Exat Solution of Asymmetri Diusion With N
Classes of Partiles of Arbitrary Size
and Hierarhial Order
F. C. Alaraz
Departamentode Fsia, UniversidadeFederalde S~aoCarlos, 13565-905,S~aoCarlos,SPBrazil
R. Z.Bariev
Departamentode Fsia, UniversidadeFederalde S~aoCarlos, 13565-905,S~aoCarlos,SPBrazil
The KazanPhysio-TehnialInstitute oftheRussianAademyofSienes,Kazan 420029,Russia
Reeivedon5August,2000
Theexatsolutionof theasymmetriexlusionproblem withN distint lassesof partiles( =
1;2;:::;N),withhierarhialorderispresented. Inthismodelthepartiles(size1)areloatedat
lattiepoints,anddiusewithequalasymmetrirates,butpartilesinalassdonotdistinguish
thoseinthelasses 0
>fromholes(emptysites). Wegeneralizeandsolveexatlythismodelby
onsideringthe moleulesineahdistint lass =1;2;:::;N with sizess (s =0;1;2;::: ), in
unitsofthelattiespaing. ThesolutionisderivedviaaBetheansatzofnestedtype.
I Introdution
Thesimilaritybetweenthemaster equationdesribing
time utuationsin nonequilibrium problems and the
Shrodinger equation desribing the quantum
utua-tions of quantum spin hains turns out to be fruitful
for both areas of researh [1℄-[15℄. Sine many
quan-tum hainsareknowntobeexatlyintegrablethrough
the Bethe ansatz, this provides exat information on
the related stohasti model. At the same time
las-sial physial intuitionand probabilisti methods
su-essfully applied to nonequilibrium systems give new
insights into the physial and algebrai properties of
quantum hains.
Anexampleofthisfruitfulinterhangeisthe
prob-lem of asymmetri diusion of hard-ore partiles on
the one dimensional lattie ( see [16, 17, 18℄ for
re-views). This model is related to the exatly
inte-grableanisotropiHeisenberghaininitsferromagneti
regime [19℄ (XXZ model). Howeverif wedemand this
quantumhaintobeinvariantunderaquantumgroup
symmetry U
q
(SU(2)), we have to introdue, for the
equilibrium statistial system, unusual surfae terms,
whihonthe otherhandhaveanieandsimple
inter-pretationfortherelatedstohastisystem[3,4℄.
In the area of exatly integrable models it is well 1
2
XXZ hain to higherspins is theanisotropi spin-S
Sutherlandmodel(grading
1 =
2
=:::=
2s+1 =1)
[20℄. Ontheotherhand intheareaofdiusionlimited
reationsa simple extension of the asymmetri
diu-sionproblemis theproblemof diusionwith partiles
belonging to N distint lasses( =1;2;:::;N) with
hierarhial order [22℄-[24℄ . In this problem a
mix-tureofhard-orepartilesdiusesonthelattie.
Par-tiles belonging to alass ( =1;:::;N) ignore the
preseneof those in lasses 0
>, i.e., they see them
in the same way as they see the holes (empty sites).
In[3℄itwasshownthat foropen boundaryonditions
the anisotropispin-1 Sutherland model and this last
stohastimodel,intheaseN =2,areexatlyrelated.
TheHamiltoniangoverningthe quantum ortime
u-tuationsofbothmodelsisgivenintermsofgenerators
ofaHekealgebra,invariantunderthequantumgroup
U
q
SU(3). In fat this relation anbeextended to
ar-bitraryvaluesofN,andthequantumhain assoiated
tothestohastimodelisinvariantunderthequantum
U
q
(SU(N+1))group. Inthispaperwederivethrough
the Bethe ansatz the exat solution of the assoiated
quantum hain, onalosed lattie. Reently [15℄ (see
also [14℄) we have shown that without losing its
ex-atintegrability,weanonsidertheproblemof
asym-metridiusionwithanarbitrarymixtureofmoleules
in-terhange positions, that is, there is no reations. In
thispaperweextendtheasymmetridiusionproblem
withN typesofpartileswithhierarhialorder,tothe
asewherethepartilesineahlasshaveanarbitrary
size,in units ofthe lattiespaing. Unliketheaseof
asymmetri diusion problem, we have in this ase a
nested Bethe ansatz [25℄. A pedagogial presentation
forthesimplestaseN=2waspresentedin [26℄.
Thepaperisorganizedasfollows. Inthenext
se-tion we introdue the generalized asymmetri model
with N types of partiles with hierarhial order and
derivetheassoiated quantum hain. Insetion3the
Betheansatzsolutionofthemodelispresented. Finally
insetion4wepresentouronlusions,withsome
possi-blegeneralizationsofthestohastiproblemonsidered
inthis paper,andsomeperspetivesonfuture work.
II The generalized asymmetri
diusion model with N lasses
of partiles with hierarhial
order
Asimpleextensionoftheasymmetriexlusionmodel,
in whih hard-ore partiles diuse on the lattie, is
theproblem where amixture ofpartiles belonging to
dierent lasses ( = 1;2;:::;N) diuses on the
lat-tie.. This problem in the ase where we have only
N = 2 lasses was used to desribe shoks [22℄-[24℄
in nonequilibrium and also has a stationary
probabil-itydistribution that an be expressedvia the
matrix-produt ansatz [27℄. In [28℄ it was also shown that
thestationary stateofthe aseN =3analso be
ex-pressedbythematrix-produtansatz. Inthismodelwe
have n
1 ;n
2 ;:::n
N
moleules belonging to the lasses
= 1;2;:::;N, respetively. All lasses of moleules
diuse asymmetrially, but with thesame
asymmetri-al rates,whenevertheyenounteremptysites (holes)
at nearest-neighborsites. However,whenmoleules of
dierentlasses,and 0
(< 0
),areattheirminimum
separation, the moleules of lass exhange position
with the same rate as they diuse, and onsequently
the moleules in the lass see no dierenebetween
moleulesbelongingtothelasses 0
> andholes.
We now introdue a generalization of the above
model,whereinsteadofhavingunitsize,themoleules
in eah distint lass = 1;2;:::;N have in general
distint sizes s
1 ;s
2 ;:::;s
N (s
1 ;:::;s
N
= 1;2;:::),
re-spetively, in units of lattie spaing. In Fig. 1 we
show some examples of moleules of dierent sizes.
