ergy modes of the dressed energy. Qualitatively, these
transitions occur at magnetic fields h coinciding with
extremal values of the dressed energy at zero field as plot-
ted in Fig. 2. This is very much like the discussion of
van-Hove singularities of free particle systems. However,
here we deal with an interacting system for which the
chemical potential is not identical to h but equal to | |h . So
only the negative extremal values of the dressed energy
are relevant. Although the analytic solution (27) is strictly
valid only for | |� 2 1 we use these formulas for slightly
larger values of | |� where they should be good approxima-
tions to the true solution.
Depending on the longer range coupling �, the model
with NNN interactions has one or two Dirac seas. For
� � 0 2. only a second order phase transition into the ferro-
magnetic phase and a first order transition line with non
vanishing spontaneous magnetization for � 2 �1, h � 0 ex-
ist (Fig. 2,c and 2,d). For � � 0 2. (Fig. 2,b) the dressed en-
ergy possesses two local minima and one local maximum.
The two local minima have identical value as . �( ) is an
even function. So here a phase transition between two
phases exists (denoted «commensurate» and «incommen-
surate»). The tricitical point can be determined to
� '� 2 48 0 2056� . and h Jf � 5 3/ by analyzing the so-
lution of the dressed energy in the ferromagnetic phase.
1216 Fizika Nizkikh Temperatur, 2007, v. 33, No. 11
Christian Trippe and Andreas Klümper
The phase diagram of the model with NN interactions
is symmetric under a sign change of the coupling �, as in
the NLIE this can be compensated for by a sign change of
the spectral parameter �. Hence for the model with NN in-
teractions it is sufficient to look at couplings � + 0. This
model has always only one Dirac sea (Fig. 2,a), hence
there is only a second order phase transition between the
antiferromagnetic and the ferromagnetic phase and a first
order transition line with non vanishing spontaneous
magnetization for | |� 0 1, h � 0.
Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions
Fizika Nizkikh Temperatur, 2007, v. 33, No. 11 1217
0
1
2
3
4
5
6
0.2 0.4 0.6 0.8 1.0 1.2 1.4
h
�
ferromagnetic phase
antiferromagnetic phase
first order phase transition
second order phase transition
0
1
2
3
4
5
6
7
8
9
–1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0
h
�
ferromagnetic phase
commensurate phase
incommensurate phase
first order
phase transition
first order
phase transition
second order phase transition
second order
phase transition
tricritical point
a b
( ) NN interactionsa Model with ( ) NNN interactionsb Model with
Fig. 1. Phase diagrams of the model with NN and NNN interactions. Both phase diagrams are symmetric with respect to a change
of sign of the magnetic field h, and for the model with NN interactions (a) also with respect to a change of sign of the coupling �. In
these pictures the normalization J � 2 in �NN and �NNN is used. The first order phase transitions are exactly at h � 0.
–5
–4
–3
–2
–1
0
–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0
� = 0 � = 0.7
� = 1.3 � = 1.1
� = 1.0 � = 1.0
� = 0.7
� = 1.1
� = 1.1
NN model with positive �
–2.5
–2.0
–1.5
–1.0
–0.5
0
0.5
–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0
NNN model with positive �
–7
–6
–5
–4
–3
–2
–1
0
–1.0 –0.5 0 0.5 1.0
–0.5
–0.4
–0.3
–0.2
–0.1
0
0.1
–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0
Enlarged case = –1.1 of ( )� cNNN model with negative �
a b
c d
�
� �
�
� = –1.0
.
��
.
��
.
��
.
��
Fig. 2. The dressed energies according to equation (27) for h � 0 and normalization J � 2. The data are approximate for | |� 01.
The critical field corresponding to the phase transition
from the antiferromagnetic into the ferromagnetic phase
can be determined as usual [14] from the bare energy (26).
For the model with NNN interactions one obtains
h
J J
Jf �
� 2
�
�
�
�
� �
2
16
48
2
24 24 4 24
2
2
2 2 2
�
'
� '
�
'
�
'
�
'
, ,for
�
�
�
�
�
� � �
�
�
�
�
16
3 24 24 4 4 72
24 24 4
2
2 2 2
2 2
J
�
'
�
'
�
'
�
'
�
'
�
'
� �
�
�
�
�
�
�
+
"
#
$
$
$$
%
$
$
$
$
24
48
2
3
2
�
'
� ', .for
(28)
For � '2 2 48 this is a linear relation between the critical
field h f and the coupling �. For � '+ 2 48 the relation is
nonlinear, however with linear asymptotic behavior for
large values of �
h J Of � �
�
�
�
� � �
�
�
�1 4
1
2
�
' �
. (29)
The applicability of (28) is restricted to really large val-
ues of �, e.g. the error gets smaller than 1% for � � 58. .
