Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions
The one-dimensional spin-1/2 model, in which the alternation of the exchange interactions between neighboring spins is accompanied by the next-nearest neighbor (NNN) spin exchange (zig-zag spin ladder with the alternation) is studied. Thermodynamic characteristics of the considered quantum spin chai...
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irk-123456789-1171962017-05-21T03:02:55Z Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions Zvyagin, A.A. Cheranovskii, V.O. Низкотемпеpатуpный магнетизм The one-dimensional spin-1/2 model, in which the alternation of the exchange interactions between neighboring spins is accompanied by the next-nearest neighbor (NNN) spin exchange (zig-zag spin ladder with the alternation) is studied. Thermodynamic characteristics of the considered quantum spin chain are obtained in the mean-field like approximation. Depending on the strength of the NNN interactions, the model manifests either the spin-gapped behavior of low-lying excitations at low magnetic fields, or the ferrimagnetic ordering in the ground state with gapless low-lying excitations. The system undergoes second order or first order quantum phase transitions, governed by the external magnetic field, NNN coupling strength, and the degree of the alternation. Hence, NNN spin–spin interactions in a dimerized quantum spin chain can produce a spontaneous magnetization. On the other hand, for quantum spin chains with a spontaneous magnetization, caused by NNN spin–spin couplings, the alternation of nearest-neighbor (NN) exchange interactions can be the reason for destroying of that magnetization and the onset of a spin gap for low-lying excitations. Alternating NN interactions produce a spin gap between two branches of low-energy excitations, and the NNN interactions yield asymmetry of dispersion laws of those excitations, with possible minima, corresponding to incommensurate structures in the spin chain. 2009 Article Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions / A.A. Zvyagin, V.O. Cheranovskii // Физика низких температур. — 2009. — Т. 35, № 6. — С. 578-592. — Бібліогр.: 25 назв. — англ. 0132-6414 PACS: 75.10.Pq, 75.40.Cx http://dspace.nbuv.gov.ua/handle/123456789/117196 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм |
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Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм Zvyagin, A.A. Cheranovskii, V.O. Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions Физика низких температур |
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The one-dimensional spin-1/2 model, in which the alternation of the exchange interactions between neighboring spins is accompanied by the next-nearest neighbor (NNN) spin exchange (zig-zag spin ladder with the alternation) is studied. Thermodynamic characteristics of the considered quantum spin chain are obtained in the mean-field like approximation. Depending on the strength of the NNN interactions, the model manifests either the spin-gapped behavior of low-lying excitations at low magnetic fields, or the ferrimagnetic ordering in the ground state with gapless low-lying excitations. The system undergoes second order or first order quantum phase transitions, governed by the external magnetic field, NNN coupling strength, and the degree of the alternation. Hence, NNN spin–spin interactions in a dimerized quantum spin chain can produce a spontaneous magnetization. On the other hand, for quantum spin chains with a spontaneous magnetization, caused by NNN spin–spin couplings, the alternation of nearest-neighbor (NN) exchange interactions can be the reason for destroying of that magnetization and the onset of a spin gap for low-lying excitations. Alternating NN interactions produce a spin gap between two branches of low-energy excitations, and the NNN interactions yield asymmetry of dispersion laws of those excitations, with possible minima, corresponding to incommensurate structures in the spin chain. |
| format |
Article |
| author |
Zvyagin, A.A. Cheranovskii, V.O. |
| author_facet |
Zvyagin, A.A. Cheranovskii, V.O. |
| author_sort |
Zvyagin, A.A. |
| title |
Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions |
| title_short |
Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions |
| title_full |
Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions |
| title_fullStr |
Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions |
| title_full_unstemmed |
Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions |
| title_sort |
elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2009 |
| topic_facet |
Низкотемпеpатуpный магнетизм |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117196 |
| citation_txt |
Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions / A.A. Zvyagin, V.O. Cheranovskii // Физика низких температур. — 2009. — Т. 35, № 6. — С. 578-592. — Бібліогр.: 25 назв. — англ. |
| series |
Физика низких температур |
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AT zvyaginaa elementaryexcitationsandthermodynamicsofzigzagspinladderswithalternatingnearestneighborexchangeinteractions AT cheranovskiivo elementaryexcitationsandthermodynamicsofzigzagspinladderswithalternatingnearestneighborexchangeinteractions |
| first_indexed |
2025-07-08T11:48:39Z |
| last_indexed |
2025-07-08T11:48:39Z |
| _version_ |
1837079273544876032 |
| fulltext |
Fizika Nizkikh Temperatur, 2009, v. 35, No. 6, p. 578–592
Elementary excitations and thermodynamics of zig-zag
spin ladders with alternating nearest neighbor exchange
interactions
A.A. Zvyagin
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
IFW Dresden, Institute for Solid State Research, P.O. 270116, Dresden D-01171, Germany
E-mail: zvyagin@ilt.kharkov.ua
V.O. Cheranovskii
V.N. Karazin Kharkov National University, 4 Svobody Sq., Kharkov 61077, Ukraine
Received January 12, 2009, revised January 20, 2009
The one-dimensional spin-1/2 model, in which the alternation of the exchange interactions between
neighboring spins is accompanied by the next-nearest neighbor (NNN) spin exchange (zig-zag spin ladder
with the alternation) is studied. Thermodynamic characteristics of the considered quantum spin chain are ob-
tained in the mean-field like approximation. Depending on the strength of the NNN interactions, the model
manifests either the spin-gapped behavior of low-lying excitations at low magnetic fields, or the fer-
rimagnetic ordering in the ground state with gapless low-lying excitations. The system undergoes second or-
der or first order quantum phase transitions, governed by the external magnetic field, NNN coupling
strength, and the degree of the alternation. Hence, NNN spin–spin interactions in a dimerized quantum spin
chain can produce a spontaneous magnetization. On the other hand, for quantum spin chains with a spontane-
ous magnetization, caused by NNN spin–spin couplings, the alternation of nearest-neighbor (NN) exchange
interactions can be the reason for destroying of that magnetization and the onset of a spin gap for low-lying
excitations. Alternating NN interactions produce a spin gap between two branches of low-energy excita-
tions, and the NNN interactions yield asymmetry of dispersion laws of those excitations, with possible min-
ima, corresponding to incommensurate structures in the spin chain.
PACS: 75.10.Pq Spin chain models;
75.40.Cx Static properties.
Keywords: spin chain, spin frustration, zig-zag spin ladders.
1. Introduction
Last decade the interest in quasi-one-dimensional
quantum spin systems has grown considerably. In these
systems spin–spin interactions along one space direction
are much stronger than couplings along other directions.
The interest of physicists to low-dimensional quantum
spin systems is motivated, first of all, by the progress in
the preparation of substances with well defined one-di-
mensional subsystems. Another reason for studying qua-
si-one-dimensional quantum spin systems is the possibil-
ity to compare experimental data with exact solutions for
one-dimensional models [1]. In one-dimensional systems
quantum fluctuations are strongly enhanced due to pecu-
liarit ies in densities of states. According to the
Mermin–Wagner theorem [2], low-dimensional spin sys-
tems cannot have magnetic ordering at any nonzero tem-
perature, due to mentioned quantum fluctuations. On the
other hand, such systems often manifest quantum phase
transitions, which take place in the ground state, and
which are governed by other than the temperature param-
eters, like an external magnetic field, external or internal
(caused by chemical substitutions) pressure, etc.
During last years more attention was paid to theoreti-
cal studies of one-dimensional quantum spin models with
not only nearest-neighbor (NN) spin–spin interactions,
© A.A. Zvyagin and V.O. Cheranovskii, 2009
but also with next-nearest neighbor (NNN) ones. Those
models are closer to real quasi-one-dimensional magnets
comparing to standard ones with only nearest-neighbor
couplings [3]. Such interactions are often present in
oxides of transition metals, where the direct exchange
between magnetic ions is complimented by the super-
exchange between magnetic ions via nonmagnetic ones.
