Bond Counting Monte Carlo for the 2-D
Ising Model in an External Field
H. G.Dias and J. Florenio
Departamento deFsia,Universidade Federalde MinasGerais,
30.161-970 BeloHorizonte,MG,Brazil
Reeivedon15August,2000
WeproposeanimprovementofaMonteCarlomethoddesignedtotreattheIsingmodelinaeld
[C.LieuandJ.Florenio,J.LowTemp. Phys.89,565(1992)℄. Themethodinvolvestheounting
of bonds linking neighboring like-spins and yields the degeneray of the system's energystates,
hene the partition funtion. There is no aeptane-rejetion proedure and all the randomly
generatedongurations are kept. Thesamplingdepends ongeometryonly,so results ofagiven
runanbe usedfor all temperatures and energyparameters. Inorderto understandthe virtues
andinadequaiesofthemethod,weobtainedexatresultsforsmalllatties. WendthataMonte
Carlorunmustbefollowedby aGaussiantinorderto aountproperlyforthe rareeventsnot
reordedinthesampling. Finally,wealsoestablished boundsfor theloationofthe peakfor the
speifheatoftheIsingmodelinamagnetield intwodimensionsfor severalvaluesoftheeld
inthethermodynamilimit.
ThethermodynamisoftheIsingmodelinoneand
two dimensions has been known for quite some time
[1, 2℄. In 1D, there is no long-range order exept at
zerotemperature. Ontheotherhand,in2Dthemodel
showsatransiton fromaparamagneti to a
ferromag-neti state as the temperature is loweredthrough T
,
the ritial temperature [3℄. The model an help the
understanding of many phenomena observed in some
magnetisystems. Inaddition,itonstitutesthe
start-ingpointformanyother systemsfoundinnature.
MonteCarloomputersimulationhasbeenusedin
thelast fewdeadesto studythemodelin dimensions
D > 2[4℄. Inspite of the inherent limitations of the
lattiessizesusedin omputersimulations,resultsan
often beextended to the thermodynami limit. Most
Monte Carlo simulationsare basedon the Metropolis
algorithm, whih employs a rejetion-aeptane
pro-edure toenfore detailedbalane [5℄. Despiteits
su-esses,those simulationsfaesomediÆulties. For
in-stane,near thetransitiontemperatureittakesalong
omputertimetoequilibratethesystemduetoritial
slowingdown. Anotherpointtoonsideristhatdistint
runsareneededtoobtainthetemperaturedependene
of a given thermodynami quantity. The
determina-tionoftheloationofnarrowpeaksofthermodynami
quantitiesisnotveryaurate,whihistheaseinthe
viinity of the phase transition. Even though several
methods havebeen proposed to dealwith these
prob-lems, we present in this work an improvement of the
doesnotinvolvetherejetion-aeptanefeatureofthe
Metropolismethod,andthatmightproveusefulinthe
studyof theIsingmodel.
TheMonte Carlomethod onsidered hereinvolves
randomsampling ofspinongurations,withallthem
takeninto aount. The method allowsforan
estima-tion of the degeneraies of the energy states, leading
diretlytothepartitionfuntionandtheensuing
ther-modynami funtions. Its main advantageis that the
sampling depends on geometry only. A single run is
validforalltheenergyparametersJ andB,as wellas
the temperature T. The method yields analytial
ex-pressions, sothere is nodiÆultyin dealingwith
nar-row peaks. Theongurationsare generated
indepen-dently, thus ritialslowingdown isnot afator. We
shallpresenthereresultsforthe2D Isingmodelin an
externaleld. Howeverthemethodanbeusedtodeal
withhigherdimensionallatties.
Considerthe Ising model in a magneti eld on a
squarelattieofN =LLsites,
H= J X
i;j (
i;j
i;j+1 +
i;j
i+;j ) B
X
i;j
i;j (1)
with periodi boundary onditions
i+L;j+L
=
i;j ,
where
i;j
antakeoneofthetwovalues,1(up-spin)or
1(down-spin),B istheexternalmagnetieldandJ
(FMorAFM)istheouplingparameterbetween
neigh-boringspins.
m nearestneighbor pairs(bonds)betweenthemselves.
TheenergyofsuhstateisthenE
0 +p 1 +m 2 ,where E 0
=N(B 2J)istheenergyofthestatewhereallthe
spins are pointingdown,
1
= 8J 2B is the energy
needed to iponedown-spinoriginally surrounded by
neighboring down-spins, and
2
= 4J is the
orre-tion in ase there exists a single up-spin loated at a
siteneighboringtheippedspin. Hene,thepartition
fution anbeexpressed as
Z=e
( 2NJ+NB) N X p=0 2p X m=0 C m p x m p ; (2)
where = e 1
and x = e 2
. The quantity C m
p
givesthenumberofstatesontainingpup-spins
form-ing m bonds. Thedetermination of C m
p
, theonly
un-knowninthepartitionfuntion,isaverydiÆult
om-binatoris problem. TheknowledgeofC m
p
forany
lat-tie size ompletely solvesthe thermodynamis of the
2D Isingmodelin aeld.
Theoreofthemethod isto determineanaverage
of these oeÆientsC m
p
by meansof sampling. Thea
priori probabilityofpikingupany givenstatewithp
up-spins is ( N
p )
1
, regardlesstheenergy of that state.
ThereforethedeterminationofC m
p
ispurelygeometri.
Sine eah onguration has the same probability to
ourand theyare statistially independent, the
vari-ane of C m
p
dereases as(Np MC
) 1
, where Np MC
is
the numberofelementsinthe sampleof stateswithp
up-spins.
