Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds
The sawtooth chain with pairs of S = 1/2 spins interacting with XXZ-interactions placed on each second tooth is considered. All other interaction bonds are taken to be of Ising type. Exact statistical mechanical solution of the model within the direct transfer-matrix technique is obtained. The sol...
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| Zitieren: | Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds / V. Ohanyan // Condensed Matter Physics. — 2009. — Т. 12, № 3. — С. 343-351. — Бібліогр.: 26 назв. — англ. |
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Ohanyan, V. 2017-06-10T17:13:36Z 2017-06-10T17:13:36Z 2009 Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds / V. Ohanyan // Condensed Matter Physics. — 2009. — Т. 12, № 3. — С. 343-351. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 75.10.Pq DOI:10.5488/CMP.12.3.343 https://nasplib.isofts.kiev.ua/handle/123456789/120000 The sawtooth chain with pairs of S = 1/2 spins interacting with XXZ-interactions placed on each second tooth is considered. All other interaction bonds are taken to be of Ising type. Exact statistical mechanical solution of the model within the direct transfer-matrix technique is obtained. The solution allows one to obtain exact analytic expressions for all thermodynamic functions of the model. Ground state properties are also investigated, the corresponding ground state phase diagram is presented. Розглядається пилкоподiбний ланцюжок з парами S = 1/2 спiнiв, що взаємодiють з XXZ взаємодiєю i розташованi на фрагментi кожного другого зубця пилки. Всi iншi взаємодiї взятi типу взаємодiї Iзинга. Отримано точний статистико-механiчний розв’язок моделi прямим методом матрицi переносу. Розв’язок дозволяє отримати точнi аналiтичнi вирази для всiх термодинамiчних функцiй моделi. Дослiджено також властивостi в основному станi i представлена вiдповiдна фазова дiаграма в основному станi. This work was partly supported by the grants CRDF{UCEP { 06/07 and ANSEF{1586{PS. The author would like to thank Oleg Derzhko for useful discussions and Lev Ananikian and Diana Antonosyan for help in preparing the gures. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds Антиферомагнiтний пилкоподiбний ланцюжок зi зв’язками Гайзенберга i Iзинга Article published earlier |
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Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds |
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Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds Ohanyan, V. |
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Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds |
| title_full |
Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds |
| title_fullStr |
Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds |
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Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds |
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antiferromagnetic sawtooth chain with heisenberg and ising bonds |
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Ohanyan, V. |
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Ohanyan, V. |
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2009 |
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Інститут фізики конденсованих систем НАН України |
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Антиферомагнiтний пилкоподiбний ланцюжок зi зв’язками Гайзенберга i Iзинга |
| description |
The sawtooth chain with pairs of S = 1/2 spins interacting with XXZ-interactions placed on each second
tooth is considered. All other interaction bonds are taken to be of Ising type. Exact statistical mechanical
solution of the model within the direct transfer-matrix technique is obtained. The solution allows one to obtain
exact analytic expressions for all thermodynamic functions of the model. Ground state properties are also
investigated, the corresponding ground state phase diagram is presented.
