Wang–Landau simulation

(1)

Contents lists available atScienceDirect

Physics Letters A

www.elsevier.com/locate/pla

Magnetization plateaus of the frustrated Ising Shastry–Sutherland system: Wang–Landau simulation

W.S. Lin

^a

, T.H. Yang

^a

, Y. Wang

^a

, M.H. Qin

^a^,^b^,^∗

, J.-M. Liu

^c

, Zhifeng Ren

^b^,^∗

aInstituteforAdvancedMaterialsandLaboratoryofQuantumEngineeringandQuantumMaterials,SouthChinaNormalUniversity,Guangzhou510006,China bDepartmentofPhysicsandTcSUH,UniversityofHouston,Houston,TX 77204,USA

cLaboratoryofSolidStateMicrostructures,NanjingUniversity,Nanjing210093,China

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received9May2014

Receivedinrevisedform30June2014 Accepted30June2014

Availableonline5July2014 CommunicatedbyR.Wu

Keywords:

Magnetizationplateaus Wang–Landaualgorithm Shastry–Sutherlandmagnets

TheWang–LandaualgorithmisusedtostudythemagneticpropertiesoftheIsingmodelontheShastry–

SutherlandlatticeinordertounderstandtheinterestingmagnetizationplateausobservedinTmB4.The simulated resultsdemonstratethat theequilibriumstateofthemodelproducesonly the1/3 and1/2 magnetizationplateaus atlow temperaturesevenwhen therandom-exchange termorthe long-range interactionsare takenintoaccount.Thisconﬁrmsourearlierconclusion(Huangetal.,2013)[20] that thosefractionalplateausobservedinexperimentsmaybeduetothemagnetizationdynamics.

1. Introduction

During the past decades, a number of frustrated spin sys- temswhichexhibit interesting magneticbehaviorshaveattracted widespread interest. In special, amazing magnetization (M) plateaus have been experimentally reported in these systems such asthe triangular spin-chain system Ca3Co2O6 andShastry–

Sutherland (S–S) magnets[1,2].Up to now, it iswidely accepted that the multi-step magnetization behaviors in Ca3Co2O6 are caused by non-equilibrium dynamics, but those in S–S systems arestillcontroversial[3–5].

In1981,theS–Slattice(Fig. 1,only J1and J2areincluded)was theoreticallyintroducedbyShastryandSutherlandasa frustrated quantum antiferromagnetic (AFM) model with an exact ground state [6]. Subsequently, thismodel has attractedextensive atten- tion because a sequence of magnetization plateaus at fractional valuesofthesaturatedmagnetization(MS)havebeenexperimen- tallyobservedinSrCu2(BO3)2[7].Furthermore,afewofrare-earth tetraboridesRB4(R=^Tm,^Dy,^Tb,^etc.),^were^found^to^be^S–S^magnets exhibitingcomplex magnetic structures underapplied magnetic ﬁelds (B) [8–10]. For example, magnetization plateaus at M/MS=¹/2,1/7,1/9, etc., were reported in TmB4, and several

*

Correspondingauthors at: Department of Physics and TcSUH, University of Houston,Houston,TX77204,USA.

E-mailaddresses:[email protected](M.H. Qin),[email protected](Z. Ren).

theoreticalattemptswere reportedto uncoverthephysicalmech- anismsforsuchaphenomenon.

InTmB4,the largemagnetic momentsTm³⁺ (∼⁶.0

μ

B) arelo- catedon a latticethat is topologicallyequivalent to theS–S one.