Some transition metal-oxide compounds exhibiting prop-
erties of quasi-one-dimensional spin systems and spin-
frustrated systems have been the topic of recent studies
because of their large variety of very interesting low tem-
perature properties. The low-energy physics of these sys-
tems is mostly determined by a strong exchange inter-
actions between spin-1/2 double chains, formed by
edge-sharing CuO 6 or VO 6 octahedra, in which neigh-
boring and next-neighboring ions of Cu 2� or V 4� with
spins 1/2 are surrounded by oxygen ions. Such a configu-
ration of those oxides produces a very special tempera-
ture and magnetic behavior of many low energy cha-
racteristics of those compounds. For the consistent
explanation of several experiments [1] by means of in-
elastic neutron scattering, optical conductivity and nu-
clear magnetic resonance one needs to account for rela-
tively large values of NNN spin–spin interactions.
Classical counterparts of these models with anti-
ferromagnetic NNN interactions manifest the spin frus-
tration. The classical lowest energy state is highly degen-
erate [5]. For such models the standard quasi-classical
theoretical description based on the quantization of small
deviations of classical vectors of magnetization of mag-
netic sublattices often cannot be applied. This is why,
quantum spin models with NNN spin–spin interactions
are of principal importance. Also, one-dimensional quan-
tum spin models with NN and NNN spin–spin couplings
often manifest quantum phase transitions: several one-di-
mensional quantum models, belonging to that class, see,
e.g., Refs. 6–12, can be solved exactly, using Bethe an-
satz. However, exact solutions were obtained for models
with additional, perhaps non-realistic spin-spin cou-
plings. Many transition metal compounds, like copper
oxides, are believed to reveal features, characteristic for
quantum phase transitions. On the other hand, it turns out
that some quasi-one dimensional spin systems manifest
features, which are characteristic for doubling of mag-
netic elementary in-chain cells, e.g., the alternation of
effective g-factors and/or dimerization of the nearest-
neighbor exchange constants.
In this paper we study a one-dimensional model of
quantum spins with NN and NNN interactions (or, in
other words, the spin zig-zag ladder) and dimerization of
the NN coupling. The main goal of our work is to con-
struct a theory of a realistic model of a quantum spin
chain, which permits to analytically describe quantum
phase transitions caused by spin-spin interactions and
the external magnetic field. That model, on the one hand,
has to contain spin-frustrating NNN spin exchange inter-
actions, which usually produce incommensurate mag-
netic structures [10]. On the other hand, the model has to
have alternating NN exchange interactions, which can
be the reason for the formation of a spin gap for low-ly-
ing excitations there [1]. Notice, that antiferromagnetic
NN and NNN interactions themselves also can produce a
gap [13]. In recent experiments on quasi-one-dimen-
sional quantum magnets [14] it was pointed out that for
some range of parameters it is possible that incommen-
surate magnetic structures co-exist with the doubling of
elementary cells for one-dimensional spin subsystems,
see also Ref. 15. The Hamiltonian of our model contains
both of mentioned factors. As it will be clear from our re-
sults, taking into account these two factors (NNN inter-
actions and dimerization), our theory really reveals
quantum phase transitions (governed also by the mag-
netic field) between magnetic commensurate and/or in-
commensurate phases, with spin-gapped and gapless ex-
citations.
2. The model
The Hamiltonian of the quantum spin model with alter-
nating NN couplings and NNN exchange interactions has
the form:
� � � � � �� �J S S S S J S S Jn
x
n
n
x
n
y
n
y z
n
z
n
n
z
1 1 2 1 2 1 1 2 2( ) (, , , , , , S S S S J S Sn
x
n
x
n
n
y
n
y z
n
z
n
z
n
, , , , , ,)2 11 2 11 2 2 11� � �� �� � �
� � � �� � ��J S S S S S S SNN n
x
n
x
n
n
y
n
y
n
x
n
x
n
( , , , , , , ,1 11 1 11 2 1 2 2
y
n
y
NN
z
n
z
n
z
n
n
z
n
z
n
S J S S S S H� � �� � �� �1 2 1 11 2 1 2 1, , , , ,) ( ) (� S Sn
z
n
z
, , )1 2 2�� ,
(1)
where S
n
x y z
, ,
, ,
1 2
are the operators of the spin-1/2 projections
of the spin in the nth cell, which belongs to the sublattice
1 or 2, �1 2 0, � are effective magnetons of the sublattices,
H is the external magnetic field, directed along z axis, J 1 2,
are the alternating exchange coupling constants between
NN spins in the cell and between cells, J Jz
1 2 1 2, ,� is the
Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions
Fizika Nizkikh Temperatur, 2009, v. 35, No. 6 579
uniaxial magnetic anisotropy of the NN interaction, J NN
is the interaction constant between NNN spins in a chain,
and J JNN
z
NN� is the uniaxial magnetic anisotropy of the
NNN interaction. For the illustration, see Fig. 1.
It is known that the sign of J 1 and J 2 usually does not
play a role [1]. While for J Jz z
1 2 0, � the situation is inves-
tigated in detail, e.g., in theoretical and experimental
studies devoted to the quasi-one-dimensional spin-1/2
compound CuGeO 3, see Ref. 16, the opposite case of
positive exchange constants, or with different signs of
them is less known, however it is realized in some real
quasi-one-dimensional compounds [3,5].
After the Jordan–Wigner transformation, see, e.g., Ref. 1
S a an
z
n n n, , , ,
†
, , , , ,1 2 1 2 1 2 1 2
1
2
1
2
� � � �
S S iS an n
x
n
y
m m n
m n
, , , , , ,1 1 1 1 2 1
�
�
� �
� � ,
S S iS an n
x
n
y
n m
m n
m, , , ,
†
, , ,1 1 1 1 1 2
�
�
� �
� �
S an m
m n
m n n, , , , , ,2 1 2 1 2
�
�
�
� � �
S an n m
m n
m n, ,
†
, , , ,2 2 1 2 1
�
�
�
� � � (2)
where a
n, ,
†
1 2
and an, ,1 2 are creation and destruction opera-
tors, which satisfy fermionic anticommutation relations,
the Hamiltonian (1) obtains the form
� � � � � �� �( ) ( ) ( ) (
,
†
, ,
†
,J / a a J / a a
n
n
n
z
n
n
n1 1 2 1 1 12 4 1 2H.c. )( ) ( ) ( )
,
†
, ,
†
,1 2 2
2 2 2 2 11� � � �� �a a J / a a
n n n
n
n H.c.
� � � �� ��( ) ( )( ) ( )
,
†
, ,
†
,J / a a a a J /z
n
n
n n n NN2 11 11 2 24 1 2 1 2 2 [ ( ) (
,
†
,
†
, , ,
†
,
†
,a a a a a a a
n
n
n n n n n n1 2 2 11 2 11 11 2 1 2� � � �� � � 1 1 2) ],an� � �H.c.
� � � � �� � �( ) [( )( ) (
,
†
, ,
†
,J / a a a a aNN
z
n
n
n n n4 1 2 1 2 1 2
1 1 11 11 n n n na a a
,
†
, ,
†
,) ( )]
2 2 1 2 1 21 2� �� �
� � � ��( ) [ ( ) ( )]
,
†
, ,
†
,H/ a a a a
n
n n n n2 1 2 1 21 1 1 2 2 2� � . (3)
The Hamiltonian (3) consists of the sum of the quadratic
form of Fermi operators, the part, which contains the
products of four Fermi operators, which correspond to
the pair interactions between fermions, and the part,
which does not depend on operators.
3. Mean-field like approximation
In wha t fo l lows we use the mean- f i e ld l ike
(Hartry–Fock like) approximation. Let us introduce the
notations
m a az
n n1 2 1 2 1 2
1
2
, , ,
†
, , ,� � � �
1 1 2 2 1� � � � � �a a a a
n n n n.
