- Email: [email protected]
Generation of a quantum integrable class of discrete-time or relativistic periodic Toda chains
11 July1994 PHYSICS LETTERS A
ELSEVIER
Physics Letters A 190 (1994 ) 79-84
Generation of a quantum integrable class of discrete-time or relativistic periodic Toda chains Anjan K u n d u l Physikalisches Institut der Universitiit Bonn, Nussallee 12, 53115 Bonn, Germany
Received 14 March 1994; accepted for publication 12 May 1994 Communicated by A.R. Bishop
Abstract
A new integrable class of quantum models representing a family of different discrete-time or relativistic generalisations of the periodic Toda chain (TC), including that of a recently proposed classical model close to TC [ Lett. Math. Phys. 29 ( 1993 ) 165 ] is presented. All such models are shown to be obtainable from a single ancestor model at different realisations of the underlying quantised algebra. As a consequence the 2 X 2 Lax operators and the associated quantum R-matrices for these models are easily derived ensuring their quantum integrability. It is shown that the functional Bethe ansatz developed for the quantum TC is trivially generalised to achieve separation of variables also for the present models. 1. The Toda chain ( T C ) , which is considered to be one of the most fascinating integrable lattice systems, attracted continuous attention over the years to investigate both its classical [ 1 ] and q u a n t u m [ 2,3 ] aspects. In the recent past a discrete-time TC ( D T T C ) was proposed by Suris at the classical level [4], which was also shown to be equivalent to the relativistic TC of Ruijsenaars [ 5 ]. Following that we obtained [ 6 ] two different q u a n t u m generalisations of such D T T C along with the associated quantum R-matrices. Recently another model close to TC, which is classically integrable, has been put forward [ 7 ]. We present here a discrete-time as well as q u a n t u m generalisation of this model. However the basic aim of this Letter is to find a new quantum integrable class of different discrete-time generalisations of the periodic TC, which are also canonically related to the relativistic TC o f Ruijsenaars. We show that all such models can be generated from a single ancestor model at different realisations of the underlying quantised algebra, which is defined by the extended trigonometric Sklyanin algebra [6]. At the same time we also obtain automatically the associated quantum R-matrix, while the integrability condition given by the q u a n t u m Yang-Baxter equation (QYBE), R(2, #)L1 (2)L2 (/.t) =L2(/~)LI (2)R(,~,/z) ,
( 1)
with L~ - L ® l, L 2 = 1 ® L is satisfied by the construction. Moreover, we observe that the functional Bethe ansatz developed by Sklyanin [ 3 ] for the separation of variables in q u a n t u m TC can be applied in a parallel way also to its discrete-time generalisations. Since our aim here is to generate models, we start not from any particular model, hut directly from the quant u m R-matrix taking it in the form t Address after June 1994: Saha Institute of Nuclear Physics, AF/1 Bidhan Nagar, Calcutta 700 064, India. 0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved SSDI0375-9601 (94)00374-X
A. Kundu/PhysicsLetters A 190 (I 994) 79-84
80
sin(AZl++) -“sinAi2 e sin (Y
l
R&,2)=
sin (Y eiesinAiz ’ sin(A12 +cr) 1
(2)
with Ai2=A- p. Note that (2) is obtained from the standard trigonometric R,,-matrix [ 81 by a simple “gauge transformation” [ 91: R,=F( t?)Rd;( 0) with F( 6) =exp [ ie( a@ 1 - 180~) 1, which takes a solution of the YBE to another solution of it introducing an extra parameter 8. Now following the Yang-Baxterisation scheme for the construction of Lax operators [ 10 ] one obtains its generalised form
(
57r+
L(A)=
(l/07:
721
712
<72+(1/07:
>
(3)
’
with spectral parameter &eil and yet undefined abstract operators 7. The integrability condition on (3), that is the validity of QYBE ( 1) with R-matrix (2), dictates the algebra for T as e’e7,272,-e-ie72,712=-2isincu(7:7~-7~7:), 7i+fV..ei(*[email protected]* y I 7
7,* 7ji =ei([email protected]+ ,r I 9
(4)
with i, j= 1,2. We call this algebra extended trigonometric Sklyanin algebra, which may be reduced to the well known quantum algebra UJsu(2)) at some particular reduction, though in general, it allows more freedom necessary for the construction of integrable models [ 6 1. Thus (3 ) represents the Lax operator of a generalised quantum integrable model, entries of which are governed by algebra (4). 2. Now, in order to generate a family of DTTC related to R-matrix (2 ), we consider different realisations of algebra (4 ) in canonical operators u, p with [u, p] = ifi. First imposing the reduction 7; = 0 in (4) we simplify it to 7:~,~=e~(**+~)7,,7:,
7i272i =e-2ie72i7i2,
7f 72, =ei(*a-e)72, 7: .