For the model with NN interactions the equation deter-
mining the minimum of the bare energy is cubic, whereas
it is biquadratic for the model with NNN interactions. For
this reason we want to give only numerical values for he
critical field h f for the model with NN interactions.
These can be taken from Fig. 1,a.
6. Magnetic susceptibility, magnetization
and specific heat
In this section we present the magnetic susceptibility 3
for typical values of the coupling � in dependence on the
magnetic field and temperature. We also show evidence
that the phase transition at h � 0 is of first order. Finally,
we calculate the specific heat c at the phase transitions.
The derivatives of the free energy are obtained by differ-
entiating (19) and deriving new integral equations for the
logarithmic derivatives of the auxiliary functions involv-
ing the auxiliary functions as external parameters.
As the situation in the model with NN interactions is
very similar to the one for the model with NNN interac-
tions and negative coupling �, we will focus in the follow-
ing on the model with NNN interactions and only some-
times give comments on the model with NN interactions.
For the numerical calculations we always use the
normalization J � 2.
For the model with NNN interactions the magnetic
susceptibility is shown for typical values of �. In Fig. 3,a
3( )h is shown for � � �1and T � 0 01. . One sees that there
is a maximum at h � 7 2. corresponding to the phase tran-
sition between the antiferromagnetic and ferromagnetic
phase and another maximum at h � 0 also corresponding
to a divergence at T � 0 (Fig. 3,b). This qualitative picture
is also true for � / �1.
For larger values of � the low field maximum exists
until � � �0 2. and h h f// but it does not correspond to a
phase transition because the magnetic susceptibility does
not diverge for T � 0.
For � � 0 206. , the value corresponding to the tricritcal
point, two maxima exist (Fig. 4,a). Here also the maxi-
mum at lower magnetic field diverges for T � 0 and so
1218 Fizika Nizkikh Temperatur, 2007, v. 33, No. 11
Christian Trippe and Andreas Klümper
T = 0.01 �
�
�
= –0.999, h = 0
= –1, h = 0
= –1, h = 10
– 3
= –1�
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 1 2 3 4 5 6 7 8
h
10
–2
10
–1
10
0
10
1
10
2
10
3
10
4
10
5
T
10
–10
10
–8
10
–6
10
–4
10
–2
10
2
10
0
3
�
h
3
4
�
a b
Fig. 3. The magnetic susceptibility for the model with NNN interactions and negative coupling � � �1.
corresponds to a phase transition (Fig. 4,b). The value of
the lower critical field decreases with increasing � and
turns 0 at � �1. This fact, known from the functional be-
havior of the dressed energy is also supported by the nu-
merical data for the magnetic susceptibility at finite tem-
perature. In Fig. 4,c one clearly sees that the maximum of
the magnetic susceptibility occurs at a finite magnetic
field for � / 1.
The value of the magnetic field corresponding to the
phase transition between the commensurate and incom-
mensurate phase could not be determined analytically.
However, good numerical results are obtained by calcu-
lating the magnetic susceptibility at finite but low temper-
ature for different values of the magnetic field and deter-
mining the local maximum. Doing this for different
temperatures one can also estimate the error. The line be-
tween the commensurate and incommensurate phase in
Fig. 1,b is located in this way, where errorbars are within
linewidth.
The magnetization for h � 0 is shown for the model
with NNN interactions and � � �1in Fig. 4,d. Clearly for
� / �1and low temperature the magnetization has a finite
asymptotic limit for small magnetic fields, whereas this is
not the case for � � �1. So, a first order phase transition
exists for � / �1. This statement also holds for h � 0 and
� 0 1 and for the model with NN interactions with zero
magnetic field and | |� 0 1.
Finally, we determine the asymptotic behavior of the
specific heat for T � 0 at the phase transitions. We find
the specific heat for h � 0 and | |� �1 to very low tempera-
tures and with very high accuracy.
The specific heat vanishes as T 1 3 for the model with
NNN interactions at � � �1 (Fig. 5,a) and as T 1 2
(Fig. 5,b) for � �1. For the phase transition into the ferro-
magnetic phase and with slightly lower numerical accu-
racy for the phase transition between the commensurate
and incommensurate phase we find the asymptotic behav-
ior T 1 2 . At the tricritical point the specific heat vanishes
like T 1 4 (see Fig. 5,c) as predicted in [4]. For the first or-
der phase transitions we were not able to compute the spe-
cific heat at sufficiently low temperatures with suffi-
ciently high accuracy to find consistent results for the low
temperature asymptotics for | |� 0 1. Here further investi-
gations are necessary.
Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions
Fizika Nizkikh Temperatur, 2007, v. 33, No. 11 1219
h = 1.3
h = 1.455
h = 1.6
0
0.2
0.4
0.6
0.8
1.0
1.2
0.5 1.5 3.02.5 3.5
3
3
0.01
0.1
1
0.01 0.1 1
3
T
0.2
0.4
0.6
0.8
1.0
0 0.01 0.02 0.03 0.04
h h
h
0.01
0.001
0.1
m
� = 0.7, T = 10
–2
T = 10
–5
� = 0.7
2.01.0
10
–5
10
–3
10
–4
� = –1.01
� = –1.00
a b
c d
Fig. 4. The magnetic susceptibility for the model with NNN interactions in figures (a)–(c) and the magnetization in figure (d)
Fig. (b) clearly shows that for � � 07. the magnetic susceptibility diverges for h �1455. , which is in contrast to the cases with lower
and higher magnetic fields.
Again the situation in the model with NN interactions
corresponds to the one in the model with NNN interac-
tions and negative coupling �. In particular this means
that the specific heat at � �1 and h � 0 vanishes like T 1 3
which is shown in Fig. 5,d.
7. Conclusion
We studied the thermodynamics and ground state
phase diagrams of two integrable models containing the
standard spin-1/2 Heisenberg Hamiltonian and additional
competing interactions.
The ground state phase diagrams depending on the ex-
ternal magnetic field and the longer range coupling � are
c o n s t r u c t e d . T h e y c o n t a i n f e r r o m a g n e t i c a n d
antiferromagnetic phases. In both models there exist sec-
ond order phase transitions between these phases and first
order phase transition lines with non-vanishing spontane-
ous magnetization. Only the model with NNN interac-
tions with positive � contains a phase transition between a
commensurate and an incommensurate phase.
The NLIE describing the models at finite temperature
are solved numerically for typical values of the coupling
� and the magnetic field h.
The vicinity of the phase coexistence in both models is
difficult to investigate. For sufficiently low temperatures
the NLIE are numerically ill-posed if straight integration
contours are used. For reaching low temperatures, either
the contours have to be deformed — or as chosen in our
approach — the truncation level has to be increased.
Acknowlegment
The authors like to acknowledge support by the re-
search program of the Graduiertenkolleg 1052 funded by
the Deutsche Forschungsgemeinschaft.
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1220 Fizika Nizkikh Temperatur, 2007, v. 33, No. 11
Christian Trippe and Andreas Klümper
a b
c d
exact
exact exact
exact
asymptote
asymptote asymptote
asymptote
� = –1
h = 0
x = 0.33348 21
� '= /48
2
h = 10/3
x = 0.2499 51
� = 1
h = 0
x = 0.506 11
5 5
�
5
1
-model
= 1
h = 0
x = 0.33339 1
10
–1
10
–1
10
–1
10
–3
10
–3
10
–3
10
–5
10
–5
10
–5
10
–7
10
–7
10
–9
0.1
0.01
T T
T T
10
–4
10
–4
10
–6
10
–5
10
–3
10
–3
10
–2
10
–2
10
–20
10
–20
10
–1
10
–1
10
–10
10
–10
10
–15
10
–15
10
–5
10
–5
10
0
10
0
10
0
10
0
10
1
10
1
c
c
c
c
Fig. 5. The specific heat for the model with NNN interactions (Fig. (a)–(c)) and the model with NN interactions (Fig. (d)). The low
temperature exponent x of T (c T x6 ) is determined by a fit on the numerical data. The error refers to the last digit and only includes
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Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions
Fizika Nizkikh Temperatur, 2007, v. 33, No. 11 1221
|
| id | nasplib_isofts_kiev_ua-123456789-7712 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T17:49:41Z |
| publishDate | 2007 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Trippe, Ch. Klumper, A. 2010-04-08T12:09:19Z 2010-04-08T12:09:19Z 2007 Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions / Ch. Trippe, A. Klumper // Физика низких температур. — 2007. — Т. 33, № 11. — С. 1213-1221. — Бібліогр.: 14 назв. — англ. 0132-6414 PACS: 75.10.Jm; 75.10.Pq; 75.40.–s https://nasplib.isofts.kiev.ua/handle/123456789/7712 We study the isotropic Heisenberg chain with nearest and next-nearest neighbor interactions. The ground state phase diagram is constructed in dependence on the additional interactions and an external magnetic field. The thermodynamics is studied by use of finite sets of nonlinear integral equations resulting from integrability. The equations are solved numerically and analytically in suitable limiting cases. We find second and first order transition lines. The exponents of the low-temperature asymptotics at the phase transitions are determined. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions Article published earlier |
| spellingShingle | Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions Trippe, Ch. Klumper, A. |
| title | Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions |
| title_full | Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions |
| title_fullStr | Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions |
| title_full_unstemmed | Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions |
| title_short | Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions |
| title_sort | quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/7712 |
| work_keys_str_mv | AT trippech quantumphasetransitionsandthermodynamicsofquantumantiferromagnetswithcompetinginteractions AT klumpera quantumphasetransitionsandthermodynamicsofquantumantiferromagnetswithcompetinginteractions |