†
, ,
†
, ,
2 11 2 2 11� � � � � �� �a a a a
n n n n,
†
, ,
†
, ,
NN n n n na a a a1 2 1 2 11 2 1 1 2 1 2, , ,
†
, , , ,
†
, ,� � � � � �� � , (4)
where brackets denote the thermodynamic average, ei-
ther with the ground state wave function for zero tempe-
rature, T � 0, or with the density matrix for nonzero
temperatures. Using the Wick’s theorem and the mean
field-like approximation we can re-write the Hamil-
tonian (3) as
580 Fizika Nizkikh Temperatur, 2009, v. 35, No. 6
A.A. Zvyagin and V.O. Cheranovski
n, 1 n + 1, 1 n + 2, 1
n – 1, 2 n, 2 n + 1, 2 n + 2, 2
(J , J )2 2
z (J , J )1 1
z
(J , J )NN NN
z
(J , J )NN NN
z
Fig. 1. Illustration of the zig-zag spin ladder. Solid zig-zag
lines denote nearest neighbor couplings. Bold line denotes the
bonds with the exchange constants J1 and J z
1 , the thick one de-
notes the bonds with coupling constants J2 and J z
2. Dashed
lines denote next-nearest neighbor interactions with coupling
constants JNN and JNN
z .
� � � � � ��[ (
,
†
,a a H J m J m J m J
n n
n
z z z z
NN
z z
NN NN1 1 1 1 2 2 1 1 22 2�
) �
� � � � � �a a H J m J m J m J a
n n
z z z z
NN
z z
NN NN,
†
, ( ) (
2 2 2 2 1 1 2 2 12 2�
n n n na a a
,
†
, ,
†
, )
1 2 2 1� �
� � � � � �� �[ ( ) ] ( ) [
,
†
, ,
†
,J / J J a a a az
NN n n n n1 1 1 2 11 2 2 112 2
� � � �( ) ]J / J Jz
NN2 2 2 12 2
� � � � �� �( ) ( ) (
,
†
, ,
†
, ,
a a a a J m J a
n n n n NN
z
NN
z
NN n1 11 11 1 2 1 2
†
, ,
†
, ) ( )]a a a J m J Cn n n NN
z
NN
z
NN� �� � � �1 2 1 2 2 1 2
, (5)
where the last term is the part, which does not contain operators. Naturally, the mean field-like approximation does not
take into account correctly interactions between excitations (for example, it does not take into account bound states), but
it qualitatively describes the possibility of quantum phase transitions in the studied model. After the Fourier transform
a N an k
k
ikn
, ,
/
, ,1 2
1 2
1 2� � � e (6)
and similar for a
n, ,
†
1 2
, where N is the number of unit cells, and a unitary transformation this Hamiltonian can be
diagonalized
� � �
�
�� � k j
jk
k j k jb b C, ,
†
,
1
2
,
(7)
where
� k A B B B B k A, , [ cos( ) ]1 2 1 1
2
2
2
1 2
21
2
2� � � � � , (8)
and
A H m m J J Jz z z z
NN
z
NN NN1 1 2 1 2 1 2 1 22 2� � � � � � � �[( ) ( ) ( ) ( )� �
J NN ]�
� � � �2 1 2 1 2cos( ) [( ) ( ) ],k m m J Jz z
NN NN NN NN
z
A H m m J Jz z
NN
z
NN NN NN� � � � � � �[( ) ( ) ( ) ]� �
1 2 1 2 1 22 2
� � � �2 1 2 1 2cos( )[( ) ( ) ],k m m J Jz z
NN NN NN NN
z
B J J J B J J Jz
NN
z
NN1 1 1 1 2 2 2 2 2 12 4 2 4� � � � � � � �
, . (9)
The value of the (possible) gap between two bands is determined by
A B B B B k2
1
2
2
2
1 22� � � cos( ) .
It is then the easy task to obtain thermodynamic characteristics of the model. For example, the free energy (for T � 0) of
the quantum alternating spin chain per unit cell is equal to
f
C
N
T
jk
T
k j
� � �
�
�
�
�
�
�
�
�
�
��
�
�� ln
,
1
2
1 e
�
. (10)
The z-projection of the average magnetization of the system per cell is
M
H T
z k j
jk
k j
�
�
�
�
�
�
�
�
�
�
�
�
��1
2 2
1
2 � �, ,
tanh . (11)
We can also calculate the magnetic susceptibility per cell
Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions
Fizika Nizkikh Temperatur, 2009, v. 35, No. 6 581
�
� � �
�
�
�
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
�
�
�
1
4
2
2
1
2
2
k j k j k j
H T T H
, , ,
tanh
2 2
1
2
2
cos
,
h
� k j
jk
T
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�� , (12)
the staggered magnetic susceptibility
�
� � �
� �
� �
�st �
� �
� �
�
�
�
( )
( )
tanh
(
, , ,1 2
2 2
1 2
2 2
12 2
k j k j k j
H T �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�� � �2
2
2
1
2
1
2 2) cos ( / ),
H T Tk jjk h
(13)
and the specific heat
c
T / T
k j
k jjk
�
�
��
�
�
,
,cos ( )
2
2 2
1
2
4 2h
(14)
per unit cell. One can see from the last equation that the temperature dependence of the specific heat can manifest two max-
ima, related to two kinds of excitations in the system.
The self-consistency conditions for the mean field parameters (1) at nonzero temperature can be written as
m
A
A B B B B k
z k
1 2
2
1
2
2
2
1 2
11
2
1
2 2
,
,
cos( )
tanh� �
� � �
�
�
�
�
�
�
�
T
A
A B B B B k
k�
�
��
�
�
�� �
� � �
�
�
�
�
�
�
1
22
1
2
2
2
1 2
�
cos( )
tanh
,� 2
2T
k
�
�
��
�
�
��
�
�
�
�
�
�
�
�
�
�
� ,
1 2
1 2 2 1
2
1
2
2
2
1 2
1
4 2
,
, ,cos( )
cos( )
tanh�
�
� � �
�
B k B
A B B B B kk
� �k k
T T
, ,
tanh ,
1 2
2 2
�
�
��
�
�
�� �
�
�
��
�
�
��
�
�
�
�
�
�
NN
k
k
k
T
A
A B B
1 2
1
2
1
2
2
2
1
2 2
1,
,
cos( ) tanh� �
�
�
��
�
�
��
� � �
� �
2 1 2B B kcos( )
�
�
�
�
�
�
�
�
�
�
�
�
� �
� � �
�
�
�
�
�
�
�
�
��
�
�
1
2 22
1
2
2
2
1 2
2A
A B B B B k T
k
cos( )
tanh
,�
��
�
�
�
�
�
. (15)
From these equations one can see, in particular, that
1 is different from
2, if J J1 2� , or if J NN � 0.
3.1. High temperature limit
For high temperatures | |, ,� k T1 2 �� Eqs. (15) can be
transformed to the form
8 4 21 2 1 1 1T J J JNN
z
� � � ,
8 4 22 1 2 2 2T J J JNN
z
� � � ,
2 1 2 1 2 1 2T m m H m mz z z z( ) ( ) ( )� � � � � �� �
� � � � �( ) ( ) ,J J J Jz z
NN
z
NN NN NN1 2 1 22 2
2 21 2 1 2 1 2T m m H m m Jz z z z
NN
z( ) ( ) ( )� � � � � �� �
� �2 1 2( ) ,
NN NN NNJ
2 21 2 1T m J JNN
z
NN NN NN
� � � ,
2 22 1 2T m J JNN
z
NN NN NN
z
� � � . (16)
The solution of Eqs. (16) in the main in 1/ T approxi-
mation is m H Tz
1 2 1 2, , /� � ,
1 2 1 2 8, , /� �J T , and
NN1 2, �
� �2 1
22, HJ / TNN . Using this solution it is easy to show
that m z
1 2, go to zero with H for nonzero temperatures, in ac-
cordance with the Mermin–Wagner theorem. Also, non-
zero values of
1 2, are determined in the first order in T �1
by the NN exchange interaction J 1 2, , while the necessary
condition for
NN1 2 0, � is the nonzero value of J NN , as ex-
pected. It is interesting to notice that
NN1 2 0, � is con-
nected with the nonzero value of the external magnetic
field for high temperatures. It turns out that
NN1 2, become
nonzero in higher order in 1/ T , comparing to m z
1 2, and
1 2, .
