Time Dependent Transverse Correlations in the
Ising Model in D Dimensions
Jo~aoFlorenio 1
, Surajit Sen 2
, and M. Howard Lee 3;4
1
Departamento deFsia,UniversidadeFederaldeMinasGerais,
30.123-970 BeloHorizonte,MG, Brazil
2
PhysisDepartment,State UniversityofNewYorkatBualo,NewYork14260,USA
3
PhysisDepartment, UniversityofGeorgia,Athens,Georgia30602, USA
4
KoreaInstitute forAdvaned Study,Seoul130-012,Korea
Reeivedon10August,2000
TheIsingmodelisknownwidelyforstudyingequilibriumbehavior. Weshowthatthemodelisalso
usefulfor studyingnonequilibriumbehaviorinsomespeialsituations. Themethodofreurrene
relations has beenapplied to obtainthe time evolution ofa non-ommutingspin operator. Also
obtained are the struture funtions and their time dependent behavior. It is shown that the
transverseomponentofthestatisuseptibilityanbeobtainedfromthedynamiresults.
I Introdution
Asiswellknown,theIsingmodelhasbeentraditionally
studied for avarietyof equilibrium properties. Can it
beusedtostudynonequilibriumpropertiesalso? Isthe
Ising model asusefulperhapsforstudying
nonequilib-riumbehavior?
Ourmodel onsistsofspin-1/2 operators,loalized
at D dimensional lattie sites. The internal energyis
limitedtopairwiseouplingofthez-omponentsofthe
spins whih are nns. This is a quantum mehanial
versionofthestandardIsing model.
Sinethezomponentsofspinsommutewithone
another,allthespinstatesoftheenergyarestationary.
Thustheydonotevolveintime. Thisiswellknownas
abasipropertyoftheIsingmodel.
Ifatransverseexternaleld (betteryet just in the
x-diretion) is turned on momentarily, it will ouple
withthex-omponentsofthespinsimpartingan
exter-nal energy to the system. If theeld is turned o, it
is as ifa state of thex-omponents of spins hasbeen
\reated"arryinganextraenergy.
Thesituation that hasbeenreatedis muh likea
dynamial piturethat isdesribedbylinearresponse
theory [1℄. Sine there is no longer an external eld,
the averagingis done with respet to theinternal
en-ergyonly.
What happens now to this state of the x
ompo-nentsofthespins? Letusonsiderthespinatjustone
lattiepoint(sayat 0)sinetheyareallequivalentby
translationalinvariane. Evidently[H;S x
0
℄6=0. Hene
the state of this spin is nonstationary and must now
evolvein timeunlikethestateofS z
0 .
Thetimeevolutionheremeansthat thisspinstate
isattempting to restore itself to a stationary stateby
givingothe aquiredenergy to itsneighboring spins
towhihitis oupledbytheexhangeinterations. It
attemptstodosobypushingtheneighboringspinsinto
anonstationarystate. Sinetheneighboring spinsare
alreadyin astationarystate,theyresistaeptingthis
energyandreturnittothesoureatsite0. Thisproess
goesonindenitely. What happensis aloalization of
theenergy,trappedaboutsite0. Thisisuniquetothe
Isingmodel. IntheXY model,forexample,theenergy
wouldbeomedeloalized.
The time dependent spin-spin orrelation funtion
willthusbeperiodi, neverdeaying. Thetrajetories
(seeSe. II)willbelosed. Thenonequilibrium
behav-iorwill notbeergodiin thesense ofthe usual
mean-ingofthatterm. Theloalizationmaybedesribedin
termsofdynami modesofdierentfrequenies,muh
likethenormalmodesofvibration. Theomplexityof
thesemodesevidently willinreasewith the
oordina-tionnumber.
Howanonestudythetimedependentbehaviorin
thisquantum Ising model? Clearlywemust solvethe
Heisenbergequationofmotion. Sinethemodelis
Her-mitian,thisanbedonemostsimplybythereurrene
relationsmethod [2-4℄.
Ordinarily one requires stati properties to obtain
dynami behavior. It is thus of interest to note that
forloalizedproblemssuhasinthisIsingmodelsome
statipropertiesare impliedbydynami propertiesas
weshalldemonstrate. Weshallonentrateonthe1D
model. Thedynamisisbasiallysimilartothatin1D.
