Random-field Ising model: Insight from zero-temperature simulations

0.59 MB·English

by  P. E. Theodorakis

CondensedMatterPhysics,2014,Vol.17,No4,43003:1–14
DOI:10.5488/CMP.17.43003
http://www.icmp.lviv.ua/journal
Random-field Ising model: Insight from
zero-temperature simulations
P.E.Theodorakis1,N.G.Fytas2
5
1 1DepartmentofChemicalEngineering,ImperialCollegeLondon,SW72AZ,London,UnitedKingdom
0 2AppliedMathematicsResearchCentre,CoventryUniversity,Coventry,CV15FB,UnitedKingdom
2

n ReceivedOctober2,2014
a
J
0 Wpeerfeonrlmigehdteantszoemroetecmritpicearlaatuspree.cWtseocfothnesitdherretew-doimdieffnesrieonnta,lin(dte=rm3)sroafntdhoemfi-efildelddiIsstirnibgumtioond,evlefrrosimonssimofumlaotidoenls,
1
namelyaGaussianrandom-fieldIsingmodelandanequal-weighttrimodalrandom-fieldIsingmodel.Byimple-

mentingacomputationalapproachthatmapstheground-stateofthesystemtothemaximum-flowoptimization
]
n problemofanetwork,weemploythemostup-to-dateversionofthepush-relabelalgorithmandsimulatelarge
n ensemblesofdisorderrealizationsofbothmodelsforabroadrangeofrandom-fieldvaluesandsystemssizes
- V L L L,whereLdenoteslinearlatticesizeandLmax 156.Usingasfinite-sizemeasuresthesample-
s = × × =
i to-samplefluctuationsofvariousquantitiesofphysicalandtechnicalorigin,andtheprimitiveoperationsofthe
d push-relabelalgorithm,wepropose,forbothtypesofdistributions,estimatesofthecriticalfieldhc andthe
.
t criticalexponentνofthecorrelationlength,thelatterclearlysuggestingthatbothmodelssharethesameuni-
a
versalityclass.AdditionalsimulationsoftheGaussianrandom-fieldIsingmodelatthebest-knownvalueofthe
m
criticalfieldprovidethemagneticexponentratioβ/νwithhighaccuracyandclearoutthecontroversialissue
- ofthecriticalexponentαofthespecificheat.Finally,wediscusstheinfinite-limitsizeextrapolationofenergy-
d
andorder-parameter-basednoisetosignalratiosrelatedtotheself-averagingpropertiesofthemodel,aswell
n
o asthecriticalslowingdownaspectsofthealgorithm.
c
Keywords:random-fieldIsingmodel,finite-sizescaling,graphtheory
[

PACS:05.50.+q,75.10.Hk,64.60.Cn,75.10.Nr
1
v
8
1. Introduction
3
3
2 Therandom-fieldIsingmodel(RFIM)isoneofthearchetypaldisorderedsystems[1–3],extensively
0 studied due toits theoretical interest, as well asits close connection to experiments inhard [4, 5] and
.
1 softcondensedmattersystems[6].ItsbeautyisthatthemixtureofrandomfieldsandthestandardIsing
0 modelcreatesrichphysicsandleavesmanystillunansweredproblems.TheHamiltoniandescribingthe
5 modelis
:1 H =−J σiσj− hiσi, (1)
v i,j i
〈X〉 X
Xi whereσi 1areIsingspins,J 0isthenearest-neighbor’sferromagneticinteraction,andhi areinde-
=± >
pendentquenchedrandomfields.
r
a TheexistenceofanorderedferromagneticphasefortheRFIM,atlowtemperatureandweakdisorder,
followedfromtheseminaldiscussionofImryandMa[1],whenthespacedimensionisgreaterthantwo
(d 2) [7–11]. This has provided us with a general qualitative agreement on the sketch of the phase
>
boundary, separating the ordered ferromagnetic phase from the high-temperature paramagnetic one.
The phase-diagram line separates the two phases of the model and intersects the randomness axis at
thecriticalvalueofthedisorderstrengthhc,asshowninfigure1.Suchqualitativesketchinghasbeen
commonlyusedfortheRFIM[12–14]andclosedformquantitativeexpressionsarealsoknownfromthe
earlymean-fieldcalculations[15–17].However,itisgenerallytruethatthequantitativeaspectsofphase
diagramsproducedbymean-fieldtreatmentsprovideratherpoorapproximations.
©P.E.Theodorakis,N.G.Fytas,2014 43003-1
P.E.Theodorakis,N.G.Fytas
h / J
hc
R
P
F
0
T / J
Figure1.Schematicphasediagramandrenormalization-groupflowoftheRFIM.Thesolidlineseparates
theferromagnetic(F)andparamagnetic(P)phases.Theblackarrowshowstheflowtotherandomfixed
point(R)atT 0andh hc,asmarkedbyanasterisk.
