Monte Carlo simulation of anisotropic Shastry–Sutherland lattice in the framework of classical Heisenberg model
Monte Carlo simulation of two-dimensional Shastry–Sutherland lattice has been carried out using heat-bath method. The dependencies of magnetization M on external field H have been obtained in the framework of classical Heisenberg model. In certain interval of exchange parameters ratio the plateau of...
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irk-123456789-1188072017-06-01T03:03:51Z Monte Carlo simulation of anisotropic Shastry–Sutherland lattice in the framework of classical Heisenberg model Slavin, V.V. Krivchikov, A.A. Низкотемператуpный магнетизм Monte Carlo simulation of two-dimensional Shastry–Sutherland lattice has been carried out using heat-bath method. The dependencies of magnetization M on external field H have been obtained in the framework of classical Heisenberg model. In certain interval of exchange parameters ratio the plateau of magnetization corresponding to M = 1 / 3 has been observed. The influence of exchange anisotropy of “easy-axis” type on this plateau width is studied. It has been shown that even weak anisotropy ( ∼1 2% − ) leads to essential enlargement of the plateau. The dependence of critical temperature on exchange parameters ratio has been established. 2011 Article Monte Carlo simulation of anisotropic Shastry–Sutherland lattice in the framework of classical Heisenberg model / V.V. Slavin, A.A. Krivchikov // Физика низких температур. — 2011. — Т. 37, № 12. — С. 1264–1268. — Бібліогр.: 16 назв. — англ. 0132-6414 PACS: 75.10.Hk, 75.30.Gw, 05.50.+q http://dspace.nbuv.gov.ua/handle/123456789/118807 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкотемператуpный магнетизм Низкотемператуpный магнетизм |
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Низкотемператуpный магнетизм Низкотемператуpный магнетизм Slavin, V.V. Krivchikov, A.A. Monte Carlo simulation of anisotropic Shastry–Sutherland lattice in the framework of classical Heisenberg model Физика низких температур |
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Monte Carlo simulation of two-dimensional Shastry–Sutherland lattice has been carried out using heat-bath method. The dependencies of magnetization M on external field H have been obtained in the framework of classical Heisenberg model. In certain interval of exchange parameters ratio the plateau of magnetization corresponding to M = 1 / 3 has been observed. The influence of exchange anisotropy of “easy-axis” type on this plateau width is studied. It has been shown that even weak anisotropy ( ∼1 2% − ) leads to essential enlargement of the plateau. The dependence of critical temperature on exchange parameters ratio has been established. |
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Slavin, V.V. Krivchikov, A.A. |
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Slavin, V.V. Krivchikov, A.A. |
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Slavin, V.V. |
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Monte Carlo simulation of anisotropic Shastry–Sutherland lattice in the framework of classical Heisenberg model |
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Monte Carlo simulation of anisotropic Shastry–Sutherland lattice in the framework of classical Heisenberg model |
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Monte Carlo simulation of anisotropic Shastry–Sutherland lattice in the framework of classical Heisenberg model |
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Monte Carlo simulation of anisotropic Shastry–Sutherland lattice in the framework of classical Heisenberg model |
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Monte Carlo simulation of anisotropic Shastry–Sutherland lattice in the framework of classical Heisenberg model |
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monte carlo simulation of anisotropic shastry–sutherland lattice in the framework of classical heisenberg model |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкотемператуpный магнетизм |
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Monte Carlo simulation of anisotropic Shastry–Sutherland lattice in the framework of classical Heisenberg model / V.V. Slavin, A.A. Krivchikov // Физика низких температур. — 2011. — Т. 37, № 12. — С. 1264–1268. — Бібліогр.: 16 назв. — англ. |
| series |
Физика низких температур |
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AT slavinvv montecarlosimulationofanisotropicshastrysutherlandlatticeintheframeworkofclassicalheisenbergmodel AT krivchikovaa montecarlosimulationofanisotropicshastrysutherlandlatticeintheframeworkofclassicalheisenbergmodel |
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2025-07-08T14:40:47Z |
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1837090103762092032 |
| fulltext |
© V.V. Slavin and A.A. Krivchikov, 2011
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12, p. 1264–1268
Monte Carlo simulation of anisotropic
Shastry–Sutherland lattice in the framework
of classical Heisenberg model
V.V. Slavin and A.A. Krivchikov
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: slavin@ilt.kharkov.ua
Received June 1, 2011
Monte Carlo simulation of two-dimensional Shastry–Sutherland lattice has been carried out using heat-bath
method. The dependencies of magnetization M on external field H have been obtained in the framework of
classical Heisenberg model. In certain interval of exchange parameters ratio the plateau of magnetization corres-
ponding to = 1 / 3M has been observed. The influence of exchange anisotropy of “easy-axis” type on this pla-
teau width is studied. It has been shown that even weak anisotropy ( 1 2%−∼ ) leads to essential enlargement of
the plateau. The dependence of critical temperature on exchange parameters ratio has been established.
