Heuristi Approah to the Critial Dynamis
of the Ising Model
P.R. Silvaand V. B. Kokshenev
Departamentode Fsia,ICEx,UniversidadeFederaldeMinasGerais,
CaixaPostal702,CEP30123-970,BeloHorizonte,MinasGerais,Brazil
Reeivedon1November,2000
Wedisussorder-disorderphasetransitionsoflassial andquantumIsingmodelsonthe basisof
theGinzburg-Landau-Wilsonfree-energyationandamodiationofasalingmethodproposedby
Thompson. Thetwo-timeritialrelaxationsenario,drivenbythermalandquantumutuations,
respetively,isgivenintermsoftheslow-downritialexponentoftheorder-parameterexitations.
The lassial Ising model (IM) and the quantum
transverse Ising model (TIM) are generially used to
study ooperativephenomena. Thethree-dimensional
(d=3)TIMhasseveralappliationstorealsystems[1℄,
inluding the desription of strutural order-disorder
phase transitions observed in ferroeletris at a
rit-ial temperature T
. In the d = 1 ase, the TIM
withnearest-neighborinterationsexhibitsT =0
long-range-order forsmall enoughtransverse elds <
.
In the urrent work we propose a new approah to
the problem ofthe ritialdynamis. It isbased ona
heuristi methodintrodued byThompson[2℄and
de-velopedbyoneoftheauthors[3℄. Thompson'smethod
is basedon theideathat the ation,as wellasa
rele-vantfreeenergyofthesystemofmanysalesoflength,
are limitedfuntions. This heuristi method was
for-mulated through a set of assumptions (see
presrip-tions A,B,C in Ref.[2℄) that avoid dealing with the
renormalization-group(RG) equationsofmotion.
Theritial dynamisof agivensystem isa
mani-festationofitsdynamialinstabilityandanbetreated
intermsoftheolletiveGoldstone-typesoftmode
van-ishing atthe transition temperatureT
. More
spei-ally, order-disordertransitions display atwo-time
re-laxation senario given by relaxation times
with
=(entralmode)and=s(softmode)for,
respe-tively,diusive-typerelaxationandosillatory-type
be-havior. Thesearerelatedtothenite-timeloal
order-parameterexitations!
ofwidth
=Im!
. We
dis-usstheritialdynamisintermsoftheslowing-down
dynamial ritial exponents dened as
_j"
T j
with " = (T T )=T as " goes to zero. We take
=z,where standsfortheorrelationlength
expo-nentandzisthegrowth-lawexponentthatdistinguishes
diusionmehanisms.
If we assume a Brownian mehanism, the
d-dimensional lassial IM predits diusive-type
relax-ationdynamis. Itsslowing-downexponentis[3℄
(lass)
=z
=
(d+2)
2(d 1)
for d4 andz=2 :
(1)
For the real d = 3 ase,
=
5
4
was independently
dedued(seeEq.15inRef.[4℄)forthequantumTIMby
theGreenfuntionmethod. Thisorrespondstoforthe
universalritialbehaviorofthelassialandquantum
Isingmodels[5℄ford>1.
Besidestheritialrelaxationdynamisgivenbythe
entralmode,wedisuss theritial osillatory
behav-ior. In the ontinuum representation, the ation
in-duedby thetime-dependentorder parameterand
as-soiatedwiththefreeenergyanbegivenby
A
T =
Z
L d
1
2
jr (x)j 2
+ "
T r(L)
2
j (x)j 2
+ u(L)
4
j (x)j 4
+ M
2 j
(x)
t j
2
d d
x; (2)
where standsforthesalareldonjugatetothe
or-der parameter and the integration is performed over
a d-dimensional hyperube of volume L d
. The
o-eÆients r(L) and u(L) are not ritial. The last
term,dependingon theeetivemassM, extends the
Ginzburg-Landau-Wilsonfree-energyationtoinlude
BasedontheideasofThompson[2℄,weassumethat
eah term in Eq.(2) is of a nite magnitude. To
esti-mate theritialexponentsit isnotneessaryto
eval-uate with auray the terms of the sum. Therefore,
weansimplify ourworkassumingthatanytermisof
theorderofunity. Adoptingthisshemeand
onsider-ingastationarymotionofthesystem, =
0 e
i!t
,we
havethefollowingestimatesfortherstandthefourth
termsinEq.(2),
Z
jr (x)j 2
d d
xsj Z
( M! 2 2
(x))d d
xjs1: (3)
This an be rewritten as hj j 2 iL d 2 Mhj!j 2 ihj j 2 iL d
; where the angle brakets mean the
integration (2). In order to perform themass saling,
a harateristi length assoiated with the quantum
partilebehavioristreatedasitsComptonwavelength,
namely,
=h=M:Thisyields M/L 1 and,hene, hj!j 2 i 1=2 / L 1=2
: By introduing the soft-mode
fre-queny!
