Bäcklund Transformations as exact integrable time-discretizations for the trigonometric Gaudin model

0.25 MB·English

by  O. Ragnisco

B¨acklund Transformations as exact integrable
time-discretizations for the trigonometric Gaudin
model
Orlando Ragnisco, Federico Zullo
Dipartimento di Fisica, Universit`a di Roma Tre
1 Istituto Nazionale di Fisica Nucleare, sezione di Roma Tre
1 Via Vasca Navale 84, 00146 Roma, Italy
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E-mail: ragnisco@fis.uniroma3.it, zullo@fis.uniroma3.it
2

n
a
J

5 Abstract
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Weconstructatwo-parameterfamilyofB¨acklundtransformationsforthetrigono-
]
I
metricclassical Gaudinmagnet. Theapproach follows closely theoneintroduced
S
. by E.Sklyanin and V.Kuznetsov (1998,1999) in a number of seminal papers, and
n
i takes advantage of the intimate relation between the trigonometric and the ra-
l
n tional case. As in the paper by A.Hone, V.Kuznetsov and one of the authors
[
(O.R.) (2001) the B¨acklund transformations are presented as explicit symplectic

2 maps, starting from their Lax representation. The (expected) connection with
v thexxz Heisenbergchain isestablished andtherational (xxx) caseisrecovered in
9
8 a suitable limit. It is shown how to obtain a “physical” transformation mapping
3 real variables into real variables. The interpolating Hamiltonian flow is derived
0
and some numerical iterations of the map are presented.
.
3
0
0
1
:
v
i
X
KEYWORDS: B¨acklund Transformations, Integrable maps, Gaudin systems, Lax rep-
r
a resentation, r-matrix.
1
1 Introduction
B¨acklund transformations are nowadays a widespread useful tool related to the theory
of nonlinear differential equations. The first historical evidence of their mathematical
significance was given by Bianchi [3] andB¨acklund [2] ontheir works onsurfaces ofcon-
stant curvature. A simple approach to understand their importance can be to regard
them as a mechanism allowing to endow a given nonlinear differential equation with a
nonlinear superposition principle yielding a set of solutions through a merely algebraic
procedure [19],[1],[12]. B¨acklund transformations are indeed parametric families of dif-
ference equations encoding the whole set of symmetries of a given integrable dynamical
system. For finite-dimensional integrable systems the technique of B¨acklund transfor-
mations leads to the construction of integrable Poisson maps that discretize a family of
continuous flows [27],[25],[24],[22],[10],[9]. Actually in the last two decades numerous
results have appeared in the field of exact discretization of many-body integrable sys-
tems employing the B¨acklund transformations tools [17],[24],[16],[9],[10],[15],[22]. For
the rational Gaudin model such discretization has been obtained ten years ago in [8];
afterwards, these results have been used for constructing an integrable discretization of
classical dynamical systems (as the Lagrange top) connected to Gaudin model through
In¨onu-Wigner contractions [13],[11],[14].
The aim of the present work is to construct B¨acklund transformations for the Gaudin
model in the partially anisotropic (xxz) case, i.e. for the trigonometric Gaudin model.
We point out that partial results on this issue have already been given in [18].
The paper is organized as follows.
In Section (2) we review the main features of the trigonometric Gaudin model from
the point of view of its integrability structure. For the sake of completeness, in Section
(3) we briefly recall the preliminary results on B¨acklund Transformations (BTs) for
trigonometric Gaudin given in [18]. In Section (4) the explicit form of BTs is given; it
is shown that they are indeed a trigonometric generalization of the rational ones (see
[8]) which can be recovered in a suitable (“small angle” ) limit. The simplecticity of
the transformations is also discussed in the same Section and the proof allows us to
elucidate the (expected) link between the Darboux-dressing matrix and the elementary
Lax matrix for the xxz Heisenberg magnet on the lattice. We end the Section by
mentioninganopenquestion, namelytheconstructionofanexplicitgeneratingfunction
for these B¨acklund transformations. In Section (5) we will show how our map can lead,
with an appropriate choice of B¨acklund parameters, to physical transformations, i.e.
transformations from real variables to real variables. In the last Section we show how a
suitable continuous limit yields the interpolating Hamiltonian flow and finally present
numerical examples of iteration of the map.