It is also important to notice that in the main approximation
at high temperatures
NN NN1 2� , if � �1 2� .
3.2. The ground state
The most important properties of the one-dimensional
spin system can be, of course, seen in the ground state. The
ground state of any fermion system is organized as the total
filling of the Fermi–Dirac sea(s), i.e. all fermionic states
582 Fizika Nizkikh Temperatur, 2009, v. 35, No. 6
A.A. Zvyagin and V.O. Cheranovski
with negative energies are occupied, while all fermionic
states with positive energies are empty.
The ground state filling of two fermionic bands Eqs.
(8) depends on the relative values of the parameters of the
considered model. Let us consider how different terms in
Eqs. (8) act. First, one can see from Eqs. (8) and (9) that
both bands are symmetric with respect to k. Also, it turns
out that the term in A1, which does not depend on k, shifts
the positions of both of two bands with respect to zero of
the energy axis, i.e. regulates the ground state filling of
bands. The term in A, which does not depend on k, shifts
two bands with respect to each other, in particular, its ab-
solute value governs the magnitude of the gap between
two bands. Hence, in particular, the external magnetic
field, on the one hand, governs the filling of bands, and,
on the other hand, the staggered component of the action
of the field, caused by the difference � �1 2� , changes the
value of the gap between bands. Also, the gap is de-
termined by the absolute value of the difference between
B1 and B2, | |
NN NN1 2� , and | |m mz z
1 2� . In particular, for
A � 0, B B1 2� , the gap between two bands becomes zero.
One can see that the gap is zero if J J1 2� , J Jz z
1 2� , even
for � �1 2� for J JNN
z
NN� � 0. However, nonzero NNN
interactions can also produce a gap even if J J1 2� ,
J Jz z
1 2� . On the other hand, the terms in A1 and A, which
are proportional to cos( )k , i.e. those, which are caused
by nonzero NNN couplings, are responsible for the asym-
metry of the bands with respect to each other. Without
those terms the bands are symmetric, see, the illu-
s t r a t i o n i n F i g . 2 . F o r m mz z
1 2� a n d
NN NN1 2� ,
if ( )m m Jz z
NN1 2� � ( )
NN NN NN
zJ1 2� , the lower band
has only one minimum at k � 0, while the upper band can
have one minimum at k � 0 and two maxima, situated
symmetrically with respect to k � 0, cf. Fig. 3 (see the dis-
cussion below). On the other hand, for m mz z
1 2� and
NN NN1 2� , if ( ) ( )m m J Jz z
NN NN NN NN
z
1 2 1 2� ! �
, the
upper band has only one maximum at k � 0, while the
lower band can have one maximum at k � 0 and two min-
ima, situated symmetrically with respect to k � 0 point, cf.
Fig. 4. Nonzero values of m mz z
1 2� and
NN NN1 2� , on
the other hand, can produce (being also the gap-yielding
reason) several extremal points of both bands (together
with k � �0, ").
In the thermodynamic limit N # $ the self-consis-
tency conditions can be written for the simple case of two
Dirac seas, each connected with the upper or lower band,
respectively, of Eq. (8), in the ground state as
m
k k
dz c c
k
k
k
k
c
c
c
c
1 2
1 21
2 2
1
4
2
2
1
1
, � �
�
�
�
�
�
�
�
�
�
��
%%" "
� k �
�
� � �
A
A B B B kB
2
1
2
2
2
12
2
cos( )
,
Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions
Fizika Nizkikh Temperatur, 2009, v. 35, No. 6 583
–1
– 0.5
0.5
1
–3 –2 –1 1 2 3
k
�k,1,2
Fig. 2. Symmetric dispersion laws �k , ,1 2 of the studied model
(e.g., at J JNN
z
NN� � 0 with the nonzero spin gap, caused by
J J1 2� ).
0.5
1
1.5
2
0 1 2 3–3 –2 –1
k
�k,1,2
Fig. 3. Asymmetric dispersion laws �k , ,1 2 of the studied model
with one minimum of the lower band and two maxima of the
upper band. The asymmetry is caused by NNN interactions.
0.5
1
1.5
0 1 2 3–3 –2 –1
k
�k,1,2
Fig. 4. Asymmetric dispersion laws �k , ,1 2 of the studied model
with one maximum of the upper band and two minima of the
lower band.
"1 2
1 2 21
8
1
1
2
2
,
, cos ( )
� �
�
�
�
�
�
�
�
�
� �
% %
k
k
k
k
c
c
c
c
dk
B k B ,
cos( )
,
1
2
1
2
2
2
1 22A B B B B k� � �
" "NN c c
k
k
k
k
k k
c
c
c
c
1 2 1 2
1
2
1
4
1
1
2
2
, [sin( ) sin( )]� � � �
� �
% %
�
�
�
�
�
�
�
�dk kcos( )
�
� � �
A
A B B B B k2
1
2
2
2
1 22 cos( )
, (17)
where 0 1& &k c " is the solut ion of the equat ion
� k ck, ( )1 1 0� , and 0 2& &k c " is determined from the con-
dition � k ck, ( )2 2 0� . This situation happens if both � k , ,1 2
have extremal values as a function of k only for k � �0, ".
It turns out that
1 2, are nonzero only when the difference
between the populations of the Fermi–Dirac seas, con-
nected with upper and lower bands, exists.
3.3. Analysis of the spectral features
Equations (8) imply several different possibilities of
the ground state behavior of the studied model. We know
that the mean-field approximation often does not provide
correct values of critical constants (critical values of ex-
change integrals or magnetic field, at which quantum
phase transitions take place), this is why, we limit our-
selves in what follows mostly with the qualitative mean
field analysis of the situation in the considered model in
the ground state. In what follows we shall consider the
properties of the bands Eqs. (8), (9). In such analysis, as
usually for one-dimensional systems, the edges of the
bands produce van Hove singularities, which manifest
themselves in quantum phase transitions. They are gov-
erned, e.g., by the magnetic field, which regulates the
filling of the bands.
The case without dimerization, i.e. with � �1 2� and
J J1 2� , J Jz z
1 2� , at H � 0 was considered, e.g., in
Refs. 17,18. This case in the magnetic field was studied in
Refs. 19. The isotropic dimerized case at H � 0 for
J JNN NN
z� � 0 was studied in the antiferromagnetic case
in Refs. 20,21. On the other hand, in the absence of the
magnetic field, H � 0, the magnetically isotropic case per-
mits the exact solution for the ground state [22,23] for
special choices of exchange constants J 1, J 2 and J NN .
If � � �1 2� � and J JNN NN
z� � 0 (i.e. in the absence
of the NNN interactions), one has A � 0, and, therefore,
m m k kz z
c c1 2 1 21 2 2, ( / ) ( ) /� � � � ", a n d
NN NN1 2, � �
� �( / )[sin ( ) sin ( )]1 2 1 2" k kc c . In this case the upper band
has one maximum at k � 0 and two minima at k � �",
while the lower band has one minimum at k � 0 and two
maxima at k � �". The gap between bands is caused by
the difference between J 1 and J 2. Both bands are totally
filled in the ground state (i.e. k c1 2, � ", hence
1 2, �,
� � � �
NN
zm /0 1 2, ) for
H H J J J J /s
z z� � � � � �1 1 2 1 21 2( / ) [ | | ( ) ].�
In this spin-saturated phase all excitations are gapped. The
low-temperature magnetic susceptibility and specific heat
are exponentially small for a gapped phase. Both bands are
empty in the ground state (i.e. k c1 2 0, � , thus
1 2, �
� �
NN 0, m z �1 2/ ) f o r H H s! �2 ( / )1 � [| |J J1 2� �
� �( ) / ]J Jz z
1 2 2 . In this spin-saturated phase all excitations
are also gapped.