s / "
=2
T
and itswidth
s / "
s
through
therelationshiph!i=!
s +i
s
;aswell asthe
orrela-tionlength / "
neartheritialtemperature,we
have! 2 s + 2 s / " T
. Usingthe known Fishersaling
equation( =(2 )),andtheinequality>=2;we
naturallyarriveat the overdamped soft-mode regime,
i.e.
s >> !
s
, that gives
s
==2. In ombination
withEq.(1)fortheorrelation-lengthritialexponent
; this provides the slowing-down ritial exponent
forthe overdamped osillatory motion ofthe Ising-like
system[5℄, (lass) s = 2 =
(d+2)
8(d 1)
; for d4: (4)
Oneanseethat,unliketheaseoftherelaxation
mo-tion regime given in Eq.(1), the osillatory motion is
haraterized by non-Browniansuperdiusion
dynam-isthatimpliesz=1=2. Inreald=3spae,itfollows
from Eq.(4) that
s
= 5=16 = 0:313. It should be
pointed out that this is lose to 0 s = 1 4 assoiated
with the soft-mode instability of the high-symmetry,
disordered paraeletriphase dedued from the
quan-tum TIMbytheGreen funtion method (seeEq.15 in
Ref.[4℄). TheT
-universalityofthelassialIMandthe
quantumTIMmodels,alsoknownfortheaseof
lassi-alandquantum Heisenbergspinmodels,isduetothe
fatthatthermalutuationsdrivethesystemthrough
thephasetransition. As aonsequene,therelaxation
mehanism doesnotdepend onthedetails introdued
ussions,see Ref.[5℄ ). Meanwhile, this isrestritedto
dimensionsd>1:
Inthed=1asetheTIMwithnearest-neighbor
in-terationsexhibitsT =0long-range-ordernearthe
rit-ialeld
. Thetransitionisdrivenbyquantum
u-tuations andanbemappedinto theritialbehavior
oftheIMwiththedimensioninreasedbyone[6,7,8℄.
By studying the T =0 xed-point saling properties,
Continentino demonstrated[10℄ that the TIM exhibits
adimensionalshiftfromdtod+zwithrespetto the
IM,where zisthegrowth-lawexponent[9,10℄.
Oneofthe substantialharateristisofthe ation
introdued in Ref.[3℄ is its Galilean invariane. The
eetivity of the lassial ation (2) to nd the
riti-al behavior driven by thermal utuations has been
demonstratedabove. Generally,thereisalengthL
min
orrespondingtothelowerwavelength(ultraviolet)
ut-o. Forthelassialase(2)oneanshowthat L
min is
large enough to avoid a relativisti treatment. On
the ontrary, the quantum behavior requires the low
wavelengthregimethat anbeintroduedthroughthe
Lorentz invariant ation (see Eq.(5) below). Wethen
write the following quantum ation endowed with a
Lorentzianinvarianeforthesalarorderparameter,
A = Z d d+1 x 1 2 ( (x))( (x)) " r(L) 2 2 (x) + u(L) 4 4 (x) ; with = 0 0 1 1 2 2 3 3 ; (5)
where (x)standsforthesalareld,andthe
integra-tionisperformedoversome(d+1)-dimensional
spae-time hyperubeofvolumeL d+1
. Thejointintegration
in time-like( = 0) and spae-like( = 1;2;3)
oor-dinatesisperformedinaordanewiththerelativisti
treatmentoftheproblem. Asbefore,r(L)andu(L)are
notritial,but " =j
j=
measuresadeviation
fromtheritialeld.
Torevealtheritialbehaviorination(5),wealso
adopttheThompsonmethod[2℄: (A)theabsolutevalue
oftheeahintegratedterminEq.(5)isoforderofunity;
(B) in the limit L! 1; thefuntions r(L) and u(L)
remain nite, and (C) the utuation part of the
a-tion (5) is of order (d+1) L ; where 1 L (= p
r(L) ) is
theharateristiorrelationlength.
WithinthespiritoftheLandauapproahto
2
=" 2
L
; for <
; and 0 otherwise;
with
2
L
=r(L)=u(L) and =(" r(L)) 1=2
(" ) 1=2
L
: (6)
d
Within the Landau sheme, r(L) and u(L) are
onstants and the ritial exponents assoiated with
the order parameter, , and the orrelation length,
; are equal, i.e., = = 1
2
: Aording to
the renormalization-group reipe, the ritial
utu-ations are aounted through the orrelation length.
