1. Introduction
During recent decades, the frustrated Ising model on the two- dimensional triangular lattice has attracted widespread interest from both theoretical and experimental approaches, because it can be applied to the description of real materials such as the triangular spin-chain system Ca3Co2O6 and the Ising magnet FeI2, which usually exhibit fascinating multi-step magnetiza- tion behaviors [1–6]. For instance, the anisotropic triangular Ising model has been successfully used to explain the fractional magnetization plateaus experimentally reported in FeI2 [7, 8].
As a matter of fact, the triangular Ising model with the antiferromagnetic (AFM) interaction of only nearest neighbor J1 was exactly solved as early as 1950 [9]. The spin config- urations UUD (spin-up, spin-up, and spin-down) and DDU (down-down-up) appear with the same probability in a trian- gular sublattice, and no spontaneous long-range spin order can be developed at any finite temperature (T ) [10]. The infinite degeneracy can be broken by the introduction of an exchange anisotropy or AFM interaction of second neighbors
J2, leading to the stabilization of the collinear AFM state (the spin structure is shown in figure 1(a)) [11, 12]. Furthermore, the UUD state (figure 1(b)) is developed when a magnetic field (h) is applied, giving rise to the magnetization (M ) pla- teau at 1/3 of the saturation magnetization, M = MS/3. The MS /3 plateau persists up to the saturation field, resulting in the two-step magnetization behaviors [13] which have been exper- imentally reported in several classical and quantum triangular spin systems [14–17]. In our earlier work, the ground-state two-step magnetization behavior has been further confirmed by the Wang–Landau simulation of the model, even with the random-exchange interaction [18].
Interestingly, J2 interaction is proved to be very efficient in modulating the magnetization behaviors in such a frustrated spin system, because of the effective reduction of the nearest neighboring interaction due to the frustration [19]. In detail, the inclusion of the J2 term produces the MS/2 magnetization plateau, in addition to the MS/3 plateau. The MS/2 plateau is caused by the ferrimagnetic (FI) state with the spin arrange- ment consisting of alternating AFM and ferromagnetic (FM)
Journal of Physics: Condensed Matter
Effect of further-neighbor interactions
on the magnetization behaviors of the Ising model on a triangular lattice
J Chen1, W Z Zhuo1, M H Qin1, S Dong2, M Zeng1, X B Lu1, X S Gao1 and J-M Liu3
1 Institute for Advanced Materials and Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, South China Normal University, Guangzhou 510006, People’s Republic of China
2 Department of Physics, Southeast University, Nanjing 211189, People’s Republic of China
3 Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, People’s Republic of China
E-mail: [email protected] and [email protected] Received 5 February 2016, revised 14 April 2016 Accepted for publication 25 May 2016
Published 29 June 2016 Abstract
In this work, we study the magnetization behaviors of the classical Ising model on the triangular lattice using Monte Carlo simulations, and pay particular attention to the effect of further-neighbor interactions. Several fascinating spin states are identified to be stabilized in certain magnetic field regions, respectively, resulting in the magnetization plateaus at 2/3, 5/7, 7/9 and 5/6 of the saturation magnetization MS, in addition to the well-known plateaus at 0, 1/3 and 1/2 of MS. The stabilization of these interesting orders can be understood as the consequence of the competition between Zeeman energy and exchange energy.
Keywords: frustration, spin orders, magnetization plateau, triangular antiferromagnets (Some figures may appear in colour only in the online journal)
J Chen et al
Printed in the UK 346004
JCOMEL
© 2016 IOP Publishing Ltd 2016
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J. Phys.: Condens. Matter
CM
0953-8984
10.1088/0953-8984/28/34/346004
Paper
34
Journal of Physics: Condensed Matter IOP
doi:10.1088/0953-8984/28/34/346004 J. Phys.: Condens. Matter 28 (2016) 346004 (6pp)
stripes, as shown in figure 1(c) [20]. It is noted that every spin- down in the UUD state is antiparallel to its nearest neighbors, and these local J1 interactions are satisfied, while all the J2
interactions in the whole system are unsatisfied. On the other hand, all the AFM J1 and J2 interactions between every down- spin site and its neighbors are satisfied in the FI state. Thus, more local AFM J2 interactions are satisfied in the FI state, leading to the replacement of the MS/3 plateau by the MS/2 one with the increasing J2. In addition, the energy loss from the J2 interaction due to the phase transition from the FI state to the FM state linearly increases with J2, leading to the fur- ther increase of the width of the 1/2 plateau.