(5)
Observing that only Weyl type relations are involved in the above algebra we easily find a realisation 7: =e(V+c)P>
7,
=
_,-ol-C,P
7
72,
T12 =
=qe-ep+q,
_ve(2+c)cP-17,
(6)
consistent with (5 ), where (Y= r,&,8= efi. Since we are dealing here with operators, a certain ordering should be maintained in (6) and in all other relevant expressions in what follows, and we also consider only the periodic boundary conditions for all models. Inserting (6) with the reduction 7; = 0 in (3) one gets the explicit form ( 1/<)e(4+C)p__&-(tl-e)P L(,,C)
(2)
=
(
_
tle(2+c)~P-_q
qe-eP+q 0
>
*
(7)
Therefore Lax operator (7 ) associated with the quantum R-matrix (2 ) represents an integrable model, which depends on additional parameters E and c. Defining I+ = - C,,_,( C,,)-’ and I- = - C_ o,_2J (C_,)-‘, where C’*n are conserved quantities obtained as the expansion coefficients of the related transfer matrix,
one arrives at the Hamiltonian H= f (I+ + I- ) and the momentum P= f (I+ - Z- ) of the system as H= C Icosh2tlpi+tt2cosh?(Pi+Pi+,) i
exp[t(1+c)(Pi+,-Pi)+(qi-qi+l)]},
(9)
A. Kundu ~Physics Letters A 190 (1994) 79--84
P= ~ {sinh D/pi + ~/2 sinh ~/(pi +p~+ 1) exp [ - ¢ ( 1 + c ) (p~ -p~+ 1) + (q~ - q~+ ~) ] }.
81
(10)
To clarify the meaning of the parameters entering into the system, notice that the Lax operator (7) and the Hamiltonian (9) depend on the deformation parameters ~/, ~ along with c coming from the realisation (6), but not on the parameter h. On the other hand R-matrix (2) and the quantised algebra (4) are free from c and depend on ol= rift, 0= ¢h, i.e. they depend not only on ~/, ¢, but also on the quantum parameter fi coming from the commutators. When the deformation parameter q=C-, 1, i.e. ~/--,0 with finite h, one has to scale also the spectral parameters like 2 = ~/u, which would reduce the R-matrix to its rational form and the L operator to the time-continuous quantum TC model, though more generalised to include the parameters ¢ and c. However we notice that for general values of 0 (2) does not give the classical r as in ( 12 ). Therefore such integrable models interestingly seem to be living only at the quantum level. I n the case when c = - 1 or ~ 0 one recovers at ~/~0 the standard Toda chain and therefore we concentrate on them now. Consider first the choice ~= Co~/yielding from (6) z~ =exp[r/(l +Co)p],
z21 =~lexp(-~lCoCp+q),
z7 = - c x p [ - ~ / ( l - c o ) p ]
,
z12=_~lexp[tl(2+C)Cop-q] ,
(ll)
with z~ =0, which readily gives the Lax operator from (3) associated with Ro-matrix (2) with O=Coa. The classical limit exists for this model, since the R-matrix at h ~ 0,
R(coOt, a, 212) =I+fir(~, ~l,;t12) + O ( h ) ,
(12)
yields the classical r-matrix and algebra (4) which with 0= Coa reduces to a Poisson bracket algebra. The relevant conserved charges of the model may be given by I + = ~ {exp( _ 2r/p,) +~/2 exp[ +-~l(p~+P,+l)+~ICo(1 + c ) (p,+l - P , ) + (q,-q~+l) ]} •
(13)
i
Note that at the continuous-time or equivalently at the nonrelativistic limit: ~/-~0, the corresponding Hamiltonian H = ½(I + + I - ), the Lax operator and the quantum R-matrix yield exactly those belonging to the standard quantum TC. Thus this model for different values of the parameters Co and c generates a family of discrete-time or relativistic quantum integrable Toda chains. Interestingly, since the R-matrix is independent of the parameter c, all the models with different values of c will share the same quantum R-matrix. Some particular models of the above integrable class ( 11 ), (13) deserve special attention and we will look more closely at them. For example, at Co= 1 and c=0, ( 11 ) reduces to zi~ = e 2~p,
zi- = - 1,
321 = Y / e q ,
Zl2 = --r/¢2~n-q ,
(14)
and the conserved quantities (13) take a simpler form as I + = ~ {exp(2~/p,)[1 +~/2 exp(q,_~-q~)]}, i
I - = ~ {exp(-E~lP~)[l+rl2exp(q,-q~+l)]}.
