Dynamial Behavior of the Four-Body Transverse
Ising Model with Random Bonds and Fields
Beatriz Boehat, ClaudetteCordeiro,
InstitutodeFsia,UniversidadeFederalFluminense,
24210-340Niteroi,RiodeJaneiro,Brazil
O.F. de Alantara Bonm,
Department ofPhysis,ReedCollege,Portland, Oregon97202, USA
J.Florenio, and F.C. SaBarreto
Departamento deFsia-ICEX,UniversidadeFederaldeMinasGerais,
30123-970BeloHorizonte,MinasGerais,Brazil
Reeivedon10August,2000
Westudytheeetofrandombondsandeldsonthedynamialbehavioroftheone-dimensional
transverseIsing modelwith four-spininterations. We onsidernitehainsof inreasingsize to
determinethetime-dependentorrelationfuntionandthelongitudinalrelaxation funtionofthe
innitehain. Inthisfullydisorderedsystemweobservearossoverfromaolletivemodetypeof
dynamistothatofaentralregime.
Thedynamialbehaviorofpurequantumspin
sys-temshasbeenthesubjetofmuhresearhativityin
the pastdeades[1℄. Onlyreently, moreattentionhas
been devoted to the investigation of the time
evolu-tionofdisorderedsystems. Anumberofstudiesabout
the phase diagram and thermodynami funtions has
demonstrated that thebehaviorofmagneti materials
maybedrastiallyaetedbythepreseneof
random-ness. Amongthesystemsstudied,thetransverseIsing
model(TIM) hasattratedonsiderableinterestin
re-entyears[2,3,4, 5℄andisregardedasoneofthe
sim-plestmodelswith non-trivialdynamis.
Inthisworkweareinterestedin thetimeevolution
oftheone-dimensionalTIMwithfour-spininterations.
ThemodelHamiltonianis
H = 8 L
X
i=1 J
i S
z
i S
z
i+1 S
z
i+2 S
z
i+3 L
X
i=1 B
i S
x
i
; (1)
where S
i
( = x;y;z) is the spin-1/2 operator at
site i and L is the number of sites of the lattie.
The four-spin interations J
i
and the magneti eld
B
i
are unorrelated variables hosen at random from
the probability distributions P(J
i
) and P(B
i ),
re-spetively. There have been several studies on the
modelwithuniformexhangeinterationandzero
mag-malization group[9℄, series expansions[10℄ and Monte
Carlo simulations[8, 11℄. The resulting phase
dia-gram has a triritial point whih separates regions
of rst and seond-order transitions. The pure TIM
was also employed to desribe the phase transition
in poly(vinylidene uoride-triuoroethylene)
[P(VDF-TrFE)℄opolymers[12℄. Morereently,somestudieson
the dynamis of this model in the presene of
disor-dershowedthatthesystemundergoesarossoverfrom
aolletive mode exitation regime to aentral mode
type of dynamis asa funtion of the bond and eld
energies[4, 13℄. Dierently from the usual two-body
model[14,15℄,thefour-spininterationmodeldoesnot
allowaGaussiandeayfortheorrelationfuntion.
The problem of disorder in magneti systems is a
fasinatingphenomenonthathasbeometheobjetof
intensestudy. Inmostmaterials,thepreseneof
mag-neti and nonmagneti ions leads to a distribution of
exhangeouplingsbetweenpairsofspins. Intheideal
ase,therandominterations J
i
fordierentpairsare
totallyunorrelated. Thesimplestaseourswhenall
theouplingsareofthesamesignbutvaryinstrength.
Other interestingbehaviorours when the model
in-ludes random magneti elds. In this ase, there is
J
i
or to followthe loal eld B
i
. The interation
en-ergies an be quite smaller than the thermal energy
even, say, at room temperature. In those aseswhere
J
i
=kT << 1 and B
i
=kT << 1 the dynamis an be
studiedintheinnite-temperaturelimit. Inthatlimit,
itisawellknownfatthat thedynamis isinsensitive
to the sign of the interation, sine only even powers
oftheinterationouplingandeldome intothe
mo-mentsoftheautoorrelationfuntion[4,15,16℄.
Thedynamisofthefour-spinTIManbeobtained
throughthetime-dependentautoorrelationfuntionof
S z i denedby C(fJ i g;fB i g;t)=
<S z i S z i (t)> <S z i S z i > : (2)
In the high temperature limit it takes the following
form, <S z i S z i (t)>= 1 2 L TrS z i e iHt S z i e iHt : (3)
We shall peform our alulations in nite rings, i.e.,
linear hains with periodi boundary onditions. The
exatbehaviorofthethermodynamisystems,atshort
times, is inferred from the dynamis obtainedfor
sys-temsofseveralsizes(L=7,9and11). Numerialresults
willbepresentedonlyforthelargestring(L=11),whih
showssignsofhavingattainedtheasymptotibehavior
ofthethermodynamisystemat largetimes.