To some extent, the same mechanism for the stabilization of the FI state by AFM J2 may also work for the system with further-neighbor interactions, and more interesting plateau states may be available. The importance of the study on this subject can be explained by the following two aspects. On one hand, several interesting plateaus have been reported in some of the triangular antiferromagnets [21, 22]. For example, the magnetization plateaus at 1/3, 1/2, and 2/3 of MS are exper- imentally reported in Ba3CoNb2O9. The first plateau state is believed to originate from the easy-axis anisotropy, while the other two plateau states are still not very clear [21]. It is noted that additional interactions may be available in real materials and may play an important role in modulating the magneti- zation behaviors. In fact, a narrow 2/3 magnetization plateau has been predicted when the dipole–dipole interaction is con- sidered in the model for Shastry–Sutherland magnets TmB4, and one may question if the same mechanism still holds true for triangular antiferromagnets [23]. On the other hand, the study of these nontrivial magnetic orders also contributes to the development of statistical mechanics. For example, the
disordering of the collinear phase in the triangular Ising model with distant neighboring interactions has been proved to occur via a first-order phase transition [24, 25]. However, the role of further-neighbor interactions in modulating the magnetization behaviors in triangular Ising antiferromagnets is far from being completely understood, which deserves to be checked in detail.
In this work, as the first step, we study the classical Ising model with further-neighbor interactions on the triangular lat- tice, as shown in figure 1(d). Several interesting spin orders are identified to be developed in certain h regions, respectively, leading to the emergence of the plateaus at 2/3, 5/7, 7/9 and 5/6 of MS, in addition to the plateaus at 0, 1/3 and 1/2 of MS. Furthermore, the magnetic structures of the 1/2 and 2/3 pla- teau states uncovered in this work are consistent with the ear- lier predictions for Cs2CuBr4, demonstrating the generality of these states, at least, partially [19].
The rest of this paper is organized as follows. In section 2, the model and the simulation method will be presented and described. Section 3 presents the simulation results and dis- cussion. The conclusion is presented in section 4.
2. Model and method
In the presence of h and further-neighbor interactions, the model Hamiltonian can be described as follows:
∑ ∑ ∑
∑ ∑ ∑
= ⋅ + ⋅ + ⋅
+ ⋅ + ⋅ −
H J S S J S S J S S
J S S J S S h S,
ij
i j
ij
i j
ij
i j
ij
i j
ij
i j
i i
1 2 3
4 5
1 2 3
4 5
(1) where J1 = 1 is the unit of energy, Si represents the Ising spin with the unit length on site i, 〈ij〉n (n = 1, 2, 3, 4, 5) denotes the summation over all pairs on the bonds with Jn coupling as shown in figure 1(d). Besides, all the distant neighbors are antiferromagnetic coupled and with Jn/Jn+1 > 1.4 (n = 1, 2, 3, 4), to be qualitatively consistent with the dipole–dipole.
Furthermore, such a frustrated spin system may be easily trapped into metastable states at low T and is hard to relax to the equilibrium state. To overcome this difficulty, one may appeal to the parallel tempering exchange Monte Carlo (MC) method which efficiently prevents the system from trapping in metastable free-energy minima caused by the frustration [26, 27]. Our simulation is performed on an N = L × L (L is reasonably chosen to be consistent with the size of the unit cell) triangular lattice with period boundary conditions. We take an exchange sampling after every 10 standard MC steps.
The simulation is started from the FM state at a high h, and the M(h) curves are calculated upon h decreasing. Typically, the initial 2 × 105 MC steps are discarded for equilibrium consid- eration and another 2 × 105 MC steps are retained for statis- tical averaging of the simulation.
3. Simulation results and discussion
First, we study the effect of J3 on the magnetization behaviors of the model. Figure 2(a) shows the simulated magnetization curves for various J3 with L = 42 at (J2, J4, J5) = (0.1, 0, 0)
Figure 1. Spin configurations in (a) the collinear state, (b) the UUD state, and (c) the 1/2 state. Solid and empty circles represent the spins-down and the spins-up, respectively. All the spins-down in the UUD (1/2) state form the hexagonal structure to save the exchange energy. (d) Simulated model on the triangular lattice with the first-, second-, third-, fourth- and fifth-nearest neighbor exchange interactions.
and T = 0.01. For J3 = 0, the magnetization plateaus at M/MS = 0, 1/3, and 1/2 are observed [28]. The first col- linear-state plateau is gradually melted when J3 increases from zero, demonstrating that the collinear state is not favored by J3 [29]. In addition, the 1/3 plateau is broad- ened at the expense of the 1/2 one with the increase of J3. More interestingly, an additional plateau at M/MS = 5/7 that resulted from a particular state with the spin configuration shown in figure 2(b) is developed. In the 5/7 state, for every spin-down, all the AFM J1, J2, and J3 interactions are sat- isfied. Furthermore, all the spins-down form a hexagonal closely packed structure to save the exchange energy. The central spin-down may flip as the magn etic energy increases to be comparable with the interaction energy, and the satur- ation h is estimated to be 6(J1 + J2 + J3), well consistent with our simulation.