PACS: 75.10.Hk Classical spin models;
75.30.Gw Magnetic anisotropy;
05.50.+q Lattice theory and statistics (Ising, Potts, etc.).
Keywords: Shastry–Sutherland lattice, Monte Carlo simulation, classic Heisenberg model.
1. Introduction
Recently the physical properties of two-dimensional
compounds with “Shastry–Sutherland Lattice” (SSL) mag-
netic structure have attracted great interest. There are many
theoretical and experimental, fundamental and applied
works dedicated to the given subjects. These systems are
interesting due to a number of unusual magnetic properties,
which exhibit various kinds of compounds. The structure
of SSL can be described as a square lattice with four anti-
ferromagnetic couplings J and one additional diagonal
antiferromagnetic coupling J ′ [1]. It is interesting, that
SSL has been considered initially by Shastry and Suther-
land as an abstract model of a frustrated quantum spin sys-
tem with an exact ground state in some region of parame-
ters. Later, it has been established that a number of quasi-
two-dimensional compounds have magnetic structures
which are close to SSL. These are 2 3 2SrCu (BO ) [2–4] and
rare earth tetraborides [5–7]. The experiments with these
compounds have shown a number of interesting features.
For example, the dependence of magnetization M on exter-
nal magnetic field H contains a series of plateaux. These
plateaux correspond to rational values of the ratio sat/M M
where satM is saturated magnetization. The plateau cor-
responding to sat/ = 1/8M M , 1/4 , 1/3 , 1/2 were observ-
ed in different compounds [6–9]. As indicated above, the
first measurements of magnetization peculiarities were
carried out on 2 3 2SrCu (BO ) compound. Due to pro-
nounced quantum magnetic properties of Cu ions, quantum
SSL model were studied intensively. At the same time,
such plateaux were discovered in rare-earth tetraborides
RB 4 in which the rare-earth ions are placed in the (001)
plane according to a lattice which is topologically equiva-
lent to the SSL. These compounds present large total angu-
lar momenta that justify a classical description of the SSL
[10–13]. Is was shown that even in classical limit and in
the framework of isotropic SSL model some peculiarity on
( )M H dependence at sat/ = 1/3M M takes place. This
peculiarity can be identified as plateau “nucleus” (or pseu-
do-plateau). Spin structure corresponding to this pseudo-
plateau has been established in the same paper. Later, in
[12] was shown that anisotropy of exchange interaction
affects the pseudo-plateau.
The main goal of our paper is to investigate in details an
influence of exchange constants ratio /J J′ and easy-axis
anisotropy on the thermodynamic properties of two-
dimensional SSL using Monte Carlo simulation in the
framework of classical Heisenberg model.
Monte Carlo simulation of anisotropic Shastry–Sutherland lattice in the framework
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12 1265
2. Hamiltonian and method
The Hamiltonian of the system under consideration has
the form
< , > < , >
ˆ ˆ= i i i i i
i i i
J J ′+δ +δ
′δ δ
′+ −∑ ∑ ∑s s s s H sH . (1)
Here is are the classical vectors with unity length
(| | = 1).is Symbol < , >i δ in the first term means that the
summation is taken over four near neighbours and < , >i ′δ
in the second term means the summation with one diagonal
neighbour. The last term is the interaction of the spins sys-
tem with the external magnetic field H . The elements of
diagonal matrixes
0 0
ˆ = 0 0
0 0
xx
yy
zz
J
J J
J
⎛ ⎞
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
and
0 0
ˆ = 0 0
0 0
xx
yy
zz
J
J J
J
⎛ ⎞′
⎜ ⎟
′ ′⎜ ⎟
⎜ ⎟′⎝ ⎠
are the exchange constants ( , > 0, = , ,J x y zα α α ).