Розглядається пилкоподiбний ланцюжок з парами S = 1/2 спiнiв, що взаємодiють з XXZ взаємодiєю i розташованi на фрагментi кожного другого зубця пилки. Всi iншi взаємодiї взятi типу взаємодiї Iзинга. Отримано точний статистико-механiчний розв’язок моделi прямим методом матрицi переносу. Розв’язок дозволяє отримати точнi аналiтичнi вирази для всiх термодинамiчних функцiй моделi. Дослiджено також властивостi в основному станi i представлена вiдповiдна фазова дiаграма в основному станi.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120000 |
| citation_txt |
Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds / V. Ohanyan // Condensed Matter Physics. — 2009. — Т. 12, № 3. — С. 343-351. — Бібліогр.: 26 назв. — англ. |
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AT ohanyanv antiferromagneticsawtoothchainwithheisenbergandisingbonds AT ohanyanv antiferomagnitniipilkopodibniilancûžokzizvâzkamigaizenbergaiizinga |
| first_indexed |
2025-11-27T03:09:03Z |
| last_indexed |
2025-11-27T03:09:03Z |
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1850793775123660800 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 3, pp. 343–351
Antiferromagnetic sawtooth chain with Heisenberg and
Ising bonds
V. Ohanyan
1 Department of Theoretical Physics, Yerevan State University, Al. Manoogian 1, 0025 Yerevan, Armenia
2 Yerevan Physics Institute, Alikhanian br. 2, 0036, Yerevan, Armenia
Received May 28, 2009, in final form June 18, 2009
The sawtooth chain with pairs of S = 1/2 spins interacting with XXZ-interactions placed on each second
tooth is considered. All other interaction bonds are taken to be of Ising type. Exact statistical mechanical
solution of the model within the direct transfer-matrix technique is obtained. The solution allows one to obtain
exact analytic expressions for all thermodynamic functions of the model. Ground state properties are also
investigated, the corresponding ground state phase diagram is presented.
Key words: sawtooth chain, exact solutions, magnetization plateaus
PACS: 75.10.Pq
1. Introduction
The sawtooth chain or delta chain is a one-dimensional lattice spin system with a topology
of corner-sharing triangles (figure 1). This system is famous for a number of important features.
Physically, magnetic lattices corresponding to the sawtooth chain are found in a number of com-
pounds, in delafossite YCuO2.5 [1,2] and olivines with structures ZnL2S4 (L=Er,Tm,Yb) [3] to
cite a couple of examples. Antiferromagnetic Heisenberg model on sawtooth chain is strongly frus-
trated, however, in contrast to many other frustrated systems, the ground state of sawtooth chain
is known exactly. Namely, the ground state in case of cyclic boundary conditions is a two–fold
degenerated dimerised structure where either each left pair of spin at each triangle or each right
pair forms spin-singlet states (dimers) [4]. The excitations of the sawtooth chain are gapped and
are of topological origin. They are found to be “kink”–“antikink”-type domain wall structures [5,6].
Anisotropic variants of sawtooth chain have also been investigated revealing additional features
[7,8]. Another important feature of the sawtooth chain as well as of some other frustrated spin
and Hubbard lattices revealed quite recently is the appearance of localized magnon states (dis-
persionless excitation bands) [9–12]. The localized low energy excitations in sawtooth chain which
exist due to frustrated geometry, more precisely, due to the triangular plaquette, affect the low–
temperature thermodynamics. The corresponding eigenstates have been constructed for sawtooth
chain in case of Heisenberg model in [12] and for Hubbard electrons in [10]. Calculating the ground
state degeneracies for the localized magnon (or electron) states one can obtain explicit expressions
for thermodynamic quantities for low temperatures and near saturation field [10,12]. However, the
problem of exact description of whole thermodynamics for sawtooth chain as well as for many
other interacting quantum spin systems is still an open issue.
Figure 1. The sawtooth chain.
c© V. Ohanyan 343
V. Ohanyan
One can mention a formal approach to these problem which is not justified properly yet but in
some particular cases demonstrates rather good agreement with experimental data and numerical
calculations. The approximation consists in replacement of some or all interaction bonds with Ising
ones [13–24]. As a result one can obtain an interacting spin system which allows one to calculate
all thermodynamic functions analytically. In this paper we use this approach for exact solution
of the system with Ising and Heisenberg interaction bonds on sawtooth chain. To make possible
exact calculation of the partition function within transfer-matrix method we consider all interaction
bonds of sawtooth chain as Ising ones except left bonds on each second triangle. Spins connected by
these bonds interact by Heisenberg XXZ interaction. Such a construction allows one to represent
the Hamiltonian in the form of sum of block Hamiltonian commutating to each other, thus, the
exponent in partition function can be expanded to the product of exponents corresponding to
each block. Due to this fact, the method of transfer-matrix can be exploited leading to an exact
calculation of all thermodynamic functions. In the first section we formulate the model and give
its exact solution, then, in the next section, we analyze various ground states of the system and
draw a corresponding phase diagram. The last section contains concluding remarks.