Furthermore, subjected to strong crystalfield effects, TmB4 is of strong easy-axis anisotropy and can be reasonably described by classical Ising model. Theoretically, the magnetization process of the classicalS–S Ising modelwas extensivelystudied, andonlya singlemagnetization plateauatM/MS=¹/3 wasconfirmedinan intermediate Brangeforacertaintemperature(T)rangeandcou- plingconstants[11,12].Itwasstronglysuggestedthattheeffectof additionallong-range interactions should beconsidered to finally explaintheexperimentalresultsinTmB4 [13].Asamatteroffact, the Ising-like X X Z (the in-plane andout-of-plane exchange cou- plingsaredifferentfromeachother)modelontheS–Slatticewith theadditionallong-rangeinteractions wasvisitedusingthequan- tum Monte Carlo Method [14,15]. The experimentally reported M/MS=¹/2 plateauintheabsenceofthe M/MS=¹/3 onewas reproduced.Inourearlierwork,theeffectofthelong-rangeinter- actionswasextensivelystudiedbasedontheclassicalIsingmodel forTmB4[16].ItwasdemonstratedthattheAFM J3andtheferro- magnetic(FM) J4 couplings(depictedinFig. 1)resultedfromthe RKKY interactions maybe responsible forthe stabilizationofthe M/MS=¹/2 state[17].Inaddition,amodelbasedonthecoexis- tenceofspinandelectronsubsystemswasinvestigated,suggesting that the interaction between the electron and spin may play an importantroleinthemagnetizationprocessesin RB4[18,19].

(2)

Fig. 1.(Color online.)EffectivemodelontheShastry–Sutherland latticewiththe diagonalcouplingof J1, J2alongtheedgesofthesquares,andtheadditionalin- teractionsJ3and J4.

On the other hand, in frustrated spin systems, magnetization dynamics may play an essential role in the stabilization of the stateswithfractionalMsteps.Actually,thedynamicmagnetization processinTmB₄wasstudiedbymeansoftheGlauberdynamicsin ourearlierwork [20].The experimental 1/7,1/9 and 1/11 magnetizationplateaus inadditiontothe1/2 onecanbereproduced, suggesting that the magnetization dynamics may be responsible for the emergence of those plateaus. Furthermore, the effect of inhomogeneity on thestep-like magnetization feature was inves- tigated usingtherandomﬁeld method[21]. Itwas observedthat the plateauat M/MS=¹/3 split into three plateaus ina proper rangeoftheinhomogeneity.Thus,thereis anurgent needinun- derstandingthe groundstate orequilibrium statesbecause inho- mogeneityisalways inevitable inrealistic materials. However, an exactsolutionofthegroundstateproblemforthismodelwiththe random-exchange termisnotavailablesofar.Inaddition,thecon- ventionalMetropolisalgorithm mayalsofail torelax a trailstate into theequilibrium one atlow T in such a frustrated spin system[22].Therefore,itisstillanunsolvedissuedemonstratingthe equilibriumstateoftheS–SIsingmodelinordertofurtherunder- standthemagneticbehaviorsofTmB₄.

Fortunately, the Wang–Landau algorithm that is very power- fultoreachtheequilibriumstate hasbeensuccessfullyappliedto variousfrustrated systems sinceitwas proposed in2001[23,24].

For example, the equilibrium state of the AFM triangular Ising modelforCa3Co2O6hasbeenobtainedbytheWang–Landausimu- lation.ThecooperationbetweentheWang–Landausimulationand theGlauber dynamicsgavea ﬁrmconclusionthat themagnetiza- tionplateausinCa₃Co₂O₆ mustbecausedbythenon-equilibrium magnetization[5,25].Naturally,onemayquestionthatifthesame mechanismalsoworksforTmB₄.However, noWang–Landau cal- culationforTmB4 hasbeenreported,asfarasweknow.

In order to clarify this critical issue, we investigate the classical Ising model with the random exchange term on the S–S lattice using the Wang–Landau algorithm. The calculated results demonstrate that the equilibrium state of the model only pro- ducesthe1/3 and1/2 magnetizationplateaus,conﬁrmingtheear- lier conclusionthat the other stepsin TmB4 magnetization curve may arise from non-equilibrium states.In addition, the effect of the additional long-range interactions is also examined in this work.