2 Gaudin magnet in the trigonometric case
For a full account of the integrability structure of the classical and quantum Gaudin
model we refer the reader to the fundamental contributions by Semenov-Tian-Shanski
[26] and Babelon-Bernard-Talon [4]. In this section we briefly recall the main features
2
of the trigonometric Gaudin magnet.
The Lax matrix of the model is given by the expression:
A(λ) B(λ)
L(λ) = (1)
C(λ) A(λ)
(cid:18) − (cid:19)
N N s− N s+
A(λ) = cot(λ λ )s3, B(λ) = j , C(λ) = j .
− j j sin(λ λ ) sin(λ λ )
j j
j=1 j=1 − j=1 −
X X X
(2)
In (1) and (2) λ C is the spectral parameter, λ are arbitrary real parameters of
j
∈
the model, while s+,s−,s3 , j = 1,...,N, are the dynamical variables of the system
j j j
obeying to Nsl(2) algebra, i.e.
⊕ (cid:0) (cid:1)
s3,s± = iδ s±, s+,s− = 2iδ s3, (3)
j k ∓ jk k j k − jk k
By fixing the N Cas(cid:8)imirs (cid:9)s3 2 +s+s− =. s2(cid:8)one ob(cid:9)tains a symplectic manifold given
j j j j
by the direct sum of the correspondent N two-spheres.
(cid:0) (cid:1)
Reformulating the Poisson structure in terms of the r-matrix formalism amounts to
state that the Lax matrix satisfies the linear r-matrix Poisson algebra (see again [26],
[4]) :
L(λ) 1,1 L(µ) = r (λ µ),L(λ) 1 +1 L(µ) , (4)
t
⊗ ⊗ − ⊗ ⊗
(cid:8) (cid:9) (cid:2) (cid:3)
where r (λ) stands for the trigonometric r matrix [5]:
t
cos(λ) 0 0 0
i 0 0 1 0
r (λ) =  , (5)
t sin(λ) 0 1 0 0
 0 0 0 cos(λ) 
 
 
Equation (4) entails the following Poisson brackets for the functions (2):
A(λ),A(µ) = B(λ),B(µ) = C(λ),C(µ) = 0,
{ } { } { }
cos(λ µ)B(µ) B(λ)
A(λ),B(µ) = i − − ,
{ } sin(λ µ)
−
C(λ) cos(λ µ)C(µ)
A(λ),C(µ) = i − − ,
{ } sin(λ µ)
−
2(A(µ) A(λ))
B(λ),C(µ) = i − . (6)
{ } sin(λ µ)
−
Thedeterminant oftheLaxmatrixisthegenerating functionoftheintegralsofmotion:
N s2
det(L) = A2(λ)+B(λ)C(λ) = i +H cot(λ λ ) H2 (7)
− sin2(λ λ ) i − i − 0
i=1 (cid:18) − i (cid:19)
X
3
where the N Hamiltonians H are of the form:
i
N 2cos(λ λ )s3s3 +s+s− +s−s+
H = i − k i k i k i k (8)
i
sin(λ λ )
i k
k6=i −
X
NotethatonlyN 1amongtheseHamiltoniansareindependent, becauseof H = 0.
− i i
Another integral is given by H , the projection of the total spin on the z axis:
0
P
N
.