2. The model and its exact solution
Let us consider the sawtooth chain in which the left pair of spins on every second triangle
interacts with the XXZ-Heisenberg interaction, while all other interaction bonds are of Ising type,
i.e. the interactions include only z-component of spin operators. The corresponding Hamiltonian
Σi Σi+1
Τi Τi+1
Si1
Si2
JK3K2 K4
K1 K1
Figure 2. The sawtooth chain with Heisenberg bond on every second triangle. The Heisenberg
bond is marked with bold line. Group of spins included into the dotted rectangle correspond to
one block with Hamiltonian Hi .
is suitable to be written as a sum of block Hamiltonians where each block contains 5 spins: a pair
of spins connected with quantum bond, two adjacent spins from the basement and one spin at the
top of the triangle from the left hand side (see figure 2):
H =
N
∑
i=1
(
Hi + K2σiτi − H1τi −
H2
2
(σi + σi+1)
)
,
Hi = J (∆ (Sx
i1S
x
i2 + Sy
i1S
y
i2) + Sz
i1S
z
i2) + K1S
z
i1 (σi + σi+1)
+ K3S
z
i1τi + K4S
z
i2σi+1 − H(Sz
i1 + Sz
i2), (1)
where Sα
i,a, a = 1, 2, α = x, y, z are S = 1/2 spin operator components of the spin connected by
quantum bond from i − th block, σi and σi+1 are two adjacent spins from the basement and τi is
the spin on the top of left triangle; as only z-components of all σ and τ spins are included in the
interaction, one can consider them just as a classical variable, taking values ±1/2. Here we also
choose the most general form of interaction with different coupling constants J, K1, K2, K3, K4 for
different kinds of bonds and different g-factors for Zeeman term corresponding to different kinds
of spins. The partition function of the system then reads:
Z =
∑
(σ)
∑
(τ)
SpSe−βH . (2)
Here one should sum over all values of classical variables σ and τ and take a trace over all states
of spin operators S. Due to commutativity of Hamiltonians for different blocks one can expand the
344
Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds
exponent and get the product of the terms corresponding to different blocks. After that the traces
for each block can be taken separately:
Z =
∑
(σ)
N
∏
i=1
∑
τi=±1/2
Ω (σi, σi+1|τi) eβH1τi−βK2τiσi+β
H2
2
(σi+σi+1), (3)
where
Ω (σi, σi+1|τi) = Spe−βHi =
4
∑
n=1
e−βλn(σi,σi+1|τi), (4)
and λn are four eigenvalues of the Hi:
λ1,2 (σi, σi+1|τi) =
1
4
J ∓
1
2
(2H − K1 (σi + σi+1) − K3τi − K4σi+1) ,
λ3,4 (σi, σi+1|τi) = −
1
4
J ∓
1
2
√
(K1 (σi + σi+1) + K3τi − K4σi+1)
2 + J2∆2. (5)
In equation (3) the sum over all states of spin τ in each block can be taken independently of the
other, then yielding
Z =
∑
(σ)
N
∏
i=1
Z (σi, σi+1) eβ
H2
2
(σi+σi+1),
Z (σi, σi+1) = Ω (σi, σi+1|1/2) eβ 1
2
(H1−K2σi) + Ω (σi, σi+1| − 1/2) e−β 1
2
(H1−K2σi). (6)
Thus, the partition function takes the form similar to that of the single chain with N sites with
classical variables σi in them. The partition function can be calculated within the standard transfer-
matrix technique (see for example [25]):
Z =
∑
(σ)
N
∏
i=1
T (σi, σi+1) = TrTN = ΛN
+ + ΛN
− , (7)