We ask the omputer to generate a onguration
with p up-spins distributed randomly on the lattie,
ountthenumberof bonds mformed in suh
ongu-ration, andstorethe result. Theproessis repeateda
number of times and, by ountinghow many
ongu-rationsyieldedmbonds,weobtaintheongurational
average C m
p
. The proper numberof states is ensured
by thenormalization ondition P m C m p =( N p ). Thus,
the determination of C m
p
is simply geometrial in
na-ture,henethenumerialvaluesofthemodel
parame-tersandofthetemperaturearenotneededinaMonte
Carlorun.
Byexatlyanalyzing the m-dependene of C m
p for
severalvaluesofpforsmalllattiesizes,wendthatit
anbettedwith aGaussian. As anillustration,Fig.
1depits aGaussian t (dashedline)to thetheexat
valuesofC m
p
(irles)ona77-lattieintheasewhere
p=21. WealsondthataGaussiantbeomesbetter
asthelattiesizeinreases. (Itisnotsogood forvery
smalllattiesizes,suhas22,33,et.) Giventhat
theoverwhelmingmajorityofthestateshavevaluesof
mdistributedaroundtheGaussianpeak,random
sam-plingwillpikupmostlythosestates. Thestatesinthe
Gaussiantailswillbeunderounted,ifnotnegleted
en-ontributionsfromthosestatesandsimilarstatesfrom
Gaussian tails of dierent p's may go unounted. In
short,simplerandomsampling favorsountingenergy
statesfrom around thepeaks of C m
p
, while negleting
statesfromitstails.Inordertoaountforthosestates,
whih are unlikely to be hosen byrandom sampling,
weproposetoimprovethemethodbyusingaGaussian
ttotheresultsobtainedfromrandomsamplingofthe
ongurations. Anotherpointtoonsider isthatsome
statesloatedinthetailoftheGaussiantneverour.
Forexample,intheaseofp=4andL=5orgreater,
ongurationswithm=5,6,7and8neverour,
be-ause we an have at most four bonds with just four
up-spinsinalattiesizewhereL5. Sinethe
parti-tionfuntion,givenbyEq. (2), involvesa summation
over m up to 2p, we have to exlude suh impossible
states. We have found a way to trak these
ongu-rationsand itisatually asimplematter to getrid of
them. Hene,after thesimulationisdone,weperform
aGaussiantandnormalizeC m
p
aftertheexlusionof
thoseimpossiblestates(wesetC m
p
=0forsuhstates).
Figure1. Exatnumberof spinongurations ona 77
lattieversusthenumberthebondsformedbyneighboring
up-spins. In the gure shown, the number of up-spins is
p=21.
Theseimprovementsofthemethodturnoutto
pro-dueamoreaurateresults. Wetesteditforthe66
and the 8x8 lattieswith J =1 and B =0. Forthe
66lattie, weperformed96 MonteCarloruns,eah
with 100000 samples pervalue of p. The exatvalue
forthetemperaturewhihmaximizesthespei heat
is 2:389709. We obtaina value of 2:330:06 for the
peaktemperature. Wendequallysatisfatoryresults
Figure2. Speiheat usingour MonteCarlo methodfor
the66and88latties,whereJ=1andB=0.
Finally, we determined bounds on the loation of
thepeaksofthespeiheatofthemodelin the
pres-ene of aeld, in thethermodynami limit. It is well
known that unlike the ase where B = 0, where the
speiheatdivergesattheritialtemperature,there
isnodivergeneofthespei heatwhenB6=0.
Nev-ertheless, the spei heat has a peak entered at a
given temperature, whih we denote by T
p
, the peak
temperature. First, we determineT
p
for small lattie
sizes, upto thesizeN =66. Fig. 3showsthepeak
temperatureasafuntionof1=N,thereiproalofthe
lattie size. The ases (J;B) = (1;0) and (1;0:5) are
displayedinthegure. Byextrapolation,weaninfer
boundsfortheloationofthepeaktemperaturesinthe
thermodynamilimit. Theresults,forseveralvaluesof
B are givenin TableI. We also nd that asthe eld
inreases the upper and lower bounds approah eah
other.
TableI.BoundsforT
p
,theloationofthespei
heatpeak,forseveralvaluesofB.
B lowerbound upperbound
0.0 -x- 2.34322
0.1 2.53993
-x-0.2 2.76417
-x-0.5 3.08842 3.13671
1.0 3.61675 3.63054
1.5 4.09641 4.10180
Figure3. Temperatureatwhihthespeiheatispeaked
vsthereiproalofthesystemsize,forB=0andB=0:5.
The oupling energy J = 1. The dashed lines are justa
guidetotheeye.
We need, of ourse, to test the method on larger
latties in order to make more preise preditions on
theinueneoftheeldBonthethermalpropertiesof
theIsingmodel. Theextensionofthemethodtohigher
dimensionsisstraightforward.
Aknowledgments
This work was partially supported by CNPq,
FAPEMIG,andMCT(Brazilianagenies).
Referenes
[1℄ E.Ising,Z.Physik31,253(1925).
[2℄ L.Onsager,Phys.Rev.64,117(1944).
[3℄ B.M. MCoyand T.T. Wu,Thetwo dimensionalIsing
model (HarvardUniv.Press,Cambridge,Mass.,1973).
[4℄ See, e.g., Computer Simulation in Condensed Matter
Physis, Eds. D.P. Landau, S.P. Lewis, and H.-B.
Shlutter(Springer-Verlag,NewYork,2000).
[5℄ N. Metropolis, W.W. Rosenbluth, M.N. Rosenbluth,
A.H. Teller, and E. Teller, J. Chem. Phys. 21, 1087
(1953).
[6℄ C.Lieu and J.Florenio, J.LowTemp. Phys.89, 565