If the lower band is totally filled and the upper band is
empty in the ground state (i.e. k c1 0� , k c2 � ", hence
m z � 0,
NN � 0) excitations are also gapped (except of
the case, when H H B Bc� � �( / )| |1 1 2� ). In all other
cases either upper band, or lower one, is partly filled in
the ground state, with gapless excitations (holes in the
Fermi–Dirac sea, related to the corresponding partly
filled band). Gapless excitations imply finite low-tem-
perature magnetic susceptibility, for m z � 0, or divergent
low-T magnetic susceptibility for m z � 0, and the
low-temperature specific heat proportional to T . If
( )J Jz z
1 2 0� ! and m z � 0 (or ( )J Jz z
1 2 0� � and m z & 0),
then the energies of states in the upper band are always
positive (i.e. k c1 0� ), and, hence, the upper band is
empty in the ground state. The lower band, depending on
H, can be totally filled (i.e. k c2 � ", therefore, m z � 0,
NN � 0) , empty ( i .e . k c2 0� , therefore , m z �1 2/ ,
NN � 0,
1 2 0, � ), and partly filled in the ground state.
The lower band is totally filled for H H c� at T � 0. In
this phase all excitations are gapped, with exponentially
small low-temperature magnetic susceptibility and spe-
cific heat.
The lower band is empty at T � 0 for H H s! �
� � �( / )[| | ( ) ]1 21 2 1 2� J J J J /z z
� , where the upper sign cor-
responds to the case with m z ! 0, and the lower one is re-
lated to the situation with m z � 0. Obviously, this phase ex-
ists if | | ( ) /J J J Jz z
1 2 1 2 2� ! � � . In this spin-saturated
phase all excitations are gapped, too. For H H Hc s� �
low-energy excitations (holes in the Dirac sea, related to
the lower band) are gapless, and the low-temperature spe-
cific heat is proportional to T , while the low-T magnetic
susceptibility is finite for m z � 0 and divergent for m z � 0.
At H c and H s second order quantum phase transitions take
place. If ( )J Jz z
1 2 0� & and m z � 0 (or ( )J Jz z
1 2 0� � and
m z & 0), then energies of states in the upper band can be
negative (i.e. k c1 0� ). In this case up to four second order
quantum phase transitions can take place. For H H s� 1 the
system is in the spin-polarized phase with m z � �1 2/ . For
H H Hs c1 1& & � ( / )1 � [ ( ) | |]� � � �m J J B Bz z z
1 2 1 2 (for
� � ! �m J J B Bz z z( ) | |1 2 1 2 ) a n d f o r H c2 1� �( / )�
� � � � � & &[ ( ) | | ]m J J B B H Hz z z
s1 2 1 2 2 the system is in the
gapless phases. For H H Hc c1 2& & the system is in the
gapped phase with empty upper band and totally filled
lower band, where m z � 0. Finally, for H H s! 2 the model
is in the spin-polarized phase with m z �1 2/ .
584 Fizika Nizkikh Temperatur, 2009, v. 35, No. 6
A.A. Zvyagin and V.O. Cheranovski
The general situation with J NN � 0, J NN
z � 0, is more
complicated, and, therefore, more interesting.
(i) This case takes place when both upper and lower
bands Eq. (8) empty in the ground state, i.e. k kc c1 2 0� � .
This case corresponds to the spin-saturated (ferromag-
netic) phase, with m mz z
1 2 1 2� � / ,
1 2 1 2 0, ,� �NN . It
happens even in the absence of the external magnetic field
for the case J J J J J Jz z
NN
z
NN1 2 1 22 2� � � ! �| |, for which
values of the exchange constants the spontaneous ferro-
magnetic order takes place in the ground state of the
model. For that situation, obviously, the line H � 0 is the
line of the first order quantum phase transition (the mag-
netic field removes the two-fold degeneracy of the ferro-
magnetic ground state). If the above mentioned condition
for exchange constants is not fulfilled, i.e. there is no
spontaneous ferromagnetism in the model, the case with
totally empty Fermi–Dirac seas can exist for large enough
values of the external magnetic field. Here, for H � 0, the
spin-saturated ground state situation is related to H H s! ,
the critical field of the second order quantum phase tran-
sition to the spin-saturated phase, which is determined by
one of the values of
H J J J Js
z z
NN
z
NN1 2
1 2
1 2 1 2
1
4
2 2, | ( ) ( )� � � � � � �
� �
� �
� � � � � �[( ) ( )� �1 2
2
1 2
22 2J J J Jz z
NN
z
NN
� �4 1 2 1 2
2 1 2� � ( ) ] | ./J J (18)
For H H s! excitations, related to both bands, are gap-
ped, which reveals itself, e.g., in exponentially small
values of the low-temperature magnetic susceptibility
and specific heat.
(ii) The low-energy band can be totally filled, and the
upper in energy band can be empty at T � 0, i.e. k c1 0� ,
k c2 � ", like in Figs. 2 and 4. Then it is seen from
Eqs. (17) that one has
NN NN1 2� � . One can also see
that the z-projection of the total spin of the model is zero
in this phase, m mz z
1 2 0� � (but not the z-projection of the
total magnetization, which can be nonzero in the general
case � �1 2� ). This situation is characterized by the gap
for the excitations, related to the upper band, and also can
be characterized by the gap for the lower-band excitations
(holes in the Dirac sea), which exists for � k ,2 0� for all k.
This lower-band gap can be closed for some values of pa-
rameters of the Hamiltonian. For example, it can be re-
lated to the critical value of the field, H H c� , at which the
second order quantum phase transition to the phase with
lower-band gapless excitations takes place. For H H c!
the excitations, related to the lower band, become
gapless, and, therefore, the low-temperature magnetic
susceptibility is finite, and the low-temperature specific
heat becomes linear in T . On the other hand, the lower
band can be totally filled at T � 0, but excitations, related
to this band (holes in the Dirac sea) can be gapless (in this
case H c � 0). Here the low-temperature magnetic suscep-
tibility of the model is finite, and the low-temperature
specific heat is proportional to T . Further growth of H in
the ground state can produce the second order quantum
phase transition to the spin-saturated phase for H H s� ,
which corresponds to the situation, at which the lower
band becomes totally empty, and excitations, related to it,
become gapped. In this case for � � �1 2� � , we have
A H1 2� � . In the case of small NNN interactions, we get
A � 0, m z
NN1 2 1 2 0, , ,� �
. The parameters
1 2, are obtained
as the solution of equations
'
"
'1 2
1
2
1� � � �( ) ( ) ,E x
'
"
'1 2
1
2
1� � � �( ) ( )K x , (19)
where ' � B /B2 1,
x �
�
4
1 2
'
'( )
, (20)
and K x( ) and E x( ) are the complete elliptic integrals of
the first and second kinds, respectively. 2�H c is deter-
mined by the value of the gap between bands. The gap, in
turn, is determined by the value
| | |( ) ( ) ( )| .B B J J J J Jz z
NN1 2 1 2 1 1 2 2 1 22 4� � � � � � �
(21)
The solution for the case with gapped excitations corre-
sponds to
1 2� , existing even for J J1 2� , J Jz z
1 2� . For
small values of J NN one has
'
�
�
�
�
�
�
�
�J J
J J
J
J J J J
z
z NN z z
2 1 2
1 1 1
1
2 2 2
2
1 1 1
2
2
1 4
2 2�
�
�
�
�
�
�
�
�
�
�
�
�
.