here the cyclic boundary conditions σi+N = σi are imposed and 2 by 2 transfer-matrix T with
eigenvalues Λ± has the following form:
T =
(
Z(1/2, 1/2)eβ
H2
2 Z(1/2,−1/2)
Z(−1/2, 1/2) Z(−1/2,−1/2)e−β
H2
2
)
. (8)
Thus, for free energy per one block of the sawtooth lattice with mixed Heisenberg and Ising
bonds described above in the thermodynamic limit when only maximal eigenvalue survives from
equation (7) one obtains
f = −
1
β
log
(
1
2
(
Z+eβ
H2
2 + Z−e−β
H2
2 +
√
(
Z+eβ
H2
2 − Z−e−β
H2
2
)2
+ 4Z0Z̄0
))
, (9)
where the following notations are introduced: Z+ = Z(1/2, 1/2), Z− = Z(−1/2,−1/2), Z0 =
Z(1/2,−1/2) and Z̄0 = Z(−1/2, 1/2). Then, one can obtain analytic expressions for all thermo-
dynamic quantities of the systems by taking derivatives of equation (9). Thus, equations (3)–(9)
exactly solve the problem of thermodynamics of the systems under consideration. Let us finally
list the explicit expressions for the entries of transfer-matrix for a particular choice of couplings,
mentioned in the beginning of the next section, K1 = J1 and K2 = K3 = K4 = J and for the same
345
V. Ohanyan
value of g-factor for all spins, which imply H1 = H2 .
Z+ = 2
(
eβ 1
2
(H−1/2J)
(
e−β J
4 cosh (β (H − 1/2(J1 − 1/2J))) + eβ J
4 cosh
(
β1/2
√
J2
1 + J2∆2
))
+ e−β 1
2
(H−1/2J)
(
e−β J
4 cosh (β (H − 1/2(J1 − 3/2J)))
+ eβ J
4 cosh
(
β1/2
√
(J1 − J)2 + J2∆2
)))
,
Z− = 2
(
eβ 1
2
(H+1/2J)
(
e−β J
4 cosh (β (H + 1/2(J1 − 1/2J)))+eβ J
4 cosh
(
β1/2
√
(J1 − J)2+J2∆2
))
+ e−β 1
2
(H+1/2J)
(
e−β J
4 cosh (β (H + 1/2(J1 + 3/2J)))
+ eβ J
4 cosh
(
β1/2
√
J2
1 + J2∆2
)))
,
Z0 = 2
(
eβ 1
2
(H−1/2J)
(
e−β J
4 cosh (β (H − 1/4J))) + eβ J
4 cosh
(
β1/2J
√
1 + ∆2
))
+ e−β 1
2
(H−1/2J)
(
e−β J
4 cosh (β (H + 3/4J)) + eβ J
4 cosh (β1/2J∆)
))
,
Z̄0 = 2
(
eβ 1
2
(H+1/2J)
(
e−β J
4 cosh (β (H − 3/4J))) + eβ J
4 cosh (β1/2J∆)
)
+e−β 1
2
(H+1/2J)
(
e−β J
4 cosh (β (H + 1/4J)) + eβ J
4 cosh
(
β1/2J
√
1 + ∆2
)))
. (10)
3. Ground states phase diagram for antiferromagnetic couplings
Let us analyze different ground states of the model. For the sake of simplicity we restrict
ourselves to the case of only two different coupling constants, one along the basement J1 and
another one between spins on the top and spins on the basement J . This means one should put
K1 = J1 and K2 = K3 = K4 = J . We also consider only antiferromagnetic coupling J > 0, J1 > 0,
the rest cases are to some extent trivial ones. Solving the eigenvalues and eigenvectors problem for
block Hamiltonian Hi and taking into account all combinations of spins σi, σi+1 and τi, one can
get the following periodic eigenstates for the whole chain with period equal to the period of the
lattice up to the inversion of all spins: three antiferromagnetic states with M = 0, two of which are
degenerated and differ from each other by the flip of all σ and τ spins(the corresponding energies
per one block are presented)
|AM+〉 =
N
∏
i=1
|Sz = 0, +〉i
⊗
|σi =↑, τi =↓〉
|AM−〉
N
∏
i=1
= |Sz = 0,−〉i
⊗
|σi =↓, τi =↑〉,
εAM = −
1
2
(
J +
√
(J − J1)
2
+ J2∆2
)
, (11)