(15)
i
We immediately recognise this case to be the quantum generalisation of the DTTC due to Suris [ 4 ]. The associated quantum R~-matrix is obtained easily by putting 0= a in (2), which through ( 12 ) recovers exactly the asymmetric classical r-matrix of Ref. [ 4 ]. A special case having the most simplification is obtained when c = Co= 0. As is seen from ( l 1 ) and ( 13 ) the Lax operator of this model is given by Lo(2) = ( ( 1 / ~ ) e ' ~ - ~e-'W - r/e-q
T/eq
)
0 ,,'
yielding the Hamiltonian and momentum in more symmetric form
(16)
A. Kundu /Physics LettersA 190 (1994) 79-84
82
• H = Z [cosh(2t/p;) "Jt-~]2 cosh tl(Pi+Pi+l) exp(qi-qi+l) ] ,
(17)
i
P = Y. [sinh(2t/p,) +t/: sinh rl(Pi+Pi+l ) exp(q/-qi+~ ) ] •
(18)
i
The corresponding R-matrix is clearly the 0 = 0 expression of (2), i.e. the standard one [ 8 ], (sin(,~12 + or) sin 2 ~2 Ro(;t12)=i sin a
/ sin a sinAiE
]"
(19)
sin(212 +or)/ Note that (14) and ( 16 ) are the cases presented in Ref. [ 6 ]. Another situation of interest appears when Co= 1 but c# 0, which generalises the quantum Suris model ( 15 ) to give I + = ~ (exp (2t/p~) { 1 +t/2 exp[tlc(p ` - p i _ l ) + (qi-~ - q i ) ]}) , I - = ~ (exp( -2t/pi){ 1 +t/2 exp[t/c(pi+l -Pi) + (qi-q,+~)]}),
(20)
i
though sharing the same quantum R-matrix with it. The Lax operator is obtained form (7) by putting e = t/. Coming now to the c= - 1 case for arbitrary values of E, it is seen clearly from (9), (10) that it yields the same symmetric conserved quantities (17), (18), though the corresponding R-matrix is given in a general form (2) and with a more involved Lax operator. Finally, it is important to observe that the Hamilton,an and momentum (9), (10) of the generating model is transformed exactly to the Suris form ( 15 ) under canonical transformation (p, q) --, (P,. Q) as
p=P,
q+ [~l-¢( l +c) ]p=Q.
(21)
Therefore, since the other models are obtained from the generating model (9) at different possible values of the parameters, all of them are naturally canonically equivalent to the Suris model (15). On the other hand canonical equivalence of the Suris model with the relativistic Toda chain of Ruijsenaars is already demonstrated in Ref. [ 4 ]. This establishes that the whole family of different quantum integrable discrete-time generalisations of TC presented here are canonically equivalent also to the relativistic generalisations of the Toda chain, though they represent as such distinct systems with different Lax operators and R-matrices. 3. Now we switch over to a recently proposed integrable classical lattice model close to TC, given in canonical variables Pi, q, by the Lax operator [ 7 ]
L,(2)=().l-TPiqi-koi \ otiqi
fliP, ~ Otifli/y]'
(22)
and find below a quantum as well as a discrete-time general,sat,on of it. obtained again from the ancestor L operator (3). Consider a real,sat,on of (4) with 0 = 0 as z ~ = ~ 2 sin(ht/)h}'exp[~:r/(Ni+oJi) ],
z21,=fliA +,
ot,fl; r ~ = - ~ exp(+t/Ni)
zl2i=oti[cos(hrl) ]A;" ,
(23)
where A +- are general,sat,on q-oscillators [ 11 ] given by the commutation relations
[Nk,A?] =_ltikt~4g, +" +
A~A~- -exp(-T- 2ifi~/)A~A~- =itiklfi exp[ + ~ / ( 2 N + t o - i h ) ] .