We may think of a moleuleof size s as formed by s
monomers (size 1), and for simpliity, we dene the
position of the moleule as the enter of its leftmost
monomer. The moleules havea hard-ore repulsion:
theminimumdistaned
,in unitsofthelattie
spa-ing, between moleules and , with in the left,
is given by d
= s
. In order to desribe the
o-upany of a given onguration of moleules we
de-ne at eah lattie site i (i = 1;2;:::;L) a variable
i
(i=1;2;:::;L), taking thevalues
i
=0;1;:::;N.
Thevalues =1;2;:::;N representsites oupied by
moleules oflass=1;2;:::;N,respetively. Onthe
otherhandthevalue=0representsanemptysiteor
anexludedone,duetothenitesizeofthemoleules.
Asanexample,inahainofL=8sites,the
ongura-tioninwhihapartileoflass1,withsizes
1
=2isat
site1,andanotherpartile,oflass2,withsizes
2 =3
isatsite3,isrepresentedbyfg=f1;0;0;2;0;0;0;0g.
Thus the allowed ongurations are given by the set
f
i
g (i = 1;:::;L), where for eah pair (
i ;
j ) 6= 0
withj>iweshould havej is
i .
000000
000000
000000
111111
111111
111111
000000
000000
000000
111111
111111
111111
00000000000
00000000000
00000000000
11111111111
11111111111
11111111111
00000000000
00000000000
00000000000
11111111111
11111111111
11111111111
000000000000000000000
000000000000000000000
000000000000000000000
111111111111111111111
111111111111111111111
111111111111111111111
s = 1
s = 2
s = 4
Figura.1 Example of ongurations of moleuleswith
dis-tint sizess inalattie of sizeL =6. Theoordinates of
themoleulesaredenotedbytheblaksquares.
The time evolution of the probability distribution
P(fg;t),of agivenonguration fgis givenby the
masterequation
P(fg;t)
t =
X
f 0
g
[ (fg!f 0
g)P(fg;t)+ (f 0
g!fg)P(f 0
g;t);℄ (1)
where (fg!f 0
g)isthetransitionratefor
ongu-ration fgtohange tof 0
g. Inthepresentmodel we
onlyallow,whenevertheonstraintofexludedvolume
is satised,thepartiles todiuse to nearest-neighbor
sites, or to exhange positions. The possible motions
arediusion totheright
i ;
i+1 !;
i
i+1
; ( =1;:::;N) (rate
R )
(2)
diusion totheleft
;
i
i+1 !
i ;
i+1
; (=1;:::;N) (rate
L )
(3)
andinterhangeofpartiles
i
0
i+s
!
0
i
i+s
0
; (< 0
=1;:::;N) (rate
R )
i
0
i+s
!
0
i
i+s
0
; (> 0
=1;:::;N) (rate
L ): (4)
Asweseefrom(4),partilesbelongingtoagivenlass
interhangepositionswiththoseoflass 0
>withthe
samerateastheyinterhangepositionswiththeempty
sites (diusion). We should remark however that
un-lessthepartilesinlass 0
haveunit size(s
=1),the
neteetofthesepartilesinthoseoflassisdistint
fromthe eet produedby theholes,sineasthe
re-sult of theexhange the partiles in lass will move
bys
0
lattiesizeunits,aeleratingitsdiusion.
The master equation (1) an be written as a
Shrodinger equation in Eulidean time (see Ref. [3℄
forgeneralappliationfortwobodyproesses)
jP >
t
= HjP >; (5)
if we interpret jP > P(fg;t) as the assoiated
wavefuntion. If we represent
i
asj >
i
the vetor
j >
1
j >
2
j >
L
will give us the
assoi-ated Hilbert spae. The proess (2)-(4) gives us the
Hamiltonian(seeRef. [3℄forgeneralappliations)
H = D
X
j H
j
H
j
= Pf
N
X
=1
+ (E
0
j E
0
j+1 E
j E
00
j+1
)+ (E 0
j E
0
j+1 E
00
j E
j+1 )
+ N
X
=1 N
X
=1
; (E
j E
0
j+s
E
0
j+s E
j E
00
j+s
E
j+s
)gP (6)
with
D=
R +
L ;
+ =
R
R +
L
; =
L
R +
L (
+
+ =1); (7)
d
=
(
+
<
0 =
>
(8)
and periodi boundaryonditions. ThematriesE ;
are (N +1)(N+1) matrieswith asingle nonzero
element(E ;
)
i;j =Æ
;i Æ
;j
(;;i;j=0;:::;N). The
projetor P in (6), projets out from the assoiated
Hilbert spae the vetorsjfg> whih represent
for-bidden positions of the moleules due to their nite
size,whihmathematiallymeansthatfor alli;j with
i ;
j
6= 0; ji jj s
i
(j > i). The
on-stant D in (6) xes the time sale; for simpliity we
hose D=1. A partiularsimpliation of (6)ours
when the moleules in all lasses have the same size
s
1 =s
2
=:::=s
N
=s. Inthis asethe Hamiltonian
anbeexpressedasananisotropinearest-neighbor
in-teration spin-N=2SU(N+1) hain. Moreoverin the
asewheretheirsizesareunity(s=1)themodelanbe
relatedtotheanisotropiversion[21℄oftheSU(N+1)
Sutherland model [20℄ with twisted boundary
ondi-tions.
III The Bethe ansatz equations
Wepresentinthissetiontheexatsolutionofthe
gen-eralquantumhain(6). Apedagogialpresentationfor
thepartiularasewhereN =2waspresentedin[26℄.