II Reurrene relations method
Inthissetionweshallbrieyreviewthereurrene
re-lations method for solving theHeisenberg equation of
motionifasystemisHermitian. This generalmethod
wasintroduedintheearly1980sandtherehavebeen
anumberofappliationsmadesinetostudythetime
evolution in avarietyof Hermitian many-partile
sys-tems[5-16℄.
We are interested in obtaining the time
depen-dentbehaviorofatransversespin-spinorrelation
fun-tion < S x
0 (t)S
x
0
(0) >, where S
i
is the omponent
( = x, y, or z) of the spin-1/2 operator at the i th
lattie site, the brakets denote the anonial
ensem-bleaveragewithrespettotheIsingHamiltonianH =
J P
(ij) S
z
i S
z
j
,where(ij)denotesnnpairsinalattie.
A moregeneraltime dependentorrelationfuntion is
therelaxationfuntion(S x
0 (t);S
x
0
(0)),where theinner
produt meansthe Kubosalar produt. Notethat if
t=0, thisinner produt denesthe xxomponentof
the suseptibility. As notedin Setion I,[H;S x
0 ℄ 6= 0.
Hene S x
0
(t)=exp(iHt)S x
0
(t=0)exp( iHt)6=S x
0 (t=
0),wherewehaveset ~=1. Thetimeevolutionofthe
operatorS x
0
angiveaompletepitureofthetime
de-pendent behavior. Hene weshall onentrate onthis
quantityratherthanthestruturefuntions.
LetusdenotebyAthedynamialvariableatt=0,
i.e., S x
0
(t =0) =A. Sine weare interested in t 0
only, we an dene A(t) = 0 if t < 0. Aording to
thereurrenerelations method A(t) att 0maybe
regardedasavetorinarealized Hilbertspaeofd
di-mensions provided that H is Hermitian (as is for the
Ising Hamiltonian). Thus, A(t) may be given an
or-thogonalexpansionsinthisspaeas
A(t)= d 1
X
k =0 a
k (t)f
k
: (1)
Hereff
k
gisaompletesetofbasisvetorswhihspan
therealizedspae. That is,
(f
k ;f
k 0
)=0 if k 0
6=k; (2)
where theinner produtmeanstheKubosalar
prod-ut[3℄. Alsofa
k
(t)gisaompleteset oflinearly
inde-pendent funtions oftime. Theyrepresentthe
magni-tudes oftheprojetionof A(t) onto thebasisvetors
at time t. The Hermitiity requirement implies that
jjA(t)jj = jjA(0)jj for all t > 0, known as the Bessel
equality. It meansthatthemagnitudeorthelengthof
thevetorisaonstantofmotion. Thisisauseful
prop-onwhihweareoperatingisnotabstratbutrealized.
Hene dmaybeniteas weshallseeinthisproblem.
Sinethereisalwaysonedegreeoffreedomin
hoos-ing one basis vetorinitially, wehoosef
0
=A. This
hoie yieldsboundaryonditionsona
k (t)'s:
a
k
(t=0)=
1 ifk=0
0 ifk=1;2;:::
(3)
Observealsothattheorthogonalitypropertygives
a
0 (t)=
(A(t);A)
(A;A)
; (4)
sometimes known astherelaxationfuntion, themost
basitimedependentfuntion asweshallsee.
GivenEqs. (1),(2)and(3),thereurrenerelations
methodrestsonthefollowingfat: Iftheinnerprodut
meanstheKubosalarprodut,bothf
k anda
k satisfy
ertain unique reurrene relations. They are
three-termreurrenerelationsexeptthebasalones(k=0)
whihhaveonlytwoterms. Theyaregivenbelow: For
k=0;1;:::;d 1,
f
k +1 =
_
f
k +
k f
k
; (5)
k +1 a
k +1
(t)= a_
k (t)+a
k 1
(t); (6)
where
k = jjf
k jj=jjf
k 1 jj, f
1 = a
1
0,
0 1.
Herejjfjj=(f;f). Theseratiosofsuessivenormsare
alled reurrants. Notiethatourdenition ofanorm
is nonstandard but moreonvenient for ourpurposes.
Therewillbenoonfusionbythisusage. Equations(5)
and(6)arereferredtointheliteratureastheRR1and
RR2,respetively.