= =
Thecriteriafordeterminingtheorderofthelow-temperaturephasetransitionanditsdependence
on the form of the field distribution have been discussed throughout the years [15–19]. Although the
view that the phase transition of the RFIM is nowadays considered to be of second order [20–25], the
extremelysmallvalueoftheexponentβcontinuestocastsomedoubts.Moreover,aratherstrongdebate
withregardtotheroleofdisorder,i.e.,thedependence,ornot,ofthecriticalexponentsontheparticular
choiceofthedistributionfortherandomfieldsandthevalueofthedisorderstrength,analogouslytothe
mean-fieldtheorypredictions[15–17],wasonlyrecentlyputonadifferentbasis[26].Currently,eventhe
well-knowncorrespondenceamongtheRFIManditsexperimentalanalogue,thedilutedantifferomagnet
inafield (DAFF),hasbeenseverelyquestioned byextensivesimulations performedonbothmodels at
positive-andzero-temperature[27].Inanycase,thewholeissueofthemodel’scriticalbehaviorisstill
underintenseinvestigation[20–25,28–43].
AlreadyfromtheworkofHoughtonetal.[44],theimportanceoftheformofthedistributionfunction
inthedeterminationofthecriticalpropertiesoftheRFIMhasbeenemphasized.Infact,differentresults
havebeenproposedfordifferentfielddistributions,liketheexistenceofatricriticalpointatthestrong
disorder regime of the system, present only in the bimodal case [15–17, 44]. Following the results of
Houghtonetal.[44],Mattis[45]reexaminedtheRFIMintroducinganewtypeofatrimodaldistribution
1 p
P(trimodal)(hi) pδ(hi) − [δ(hi h) δ(hi h)], (2)
= + 2 − + +
µ ¶
whereh defines thedisorder (field) strength and p (0,1). Clearly, for p 1 one switches tothe pure
∈ =
Isingmodel,whereasforp 0thewell-knownbimodaldistributionisrecovered.Ingeneralterms,the
=
trimodaldistribution(2)permitsaphysicalinterpretationasadilutedbimodaldistribution,inwhicha
fraction p of the spins are not exposed to the external field. Thus, it mimics the salient feature of the
Gaussiandistribution
1 h2
P(Gaussian)(hi)= p2πh2expÃ−2hi2!, (3)
forwhichasignificantfractionofthespinsareinweakexternalfields.Mattissuggestedthatforapar-
ticularcase,p 1/3,equation(2)maybeconsideredtoagoodapproximationastheGaussiandistribu-
=
tion[45]. Thisin turnindicated that the two models should be inthe same universality class. Further
studiesalongtheselines,usingmean-fieldandrenormalization-groupapproaches,providedcontradict-
ingevidenceforthecriticalaspectsofthep 1/3modelandalsoproposedseveralapproximationsofits
=
phasediagramforarangeofvaluesofp[46–48].However,noneofthesepredictionshasbeenconfirmed
bynumericalsimulationsuptillnow,thusremainingambiguous,duetotheapproximatenatureofthe
mean-field-typeofthemethodsused.
Thescopeofthepresentworkistoshedsomelighttowardsthisdirectionbyexaminingseveralcriti-
calfeaturesofthephasediagramoftheRFIMatd 3,usingbothdistributionsdescribedaboveinequa-
=
43003-2
Random-fieldIsingmodel:Insightfromzero-temperaturesimulations
tions(2)and(3).Inparticular,inthefirstpartofourstudyweprovidenumericalevidencethatclarify
thematchingbetweenthetrimodal(p 1/3)andGaussianmodelsandwegiveestimatesforthecritical
=
fieldhc andcriticalexponentνthatcompareverywelltothemostaccurateonesinthecorresponding
literatureof theRFIM.Inthe second partof our studyweconcentrate onthemost studied caseofthe
GaussianRFIM,forwhichwepresentascalinganalysisofcriticaldatafortheorderparameterandthe
specificheat,i.e.,dataobtainedatthebestknownestimateofthecriticalfieldhc.Ouranalysispointsto
averysmall,butnon-zero,valueforthemagneticexponentratioβ/ν,andacriticalexponentα 0−,in
→
goodagreementwithexperimentalpredictions[49,50].Wealsodiscusstheinfinite-limitsizeextrapola-
tionofenergy-andorder-parameter-basednoisetosignalratiosrelatedtotheself-averagingproperties
ofthemodel,aswellassometechnicalaspectsoftheimplementednumericalmethod.
