Numerical Studies on Antiferromagnetic Skyrmions in Nanodisks by Means of A New Quantum Simulation Approach

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by  Zhaosen Liu

Numerical Studies on Antiferromagnetic Skyrmions in Nanodisks by Means of A New
Quantum Simulation Approach
Zhaosen Liua,b∗, Hou Ianb†
aDepartment of Applied Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China
bInstitute of Applied Physics and Materials Engineering, FST, University of Macau, Macau
We employ a self-consistent simulation approach based on quantum physics here to study the
magnetism of antiferromagnetic skyrmions formed on manolayer nanodisk planes. We find that if
6
thedisk is small and theDzyaloshinsky-Moriya (DM) interaction is weak, a single magnetic vortex
1
may be formed on the disk plane. In such a case, when uniaxial anisotropy normal to the disk
0
2 plane is further considered, the magnetic configuration remains unchanged, but the magnetization
isenhancedinthatdirection,andreducedinothertwoperpendicularorientations. Verysimilarly,a
n
weak external magnetic field normal to thedisk plane cannot obviously affect the spin structure of
a
thenanodisk; however, when it is sufficiently strong, it can destroy theAFM skyrmion completely.
J
Ontheotherhand,byincreasingDMinteractionsothatthediskdiameterisafewtimeslargerthan
0 the DM length, more self-organized magnetic domains, such as vortices and strips, will be formed
2
in the disk plane. They evolve with decreasing temperature, however always symmetric about a