The remainder of this paper is organized as follows: in Sec- tion 2the model andthe simulationmethod are described. Sec- tion 3 is attributed to the simulation results and discussion. At last,theconclusionispresentedinSection4.

i,j

=

A

·

RAMi,j

,

(2)

wheretheAFMexchangecouplings J1=^{1 and} ^J2=¹/2,Sirepre- sentstheIsingspinwiththevalue ±^1, ^B îsâppliedâlong^the+^z axis,RAMi,j isarandomnumberin[−¹,1]^,ând Â ^represents^the magnitudeof therandom exchangeterm. J₁ and J₂ are fixed to thevaluesthesameasthoseusedpreviously,makingouranalysis simple[21].

Ontheotherhand,whentheadditionallong-rangeinteractions areconsidered,themodelcanbedescribedas

H

=

^J1

diagonal

S_i

·

^Sj

+

^J2

edges

S_i

·

^Sj

+

^J3

ⁱ,j S_i

·

^Sj

+

J4

i,j

Si

·

Sj

−

B

i

Si

,

(3)

whereⁱ,jândⁱ,j^denote^the^summationsôverâll^pairsôn^the bondswith J3 and J4 couplingsasshowninFig. 1,respectively.

Our simulation is performed on a 12×^{12 S–S} ^lattice ^with period boundary conditions. In the Wang–Landau algorithm, the densityofstate(DOS) g(E,M)inenergyandmagnetizationspace (E istheenergyofagivenspin configurationoftheHamiltonian intheabsenceof B)needs tobeaccuratelyestimatedinorderto investigatethemagneticpropertiesofasystem.Inthisreport,the simulationprocedureischosentobethesameasthatstatedinthe pathbreakingworkofWangandLandau.Foreverypossible(E,M) states,wesetall entriestotheDOS g(E,M)=^{1 and}â^histogram RH(E,M)=^{0 at}^the^very ^beginning.^Then^we ^flip^spins^randomly tobeginrandomwalkintheenergyandmagnetizationspace.The transitionprobabilityfromstate(E1,M1)tostate(E2,M2)is p

(

E₁

,

M₁

→

^E2

,

M₂

) =

^min

g

(

E₁

,

M₁

)

g

(

E₂

,

M₂

) ,

1

.

(4)

Each time a state (Ei,Mi) is visited or retained, the histogram RH(Ei,Mi) (the number of visits) is accumulated and the exist- ingDOSismodifiedby g(Ei,Mi)=^g(Ei,Mi)f0.Inthisreport,the initialmodificationfactorissetto f0=exp(1),andthefactorisre- ducedtoafineroneaccordingtotherecipe f_i₊₁= ^f_i¹^/²^when^the histogrambecomes“flat”.Afterfinishingtheinitialrunweperform 27cycles,resultinginafinalmodificationfactorof1.00000000745.

Forsimplicity,the“ﬂat”histogramisveriﬁedwhenthehistogram for all possible (E,M) is not less than 80% of the average histogram.

Afterg(E,M)hasbeenobtained,wemaycalculatethethermo- dynamicandmagneticquantitiesatanyT andB.Forexample,the magnetization M(T,B)canbecalculatedfrom

M

(

T

,

h

) =

E,MM g

(

E

,

M

)

exp

(−

H

/

kBT

)

E,Mg

(

E

,

M

)

exp

( −

^H

/

kBT

) .

(5)

(3)

Fig. 2.(Color online.) (a) Evaluated DOS ln[^g(E,M)]of the model (1) withA=0 and curves in (b) are DOS at various negative energy valuesE.

Fig. 3.(Color online.)(a)MagnetizationM/MSasafunctionofTandB forA=0.

Spinconﬁgurationin(b)theup–up–downstate,and(c)the1/2 plateaustate.Solid andemptycirclesrepresenttheup-spinsandthedown-spins,respectively.