Ourattemptbenefitsfrom:(i)theexistenceofrobustcomputationalmethodsofgraphtheoryatzero
temperature(T 0)thatallowustosimulateverylargesystemsizesanddisorderensembles,necessary
=
for an accurate investigation of the delicate propertiesdiscussed above, (ii)classical finite-size scaling
(FSS)techniques,and(iii)anewscalingapproachthatinvolvestheanalysisofthesample-to-samplefluc-
tuationsofvariouswell-definedquantities.Inparticular,sample-to-samplefluctuationsandtherelative
issue of self-averaging have attracted much interest in the study of disordered systems [51]. Although
ithasbeenknownformanyyearsnowthatfor(spinandregular)glassesthereisnoself-averagingin
the ordered phase [52], for random ferromagnets such a behavior was first observed for the RFIM by
Dayanetal.[53]andsomeyearslaterfortherandomversionsoftheIsingandAshkin-Tellermodelsby
WisemanandDomany[54].Theselatterauthorssuggested aFSSansatzdescribingtheabsenceofself-
averaging and the universal fluctuations of randomsystems near criticalpoints that was refined on a
morerigorousbasisbyAharonyandHarris[55].Eversince,thesubjectofbreakdownofself-averagingis
animportantaspectofseveraltheoreticalandnumericalinvestigationsofdisorderedspinsystems[56–
68].Infact,EfratandSchwartz[69]showedthatthepropertyoflackofself-averagingmaybeturnedinto
ausefultoolthatcanprovideanindependentmeasuretodistinguishtheorderedanddisorderedphases
ofthesystem.Inviewofthisobservation,wediscusshereanotherusefulapplicationofthefluctuation
propertiesofseveralquantitiesofthesystemtoobtaininformationontheground-statecriticalityofthe
RFIM.
Therestofthepaperisorganizedasfollows:Inthenextsectionwedescribethegeneralframework
behind the mapping of the RFIM to the corresponding network, outline the numerical approach, and
provideallthenecessarydetailsofourimplementation.TherelevantFSSanalysisthatshowstheequiva-
lenceofbothdistributionsunderstudyintermsofthecriticalexponentνofthecorrelationlength,using
anapproachbasedonthesample-to-samplefluctuationsofthemodel,ispresentedinsection3.Then,in
section4wefocusourattentiononthemoststudiedcaseoftheGaussianmodelandweprovideestimates
forthemagneticexponentratioβ/νandthecriticalexponentαofthespecificheat,viathescalingofthe
order parameter and bond energy, respectively, at the best known estimate of the critical field value.
Wealsoinvestigatetheself-averagingpropertiesofthemodelatcriticality,usingproperlydefinednoise
tosignalratiosandweprovideanestimatefortheexponentz thatdescribesthecriticalslowingofthe
algorithmused.Finally,wesynopsizeourfindingsinsection5.
2. Simulation protocol
Asalreadydiscussedextensivelyintheliterature(seereference[70]andreferencestherein),theRFIM
capturesessentialfeaturesofmodelsinstatisticalphysicsthatarecontrolledbydisorderandhavefrus-
tration.Suchsystemsshowcomplexenergylandscapesduetothepresenceoflargebarriersthatseparate
severalmeta-stablestates.Whensuchmodelsarestudiedusingsimulationsmimickingthelocaldynam-
icsofphysicalprocesses,ittakesanextremelylongtimetoencountertheexactgroundstate.However,
therearecaseswhereefficientmethodsforfindingthegroundstatecanbeutilizedand,fortunately,the
RFIMisonesuchcase.Thesemethodsescapefromthetypicaldirectphysicalrepresentationofthesys-
tem,inawaythatextradegreesoffreedomareintroducedandanexpandedproblemisfinallysolved.
Byexpanding the configuration spaceand choosing properdynamics, thealgorithm practicallyavoids
theneedofovercominglargebarriersthatexistintheoriginalphysicalconfigurationspace.Anattractor
state in the expended space is found in time polynomial in the size of the system and when the algo-
43003-3
P.E.Theodorakis,N.G.Fytas
rithmterminates,therelevantauxiliaryfieldscanbeprojectedontoaphysicalconfiguration, whichis
theguaranteedgroundstate.
Therandomfieldisarelevantperturbationatthepurefixed-point,andtherandom-fieldfixed-pointis
atT 0[7–10].Hence,thecriticalbehavioristhesameeverywherealongthephaseboundaryoffigure1,
=
and we canpredictitsimply bystaying at T 0and crossing the phaseboundary ath hc. Thisis a
= =
convenientapproach,becausewecandeterminethegroundstatesofthesystemexactlyusingefficient
optimizationalgorithms[20,21,25,65,66,71–76]throughanexistingmappingofthegroundstatetothe
maximum-flowoptimizationproblem[77].Aclearadvantageofthisapproachistheabilitytosimulate
largesystemsizesanddisorderensemblesinrathermoderatecomputationaltimes.Weshouldunderline
herethat,eventhemostefficientT 0MonteCarloschemesexhibitextremelyslowdynamicsinthelow-
>
temperaturephaseofthesesystemsandareupperboundedbylinearsizesoftheorderofLmax 32[70].
É
FurtheradvantagesoftheT 0approacharetheabsenceofstatisticalerrorsandequilibrationproblems,
=
which,onthecontrary,arethetwomajordrawbacksencounteredintheT 0simulationofsystemswith
>
roughfree-energylandscapes[5].