Observethat givenf
0
, theother orhigherf
k 'san
beobtainedquitesystematially. Thebasisvetorsare
thushierarhi,suessivelyrepresentinginreasing
di-mensions of the realized spae. The projetion
oeÆ-ients annot enjoy this property sine the basal one
a
0
(t)isnotknown.
Thereurrenerelations methods meansthat Eqs.
(5)and(6),orRR1and2,aretobesolved. Their
solu-tionsarethenthesolutionsforEq. (1). Thishasbeen
alreadydemonstratedinnumerousexamples[5-15℄.
Thereisonepartiularlyimportantrelation, whih
followsfrom Eq. (6),
1 a
1
(t)= a_
0
(t): (7)
This is one of the basal relations, whih follows from
Eq. (6)bysetting k=0. Therighthand side(rhs)of
Eq. (7) denotes therelaxationfuntion (see Eq. (4)).
Theleft-hand-side(lhs)ofEq. (7)denotestheresponse
funtion orutuations,e.g.,<A(t)A>. The
onne-tionbetweenthesetwophysialquantitiesisentralto
Letusdenea~
k
(z)=La
k
(t),whereListheLaplae
transform operator. If Lis appliedto Eq. (6),we
ob-tain
~ a
0
(z)=1=z+
1 =z+
2
=z+:::+
d 1
=z1=z+
1 ~ b 1 (z); (8)
a ontinued fration of order d 1. The seond term
on the rhs of Eq. (8), a ontinued fration of order
d 2,is knownas the memory funtion in the
gener-alized Langevinequation [18℄. There is a onvolution
relationbetweena
1 andb
1 .
III Appliation to the Ising
model
Heneforth weshall adoptthenotationS x i =x i ,S y i = y i , and S
z
i =z
i
, whih will simplify ourpresentation.
We aution that thesubsripts on f
k
(k =0;1;2;:::)
arenottobeonfusedwiththesubsriptson,e.g.,x
j .
There arenorelationsbetweenthetwo.
In this setion we will onsider theIsing model in
1D (linear hain) with periodi boundary onditions
imposed. This1D modelisfound toontainthebasi
struture of the Ising dynamis. Hene we will show
expliit details of our alulation. Afterward we will
remarkhowtheonlusionisgeneralizedtohigherD's.
III.1Reurrenerelationsanalysisfor1D
Ising model
Intermsofournewnotation,for1D
H= J X i S z i S z i+1 = J X i z i z i+1 ; (9) f 0
=A=S x
0 =x
0
: (10)
UsingtheRR1(Eq. (5)),
f
1 =J(y
0 z 1 +z 1 y 0 ): (11) Hene, jjf 1 jj= J 2 2 [(x 0 ;x 0 )+4(x
0 ;z 1 x 0 z 1 )℄; (12) 1 = jjf 1 jj jjf 0 jj = J 2 2
(1+4 ); (13)
where = (x 0 ;z 1 x 0 z 1 ) (x 0 ;x 0 ) : (14)
Notethatthexxomponentofthesuseptibility xx
=N(x
0 ;x
0
),whereNisthetotalnumberofspinsin
thesystem. Weanalsoalulate Eq. (13)byKubo's
theorem [1℄,
jjf
1 jj=(f
1 ;f
1 )=i
1 <[f 1 ;f 0 ℄>= 2J <z 0 z 1 >; (15) where <z 0 z 1 >= 1 4
tanhK ; (16)
where K =J=4and =1=k
B
T [19℄. Weshall later
exploit the equality betweenEqs.(12) and (15) to
ob-tainanexpliitform for ,animportant quantityfor
thedynami analysis.
Given
1
byEq. (13), we arenowin the position
toobtainf
2
bytheRR1,
f 2 = _ f 1 + 1 f 0 : (17)
UsingEqs. (11)and(13)inEq. (17),weobtain
f 2 =2J 2 ( x 0 z 1 x 0 z 1 ); (18) jjf 2 jj= J 2 4 (1 16 2 ); (19) and 2 = J 2 2 (1 4 2 ): (20)
Continuingthiswaywenextlookat f
3
bytheRR1
f 3 = _ f 2 + 2 f 1 : (21)
UsingEqs. (11),(18),and(20)inEq. (21)wendthat
f
3
=0; (22)
henealso
3 =0..