3. Simulationresultsanddiscussion

Fig. 2(a) showsthe simulated g(E,M) of the model (1) with A=^0. ^ln[^g(E,M)] ^shows ^a ^parabolic ^shape ⁱⁿ ^the ^low-energy range (E <0) and reaches its single maximum value at M=⁰ foragivenenergy.Furthermore,thegroundstate forsuch afrus- tratedsystemis highly degenerate,leading to a ratherlarge DOS atthelowest energy Emin (∼−^{72 in} ^this^case),^as^clearly ^shown inFig. 2(b).On theother hand,only two FM states(M=¹,−¹) arepossibleatthehighestenergy.Similarbehaviorshavebeenre- portedforsome other systemssuch asthetwo-dimensional AFM triangularmagnets[5].

Subsequently,MasafunctionofT andBiscalculated,andthe simulatedresultsareshownin Fig. 3(a).At low T (T<0.4), two stepsareobserved.AtT=⁰.02,asanexample,Mrapidlyreaches thefirstplateauatM/M_S=¹/3 whenB increasesfromzero,and thenswitchesto MS above B∼^3.^The ^sameâsêarlier^report,^the MS/3 plateau results from the up–up–down state (each triangle containstwoup-spinsandonedown-spin,asdepictedinFig. 3(b)) [11]. With the increasing of T, the steps can be progressively washedoutduetotheenhancedthermalactivation.WhenT isin- creasedtoabout0.5,theM₀/3 stepdisappearscompletelyandthe systemexhibits the paramagnetic property with the linear M–B relation. Thus, our simulation results show that the equilibrium state ofthe Ising modelfor the perfectS–S lattice produce two- step magnetization curve at low T, the same as those obtained fromothermethodssuchastherenormalization-groupapproach.

However,earliersimulationofthemodelincludingtherandom exchange term based on the Metropolis algorithm gave a rather differentmulti-stepmagnetizationcurve.Similarly,onemaydoubt

Fig. 4.(Color online.)Comparisonof(a)M/MSand (b)internalenergiesU asa functionofB calculatedfromthe Wang–Landaualgorithmwith computedusing theMetropolisalgorithmatT=⁰.02 forA=⁰.2.

Fig. 5.(Color online.)SimulatedcurvesofM/MSasafunctionofBforvariousAat T=⁰.02.

that this algorithm may fail to relax the system into the equilibrium state. This argumenthas been verified in oursimulation by comparing the magnetizationsand internal energies obtained by theWang–Landau andMetropolisalgorithms. IntheMetropo- lis simulation,theinitial1×¹⁰⁵ ^stepsâre ^discardedândânother 1×¹⁰⁵ ^steps âre ^retained^for^statistic âveraging. ^{Fig. 4(a)} ^shows

(4)

Fig. 6.(Color online.) MagnetizationM/MSas a function ofTandBat (a) J3=0.10 and J4= −0.15, and (c) J3=0.18 and J4= −0.25.

thesimulatedmagnetizationcurvesforA=⁰.2 atT=⁰.02,which exhibits a big discrepancy between the two simulations. In de- tails, the MS/3 step which is stable below B∼^{3 in} ^the ^Wang–

LandausimulationsplitsintothreestepsintheMetropolissimula- tion.Importantly,therelevantinternalenergiesobtainedfromthe Metropolis simulation are obviouslyhigher than the correspond- ingWang–Landausimulationresults,asclearlyshowninFig. 4(b).

Furthermore,itisnotedthatthetemperatureconcernedhereisso low that contribute littleto thevalue of thefree energies ofthe system.Asaconsequence,theaboveresultsallowustoarguethat theequilibriumstateofthemodelproducesonlythe1/3 magneti- zationplateauatlowT evenwhentherandom-exchangetermare takenintoaccount.

Moreover,theeffectoftherandom-exchangetermonthestep- likemagnetization feature hasalso beenexamined, andthesim- ulatedresultsarepresentedinFig. 5,wheremagnetization curves uponvarious A atT=⁰.02 aregiven. Itis obviousthatno addi- tionalstepscanbegeneratedbytherandom-exchangeterm.How- ever,theperfectup–up–downstatewiththe M/MS=¹/3 plateau canbe partiallydestroyednearthecritical B (0 and3)whenthe random-exchangetermisconsidered,leadingtothesmoothnessof themagnetizationcurve.Thus,thebordersbetweenthetwosteps becomemoreandmorefaintwiththeincreasingofA,asrevealed inoursimulation.