A number of interesting features has been discovered in
SSL in the framework of classical Monte Carlo method
with Metropolis test. It should be noted that along with
evident advantages (simplicity and universality) the me-
thod possess a number of disadvantages. The most impor-
tant of them is low efficiency in low-temperature region. It
is essentially because the specific characteristics of SSL
manifest itself in full measure just in this temperature re-
gion. Further, low efficiency of the method leads to limita-
tion on the system size (as far as we know, the maximal
system size studied in the previous works is about 40×40).
As the result the question about an influence of boundary
effects is open yet. In this work we propose algorithm of
Monte Carlo simulation based on the so-called “heat-bath”
method. The method significantly improves computing
efficiency at low-temperatures region and for systems with
continuous degrees of freedom. Moreover, we adopted the
method for parallel (cluster) calculations, which allows us
to improve the efficiency also.
In the framework of “heat-bath” method the transition
probability from “old” configuration s to a “new” one s′
has the form [14]
{ }
( , )
( , ) =
( , )
a
a
s
f s s
w s s
f s s
′
′
′
′∑
. (2)
Symbol { }s′ means summation over all possible states of
spin s′ . For our calculation the acceptance probability af
has been chosen in Metropolis form:
( , ) = exp ( [ ( ) ( )])af s s E s E s′ ′−β −
where = 1/ Tβ is inverse temperature (T is in energy
units) and ( )E s is the system energy in configuration s. In
such a case (2) depends on s′ only and acquires the form
( )
( )
{ }
exp ( )
( , ) =
exp ( )
s
E s
w s s
E s
′
′−β
′
′−β∑
. (3)
According to Monte Carlo approach the configurations s
and s′ differ by the state of one particle which we will in-
dicate by index i . Let us introduce
ˆ ˆ=i i iJ J ′+δ +δ
δ
′+ −∑S s s H . (4)
Then
ˆ ˆ( ) = ( ) = =i i i i i i i iE s E J J ′+δ +δ
δ
′ ′+ −∑s s s s s Hs s S (5)
and
( )exp ( )
( , ) = ( , ) = i
i
i
E
w s s w S
Z
−β
′ ξ =
s
exp ( ) exp ( )
= =i i i
i i
S
Z Z
−β ξs S
. (6)
Here = | |i iS β S and = cos( )ξ π− θ , where θ is the angle
between is and iS . This choice of ξ is determined by
antiferromagnetic interaction among the spins. iZ is nor-
malization constant in denominator (3). In our case
1
1
2= exp ( ) = sinh ( )i i i
i
Z S d S
S
−
ξ ξ∫ . (7)
Thus,
( , ) = exp ( )
2sinh ( )
i
i i
i
S
w S S
S
ξ ξ . (8)
Performing non-linear white noise selection
1
exp ( ) exp ( )
( ) = ( , ) =
2sinh ( )
i i
i
i
S S
w S d
S
ξ
−
ξ −′γ ξ ξ ξ∫ (9)
we obtain finally
cos ( )rξ = π − θ =
( )0
1 ln exp ( ) [exp ( ) exp ( )]r
i i i
i
S S S
S
= − γ − − (10)
where 0
rγ is uniformly distributed random number
0(0 < 1r≤ γ ). This expression has the following sense. Let
us introduce local polar coordinate system with || iz −S and
arbitrary direction of x and y in the plane perpendicular
to z. In such a case (10) gives the transition probability of
spin is to a new state as the function of polar angle θ . As
far as (5) depends on θ only one can choice azimuthal
angle φ in xy plane as 1= 2 rφ πγ , where 1
rγ is uniformly
distributed random number ( 10 < 1r≤ γ ).