H = s3 = J3 (9)
0 j
j=1
X
The Hamiltonians H are in involution for the Poisson bracket (3):
i
H ,H = 0 i,j = 0,...,N 1 (10)
i j
{ } −
The corresponding Hamiltonian flows are then given by:
ds3 ds±
j = H ,s3 j = H ,s± (11)
dt { i j} dt { i j }
i i
In the xxx model a remarkable Hamiltonian is found by taking a linear combination
of the integrals corresponding to (8) in the rational case [6]. It describes a mean field
spin-spin interaction:
N
1
= s s
r i j
H 2 ·
i6=j
X
Where the notation for the bold symbol s is s = (s1,s2,s3) with s+ = s1 + is2 and
i i i i i i i i
s− = s1 is2. The natural trigonometric generalization of this Hamiltonian can be
i i − i
found by taking the linear combination of 8:
N
sin(2λ )
i
H
i
2
i=1
X
giving
N
1
= cos(λ +λ ) s1s1 +s2s2 +cos(λ λ )s3s3 (12)
Ht 2 i j i j i j i − j i j
i6=j
X (cid:0) (cid:1)
3 A first approach to Darboux-dressing matrix
In this Section, for the sake of completeness, we recall the results already appeared in
[18]. The leading observation is that by performing the “uniformization” mapping:
.
λ z = eiλ
→
the N-sites trigonometric Lax matrix takes a rational form in z that corresponds to
the 2N-sites rational Lax matrix plus an additional reflection symmetry (see also [7]);
in fact, by performing the substitution (3), the Lax matrix (1) becomes:
4
N Lj Lj
L(z) = iJ3 + 1 σ 1 σ , (13)
3 3
z z − z +z
j j !
j=1 −
X
j
where σ is the Pauli matrix diag(1, 1) and the matrices L , j = 1,...,N, are given
3 − 1
by:
s3 s−
Lj = iz j j
1 j s+ s3
j − j !
So, equation (13) entails the following involution on L(z):
L(z) = σ L( z)σ (14)
3 3
−
Constructing a B¨acklund transformation for the Trigonometric Gaudin System (TGS)
amounts to build up a Poisson map for the field variables of the model (2) such that the
integrals of motion (8) are preserved. At the level of Lax matrices, this transformation
is usually seeked as a similarity transformation between an old, or “undressed”, Lax
matrix L, and a new, or “dressed” one, say L˜:
L(z) D(z)L(z)D−1(z) L˜(z) (15)
→ ≡
But L and L˜ have to enjoy the same reflection symmetry (14) too: to preserve this
involution the Darboux dressing matrix D has to share with L the property (14);
the elementary dressing matrix D is then obtained by requiring the existence of only
one pair of opposite poles for D in the complex plane of the spectral parameter. We
will show in the next Section that, thanks to this constraint, one recovers the form
of the Lax matrix for the elementary xxz Heisenberg spin chain: on the other hand,
this is quite natural if one recalls that for the rational Gaudin model the elementary
Darboux-dressing matrix is given by the Lax matrix for the elementary xxx Heisenberg
spin chain [8],[10]. The previous observations lead to the following Darboux matrix:
D D
1 1
D(z) = D + σ σ (16)
∞ 3 3
z ξ − z +ξ
−
By taking the limit z in (16) it is readily seen that D has to be a diagonal
∞
matrix. In order to en→sur∞e that L and L˜ have the same rational structure in z, we
rewrite equation (15) in the form:
L˜(z)D(z) = D(z)L(z) (17)
Now it is clear that both sides have the same residues at the poles z = z , z = ξ (it
j j
is unnecessary to look at the poles in z = z and z = ξ because of the symmetry
j j
− −
(14), so that the following set of equations have to be satisfied:
L˜(j)D(z ) = D(z )L(j), (18)
1 j j 1
L˜(ξ)D = D L(ξ). (19)
1 1
5
In principle, equations (18), (19) yield a Darboux matrix depending both on the old
(untilded) variables and the new (tilded) ones, implying in turn animplicit relationship
between the same variables. To get an explicit relationship one has to resort to the so-
called spectrality property [10] [9]. To this aim we need to force the determinant of the
Darboux matrix D(z) to have, besides the pair of poles at z = ξ, a pair of opposite
±
nondynamical zeroes, say at z = η, and to allow the matrix D to be proportional to
1
±
a projector [18]. Again by symmetry it suffices to consider just one of these zeroes. If
η is a zero of detD(z), then D(η) is a rank one matrix, possessing a one dimensional
kernel K(η) ; the equation (17) :
| i
L˜(η)D(η) = D(η)L(η) (20)
entails
D(η)L(η) K(η) = 0. (21)
| i
This equation in turn allows to infer that K(η) is an eigenvector for the Lax matrix
| i
L(η):
L(η) K(η) = µ(η) K(η) , (22)
| i | i
This relations gives a direct link between the parameters appearing in the dressing
matrix D and the old dynamical variables in L. Because of (19) we have another one
dimensional kernel K(ξ) of D , such that:
1
| i
L(ξ) K(ξ) = µ(ξ) K(ξ) . (23)
| i | i
In [18] we have shown how the two spectrality conditions (22), (23) enable to write D
in terms of the old dynamical variables and of the two B¨acklund parameters ξ and η.
