Two-Dimensional Ising Model and Loal
Nonuniversality of Critial Exponents
R. Z.Bariev 1;2
1
Departamento deFsia,Universidade FederaldeS~aoCarlos,13565-905, S~aoCarlos,SPBrazil
2
TheKazanPhysio-TehnialInstituteof theRussianAademy ofSienes,Kazan420029, Russia
Reeivedon15August,2000
WeobtaintheloalmagnetizationofaplanarIsingmodelwithdefetsofdierenttypes. Itisshown
thatneartheritialpointtheloalmagnetizationhasanonuniversalbehaviorthatmanifestsitself
inthefatthat itsritial exponentis aontinuousfuntionofthemirosopiparametersofthe
system.
TheIsingmodelisthesimplestmodelofmagnetism
andwasproposedbyIsing[1℄in 1925. Themost
inter-estingpropertyofthismodelwasdisoveredbyOnsager
[2℄forthetwo-dimensionalmodelin 1944. Thismodel
hasaphasetransition and wasthe basisforthe
mod-ern theory of phase transitions. The 8-vertex model
wassolvedbyBaxter[3℄andhasanonuniversal
behav-iorthatmanifestsitselfinthefatthatitsritialindex
is aontinuous funtion ofthe mirosopi
parame-tersofthesystem. Inthisreportwepresentarigorous
andphenomenologialanalysis ofdierenttypesof
lo-al magnetization ofthe two-dimensional Ising model.
Firstlyweonsiderlineardefetsoftwodierenttypes
[4℄. The loal magnetization is alulated exatly for
this model, anda study ismade of its dependene on
the distane to the defet and on the size of the
de-fet. Themostinterestingfeatureofthesolutionisthe
dependeneoftheritialexponentoftheloal
magne-tizationontheinterationparameter.
Theinterationenergyofthelattiesonsideredan
berepresentedasasumoftwoterms
E=E
0
+E; (1)
whereE
0
istheenergyof thedefet-free lattie,
E
0 =
M
X
m=1 M N
X
n=1 N (J
1 s
mn s
m;n+1 +J
2 s
mn s
m+1;n ); (2)
and E is the energy of the perturbation due to the
preseneofthedefets. Forthelattieofthersttype,
E = (J 0
1 J
1 )
M
X
m=1 M s
mo s
M1
; (3)
andforthelattieoftheseondtype,
E = (J 0
2 J
2 )
M
X
s
mo s
N1
: (4)
Intheseexpressionss
mn
isthespinvariableonneted
with a lattie site having oordinates m and n , and
takesvalues1;J
1 ;J
0
1 andJ
2 ;J
0
2
aretheenergiesofthe
interationbetweenthehorizontalandvertialpairsof
neighboring spins, respetively. Here both diretions
M;N !1.
Wedenethe loalmagnetization ofalattiewith
defetsinanalogywiththedenitionofthe
magnetiza-tionofthespin-spinorrelationfuntion
<s
n
>= lim
L!1 <s
mn s
m;n+L
>=<s
m;n+L
>: (5)
We haveommitted from theleft-hand side of this
ex-pressiontherowindex beauseofthetranslational
in-varianeofthelattiein questionin thevertial
dire-tion.
Forspinswithlargeolumnnumbertheloal
mag-netization does not dier from the magnetization <
s>
0
ofadefet-freelattie
<s
0
> = [1 (sinh2K
1
sinh2K
2 )
2
℄ 1=8
;
= j1 T=T
j; (6)
K
i =J
i
=kT;wherekisBoltzmann'sonstant,Tisthe
temperature,andthesubsriptmarksthevalueofthe
funtion at theritialpointT
. Wehaveobtainedan
exatsolutionfortheloalmagnetizationandonsider
its behaviorin the most interesting- saling - region,
dened bythefollowingrelations:
1
!0;n!1; (7)
where istheorrelationradius ofthedefet-free
lat-tie
1
Ananalysisoftheexatexpressionshowsthattheloal
magnetization inthisregionhasthesalingform
<s
n
>=<s>
0
f(x;) (9)
Forthelattiewithdefetsofthersttype
x = (2n 1) 1
S 1=2
1 ;
= (C
1 C
0
1 )(C
1 C
0
1 )
1
; C 1
1
<1;
x = 2(n 1) 1
S 1=2
1 ;
= tanh2(K
2 K
0
2
); 1<<1;
C
i
= osh2K
i ;S
i
=sinh2K
i
: (10)
Themost interestingfeature ofthis solutionis the
nonuniversalbehavioroftheloalmagnetizationat
dis-tanesnsmallerthanhalftheorrelationradiusofthe
defet-free model. This nonuniversalityonsistsin the
fatthattheritialexponent
lo
ofthemagnetization
isaontinuousfuntionoftheinterationparameters
<s
n >
lo
forn; (11)
where
lo
=[aros( )℄ 2
: (12)
Thus,theloalmagnetizationofalattiewithalineof
defets,atdistaneslargerthan tothedefet,hardly
diers from themagnetization of adefet-free lattie.