geometricaxisofthesquareunitcell. Wefurtherfindthatinthiscaseintroducingnormalmagnetic
]
l anisotropy givesrisetothere-construction ofAFMsingle-vortex structureorskyrmion onthedisk
al plane, which provides a way to create and/or stabilize such spin texturein experiment.
h
- PACSnumbers: 75.40.Mg,75.10.Jm
s
e
m
I. INTRODUCTION Skyrmionshavebeenpredictedtoappearintheground
. stateofdopedantiferromagneticinsulators[18–20]. How-
t
a ever, it is difficult to identify these isolated skyrmions.
m The concepts of skyrmions were originally introduced
Neutron scattering, for example, would not be an effec-
by a particle physicist, Tony Skyrme, to describe the lo-
- tive probe, since these skyrmions do not form a lattice,
d calized, particle-like structures in the field of pion parti-
whereas their signatures on transport may be screened
n cles in the early 1960s [1]. About 30 years later, Bog-
o danov and Yablonskii theoretically predicted [2] that by the insulating characterof the carriers[21]. Basedon
c their experimental observation, Raiˇcvi´c et al. concluded
theycouldexistinmagnetswhenachiralDzyaloshinsky-
[ that the low-temperature magnetic and transport prop-
Moriya (DM) interaction [3–5] is present. Indeed, it was
1 laterfoundinexperimentsthatmagneticskyrmionsexist erties of the AFM La2Cu1−xLixO4 providedthe first ex-
v perimentalsupportforthepresenceofskyrmionsinAFM
0 in helical magnets, such as MnSi and Fe1−xCoxSi [6–8], insulators.
and DM interaction favors canted spin configuration [6–
7
16]. Most skyrmions found in helimagnets were induced Recently,Huangetal. [22]simulatedthecreationpro-
1
5 by an external magnetic field at low temperatures [6– cessofskyrmioninatwo-dimensional(2D)antiferromag-
0 8,17]. Forexample,Heinzeetal. [10]observedasponta- netic system to investigate the dynamics of the created
. neous atomic-scale magnetic ground-state skyrmion lat- skyrmions, and observed stable skyrmions even at long
1
tice in a mono-layer Fe film at a low temperature about time scales. So far, many researchers have done exten-
0
6 11K.However,Yu et al. [7]obtaineda skyrmioncrystal sivestudiesonthestaticpropertiesof2DFMskyrmions.
1 near room-temperature in FeGe with a high transition Therefore,itisobviouslynecessaryandmeaningfultoin-
: temperature (280 K) by applying a magnetic field. vestigate how the magnetism of the 2D AFM skyrmions
v
areinfluenced by externalmagnetic field, Heisbenerg ex-
i Sofar,ferromagnetic(FM)skyrmionshavebeeninten-
X change, anisotropic and DM interactions, so as to find
sivelyinvestigatedboththeoreticallyandexperimentally.
r However, the DM interaction is more generally found in ways to create or stabilize the AFM skyrmion in experi-
a ments. For the purpose, this work has been done.
antiferromagnetic (AFM) materials than ferromagnetic
materials. Most recent experiments on FM skyrmions In a just finished work [23], we investigated the mag-
rely on the presence of the interfacial DM interaction to netic and thermodynamic properties of mono-layer nan-
stabilize skyrmions. In contrast, bulk DM interaction is odisks with the co-exitance of DM and FM Heisenberg
more prevalent in antiferromagnets [3, 5], and they are interactions by means of a new quantum simulation ap-
considerably more abundant in nature than ferromag- proach we develop in recent years [24, 25]. We found
nets. there that the chirality of the single magnetic vortex on
a small nanodisk is only determined by the sign of DM
interaction parameter, no matter an external magnetic
field is absent or applied normal to the disk plane, how-
∗Email: [email protected] evertheappliedmagneticfieldperpendiculartothedisk-
†Email: [email protected] plane is able to stabilize the vortex structure and induce
2
skyrmions [6–8, 17]. energy [24, 25]. Thus, as a computational code imple-
In the present work, the new quantum simulation ap- mented with this algorithm runs, all magnetic moments
proachisappliedtoAFMmono-layernanodiskswiththe in the sample are rotatedand their magnitudes adjusted
co-existence of Heiseinberg and DM interaction as well. bythelocaleffectivemagneticfieldtominimizethetotal
We find that for small disk of weak DM interaction, sin- (free) energyofthe whole nanosystemspontaneouslyac-
gle AFM skyrmion is always formed on the disk plane. cordingto the lawofleast(free)energy,sothatthe code
Further inclusion of uniaxial anisotropy or weak exter- can finally converge down to the equilibrium state auto-
nal magnetic field normal to the disk plane causes no matically without the need to minimize the total (free)
obvious change in the spin configuration, but they do energy elaborately in every simulation step.
enhance the magnetization in that direction and reduce All of our recent simulations are started from a ran-
the other two in-plane components. However, if this ap- dommagneticconfigurationandfromatemperaturewell
plied magnetic field is strong enough, the in-plane AFM abovethemagnetictransitiontemperatureT ,thencar-
M
Skyrmion will be completely destroyed. Moreover, by riedoutstepwisedowntoverylowtemperatureswithan
increasing the DM interaction, more self-organizedmag- iteration step ∆T < 0. At any temperature, if the dif-
neticdomainswillappearonthediskplane. Theyevolve ference (|hS~′i−hS~ i|)/|hS~ i| between two successive iter-
i i i
with varying temperature, but always symmetric about ationsfor everyspinis lessthan averysmallgivenvalue
a geometric axis of the square unit cell. In this case, an τ0, convergency is believed to be reached.
uniaxialmagnetic anisotropynormalto the disk plane is
able to force the multi-domain structure merge to form
a single AFM vortex. In another word, the anisotropy III. CALCULATED RESULTS
can induce and/or stabilize the AFM skyrmion, which
the experimentalists may be very interested.
A. Simulations for Nanodisk without Uniaxial
Anisotropy and External Magnetic Field
II. MODELING AND COMPUTATIONAL
To investigate the effects of DM interaction, the mag-
ALGORITHM
netic anisotropy was neglected in simulations at the be-
ginning. And to visualize the spin configuration clearly,
The Hamiltonian of this sort of nanosystems can be weconsideredaverytinyroundmono-layernanodisk,its
written as [10, 26–34] radius R = 10a, where a is the side length of the square
crystal unit cell, and the spins on the disks are assumed
H= −1 J S~ ·S~ −D ~r ·(S~ ×S~ ) to be antiferromagnetically coupled uniformly. We per-
2Pi,j6=ih ij i j ij ij i j i formed simulations with the SCA approachby assigning
2
−K S~ ·nˆ −µ g B~ · S~ , (1) J to -1K andD to 0.1 K,respectively. To avoidmisun-
APi(cid:16) i (cid:17) B S Pi i derstanding,weindicateherethatallparametersusedin
this paper are scaled with Boltzmann constant k .
where the first and second terms represents the Heisen- B
berg exchange and DM interactions with strength of J Under the DM interaction, magnetic vortex is formed
ij
andD betweeneverypairofneighboringspinssittingat on the nanodisk, and owing to the antiferromagnetic
ij
thei-andj−thsites,respectively,thethirdtermdenotes Heisenberg interaction, each pair of neighboring spins
theuniaxialanisotropyalongnˆ,assumedtobenormalto order oppositely both in-plane and out-plane below the
the disk plane here, and the last one is the Zeeman en-
transition temperature T ≈ 2.65 K as shown in Figure
M
ergy ofthe system within externalmagnetic field B~. For 1(a,b).
simplicity,weconsiderinthecurrentworkaroundmono-
Figure 2(a,b) display our calculated thermally aver-
layer nanodisk consisting of S = 1 spins which interact
antiferromagnetically only with their nearest neighbors aged hSzi, hSxi and hSyi for the nanodisk in the ab-
uniformly, that is, J = J and D = D, across the senceofexternalmagneticfield. TheDMinteractionhas
ij ij
whole disk plane. In our model, the spins are quantum induced out-plane magnetic moments [35–38], which is
operators instead of the classical vectors. Since S = 1, muchstrongerthantheothertwocomponentsatalltem-
the matrices of the three spin components are given by
peratures. The three components decay monotonously
0 √2 0 0 √2 0 with increasing temperature until the transition point
Sx= 21 √2 0 √2 , Sy = 21i −√2 0 −√2(2) TM ≈ 2.65 K, and the saturated value of hSzi is approx-
 0 √2 0   0 √2 0  imately 0.85 at very low temperatures, much less than
the maximum value S = 1.
1 0 0
 
