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One-andtwo -dimensio nal

Isingmodel

StatisticalPhysicsSeminarbyProf .Wolschin

AntonKabelac

08.11.2021

Abstract

TheIsingm odelisoneof themostimportant modelsi nstatist icalp hysics.Itisanal ytically exactlysolvableinone andtwodimensions.Inthisex tended summaryof aseminar presentation,theone-andtwo-dimensional Ising modelsarepre sen tedandmainaspects suchasphasetr ansit ionsarediscussed .Furtherthehistoricalbackground andmoder n applicationsoftheIsingmodelareoutlin ed. 1

1Ba sicideaofthemodel

TheIsingmo delisatheoreti calmodelins tatist ica lphysicsthatwasoriginallydevelop ed todes cribeferromagnetism.Asystemof magneticparticlescanbemodeledas alinear chaininonedi mensio noralatti ceintwodimension,withonemol eculeo ra tomateach inthemo delb yadiscretevariable ! i .Each'spin'!canonl yhaveavalu eof! i =±1.

Thetwop ossiblevaluesi ndicatewhethertwosp ins!

i and! j arealigna ndthus parallel i j =+1)oranti-parallel(! i j =!1). momentsarealigned.If themagnetic momentspoints inoppositedirec tionsthey are considertobeinah igherenerget icstate. Duetot his interactionthesy stemtendsto alignallmagneti cmoment sinonedirectioninordert ominimiseenergy.Ifn early all magneticmomentsp ointinthesamedirec tionthearrangementofmolec ulesb ehav eslike amacroscopicmagnet. toadis ordered state.Aferromagnetabove thecriticaltemperature T C isina disordere d state.IntheIsin gmo delthiscorrespon dstoarand omdistributionof thespinvalues.

Belowthecritical temperatureT

C (nearly)allspinsare aligned,e venintheabs ence ofan externalappliedmagnetic fieldH.Ifweheatupacool edferromagnet,themagnet ization vanishesatT C andtheferromagnet switchesfrom anorderedto adisorderedstate.This isaphas etransition ofsecondorde r.

2Hi storicalBackground

TheIsing modelwasinvente din1920byWilhel mLenz,whi chiswhyitsalsoreferred toasthe Lenz-I singor Ising-Lenzmodel.LenzwasaGerman phys icist,alsonotablefor hisapplic ationoftheLaplace-Runge-Lenzvect or.Hewa satRosto ckUniversityin1920, butthefol lowingye arhewasappointedordinarypr ofessoratHamb urg.O neofhisfir st studentswasErnstIsing. 2

Figure1:ErnstIsi ng

Isingstartedhi sdissertationonthe investigation offer- romagnetism,summarizedinashortpaper writtenin

1924andpublished [1]in1925. Isingcarried outanex-

actcalculationfor thespecial caseofa one-dimensional lattice.Hisanalysisshowed thatt herewasnophase transitiontoaferromagneticord ereds tat eatanyfi- nitetemp erature.Isingwronglypredicted,thataphase transitionalsodoesnotoccurinh igherdimensio ns.

Thisleadtoin itialreject ionoft heLenz-Isingmodel

formthephysi calcommun ity,includingIsinghim- self.

WhenWerner Heisenbergproposed hisowntheoryof

ferromagnetismin1928,hesaid: "Isingsucceededin showingthatalsotheassumptio nofdir ectedsu ciently greatforcesb etween twoneighboringatomsofachain isnotsu cienttoexplain ferromagnetism." [2] TheLenz-Isin gmodelbecamemorerelevan tin1936,whenRudolfP eierlshowedt hatthe

2dvers ionmusthaveaphasetra nsitionatfini tetemperatu re[ 3].Finallyin1944 thetwo-

dimensionalIsingmode lwithoutanexternalfie ldwassolvedanalytica llybyLarsOns ager byat ran sfer-matrixmethod.

3One dimensio nalIsingmodel

Theone-di mensionalIsingmodelisanchainofspins.Eachspin !canonl yhaveadiscr ete valueof! i =±1.Thei ndeximar ksthepositionof thes pininthechain. 3

Figure2:Isingcha in

LikeIsingdid in1924[1]wewilltake alook atthes imples tpos siblecaseof theone- dimensionalIsingmodel. Ourgoalistoinvestigate ifaphasetransitionoc cures ,explaining spontaneousmagnetisationandthu sferromagnetism.Wewillintrod ucetwocon ditions. •Noex ternalmagneticfieldH •Eachspin canonlyinteract with itsneighbori ngspin. Wewilll aterrefer tothesecondcon ditionasonlynea rsnei ghboringi nteractions(NN).