(24)
A. Kundu ~PhysicsLettersA 190 (1994) 79-84
83
Through canonical variables these operators may be expressed as
N=pq,
A-=qf(N),
A+ = f ( N ) p ,
(25)
with
f 2 ( N ) = 2 cos(hq) [ih].
( [ 2 N + t o - i h ] ~ - [og-ih],)
and Ix ]. = sinh ~lx/sinh r/. Inserting (23) into (3) and assuming ~= e"a we get the Lax operator of the discretetime quantum model as
((iTh/[ih].) [2+N~ + oJ~],~ '8~A+ ) oli cos(hq)A7 (olifli/~)) c o s h [ q ( N - 2 ) ] '
Li(2)--\
(26)
associated with the quantum Ro-matrix (19). In the time-continuous limit q--. 0, (26) reduces clearly to the form (22), though giving a quantum version of it with quantum R-matrix, R = I + ih ~ / ( 2 - #), obtainable from (19) at n~0. 4. Though the explicit R-matrix and L operators are found for all the above quantum DTTC, the standard quantum inverse scattering method is not applicable to them. The reason of this, as is also true for the TC, is the absence of a pseudovacuum for such models. However, with the use of the functional Bethe ansatz (FBA), Sklyanin was able to transform the eigenvalue of the quantum TC from an n-particle to a single particle eigenfunction [ 3 ]. It is interesting to observe that the same scheme for separation of variables is generalised trivially for the present discrete-time models. Therefore, referring to Ref. [3] for details, we mention only its main features sticking to the Ro-matrix ( 19 ). Using QYBE ( 1 ) with L ~ T for the monodromy matrices T(~) (see (8) for the definition) we find [C(~), C(() ] =0, which is crucial for the application of FBA, along with functional relations between C(~), A (() and C(~), D ((). Defining commuting operators 0. as C(2 = 0.) = 0 and ~=exp(i2), ~ = e x p ( i 0 . ) , we construct conjugate operators A7 =A(A~Oj), A+ =D(2~Oj) with the proper operator ordering prescription [ 3 ]. Parallel to Ref. [ 3 ] the relations between C, A and C, D show that the operators A~ shift the operators Ojby -T-a. Since the expansion of C(~, ~.) describes symmetric polynomials in ~. and the Af have a shifting effect only on ~j, we achieve a separation of variables for the N-particle eigenfunction 0 ( ~ , ~2, ..., ~ ) = ]78 0. (~.) symmetric in ~. as
t(~.)fb.(~.) = [A (~.) + D ( ~ ) ]¢~(~.) = i - N ¢ . ( e - i ~ ) + iN~. ( e i ~ . ) .
(27)
The above described method of separation of variables is applicable directly to the discrete-time models associated with Ro-matrix ( 19 ), while for other models related to (2) some more effort is needed. We have considered here the periodic boundary condition for the discrete-time family of quantum TC, which corresponds to the root system of AN_ ~. The extension of these models for other types of boundary conditions related to other classical algebras would be an interesting problem. The author likes to thank Orlando Ragnisco of Rome University for many valuable discussions and Professor Vladimir Rittenberg for encouragement and for creating a stimulating atmosphere in his research group. The support of the Alexander yon Humboldt Foundation is thankfully acknowledged.
References
[ 1] S.v. Manakov,Zh. Eksp.Teor. Fiz. 67 (1974) 269; H. Flaschkaand D.W.McLaughlin,Pro~ Theor. Phys.55 (1976) 438.