Thuswehavearrivedatanessentialresultthatthe
realizedHilbertspaeforA(t)=x
0
(t)hasbutthree
di-mensions,spannedbyf
0 ,f
1 ,andf
2
only. Theshapeof
thespaeis determinedby thetworeurrants
1 and
2
. Thetrajetoryofthisvetor,whihisonstrained
tothesurfaeofthisspae,islosed.
Giventhetworeurrants,weannowobtaina
0 ,a
1
anda
2
bytheRR2(Eq. (6))oralso~a
0
byEq. (8)and
thenbyinversetransformL 1
. Theyare
a 0 (t)= 1 ! 2 ( 2 + 1
os!t); (23)
a 1 (t)= sin!t ! ; (24) a 2 (t)= 1 ! 2
(1 os!t); (25)
where! =J=~(but ~=1), and
1 and
2
givenup
tothe funtion by Eqs. (13)and (20),respetively.
Observe that Eqs. (23) to (25) satisfy the boundary
onditions(see Eq. (3)). In addition, Eqs. (23) and
(25)satisfythebasalRR2(seeEq. (7)).
Finallyweanwrite downthetotaltime evolution
where for eah term on therhs wehavefound an
ex-pliit expression. The validity of Eq. (26) anbe
fur-thertestedthroughtheBesselequalityjjx
0
(t)jj=jjx
0 jj.
Notingtheorthogonality,weobtain
jjx
0 (t)jj
jjx
0 jj
=(a
0 )
2
+(a
1 )
2
1 +(a
2 )
2
1
2
=1; (27)
where thenal resultis obtainedbysubstituting
vari-ousidentitiesalreadyobtainedabove.
III.2 Dynamial impliations
UsingEq. (26)weanobtainanumberof
dynami-alresults. Forexample,weanimmediatelydetermine
thedynamistruturefuntion<x
0 (t)x
0
>asfollows:
FromEq. (26)
S(t)
N =<x
0 (t)x
0 >=
1
4 a
0
(t) iJ<z
0 z
1 >a
1 (t)+
J 2
2
( <z
1 z
2 >)a
2
(t); (28)
where we have used < f
1 x
0
>= iJ < z
0 z
1 > and
<f
2 x
0 >=J
2
=2( <z
0 z
2
>). Ifwedene
~
S(z)=LS(t)= ~
R(z)+i ~
I(z); (29)
the real and imaginary parts ~
R and ~
I (whih an be
read o from Eq. (28)) are related through
Kramers-Kronigrelations[1℄.
Asasatteringproblem,theterm ~
I(z)woulddenote
the absorptive part. Thus the dynami suseptibility
~
(z) is ontained in ~
I(z) whih weshall prove below:
Fromthereurrenerelationstheory[10℄
1 ~ a
1 (z)=
~ (z)
; (30)
where=N(x
0 ;x
0 ). Now
1 =
J 2
2
(1+4 )= 2J
<z
0 z
1 >
(x
0 ;x
0 )
; (31)
where theseond equalityis obtainedbyapplyingthe
Kubotheoremto jjf
1
jj(seeEq. (15)). Hene,
2J<z
0 z
1
>=
1 (x
0 ;x
0
): (32)
Observethatthelhsoftheaboveisthestatiterm
on-jugatetoa
1
(t),theseond termintherhsofEq. (28),
thustogetherorrespondingto(t)=L 1
~ (z).
Oneanalso obtainthedynami suseptibility
us-ingEq. (26)in thedenition[1℄: Fort>0,
(t)=i<[x
0 (t);x
0 ℄>=ia
1 (t)<[f
1 ;x
0
℄>=jjf
1 jja
1 (t);
(33)
where jjf
1
jj = (2J=) < z
0 z
1
>= (J=2)tanhK.
Hene,
~
(z=0)=
jjf
1
jj: (34)
Weshall seein Setion IVthat (z~ =0)<
T , where
T
meanstheisothermal suseptibility.