Duetotheonservationofpartilesinthediusion
andinterhangeproessesthetotalnumberofpartiles
n
1 ;n
2 ;:::;n
N
ineahlassaregoodquantumnumbers
and onsequently we an split the assoiated Hilbert
spaeintoblokdisjointsetorslabeledbythenumbers
onsidertheeigenvalueequation
Hjn
1 ;n
2 ;:::;n
N
>=Ejn
1 ;n
2 :::;n
N
>; (9)
where
jn
1 ;n
2 ;:::;n
N >=
X
fQg X
fxg f(x
1 ;Q
1 ;:::;x
n ;Q
n )jx
1 ;Q
1 ;:::;x
n ;Q
n
>; (10)
d
and n = P
N
i=1 n
i
is the total number of
parti-les. In (10) jx
1 ;Q
1 ;:::;x
n ;Q
n
> means the
ong-urationwhereapartile oflass Q
i (Q
i
=1;2;:::;N)
is at position x
i (x
i
= 1;:::;L). The summation
fQg =fQ
1 ;:::;Q
n
g extendsoverall permutations of
then integernumbersf1;2;:::;Ngin whih n
i terms
havethevaluei(i=1;2;:::;N),whilethesummation
fxg=fx
1 ;:::;x
n
gruns,foreahpermutation fQg,in
thesetofthennondereasingintegerssatisfying
x
i+1 x
i +s
Q
i
; i=1;:::;n 1;
s
Q1 x
n x
1
N s
Qn
: (11)
Beforegettingtheresultsforgeneralvaluesofnletus
onsiderinitially theaseswhere wehave1or2
parti-les.
n= 1. For onepartile onthehain,in anylass
= 1;2;:::;N, as a onsequeneof the translational
invarianeof(6)itissimpleto verifydiretlythatthe
eigenfuntionsarethemomentum-k eigenfuntions
j0;:::;0;1
;0;:::;0>= L
X
x=1
f(x;)jx;>; =1;:::;N
(12)
with
f(x;)=e ik x
; k= 2l
L
; l=0;1;:::;L 1; (13)
andenergygivenby
E=e(k) ( e ik
+
+ e
ik
1): (14)
n =2. For two partiles of lasses Q
1
and Q
2
(Q
1 ;Q
2
= 1;2;:::;N) on the lattie, the eigenvalue
equation (9) gives us twodistint relationsdepending
ontherelativeloationofthepartiles. Therst
rela-tionappliesto theasein whih apartileoflass Q
1
(sizes
Q1
)isatpositionx
1
andapartileQ
2 (sizes
Q2 )
is at position x
2
, where x
2 >x
1 +s
Q
1
. Weobtain in
thisasetherelation
Ef(x
1 ;Q
1 ;x
2 ;Q
2
) =
+ f(x
1 1;Q
1 ;x
2 ;Q
2
) f(x
1 ;Q
1 ;x
2 +1;Q
2 )
f(x
1 +1;Q
1 ;x
2 ;Q
2
)
+ f(x
1 ;Q
1 ;x
2 1;Q
2
)+2f(x
1 ;Q
1 ;x
2 ;Q
2
); (15)
wherewehaveusedtherelation
+
+ =1. Thislastequationanbesolvedpromptlybytheansatz
f(x
1 ;Q
1 ;x
2 ;Q
2 ) =
X
P A
Q1;Q2
P
1 ;P
2 e
i(k
P
1 x
1 +k
P
2 x
2 )
= A
Q
1 ;Q
2
1;2 e
i(k
1 x
1 +k
2 x
2 )
+A Q
1 ;Q
2
2;1 e
i(k
2 x
1 +k
1 x
2 )
(16)
withenergy
E=e(k
1 )+e(k
2
); (17)
where k
1 ;k
2 ;A
Q1;Q2
1;2
and A Q1;Q2
2;1
are freeparametersto bexed. In(16)thesummation isoverthepermutations
P =P
1 ;P
2
of(1,2). Theseondrelationapplieswhenx
2 =x
1 +s
Q1
. Inthisaseinsteadof(15)wehave
Ef(x
1 ;Q
1
; x
1 +s
Q
1 ;Q
2 )=
+ f(x
1 1;Q
1 ;x
1 +s
Q
2 ;Q
2
) f(x
1 ;Q
1 ;x
1 +s
Q
1 +1;Q
2 )
~
Q
2 ;Q
1 f(x
1 ;Q
2 ;x
1 +s
Q
2 ;Q
1
)+(1+~
Q
1 ;Q
2 )f(x
1 ;Q
1 ;x
1 +s
Q
1 ;Q
2
): (18)
If we nowsubstitute the ansatz (16) with the energy (17), theonstants A Q1;Q2
12
and A Q1;Q2
21
X
P f
D
P
1 ;P
2 +e
ikP
2
(1 ~
Q
1 ;Q
2 )
e ikP
2 (sQ
1 1)
A Q1;Q2
P
1 ;P
2 +~
Q
2 ;Q
1 e
ikP
2 sQ
2
A Q2;Q1
P
1 ;P
2
g=0 (19)
where
D
l;m
= (
+ + e
i(k
l +km)
): (20)
At this point it is onvenient to onsider separately the ase where Q
1 = Q
2
from those where Q
1 6= Q
2 . If
Q
1 =Q
2
=Q(Q=1;:::;N)eq. (19)gives
X
P D
P1;P2 +e
ikP
2
e ikP
2 (sQ 1)
A Q;Q
P1;P2
=0 (21)
andtheasesQ
1 6=Q
2
giveus theequations
X
P
D
P
1 ;P
2 +e
ikP
2
Q
2 ;Q
1
Q
2 ;Q
1 e
ikP
2
Q
1 ;Q
2 e
ikP
2
D
P
1 ;P
2 +
Q
1 ;Q
2 e
ikP
2
"
e ikP
2 (sQ
1 1)
A Q1;Q2
P1;P2
e ikP
2 (sQ
2 1)
A Q2;Q1
P
1 ;P
2 #
=0:
Performingtheabovesummationweobtain,afterlengthybutstraightforwardalgebra,thefollowingrelationamong