84
A. Kundu / Physics Letters A 190 (1994) 79-84
[2 ] M.A. Olshanetsky and A.M. Perelomov, Lett. Math. Phys. 2 (1977 ) 7; M.C. Gutzwiller, Ann. Phys. 133 (1981) 304; V. Pasquier and M. Gaudin, J. Phys. A 25 (1992) 5243. [ 3 ] E.K. Sklyanin, Springer Lecture Notes on Physics, Vol. 226 (Springer, Berlin, 1985 ) p. 196. [4] Yu.B. Suds, Phys. Lett. A 145 (1990) 113. [ 5 ] S.N.M. Ruijsenaars, Comm. Math. Phys. 133 (1990) 217. [6] A. Kundu and B. Basu Mallick, Mod. Phys. Lett. A 7 (1992) 61. [7] P.L. Cristiansen, M.F. Jorgensen and V.B. Kuznetsov, Lett. Math. Phys. 29 (1993) 165. [8] E.K. Sklyanin, L.A. Takhtajan and L.D. Faddeev, Teor. Mat. Fiz. 40 (1979) 194. [ 9 ] IC Sogo, M. Uchinami, Y. Akutsu and M. Wadati, Prog. Theor. Phys. 68 (1982) 508. [ 10] B. Basu Mallick and A. Kundu, J. Phys. A 25 (1992) 4147. [ 11 ] A.J. Macfarlane, J. Phys. 22 (1989) 4581; L.C. Biederham, J. Phys. A 22 (1989) L873.
ELSEVIER
Physics Letters A 190 (1994 ) 79-84
Generation of a quantum integrable class of discrete-time or relativistic periodic Toda chains Anjan K u n d u l Physikalisches Institut der Universitiit Bonn, Nussallee 12, 53115 Bonn, Germany
Received 14 March 1994; accepted for publication 12 May 1994 Communicated by A.R. Bishop
Abstract
A new integrable class of quantum models representing a family of different discrete-time or relativistic generalisations of the periodic Toda chain (TC), including that of a recently proposed classical model close to TC [ Lett. Math. Phys. 29 ( 1993 ) 165 ] is presented. All such models are shown to be obtainable from a single ancestor model at different realisations of the underlying quantised algebra. As a consequence the 2 X 2 Lax operators and the associated quantum R-matrices for these models are easily derived ensuring their quantum integrability. It is shown that the functional Bethe ansatz developed for the quantum TC is trivially generalised to achieve separation of variables also for the present models. 1. The Toda chain ( T C ) , which is considered to be one of the most fascinating integrable lattice systems, attracted continuous attention over the years to investigate both its classical [ 1 ] and q u a n t u m [ 2,3 ] aspects. In the recent past a discrete-time TC ( D T T C ) was proposed by Suris at the classical level [4], which was also shown to be equivalent to the relativistic TC of Ruijsenaars [ 5 ]. Following that we obtained [ 6 ] two different q u a n t u m generalisations of such D T T C along with the associated quantum R-matrices. Recently another model close to TC, which is classically integrable, has been put forward [ 7 ]. We present here a discrete-time as well as q u a n t u m generalisation of this model. However the basic aim of this Letter is to find a new quantum integrable class of different discrete-time generalisations of the periodic TC, which are also canonically related to the relativistic TC o f Ruijsenaars. We show that all such models can be generated from a single ancestor model at different realisations of the underlying quantised algebra, which is defined by the extended trigonometric Sklyanin algebra [6]. At the same time we also obtain automatically the associated quantum R-matrix, while the integrability condition given by the q u a n t u m Yang-Baxter equation (QYBE), R(2, #)L1 (2)L2 (/.t) =L2(/~)LI (2)R(,~,/z) ,
( 1)
with L~ - L ® l, L 2 = 1 ® L is satisfied by the construction. Moreover, we observe that the functional Bethe ansatz developed by Sklyanin [ 3 ] for the separation of variables in q u a n t u m TC can be applied in a parallel way also to its discrete-time generalisations. Since our aim here is to generate models, we start not from any particular model, hut directly from the quant u m R-matrix taking it in the form t Address after June 1994: Saha Institute of Nuclear Physics, AF/1 Bidhan Nagar, Calcutta 700 064, India. 0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved SSDI0375-9601 (94)00374-X
A. Kundu/PhysicsLetters A 190 (I 994) 79-84
80
sin(AZl++) -“sinAi2 e sin (Y
l
R&,2)=
sin (Y eiesinAiz ’ sin(A12 +cr) 1
(2)
with Ai2=A- p. Note that (2) is obtained from the standard trigonometric R,,-matrix [ 81 by a simple “gauge transformation” [ 91: R,=F( t?)Rd;( 0) with F( 6) =exp [ ie( a@ 1 - 180~) 1, which takes a solution of the YBE to another solution of it introducing an extra parameter 8. Now following the Yang-Baxterisation scheme for the construction of Lax operators [ 10 ] one obtains its generalised form
(
57r+
L(A)=
(l/07:
721
712
<72+(1/07:
>
(3)
’
with spectral parameter &eil and yet undefined abstract operators 7. The integrability condition on (3), that is the validity of QYBE ( 1) with R-matrix (2), dictates the algebra for T as e’e7,272,-e-ie72,712=-2isincu(7:7~-7~7:), 7i+fV..ei(*[email protected]* y I 7
7,* 7ji =ei([email protected]+ ,r I 9
(4)
with i, j= 1,2. We call this algebra extended trigonometric Sklyanin algebra, which may be reduced to the well known quantum algebra UJsu(2)) at some particular reduction, though in general, it allows more freedom necessary for the construction of integrable models [ 6 1. Thus (3 ) represents the Lax operator of a generalised quantum integrable model, entries of which are governed by algebra (4). 2. Now, in order to generate a family of DTTC related to R-matrix (2 ), we consider different realisations of algebra (4 ) in canonical operators u, p with [u, p] = ifi. First imposing the reduction 7; = 0 in (4) we simplify it to 7:~,~=e~(**+~)7,,7:,
7i272i =e-2ie72i7i2,
7f 72, =ei(*a-e)72, 7: .