where |Sz = 0,±〉i stands for the following state of two S spins form i-th block with total projection
equal to zero
|Sz = 0,±〉 =
1
√
1 + γ2
±
(| ↑↓〉 − γ±| ↓↑〉) ,
γ± =
± (J − J1) +
√
(J − J1)
2 + J2∆2
J∆
. (12)
346
Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds
This eigenstate corresponds to the λ3 eigenvalue from equation (5). For arbitrary values of σ and
τ surrounding pairs of S spins, coefficient γ from equation (12) is
γ (σi, σi+1|τi) =
J (σi+1 − τi) − J1 (σi + σi+1) +
√
(J (σi+1 − τi) − J1 (σi + σi+1))
2 + J2∆2
J∆
. (13)
Figure 3. Spin configuration corresponding to the |AM
−
〉 antiferromagnetic state. Ovals mark
the pairs of spin connected with Heisenberg bonds in |Sz = 0,−〉 state. One should inverse all
arrows to obtain |Sz = 0, +〉 state.
Figure 4. Spin configuration corresponding to the |F1〉 ferrimagnetic state.
Thus, in these configurations the pair of spins connected with the Ising bonds are aligned
opposite to each other, while each pair of spins connected with the Heisenberg bond is in a special
state with Sz
tot = 0, see figure (3). Another antiferromagnetic ground state consists of triplet spin
configurations on each Heisenberg bond and pairs of spins connected with Ising bond both pointed
opposite to the direction of S = 1 pairs. However, this ground state is never realized at T = 0
for antiferromagnetic region of values of J and J1. There are also three ferrimagnetic states of the
chain when external magnetic field is turned on. All these states have spatial period corresponding
to the period of the chain and magnetization equal to 1/2. In the first ferrimagnetic state all pairs
of S spins are in a singlet state, while the rest σ and τ spins are pointed along the field (figure 4):
|F1〉 =
N
∏
i=1
|Sz = 0〉i
⊗
|σi =↑, τi =↑〉, εF1 = −
1
2
√
J2
1 + J2∆2 − H, (14)
where eigenstate |Sz = 0〉i is given by the same formula as equation (12) but with another value
of the coefficient gamma, which is given by equation (13)
γ =
−J1 +
√
J2
1 + J2∆2
J∆
. (15)
In the rest two ferrimagnetic states all pairs of S spins are in the triplet polarized state with
Sz = 1, while σ and τ spins surrounding them are pointed opposite to each other:
|F2〉 =
N
∏
i=1
| ↑↑〉i
⊗
|σi =↑, τi =↓〉, εF2 =
1
4
J1 − H,
|F3〉 =
N
∏
i=1
| ↑↑〉i
⊗
|σi =↓, τi =↑〉, εF3 = −
1
4
J1 − H. (16)
347
V. Ohanyan
And finally in the spin polarized phase all spins are pointed along the field,
|SP 〉 =
N
∏
i=1
| ↑↑〉i
⊗
|σi =↑, τi =↑〉, εSP = J +
1
2
J1 − 2H. (17)
Figure 5. Spin configuration for corresponding to |SM〉 phase.
Besides the states with non-broken lattice symmetry the system under consideration can possess
another ground state with a spatial period equal to two blocks. This state can be realized at T = 0
only for η = J1
J 6 1. This state can be obtained from |AM−〉 by flipping every second σ spin in
the basement of the chain (figure 5). This state with broken translational symmetry corresponds
to magnetization of the system equal to 1/4. We will refer to it as a spin modulated phase, |SM〉.