A short directsketching of how this mapping may in principleoccur through some simple consid-
erationsisasfollows:LetG (V,E)beadirected,weightedgraphconsistingofasetV ofnodesanda
=
setE ofedges,eachofthelatterconnectingtwonodes.Inadirectedgraph,foreachedgeadirectionis
specified.Thepropertyofbeingweightedmeansthattoeachedgefromnodei tonode j acapacitycij is
assigned.Letthenumberofnodesben 2.WeenumeratethenodesV {0,1,2,...,n,n 1}anddefine
+ = +
thefirstnode0assources andthelastnoden 1asthesinkt.Theremainingnodeswillbeassociated
+
tothelatticesitesoftheRFIM.Wecalladirected,weightedgraphG withsources,sinkt,andcapacities
c,asnetworkN (G,c,s,t).Now,inanetworkN (G,c,s,t),an(s,t)-cut(S,S)isdefinedasapartition
ofthesetofnode=sV intotwodisjointsetsSandS(S= S ∅andS S V)withs Sandt S.Inother
∩ = ∪ = ∈ ∈
words,onecanimagineacutasapartitionthatdividesthenetworkintotwoparts,onepartbelongingto
thesourceandtheothertothesink.Generally,therearemanydifferentpossiblecutsinanetwork.We
canassigntoeachofthemacapacityC(S,S),definedasthesumofthecapacitiesoftheedgesthatthecut
crosses
C(S,S) cij. (4)
=
i S,j S
∈X∈
Notethatedgesaredirected,thatiswhyonlyedgesthatstartatthesourcesideofthecutandendatthe
sinksidecontributetothecapacityofthecut.
Now, thecentralideathatallowsustomaptheRFIMintoanetwork definedabove, consistsofde-
scribingacutbyavector X withtheproperty: Xi 1ifi S and Xi 0otherwise, i.e.,ifi S.Then,
= ∈ = ∈
bydefinition, X0 1 and Xn 1 0. Using thisrepresentation, the formula for the cutcapacity maybe
= + =
writteninthefollowingform
n 1n 1
+ +
C(S,S) cijXi(1 Xj). (5)
= −
i 0 j 0
X= X=
Anexpansionofequation(5)leadsto
C(S,S) cijXiXj cij Xi, (6)
=−i,j + i à j !
X X X
andalreadyastructuralsimilaritytothefundamentalHamiltoniandefinitionoftheRFIM[equation(1)]
isclearlyseen.Furtherinformationonthisstructuralsimilarities,includingadetailedalgebra,maybe
foundfortheinterestedreaderintherelevantliterature(seeforinstancereference[70]andreferences
therein).
Theapplicationofmaximum-flowalgorithmstotheRFIMisnowadayswellestablished[75].Themost
efficientnetworkflowalgorithmusedtosolvetheRFIMisthepush-relabel(PR)algorithmofTarjanand
Goldberg[78].Fortheinterestedreader,generalproofsandtheoremsonthePRalgorithmcanbefoundin
standardtextbooks[77].Theversionofthealgorithmimplementedinourstudyinvolvesamodification
proposedbyMiddletonetal.[21,79,80]thatremovesthesourceandsinknodes,reducingmemoryusage
andalsoclarifyingthephysicalconnection[79,80].
43003-4
Random-fieldIsingmodel:Insightfromzero-temperaturesimulations
Thealgorithmstartsbyassigning anexcess xi toeach latticesitei,with xi hi.Residualcapacity
=
variablesrij betweenneighboringsitesareinitiallysettoJ.Aheightvariableui isthenassignedtoeach
nodeviaaglobalupdatestep.Inthisglobalupdate,thevalueofui ateachsiteinthesetT j xj 0
= | <
of negative excess sites is set to zero. Sites with xi Ê0 have ui set to the length of the shor©test pathª,
viaedgeswithpositivecapacity,fromi toT.Thegroundstateisfoundbysuccessivelyrearrangingthe
excessesxi,viapushoperations,andupdatingtheheights,viarelabeloperations.Theorderinwhichsites
areconsideredisgivenbyaqueue.Inthispaper,weconsiderafirst-in-first-out(FIFO)queue.TheFIFO
structureexecutesaPRstepforthesitei atthefrontofalist.Ifanyneighboringsiteismadeactiveby
thePRstep,itisaddedtotheendofthelist.Ifi isstillactiveafterthePRstep,itisalsoaddedtotheend
ofthelist.Thisstructuremaintainsthecyclesthroughthesetofactivesites.
When no more pushes or relabels are possible, a final global update determines the ground state,
so that sites which are path connected by bonds with rij 0 to T have σi 1, while those which
> =−
aredisconnected from T have σi 1. A push operationmoves excess from a site i to a lower height
=
neighbor j,ifpossible,thatis,wheneverxi 0,rij 0,anduj ui 1.Inapush,theworkingvariables
> > = −
aremodifiedaccordingtoxi xi δ,xj xj δ,rij rij δ,andrji rji δ,withδ min(xi,rij).
→ − → + → − → + =
PushoperationstendtomovethepositiveexcesstowardssitesinT.Whenxi 0andnopushispossible,
>
tishoelasitteedi,swrietlhabueilledu,jwi1t,hfouriainllcireaUsedantod1al+lmj inU{j,|rtihj>e0h}ueijg.hItnuaidfdoirtiaolnli,ifaUseistoinfchriegahseesdttsoitietssUmabxeicmoumme
> + ∈ ∉ ∈
value,V,asthesesiteswillalwaysbeisolatedfromthenegativeexcessnodes.