(22)
(iii) The lower band can be partly filled, while the upper
band is empty in the ground state, i.e. k c1 0� , like in Fig. 3.
This case is characterized by the nonzero value of the
z-projection of the total spin of the model per cell (differ-
ent, generally speaking, from the nominal value, 1/2),
m m kz z
c1 2 21� � � / ", which can be spontaneous, i.e. it can
exist even for H � 0. Low-energy excitations (holes in the
lower band) are gapless, and, therefore, the low-T specific
heat is proportional to the temperature. For nonzero spon-
taneous spin moment the low-temperature magnetic sus-
ceptibility is divergent, while if the moment is caused by
the nonzero value of the field H, the low-T susceptibility is
finite. At H � 0 in the ground state for the case with non-
zero spontaneous magnetization the first order quantum
phase transition takes place, because the magnetic field re-
moves the degeneracy of the ground state. The growth of
the magnetic field yields a second order quantum phase
Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions
Fizika Nizkikh Temperatur, 2009, v. 35, No. 6 585
transition at H H s� to the spin-polarized state with the
nominal value of M z with only gapped excitations. In the
case of small k c2 one has the solution of self-consistency
equations A � 0, m kz
c1 2 21 2 2, ( / ) ( / )� � " (equal to the
ground state spontaneous magnetization),
NN , ,1 2 �
� k c2 2/ ", and
"1 2 2 4, ( )� � k /c . Here the critical value of
k c2 is determined at H � 0 by
k J J J J J J
J J J J
c
z z
NN
z
NN
z z
NN
z
N
2 1 2 1 2
1 2
2
2 2
3 8 4"
�
� � � � �
� � �( ) N
, (23)
or
k J J J J J J
J J J J
c
z z
NN
z
NN
z z
NN
z
NN
2 1 2 1 2
1 2
2
2 2
8 12"
�
� � � � �
� � �
, (24)
for ( )J J1 2� , being negative or positive, respectively.
Such solutions must exist only for small values of k c2.
(iv) At T � 0 the lower band can be totally filled, i.e.
k c1 � ", while the upper band is partly filled. In this case
one has also, as in the previous case, possibility of the
nonzero value of the z-projection of the total spin of the
model, spontaneous (generally speaking different from
1/2), or caused by the external magnetic field. Depending
on that fact (i.e., whether the moment is spontaneous, or
not), the low-temperature magnetic susceptibility is di-
vergent, or finite. Again, lowest in energy excitations
(holes in the Fermi–Dirac sea, connected with the upper
band) are gapless, and the low-temperature specific heat
is proportional to T . Also, at H H s! the model appears in
the spin-polarized ground state, as in the previous case. In
the situation with nonzero spontaneous magnetization in
the ground state, the first order phase transition takes
place, because the field removes the degeneracy of the
ground state. However, this situation is different from the
previous one, because, if the bottom of the upper band is
lower than the top of the lower band, at H H Hc c s1 2, �
two quantum second order phase transitions take place,
with the square root feature in the behavior of the mag-
netic susceptibility. It turns out that these phase transi-
tions are between phases with gapless excitations. On the
other hand, if the bottom of the upper band is higher than
the top of the lower band, one has two second order quan-
tum phase transitions, and, hence, two critical values of
the magnetic field H c1 and H c2 with the values smaller
than H s . Between those field values there are no magnetic
eigenstates of the Hamiltonian, and, therefore, the field
dependence of the magnetization has to manifest a
magnetic plateau for H H Hc c1 2� � .
(v) Then, both, lower and upper bands can be partly
filled in the ground state. The properties of this situation
are similar to the ones of the previous case with two criti-
cal values of the field, H c and H s . The transition at H c is
between phases with gapless excitations. At H � 0 one has
to observe a first order phase transition, because the field
removes the degeneracy of the ground state. Cases (iv)
and (v) are related to large values of the NNN interac-
tions.
(vi) Finally, the case with both upper and lower band
filled (i.e. k kc c1 2� � ") at T � 0 formally corresponds to
the solution of self-consistency equations with m z
1 �
� � �m z
2 1 2/ ,
1 2 1 2 0, ,� �NN .
The situation becomes more complicated for the case,
where � k , ,1 2 as functions of k have additional, with respect
to k � �0, ", extremum values. Such situation corresponds
to more than one Fermi–Dirac sea per band, which are sit-
uated symmetrically with respect to k � 0. Obviously, it
exists only for nonzero J NN and J NN
z . It implies that in
Eqs. (17) one has to replace integration from �k c1 2, to
k c1 2, to the integration over the intervals, where � k , ,1 2 0� .
Here the system can be in incommensurate structures in
the ground state (it is in the incommensurate phase if the
lowest value of the energy corresponds to some k � �0, ").
For the cases (i) and (vi) there are no incommensurate
structures, obviously. For (ii) additional extremum values
of only lower branch matter, see Fig. 4. In the gapped case
(or in the gapless one, if H c � 0), additional field-gov-
erned second order phase transition between commensu-
rate and possible incommensurate gapless phases takes
place for T � 0, cf. Fig. 4. For (iii), as in the previous case,
additional extremum values of only lower branch matter.
Again, additional field-governed second order quantum
phase transition between commensurate and possible in-
commensurate gapless phases takes place, see Fig. 5. In
the cases (iv) and (v) additional extremum values of both,
upper and lower bands matter, see Fig. 6. In the case (iv),
with a magnetic plateau, in addition to H c1 2, two more
quantum phase transitions can take place. Both of those
second order quantum phase transitions are between com-
mensurate and possible incommensurate phases with
gapless excitations. For (iv) with the bottom of the upper
586 Fizika Nizkikh Temperatur, 2009, v. 35, No. 6
A.A. Zvyagin and V.O. Cheranovski
0
0.5
1
1.5
–3 –2 –1 1 2 3
k
�k,1,2
Fig. 5. Dispersion laws �k , ,1 2 of the studied model with empty
upper band and partly filled lower band.
band being higher than the top of the lower band, as well
as for (v), again, two additional ground state phase transi-
tions can take place, also between commensurate and pos-
sible incommensurate phases with gapless excitations.
Notice that low-energy excitations correspond to the
states with positive energies, i.e. the dispersion laws for
those excitations are related to � k , ,1 2 for branches Eqs.
(8) with positive energies and to | |, ,� k 1 2 for holes for the
branches with negative energies.
It is interesting to observe that proposed recently ex-
actly solvable model of a spin-1/2 chain with alternating
NN interactions and multiple spin exchange [24] mani-
fest properties, similar to the ones of the present model.
4. Exact diagonalization: thermodynamics
Now let us check the features of the behavior of the
spin-1/2 chain with alternating NN and (homogeneous)
NNN interactions, obtained with our mean field analysis
with the results of exact diagonalization of small clusters
(short chains). Most of our calculations were performed for
the chains containing 14 spins (even numbers). The number
of spins in the chain was limited by the exponential growth
of the computation time, however the accuracy of our calcu-
lations was sufficient to reproduce the features of the
magnetization, magnetic susceptibility and specific heat of
the considered model down to temperatures of order of 0.01
of the smallest absolute value of the NN exchange coupling.
In our calculations we used arbitrary units with Boltzmann’s
constant, Bohr’s magneton and g-factor being equal to unity
(we studied the case of � � �1 2� � ). Figs. 7–15 present
results of those calculations for S M Hz z ( ) / � at
T J� 0 1 1. | |, � z T( ) and c T( ) for H � 0 for the model with
magnet ica l ly iso tropic NN ( ), ,J J z
1 2 1 2� and NNN
(J JNN NN
z� ) interactions for J 1 1� � , J 2 1 2� � . , J NN �
� � � �0 1 0 3 0 8. , . , . (except of the trivial case with all ex-
change constants being ferromagnetic).
