Berezinskii-Kosterlitz-Thouless transition in two-dimensional lattice gas models

0.16 MB·English

by  H Chamati

PHYSICALREVIEWB73,184424(2006)
Berezinskiˇı–Kosterlitz–Thouless transitionintwo–dimensional latticegasmodels
Hassan CHAMATI and Silvano ROMANO
∗ †
Unita´ CNISMeDipartimentodiFisica”A.Volta”,Universita` diPavia,viaA.Bassi6,I-27100Pavia,ITALY
7
0
Wehaveconsideredtwoclassicallattice–gasmodels,consistingofparticlesthatcarrymulticomponentmag-
0
neticmomenta,andassociatedwithatwo–dimensionalsquarelattices;eachsitecanhostoneparticleatmost,
2
thusimplicitlyallowingforhard–corerepulsion; thepairinteraction, restrictedtonearestneighbors, isferro-
n magnetic and involves only two components. The case of zero chemical potential has been investigated by
a
Grand–Canonical Monte Carlo simulations; the fluctuating occupation numbers now give rise to additional
J
fluid–likeobservablesincomparisonwiththeusualsaturated–latticesituation;thesewereinvestigatedandtheir

4 possibleinfluenceonthecriticalbehaviourwasdiscussed. Ourresultsshowthatthepresentmodelsupportsa
2 Berezinskiˇı–Kosterlitz–Thoulessphasetransitionwithatransitiontemperaturelowerthanthatofthesaturated

latticecounterpartduetothepresenceof“vacancies”;comparisonswerealsomadewithsimilarmodelsstudied
]
h intheliterature.
c
e PACSnumbers:75.10.Hk,05.50.+q,64.60.–i
m Keywords:latticegases,classicalspinmodels,Berezinskiˇı–Kosterlitz–Thoulesstransition.
-
t
a I. INTRODUCTION Here ǫ is a positive quantity setting energy and temperature
t scales(i.e. T = k t/ǫ, wheretdenotestheabsolutetempera-
s B
. ture);thecorresponding(scaled)Hamiltonianreads
t Planar rotators are models with potential interactions in-
a
m volvingtwo–componentspins; on the otherhand, in the XY Λ= Ω , (3)
jk
model, the spins have three components, only two of which −
- Xj<k
are involvedin the interaction, andthe XY modelcanbe re- { }
d
where isrestrictedtonearestneighbours,witheachdis-
n garded as an extremely anisotropic (easy–plane) Heisenberg j<k
tinctpair{be}ingcountedonce. Whend = 1,2,potentialmod-
o model. Actually,theterminologicalconventionadoptedhere P
c as well as by other Authors is not always followed, and the els defined by Eq. (2) produce orientational disorder at all
[ finite temperatures (thus no orientational ordering transition
name“XYmodel”is sometimesusedintheLiteraturetoin-
atfinitetemperature)4,5;inthefollowing,thediscussionshall
1 dicate planar rotators; on the other hand, both models are
be essentially restricted to the two cases n = 2, a = b = 1
v knowntoproducethesameuniversalityclass. Thetwonamed
4 modelshavebeenextensivelystudied,andpossessarichvari- (PR)andn=3, a=0, b=1(XY),respectively.
8 etyofapplicationsinStatisticalaswellasCondensedMatter It is by now well known that the PR model defined by
5 Physics1,2,3. d = 2 supportsa transitionto a low–temperaturephase with
1 slow (inverse–power–law) decay of the correlation function
These models have been especially studied in their
0
andinfinitesusceptibility;thisistheveryextensivelystudied
7 saturated-lattice(SL)version,whereeachsiteisoccupiedby
Berezinskiˇı–Kosterlitz–Thouless (BKT) transition (see, e.g.