Sz = 0 0 0 Todescribethedetailedspinconfigurationonthenan-
0 0 1  odisk, two new quantities are introduced and defined as
−
A = |hS i|/N and A = hS i2+hS i2/N for the
z z c xy x y c
p
respectively. out- and in-plane components, respectively. Here N (r)
c
Oursimulationapproach,whichisfacilitatedbyaself- is the spin number on the circle of radius r around the
consistentalgorithm,socalledastheSCAapproach,was disk center. Figure 2(c,d) display their variations with
assumed to be based on the principle of the least (free) changing r = |N | at four different temperatures. Natu-
x
3

12 (a) T <= 2.5 K, J = -1 K, D = 0.1 K 12 (b) T <= 2.5 K, J = -1 K, D = 0.1 K
8 8
4 4
Ny Ny
0 0
-4 -4
-8 -8
-12 -12
-10 -8 -6 -4 -2 0 2 4 6 8 10 -12 -8 -4 0 4 8 12

Nx Nx
Figure 1. The (a) xy,and (b) z components of calculated spin configurations projected onto the
nanodisk plane. Here J = -1K, D =0.1 K,and R=10a, respectively.
rally, the larger the radius, the more spins on the circle. to a temperature well above T as observed in experi-
M
In the inner region of the disk, A is very weaker than ments [6–8, 17]. So we naturally wonder if this rule still
xy
A . Thatis,thereinthespinaremainlyorientedantifer- holds true in the case of AFM nanodisks.
z
romagneticallyoutofthe plane,butslightlycantedfrom For the purpose, by assuming a magnetic field exerted
the normal. Until r < 6a, while the radius increases, Az normal to the disk plane at T = 0.25 K, we did sim-
decreasesbutAxy growsgradually. Thatis,asrincreases ulations for the AFM nanodisk with the spin structure
the spins are rotated by the effective magnetic field to- calculatedatthattemperatureintheabsenceofexternal
wards the plane, so that finally, within the marginal re- magnetic field as the input data. In this circumstance,
gion of the disk the magnitudes of Az and Axy become the nanosystem is able to sustain the external influence
comparable. tomaintainthevorticalstructureuntilB =0.3Teslaas
z
The total free energy F, total energy E, magnetic en- showninFig.4(a). However,whenB isfurtherincreased
z
tropy SM and specific heat CM of this sort of canonical to 0.4 Tesla,the spiralstructure is thus completely over-
magnetic systems can be calculated with following for- come, whereas the in-plane components of the spins still
mulas orderantiferromagneticallyinthe[-1,1,0]directionasde-
picted in Fig.4(b). This behavior of the nanosystem is
F =−kBT logZN, E =−∂∂β logZN , easy to understand. When the external magnetic field
along the z direction is strong enough, the spins are un-
E
SM = T +kBlogZN, CM =T (cid:0)∂∂STM(cid:1)B , (3) able to alignantiferromagneticallyin the z directionany
longer,as a result, the in-plane components cannot form
successively,where β =1/(k T)andZ is the partition antiferromagnetic vortices either. That is, the forma-
B N
function of the whole system. Figure 3(a,b) display the tion of an FM (AFM) vortex in the disk plane depends
F, E, S andC curvesobtainedbymeansofthe SCA strongly on the presence of a spatial region wherein the
M M
approachfortheAFMnanodisk. TheslopesofF,E and spins order ferromagnetically (antiferomagnetically) in
S curveschangesuddenlynearT ,whicharethesigns the normal direction.
M M
of phase transition. However, the C curve now varies
M
smoothly around T , in contrastto the sharp peaks ob-
M
servedintheCM curvesnearTM’sofbulkmagnets. This C. Effect of Uniaxial Anisotropy
fact suggests that the phase transition behavior of the
nanosystem has been strongly modified by its finite size
So far, we have not considered the influence of mag-
and the spiral DM interaction.
neticanisotropy. Tostudyitseffects,itisnowassumedto
be along the z-directionto do further simulations. Since
other parameters are kept unchanged, as expected, be-
B. Effect of External Magnetic Field
low T , hS i has been enhanced by the anisotropy, but
M z
both hS i and hS i are suppressed for the same reason,
x y
InourprevioussimulationsforFMnanodiskswithDM sothat the maximumof|hS i| is now increasedto 0.984,
z
interaction [23], we found that an applied magnetic field but that of |hS i| reduced to 0.100 as seen in Fig.5(a).
x,y
is able to induce or stabilize the in-plane vortical spin In addition, all these curves changes gradually, though
structures, so that the chiral configuration can maintain notsmoothlyduetorelativelyweakDMinteraction,and
4