The spatial period of the state and the value of magnetization per site are in full agreement with
Oshikawa–Yamanaka-Affleck criterion [26]. The corresponding energy per one block is
εSM = −
1
4
J
(
1 + |∆| +
√
1 + ∆2
)
−
1
2
H. (18)
0 1 2
0
1
2
h
η
SP
F1
SM
AM
Figure 6. T = 0 ground state phase diagram for the sawtooth chain with Ising and Heisenberg
bonds for J > 0, J1 > 0 in (η, h)-plane for ∆ = 0.3.
Now it is straightforward to draw the T = 0 ground state phase diagram for antiferromagnetic
couplings J > 0, J1 > 0. In figure (6) one can see the phase boundaries in (η, h) plane (h = H/J)
for ∆ = 0.3. An interesting feature of the system is that at antiferromagnetic values of couplings
with the ground state for H = 0 is unaffected by the value of η. Thus, the unique ground state
is either |AM−〉 or |AM+〉. One should distinguish two regions of η, namely, for η < 1 increasing
the magnetic field one can see transition to spin modulated phase with M = 1/4 prior to the
appearance of the ferrimagnetic structure F1 with M = 1/2. However, for η > 1, antiferromagnetic
phase changes immediately with the ferrimagnetic one, though the width of the AM phase region
(the value of the h at which the zero temperature quantum phase transition takes the place) with
η → ∞ asymptotically goes to zero. So, one can see a small closed region of SM phase in the
348
Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds
(η, h) plane. At the values η = 1, h = 0, the system demonstrates an additional feature, i. e., the
macroscopic degeneracy of the ground state. At this point the S spins in each block are in the singlet
state, whereas its adjacent σ and τ can take an arbitrary value with one restriction σ + τ = 0.
Thus, in each block, σ and τ can freely take either σ = 1/2, τ = −1/2 or σ = −1/2, τ = 1/2 values.
Thus, one obtains macroscopic two-fold degeneracy at J = J1. One can refer to this state as a
frustrated antiferromagnetic state:
|FR〉 =
N
∏
i=1
|Sz = 0〉i
⊗
|ξi〉, (19)
where |ξi〉 can be either |σi =↑, τi =↓〉 or |σi =↓, τi =↑〉 And finally the equations of phase bound-
aries between four phases presented in the phase diagram in figure (5) are listed below:
between |AM〉 and |SM〉 : h =
1
2
(
1 − |∆| −
√
1 + ∆2
)
+
√
(1 − η)
2
+ ∆2,
between |AM〉 and |F1〉 : h =
1
2
(
1 +
√
(1 − η)2 + ∆2 −
√
η2 + ∆2
)
,
between |SM〉 and |F1〉 : h =
1
2
(
1 + |∆| +
√
1 + ∆2
)
−
√
η2 + ∆2,
between |F1〉 and |SP 〉 : h = 1 +
1
2
(
η +
√
η2 + ∆2
)
. (20)
4. Conclusion
In this paper using the example of sawtooth chain we considered the possibility to obtain
exact thermodynamic solution for the one-dimensional quantum spin systems by considering their
counterparts where some interaction bonds are changed with the Ising ones. Namely, we considered
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
M
/M
sa
t
H/J
∆=0.3
J1/J=0.5
T=0.001J
Figure 7. Magnetization process for η = 0.5 at extremely low temperature T = 0.001J and
∆ = 0.3. Three magnetization plateaus at M = 0, M = 1/4 and M = 1/2 correspond to AM ,
SM and F1 phases.
the pairs of S = 1/2 quantum spins interacting with XXZ interaction arranged in the sawtooth
chain in such a way which allows one to use transfer-matrix technique for exact calculations.