0 a particle; as for symbols, classical SL spin models are first
Refs.1,2,3,6,7,8andothersquotedtherein),whoseexistencewas
/ defined here: we consider a classical system, consisting of
at n component unit vectors u (usually n = 2,3), associated proven rigorously by Fro¨hlich and Spencer6: the transition
m w−ithad−dimensional(bipartkite)latticeZd (mostlyd =2);let tmemorpeerreactuernetwanads emstoirmearteefidnteodbreesΘuPltRis=Θ0.88=±00..809127,98a;ndthae
- x denote dimensionless coordinates of the lattice sites, and PR
d k specific heat exhibitsa maximum at a higher temperature7,9.
let u denote Cartesian spin componentswith respect to an
n k,α Moreover,anisotropicmodelsdefinedbyd = 2, n = 3, 0
o orthonormalbasiseα; particleorientationsare parameterized ≤
:c (bny=us3u)a;lpolarangles{ϕj}(n = 2)orsphericalones{(θj,φj)} tara<nsbithioanvewbheeenntshtuedriaetdioasa/wbelilsasnudffipcrioevnetlnytosmsuapllp10o;rtaanBeKstTi-
v mateofthetransitiontemperaturefortheXYcaseisgivenby
The interaction potentials considered here are assumed to
Xi be translationally invariant, restricted to nearest–neighbours, ΘXY =0.700±0.00511,12,13.
Lattice–gas (LG) extensions of the SL models can be de-
r ferromagnetic(FM);they are, in general, anisotropicin spin
a finedaswell,whereeachlatticesitehostsoneparticleatmost,
space,andpossessO(2)symmetryatleast:
andsite occupationisalsocontrolledbythechemicalpoten-
tial µ; such modelshave often been used in connectionwith
Ψ = ǫΩ , (1)
jk − jk alloysandabsorption;thismethodologysomehowallowsfor
pressure and density effects. Lattice–gas(LG) extensionsof
where
the continuous–spinpotential modelconsidered here are de-
finedbyHamiltonians
Ω = au u +b u u ;
jk j,n k,n j,α k,α
Xα<n Λ= ν ν (λ Ω ) µN, N = ν , (4)
j k jk k
ǫ >0, 0 a b, b>0, max(a,b)=1. (2) Xj<k − − Xk
≤ ≤ { }
2
whereν =0,1denotesoccupationnumbers;noticethatλ 0 inspinspace;moreover,anevengreatervarietyofanchorings
k
≤
reinforcestheorientation–dependentterm,whereasλ>0op- can be realized via the recently introduced generalized XY
posesit,andthatafinitevalueofλonlybecomesimmaterial models26,27.
intheSLlimitµ + . The rest of the paper is organized as follows: in Section
→ ∞
For a square lattice, the SL–PR model produces a low– IIwe discussdetailsoursimulationprocedure. SectionIIIis
temperatureBKTtransition;theexistenceofsuchatransition devotedtothediscussionofsimulationresults:wehavefound
fortheLGcounterparthasbeenprovenrigorouslyaswell14. evidencepointingto a BKT transition, andused the relevant
Inadditionto thenamedrigorousresults, still comparatively finite–size scaling theory to locate it. Possible effects of the
littleisknownabouttheabovemagneticLGmodelswithcon- chemicalpotentialonthenatureofthetransitionarediscussed
tinuousspins,inmarkedcontrasttothevastamountofinfor- inSectionIV,whichsummarizesourresults.
mation available for their SL counterparts; for example, as
faraswecouldcheckintheLiterature,theirsimulationstudy
onlystartedattheendofthe1990’s15,16. II. MONTECARLOSIMULATIONS
A few yearsbefore the existence of a BKT transition was
provenfortheLG–PRmodel14,thesamemodelhadbeenin- A detailed treatment of Grand–Canonicalsimulations can
vestigatedbyMonteCarlosimulationintheGrand–Canonical be found in or via Refs.15,28,29; the method outlined here
ensemble, and in the absence of a purely positional interac- has already been used in our previous studies of other LG
tion, i.e. λ = 015. Simulationswere carried out for µ = 0.1 models16,28,30. Toavoidsurfaceeffectssimulationswerecar-
and µ = 0.2, and showed that the BKT transition survives, ried outon periodicallyrepeatedsamples, consisting of V =
−
evenformildlynegativeµ,andthatthetransitiontemperature L2sites,L=40,80,120,160;calculationswerecarriedoutin
isanincreasingfunctionofµ,inbroadqualitativeagreement cascade,inorderofincreasingreducedtemperatureT.