Periodicglobal updatesareoften crucialto thepracticalspeed of thealgorithm [79, 80]. Following
thesuggestions ofreferences[21,79,80],wehavealsoappliedglobalupdateshereeveryV relabels,a
practicefoundtobecomputationallyoptimum[25,76,79,80].
Usingthisschemeweperformedlarge-scalesimulationsoftheRFIMwithbothtypeofdistributions
discussedaboveinsection1.Letusnoteherethatpriortothecommencementoftheselarge-scalesimu-
lations,asetofpreliminaryrunswithsmallersystemsizesrevealedthecriticalh-regimethatweshould
workon.Inparticular,forthetrimodal(p 1/3)RFIMsimulationshavebeenperformedforlatticesizes
=
L {24,32,48,64,96,128}anddisorderstrengthsh [2.7 3.3].FortheGaussianmodellatticesizesinthe
∈ ∈ −
rangeL {Lmin Lmax},whereLmin 24andLmax 156,wereusedanddisorderstrengthsh [2.0 3.0].
∈ − = = ∈ −
Inbothcasesadisorder-strengthstepofδh 0.02wasused.Regardingthedisorderaveragingprocedure,
=
whichisofparamountimportanceinthestudyoftheRFIM,foreachpair(L,h)anextensiveaveraging
overNs 50 103independentrandom-fieldrealizationshasbeenundertaken,muchlargerthaninpre-
= ×
viousrelevantstudiesofthemodel[21,65,66,72,73].Additionally,fortheGaussianRFIMweperformed
some further and even more extensive simulations, at the bestknownestimate ofthe criticalfield hc,
usinginthiscaseanensembleofNs 200 103 randomrealizations.
= ×
3. Universality aspects
As the outcome of the PR algorithm is the spinconfiguration of the ground state, we can calculate
foragivensampleofalatticewithlinearsizeL themagnetizationviam=V−1 iσi.Takingtheaver-
ageoverdifferentdisorderconfigurationswemaydefinetheorderparameterofthesystemM [m ],
P = | |
where[ ]denotesdisorderaveraging.Another physicalparameterofinterestisthebondenergyper
···
sapvienrathgea,tdceofirrneesdpohnedrseatoftethreasfirEsJtter[emJ].oOftuhreaHnaamlyislitsoniniatnhe(1s)e,qi.ue.eelJw=il−l bVe−1ma〈iin,jl〉yσbiaσsje,danonditthsedseisothrdreeer
= P
thermodynamicquantities,aswellasarelevantalgorithmicquantity,namelythenumberofprimitive
operationsofthePRalgorithm,thatisthenumberofrelabelsperspinR.
Atthispoint,letusstartthepresentationofourFSSapproachwithfigures2(a)and(b),whereweplot
thesample-to-samplefluctuationsoverdisorderoftwoquantities,ofphysicalandtechnicalorigin,forthe
caseofthetrimodalRFIM.Inparticular,weplotthefluctuationsofthebondenergyEJ[figure2(a)]and
thenumberofprimitiveoperationsofthePRalgorithm[figure2(b)].Allthesefluctuationsareplottedas
afunctionofthedisorderstrengthh forthecompletelatticesize-rangeL 32 128. Itisclearthatfor
= −
everylatticesizeL,thesefluctuationsappeartohaveamaximumvalueatacertainvalueofh,denoted
hereafterashL∗,thatmaybeconsideredinthefollowingasasuitablepseudo-criticaldisorderstrength.
Byfittingthedatapointsaroundthemaximum firsttoaGaussian,andsubsequentlytoafourth-order
43003-5
P.E.Theodorakis,N.G.Fytas
Figure2.(Coloronline)(a)Sample-to-samplefluctuationsofthebondenergyVEJ ofthetrimodalRFIM
asafunctionofthedisorderstrengthforvariouslatticesizesintherangeL 32 128.Linesaresimple
= −
guides tothe eye. (b) Sameas in panel (a) but now thesample-to-sample fluctuations of the number
ofprimitive operations ofthePRalgorithm,VR,areshown.(c)Simultaneousfittingoftheform(7)of
thepseudo-criticaldisorderstrengthshL∗,obtainedfromthepeakpositionsofthefluctuationsshownin
panels(a)and(b).Thesharedparametersofthethreedatasetsofthefitarethecriticalstrengthhcand
thecorrelationlength’sexponentν.
polynomial,wehaveextractedthevaluesofthepeak-locations(hL∗)bytakingthemeanvalueviathetwo
fittingfunctions,aswellasthecorrespondingerrorbars.UsingnowthesevaluesforhL∗ weconsiderin
thepanel(c)offigure2asimultaneouspower-lawfittingattemptoftheform
hL∗=hc+bL−1/ν, (7)
simultaneous meaning that the values ofhc and νfor all datasets inthe fitting procedureare shared
during the fit. The quality of the fit is fair enough, with a value of χ2/dof of the order of 0.6, where
dofreferstothe degrees offreedom, and producesthe estimates hc 2.745(7) and ν 1.37(2) for the
= =
criticaldisorderstrength andthecorrelationlength’s exponent, inagreement withrecentestimates in
theliterature[76].