1.0 0.4
(b)
0.8 (a) 0.3
0.6
0.2
0.4
<S>z00..02 J = -1 K, D = 0.1 K <S>x,y 00..01 J = -1 K, D < =S x0>.1 K
<Sy>
-0.2 -0.1
-0.4
-0.2
-0.6
-0.3
-0.8
-1.0 -0.4
0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
T(K) T(K)

0.8
1.0 0.25 K 1.5 K 0.7 (d)
0.8 (c) 0.6 J = -1 K, D = 0.1 K
0.25 K 1.5 K
0.5
2 K 2.5 K
A 0z.6 2 K 2.5 K Axy 0.4
0.4 0.3
0.2
0.2 J = -1 K, D = 0.1 K
0.1
0.0 0.0
-0.1
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10 -8 -6 -4 -2 0 2 4 6 8 10
Nx Nx
Figure 2. Calculated spontaneous (a) hSzi, and (b) hSx,yi for themono-layer AFM nanodisk as
functions of temperature; (c) Az, and (d) Axy as functions of the distances from thecenterof the
nanodisk at four different temperatures. HereR = 10a, J = -1K, and D = 0.1 K, respectively.
asinglemagneticvortexisfoundonthe nanodisk,which 0.3 K, but keeping other parameters unchanged. The
prevails in the whole magnetic phase as depicted in Fig- calculatedhS i, hS i and hS i in the absence of external
z x y
ure 5(b). magnetic field are plotted in Figure 6. These magneti-
zation curves are not smooth in the whole low tempera-
ture range, reflecting the fierce competition between the
D. Effect of DM Interaction Strength Heisenberg and DM interactions. The sudden changes
appearingaroundT ≈1.25Kand0.7Kespeciallyinthe
hS iandhS icurvesindicatethatphasetransitionshap-
To describe the multi-domain structures, a new quan- x y
pen nearby, leading to formations of different magnetic
tity named DM length has been introduced and defined
structures. According to the theory just described, now
as ζ =J/D which is related to the size of self-organized
ζ = J/D ≈ 3.333, and 2R > ζ, so it is expected more
structures, where the distance between two unit grids is
self-organized magnetic domains will appear in the low
defined as the unity [26]. When the Monte carlomethod
temperature region. Above1.4 K,a single magnetic vor-
is employed, each grid contains n×n atomic sites. We
tex, as shown in Figure 7(a), occupies the whole disk.
adopt this theory by replacing the grid with a spin, and
However, below T = 1.2 K, a few magnetic domains
will see how the theory works. Thus, as the disk scale is
appear. The spin configuration evolves with decreasing
a few times lager than ζ in the unit of lattice parameter
temperature until T = 0.6 K, where we observe a very
a, more magnetic structures, such as strips and vortices,
symmetricmagneticstructure: averticalstripappearing
will be formed in the disk plane. This condition can re-
exactlyinthe middle oftwoAFM vortices,andthis pat-
alized by either increasing DM interaction or the lattice
tern remains unchanged down to very low temperature,
size.
as displayed in Figure 7(b). The two vortex centers are
To test the idea, we then carried out simulations for
approximately 11a apart, three self-organized domains
the nanodisk by increasing the DM interaction to D =
5