theamplitudes
"
A Q
1 ;Q
2
1;2 e
ik
2 (s
Q
1 1)
A Q2;Q1
1;2 e
ik2(sQ
2 1)
#
= D
1;2 +e
ik1
D
1;2 +e
ik
2
1 (k
1 ;k
2 )
Q
1 ;Q
2
Q
2 ;Q
1
Q
1 ;Q
2
Q
2 ;Q
1
"
A Q
1 ;Q
2
2;1 e
ik1(sQ
1 1)
A Q2;Q1
2;1 e
ik
1 (s
Q
2 1)
#
;
where
(k
1 ;k
2 )=
e ik1
e ik2
D
1;2 +e
ik1
: (22)
Equations(21)and(22)anbewritteninaompatform
A Q1;Q2
P1;P2
=
P1;P2 N
X
Q 0
1 ;Q
0
2 =1
S Q
1 ;Q
2
Q 0
1 ;Q
0
2 (k
P1 ;k
P2 )A
Q 0
2 ;Q
0
1
P2;P1 ; (Q
1 ;Q
2
=1;:::;N) (23)
with
l;j =
D
l;j +e
ik
l
D
l;j +e
ikj =
+ + e
i(k
l +k
j )
e ik
l
+ + e
i(k
l +kj)
e ikj
; (24)
where wehaveintroduedtheS matrix. From(21)and(22)thisSmatrixhasonlyN(2N 1)nonzeroelements,
namely
S Q1;Q2
Q
2 ;Q
1 (k
1 ;k
2
) = [1
Q
1 ;Q
2 (k
1 ;k
2 )℄e
i(k1 k2)(sQ
1 1)
(Q
1 ;Q
2
=1;:::;N);
S Q
1 ;Q
2
Q1;Q2 (k
1 ;k
2
) =
Q2;Q1 (k
1 ;k
2 )e
ik
1 (s
Q
2 1)
e ik
2 (s
Q
1 1)
(Q
1 ;Q
2
=1;:::;N;Q
1 6=Q
2
): (25)
Equations(23)donotxthe\wavenumbers"k
1 andk
2
. Ingeneral,thesenumbersareomplex,andarexeddue
to theyli boundaryondition
f(x
1 ;Q
1 ;x
2 ;Q
2 )=f(x
2 ;Q
2 ;x
1
+N;Q
1
); (26)
whihfrom (16)givetherelations
A Q
1 Q
2
1;2 =e
ik
1 N
A Q
2 ;Q
1
2;1
; A
Q
1 ;Q
2
2;1 =e
ik
2 N
A Q
2 ;Q
1
2;1
: (27)
Thislastequation,whensolvedbyexploiting(23)-(25),givesusthepossiblevaluesofk
1 andk
2
,andfrom(17)the
Generaln. Theabovealulationanbegeneralizedforarbitraryoupation
fn
1 ;n
2 ;:::;n
N
gof partilesin lasses1;2;:::;N,respetively. Theansatzforthewavefuntion (10)beomes
f(x
1 ;Q
1 ;:::;x
n ;Q
n )=
X
P A
Q
1 ;;Q
n
P1;:::;Pn e
i(k
P
1 x
1 ++k
Pn x
n )
; (28)
wherethesumextendsoverallpermutationsP oftheintegers1;2;:::;n,andn= P
N
i=1 n
i
isthetotalnumberof
partiles.
Appliationofthetranslationoperatortotheabovewavefuntionsimpliesthat(10)arealsoeigenfuntionsof
themomentumoperator witheigenvalues
p= n
X
j=1 k
j =
2l
L
; (l=0;1;:::;L 1): (29)
For the omponentsjx
1 ;Q
1 ;:::;x
n ;Q
n
> where x
i+1 x
i > s
Q
i
for i = 1;2;:::;n, it is simpleto see that the
eigenvalueequation(9)is satisedbytheansatz(28)withenergy
E= n
X
j=1 e(k
j
): (30)
Ontheother handifapairofpartilesoflass Q
i ;Q
i+1
isat positionsx
i ; x
i+1
, wherex
i+1 =x
i +s
Q
i
,equation
(9)withtheansatz(28)andtherelation(30)giveusthegeneralizationofrelation(23),namely
A
;Qi;Qi+1;
:::;Pi;Pi+1;:::
=
Pi;Pi+1 N
X
Q 0
1 ;Q
0
2 S
Qi;Qi+1
Q 0
1 ;Q
0
2 (k
Pi ;k
Pi+1 )A
;Q 0
2 ;Q
0
1 ;
:::;Pi+1;Pi;::: (Q
i ;Q
i+1
=1;2;;:::;N); (31)
withS givenbyeq. (25). Insertingtheansatz(28)in theboundaryondition
f(x
1 ;Q
1 ;:::;x
n ;Q
n )=f(x
2 ;Q
2 ;:::;x
n ;Q
n ;x
1 +N;Q
1
) (32)
weobtaintheadditionalrelation
A Q
1 ;;Q
n
P1;:::;Pn =e
ik
P
1 N
A Q
2 ;;Q
n ;Q
1
P2;:::;Pn;P1
; (33)
whihtogetherwith(31)should giveustheenergies.
Suessive appliations of (31) give us in general distint relations between the amplitudes. For example
A
:::;;;;:::
:::;k1;k2;k3;:::
relate to A
:::;;;;:::
:::;k3;k2;k1;:::
by performing the permutations ! ! ! or !
!!,andonsequentlytheS-matrixshouldsatisfytheYang-Baxter[19,29℄equation
N
X
; 0
; 00
=1 S
; 0
; 0
(k
1 ;k
2 )S
; 00
; 00
(k
1 ;k
3
) S
0
; 00
0
; 00
(k
2 ;k
3 )=
N
X
; 0
; 00
=1 S
0
; 00
0
; 00
(k
2 ;k
3 )S
; 00
; 00
(k
1 ;k
3 )S
; 0
; 0
(k
1 ;k
2
); (34)
for ; 0
; 00
;; 0
; 00
=1;2;:::;N and S given by (25). Atuallythe relation (34) is a neessaryand suÆient
ondition[19,29℄toobtainanon-trivialsolutionfortheamplitudesin Eq. (31).