This result continues the series of investigations of the subject performed earlier for other one-
dimensional spin systems with Ising and Heisenberg bonds [13–24]. These results are not only of
academic interest. There are many one–dimensional spin systems which are the models of a real
magnetic material and which are not integrable. Until now, the only truly reliable way of describing
349
V. Ohanyan
the thermodynamic and magnetic properties of such a system is laborious numerical calculations.
However, the results of recent investigations provide evidence that changing some bonds of the
system with Ising ones, on the one hand, makes the thermodynamical problem exactly solvable
and, on the other hand, does not drastically change the magnetic properties of the system at
least in case when only ferromagnetic bounds have been changed. One can look at the plots of
low–temperature magnetization curves for the system considered in the paper (figures (7), (8)).
As it can be seen from phase diagram (figure 6), there is qualitative difference between η < 0 and
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
M
/M
sa
t
H/J
T=0.001J
∆=0.3
J
1
/J=1
Figure 8. Magnetization process for η = 1 at extremely low temperature T = 0.001J and
∆ = 0.3. Spin modulated phase corresponding to the plateau at M = 1/4 is absents here. One
can see only two plateaus at M = 0 and M = 1/2 corresponding to FR and F1 phases.
η > 1 cases. For η < 1, one obtains the curve with three intermediate magnetization plateaus, at
M = 0, M = 1/4 and M = 1/2, which corresponds to the stability regions of AM , SM and F1
structures, respectively. In figure (7) one can see the corresponding plot for ∆ = 0.3 and η = 0.5.
For η = 1 and ∆ = 0.3 one can see in figure (8) only one plateau at M = 1/2. This result
qualitatively corresponds to that obtained in purely quantum sawtooth chain [12]. Numerically
obtained magnetization curves for sawtooth Heisenberg chain contain only plateaus at M = 0
and M = 1/2 for the whole range of parameters. Though there is a rather large discrepancy in
quantitative characteristics, such as positions of the terminal points of the plateau, this can be
explained by the fact that in our case we have changed antiferromagnetic bonds with Ising ones,
which generally speaking makes much more changes into the thermodynamics of the system than
changing ferromagnetic bonds. An additional feature which is absent in case of purely quantum
antiferromagnetic sawtooth chain is the plateau at M = 1/4 for η < 1. In order to obtain a more
satisfactory correspondence between numerical results for a purely quantum system and its mixed
counterpart,one can consider a larger quantum cluster, containing three or more spins.
Acknowledgements
This work was partly supported by the grants CRDF–UCEP – 06/07 and ANSEF–1586–PS.
The author would like to thank Oleg Derzhko for useful discussions and Lev Ananikian and Diana
Antonosyan for help in preparing the figures.
350
Antiferromagnetic sawtooth chain with Heisenberg and Ising bonds
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Антиферомагнiтний пилкоподiбний ланцюжок зi зв’язками
Гайзенберга i Iзинга
В. Оганян
1 Вiддiл теоретичної фiзики, Єреванський державний унiверситет, алея Манугяна 1, 0025 Єреван,
Вiрменiя
2 Єреванський iнститут фiзики, бульвар Алiханяна 2, 0036 Єреван, Вiрменiя
Отримано 28 травня 2009 р., в остаточному виглядi – 18 червня 2009 р.
Розглядається пилкоподiбний ланцюжок з парами S = 1/2 спiнiв, що взаємодiють з XXZ взаємодi-
єю i розташованi на фрагментi кожного другого зубця пилки. Всi iншi взаємодiї взятi типу взаємодiї
Iзинга. Отримано точний статистико-механiчний розв’язок моделi прямим методом матрицi пере-
носу. Розв’язок дозволяє отримати точнi аналiтичнi вирази для всiх термодинамiчних функцiй мо-
делi. Дослiджено також властивостi в основному станi i представлена вiдповiдна фазова дiаграма в
основному станi.
Ключовi слова: пилкоподiбний ланцюжок, точнi розв’язки, плато намагнiченостi
PACS: 75.10.Pq
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