The deviation beomes so substantialthat it leadsto
a hange of the ritial exponent
lo
that
harater-izes thedependene of theloal magnetization on the
relativetemperature.
We now disuss the reasonfor the nonuniversality
oftheritialbehavioroftheloalmagnetizationfrom
the point of viewof the generaltheory of phase
tran-sitions [5℄. Let us investigate ad-dimensional system
near aritialpointdeterminedbytheHamiltonian
H =H
0 +H
1 =
Z
(r)dr+ Z
!(r)dr (13)
withasmallparameter. Theenergydensity(r)and
perturbationdensity!(r)are loal quantities that
de-pend on the mirosopi variables whih desribe the
system.
Weinvestigatead-dimensionalsystemnear a
rit-ial pointdetermined bytheHamiltonian (1). We
as-sumethattheperturbationisausedbythepreseneof
ahomogeneousdefetofdimensionalityd
. Inthisase
the perturbationitself hasdimensionality d 0
(d
<d 0
).
Dene thefuntion (r) inthefollowingway
(r) = z(r);
z(r) = [(x
d
+1 )
2
+(x
d
+2 )
2
+:::+(x
d 0)
2
℄ =2
;
wherex
1 ;:::;x
d
are theaxesofaCartesianoordinate
system. The oordinate axes x
1 ;:::;x
d
are hosen in
suh a way that axes x
1 ;:::;x
d
0 span the subspae of
theperturbationinwhihthedefetsubspaewithaxes
x
1 ;:::;x
d
is embedded. The integration in the rst
term of (13) is arried out in a d-dimensional spae,
while theseond term involvesa d' -dimensional
sub-spaeintegration.
Theresultsofouranalysisare thefollowing[6℄. In
thease
d
eff =
! ;
where
d
eff
=max(d 0
;d); (15)
weanalulatethehangeintheritialexponentsto
linearorderin ,
' =
0
' +a
1 S
d 0
d ()
0
;
! =
0
! +b
1 S
d 0
d
(): (16)
Thus in this aseloal ritial exponents are
ontinu-ousfuntionsofthemirosopipertubationparameter
j
. Thisindiates thatthere isaloal violationofthe
universalityhypothesis.
Suhasituationanourinthegeneralase,and
notonlyfortheaseofaspeial symmetry-a
hara-teristiof aspeial symmetry -aharateristiof the
defetmodels investigated earlierwhose
dimensionali-tieswereintegers.
Reent developments in this eld of investigation
anbefoundin Refs. [6-8℄.
This work wassupported in part byConselho
Na-ional de Desenvolvimento Ciento e Tenologio
-CNPq-Brazil,andbyFunda~aodeAmparoaPesquisa
doEstadodeS~aoPaulo-FAPESP-Brazil.
Referenes
[1℄ E.Ising,Zeit.Phys.31,253(1925).
[2℄ L.Onsager,Phys.Rev.65,117(1944).
[3℄ R.Baxter,Phys.Rev.Lett.26,832(19710.
[4℄ R.Z.Bariev,JETP77,1217(1979).
[5℄ R.Z.Bariev,JETP94,374(1988).
[6℄ B.N.Shalaev,Phys.Rept.237,129(1994).
[7℄ F. Igloi, I. Peshel and L. Turban Adv. Phys. 42,
683(1993)