One can see that for the cases with antiferromagnetic
NN interactions for any sign of NNN interactions, and for
alternating in signs NN interactions with antiferro-
magnetic NNN interactions low-lying excitations are
gapped, and, therefore, the magnetization is small for
small values of the field (cf. Figs. 7, 10, and 13), and the
magnetic susceptibility (cf. Figs. 8, 11, and 14) and spe-
cific heat (cf. Figs. 9, 12, and 15) are exponentially small
at low temperatures. The same is true for larger values of
NNN exchange constants for the case with ferromagnetic
NN and antiferromagnetic NNN interactions, and for al-
ternating in signs NN interactions with ferromagnetic
NNN couplings. It implies that the mean field like ap-
proximation correctly (at least in these cases) reproduces
Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions
Fizika Nizkikh Temperatur, 2009, v. 35, No. 6 587
–2.5
–2
–1.5
–1
– 0.5
0
0.5
–3 –2 –1 1 2 3
k
�k,1,2
Fig. 6. Dispersion laws �k , ,1 2 of the studied model with partly
filled upper band with two minima and totally filled lower
band with two minima.
0.5
0.4
0.3
0,2
0.1
0
S
z
H/|J1|
1 2 3 4 5
1
2
34
5
Fig. 7. The magnetization per site of the considered model for
J JNN � �01 1. | | as a function of H at T J� 01 1. | |. Notations here
and in Figs. 7–14 are as follows. J1 1� , J2 12� . , J JNN � 01 1. | |
(1); J1 1� , J2 12� . , J JNN � �01 1. | | (2); J1 1� , J2 12� � . ,
J JNN � � 01 1. | | (3); J1 1� , J2 12� � . , J JNN � 01 1. | | (4); J1 1� � ,
J2 12� . , J JNN � 01 1. | | (5).
1.0
0.8
0.6
0.4
0.2
0
1 2 3 4 5
1
2
3
4
5
T/|J |1
Fig. 8. The magnetic susceptibility per site of the considered
model for J JNN � �01 1. | | as a function of T at H � 0.
such a behavior, cf. (ii), like in Fig. 2, or (ii) with possible
additional Fermi–Dirac seas of the lower band, see Fig. 4.
The large region of an exponentially small magnetization
implies a large value of the spin gap.
On the other hand, for the case with alternating in
signs NN interaction for J JNN � 0 1 1. | |, and for ferromag-
netic NN interactions for J J JNN � � �0 1 0 31 1. | | , . | | we see
the divergency of the low-temperature magnetic suscepti-
bility, see Figs. 8 and 11 (which manifests the spontane-
ous magnetization in the ground state), additional
low-temperature narrow maxima for the specific heat, see
Figs. 9 and 12, and a metamagnetic-like transition at
small values of the field of the low-temperature magneti-
zation, see Figs. 7, 10, and 13. This situation is, probably,
related to the cases (i) or (iii), cf. Fig. 3, with one
Fermi–Dirac sea, or with possible additional Fermi–Dirac
seas of the lower band in the mean field like analysis. No-
tice that in those cases the bandwidth of the lower band is
narrow, which produces the metamagnetic-like transition
to the spin-polarized phase. Our exact diagonalization re-
sults imply that at least for the calculated cases the situa-
tions like in (iv), (v) and (vi) are not realized in the
studied spin chain (zig-zag ladder).
In order to get more information about quantum phase
transitions with respect to external field, we calculated
the lowest energy levels of finite chain fragments formed
up to 16 spins for each value of total spin of clusters for
the isotropic case | |, ,J Jz
1 2 1 2� , | |J JNN
z
NN� . For this pur-
pose we used branching diagram technique and the
Davidson’s method (the modification of the Lanczos
scheme for the determination of lowest eigenvalues of
large matrices). We found the jumps of the ground state
total spin S due to change of coupling parameter J NN in
rather narrow region. This gives us the idea to find simple
approximate estimate for critical value of this parameter,
which corresponds to the beginning of the destruction of
ferromagnetic ground state of the chain, by means of
study the stability of ferromagnetic state with respect to
the inversion of a single site spin. It is easily shown, that
the energy spectrum of the states with one inverted spin is
defined by formula
588 Fizika Nizkikh Temperatur, 2009, v. 35, No. 6
A.A. Zvyagin and V.O. Cheranovski
1 2 3 4 5
0.40
0.30
0.20
0.10
0
T/|J |1
1
2
3
4
5
c
Fig. 9. The specific heat per site of the considered model for
J JNN � �01 1. | | as a function of T at H � 0.
S
z
0.5
0.4
0.3
0.2
0.1
0
1 2 3 4 5
H/|J |1
1
2
3
4
5
Fig. 10. The same as in Fig. 7, but for J JNN � �0 3 1. | |.
1
23
4
5
1.0
0.8
0.6
0.4
0.2
1 2 3 4 5
0
T/|J |1
Fig. 11. The same as in Fig. 8, but for J JNN � �0 3 1. | |.
1 2 3 4 5
0
T/|J |1
0.50
0.40
0.30
0.20
0.10
1
2
3
4
5
c
Fig. 12. The same as in Fig. 9, but for J JNN � �0 3 1. | |.
( � � � � � � �E k E J J J kf NN( ) [ cos ( )](
1
2
2 11 2
� � �J J J J k1
2
2
2
1 22 cos( )) , (25)
where E f is the energy of the ferromagnetic (totally
spin-polarized) state, and E k( ) is the energy of the single
inverted spin with respect to the spin-polarized state. Let
J 1 2 0, � and J NN ! 0. In this case for the infinite chain it
can be shown that the ferromagnetic state is unstable
(non-positive values of (), if the coupling J NN takes
values from the interval
J J
J J
J
J J
J J
NN
1 2
1 2
1 2
1 22 2| | | |�
� �
�
. (26)
In other words, the expression
J
J J
J J
NN �
�
1 2
1 22| |
(27)
may be a good estimate the for critical value of J NN . Our
numerical calculations for linear chain fragments support
this conclusion. Our calculations, compare with the re-
sults of Ref. 22, for chain fragments with periodic bound-
ary conditions (Fig. 16) demonstrate that this estimate
corresponds to the exact degeneracy of the ferromagnetic
and the lowest singlet states regardless on the size of frag-
ment.
Simultaneously, all the lowest states from other
subspaces with the given total spin have larger energy in
the vicinity of this point. It means that Eq. (27) deter-
mines the critical surface in the space of three coupling
parameters, that corresponds to the first order transition
between the ferromagnetic and the singlet states.
On the other hand, we see that for antiferromagnetic
NN interactions additional field-induced ground state
phase transitions can take place. For example, Figs. 7, 10,
and 13 manifest features of the behavior at the values of
Elementary excitations and thermodynamics of zig-zag spin ladders with alternating nearest neighbor exchange interactions
Fizika Nizkikh Temperatur, 2009, v. 35, No. 6 589
S
z
1 2 3 4 50
H/|J |1
0.5
0.4
0.3
0.2
0.1
1
2
3
4
5
Fig. 13. The same as in Fig. 7, but for J JNN � �0 8 1. | |.
1 2 3 4 5
0
T/|J |1
0.6
0.5
0.4
0.3
0.2
0.1 1
2
3
4
5
Fig. 14. The same as in Fig. 8, but for J JNN � �0 8 1. | |.
1 2 3 4 5
0
T/|J |1
0.5
0.4
0.3
0.2
0.1
1
2
3
4
5
c
Fig. 15. The same as in Fig. 9, but for J JNN � �0 8 1. | |.