Theinter actionstrengthbetweentwospins!

i and! i+1 ischarac terisedbythecoupling strengthJ.Th eHamiltoni anHofsuch asystemisthangiv enby (1)H=!J i j withthenears neighboringsum .ForasystemwithN tot latticesites andtwo possible! i -valuesateachlatti cesite ,atotalnumb erof2 Ntot possibleconfigurationso fthe arrangementofparticles exists.Summingoverall possibleconfigurations ithenyieldsthe partitionsum Z: Z= {i} e !!H (2) 1 =±1 2 =±1 N =±1 e J(" 1 2 2 3 3 4 Inorde rtosimplifyeq.(2 )wein troduceanewvariableµ i i i+1 ,des cribingwhether twoneighb ouringspinsareparalleloranti-parall el.TheHamiltoni an(1)andt hepartition sum(2)can nowberewr ittenwit houtaNNsum : (3)H=!J i i "Z=2· e J N i=1 i 4 Thefactor of2inthepartition functi on arises fromthetw opossibleconfigurationsforth e firstspininthe chain. Inth ethermody namiclimit(N>>1)weca nsimpl ifythepa rtitionfunction:

Z=2·

e J N!1 i=1 i (4) =2· 1 =±1 2 =±1 N!1 =±1 e

J(µ

1 2 N!1 =2· 1 =±1 2 =±1 N!2 =±1 e

J(µ

1 2 N!2 N!1 =±1 e Jµ N!1 (e J +e !!J

Withtherel ation

e J +e !!J =2cosh("J)itfollows:

Z=2·

1 =±1 2 =±1 N!2 =±1 e

J(µ

1 2 N!2

2cosh("J)(5)

=2[2cosh("J)] N!1 N"1 #[2cosh("J)] N Thisisourfin alresu ltforthep artitionfunc tionoftheone-dimen sionalIs ingmodelw ithout anexte rnalfield. Newtwewan ttosho wthatinthiss imple casenophaset ransitionata finite temperature occurs.Theaverages pininthec hainisgivenby: (6)Inor dertosimplefyeq.( 7)weint roduceadi erentcoupling constantJ i foreach spin pair. i i+j 1 Z i i+j e !!H (8) 1 Z 1 =±1 i =±1 i+1 =±1 N =±1 i i+j e !(J 1 1 2 +J 2 2 3 +J 3 3 4 5

Nextwerew riteth eproduct!

i i+j interm sofbondsrathert hansp ins.Notethatthe productofanyspinwith itself( ! i i =1)isalwaysequaltoone. i i+j i

·1·...·1·!

i+j (9) i i+1 i+1 i+2 i+j!2 i+j!1 i+j!1 i+j i i+1 i i+1 i+2 i+1 i+j!2 i+j!1 i+j!2 i+j!1 i+j i+j!1

Combiningeq.(8)andeq.(9)yield s:

i i+j 1 Z i i+j e !!H (10) 1 Z [2cosh("J 1 )·...·2sinh("J i )·...·2sinh("J i+j!1 )·...·2cosh("J N!1

Theparti tionfunctionZfordi

erentcoupling constantJ i foreach spinpair canbecalcu- latedanaloguetoeq.(1- 5): i i+j cosh("J 1 )·...·sinh("J i )·...·sinh("J i+j!1 )·...·cosh("J N!1 cosh("J 1 )·...·cosh("J i )·...·cosh("J i+j!1 )·...·cosh("J N!1 (11) j m=1 tanh("J i+m!1

Ifwe gobackto aconst antcouplingcon stantJ

i =Jtheres ultbecomes: (12)=[tanh("J)] j Allthat 'slefttodoistoloo katthetemper atured epend entmagn etisat ionMofthesystem

M=mN(13)

M 2 =m 2 N 2 2 =m 2 N 2 lim j#$ i i+j >=0$T>0 withthemagnet icmomento feachspinm,thenumberofspinsinthesystemNandthe averagespin. Becausetanh("J)%1the expres sionineq.(12)becomeszeroforla rgej.Theonly exceptionisatT=0,wherethetanh("J)diverges.Sotobepreciseonehav etosay thataphas etransition intheone-dimens ionalIsingmode ldoe snoto ccuratafinite temperature. 6

4Tr ansferMatrix

Thenextq uestionweare goingtoansweriswhathapp enstoou rsystemi fweapplyan externalmagneticfieldHth atcaninteractwiththemag neticmomentmofeach spin.The

Hamiltonianofsuchasystembecom es:

(14)H=!J i j !mH i i Itis helpfult oassumeperiodicboun darycondi tions(! N+1quotesdbs_dbs20.pdfusesText_26

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