The explicit expression of the Darboux dressing matrix is given by:
z(p(η)η−p(ξ)ξ) + (p(ξ)η−p(η)ξ)ηξ ξ2−η2
βz b bz b
D(z) = . (24)
z2 ξ2  bp(ξ)p(η)(ξ2−η2) b(p(η)η−p(ξ)ξ) + bz(p(ξ)η−p(η)ξ) 
−
ηξ z ηξ
 
In this expression β is a global multiplicative factor, inessential with respect to the
form of the BT, b is an undeterminate parameter that in Section (4) we will fix in
order to recover the form of the Lax matrix for the discrete xxz Heisenberg spin chain.
The functions p(η) and p(ξ) characterize completely the kernels of D(η) and D(ξ): in
fact we have the following formulas [18]:
1 1
K(ξ) = K(η) = (25)
| i p(ξ) | i p(η)
(cid:18) (cid:19) (cid:18) (cid:19)
As K(ξ) and K(η) are respectively eigenvectors of L(ξ) and L(η), p(ξ) and p(η)
| i | i
must satisfy:
µ(ξ) A(ξ) µ(η) A(η)
p(ξ) = − , p(η) = − (26)
B(ξ) B(η)
with A(z), B(z), C(z) given by (2) and µ2(z) = A2(z)+B(z)C(z).
6
4 Explicit map and an equivalent approach to Darboux-
dressing matrix
The matrix (24) contains just one set of dynamical variables so that the relation (15)
gives now an explicit map between the variables s˜+,s˜−,s˜3 and s+,s−,s3 . The map
j j j j j j
is easily found by (18); it reads:
(cid:0) (cid:1) (cid:0) (cid:1)
p(ξ)p(η)(ξ2 η2)((z 2 η2)p(ξ)ξ (z 2 ξ2)p(η)η)s−z
s˜3 = − k − − k − k k+
k ∆
k
(ξ2 η2)((z 2 ξ2)p(ξ)η p(η)ξ(z 2 η2))s+z
− k − − k − k k+
∆
k
s3 p(ξ)p(η)((ξ2 +z 2)(η2 +z 2) (η2 +ξ2) 8η2ξ2z 2) (27a)
k k k − − k
+
h ∆
k
ηξ(ξ2 z 2)(η2 z 2) p(ξ)2 +p(η)2
k k
− −
−(cid:0) ∆k (cid:0) (cid:1)(cid:1)i
b2p(ξ)2p(η)2(η2 ξ2)2s−z2 b2((z 2 ξ2)p(ξ)η p(η)ξ(z 2 η2))2s+
s˜+ = − k k + k − − k − k +
k − ξη∆ ηξ∆
k k
2b2p(ξ)p(η)(ξ2 η2)((z 2 ξ2)p(ξ)η p(η)ξ(z 2 η2))s3z
− k − − k − k k
ηξ∆
k
(27b)
(η2 ξ2)2s+z2ξη ((z 2 η2)p(ξ)ξ (z 2 ξ2)p(η)η)2s−ξη
s˜− = − k k + k − − k − k +
k − b2∆ b2∆
k k (27c)
2(ξ2 η2)((z 2 η2)p(ξ)ξ (z 2 ξ2)p(η)η)s3z ξη
− k − − k − k k
b2∆
k
where ∆ is proportional to the determinant of D(z ), i.e.
k k
∆ = (z2 ξ2)(z2 η2)(p(ξ)η p(η)ξ)(p(η)η p(ξ)ξ) (28)
k k − k − − −
Formulas (27a), (27b), (27c) define a two-parameter B¨acklund transformation, the
parameters being ξ and η: as we will show in the next section, it is a crucial point to
have a two-parameter family of transformations when looking for a physical map from
real variables to real variables. As mentioned in the previous Section, we now show
that indeed, by posing:
b = i ηξ (29)
the expression (24) of the dressing matrix pgoes into the expression of the elementary
Lax matrix for the classical, partially anisotropic, Heisenberg spin chain on the lattice
[5].