withpreviousRenormalizationGroup(RG)studies17,18,19. The two basic MC steps used here were Canonical and
TheHamiltonian(Eq. (4))canbeinterpretedasdescribing semi–Grand–Canonical attempts; in addition two other fea-
atwo–componentsystemconsistingofinterconverting“real” tureswereimplemented31,32: (i)whenalattice siteswasvis-
(νk =1)and“ghost”,“virtual”orideal–gasparticles(νk =0); ited, Canonical or semi–Grand–Canonical steps were ran-
bothkindsofparticleshavethesamekineticenergy,µdenotes domly chosen with probabilities and , respectively;
can GC
the excesschemicalpotentialof “real” particlesover“ideal” we used / = n 1, siPnce spinPorientation is de-
can GC
P P −
ones, and the total numberof particlesequalsthe numberof fined by (n 1) angles, versus one occupation number and
−
availablelatticesites(semi–Grand–Canonicalinterpretation). (ii)sublatticesweeps(checkerboarddecomposition)31,32;thus
The semi–Grand–Canonical interpretation was also used in each sweep (or cycle) consisted of 2V attempts, first V at-
early studies of the two–dimensional planar rotator, carried temptswherethelatticesiteswaschosenrandomly,thenV/2
out by the Migdal–Kadanoff RG techniques, and aiming attemptsonlatticesitesofoddparity,andfinallyV/2attempts
at two–dimensional mixtures (films) of 3He and 4He17,18,19, on lattice sites on even parity. Equilibration runs took be-
where non–magneticimpurities correspondto 3He. In addi- tween 25000 and 200000 cycles, and production runs took
tiontophaseseparations,theresultsinRefs.17,18,19showthat, between250000and1000000;macrostepaveragesforeval-
in a re´gime of low fraction of non–magnetic impurities, the uatingstatisticalerrorsweretakenover1000cycles.Different
paramagneticphaseundergoesaBKTtransition. random–numbergeneratorswereused,asdiscussedinRef.32.
NoticealsothattheaboveHamiltonian(Eq. (4))describes ComputedobservablesincludedmeanHamiltonianpersite
asituationofannealeddilution;ontheotherhand,modelsin anditstemperaturederivative(specificheatatconstantµand
thepresenceofquencheddilution,andhencetheeffectofdis- V),densityanditsderivativeswithrespecttotemperatureand
orderontheBKTtransition,havebeeninvestigatedusingthe chemicalpotential,definedby
PR model20,21,22,23,24,25 and veryrecently the XY model25; it
was found that a sufficiently weak disorder does not destroy H = 1 Λ , (6)
∗
thetransition,whichsurvivesuptoaconcentrationofvacan- V h i
ciesclosetothepercolationthreshold.
Inthispaper,wepresentanextensivestudyoftheferromag- 1
ρ= N , (7)
neticLG–PRandLG–XYmodels,whoseHamiltoniancanbe V h i
explicitlywrittenas
andbythefluctuationformulae(seee.g.29)
Λ= ν ν λ (u u +u u ) µN, (5)
j k j,1 k,1 j,2 k,2 ∂ρ 1
{Xj<k} h − i− ρT = ∂T(cid:12)(cid:12)µ,V = VT2 [hNΛi−hNihΛi], (8)
inordertogaininsightsintotheircriticalbehaviour.Themod- (cid:12)(cid:12)
(cid:12)
elsarefurthersimplifiedbychoosingλ = 0,i.e. nopurepo- ∂ρ 1
sitionalinteractions;Letusalso remarkthattwo–component ρµ = = N2 N 2 , (9)
∂µ(cid:12) VT −h i
spinsareinvolvedinthePRcase,whereasXYinvolvesthree– (cid:12)T,V hD E i
(cid:12)
(cid:12)
component spins but only two of their components are in- (cid:12)
volved in the interaction: in this sense the two models en- Cµ,V = 1 1 ∂hΛi = 1 Λ2 Λ 2 . (10)
tail different anchorings with respect to the horizontalplane kB kBV ∂T (cid:12)(cid:12)µ,V VT2 hD E−h i i
(cid:12)
(cid:12)
(cid:12)
3
We alsocalculatedmeanin–planemagneticmomentpersite 9
andin–planesusceptibilitydefinedby
1 C /k
M = √F F (11) µV B
V ·
D E 6
and
1
χ= F F , (12)
VT h · i
3
−ρ
where T
F= ν u e +u e , (13)
k k,1 1 k,2 2
Xk (cid:0) (cid:1) 0
0.40 0.50 0.60 0.70 0.80
takingintoaccountonlythein–planecomponentsofthevec-
T
torspin.