Wean verifyby alongand straightforwardalulationthat forarbitrary number oflasses N and valuesof
thesizes s
1 ;s
2 ;:::;s
N
, theS matrix(25), satisesthe Yang-Baxterequation(34), and onsequentlywemay use
relations(31)and (33)to obtaintheeigenenergiesof theHamiltonian(6). Applyingrelation(31)n timeson the
rightofequation(33)weobtainarelationbetweentheamplitudeswiththesameorderinginthelowerindies:
A Q
1 ;:::;Q
n
P1;:::;Pn =e
ik
P
1 N
A Q
2 ;:::;Q
n ;Q
1
P2;:::;Pn;P1 =
n
Y
i=2
Pi;P1 !
e ik
P
1 N
X
Q 0
1 ;:::;Q
0
n X
Q 00
1 ;:::;Q
00
n
S Q1;Q
00
2
Q 0
1 ;Q
00
1 (k
P
1 ;k
P
1 )S
Q2;Q 00
3
Q 0
2 ;Q
00
2 (k
P
2 ;k
P
1 )S
Qn 1;Q 00
n
Q 0
n 1 ;Q
00
n 1 (k
P
n 1 ;k
P
1 )S
Qn;Q 00
1
Q 0
n ;Q
00
n (k
P
n ;k
P
1 )A
Q 0
1 ;:::;Q
0
n
P
1 ;:::;P
n
; (35)
1= N
X
Q 00
1 ;Q
00
2 =1
Æ
Q 00
2 ;Q
0
1 Æ
Q 00
1 ;Q
1 =
N
X
Q 00
1 ;Q
00
2 =1
S Q1;Q
00
2
Q 0
1 ;Q
00
1 (k
P1 ;k
P1 )
(36)
(see [26℄ for illustrations of the above equations). In
ordertoxthevaluesoffk
j
gweshouldsolve(35),i.e.,
weshould ndtheeigenvalues(k)ofthematrix
T(k) fQg
fQ 0
g =
N
X
Q 00
1 ;:::;Q
00
n =1
n
Y
l=1 S
Ql;Q 00
l+1
Q 0
l ;Q
00
l (k
P
l ;k)
!
; (37)
withperiodiboundaryondition
S Qn;Q
00
n+1
Q 0
n ;Q
00
n (k
Pn
;k)=S Qn;Q
00
1
Q 0
n ;Q
00
n (k
Pn
;k): (38)
TheBethe-ansatzequationswhihxtheset fk
l gwill
begivenfrom(35)by
e ikjN
=( 1) n 1
n
Y
l=1
l;j !
(k
j
); j =1;:::;n:
(39)
The matrixT(k)has dimensionN n
N n
and anbe
interpretedasthetransfermatrixofaninhomogeneous
N(2N 1)-vertex model in a twodimensional lattie
withperiodiboundaryonditionsinthehorizontal
di-retion(nsites). DuetothespeialformoftheSmatrix
(25)theeigenvaluesof (37)areinvariantunder aloal
gaugetransformationwhereforeahfatorS(k
P
l ;k)in
(37):
S Q
l ;Q
00
l+1
Q 0
l ;Q
00
l (k
Pl
;k)!S Q
l ;Q
00
l+1
Q 0
l ;Q
00
l (k
Pl ;k)
(l)
Q 00
l+1
(l)
Q 00
l
; (40)
where (l)
(l =1;:::;L; =1;:::;N)are arbitrary
funtions. If we perform thetransformation(40) with
thespeialhoie
(l+1)
(l)
=e
ikP
l (s 1)
; (l=2;3;:::;N); (41)
the equivalent transfer matrix to be diagonalized is
givenby
~
T(k) fQg
fQ 0
g =e
ik P
n
i=1 (sQ
i 1)
T
0 (k)
fQg
fQ 0
g
(42)
where
T
0 (k)
fQg
fQ 0
g =
N
X
Q 00
1 ;:::;Q
00
n =1
n
Y
l=1 ~
S Ql;Q
00
l+1
Q 0
l ;Q
00
l (k
P
l ;k)
!
; (43)
withthetwistedboundaryondition
~
S Qn;Q
00
n+1
Q 0
n ;Q
00
n (k
P
n ;k)=
~
S Qn;Q
00
1
Q 0
n ;Q
00
n (k
P
n ;k)
Q 00
1
(44)
withtwistedphase
l =e
i(s
l 1)
P
n
j=1 kj
; l=1;:::;N: (45)
Thematrix ~
Sin(43)and(44)isobtainedfromthosein
(25)bytakingthesizeofallpartilesequaltounity. In
thiswaytheproblemistransformedintotheevaluation
oftheeigenvaluesofaregular(allpartileswithsize1)
inhomogeneoustransfermatrixT
0
withn(2N 1)
non-zerovertexand twistedboundaryondition.
DiagonalizationofT
0 (k)
Thesimplest wayto diagonalizeT
0
isthrough the
in-trodutionofthemonodromymatrixM(k)[25℄,whih
isatransfermatrixoftheinhomogeneousvertexmodel
underonsideration,wheretherstandlastlinkinthe
horizontaldiretionarexedtothevalues
1 and
n+1
(
1 ;
n+1
=1;2;:::;N),that is
M fQg;
n+1
fQ 0
g;
1
(k) =
1 N
X
2;:::;n=1 ~
S Q1;2
Q 0
1 ;1
(k
P
1 ;k)
~
S Q2;3
Q 0
2 ;2
(k
P
2 ;k)
~
S Q
n 1 ;
n
Q 0
n 1 ;
n 1 (k
P
n 1 ;k)
~
S Q
n ;
n+1
Q 0
n ;
n (k
P
n
;k): (46)
ThemonodromymatrixM
fQg;n+1
fQ 0
g;1
(k)hasoordinatesfQg;fQ 0
ginthevertialspae(N n
dimensions) and
oor-dinates
1 ;
n+1
inthehorizontalspae(N 2
dimensions). Thismatrixsatisesthefollowingimportantrelations
N
X
0
1 ;
0
1 =1
~
S
0
1 ;
0
1
1;1 (k
0
;k)M f
l g;n+1
f
l g;
0
1
(k)M f
l g;n+1
f
l g;
0
1 (k
0
) =
N
X
0
n+1 ;
0
n+1 =1
M f
l g;
0
n+1
flg;1 (k
0
) M
f
l g;
0
n+1
flg;1 (k)
~
S
n+1;n+1
0
n+1 ;
0
n+1 (k
0
for
1 ;
1 ;
n+1 ;
n+1
= 1;2;:::;N.This relation
fol-lowsdiretlyfromsuessiveappliationsofthe
Yang-Baxter equations (34) (see [26℄, for agraphial
repre-sentationofthese equations).