WenowturnourdiscussionontheGaussianRFIM.Forthiscaseweshowinfigure3(a)thenumberof
relabelsperspinRasafunctionofthedisorderstrengthforvariouslatticesizesintherangeL 24 156.
= −
Again,weobservethatforeverylatticesizeL,Rhasamaximumatacertainvalueofh,denotedasbefore
withhL∗,thatmaybeconsiderednowasarelevantpseudo-criticaldisorderstrength.Followingasimilar
procedure,we extracted the values of the peak-locations(hL∗)as well asthe correspondingerrorbars,
whoseshift-behaviorisnowplottedinpanel(b)offigure3.Thestraightlineispower-lawfittingattempt
ofthesameform(7)andtheoutcomeforhc andνis2.274(4)and1.37(1),respectively.Thequalityofthe
fitisalsointhiscasegood,withavalueofχ2/dofoftheorderof0.4.
Afewcomments onthescalinganalysisarenowinorder:Havingsimulated morethanfivelattice
size-pointsineachcase,wealsotriedtoperformtheaboveanalysisincludinghigher-orderscalingcor-
rections of the form (1 b′L−ω), where ω is the well-known correction-to-scaling exponent, obtained
+
veryrecentlytobeω 0.52(11) for thismodel[26],usingthescalingbehavior ofuniversalquantities.
=
However,noimprovementhasbeenobservedinthequalityofthefitofourdata.Onthecontrary,the
43003-6
Random-fieldIsingmodel:Insightfromzero-temperaturesimulations
Figure3.(Coloronline)(a)ThenumberofrelabelsperspinRoftheGaussianRFIMasafunctionofthe
disorderstrengthforvariouslatticesizesintherangeL 24 156.Linesaresimpleguidestotheeye.(b)
= −
Fittingoftheform(7)ofthepseudo-criticaldisorderstrengthshL∗,obtainedfromthepeakpositionsof
panel(a).
correctedscalingassumptionresultedinanunstablefittingprocedurewithsignificantlylargeerrorsin
thevaluesoftheexponentν,thecoefficientb′,aswellastheexponentω.
Our suggestion of choosing these newly defined pseudo-critical disorder strengths hL∗ as a proper
measureforperformingFSS,closelyfollowstheanalogousconsiderationsofHartmannandYoung[72]
and Dukovski and Machta [73], also for the Gaussian RFIM. The first authors [72] considered pseudo-
criticaldisorderstrengthsatthevaluesofhatwhichaspecific-heat-likequantityobtainedbynumerically
differentiatingthebondenergywithrespecttoh attainsitsmaximum.Ontheotherhand,theauthors
of reference [73] identified the pseudo-critical points as those in the H h plane (with H a uniform
−
externalfield), where threedegenerateground states ofthe system show thelargest discontinuitiesin
themagnetization.
Respectively,MiddletonandFisher[21]usingsimilarreasoningontheGaussianRFIM,characterized
thedistributionoftheorderparameterbytheaverageoversamplesofthesquareofthemagnetization
perspinandtheroot-mean-squaresample-to-samplevariationsofthesquareofthemagnetization.They
identifiedasimilarbehaviortothatoffigures2(a)and(b),i.e.,withincreasingL,thepeakmagnitudeof
thisquantitymoveditslocationtosmallervaluesofh,defininganotherrelevantpseudo-criticaldisorder
strength.However,inreference[21]theauthorswereonlyinterestedinthescalingbehavioroftheheight
ofthesepeaks.Thepracticefollowed inthecurrentpaper,employingtheFSSbehaviorofthepeaksof
thesample-to-sample fluctuationsofseveralquantitiesofphysical(M andEJ)andtechnical(R)origin,
wasinspiredbytheintriguinganalysisofEfratandSchwartz[69].Theseauthors,studyingalsothed 3
=
RFIM, showed that the behavior of the sample-to-sample fluctuations in a disordered system may be
turnedintoausefultoolthatcanprovideanindependentmeasuretodistinguishbetweentheordered
anddisorderedphasesofthesystem.Theanalysisoffigures2(a)and(b)aboveverifiestheirprediction,
andtheaccuracyintheestimationofrelevantphasediagramfeatures,likethecriticalfieldhc andthe
criticalexponentν,turnsouttobeacleartestinfavoroftheoverallscheme.
Letusmakeatthispointasmallcommentconcerningtheerrorsinherentinthesetypesofapprox-
imations.Theerrorsinducedintheschemebasedonthesample-to-samplefluctuationsoffigures2(a),
2(b),ortheprimitiveoperationsofthePRalgorithmshowninfigure3(a),havetheiroriginintheapplica-
tionofsomepolynomial,orpeak-like,functioninordertoextracttherelevantpositionofthemaximum
intheh-axis.Onthecontrary,insimilardefinitionsofpseudo-criticalpoints,suchasthroughtheuseof
someproperlydefinedspecific-heat-likequantityatT 0[72],oneshouldnumericallydifferentiatethe
=
dataofthebondenergyEJ,andthenconsiderasmoothingfunctiontolocatethepositionofthemaxi-
mum.Thisschemeissubjectedtotwosuccessivefittingapproximations,thusincreasingtheerrorsinthe
estimationofthepseudo-criticalpoints.