A square sample of V sites contains 2V distinct nearest– FIG. 1: Simulation estimates for the specific heat per site and for
neighbouring pairs of lattice sites; we worked out pair oc- ρ ,obtainedwithdifferentsamplesizes: circles: L = 40;squares:
cupation probabilities, i.e. the mean fractions R of pairs −L =T 80; triangles: L = 120; diamonds: 160. The value µ = 0
JK
being both empty (R = (1 ν )(1 ν ) ), both occupied wasusedinthepresentcalculations. StatisticalerrorsforCµ,V, not
ee j k
− − reported,rangebetween1and3%;otherwise,hereandinthefollow-
(Roo = νjνk ), or consistiDng of an emptyEand an occupied ingFigures,statisticalerrorsfallwithinsymbolsizes.
site (R D= (E1 ν )ν +(1 ν )ν ). Itshouldbenotedthat
eo j k k j
Ree+Roo+DReo−=1. − E
withincreasingsamplesize, i.e. theywere wellfittedby the
Short– and long–range positional correlations, were com-
relation
paredbymeansoftheexcessquantities
lnM = b lnL+b , b >0, (15)
R =ln Roo , R =R ρ2, (14) − 1 0 1
′oo ρ2 ! ′o′o oo− wheretheratiob (T)/T wasfoundtoincreasewithtempera-
1
ture;aspin–waveanalysisoftheSL–PR(Refs.33,34andthose
collectivelydenotedbyR (noticethatthesetwo definitions
∗oo quotedtherein)predictsasimilarsample–sizedependence,in
entailcomparablenumericalvalues).
agreementwithsimulationresultsforthepresentmodel;fur-
Quantitiessuchasρ,ρ ,ρ andtheabovepaircorrelations
T µ thermore, in the low–temperature limit b becomes propor-
1
R orR canbedefinedas“fluid–like”,inthesensethatthey
JK ∗oo tionaltoT.
allgooverthetrivialconstantsintheSLlimit. Letusalsore-
Thebehavioursofspecificheatandsusceptibility(FIG.2)
mark that some of the above definitions (e.g. C and ρ )
µ,V T suggestthattheLG–XYmodelundergoesa BKT–likephase
involvethe total potentialenergyboth in the stochastic vari-
transition. According to the BKT theory, in the thermody-
able and in the probability measure (“explicit” dependence),
namiclimit,thein–planesusceptibility,χ,divergesexponen-
whereassomeotherdefinitions,e.g. ρµ orthequantitiesRJK, tially while approaching the transition temperature Θ in the
involvethetotalpotentialenergyonlyintheprobabilitymea-
hightemperatureregion2i.e.
sure(“implicit”dependence).
χ aχexp bχ(T Θ)−21 ; T Θ+ (16)
∼ − →
h i
III. SIMULATIONRESULTS andremainsinfiniteinthelowtemperatureregionT <Θ.
ForafinitesamplewithvolumeV = L2,thesituationisdif-
Westartbydiscussingtheoutcomeofthesimulationsper- ferent,sincethesusceptibilityobeystheconstraintχ V/T;
≤
formed for the LG–XY model. Results for a number of ob- thus χ is always finite, and its exponentialdivergencein the
servables, such as the mean energy per site H , and density transitionregionis rounded,becauseof theimportantfinite–
∗
ρ (notreportedhere) were foundto evolve with temperature size effects. Actually, results for lnχ versus temperature
in a smoothway, and to be independentof samplesizes; the (FIG. 2) were found to be independentof sample size when
derivatives C and ρ (both plotted on FIG. 1) showed a T & 0.62, and showed a recognizable increase with it (a
µ,V T
rather smooth trend with temperature, and a “sharp” peak at power–lawdependenceofχonL)whenT .0.6.
