Spin Pair Correlation of the ANNNI
Chain in a Field
Nelson AlvesJr. and Carlos S.O. Yokoi
Instituto deFsia,UniversidadedeS~aoPaulo,
CaixaPostal66318,05315-970 S~aoPaulo,SP, Brazil
Reeivedon10Otober,2000
We study the spin pair orrelation funtion of the one-dimensional Ising model with ompeting
nearest and next-nearest neighbor interations, or ANNNIhain, in the presene of anexternal
eld. Ofpartiularinterestarethedisorderlineswhereexponentialdeayofthespinpairorrelation
hangesfrommonotonitoosillatory. Weextendpreviousstudiesforhighereldvaluesandobtain
asymptotiexpressions for disorderlinesatlowtemperatures. Wealsoobservereentrant disorder
lines.
I Introdution
Theaxial next-nearestneighborIsing(ANNNI)model
isanexampleofafrustratedsystem[1℄displaying
mod-ulated orderat lowtemperatures in twoandthree
di-mensions [2, 3℄. Inonedimensionthemodelisexatly
solvableandhasbeeninvestigatedinzeroeld[4-10℄,in
thepreseneofaeld[11,12℄andatzerotemperature
[13,14℄. Ofpartiularinterestistheexponentialdeay
ofthespinpairorrelationfuntionwhihmaybe
mod-ulatedbyanosillatoryfatorwithatemperature-and
eld-dependent wave number. The line in the phase
diagramthatseparatestheregionwithzerowave
num-ber from non-zero wave number has been alled the
disorderline [7,8℄. The mainpurpose ofthis paperis
to further investigate the ANNNI hain in aeld and
examine someaspetsof the spin pairorrelation not
onsidered previously.
Theenergyofthemodel anbewritten
E= N
X
i=1 (J
1
i
i+1 +J
2
i
i+2 +H
i )+E
0 ; (1)
where
i
= 1 and the periodi boundary ondition
i+N
=
i
is assumed. J
1 and J
2
are the
near-est andthenext-nearest neighbor interations,
respe-tively, and H is the magneti eld. For low
tempera-turenumerialalulationsit isonvenientto takeE
0
to bethegroundstateenergy, sinethishoieavoids
the divergene of the eigenvalues of the transfer
ma-trix. We will restrit our disussion to ferromagneti
nearest neighbor interation (J
1
> 0),
antiferromag-neti next-nearest neighbor interation (J
2
< 0) and
positiveeld(H >0). Theratio J
2 =J
1
measuresthe
ompetition between nearest and next-nearest
neigh-bor interations. Thezerotemperaturephasediagram
intheplaneHversus J
2 =J
1
isshowninFig. 1. There
areonlytwophases: Intheferromagnetiphaseallthe
spinspoint up,whereas in theantiphasestatetwoup
spinsalternate withtwodown spins[13, 14℄. Weshall
prinipally onsider parametervaluesyielding a
ferro-magnetigroundstatewith
E
0 =
N
X
i=1 (J
1 +J
2
+H): (2)
Thesolutionofthisproblemanbeformulatedinterms
ofthetransfermatrixVwithelements[9,12,15℄
h
1
2 jV j
3
4 i=Æ
2;3 x
1
1
2
y 1
1
4
z 1
1
; (3)
where
x=e J
1
; y=e J
2
; z=e H
; (4)
with =1=k
B T, k
B
beingBoltzmann's onstant and
T theabsolutetemperature. Theeigenvaluesaregiven
bytheseularequation
f()=a
0
4
+4a
1
3
+6a
2
2
+4a
3 +a
4
=0; (5)
where
a
0
=1; a
1 =
1
4 (1+z
2
); a
2 =
1
6 (1 x
4
)z 2
;
a
3 =
1
4 x
4
z 2
(1+z 2
)w; a
4 = x
4
z 4
w 2
; (6)
with
w=y 4
1: (7)
The nature of these eigenvalues is determined by the
disriminant[16℄ofthequartiequation(5),
=(a
0 a
4 4a
1 a
3 +3a
2
27(a
0 a
2 a
4 a
0 a
2
3 a
2
1 a
4 +2a
1 a
2 a
3 a
3
2 )
2
: (8)
The ondition for two real and two omplex roots is
<0, andthat for fourreal or four omplexrootsis
>0[16℄. Sinef(1)=1andf(0)<0,therewill
alwaysbetworealrootswiththelargestrootbeingreal
and positive. Moreover, sine f 0
() < 0for < 0, if
theseondlargestrootinmagnitudeisrealthenitwill
bepositive. Let
1
denote the largest eigenvalueand
2
the seondlargesteigenvaluein magnitude. Inthe
thermodynamilimitN !1thespinpairorrelation
atlargedistanesr1hastheasymptotiform[12℄
h
i
i+r im
2
+Ae r=
osqr; (9)
where m = h
i
i is the magnetization, A is the
am-plitude, is the orrelation length and q is the wave
number,where
=ln
1
j
2 j
; q=arg
2
: (10)
If
2
is real then q = 0 and the orrelation will have
a monotoni exponentialdeay, whereas if
2
is
om-plextherewill beanosillatorydeaywithwave
num-ber q. The hangeover from monotoni to osillatory
deaydenes thedisorderline [7, 8℄orLifshitz
ondi-tion[10℄. Therefore theonditionforthedisorderline
is the vanishing of the disriminant given by (8).