(5)
Observing that only Weyl type relations are involved in the above algebra we easily find a realisation 7: =e(V+c)P>
7,
=
_,-ol-C,P
7
72,
T12 =
=qe-ep+q,
_ve(2+c)cP-17,
(6)
consistent with (5 ), where (Y= r,&,8= efi. Since we are dealing here with operators, a certain ordering should be maintained in (6) and in all other relevant expressions in what follows, and we also consider only the periodic boundary conditions for all models. Inserting (6) with the reduction 7; = 0 in (3) one gets the explicit form ( 1/<)e(4+C)p__&-(tl-e)P L(,,C)
(2)
=
(
_
tle(2+c)~P-_q
qe-eP+q 0
>
*
(7)
Therefore Lax operator (7 ) associated with the quantum R-matrix (2 ) represents an integrable model, which depends on additional parameters E and c. Defining I+ = - C,,_,( C,,)-’ and I- = - C_ o,_2J (C_,)-‘, where C’*n are conserved quantities obtained as the expansion coefficients of the related transfer matrix,
one arrives at the Hamiltonian H= f (I+ + I- ) and the momentum P= f (I+ - Z- ) of the system as H= C Icosh2tlpi+tt2cosh?(Pi+Pi+,) i
exp[t(1+c)(Pi+,-Pi)+(qi-qi+l)]},
(9)
A. Kundu ~Physics Letters A 190 (1994) 79--84
P= ~ {sinh D/pi + ~/2 sinh ~/(pi +p~+ 1) exp [ - ¢ ( 1 + c ) (p~ -p~+ 1) + (q~ - q~+ ~) ] }.
81
(10)
To clarify the meaning of the parameters entering into the system, notice that the Lax operator (7) and the Hamiltonian (9) depend on the deformation parameters ~/, ~ along with c coming from the realisation (6), but not on the parameter h. On the other hand R-matrix (2) and the quantised algebra (4) are free from c and depend on ol= rift, 0= ¢h, i.e. they depend not only on ~/, ¢, but also on the quantum parameter fi coming from the commutators. When the deformation parameter q=C-, 1, i.e. ~/--,0 with finite h, one has to scale also the spectral parameters like 2 = ~/u, which would reduce the R-matrix to its rational form and the L operator to the time-continuous quantum TC model, though more generalised to include the parameters ¢ and c. However we notice that for general values of 0 (2) does not give the classical r as in ( 12 ). Therefore such integrable models interestingly seem to be living only at the quantum level. I n the case when c = - 1 or ~ 0 one recovers at ~/~0 the standard Toda chain and therefore we concentrate on them now. Consider first the choice ~= Co~/yielding from (6) z~ =exp[r/(l +Co)p],
z21 =~lexp(-~lCoCp+q),
z7 = - c x p [ - ~ / ( l - c o ) p ]
,
z12=_~lexp[tl(2+C)Cop-q] ,
(ll)
with z~ =0, which readily gives the Lax operator from (3) associated with Ro-matrix (2) with O=Coa. The classical limit exists for this model, since the R-matrix at h ~ 0,
R(coOt, a, 212) =I+fir(~, ~l,;t12) + O ( h ) ,
(12)
yields the classical r-matrix and algebra (4) which with 0= Coa reduces to a Poisson bracket algebra. The relevant conserved charges of the model may be given by I + = ~ {exp( _ 2r/p,) +~/2 exp[ +-~l(p~+P,+l)+~ICo(1 + c ) (p,+l - P , ) + (q,-q~+l) ]} •