To understand the simulated results, we calculate the h-dependence of the individual energy components of the Jn coupling En (n = 1, 2, 3) and the Zeeman energy Ezee for J3 = 0.03 (figure 2(c)). It is noted that the J3 interactions between the down-spin site and its third neighbors in the UUD state are satisfied, while those in the collinear/FI state are not satisfied, leading to the broadening of the 1/3 plateau with the increasing J3, accompanied by the destabilizations of the collinear and 1/2 plateaus. In the 5/7 state, totally 6N/7 J3 bonds are satisfied, and the enhancement of this state with J3 can be understood from the following two aspects. On one hand, within a certain h range, the energy loss from E1 and E2
due to the phase transition from the FI state to the 5/7 state can be covered by the energy gain from E3 and Ezee, resulting in the stabilization of the 5/7 state. Furthermore, the energy gain from E3 linearly increases with J3, leading to the gradual replacement of the 1/2 plateau by the 5/7 one. On the other hand, the energy loss from E3 due to the phase transition from
the 5/7 state to the FM state increases when J3 is increased, and a larger h is needed to flip spins-down in the 5/7 state. As a result, the h-region with the 5/7 state is further enlarged at the expense of that with the FM state.
One may note that the configuration of the 5/7 state is filled by two kinds of hexagonal clusters (connected by red lines in figure 2(b)), in which every site belongs to one hexagon as the central one and to six other hexagons as the lateral one.
As a matter of fact, the configuration has been exactly proved to be the unique ground state by the ‘m-potential methods’, while a detail ground-state phase diagram is not available [30]. Figure 3 gives the calculated ground-state phase dia- gram at J2 = J1/10. Specifically, at the six multi-phase points (A, B, C, D, E, F) confirmed by the MC simulations, we deduce that the hexagonal clusters have minimal energy (equations of the calculation are presented in the appendix). In each trian- gular region (ABC, BCD, etc), the configuration of the ground state is constructed by the hexagonal clusters, which exist simultaneously in all three points. Thus, the collinear state, UUD state, FI state, and 5/7 state are exactly proved to be the ground states, respectively. The phase diagram is well con- sistent with the MC simulated one (the transition points are estimated from the magnetization jumps) at low T (T = 0.01) [31], although a tiny bifurcation is noticed due to the una- voidable thermal fluctuations in MC simulations, as shown in figure 2(d).
Subsequently, we paid attention to the effect of AFM J4
on the magnetization behaviors. The simulated magnetization curves for various J4 with L = 36 at (J2, J3, J5) = (0.2, 0.1, 0) and T = 0.01 are given in figure 4(a), and the corresp onding phase diagram in the J4-h plane for J4 > 0.025 is shown in figure 4(b) (the transition h at zero T obtained by comparing the energies of these phases are also given with the dotted lines). The phase diagram shows that the width of the 1/3 pla- teau is increased with the increase of J4, demonstrating that J4 Figure 2. (a) Magnetization curves for various J3. The parameters
are L = 42, T = 0.01 and (J2, J4, J5) = (0.1, 0, 0). (b) Spin configuration for the state with the plateau at 5/7 of MS. (c) The calculated E1, E2, E3 − 2, and Ezee + 3 as a function of h at T = 0.01 for J3 = 0.03. (d) MC-simulated (empty cycles) phase diagram at T = 0.01 in the h-J3 plane, and the exact ground-state boundaries are shown with the dotted lines.
Figure 3. Ground-state phase diagram in the (h, J3) plane at J4 = J5 = 0 obtained by the ‘m-potential methods’. The spin configurations of hexagons are also presented.
favors the UUD state. Furthermore, two additional plateaus at M/MS = 2/3 and 7/9 are subsequently stabilized with the increasing h for J4 > 0.025. The spin configurations of the 2/3 and 7/9 states are shown in figures 4(c) and (d), respectively.