In the general problem of quenhed disorder, the
physialpropertiesofasystemareobtainedby
perform-ingaongurationalaverageinthestatistialensemble
of realizations of the random variables. Formally, the
averagedautoorrelationfuntion isdenedas,
C(t)= Z Z C(fJ i g;fB i
g;t)P(J
i )P(B i )dJ i dB i : (4)
InourapproahthemeanautoorrelationfuntionC(t)
is obtained as follows. We rst diagonalize the full
Hamiltonian to determine the energies
n
and
eigen-vetorsjn > for alarge number of realizations in the
statistialensembleofenergyouplingrandomness.We
employed1000 to10000ongurationstoobtain
aver-agesovertherandomvariables; sothat the errorbars
fall within the thiknessof the urvespresented. The
results are then used to determine the mean
autoor-relation funtion, whih is ast in the following form
[17℄, C(t)= 4 2 L X n;m os( n m
)tj<njS z
i jm>j
2
; (5)
Anotherquantity of interestis the averaged
longi-tudinal relaxationshapefuntion,denedas
(!)= Z 1 0 C(t)e i!t dt; (6)
where C(t) isgivenbyEq. (5). Therealpartof (!)
givesdiretly aphysially aessiblequantity,the
lon-gitudinal relaxation funtion,F(!) =Re (!), whih
anbemeasureddiretlyinnulearmagnetiresonane
(NMR) experiments |the so-alled NMR line shape
[18℄. Thelongitudinal relaxationfuntion is also very
usefulintheunderstandingofthedynamisofthe
sys-tem,sinedierentdynamibehaviorshavedistint
sig-naturesin thatquantity[19℄.
Weinvestigatedthefour-bodyTImodelhain
on-sidering bimodal probability distributions for the
ex-hange ouplingandmagnetield,
P(J
i
)=(1 p)Æ(J
i J
1
)+pÆ(J
i J 2 ); (7) and P(B i
)=(1 p 0
)Æ(B
i B
1 )+p
0 Æ(B i B 2 ); (8)
where p is the onentration of ouplings of type J
2 ,
and p 0
is the onentration of magneti eld of type
B
2
. We determined the time-dependent
autoorrela-tion funtion and the longitudinal relaxationfuntion
fortheset(J
i ,B
i
)withseveralvaluesofpandp 0
.
InouralulationsweonsideredthevaluesJ
1 =1
and J
2
= 0:5 in the bimodal probability distribution
for theexhangeinteration, andseveral valuesof the
onentration pof J
2
-ouplings. Theoupling J
1 =1
is kept xed and sets the energy and time sales. As
to the eld probability distribution weonsidered the
ases where B
1
=1:5 and B
2
= 0:5, alsowith several
onentrationsp 0
. Forsimpliity, wewill present here
onlythepartiularasewherep=p 0
. Theonvergene
ofourresultsasthehainsizeinreasesisreahedwith
theL=11hainin thetimedomain shownin the
g-ures, 0t 8. In all asesstudied, our alulations
alsoreovertheresultsforthepureTImodelwith
four-spininterations(p=0) [4℄.
Theresultsforthetime-dependentorrelation
fun-tionare shownin Figs.1and 2. Theenergyouplings
are J
1
= 1 and J
2
= 0:5 with probability p, and the
elds areB
1
=1:5andB
2
=0:5, thelatterwith
prob-ability p 0
= p = 0, 0:2, 0:5 and 0:8. The urves for
p= 0orresponds to the pure asedominated by the
stronger eld energy. The orrelation funtion is
os-illatoryand damped, falling oquiklytowardszero.
Thelongitudinalshapefuntionshowsapeakata
non-zero frequeny. Hene, at p = 0the system is at the
eld axis. As we inrease p from 0 to 0.2, the
or-relation funtion still osillates. However,the peak in
thelongitudinalshapefuntionhasnowmovedtowards
lowerfrequenies. Whenp=0:5, theorrelation
fun-tiondeaysmonotoniallyto zero. Theshapefuntion
isnowpeakedatzerofrequeny,andallthesystemhas
undergonearossoverto aentral modebehavior. As
p is inreased further, theorrelation funtion deays
more slowly, and theentralpeak is enhaned,asan
beseenfrom theasep=0:8.
Figure1. Time-dependentorrelationfuntionforthe
four-bodytransverseIsingmodelwithrandombondsandelds.
The ouplings and the magneti eldsare distributed
a-ording to bimodal probability distributions, and an
as-sumethevaluesJ1 andB1 withprobability(1 p)andJ2
andB2 withprobabilityp. Theenergyandtimesalesare
setbytheouplingJ1 =1:0. Theurvesrepresenttheases
J
2
=0:5, B
1
=1:5 and B
2
=0:5 for variousvaluesofthe
probabilityp.
Figure 2. Longitudinal relaxation funtion (in arbitrary
units)vsfrequeny. Thesystemundergoesarossoverfrom
olletivemodebehaviortoaentralmodeaspisinreased.
In onlusion, we have studied the dynamial
be-haviorofthetransverseIsing model withfour-spin
in-viaexatdiagonalizationofnitehains. Wehave
al-ulated the time-dependent orrelation funtions and
thelongitudinalrelaxationfuntionsforsystemofsizes
L =7, 9and 11. We present the resultsonly for the
size L = 11, for whih the dynami orrelation
fun-tionshavealreadyonvergedtothoseatthe
thermody-namilimit. Our resultsshow that disorderinduesa
rossoverfromaolletivemodetypeofdynamistoa
entralmodebehaviorasafuntion ofdisorder.
This work was partially supported by FAPERJ,
FAPEMIG,CNPqandMCT (Brazilianagenies).
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