14 10
-24 (a) 0 12 (b)
-25 J = -1 K, D = 0.1 K -5 J = -1 K, D = 0.1 K 8
F(J/MOL)---222876 F E ---211050 E(J/MOL) C(J/MOL K)M1068 CSMM 46 S(J/MOL K) M
-29 4
-25 2
-30
2
-31 -30 0 0
-32 -35
0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
T( K) T(K)
Figure 3. Calculated (a) free energy and energy, (b) magnetic entropy and spefic heat per mole
of thespins. Here R = 10a, J = -1 K,and D = 0.1 K,respectively.

12 (a) T = 0.25 K, B = 0.3 Tesla, J = -1 K, D = 0.1 K 12 (b) T = 0.25 K, B = 0.4 Tesla, J = -1 K, D = 0.1 K
8 8
4 4
Ny Ny
0 0
-4 -4
-8 -8
-12 -12
-12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12
Nx Nx
Figure 4. Calculated spin configurations projected onto thedisk plane at T = 0.25 K, when an
external magnetic field (a) Bz = 0.3 Tesla, and (b) Bz = 0.4 Tesla, is applied normal to thedisk
plane. HereJ = -1 K, D = 0.1 K and R=10a, respectively.
are involved between, thus the averaged distance of two axialanisotropynormaltothediskplanewillrecoverthe
neighboringstructuresisabout3.67a,slightlylargerthan single vortex structure.
ζ due to the influence of the disk boundary. Therefore,
our simulated results agree well with the adopted gird
theory.
E. Joint Effects of Uniaxial Anisotropy and Strong
DM Interaction To test this idea, we carried out simulations by us-
ing the parameters given in Figure 7, but increasing
As described above, a strong DM interaction usually the anisotropy strength K from zero to 0.1 K. This
A
leads to a multi-domain structure on the disk plane. On anisotropic interaction effectively suppresses the strong
the other hand, as described above, the formation of an DM interaction, consequently, the three components of
in-plane AFM skyrmion requires the spins to order also the magnetization change smoothly and fade gradually
antiferromagnetically in the normal direction, and the with increasing temperature below T as shown in Fig-
N
magnetic anisotropy perpendicular to the disk-plane has ure 8(a), foretelling the appearance of a single magnetic
such an effect. Therefore, we expect that when a strong vortexonthediskplane. Thispredictionisconfirmedby
DMinteractionispresentinthenanosystem,whichgives the spin structure that is stable in the whole magnetic
rise to multi-domain configuration, introducing the uni- phase, as displayed in Figure 8(b).
6

1.2 0.15
(a) 12 (b) T < TM, J = -1 K, D = 0.1 K, KA = 0.1 K
0.8 0.10
8
0.4 J = -1 K, D = 0.1 K, KA = 0.1 K 0.05
<S>z0.0 <<SSxz>> <Sy> 0.00 <S> x,y Ny 04
-0.4 -0.05 -4
-0.8 -0.10 -8
-1.2 -0.15 -12
0.0 0.5 1.0 1.5 2.0 2.5 3.0 -10 -5 0 5 10
T(K ) Nx
Figure 5. Calculated (a) spontaneous magnetization for themono-layer AFMnanodisk as the
functions of temperature, and (b) spin configurations projected onto thexy-planein the
temperatureregion belowTM. HereR=10a, J =-1K,D =0.1KandKA =0.1K,respectively.