Actually, the 2/3 magnetization step has been observed in sev- eral triangular antiferromagnets, and the spin configurations remain to be checked. Here, the spin structure of the 2/3 state uncovered in this work is the same as that predicted for the spin-1/2 Heisenberg system Cs2CuBr4 [19], demonstrating the common feature of this state, to some extent. Furthermore, for every spin-down in the 7/9 state, all the AFM J1, J2, J3, and J4 interactions are satisfied, and the saturation field h = 6 (J1 + J2 + J3 + 2J4) increases with the increase of J4. In addition, the slopes of the transition fields to the FI, 2/3, and 7/9 states versus J4 are with a fixed value, leading to the fact that the widths of the plateaus at 1/2, 2/3, and 7/9 are invariant for various J4 (J4 > 0.025).
Finally, the effect of AFM J5 is investigated. Figure 5(a) gives the magnetization curves for various J5 for L = 48 at (J2, J3, J4) = (0.2, 0.1, 0.07) at T = 0.01. In the h region just below the saturation field, the 7/9 plateau is gradually replaced by the 5/6 plateau when J5 is increased from 0.01. Similarly, for every spin-down in the 5/6 state (figure 5(b)), all the AFM J1, J2, J3, J4, and J5 interactions are satisfied. The widths of the 5/6 and 2/3 plateaus are almost the same, and show little dependence on J5. Furthermore, the width of the 1/2 plateau is broadened at the expense of the 1/3 plateau, indicating that J5 favors the 1/2 state more than the UUD state.
Following our earlier work, the transition fields at zero T between the different phases uncovered in our simulations can also be quantitatively obtained by comparing the energies of these phases [32]. In detail, the energy per site of the collinear state, the UUD state, the 1/2 state, the 2/3 state, the 5/7 state, the 7/9 state, the 5/6 state, and the FM state can be exactly
calculated based on their spin configurations of the unit cells, respectively, and are stated as follows:
= − − + − − Hcollinear J1 J2 3J3 2 J4 J5,
(2)
= − + − − + −
H J 3J J 2J 3J 1h
3 ,
UUD 1 2 3 4 5
(3)
= − H 3J 1 h
2 ,
FI 3
(4)
= + + + + − H J J 2J 2J J 2 h
3 ,
2/3 1 2 3 4 5
(5)
= + + + + −
H 9 J J J J J h
7 9 7
9 7
30 7
9 7
5 7 ,
5/7 1 2 3 4 5
(6)
= + + + + −
H 5J J J J J h
3 5 3
5 3
10
3 3 7
9 ,
7/9 1 2 3 4 5
(7)
= + + + + −
H 2J 2J 2J 4J 2J 5 h 6 ,
5/6 1 2 3 4 5
(8)
= + + + + −
HFM 3J1 3J2 3J3 6J4 3J5 h,
(9) Figure 5(c) shows the calculated local energies as a func- tion of h for these states at (J2, J3, J4, J5) = (0.2, 0.1, 0.07, 0.02). Both the 5/7 and 7/9 states are not stabilized in the whole h region, and H5/7 and H7/9 are not shown for simplicity.
Clearly, five transition fields are recognized and exactly cal- culated, well consistent with the simulated ones, respectively.
The width of the 2/3 plateau is calculated to be 6J3, which is irrelevant to J4 and J5, as confirmed in our simulations. To some extent, it is suggested that the 2/3 plateau has a special configuration and can be stabilized by further-neighbor AFM interactions.
Figure 4. (a) Magnetization curves for various J4 (J4 > 0.025). The parameters are L = 36, T = 0.01 and (J2, J3, J5) = (0.2, 0.1, 0). (b) MC-simulated (empty cycles) phase diagram at T = 0.01 in the h-J4
plane (J4 > 0.025), and the exact ground-state boundaries are shown with the dotted lines. Spin configurations in the (c) 2/3 plateau state, and (d) 7/9 plateau state.
Figure 5. (a) Magnetization curves for various J5. The parameters are L = 48, T = 0.01 and (J2, J3, J4) = (0.2, 0.1, 0.07). (b) Spin configurations in the 5/6 plateau state. (c) The local energies as a function of h for J5 = 0.02. (d) MC-simulated (empty cycles) phase diagram at T = 0.01 in the h-J5 plane, and the exact ground-state boundaries are shown with the dotted lines.
Despite many years of fruitful research, triangular anti- ferromagnets continue to offer us novel spin states and mag- netization behaviors which can be efficiently modulated by further-neighbor interactions, as revealed again in this work.
Some of these states have very close local energies, and are very sensitive to external fluctuations. As a result, even a weak additional interaction may have a significant influence on the magnetization behaviors of the system. Furthermore, all the neighboring interactions studied in this work are anti- ferromagnetic, which may be available in real materials with strong dipole–dipole interaction4. Thus, this work strongly suggests that the interactions between distant neighbors may contribute to the magnetization plateaus observed in real materials, although not all the spin states predicted here have been experimentally reported.