E
(S
)
–
E
m
in
f
0.012
0.010
0.008
0.006
0.004
0.002
0
1 2 3 4 5 6 7 8 9
S
1
2
3
4
5
6
7
8
9
Fig. 16. Lowest energies with the specified value of the total spin
S of spin-1/2 chain fragments consisting of N � �12 16 spins with
J1 1� � for the cases: J2 1� � , JNN � 0 25. , N �12 (1), N �14 (2),
N �16 (3); J2 12� � . , JNN � 3 11/ , N �12 (4), N �14 (5), N �16
(6); J2 15� � . , JNN � 0 3. , N �12 (7), N �14 (8), N �16 (9).
the magnetization per spin equal to approximately 0.08,
0.16, 0.25, and/or 0.34. We performed calculations for
J JNN � �2 5 1. | | (not shown here), and they also reveal ad-
ditional features, clearly demonstrating magnetic pla-
teaux at approximately 0.16 of the magnetization per
spin. Those features in the behavior of the magnetization
are followed, naturally, by the features in the magnetic
field behavior of the magnetic susceptibility and specific
heat at low temperatures at the same values of the mag-
netic field, at which features of the magnetization take
place. It is not clear, however, whether all those addi-
tional field-induced features are related to real quantum
phase transitions, or they are connected with the fi-
nite-size effects. At least, most of such a features are re-
produced in all performed calculations (up to 18 spins).
Obviously, additional plateaux in the magnetization be-
havior imply additional bands of low lying excitations of
the considered chain, and, therefore, our mean field like
analysis, based on two bands for magnetic excitations, is,
probably, limited to relatively small values of NNN inter-
actions.
We also performed calculations of the ground state
magnetization of chains, containing 14, 16, and 18 spins
using the Davidson’s method. The results of calculations
with periodic boundary conditions for the cases with
J NN � 0 3. with J 1 1� , J 2 1 2� . , and J 1 1� � , J 2 1 2� � . , are
given in Figs. 17 and 18, respectively. One can see the
qualitative agreement between these calculations and the
ones, using the exact diagonalization scheme.
590 Fizika Nizkikh Temperatur, 2009, v. 35, No. 6
A.A. Zvyagin and V.O. Cheranovski
Sz
0.5
0.4
0.3
0.2
0.1
0
0 0.5 1.0 1.5 2.0 2.5 3.0
H/|J |1
1
2
3
Fig. 17. The magnetization of the chain consisting of 14 (1),
16 (2), and 18 spins 1/2 (3) with antiferromagnetic alternating
NN interactions and antiferromagnetic NNN couplings as a
function of the magnetic field at T � 0 calculated using
Davidson’s method.
Sz
H/|J |1
1
23
0 0.02 0.04 0.06 0.08 0.10
0.5
0.4
0.3
0.2
0.1
0
Fig. 18. The same as in Fig. 17, but for the chain with ferromag-
netic alternating NN and antiferromagnetic NNN interactions.
Sz
H/|J |1
1
2
3
4
5
6
0.5
0.4
0.3
0.2
0.1
0
1 2 3 4 5
Fig. 19. The magnetization of the spin-1/2 chain with anti-
ferromagnetic alternating NN (J1 1� , J2 12� . ) interactions and
antiferromagnetic NNN couplings as a function of the mag-
netic field at low temperatures for the isotropic case, and for
the weak easy-axis and easy-plane magnetic anisotropy. The
notations are as follows. For JNN � 0 3. the isotropic case (1);
the easy-axis case (2); the easy-plane case (3). For JNN � 0 8.
the isotropic case (4); easy-axis case (5); the easy-plane case
(6).
S
z
0.5
0.4
0.3
0.2
0.1
0
1 2
H/|J |1
1
2
3
4
5
6
Fig. 20. The same as in Fig. 19 (with the same notations), but
for the spin chain with ferromagnetic alternating NN interac-
tions (J1 1� � , J2 12� � . ).
5. Magnetic anisotropy
Figures 19 and 20 show the behavior of the low-tem-
perature (T J� 0 1 1. | |) magnetization as a function of H
for the model with J 1 1� , J 2 1 2� . and J 1 1� � , J 2 1 2� � . ,
respectively, for J NN � 0 3 0 8. , . , for the case of a weak
uniaxial magnetic anisotropy, the axis of which is di-
rected along the magnetic field (the data are presented
for the isotropic case, and for the easy-axis and
easy-plane cases, ( ) / ., , , , , ,J J JNN
z
NN NN1 2 1 2 1 2 0 1� � � . One
can see that the weak magnetic anisotropy of any sign
(easy-axis or easy-plane) does not produce drastic dif-
ferences, comparing to the isotropic case.
6. Concluding remarks
In conclusion, motivated by recent experiments on
quasi-one-dimensional quantum spin systems, we have
studied the one-dimensional model, in which the alterna-
tion of the exchange interactions between neighboring
spins is accompanied by the next-nearest neighbor spin
exchange (zig-zag spin ladder with the alternation). The
model permitted to obtain thermodynamic characteristics
of the considered quantum spin chain in the mean-field
like approximation. The most important behavior of the
model is in the ground state. Depending on the strength of
the NNN interactions, the model manifests either the
spin-gapped behavior of low-lying excitations at low
magnetic fields, or the ground state ferrimagnetic order-
ing with gapless low-lying excitations, or the incommen-
surate magnetic phase. The system undergoes second or-
der or first order quantum phase transitions, governed by
the external magnetic field, NNN coupling strength (the
later can be caused by an external or internal pressure),
and the degree of the alternation. Hence, on the one hand,
NNN spin–spin interactions in a dimerized quantum spin
chain can produce a spontaneous magnetization. On the
other hand, for quantum spin chains with a spontaneous
magnetization, caused by NNN spin–spin couplings, the
alternation of NN exchange interactions can be the reason
for destroying of that magnetization and the onset of a
spin gap for low-lying excitations. The weak uniaxial
magnetic anisotropy does not affect the main features of
the thermodynamic characteristics of the spin chain
(zig-zag ladder). Alternating NN interactions produce a
spin gap between two branches of low-energy excitations,
and the NNN interactions yield asymmetry of dispersion
laws of those excitations, with possible minima, corre-
sponding to incommensurate structures in the chain. It is
interesting to notice that two branches of magnetic excita-
tions, with the lowest branch having minima at incom-
mensurate wave vectors, cf. Figs. 4 and 5, were recently
observed in inelastic neutron scattering experiments on a
quasi-one dimensional compound Li 2CuO 2 at tempera-
tures, higher than the N�el temperature [25]. In that com-
pound it is believed that intra-chain ferromagnetic NN
coupl ing between spins is accompanied by the
antiferromagnetic NNN coupling.
Our numerical exact diagonalization results have
shown that the mean field like approximation correctly
reproduces the features of the behavior of spin chains
with alternating nearest neighbor and next-nearest neigh-
bor interactions for the ferromagnetic NN couplings and
alternating in sign NN exchange constants, if the strength
of the NNN interaction is weak enough. On the other
hand, for large values of the NNN coupling, especially for
antiferromagnetic interactions in the chain, the mean field
apprioximation cannot reproduce the important features
of the behavior of such spin chains (zig-zag ladders) in
high magnetic fields, which produce additional bands of
excitations, and, as a consequence, additional quantum
phase transitions in the magnetic field.
A.A.Z. thanks S.-L. Drechsler and W.E.A. Lorenz for
interesting discussions. The financial support from the
DFG via Grant BU 887/7-1 (A.A.Z.), and the Ukrainian
Fundamental Research State Fund via Grant F25.4/13 is
acknowledged.
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23. H. Suzuki and K. Takano, J. Phys. Soc. Jpn. 77, 113701
(2008).
24. A.A. Zvyagin, and G.A. Skorobagatko, Phys. Rev. B73,
024427 (2006).
25. W.E.A. Lorenz, R.O. Kuzian, S.-L- Drechsler, J. Richter,
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592 Fizika Nizkikh Temperatur, 2009, v. 35, No. 6
A.A. Zvyagin and V.O. Cheranovski
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>>
>> setdistillerparams
<<
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>> setpagedevice
|