0.8 0.6
(b)
0.6 (a)
0.4
0.4
0.2
0.2 J = -1 K, D = 0.3 K > J = -1 K, D = 0.3 K
<S>z 0.0 <Sx,y0.0 <Sx>
<Sy>
-0.2
-0.2
-0.4
-0.4
-0.6
-0.8 -0.6
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 00..00 00..55 11..00 11..55 22..00 22..55 33..00 33..55
T(K) T(K)
Figure 6. Calculated spontaneous (a) hSzi, and (b) hSx,yi for themono-layer antiferromagnetic
nanoodisk asthefunctionsoftemperature. HereR=10a, J =-1K,andD =0.3 K,respectively.
IV. CONCLUSIONS AND DISCUSSION evolveswithvaryingtemperature,butisalwayssymmet-
ric abouta geometricaxisof the squareunit cell. In this
case,amoderateuniaxialmagneticanisotropynormalto
We have successfully carried out simulations for AFM the disk-plane is able to suppress the DM interaction, so
skyrmions on manolayer nanodisks by means of a new thatthemulti-domainsmergestoasingleAFMskyrmion
quantum computational approach. We find that if the that occupies the whole disk plane below the transition
disk size is small and the DM interaction weak, single temperature.
magnetic vortex, or a skyrmion is formed on the disk
We have adopted a gird theory [26] to describe the
plane. The uniaxial magnetic anisotropy normal to the
multi-domain structures on the nanodisks. The sizes of
diskplanedoesnotaffectthissinglemagnetictextureevi-
the magnetic domains and the averagedistance between
dently,itcanonlyenhancethemagneticmomentsinthat
a pair of them agree approximately with this modified
direction,butreducetheothertwoin-planecomponents.
theory, as already achieved in our recent simulations for
Aweakexternalmagneticfieldappliednormaltothedisk
FM nanodisks [23].
planeproducessimilareffects;however,ifitissufficiently
strong, it will completely destroy the magnetic vortex. We would like to stress finally that our simulation ap-
By increasing the DM interaction strength so that the proachisbasedonquantumphysics–thespinsappearing
disk diameter is a few times larger than the DM length, in the Hamiltonian aretreatedas quantum operatorsin-
more self-organizeddomains,such asvorticesand strips, steadof classicalvectors,the thermalexpectation values
can be formed on the disk plane. The spin configuration of all physical quantities are calculated with quantum
7

12 (a) T = 2.7 K, 1.8 K, 1.5 K, J = -1 K, D = 0.3 K 12 (b) T = 0.3 K, 0.6 K, J = -1 K, D =0.3 K
8 8
Title 4 4
Y Axis 0 Ny 0
-4 -4
-8 -8
-12 -12
-12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12
Nx Nx
Figure 7. Spin configurations projected onto thexy-planecalculated at (a) T = 2.7, 1.8, 1.5 K,
and (b) T = 0.6, 0.3 K.Here R = 10a, J = -1 K, and D = 0.3 K, respectively.

1.0
(a) 0.3 (b) T < = 2.7 K, J = 1 K, D = 0.3 K, K =0.1 K
0.8 12
0.6 0.2
8
0.4
<S>z00..02 J = -1 K<,S Dx> = 0.3 K, K <AS =y> 0.1 K 00..01 <S>x,y Ny 4
0
-0.2 <Sz>
-0.1
-0.4 -4
-0.6 -0.2 -8
-0.8
-0.3
-12
-1.0
-12 -8 -4 0 4 8 12
0.0 0.5 1.0 1.5 2.0 2.5 3.0
T(K) Nx
Figure 8. Calculated (a) magnetization for themono-layer AFM nanodisk as thefunctions of
temperature, (b) spin configurations projected onto the xy-planein thetemperature region below
TM,in theabsence of externalmagnetic field. HereR = 10a, J = -1K,D = 0.3 Kand KA =0.1
K,respectively.
formulas. Consequently, the computational program is Acknowledgments
able to run self-consistently, and quickly converge down
to equilibrium state of the magnetic system automati-
Z.-S. Liu is supported by National Natural Science
cally. Frequently, we find that the computational code
Foundation of China under grant No. 11274177 and
only takes a few loops, or even one loop, to converge in
University of Macau, H. Ian by the FDCT of Macau
low temperature region. And especially, the approach
under grant 013/2013/A1, University of Macau under
has produced good agreements with experimental and
grants MRG022/IH/2013/FST and MYRG2014-00052-
our theoretical results [25, 39].
FST,andNationalNaturalScience FoundationofChina
under Grant No. 11404415.
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