Figure 1. Zerotemperature phasediagram ofthe ANNNI
hainin amagnetield. The solidline J
2 =J
1
=1=2+
H=2J1separatestheferromagnetiphasefromtheantiphase
(shaded region). The numbers indiate the limiting
val-ues taken by the wave numberq= as T ! 0 inside the
regions and over the lines separating them. The
disor-der lines interset the T = 0 axis for H=J1 < 2 on the
dashedline J2=J1 =1=2 H=4J1. The dot-dashed line
J2=J1=H=2J1 1separatesregionswithdierent
limit-ingwavenumbervaluesfor H=J1>2.
II Numerial Results
Let us rst onsider eld values H=J
1
< 2. Fig. 2
showsdisorderlinesintheplaneoftemperatureT
ver-sus J
2 =J
1
forvariousxedvaluesofthemagnetield
H. Thewavenumberofthespinpairorrelation(9)is
zerototheleftofthedisorderlineandnon-zerotothe
right. The behavior of the wavenumber q as a
fun-tion of temperature T is illustrated in Fig. 3for the
eld H=J
1
= 1:5 and various xed values of J
2 =J
1 .
An interesting aspet of these graphs [12℄ is that for
0:125 J
2 =J
1
1:25thelimitingvalueofwave
num-berqasT !0isnon-zerodespitethegroundstate
be-ingferromagneti. Thereis noontradition,however,
sinein the ferromagnetiphasem !1and !0as
T !0. Thustheseondtermin therighthand sideof
Eq. (9) vanishesand thewavenumberis notrelevant
as far as the ground state ordering is onerned. On
theotherhand,in theantiphasem!0and!1as
T !0. Thereforeitisthersttermin therighthand
side of Eq. (9) that vanishes and the wave number
q==2orretlydesribestheantiphasestate[12℄. In
thezerotemperaturephasediagramofFig. 1wehave
indiatedthelimitingvaluestakenbythewavenumber
q asT ! 0. Another interesting feature is the
reen-trantbehaviorexhibited bythedisorderlinesforlarge
enoughelds: As thetemperatureis loweredthewave
numberrstvanishesbut atstilllowertemperaturesit
beomesnon-zeroagain,asshowninFig. 3forthease
J
2 =J
1
= 0:08. We found numerially that the
reen-trant behaviorof disorderlines is presentfor allelds
H=J
1
>1:333.
Figure2. Disorderlinesforvariousxedvaluesofthe
mag-neti eldH=J1 <2. Thewave numberis zero tothe left
andnon-zerototherightoftheselines.Thestraightdashed
lines orrespond tothe asymptotiresultsfor T !0. For
Figure 3. Wave numberq as afuntionof temperature T
for H=J1 = 1:5 and various xedvalues of J2=J1. The
graph for J2=J1 = 0:08 shows the vanishingof thewave
numberatintermediatetemperaturesorrespondingtothe
reentrantbehaviorofthedisorderline.
Wewillnowonsider eldvaluesH=J
1
2. Fig. 4
showsthedisorderlinesin theplaneoftemperatureT
versus J
2 =J
1
for various xed valuesof the eld H.
Thewavenumberofthespinpairorrelation(9)iszero
totheleftofthedisorderlineandnon-zerototheright.