T 0.6,aroundwhichthesample–sizedependenceofresults In the criticalregion,i.e. T Θ, the correlationlengthis
≈ ∼
becameslightlymorepronounced;noticethatneitherthespe- ofthesameorderasthelinearsize, L, ofthesystem. Inthis
cific heatnorρ are exhibitinga divergenceas a functionof case,theexponentialdivergencedisappearsinthecriticalre-
T
thetemperature. gion,buta reminiscenceof thedivergencecanstill befound
At all investigated temperatures simulation results for M in the behaviourof χ at higher temperatures, where the cor-
(not reported here) exhibited the expected power–law decay relation length is still small compared to large linear system
4
12
TABLE I: Comparison of the transition temperatures, Θ , of the
PR
LG–PRmodelobtainedinRef.15andthoseobtainedusingthefinite–
10
sizescalingpresentedinthepresentpaper.
µ Θ (Ref.15) Θ (presentwork)
8 PR PR
0.2 0.73 0.01 0.71 0.01
− ± ±
0.1 0.79 0.01 0.75 0.01
χn 6 ± ±
l
4
approximately 18% compared to the SL transition tempera-
ture,butdoesnotchangeitsnature.
2
T = 0.5700
0 0.6 T = 0.5725
0.4 0.5 0.6 0.7 0.8 T = 0.5750
T = 0.5775
T T = 0.5800
FIG. 2: Simulation estimates for the logarithm of the magnetic
0.4
ssuizsecse;pstaibmileitymeχanvienrgsuosftseymmpbeoralstuarse,inobFtiagi.ne1d. Twhiethsodliiffdecreunrvtesaismopble- -7/4L)
tainedbyfittingtoequation(16)withdatacorrespondingtoL=160. χ(
n
l
0.2
sizes and finite–size effects can be neglected. We first fitted
our MC results obtained for the largest sample size and for
temperaturesin the range 0.60 . T . 0.62 to eq. (16), and
obtainedevidenceoftheexponentialdivergenceofthesuscep- 0.0
3.5 4.0 4.5 5.0 5.5
tibility (see FIG. 2), and an estimated transition temperature
Θ=0.57 0.01,witha =0.123 0.005andb =1.55 0.01. ln(L)
χ χ
± ± ±
We checkedour results by using data on a wider rangeof
FIG.3: Plotof ln χL 7/4 versuslnL. Herethe curveclosest toa
temperaturesextendingupto0.70,andwhichyieldeda con- −
horizontalstraight(cid:16)linesig(cid:17)nalsthetransition.
sistent result. In the following we refine our results using a
moreelaboratemethod,namelythefinite-sizescalingtheory.
ThePRcounterpartofthepresentmodelhadbeeninvesti-
Detailsonthisprocedurecanbefoundinorviareference12.
gatedinRef.15,whereamaximumsamplesizecorresponding
In the temperature region where the singularity of χ is
to L = 80 was used. Moreover, in that paper the transition
rounded the finite–size scaling theory holds, i.e. the corre-
temperaturewasestimatedasthetemperaturewhereχstarted
lationlengthξisproportionaltothesizeofthesample,ξ L.
From the behaviourof the magnetic susceptibility χ ξ∼2 η, exhibiting a power–law dependency on L. Calculations had
withη=1/42,weendupwith ∼ − beencarriedoutwithtwodifferentvaluesofthechemicalpo-
tential, µ = 0.2 and µ = +0.1, respectively(see e.g. Table
−
χ(Θ) L2 η, (17) I);thetransitionhadbeenfoundtosurviveinbothcases(even
−
∼ withaslightlynegativevalue),andthetransitiontemperature
wherewehaveomittedthecorrectionsarisingfromthepres- had been found to increase with increasing µ, in qualitative
enceofthebackground(analytic)contributionstothefinite– agreementwithpreviousRGresults15.
sizescaling. Forfurthercomparisonwereexaminedtheresultsobtained
ThetemperatureregionaroundT = 0.57,wasanalyzedin inthecaseoftheLG–PRmodelintwoways.Ontheonehand,
greaterdetail,atfirstbycarryingoutalinearfitoflnχversus the above finite–size analysis was applied to the simulation
lnL and extracting η from the slope; the values were found dataproducedinRef.15. Thetransitiontemperatures,reported
to be η(T) = 0.223 0.008, 0.244 0.012, 0.255 0.007, in Table I, corresponding to the two values of the chemical
0.273 0.005,0.308±0.015forT =±0.5700,0.5725,±0.5750, potentialusedwerefoundtobelowerthantheonesreported
± ±
0.5775, 0.5800, respectively. A non–linear square fit, based previously. The behaviour of the µ–dependence of the tem-
on equation (17), was attempted as well, and yielded results peraturewassimilarforbothmethods. Ontheother,inorder
consistentwiththeseones;italsoprovedconvenientforresult toachieveabettercomparisonwiththepresentcase,simula-
visualizationtoplotln(χL 7/4)versuslnL(FIG.3). Thusthe tions were run anew for the case µ = 0, using same sample
−
transitiontemperatureisestimatedtobeΘ=0.574 0.003,in sizes as for the LG–XY counterpart (L = 40,80,120,160).