4. Conclusion
In conclusion, we have studied the magnetization behaviors and competing spin orders of the classical Ising model with interactions between distant neighbors on a triangular lattice by means of MC simulations. In addition to the well-known plateaus at 0, 1/3 and 1/2 of MS, those at 2/3, 5/7, 7/9 and 5/6 of MS are predicted when further-neighbor interactions are considered. These fascinating plateau states are discussed in detail and confirmed by the ground-state analysis. It is sug- gested that even weak distant neighboring interactions may have a significant effect on the magnetization behaviors, and may contribute to the experimentally reported magnetization plateaus.
Acknowledgments
This work was supported by the Natural Science Foun- dation of China (51322206, 11274094, 51332007), and the National Key Projects for Basic Research of China (2015CB921202), and the National Key Research Programme of China (2016YFA0300101), and the International Science
& Technology Cooperation Platform Program of Guangzhou (2014J4500016), and the Science and Technology Planning Project of Guangdong Province, China (2015B090927006), and Special Funds for the Cultivation of Guangdong Col- lege Students’ Scientific and Technological Innovation (pdjh2016b0138).
Appendix
In this section, we briefly present the process of the
‘m- potential’ calculations, and more details can be found in section 2 of [30]. Considering J1, J2 and J3 interactions, the Hamiltonian can be stated as:
( ) ( ) ( ) ( )
( ) ( ) ( )
∑ ∑
∑ ∑
σ σ σ σ
σ σ µ σ
= + + + + +
+ + + − +
H V a b a b V a b a b
V a b a b a b
ij
i j
ij
i j
ij
i j
i i
1 1 1 1 1 2 2 2 2 2
3 3 3 3 3 0 0
1 2
3
(A.1)
with an = 1, bn = 0 (n = 0, 1, 2, 3). In equation (A.1), the following notations are introduced:
= =
V J
a ,n 1, 2, 3,
n n
n2
(A.2) σ
= + =
Si an i bn, n 1, 2, 3
(A.3) and
( )
µ= + + +
a1 h a b J a b J a b J
6 6 6 .
0 1 1 1 2 2 2 3 3 3
(A.4) For an infinite size, the Hamiltonian can be rewritten as:
( )[ ( )
( )]
( )
( )
[ ( )]
∑
β β β σ σ σ σ σ σ σβ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ
µ
α α α σ α σ σ σ σ σ σ
= + + + + + +
+ + + + +
+ + + + + +
+ + +
− + + + + + + +
H V
V V 2
2
6 ,
i
i i i i i i i
i i i i i i i i i i
i i i i i i i i i i i i
i i i i i i
i i i i i i i
1
1 2 1 01
11 12
13 14
15 16
2 11 12
12 13
13 14
14 15
16 11 2 21
23 23
25 25
21 22
24 24
26 26
22
3 31 34
32 35
33 36
1 2 1 00
2 01 02
03 04
05 06
(A.5) with
σijn=an ijσ +b nn, =1, 2, 3
(A.6) In detail, the hexagonal configurations at the multiphase points can be obtained from the following equations, respectively:
(1) Point A:
α β
β
− + =
= = = =
h J J
J a
b 12 12 0, with
0, 0, 2, 3.
2 3
3 1 1
2
1 1
(A.7) (2) Point B:
α β
β
− + − = =
= = =
h J J J J
a b
6 18 24 0, with 0,
0, 3
7, 3.
1 2 3 3
1 1
2
1
1 (A.8)
(3) Point C:
α β
β
− + − = =
= = =
h J J J J
a b
6 18 24 0, with 0.03,
0, 7
3, 5
4.
1 2 3 3
1 1
2
1 1
(A.9) (4) Point D:
α α
β β
− − + =
= = = = =
h J J J
J b a
b
6 6 8 0, with
0.03, 2, 2
3, 0, 1
3.
1 2 3
3 1
2
1
2 1 2
2
(A.10)
4 For example, if one takes J1 = 0.5, J2 = 0.05, the lattice constant to be 1, and the magnitude of the dipole–dipole interaction D = 0.5, the effective parameters in this work will be (J1, J2, J3, J4, J5) = (1, 0.15, 0.0625, 0.027, 0.0185). The 2/3 plateaus should be stabilized in the spin model under these chosen parameters.
(5) Point E:
α
− − + = =
= =
h J J J J
a b
6 6 8 0, with 0,
0, 1.
1 2 3 3
1 1
1 (A.11)
(6) Infinite point F:
→ α
− − − = ∞
= =
h J J J h
a b
6 6 6 0, with ,
0, 1.
1 2 3
1 1
1 (A.12)
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