In order to exploit relation (47) let us denote the
omponentsof themonodromymatrix in the
horizon-tal spaeby
A(k)
= M
flg;
f
l g;
(k); B(k)
=M flg;
f
l g;N
(k);
C(k)
= M
f
l g;N
flg;
(k); D(k)=M f
l g;N
flg;N
(k);(48)
where ; =1;2;:::;N 1. Clearlythe transfer
ma-trix T
0
(k) of the inhomogeneous lattie with twisted
boundaryonditions, wewant to diagonalize,is given
by
T
0 (k)=
N 1
X
=1 A
(k)+D(k): (49)
Asaonsequeneof(47)thematriesA
,B
,C
and
D in (48) obey some algebrai relations. By setting
(
1 ;
1 ;
n+1 ;
n+1
)=(N;;;)in (47)weobtain
A
(k)B
(k 0
)= ~
S ;N
N; (k
0
;k)
~
S N;
N; (k
0
;k) B
(k)A
(k
0
)+ N 1
X
0
; 0
=1 ~
S ;
0
; 0
(k 0
;k)
~
S N;
N; (k
0
;k) B
0
(k 0
)A
0
(k); (50)
with(;=1;:::;N 1). Bysetting (
1 ;
1 ;
n+1 ;
n+1
)=(N;N;N;)weobtain
D(k)B
(k 0
)= ~
S N;N
N;N (k;k
0
)
~
S N;
N; (k;k
0
) B
(k 0
)D(k) ~
S N;
;N (k;k
0
)
~
S N;
N; (k;k
0
) B
(k)D(k 0
); (51)
where(=1;:::;N 1). ThediagonalizationofT
0
(k)in(49)willbedonebyexploitingtheaboverelations. This
proedureis knownin theliteratureasthealgebraiBetheansatz[25℄. Therststepin thismethod followsfrom
theidentiationofareferenestatej>,whihshouldbeaneigenstateofA
(k)andD(k),andheneT
0
(k),but
notofB
(k). Inthepresentaseasuitablereferenestateisj>=jf
l
=Ng>
l=1;:::;n
, whih orrespondsto a
statewithN-lasspartilesonly. Itissimpletoalulate
A
(k)j> = a
(k)j>; D(k)j>=d(k)j>;
C
(k)j> = 0; B
(k)j>= n
X
i=1 b
i (k)j
(i)
>; (52)
where
a
(k) = Æ
;
n
Y
i=1 ~
S N;
N; (k
Pi
;k); d(k)=
N n
Y
i=1 ~
S N;N
N;N (k
Pi ;k);
b
i
(k) =
N i 1
Y
l=1 ~
S N;N
N;N (k
Pl ;k)
n
Y
l=i ~
S N;
N; (k
Pl
;k); (53)
and j (i)
>= jf
l6=i
=Ng;
i
=>. Thematries B
(k) atasreationoperators in thereferene (\vauum")
state, by reating partiles of lass (1;2;:::;N) in asea of partiles of Nth lass j >. We then expet that
theeigenvetorsofT
0
(k) orresponding to m
1
(1;2;:::;n) partiles, belonging to lassesdistintfrom N, anbe
expressedas
jk (1)
l
;F>= X
fg F
1 ;:::;
m
1 B
1
(k (1)
1 )B
2
(k (1)
2 )B
m
1
(k (1)
m1
)j>; (54)
wherefk (1)
l
;l=1;:::;m
1
gandF
1;:::;m
1
arevariablesto bexedbytheeigenvalueequation
T
0 (k)jk
(1)
l
;F >= (0)
(k)jk (1)
l
;F >: (55)
Using(50)suessively,and(52),(53)weobtain
A
(k)B
1
(k (1)
1 )B
2
(k (1)
2 )B
m
1
(k (1)
m1 )=
N
X
f 0
1 ;:::;
0
m
1 =1g
N
X
f 0
1 ;:::;
0
m
1 =1g
~
S
1 ;
0
1 ;
0
1 (k
(1)
1 ;k)
~
S
2 ;
0
1
0
2 ;
0
2 (k
(1)
2
;k) ~
S m
1 1;
0
m
1 2
0
m
1 1
; 0
m
1 1
(k (1)
m
1 1
;k)
~
S m
1 ;
0
m
1 1
0
m
1 ;
(k (1)
m
1 ;k)
Q
n
j=1 ~
S N;
N; (k
j ;k)
Q
m1
~
S N;
(k (1)
j ;k)
B
0
1
(k (1)
1 )B
0
2
(k (1)
2 )B
0
m
1
(k (1)
m1
wherethe\unwantedterms"arethoseoneswhiharenotexpressedinthe\Bethebasis"produedbytheB
(k (1)
j )
operators. Similarly,using(51)suessivelyand(52)-(53)weobtain
D(k)B
1
(k (1)
1 )B
m
1
(k (1)
m
1
)j>=
N n
Y
l=1 ~
S N;N
N;N (k
l ;k)
!
m1
Y
l=1 ~
S N;N
N;N (k;k
(1)
l )
~
S N;l
N;
l (k;k
(1)
l )
!
B 1
(k (1)
1 )B
m
1
(k (1)
m1
)j> + "unwantedterms": (57)
Therelations(56)and (57)whenusedin (54)-(55)giveus
T
0
(k) jk (1)
;F >= Q
n
j=1 ~
S N;1
N;1 (k
j ;k)
Q
m
1
j=1 ~
S N;1
N;1 (k
(1)
j ;k)
X
fg X
f 0
g T
1 (k)
fg
f 0
g F
fg B
0
1
(k (1)
1 )B
0
m
1
(k (1)
m1 )j>
+
N n
Y
i=1 ~
S N;N
N;N (k
i ;k)
X
fg m1
Y
l=1 ~
S N;N
N;N (k;k
(1)
l )
~
S N;
l
N;l (k;k
(1)
l )
!
F
fg B
1
(k (1)
1 )B
m
1
(k (1)
m1 )j>
+"unwantedterms"; (58)
where
T
1 (k)
fg
f 0
g =
N 1
X
=1
N 1
X
0
1 ;:::;
0
m
1 =1
~
S 1;
0
1 ;
0
1 (k
(1)
1 ;k)
m
1 1
Y
i=1 ~
S i;
0
i
0
i ;
0
i+1 (k
(1)
i ;k)
!
~
S m
1 ;
0
m
1 1
0
m
1 ;
(k (1)
m
;k) (59)
isa(N 1) m
1
-dimensionaltransfermatrixofainhomogeneousvertexmodel,withinhomogenetiesfk (1)
m1 ;k
(1)
m1 1 ;:::;k
(1)
1 g
(notie the reverse order of the inohomogeneties, when ompared with (43)) and twisted boundary onditions
(boundaryphases
;=1;:::;N 1).