7
Obviously two matrices differing only for a global multiplicative factor give rise to the
same similarity transformation. So we omit the term βz in (24), and, taking into
z2−ξ2
account (29), we write for the diagonal part D of (24):
d
i
D = (p(ξ) p(η))(v w)1 +(p(ξ)+p(η))(v+w)σ (30)
d 3
2 − −
(cid:16) (cid:17)
where v(ξ,η) and w(ξ,η) are given by:
zξ η√ηξ ξ√ηξ zη
v(ξ,η) = w(ξ,η) = = v(η,ξ) (31)
√ηξ − z z − √ηξ −
We substitute:
ξ eiζ1 η eiζ2 z eiλ (32)
→ → →
and take a suitable redefinition of the B¨acklund parameters to clarify the structure of
the D matrix:
. ζ +ζ . ζ ζ
1 2 1 2
λ = µ = − (33)
0
2 2
With these positions it is simple to find that v w = 4ieiλ0 sin(λ λ )cos(µ) and
0
− −
v + w = 4ieiλ0cos(λ λ )sin(µ). So, considering equation (30) jointly with the off-
0
−
diagonal part of (24), the dressing matrix can be written as:
p(ζ )+p(ζ )
1 2
D(λ) =α sin(λ λ )1 + tan(µ)cos(λ λ )σ +
0 0 3
− p(ζ ) p(ζ ) −
1 2
h − (34)
2sin(µ) 0 1
+
p(ζ ) p(ζ ) p(ζ )p(ζ ) 0
2 − 1 (cid:18) − 1 2 (cid:19)i
where α is the global factor 2eiλ0(p(ζ ) p(ζ )). Observe that in formula (34), with
2 1
−
some abuse of notation, p(ζ ) (p(ζ )) stands of course for p(ξ) p(η) .
1 2 |ξ=eiζ1 |η=eiζ2
A last change of variables allows to identify the dressing matrix wit(cid:16)h the eleme(cid:17)ntary
Lax matrix of the classical xxz Heisenberg spin chain on the lattice, and furthermore
to recover the form of the Darboux matrix for the rational Gaudin model [8][20] in the
limit of small angles. Namely, we introduce two new functions, P and Q, by letting
2sin(µ)
p(ζ ) = Q p(ζ ) = Q. (35)
1 2
− P −
Then equation (34) becomes:
sin(λ λ µ)+PQ cos(λ λ ) P cos(µ)
D(λ) = α − 0 − − 0
Q sin(2µ) PQ2cos(µ) sin(λ λ +µ) PQ cos(λ λ )
0 0
(cid:18) − − − − (cid:19)
(36)
Obviously now we can repeat the argument made before about spectrality; indeed now
D and D are rank one matrices. So if Ω and Ω are respectively the
λ=λ0+µ λ=λ0−µ + −
(cid:12) (cid:12)
(cid:12) (cid:12)
8
kernels of D(λ +µ) and D(λ µ) one has again that Ω and Ω are eigenvectors of
0 0 + −
−
L(λ +µ) and L(λ µ) with eigenvalues γ and γ where
0 0 + −
−
γ = γ(λ)
±
λ=λ0±µ
(cid:13)
(cid:13)
and we have set (7) (cid:13)
γ2(λ) A2(λ)+B(λ)C(λ) = det(L(λ)) (37)
≡ −
The two kernels are given by:
1 P
Ω = Ω = (38)
+ Q − 2 sin(µ) PQ
(cid:18) − (cid:19) (cid:18) − (cid:19)
and the eigenvectors relations yields the following expression of P and Q in terms of
the old variables only:
A(λ) γ(λ) 1 Q(λ +µ) Q(λ µ)
0 0
Q = Q(λ +µ) = − = − − (39)
0
B(λ) λ=λ0+µ P 2 sin(µ)
(cid:13)
(cid:13)
The explicit map can be found by e(cid:13)quating the residues at the poles λ = λ in (17),