(13)
i
Note that at the continuous-time or equivalently at the nonrelativistic limit: ~/-~0, the corresponding Hamiltonian H = ½(I + + I - ), the Lax operator and the quantum R-matrix yield exactly those belonging to the standard quantum TC. Thus this model for different values of the parameters Co and c generates a family of discrete-time or relativistic quantum integrable Toda chains. Interestingly, since the R-matrix is independent of the parameter c, all the models with different values of c will share the same quantum R-matrix. Some particular models of the above integrable class ( 11 ), (13) deserve special attention and we will look more closely at them. For example, at Co= 1 and c=0, ( 11 ) reduces to zi~ = e 2~p,
zi- = - 1,
321 = Y / e q ,
Zl2 = --r/¢2~n-q ,
(14)
and the conserved quantities (13) take a simpler form as I + = ~ {exp(2~/p,)[1 +~/2 exp(q,_~-q~)]}, i
I - = ~ {exp(-E~lP~)[l+rl2exp(q,-q~+l)]}.
(15)
i
We immediately recognise this case to be the quantum generalisation of the DTTC due to Suris [ 4 ]. The associated quantum R~-matrix is obtained easily by putting 0= a in (2), which through ( 12 ) recovers exactly the asymmetric classical r-matrix of Ref. [ 4 ]. A special case having the most simplification is obtained when c = Co= 0. As is seen from ( l 1 ) and ( 13 ) the Lax operator of this model is given by Lo(2) = ( ( 1 / ~ ) e ' ~ - ~e-'W - r/e-q
T/eq
)
0 ,,'
yielding the Hamiltonian and momentum in more symmetric form
(16)
A. Kundu /Physics LettersA 190 (1994) 79-84
82
• H = Z [cosh(2t/p;) "Jt-~]2 cosh tl(Pi+Pi+l) exp(qi-qi+l) ] ,
(17)
i
P = Y. [sinh(2t/p,) +t/: sinh rl(Pi+Pi+l ) exp(q/-qi+~ ) ] •
(18)
i
The corresponding R-matrix is clearly the 0 = 0 expression of (2), i.e. the standard one [ 8 ], (sin(,~12 + or) sin 2 ~2 Ro(;t12)=i sin a
/ sin a sinAiE
]"
(19)
sin(212 +or)/ Note that (14) and ( 16 ) are the cases presented in Ref. [ 6 ]. Another situation of interest appears when Co= 1 but c# 0, which generalises the quantum Suris model ( 15 ) to give I + = ~ (exp (2t/p~) { 1 +t/2 exp[tlc(p ` - p i _ l ) + (qi-~ - q i ) ]}) , I - = ~ (exp( -2t/pi){ 1 +t/2 exp[t/c(pi+l -Pi) + (qi-q,+~)]}),
(20)
i
though sharing the same quantum R-matrix with it. The Lax operator is obtained form (7) by putting e = t/. Coming now to the c= - 1 case for arbitrary values of E, it is seen clearly from (9), (10) that it yields the same symmetric conserved quantities (17), (18), though the corresponding R-matrix is given in a general form (2) and with a more involved Lax operator. Finally, it is important to observe that the Hamilton,an and momentum (9), (10) of the generating model is transformed exactly to the Suris form ( 15 ) under canonical transformation (p, q) --, (P,. Q) as
p=P,
q+ [~l-¢( l +c) ]p=Q.