Inorder toproeedweneednowtodiagonalizethenewtransfermatrixT
1
(k), thatiswemustsolve
X
fg T
1 (k)
fg
f 0
g F
fg =
(1)
(k)F
f 0
g
(60)
andthen (58)giveus
T
0 (k)jk
(1)
;F>= (0)
(k)jk (1)
;F >+\unwantedterms"; (61)
where, usingthefatthat ~
S N;N
N;N (k
l
;k)=1,
(0)
(k)= Q
n
j=1 ~
S N;
N; (k
j ;k)
Q
m
1
j=1 ~
S N;1
N;1 (k
(1)
j ;k)
(1)
(k)+
N m1
Y
l=1 1
~
S N;1
N;1 (k;k
(1)
l )
: (62)
Inordertoprovethat (0)
andjK (1)
;F >aretheeigenvaluesandeigenvetorsofT
0
(k),weshouldxfk (1)
1
;:::;k (1)
m
1 g
byrequiringthatthe\unwantedterms"in(61)vanish. AlthoughforN=2thisalulationisnotompliated[26℄
for arbitraryN itis notsimple. Sinethe expression(62)forthe eigenvaluesshould bevalid forarbitraryvalues
of k weanobtain (1)
(k (1)
j
)in analternativeway from thefollowingtrik[31℄. At k=k (1)
j
(j=1;:::;m
1 )the
denominatorsofthefatorsin (62)vanish( ~
S N;l
n;l (k
(1)
j ;k
(1)
j
)=0;l6=N),andsineweshouldhaveaniteresult,we
havetheonditions
(1)
(k (1)
j )=
N n
Y
i=1
1
~
S N;1
N;1 (k
i ;k
(1)
j )
m1
Y
l 0
=1;l 0
6=j ~
S N;1
N;1 (k
(1)
l 0
;k (1)
j )
~
S N;1
N;1 (k
(1)
j ;k
(1)
l 0
)
;j=1;:::;m
1
: (63)
Notie that our result in (63) does not depend on the partiular ordering of the additional variables k (1)
j (j =
1;:::;m
1
). This meansthat ifinsteadoftheorderinghosenin(54),wehosethereverseorder,namely,
jk (1)
;F>= X
fg F
1;:::;m
1 B
m
1
(k (1)
m
1 )B
m
1 1
(k (1)
m
1 1
)B 1
(k (1)
1
)j>; (64)
wewouldobtainthesameresults(61)-(63),butnowT
1
isthetransfermatrix,withboundaryonditionspeiedby
thephase
,ofaproblemwith (N 1)speiesandinhomogeneitiesk (1)
;:::;k (1)
m
1
order ofthe inhomogenetiesasin (43)). Thismeans that theeigenvalue(k)= (0)
(k)of thetransfermatrixof
theproblem withN lassesandinhomogeneities (k (0)
1 ;k
(0)
2
;:::;k (0)
n
)(k
1 ;k
2 ;:::;k
n
)isrelatedto theeigenvalue
(1)
(k)oftheproblemwith(N 1)lassesandinhomogeneitiesk (1)
1 ;k
(1)
2
;:::;k (1)
m1
. Iteratingthesealulationswe
obtainthegeneralizationoftherelation(62)andtheondition(63)
(l)
(k) = m
l
Y
l 0
=1 ~
S N;1
N;1 (k
(l)
l 0
;k) !
ml+1
Y
l 0
=1
1
~
S N;1
N;1 (k
(l+1)
l 0
;k) !
(l+1)
(k)+
N l m
l+1
Y
l 0
=1
1
~
S N;1
N;1 (k;k
(l+1)
l 0
)
;l=0;1;:::;N 1; (65)
(l+1)
(k (l+1)
j
)=
N l m
l
Y
l 0
=1
1
~
S N;1
N;1 (k
(l)
l 0
;k (l+1)
j )
!
m
l+1
Y
l 0
=1;l 0
6=j ~
S N;1
N;1 (k
(l+1)
l 0
;k (l+1)
j )
~
S N;1
N;1 (k
(l+1)
j ;k
(l+1)
l 0
)
; (66)
d
whih onnets the eigenvalues of the inhomogeneous
transfermatrixT
l
(k)andT
l+1
(k),withinhomogeneities
fk (l)
j
g and fk (l+1)
j
g, related with the problem with
(N l)and(N l 1)lassesofpartiles,respetively.
Howeverfrom(39)and(42)-(43),inordertoobtain
theBethe-ansatzequationsforouroriginalproblemwe
needtheeigenvaluesofthetransfermatriesevaluated
atk
j
(j=1;:::;n),i.e., (0)
(k
j
),whiharegivenby
(0)
(k
Pj )=
N m
1
Y
l=1
1
~
S N;1
N;1 (k
Pj ;k
(1)
l )
; (67)
sine Q
n
j=1 ~
S N;
n; (k
j ;k
P
j
)=0. The onditionsthat x
thevariables(k (1)
j
;j=1;:::;m
1
)aregivenby(63). In
theleft sideofthis equationwehave (1)
(k (1)
j
), whih
are the eigenvalues of the transfer matrix T
1 of the
model with (N 1) lasses of partiles and
inhomo-geneitiesfk (1)
j
;j =1;:::;m
1
g,evaluatedatthe
partu-lar point k (1)
j
. This value an be obtained from (65)
whihgivesageneralizationof(67)
(l)
(k (l)
j )=
N l ml+1
Y
l 0
=1
1
~
S N;1
N;1 (k
(l)
j ;k
(l+1)
l 0
)
(l=0;1;:::;N 1): (68)
Theondition(63)isthenreplaedby
(1)
(k (1)
j )=
N 1 m2
Y
l 0
=1
1
~
S N;1
N;1 (k
(1)
j ;k
(2)
l 0
) =
N n
Y
i=1
1
~
S N;1
N;1 (k
(1)
i ;k
(1)
j )
m1
Y
l 0
=1;l 0
6=j ~
S N;1
N;1 (k
(1)
l 0
;k (1)
j )
~
S N;1
N;1 (k
(1)
j ;k
(1)
l 0
)
; (69)
wherenowweneedtondtherelationsthatxfk (2)
j
g. Iteratingthisproesswendthegeneralizationof(69)
N l m
l+1
Y
l=1
1
~
S N;1
N;1 (k
(l)
j ;k
(l+1)
l 0
) =
N (l 1) m
l 1
Y
i=1
1
~
S N;1
N;1 (k
(l+1)
i ;k
(l)
j )
ml
Y
l 0
=1;l 0
6=j ~
S N;1
N;1 (k
(l)
l 0
;k (l)
j )
~
S N;1
N;1 (k
(l)
j ;k
(l)
l 0
)
;(j=1;2;:::;m
l
; l=0;1;:::;N 2): (70)
Equations (67)and (70) giveus the eigenvaluesof the transfermatrix T
0
(k) evaluated at the points fk
j g, i. e.