To summarize, in this section, we have investigated the matching between the trimodal, p 1/3,
=
RFIMandtheGaussianRFIM.Clearlyenough, ourestimatesforthecriticalexponentνofbothmodels
43003-7
P.E.Theodorakis,N.G.Fytas
indicateanequivalenceamongbothdistributionswithinerrorbars,justifyingtheoriginalpredictionof
Mattis[45].Furthermore,wehavesuggestedthevaluesforthecriticalfieldhcwhichcompareverywell
tothemostaccurateestimationsoftheliterature.Forinstance,thebestknownestimatefortheGaussian
RFIMishc 2.27205 [26],veryclosetothevalue2.274(4) ofthepresentwork.Thisisalsotrueforthe
=
reportedvaluesofthecorrelation-length’sexponent, asfortheGaussianRFIM,previoushigh-accuracy
estimates suggest a value of ν 1.37 [21, 26, 72]. An interesting aspect of our analysis that led to the
=
aboveresultswastheillustrationthatquantitiesrelatedtothesample-to-samplefluctuationsofseveral
quantities of the system or simply the, originally technical, number of primitive operations of the PR
algorithm,constituteausefulalternativetoinvestigatecriticality.
4. Gaussian RFIM
Inthislastpartofourwork,weconcentrateontheGaussiandistribution,whichisthemoststudied
caseintheliteratureoftheRFIM,andpresentfurtherresultsonimportantaspectsofitscriticalbehavior.
Asalreadymentionedabove,wehaveperformedadditionalrunsatthebest-knownvalueofthecritical
field,thatisthevaluehc 2.27205[26].Thus,thedataandanalysisofthissectionarebasedonextensive
=
simulationsperformedatthisvalueofthecriticalfield.
Inprincipal,weareinterested intheextractionofanaccurateestimate forthemagneticexponent
ratioβ/ν,whosesmallvaluecastssomedoubtsontheorderofthetransitionoftheRFIM.Theroutewe
followhereisviathescalingoftheorderparameterM atthecriticalfield.Thisisshowninfigure4,and
thesolidlineisapower-lawfittingoftheformM(h=hc) L−β/ν.Theresultingestimateofthemagnetic
∼
exponent ratio, given also in the figure, is β/ν 0.0131(3), a rather small, but non-zero value, also in
=
agreementwithsomeofthemostaccurateestimationsintheliterature[21].
ThenextpartofourFSSanalysisconcernsthecontroversialissueofthespecificheatoftheRFIM.The
specificheatoftheRFIMcanbeexperimentally measured [49,50]andis,forsure, ofgreattheoretical
importance. Yet, itis well known that itis one of the most intricatethermodynamic quantities to deal
withinnumericalsimulations,evenwhenitcomestopuresystems.FortheRFIM,MonteCarlomethods
atT 0havebeenusedtoestimatethevalueofitscriticalexponentα,butwererestrictedtorathersmall
>
systemssizesandhavealsorevealedmanyseriousproblems,i.e.,severeviolationsofselfaveraging[61,
64]. A better picture emerged throughout the years from T 0 computations, proposing estimates of
=
α 0.However, evenbyusingthesamenumericaltechniques,butdifferentscalingapproaches,some
≈
inconsistencieswererecordedintheliterature.Themost prominentwasthatofreference[72],where
astronglynegativevalueofthecriticalexponentαwasestimated.Ontheotherhand,experimentson
randomfieldanddilutedantiferromagneticsystemssuggestaclearlogarithmicdivergenceofthespecific
heat[49,50].
Ingeneral,oneexpectsthatthefinite-temperaturedefinitionofthespecificheatC canbeextended
to T 0, with the second derivative of E with respect to temperature being replaced by the second
= 〈 〉
Figure4.FSSoftheorderparameteratthecriticalfieldhc.
43003-8
Random-fieldIsingmodel:Insightfromzero-temperaturesimulations
Figure5.FSSbehaviorofthebondpartoftheenergydensityatthecriticalfieldhc.Thelineisafittingof
theform(8).
derivative of the ground-state energy densityEgs with respect to the randomfield h [21, 72].The first
derivative∂Egs/∂J isthebondenergyEJ,alreadydefinedabove.ThegeneralFSSformassumedisthat
the singular part of the specific heatCs behaves asCs Lα/νC˜ (h hc)L1/ν . Thus, one may estimate
∼ −
α bystudying the behavior of EJ at h=hc [21].The computatio£n from the b¤ehavior of EJ isbased on
integratingtheabovescalingequationuptohc,whichgivesadependence
EJ(h=hc)=c1+c2L(α−1)/ν, (8)
withci constants.Alternatively,followingtheprescriptionof[72],onemaycalculatethesecondderiva-
tivebyfinitedifferencesofEJ(h)forvaluesofh nearhc anddetermineαbyfittingtothemaximumof
thepeaksinCs,whichoccurathL∗−hc≈L−1/ν.However,asalreadynotedin[21],thislatterapproach
maybemorestronglyaffected byfinite-size corrections,sincethepeaksinCs foundbynumericaldif-
ferentiationaresomewhatabovehc,andfurthermoreitiscomputationallymoredemanding,sinceone
musthavethevaluesofEJinawideandverydenserangeofh-values.