III.3 Higher dimensions [20℄
Thetime evolutionof x
0
in higherdimensionsmay
beobtainedin asimilarmannerasfor1D. The
essen-tialaspetin 1DisthattherealizedHilbert spaehas
d=3. AsalludedinSetionI,theHilbertspae
dimen-sionsturnouttobesimplyrelatedtotheoordination
number. For the Ising model in D lattie dimensions
theHilbertspaedimensionsare:
d=q+1; (35)
where qistheoordination number[20℄. Thus, for
ex-ample, inthehoneyomblattiethere isbutone more
basis vetorthan in the linear hain. The orrelation
funtions that enter depend onthe lattie dimensions
D. Otherwise the dynami strutures are determined
solelybyqalone.
Ifq !1, themodel is known asthe spinvan der
Waals model [21℄. Atthis limitthedynamial piture
hanges drastially. As d ! 1, the time orrelation
funtions are no longer periodi. It has already been
found through a reurrene relations analysis that if
T >T
a
k (t)=
t k
k! e
t 2
; (36)
where>0isaonstant.
IV Statis from Dynamis
To obtain dynami properties one ordinarily needs
stati properties as for example Eq. (26). Thus to
think that stati properties an be dedued from
dy-namipropertieswouldseemquiteunusualifnotlikely.
ButiftheHilbert spaedimensionalitydisnite asin
theIsingmodel(seeSetionIII),weanin fatobtain
ertainstatipropertiesfromdynamiresultsaswewill
illustratehere.
Considerthe xx omponent of the stati
susepti-bility,
=( X
x
i ;
X
x
j
)=N(x
0 ;x
0
): (37)
(Notethat= 1
T
,where
T
istheisothermal
sus-eptibility.) The suseptibility hasthis form,dierent
from the zz omponent, sine[H;x
0
℄6= 0. Theinner
produtappearinginEq. (37)isakindoftemperature
integral,i.e.,
(x
0 ;x
0 )=
1 Z
0 <x
0 ()x
0
>d; (38)
where
x
0
()=exp(H)x
0
Hene if we know the \temperature" evolution of x
0 ,
the rhs of Eq. (38) may be evaluated. But we
al-ready know the time evolution of x
0
(see Eq. (26)).
Ift! itherein,
x
0 ()=a
0 ( i)f
0 +a
1 ( i)f
1 +a
2 ( i)f
3 ; (40)
wherea
0 ,a
1 anda
2
anbeimmediatelyobtainedfrom
Eqs. (23),(24)and (25). If Eq. (40)issubstituted in
Eq. (38),weobtain
(x
0 ;x
0 )=
1
4 I
0
+i<f
1 x
0 >I
1 +<f
2 x
0 >I
2 ; (41)
where I
0 , I
1 and I
2
are \temperature" integrals,
eas-ily evaluated. Using < f
1 x
0
>= iJ < z
0 z
1 > and
< f
2 x
0 >= J
2
=2( <z
0 z
2
>) (see Setion III), we
obtain
(x
0 ;x
0 )=
1
8 [(1+
sinhu
u +8
(1 oshu)
u
<z
0 z
1 >
(1
sinhu
u )<z
0 z
2
>℄; (42)
where u = J. Observe that the rhs does not
on-tain . Using the results < z
0 z
1
>= 1=4tanhu=4,
<z
0 z
2
>=(1=4tanhu=4) 2
, and after some
rearrange-mentsweobtain
8(x
0 ;x
0
)=seh 2
K+ tanhK
K
; (43)
where K =u=4. Wehavereoveredthe known result
[22℄.
Theaboveresultmaybeusedto obtainan
expres-sionfor . Fromthedenition,Eq. (13)or(32),
1+4 =
2tanhK =K
tanhK =K+seh 2
K
: (44)
Hene,
=(x
0 ;z
1 x
0 z
1 )=
1
4
1 2Ksh2K
1+2Ksh2K
: (45)
Finallyomparing withthezero-frequenylimitofthe
dynamisuseptibility(z~ =0)(seeEq. (34)),wenote
theinequality[23℄
T
>(z~ =0): (46)
Inhigherdimensions oneanobtain,e.g., the
sus-eptibility. But sine the stati orrelation funtions
arenotknownexeptin2D,thesenewresultsmaynot
be as interesting as in 1D. However they an yield,
e.g., high temperature expansions muh more simply
thanthestandardmethod [20℄.