±
agreementwiththeabovementionedresultobtainedbyfitting The thermodynamicquantities of interest were found to be-
the data of the susceptibility in the high temperatureregion; havequalitativelyinasimilarfashionasthoseobtainedinthe
atthistemperatureweexpectthevalueofηtobe 1 towithin frameworkoftheXYmodel.
4
statistical errors. Thisresult shows thatthe presenceof “va- A detailedinvestigationofthesusceptibilityvia theabove
cancies” in the sample reduce the transition temperature by finite–sizescalingprocedureresultedinaBKT–liketransition
5
withatransitiontemperatureestimateof0.733 0.003,corre- 1.00
±
spondingtoaparticledensityof0.924 0.003.Thetransition
temperatureinthiscasewasfoundtob±e18%lowerthanthe Roo
caseoftheplanarrotatorSLcounterpart. 0.75
Asforotherfluid–likequantitiesobtainedwithintheframe-
workofLG–XY,theplotofthederivativeρ (FIG.4)against
µ
the temperature exhibited a smooth evolution, a broad max-
ρ
imum at T 0.625 and a comparatively weak sample–size 0.50 µ
≈
dependence;pairoccupationprobabilitiesR werefoundto
JK R
beessentiallyindependentofsamplesize,similarlytoquanti- eo
tiessuchasH andρ;theirresultsforthelargestsamplesize 0.25
∗
L = 160 are presented in FIG. 4 as well; the three quanti-
R
tiesevolvewithtemperatureinagradualandmonotonicway, ee
andtheirplotssuggestinflectionpointsbetweenT = 0.6and
0.00
T = 0.625, roughly corresponding to the maximum in ρµ. 0.40 0.50 0.60 0.70 0.80
Short–andlong–rangepositionalcorrelationshavebeencom- T
paredviathe excessquantitiesR , whose simulationresults
∗oo
for the largestsample size are reportedin FIG. 5, showinga FIG. 4: Simulation estimates for the three pair occupation proba-
broadmaximumwellabovethetransitiontemperature;notice bilitiesRJK obtainedforatwo–dimensional samplewithlinearsize
L = 160,alongwithsimulationestimatesforρ ,obtainedwithdif-
alsothatR remainsrathersmall,reflectingtheabsenceofa µ
∗oo ferentsamplesizes;samemeaningofsymbolsasinFIG.1.
purelypositionaltermintheinteractionpotential. Ingeneral
ithasbeenfoundthattheobtainedpropertiesexhibitarecog-
nizablesimilaritywithwiththecounterpartsfortheisotropic 0.04
LG–PR model15. Some other remarks are appropriate, con-
cerning the temperature dependenceof fluid–like properties;
letusrecallthatourpotentialmodelcontainsnoexplicitposi- 0.03
tionalinteraction,i.eλ=0inEq. (5),sothatpositionalprop-
ertiesareessentiallydrivenbytheorientation-dependentpair
Cpotenatniadl.ρSiomniltahreitoienseahnadnddi(ffFeIrGe.n1ce),sainndthρeoarboRve p(FloItGs.fo4r) *Roo 0.02
µ,V T µ JK
ontheotherhand,canthenbecorrelatedwiththe”implicit”or
“explicit”dependenciesofthenamedquantitiesonthepoten-
tialenergy(seeremarksattheendoftheprevioussection);the 0.01
pairinteractionenergychangesmostrapidlyatT 0.6,and
≈
this is reflected by narrowpeaks inC and ρ ; conversely,
µ,V T
quantitieswith “implicitdependence”onthe pairinteraction 0.00
energyarelessaffectedbythechange,asshownbyabroader 0.4 0.5 0.6 0.7 0.8
T
peakathighertemperatureT 0.625.