(21)
Therefore, since the other models are obtained from the generating model (9) at different possible values of the parameters, all of them are naturally canonically equivalent to the Suris model (15). On the other hand canonical equivalence of the Suris model with the relativistic Toda chain of Ruijsenaars is already demonstrated in Ref. [ 4 ]. This establishes that the whole family of different quantum integrable discrete-time generalisations of TC presented here are canonically equivalent also to the relativistic generalisations of the Toda chain, though they represent as such distinct systems with different Lax operators and R-matrices. 3. Now we switch over to a recently proposed integrable classical lattice model close to TC, given in canonical variables Pi, q, by the Lax operator [ 7 ]
L,(2)=().l-TPiqi-koi \ otiqi
fliP, ~ Otifli/y]'
(22)
and find below a quantum as well as a discrete-time general,sat,on of it. obtained again from the ancestor L operator (3). Consider a real,sat,on of (4) with 0 = 0 as z ~ = ~ 2 sin(ht/)h}'exp[~:r/(Ni+oJi) ],
z21,=fliA +,
ot,fl; r ~ = - ~ exp(+t/Ni)
zl2i=oti[cos(hrl) ]A;" ,
(23)
where A +- are general,sat,on q-oscillators [ 11 ] given by the commutation relations
[Nk,A?] =_ltikt~4g, +" +
A~A~- -exp(-T- 2ifi~/)A~A~- =itiklfi exp[ + ~ / ( 2 N + t o - i h ) ] .
(24)
A. Kundu ~PhysicsLettersA 190 (1994) 79-84
83
Through canonical variables these operators may be expressed as
N=pq,
A-=qf(N),
A+ = f ( N ) p ,
(25)
with
f 2 ( N ) = 2 cos(hq) [ih].
( [ 2 N + t o - i h ] ~ - [og-ih],)
and Ix ]. = sinh ~lx/sinh r/. Inserting (23) into (3) and assuming ~= e"a we get the Lax operator of the discretetime quantum model as
((iTh/[ih].) [2+N~ + oJ~],~ '8~A+ ) oli cos(hq)A7 (olifli/~)) c o s h [ q ( N - 2 ) ] '
Li(2)--\
(26)
associated with the quantum Ro-matrix (19). In the time-continuous limit q--. 0, (26) reduces clearly to the form (22), though giving a quantum version of it with quantum R-matrix, R = I + ih ~ / ( 2 - #), obtainable from (19) at n~0. 4. Though the explicit R-matrix and L operators are found for all the above quantum DTTC, the standard quantum inverse scattering method is not applicable to them. The reason of this, as is also true for the TC, is the absence of a pseudovacuum for such models. However, with the use of the functional Bethe ansatz (FBA), Sklyanin was able to transform the eigenvalue of the quantum TC from an n-particle to a single particle eigenfunction [ 3 ]. It is interesting to observe that the same scheme for separation of variables is generalised trivially for the present discrete-time models. Therefore, referring to Ref. [3] for details, we mention only its main features sticking to the Ro-matrix ( 19 ). Using QYBE ( 1 ) with L ~ T for the monodromy matrices T(~) (see (8) for the definition) we find [C(~), C(() ] =0, which is crucial for the application of FBA, along with functional relations between C(~), A (() and C(~), D ((). Defining commuting operators 0. as C(2 = 0.) = 0 and ~=exp(i2), ~ = e x p ( i 0 . ) , we construct conjugate operators A7 =A(A~Oj), A+ =D(2~Oj) with the proper operator ordering prescription [ 3 ]. Parallel to Ref. [ 3 ] the relations between C, A and C, D show that the operators A~ shift the operators Ojby -T-a. Since the expansion of C(~, ~.) describes symmetric polynomials in ~. and the Af have a shifting effect only on ~j, we achieve a separation of variables for the N-particle eigenfunction 0 ( ~ , ~2, ..., ~ ) = ]78 0. (~.) symmetric in ~. as
t(~.)fb.(~.) = [A (~.) + D ( ~ ) ]¢~(~.) = i - N ¢ . ( e - i ~ ) + iN~. ( e i ~ . ) .
(27)
The above described method of separation of variables is applicable directly to the discrete-time models associated with Ro-matrix ( 19 ), while for other models related to (2) some more effort is needed. We have considered here the periodic boundary condition for the discrete-time family of quantum TC, which corresponds to the root system of AN_ ~. The extension of these models for other types of boundary conditions related to other classical algebras would be an interesting problem. The author likes to thank Orlando Ragnisco of Rome University for many valuable discussions and Professor Vladimir Rittenberg for encouragement and for creating a stimulating atmosphere in his research group. The support of the Alexander yon Humboldt Foundation is thankfully acknowledged.
References
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A. Kundu / Physics Letters A 190 (1994) 79-84
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