On the other hand, additional long-rangeinteractions are be- lieved to be responsible forthe stabilization ofthe M/MS=¹/2 plateauinearlierreports.Inthiswork,thedependenceofthemag- netic propertieson theadditional long-rangeinteractions J3 and J4 isalso examinedusing theWang–Landau simulation.Fig. 6(a) showsthe simulated magnetization curves atvarious T forAFM J3 =⁰.1 and FM J4 = −⁰.15. In addition to the plateaus at M/M_S=⁰,1/3, and 1,the experimentally reported M/M_S=¹/2 plateau is reproduced at low T. Furthermore, with the increasing T, both the 1/3 and 1/2 steps are progressively washed out due to thermal ﬂuctuation. However, the disappearing T of the 1/2 step(∼⁰.6)ismuch higherthan thatofthe1/3 step (∼⁰.3), demonstratingthat the1/2 plateaustate is morestablethanthe 1/3 one.Asrevealed inearlierwork,morelocal AFM J₃ andFM J4 interactions canbe satisﬁedin the1/2 plateau state (spinar- rangementconsistingofalternativeAFMandFMstripes,asshown in Fig. 3(c)) than those in theup–up–down state, leading to the stabilization of the 1/2 plateau accompanied by the destabiliza- tionofthe1/3 plateau[16].Asaresult,withtheincreasingofthe magnitudes ofthe AFM J3 andFM J4,the 1/3 plateau are pro- gressivelyreplacedbythe1/2 one.For J3=⁰.18 and J4= −⁰.25, theup–up–downstate is completelyreplacedby the 1/2 plateau one, resulting in the stabilization of the 1/2 plateau in the ab- senceofthe1/3 one,asclearlyshowninFig. 6(b).Thus,theeffect ofthelong-rangeinteractions ontheequilibriummagnetizationis examinedinourWang–Landausimulation.However,theadditional

plateausatM/M_S=¹/7,1/9,etc.,reportedinexperimentscannot bestabilizedbytheadditionallong-rangeinteractions.

Anyway,it iswellknownthat afrustrated spinsystemcanbe easilylocalizedinoneofthesemetastablestatesatlowT,andthe timeavailableexperimentallymaynotbesuﬃcientforthesystem to relaxto theequilibriumstate.Althoughthe magneticproperty ofTmB4 hasattractedattentionsformanyyears,butthereisstill a bigchallengeindevelopinga reliableexperimental approachto uncover thegroundstatesorequilibriumstatesofthissystem. In this report, we used Wang–Landau simulation to clearly demonstrate the equilibriummagnetization ofthe theoretical modelfor TmB4. No fractional magnetization plateaus other than those at M/MS=¹/3 and1/2 canbereproducedevenwheninhomogene- ityoradditionallong-rangeinteractionsaretakenintoaccount.As a consequence,thecooperationbetweentheWang–Landau simu- lationandtheGlauberdynamicsgivesastrongsuggestionthatthe emergenceofthoseplateausinTmB4 maybecausedbythemag- netizationdynamics.

4. Conclusion

In thisreport, we havestudied theequilibriummagnetization oftheS–SsystemTmB₄basedontwo-dimensionalIsingmodelus- ing theWang–Landaumethod.Thesimulatedresultsdemonstrate thattheequilibriumstateofthemodelonlyproducesthe1/3 and 1/2 magnetization plateaus at low temperatures even when the random-exchange term or the long-range interactions are taken into account. Thus, the present simulation indicates that those fractional plateaus atsmall magnetization values reportedin ex- perimentsmaybeduetothemagnetizationdynamics.