We oughtto mentionthat the stati suseptibility
(K) (see Eq. (43)) satises the bounds due to Falk
andBruh[24℄,
tanhp=p (K)
1; (47)
where
Y(K)=N <x 2
2
>=N=4 (48)
and
p 0
tanhp 0
j
p
=2KtanhK : (49)
ThelhsofEq. (47)isknownasastrongerlowerbound.
Oursolution,Eq. (43),suggeststhatthestrongerlower
bound ofFalk-Bruhmaynotbestrongenough.
V Disussion
That the time evolution of x
0 = S
x
0
requires a nite
number of the basis vetors is perhaps most
remark-able. It impliesdynamiallythat ifenergyisimparted
tothisspinbysomeexternalperturbation, itdoesnot
beome deloalized. This energy goesbakand forth
between its neighbors, desribing in eet a periodi
motion. From the perspetive of theHilbert spae, it
delineatesalosedtrajetory. Wemustonlude
there-forethat the timeevolutionin thisaseis notergodi
inthesenseoftheusualmeaningofthisword.
Iftheinterationsontainedotherterms(e.g.,XX
interations), the dynamis would hange [25℄. The
energy would beome deloalized as there are
numer-ousothernonstationaryspinstates. Thedimensionsof
therealizedHilbertspaewouldbeomeinnitelylarge
andthetrajetorywouldbenolongerlosed,butopen.
Inertainstatilimitssomeexatsolutionshavebeen
foundbythereurrenerelationsmethod [5-16℄.
We havedemonstrated in somedetail for 1D that
thereurrenerelationsmethod isapowerfulyet
sim-pletehniqueforobtainingveryprofounddesriptions
of dynamis for Hermitian systems. There are other
propertiessuhasthememoryfuntion,subspaes,not
exploredinthispaper,whihareofspeialsigniane
todynamialproesses[26℄.
Referenes
[1℄ W.MarshallandR.D.Lowde,Rep.Prog.Phys.31,705
(1968).
[2℄ M.H.Lee,Phys.Rev.B26,2547(1982).
[3℄ M.H.Lee,Phys.Rev.Lett.49,1072(1982).
[4℄ M.H.Lee,J.Math.Phys.24,2512 (1983).
[5℄ J.FlorenioandF.C.SaBarreto,Phys.Rev.B60,9555
(1999).
[6℄ R.S. Sinkovits and S. Sen, Phys. Rev. Lett. 74, 2686
(1995).
[7℄ I.Sawada,Phys.Rev.Lett.83,1668(1999).
[8℄ S.G.Jo,K.H.Lee,S.C.Kim,andS.D.Choi,Phys.Rev.
E55,3676(1997).
[10℄ J. Hong and M.H. Lee, Phys. Rev. Lett. 70, 1972
(1993).
[11℄ J.KimandI.Sawada,Phys.Rev.E61,R2172(2000).
[12℄ V.S. Viswanath and G. Muller, Reursion Method
(Springer-Verlag,Berlin,1994).
[13℄ E.B.Brown, Phys.Rev.B45,10809 (1992);49,4305
(1994).
[14℄ D. Vitali and P. Grigolini, Phys. Rev. A 39, 1486
(1989).
[15℄ A.S.T. Pires, Helv.Phys.Ata61, 988(1988); Phys.
St.Sol.b129,136(1985).
[16℄ J.F. Annett et al., J. Phys. Cond. Matter 6, 6455
(1994).
[17℄ R.Kubo,Rep.Prog.Phys.29,235(1966).
[18℄ U.Baluani,V.Tognetti,andR.Valluauri,Phys.Rev.
A19,177(1979).
[19℄ H.E. Stanley, Introdution to Phase Transitions and
CritialPhenomena(OxfordUniv.Press,Oxford,1971).
[20℄ S.Sen,Ph.D. Thesis,UniversityofGeorgia, 1990.
[21℄ M.H.Lee,I.M.Kim,andR.Dekeyser,Phys.Rev.Lett.
52,1579(1984).
[22℄ M.E.Fisher,J.Math.Phys.4,124(1963).
[23℄ H.Falk,Phys.Rev.165,602(1968).
[24℄ H.FalkandL.W. Bruh,Phys.Rev.180,442(1969).
[25℄ J. Florenio and M.H. Lee, Phys. Rev. B 35, 1835
(1987).