We observethatin ontrastto theaseH=J
1
<2, all
disorderlinestendto J
2 =J
1
=0asT !0. InFig. 5
thegraphsofthewavenumberqasafuntion of
tem-peratureT areshownfortheeldH=J
1
=3andseveral
valuesof J
2 =J
1
. Thelimitingvalueofthewave
num-ber q asT !0 is always non-zero. Also, although it
is outsidethe range of temperaturesshown, the wave
numberfor J
2 =J
1
=0:01vanishesonlyin some
tem-perature interval, reeting the reentrant behavior of
the disorder line. Finally we observe that, no matter
the eld strengthH, allthe disorderlinestend to the
T axisasT !1.
Figure4.Disorderlinesforvariousxedvaluesofthe
mag-netieldH=J12. Thewavenumbersarezerototheleft
and non-zero to the right ofthese lines. The dashedlines
orrespond to the asymptoti results for T ! 0. All the
Figure 5. Wave number q as a funtion of temperature
T for H=J1 =3 and various xed values of J2=J1. For
J2=J1 =0:01 the wave numbervanishesat intermediate
temperaturesorrespondingtothereentrantbehaviorofthe
disorderline,butthetemperaturevalueatwhihthewave
numberassumes non-zero value again is o the sale and
annotbeseen.
III Asymptoti Results
Inthissetionweobtainasymptotiexpressionsfor
dis-orderlinesatlowtemperatures. Sinex!0andz!0
asT !0,thedisriminant(8)isgiventoleadingorder
by
z 12
6912 "
4
1 12x 4
w 2
3 x
4
w
z 2
3
2 27 x
4
w 2
z 2
9 x
4
w
z 2
2 #
: (11)
Werstassumethat w!1asT !0. Then the
dis-riminant(11)anvanishonlyifx 4
w 2
=O(z 2
)andwe
have
z 12
6912 "
4
2 27 x
4
w 2
z 2
2 #
: (12)
The disorder line ondition = 0 gives x 4
w 2
=z 2
=
4=27or
J
2
J
1
1
2 H
4J
1 k
B T
8J
1 ln
27
4
: (13)
This result,validonly for H=J
1
< 2,implies that the
disorderlinesmeet theT =0axiswith eld
indepen-dentslope ln(27=4)=8= 0:238:::at
J
2
J
1 =
1
2 H
4J
1
; (14)
in good agreement with the numerial results of Fig.
withq==1=3inthezerotemperaturephasediagram
of Fig. 1. Letus nextassume that w !0as T ! 0.
Wethenhave
z 12
6912 "
4
1 3 x
4
w
z 2
3
2 9 x
4
w
z 2
2 #
: (15)
Thedisorderlineondition=0givesx 4
w=z 2
=1=4
or
J
2
J
1
k
B T
16J
1 exp
4J
1 2H
k
B T
: (16)
Thisresult,validforH=J
1
>2,impliesexponential
ap-proahofthedisorderlines tothetemperatureaxisas
T !0, in good agreement with thenumerial results
of Fig. 4. Finally, the asewhere w remains nite as
T !0shouldorrespondtoH=J
1
=2. Thenx 4
=z 2
=1
andwehave
z 12
6912 h
4(1 3w) 3
2 27w 2
9w
2 i
: (17)
The disorder line ondition = 0 gives w = 5=27.
Therefore
J
2
J
1
1
4 ln
32
27
k
B T
J
1
: (18)
Thisresultshowsthatthedisorderlineapproahesthe
originasT !0with slopeln(32=27)=4=0:04247:::,
in goodagreementwiththenumerialresultofFig. 4.
IV Summary
Wehaveextendedpreviouswork ontheANNNI hain
in aeld [12℄byonsidering thespinpair orrelations
for all eld values. We have shown that the
disor-der lines have dierent low temperature behavior for
H=J
1
<2,H=J
1
=2andH=J
1
>2. Numerialresults
were onrmed byasymptoti alulationsforT ! 0.
Wehavealso shown that thedisorderlines have
reen-trantbehaviorforH=J >1:333.
Aknowledgments
The authors aknowledge nanial support from
BrazilianageniesFunda~aodeAmparoaPesquisa do
EstadodeS~aoPaulo(FAPESP)andConselhoNaional
deDesenvolvimentoCientoeTenologio(CNPq).
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