Acknowledgements

ThisworkwassupportedbytheTexasCenterforSuperconduc- tivity of the University of Houston (TcSUH) andthe U.S. Depart- mentofEnergy underContractNumberDOEDE-FG02-13ER46917 (DE-SC0010831),theNationalNaturalScienceFoundationofChina (11204091,11274094,51332007,and61106061),andtheNational KeyProjectsforBasicResearchofChina(2011CB922101).

References

[1]H.Kageyama,K. Yoshimura,K.Kosuge,M. Azuma,M. Takano,H.Mitamura, T. Goto,J.Phys.Soc.Jpn.66(1997)3996.

[2]H.Kageyama, K.Yoshimura, R.Stern, N.V.Mushnikov,K. Onizuka,M. Kato, K. Kosuge,C.P.Slichter,T.Goto,Y.Ueda,Phys.Rev.Lett.82(1999)3168.

[3]Y.B.Kudasov,Phys.Rev.Lett.96(2006)027212.

[4]X.Y.Yao,S.Dong,J.-M.Liu,Phys.Rev.B73(2006)212415.

[5]M.H.Qin,K.F.Wang,J.-M.Liu,Phys.Rev.B79(2009)172405.

[6]B.S.Shastry,B.Sutherland,PhysicaB&C108(1981)1069.

[7]R.W.Smith,D.A.Keszler,J.SolidStateChem.93(1991)430.

[8]K.Siemensmeyer,E.Wulf,H.J.Mikeska,K.Flachbart,S.Gabani,S.Matas,P. Pri- puten,A.Efdokimova,N.Shitsevalova,Phys.Rev.Lett.101(2008)177201.

(5)

[9]S.Yoshii,T.Yamamoto,M.Hagiwara,S.Michimura,A.Shigekawa,F.Iga,T.Tak- abatake,K.Kindo,Phys.Rev.Lett.101(2008)087202.

[10]R.Watanuki,G.Sato,K.Suzuki,M.Ishihara,T.Yanagisawa,Y.Nemoto,T.Goto, J.Phys.Soc.Jpn.74(2005)2169.

[11]M.C.Chang,M.F.Yang,Phys.Rev.B79(2009)104411.

[12]Y.L.Dublenych,Phys.Rev.Lett.109(2012)167202.

[13]H.Cencarikova,P.Farkasovsky,P.Puchala,ActaPhys.Pol.A126(2014)46.

[14]T.Suzuki,Y.Tomita,N.Kawashima,Phys.Rev.B80(2009)180405(R).

[15]T.Suzuki,Y.Tomita,N.Kawashima,Phys.Rev.B82(2010)214404.

[16]L.Huo,W.C.Huang,Z.B.Yan,X.T.Jia,X.S.Gao,M.H.Qin,J.-M.Liu,J.Appl.Phys.

113(2013)073908.

[17]J.J.Feng,L.Huo,W.C.Huang,Y.Wang,M.H.Qin,J.-M.Liu,Z.F.Ren,Europhys.

Lett.105(2014)17009.

[18]P.Farkasovsky,H.Cencarikova,ActaPhys.Pol.A126(2014)44.

[19]P.Farkasovsky,H.Cencarikova,S.Matas,Phys.Rev.B82(2010)054409.

[20]W.C.Huang,L.Huo,J.J. Feng,Z.B.Yan,X.T.Jia,X.S.Gao,M.H. Qin,J.-M.Liu, Europhys.Lett.102(2013)37005.

[21]M.H.Qin,G.Q.Zhang,K.F.Wang,X.S.Gao,J.-M.Liu,J.Appl.Phys.109(2011) 07E103.

[22]D.P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics,CambridgeUniversityPress,Cambridge,England,2008.

[23]F.Wang,D.P.Landau,Phys.Rev.Lett.86(2001)2050.

[24]F.Wang,D.P.Landau,Phys.Rev.E64(2001)056101.

[25]Y.B.Kudasov,A.S.Korshunov,V.N.Pavlov,D.A.Maslov,Phys.Rev.B78(2008) 132407.

[26]H.Shinaoka,Y.Tomita,Y.Motome,Phys.Rev.Lett.107(2011)047204.