≈
FIG.5:SimulationestimatesforthequantitiesR ;discretesymbols
∗oo
have been used for simulation results obtained with L = 160, and
IV. CONCLUDINGREMARKS havethefollowingmeanings: circles:R′oo;squares:R′o′o.
In this paper, we have investigated two magnetic lattice–
gas models living on two–dimensional lattices, i.e. PR and tainedintheframeworkofmodelsdescribingquencheddilu-
XY, respectively. We have studied both lattice–gas models, tion; the ratio of the transition temperaturesof the LG mod-
in the absence of a pure positional interaction, using semi– els to the SL ones is approximately 0.82. The peak of the
Grand–CanonicalMonteCarloSimulation.Inthecaseofzero specific heat falls closer to ΘBKT than for the SL counter-
chemicalpotentialanumberofthermodynamicquantitiesob- part; this can be interpreted as reflecting the fact that, since
servable,includingsomecharacteristicoffluidsystems,were λ = 0,thestrengtheningofshort–rangeorientationalcorrela-
computed.Ingeneralithasbeenfoundthattheobtainedprop- tionsalso bringsabouta morerapid increase of density, and
ertiesforbothmodelsexhibitarecognizablequalitativesimi- hence,inturn,ofH∗.
larity. Asforthepossibleµ–dependenceofthepresentresults,let
Results for the susceptibility yield consistent evidence of us first notice that, since (∂ρ/∂µ) > 0, it is reasonable to
T,V
the existence of a BKT transition, suggesting the estimates expectthat χ, and hence Θ , are increasing functions of µ.
LG
Θ = 0.574 0.003andΘ = 0.733 0.003forthe tran- Ontheotherhand,theground–stateenergyfortheSL coun-
XY PR
sitiontempera±tures,withcorrespondingd±ensitiesattransition terpart is V per site, where V = 2; thus the lattice is es-
0 0
ρ = 0.918 0.004andρ = 0.924 0.003,respectively. sentiallys−aturatedwhenT 1andµexceedsafewmultiples
XY PR
Theseresults±areconsistentwiththoser±eportedinRef.25 ob- of V (say µ ranging betwe≤en 5 and 10V ); at the other ex-
0 0
6
treme,whenµisnegativeandofcomparablemagnitude,one Sezione di Pavia of Istituto Nazionale di Fisica Nucleare
expectsanessentiallyemptylattice,wherenoBKTLTshould (INFN);allocationsofcomputertimebytheComputerCentre
survive.Moreover,aBKTLTmaystillsurviveifµisnegative ofPaviaUniversityandCILEA(ConsorzioInteruniversitario
butnottoolargeinmagnitude;thisconjectureissupportedby Lombardoperl’ElaborazioneAutomatica,Segrate–Milan),
RG studies for the LG–PR17,18,19, and was confirmed by the as well as by CINECA (Centro Interuniversitario Nord–Est
simulation results obtained in Ref.15; it also agrees with the diCalcolo Automatico,Casalecchio diReno - Bologna), are
conclusionsofreferences22,23,24,25 fortwo–dimensionalmag- gratefullyacknowledged. H.Chamati’sstayatPaviaUniver-
netswith quencheddilution; forlargenegativeµ, the named sitywasmadepossiblebyaNATO–CNRfellowship;financial
RGtreatmentspredictphaseseparation,i.e. afirst–ordertran- supportaswellasscientifichospitalityaregratefullyacknowl-
sition between a BKT and a paramagnetic phase, and even- edged; the authors also thank Prof. V. A. Zagrebnov(CPT–
tuallythedisappearanceoftheBKTphase. We expecttoin- CNRSandUniversite´delaMe´diterrane´e,Luminy,Marseille,
vestigatetheµ–dependenceofthetransitiontemperatureina France)forhelpfuldiscussions.
moredetailedwayinfuturework.
Acknowledgements
The present extensive calculations were carried out, on,
among other machines, workstations, belonging to the
Present andpermanent address: InstituteofSolidStatePhysics, 18 A.N.BerkerandD.R.Nelson,J.Appl.Phys.50,1799(1979).
∗
72TzarigradskoChausse´e,1784Sofia,Bulgaria. 19 A.N.BerkerandD.R.Nelson,Phys.Rev.B19,2488(1979).
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†
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