V.V. Slavin and A.A. Krivchikov
1266 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12
Performing back transformation to global Cartesian
coordinate system we obtain
2
1
2
2
1 ( )= cos ( )
1 ( )= = sin ( )
= ( ) /
r
z
r
i z
r
x y z
x n
l
y n
l
z n x n y n
⎧ − ξ⎪ φ
⎪
⎪
⎪ − ξ⎪ φ⎨
⎪
⎪
⎪ ξ − −
⎪
⎪⎩
s (11)
where
= = { , , },
| |
i
x y z
i
n n n
S
n
S
2 2
1 0 0 0 0= cos ( ) sin ( ) sin ( ) cos ( ),l a b cφ + φ − φ φ
2 2
2 0 0 0 0= sin ( ) cos ( ) sin ( ) cos ( ),l a b cφ + φ + φ φ
2 2 2 2
0= , = , = , = arctan .x z y y x y
ca n n b n n c n n
b a
⎛ ⎞+ + φ ⎜ ⎟−⎝ ⎠
The final expression (11) describes the probability of
system transition as the function of uniformly distributed
random numbers 0
rγ and 1
rγ . This Monte Carlo algorithm
was tested on exact solvable models. Beside this, the
search algorithm for the configurations corresponding to a
minimum of internal energy is realised. The analysis of the
properties of such configurations is extremely important
for studying the ground state structure.
3. Results and discussion
One of the SSL distinctive features is a step-like beha-
vior in the field dependence of magnetization M . Such
behavior takes place even in the case of isotropic spin–spin
exchange interaction. In the framework of classic Heisen-
berg model the plateau takes place for = 1/ 3M [11] (as
far as sat = 1M here and further more sat/ =M M M ). The
dependence ( )M H obtained in the framework of our ap-
proach for isotropic case and || zH is presented in
Fig. 1,a. Here = = = 1xx yy zzJ J J , = = = 2xx yy zzJ J J′ ′ ′ ,
temperature = 0.02T and system size is 48 48× . It should
be stressed, that our result is in good agreement with litera-
ture data [11,12].
On the first stage of our investigation we have studied
influence of exchange anisotropy on such a peculiarity.
Corresponding dependence for the case of exchange aniso-
tropy is presented in Fig. 1,b. Here = = 1xx yyJ J ,
= = 2xx yyJ J′ ′ = 1zzJ + α , = 2zz zzJ J′ and anisotropy
parameter = 0.05α . One can see that even weak anisotro-
py (α ∼ 5%) leads to essential growth of “step”, corres-
ponding to = 1/ 3M [12]. Another important thing is that
the curves corresponding to different system sizes ( 24 24×
and 48 48× ) are in a good agreement. It means that boun-
dary effects are small and, thus, such system size is appro-
priate for our calculations. The dependence of = 1/ 3M
magnetization plateau width, Δ , on α is presented in
Fig. 2. It should be noted that increase in α leads to Δ
growth only. The plateaux corresponding to 1 / 3M ≠ are
absent.
Spin configurations corresponding for = 1/ 3M phase
coincide with those ones described in [11]. The calculated
data (solid boxed) have been fitted by power function
1/= kAΔ α . The best fit corresponds to = 0.36A and
= 1.73k which is very close to 3 .
On the second stage we have determined the region of
the step-like peculiarity as the function of exchange con-
stants ratio = /J J′μ . The dependence of plateau width Δ
on μ for = 0.005α is presented in Fig. 3. One can see
that in the phase, corresponding to = 1/ 3M plateau exists
in the region 1.5 2.4μ .
On the next stage we have studied temperature depen-
dencies of specific heat
2 2
2
1( ) = (< > < > )C T E E
T
−
Fig. 1. The dependence of magnetization M on external magnetic field H for system size 24 24× , = / = 2J J′μ and anisotropy para-
meter = 0α (isotropic case). The vicinity of pseudo-plateau = 1 / 3M are presented in the inset (a). The same dependence for
= 0.05.α The temperature = 0.02T was used for both figures (b).