(0)
(k
j
). Insertingtheaboveresultsin (42)and thenin(39)weobtaintheBethe-ansatzequationsof ouroriginal
problem.
Theeigenenergiesof theHamiltonian(6) in thesetorontainingn
i
partilesin lass i(i=1;2;:::;N)(n=
P
N
j=1 n
j
)andtotalmomentum p= 2
L
(l=0;1;:::;L 1)aregivenby
E= n
X
( e ik
(0)
j
+
+ e
ik (0)
j
where fk (0)
j =k
j
;j =1;:::;ngareobtainedfromthesolutionsfk (l)
j
;l=0;:::;N 1;j =1;:::;m
l
gof theBethe
ansatzequations
e ik
j (L+n
P
N
i=1 n
i s
i )
= ( 1) n 1
e ip(s
N 1)
n
Y
j 0
=1(j 0
6=j)
+ + e
i(k (0)
j +k
(0)
j 0
)
e ik
(0)
j
+ + e
i(k (0)
j +k
(0)
j 0
)
e ik
(0)
j 0
m
1
Y
l=1
+ (e
ik (1)
l
e ik
(0)
j
)
+ + e
i(k (1)
l +k
(0)
j
e ik
(0)
j
j =1;2;:::;n; (72)
and
m
l
Y
=1
+ (e
ik (l+1)
e ik
(l)
)
+ + e
i(k (l+1)
+k
(l)
)
e ik
(l)
=( 1) m
l+1
e ip(s
N l s
N l 1 )
m
l+2
Y
Æ=1
+ (e
ik (l+2)
Æ
e ik
(l+1)
)
+ + e
i(k (l+2)
Æ +k
(l+1)
)
e ik
(l+1)
ml+1
Y
0
=1( 0
6=)
+ + e
i(k (l+1)
+k
(l+1)
0
)
e ik
(l+1)
+ + e
i(k (l+1)
+k
(l+1)
0
)
e ik
(l+1)
0
l=0;1;:::;N 2; =1;:::;m
l
; (73)
d
and m
l =
P
N l
j=1 n
j
, l =0;:::;N (m
0
=N;m
N =0).
It is interestingto observethat in the partiular ase
where n
2 =n
3
=:::= n
N
= 0we obtainthe
Bethe-ansatz equations, reently derived [15℄ (see also [14℄),
for theasymmetri diusion problemwith partilesof
size s
1
. Also the ase s
1 = s
2
=::: = s
N
= 1 gives
us the orresponding Bethe ansatz equations for the
standardproblem of N typesof partiles in
hierarhi-al order. The Bethe-ansatz solution in the
parti-ular ase of N=2 with a a single partile of lass 2
(n
1
= n 1;n
2
= 1) was derived reently [32℄. The
Bethe-ansatzequations forthefully asymmetri
prob-lem are obtained by setting in (72)-(73)
+
= 1 and
=0.
IV Conlusions and
generaliza-tions
WeobtainedthroughtheBetheansatztheexat
solu-tion ofthe problem in whih partiles belonging to N
distint lasses with hierarhial order diuse as well
interhangepositionswithratesdependingontheir
rel-ative hierarhy. We showthat the exatsolution an
also be derivedin the generalasewherethe partiles
havearbitrarysizes.
Someextensionsofourresultsanbemade. Arst
and quiteinteresting generalizationofour model
hap-penswhenweallowmoleulesin anylasstohavesize
s = 0. Moleules of size zero do not oupy spae
on the lattie, having no hard-ore exlusion eet.
Consequently we may have, at a given lattie point,
an arbitrarynumberof them. The Bethe-ansatz
solu-retlyinthisase(theequationsarethesame)andthe
eigenenergiesaregivenbyxingin(71)-(73)the
appro-priatesizesofthemoleules. Itisinterestingtoremark
thatpartilesofagivenlass 0
(2;3;:::;N),withsize
s
0
=0,ontrarytotheases
0
>1,wherethey
\ael-erate"the diusion of partiles in lasses < 0
, they
now \retard" the diusive motion of these partiles.
Thequantum Hamiltonianintheaseswherethe
par-tiles havesize zero is obviously notgiven by (6) but
anbewrittenintermsofspinS=1quantumhains.
Anotherfurther extensionofourmodelisobtainedby
onsidering an arbitrary mixture of moleules, where
moleulesinthesamehierarhymayhavedistintsizes.
Theresultspresentedin[15℄orrespondtothe
partiu-laraseofthisgeneralizationwhereN =1(simple
dif-fusion). ForgeneralN theS matrixweobtainin(25)
isalsoasolutionoftheYang-Baxterequation(34),but
thediagonalizationofthetransfermatrixofthe
assoi-atedinhomogeneousvertexmodelismoreompliated.
TheBethe-ansatzequationsin theaseofasymmetri
diusion, with partiles of unit size [10, 11℄, or with
arbitrary size [15℄, were used to obtain the nite-size
orretionsofthemassgapG
N
oftheassoiated
quan-tumhain. Therealpartofthesenite-sizeorretions
isgovernedbythedynamialritialexponentz,i. e.,
Re(G
N )N
z
: (74)
The alulation of the exponent z for the model
pre-sentedinthispaper,withpartilesofarbitrarysizes,is
presentlyinprogress[30℄.
Aknowledgements
Na-CNPq-BrazilandbytheRussianFoundationof
Fun-damentalInvestigation(Grant99-02-17646).
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