Inthepresentcase,wherethecriticalvaluehcisknownwithgoodaccuracy,thefirstapproachseems
to be more suitable to follow. The numerical data of the criticalbond energy and the relevant scaling
analysisarepresentedinfigure5.Thesolidlineisapower-lawfittingoftheform(8)andtheestimatefor
theexponentratio(α 1)/νis 0.799(28),asalsogiveninthefigure.Usingnowourestimateν 1.37(1),
− − =
wecalculatethecriticalexponentαofthespecificheat,resultinginanestimateα 0.095(37),whichis
=−
fairlycompatibletotheexperimentalscenarioofalogarithmicdivergence(α 0)[49,50].
=
Followingthediscussionaboveinsection1,ournumericalstudiesofdisorderedsystemsarecarried
outneartheircriticalpointsusingfinitesamples;eachsampleisaparticularrandomrealizationofthe
quenched disorder. A measurement of a thermodynamic property, say Z, yields a different value for
everysample.InanensembleofdisorderedsamplesoflinearsizeL,thevaluesofZ aredistributedac-
cordingtoaprobabilitydistribution.Thebehaviorofthisdistributionisdirectlyrelatedtotheissueof
self-averaging.Inparticular,bystudyingthebehaviorofthewidthofthisdistribution,onemayqualita-
tivelyaddresstheissueofself-averaging,ashasalreadybeenstressedbypreviousauthors[54,55,58].In
general,wecharacterizethedistributionbyitsaverage[Z]andalsobytherelativevariance
V [Z2] [Z]2
Z
RZ =[Z]2 = [Z−]2 , (9)
thatweemployheretoinvestigatetheself-averagingpropertiesoftheRFIM.
In particular, we study the behavior of the ratioRZ, in the framework of the two main quantities
typicallyused,theorderparameterM andthebondenergyEJofthemodel.Infigure6weplottheratio
RZ,estimatedatthecriticalfieldhc,forbothquantities,asafunctionoftheinverselinearsize.Thesolid
linesaresimplelinearextrapolationstotheinfinite-limitsizeL .Asitisstraightforwardfromthe
→∞
extrapolations, the order-parameter that carries the effect of the disorder — we remind here that the
43003-9
P.E.Theodorakis,N.G.Fytas
Figure6.(Coloronline)Illustrationoftheself-averagingpropertiesofthemodelintermsofthemagneti-
zationandbondenergy.Thelinesarelinearextrapolationstotheinfinite-limitsize.
randomfieldcouplestothelocalspins—isastronglynon-self-averagingquantity.Ontheotherhand,as
expected,thebondenergyrestoresself-averaginginthethermodynamiclimit.
Closing, we present some computational aspects of the implemented PR algorithm and its perfor-
manceonthestudyoftheGaussianRFIM.Althoughitsgenericimplementationhasapolynomialtime
bound, its actual performance depends on the order in which operations are performed and which
heuristics are used to maintain auxiliary fields for the algorithm. Even within this polynomial time
bound, thereisapower-law criticalslowing down ofthe PR algorithm at thezero-temperature transi-
tion [21, 71]. This critical slowing down is certainly reminiscent of the slowing down seen in local al-
gorithms for statistical mechanics at finite temperature, such asMetropolis, and even for cluster algo-
rithms[81].Infact,OgielskiwasthefirsttonotethatthePRalgorithmtakesmoretimetofindtheground
statenear thetransitioninthreedimensions fromtheferromagnetic toparamagneticphase [71].This
hasalreadybeenqualitativelyseeninfigure3(a),where,indeed,thenumberofprimitiveoperationsR
ofthePRalgorithmismaximizedinthesuitablydefinedpseudo-criticalfieldshL∗.Assumingthestandard
scalingoftheformR Lzw (h hc)−1/νL ,wherethedynamicexponentz describesthedivergencein
≈ −
therunningtimeath=hc,a£ndw(x)∼x−z¤atlargexandw(x)∼|x|−zln|x|asx→−∞,tobeconsistent
withconvergence toconstant R for h hc and R lnL for small h.Our fitting attempt of thisscaling
> ∼
formisplottedinfigure7andtheobtainedestimateforthedynamiccriticalexponentzis0.487(7).
Figure7.EstimationofthecriticalslowingdownexponentzofthePRalgorithmviatheFSSbehaviorof
thenumberofrelabelsperspinatthecriticalfieldhc.
43003-10