Thus, Eq.(6) provides theself-onsistentequation =
(" r()) 1=2
for the diverging orrelation length s
"
. Appliation of the presription A to the rst
termin Eq.(5)provides
Z
L d+1
r (x) 2
d d+1
xsL (d+1) 2
< 2
>s1; (7)
and, thus, < 2
> sL 2 (d+1)
. Forthe seond term,
onehas
Z
L d+1
" r(L) 2
(x)d d+1
xs" r(L)L d+1
< 2
>s1;
(8)
that gives" r(L)L 2
s1. Finally,thethirdtermyields
Z
L d+1
u(L) 4
(x)d d+1
xsu(L)L d+1
< 2
> 2
s1; (9)
wherewehaveadoptedaweak-orrelationassumption,
< 4
>t< 2
> 2
. Byapplyingtheresult(7),< 2
>
s L 2 (d+1)
; to Eq.(9), in ombination with the
pre-sriptionB,wehave
u(L)sL (d+1) 4
; for d+1<4; and s1; if d+14:
(10)
We see that there is a speial dimension, given by
d+1 = 4, above whih the exponents beome
las-sial. Fromrelation(7),< 2
>sL 2 (d+1)
;itfollows
that thedimension,givenbyd+1=2,isalsospeial.
Inbothases,weexpetlogarithmiorretions. Note
that, by putting L = in relation (8), weobtain the
RGresult(6).
TousethepresriptionCweintroduethe
normal-ized utuation order parameter, ' 2
= 2
= 2
L : Thus,
thequantumation(5)isgivenby
A = r(L)
2u(L) a ;
with
a = Z
L d+1
(r') 2
" r(L)' 2
+ r(L)
2 '
4
d d+1
x:
Wetreata astheutuationpartofthequantum
a-tion. Appliationof thepresriptionCmeansthat a
s (d+1)
L
=[r(L)℄ d+1
2
. Substituting intoEq.(11), and
using Eq.(10)along with the relation A s1(due to
thepresriptionA),wehave
r(L)sL 2(d+1) 8
(d+1)+2
; for d+14; and; s1; if d+14:
(12)
Solving Eq.(6) for , and taking into aount that
s"
;wearriveatourmain result,
=
(d+1)+2
4[(d+1) 1℄ ;
ford+14,and
= 1
2
; if d+14: (13)
This result an also be written in terms of the
slow-downritialexponent,
(quant)
=z = d
+2
4(d
1) ;
ford
4with
z=1 and d
=d+z: (14)
Otherwise (d
4 ), we ome bak to the lassial
result = 1
2
and z = 1: From Eqs.(1) and (14), one
anseethatthequantum-TIMritialbehavioranbe
mapped into the lassial IM ritial behavior by the
formaltransformationd!d+z:Thisisonsistentwith
thedimensional shiftpreditedbyContinentino[10℄.
Toonlude,wehavedisussedorder-disorderphase
transitionswithintheframeworkoflassialand
quan-tumIsingmodelsonthebasisofthe
Ginzburg-Landau-Wilsonationwithin theheuristisalingmethod
pro-posed byThompson. Theritial behaviorofthe
las-sialIsingmodelnearthenon-zeroritialtemperature
(fordimensionsd>1)andofthequantumIsingmodel
(d 1) near the ritial transverse eld, whih are
driven by thermal and quantum utuations,
respe-tively,aredesribedintermsoftheorrelation-length
andthegrowth-lawzritialexponents. Thetwo-time
order-parameterrelaxationsenarioisgivenintermsof
theslow-down =z ritial exponent of the
order-parameterexitations.
This work was supported by the Brazilian ageny
Referenes
[1℄ R.V.Stinhombe,J.Phys.C6,2459 (1973).
[2℄ C.J. Thompson,J.Phys.A9,L25(1976).
[3℄ P.R.Silva,Phys.,Stat.Sol.(b) 166,K103(1991).
[4℄ V.B.Kokshenev,M.C.NemesandJ.I.Kim,SolidState
Commun.98421(1996).
[5℄ V.B.KokshenevandP.R.Silva,Mod.Phys.Lett.B 12,
265(1998).
[6℄ P.Pfeuty,Ann.Phys.57,79(1970).
[7℄ A.P.Young,J.Phys.C8,L309(1975).
[8℄ M.Suzuki,Prog.Theor.Phys.56,1454(1976).
[9℄ J.A.Hertz,Phys.Rev.B14,1165(1976)