0 02 24 46 68 810 10
0.2 0.2
0.4 0.4
0.6 0.6
0.8 0.8
1.0 1.0
2 3 4
0.2
0.3
0.4
M
M
M
H
H
H
a
24 24×
48 48×
b
Monte Carlo simulation of anisotropic Shastry–Sutherland lattice in the framework
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12 1267
and magnetic susceptibility
2 21( ) = (< > < > ).T M M
T
χ −
Here < >E and 2< >E are average energy and square
average energy, respectively. These averages have been
calculated also in Monte Carlo process. It is well known
that these quantities are the most sensitive for the phase
transitions in antiferromagnetic compounds. Analysing
these curves for different μ values one can plot the depen-
dence of critical temperature NT on μ (Fig. 4). As seen
from this figure, NT goes to zero for 0 1.96μ →μ ≈ . It
should be noted, that 0μ is very close to critical point val-
ue for classical SSL model ( = 2μ ) [11]. Another interest-
ing thing is applicability of Mermin–Wagner theorem for
the system under consideration. Really, according to Mer-
min–Wagner theorem continuous symmetries cannot be
spontaneously broken at finite temperature in isotropic
two-dimensional systems with sufficiently short-range in-
teraction. In real systems there are many reasons resulting
in Mermin–Wagner theorem violation. This is quasi-two-
dimensionality, long-range interaction etc. In numerical
experiments this is, for example, pseudo random distribu-
tion of random number. The correlation length is large, but
finite. Another important reason is discreteness of real
number representation in computer. The density of num-
bers is not constant. In our case it leads to appearance of
extremely small, but finite effective anisotropy. In addi-
tion, if long-range correlations decay slow, then it is very
difficult to detect this phenomena numerically (including
Monte Carlo method). It is necessary to consider extremely
large systems and calculation time becomes huge. Even so,
boundary effects can be small, but boundary conditions can
affect on correlation functions behavior. The detailed dis-
cussion dedicated to Mermin–Wagner theorem applicabili-
ty for SSL model goes beyond the scope of this paper. We
plan to investigate this problem in near future.
Unlike experimental data, any low-temperature (below
NT ) peculiarities of ( )Tχ and ( )C T are absent in the
framework on the proposed classical model. It confirms
indirectly that the unusual low-temperature behavior of
( )Tχ inherent in SSL has quantum origin.
In addition, in order to check our results we have car-
ried out a number of computer simulation using multica-
nonical method, described in [15,16]. We have extended
this approach to the systems with continues degrees of
freedom. It should be noted that the results obtained in the
framework of both Monte Carlo methods are in good
agreement.
4. Conclusions
We have studied numerically two-dimensional Shastry–
Sutherland lattice in the framework of classical Heisenberg
model. Parallel Monte Carlo algorithm based on heat-bath
method has been developed. It has been shown, that the
influence of boundary effects on SSL magnetic properties
Fig. 2. The dependence of plateau width Δ on anisotropy para-
meter α for system size 24 24× , = / = 2J J′μ , = 0H and
= 0.02T .
0 2 4 6 8 10
0
0.5
1.0
a, %
�
Fig. 3. The dependence of plateau width Δ on exchange con-
stants ratio μ for = 0.005α , system size 24 24× , = 0H and
= 0.02T .
1.2 1.4 1.6 1.8 2.0 2.2 2.4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
�
/J J�� �
0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
/J J�� �
T
N
Fig. 4. The dependence of critical temperature NT on exchange
constants ratio μ for system size 24 24× , = 0H , and = 0α
(isotropic case). Solid boxes are calculated values, solid line is
fitting by linear function.
V.V. Slavin and A.A. Krivchikov
1268 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 12
is extremely small for the systems with sizes greater than
24 24× . The dependence of magnetization plateau width
Δ on anisotropy parameter α has been established. This
dependence can be approximated well by power function
1/ 3= .AΔ α
The plateau corresponding to 1/ 3M ≠ are absent in
the framework of this classical model. It has been deter-
mined that critical temperature NT depends linearly on
exchange constants ratio μ . The extrapolation of ( )NT μ
shows that the critical temperature goes to zero for
0 1.96.μ → μ ≈ The obtained limiting value 0μ is very
close to critical point for classical SSL model ( = 2μ ).
Acknowledgements
Computer simulations were carried out on computa-
tional cluster in B. Verkin Institute for Low Temperature
Physics and Engineering of Ukrainian Academy of
Science. The authors express one's thanks to L.F. Belous
for comprehensive assistance in this work.
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