k
that is by the relation:
L˜ D = D L (40)
k k k k
where
s3 s−
L = k k , D = D(λ = λ ) (41)
k s+ s3 k k
(cid:18) k − k (cid:19)
or by performing the needed changes of variables in (27a), (27b), (27c). Anyway now
the map reads:
2 cos2(µ) (cos2(µ)+cos2(δk))(1 2PQ sin(µ)+P2Q2)
s˜3 = − 0 − s3+
k ∆ k
k
P cos(µ)(sin(δk) PQ cos(δk))
+ + − 0 s++ (42a)
∆ k
k
Q cos(µ)(2 sin(µ) PQ)(sin(δk)+PQ cos(δk))
− − 0 s−
− ∆ k
k
(sin(δk) PQ cos(δk))2 (Q2cos2(µ)(2 sin(µ) PQ))2
s˜+ = + − 0 s+ − s−+
k ∆ k − ∆ k
k k (42b)
2Q cos(µ)(2 sin(µ) PQ)(sin(δk) PQ cos(δk))
+ − + − 0 s3
∆ k
k
(sin(δk)+PQ cos(δk))2 P2 cos2(µ)
s˜− = − 0 s− s++
k ∆ k − ∆ k
k k (42c)
2P cos(µ)(sin(δk)+PQ cos(δk))
− 0 s3
− ∆ k
k
9
where for typesetting brevity we have put:
δk = λ λ
0 k − 0 (43)
δk = λ λ µ
(cid:26) ± k − 0 ±
and we have denoted by ∆ the determinant of D(λ ), that is:
k k
∆ := sin(λ λ µ)sin(λ λ +µ)(1 2PQ sin(µ)+P2Q2)
k k 0 k 0
− − − −
At this point we can show that for “small” λ and µ one obtains, at first order, the
0
B¨acklund for the rational Gaudin model, independently found by Sklyanin [20] on one
hand and Hone, Kuznetsov and Ragnisco [8] on the other, as the composition of two
one-parameter B¨acklunds. So let us take λ hλ , µ hµ and λ hλ where h is
0 0
→ → →
the expansion parameter. One has:
1 1 1
cot(λ λ ) = +O(h) = +O(h),
k
− h(λ λ ) sin(λ λ ) h(λ λ )
k k k
− − −
so that Q = qr + O(h2), where the superscript r stands for “rational” . Thus, qr
coincides with the variable q that one finds in the rational case [8]. For the variable P
one has:
2µ
P = h(pr +O(h2)) where pr =
qr(λ +µ) qr(λ µ)
0 0
− −
Taking into account these expressions, it is straightforward to see that the matrix (36)
has the expansion:
D(λ) = hDr(λ)+O(h3) (44)
where
λ λ µ+prqr pr
Dr(λ) = − 0 − . (45)
qr(2µ prqr) λ λ +µ prqr
0
(cid:18) − − − (cid:19)
The limit of “small angles” in (27a), (27b), (27c) obviously leads to the rational map
of [8].
4.1 Symplecticity
In this subsection we face the question of the simplecticity of our map; the corre-
spondence with the rational B¨acklund in the limit of “small angles” shows that the
transformations are surely canonical in this limit. Indeed, as our map is explicit, we
could check by brute-foce calculations whether the Poisson structure (3) is preserved
by tilded variables. However we will follow a finer argument due to Sklyanin [21].
Suppose that D(λ) obeys the quadratic Poisson bracket, that is
D1(λ),D2(τ) = [r (λ τ),D1(λ) D2(τ)] (46)
t
{ } − ⊗
where as usually D1 = D 1, D2 = 1 D. Consider the relation
⊗ ⊗
L(˜λ)D˜(λ λ ) = D(λ λ )L(λ) (47)
0 0
− −
10