Coordinate Bethe Ansatz for Spin s XXX Model

0.2 MB·English

by  Nicolas Crampé

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7(2011), 006, 13 pages
⋆
Coordinate Bethe Ansatz for Spin s XXX Model
Nicolas CRAMPE´ †‡, Eric RAGOUCY § and Ludovic ALONZI §
† Universit´e Montpellier 2, Laboratoire Charles Coulomb UMR 5221,
F-34095 Montpellier, France
‡ CNRS, Laboratoire Charles Coulomb UMR 5221, F-34095 Montpellier, France
E-mail: [email protected]
§ LAPTh, CNRS and Universit´e de Savoie, 9 chemin de Bellevue,
1
BP 110, 74941, Annecy-Le-Vieux Cedex, France
1
0 E-mail: [email protected]
2
Received September 06, 2010,in final form January 05, 2011; Published online January 12, 2011

n
doi:10.3842/SIGMA.2011.006
a
J

2 Abstract. We compute the eigenfunctions andeigenvaluesofthe periodic integrablespins
1
XXX model using the coordinate Bethe ansatz. To do so,we compute explicitly the Hamil-
1
tonian of the model. These results generalize what has been obtained for spin and spin 1
] 2
h chains.
p
- Key words: coordinate Bethe ansatz; spin chains
h
at 2010 Mathematics Subject Classification: 81R12; 17B80
m
[

2 1 Introduction
v
8
The resolution of Heisenberg spin chain [1] was initiated in H. Bethe’s seminal paper [2] where
0
4 he used a method called now coordinate Bethe ansatz. Since this work, several new methods
0 appeared: algebraic Bethe ansatz [3, 4], functional Bethe ansatz (or separation of variables) [5]
.
9 or analytical Bethe ansatz [6]. These more elaborated techniques allowed one to go further: new
0
solvable models have been discovered and new results have been computed such as correlation
0
1 functions. As a consequence, the coordinate Bethe ansatz was neglected. However, this method
: is the simplest one and gives a very efficient way to construct explicitly eigenfunctions, but it
v
i is believed that it works only for simple models. In this note, we show that actually it can be
X
applied also to more complicated models as the spin s XXX model.
r
a This paper is organized as follows. In Section 2, we compute the Hamiltonian of the spin s
chain wewant tosolve. Toourknowledge, theexplicit formof theentries oftheHamiltonian are
written for the first time. We also compte the su(2) symmetry algebra and the pseudo-vacuum,
a particular (reference) eigenstate. In Section 3, we present the coordinate Bethe ansatz and
get the Bethe equations obtained previously by the algebraic or analytical Bethe ansatz. We
conclude, in Section 4, on the advantages of this method and on open problems.
2 Integrable periodic spin s chain
2.1 Hamiltonian of the spin s chain
The Hamiltonian of the periodic integrable spin s chain has been computed in [7] thanks to
a fusion procedure. This Hamiltonian has been expressed as a polynomial of the invariant
⋆ThispaperisacontributiontotheProceedingsoftheInternationalWorkshop“RecentAdvancesinQuantum
Integrable Systems”. The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2010.html
2 N. Cramp´e, E. Ragoucy and L. Alonzi
of su(2) (see for example also [8] for a review). For our purpose, we need to give an explicit
expression of the Hamiltonian entries. Namely, the Hamiltonian is the following matrix acting
on (C2s+1)⊗L
L
H = h (2.1)
j,j+1
j=1
X
with the periodic condition L + 1 = 1 and the subscript (j,j + 1) stands for the two spaces
where the (2s+1)2 ×(2s+1)2-matrix h acts non-trivially. We choose to enumerate the basis
of C2s+1 as follows: |si, |s−1i, ..., |−si, where |mi ≡ |s,mi denotes1 the spin s state with
sz-component equals to m.
The non-vanishing entries of the matrix h may be parameterized by three integer parameters
m , m , n
1 2
2s min(m2,2s−m1)
h = βn |s−m −nihs−m |⊗|s−m +nihs−m |.
m1,m2 1 1 2 2
m1X,m2=0 n=−minX(m1,2s−m2)
From the results found in [7], one may prove that βn can be factorised as:
m1,m2
M +n M
1 2
βn = (−1)n−1vu (cid:18) M1 (cid:19) (cid:18) n (cid:19) for n > 0, (2.2)
m1,m2 n u 2s−M 2s−M +n
u 1 2
u n n
u
(cid:18) (cid:19) (cid:18) (cid:19)
u
t
where we have introduced the notation
M = min(m ,2s−m ) and M =min(m ,2s−m ). (2.3)
1 1 2 2 2 1
The remaining β’s are given by the relations:
M1−1 1 M2−1 1
β0 = − − ,
m1,m2 2s−ℓ 2s−ℓ
ℓ=0 ℓ=0
X X
β−n = βn .
m2,m1 m1,m2
2.2 su(2) symmetry
At each site, we have a spin s representation, and the expression of the su(2) generators in this
representation reads:
s
s− = (s+n)(s−n+1)|n−1ihn|,
n=−s+1
X p
s−1
s+ = (s−n)(s+n+1)|n+1ihn|,
n=−s
X p
s
sz = n|nihn|.
n=−s
X
1Let us stress that since all the sites have spin s, we do not mention it, and write only the sz value of the
states.
Coordinate Bethe Ansatz for Spin s XXX Model 3
They obey
1
[s+,s−]= 2sz, [sz,s±]= ±s±, c = (sz)2+ (s+s−+s−s+)= s(s+1).
2
2
We will note sα, α = z,± and j = 1,...,L, the generators acting on site j. Let us stress that
j
these local operators do not commute with the Hamiltonian H given in (2.1). However, there is
a global su(2) symmetry. The generators of this su(2) symmetry take the form
L L
Sz = sz and S± = s±.
j j
j=1 j=1
X X
They obey the su(2) commutation relations [Sz,S±] =±S±, [S+,S−]= 2Sz. Remark that the
Casimir operator C = (Sz)2 + 1(S+S− +S−S+), although central, is not proportional to the
2 2
identity, since we are considering the tensor product of L spin s representations, a reducible
representation.
It is a simple calculation to show that [sα +sα ,h ] = 0, α = z,±. It amounts to check
j j+1 j,j+1
the following recursion relations on the coefficients βn :
m1,m2
(m +1)(2s−m )βn = (2s+n−m +1)(m −n)βn+1
1 1 m1+1,m2 2 2 m1,m2
p − (m +1)(2s−m )βn+1p + (2s−n−m )(n+m +1)βn ,
2 2 m1,m2+1 1 1 m1,m2
m (2sp−m +1)βn = (2s+np−m )(m −n+1)βn−1
1 1 m1−1,m2 2 2 m1,m2
p − m (2s−m +1)βn−1p + (2s−n−m +1)(n+m )βn .
2 2 m1,m2−1 1 1 m1,m2
p p
Hence, Sα commutes with H.
Due to this su(2) symmetry, the wave functions can be characterized by their energy, their
spin and their Sz component. In other words, one can diagonalize the Hamiltonian in a sector
where the operators Sz has a fixed value Sz = Ls−m. This is done in the next section.
2.3 Pseudo-vacuum and pseudo-excitations
We wish to present the construction of the Hamiltonian eigenfunction in the framework of
coordinate Bethe ansatz for spin s. The spin s = 1 case is the Heisenberg chain, solved in [2].
2
It gave the name to the method. For the case s = 1, the method has been generalized in [9].
The first step of the coordinate Bethe ansatz [2] consists in finding a particular eigenvector,
called the pseudo-vacuum, for the Hamiltonian. It is usually chosen as the vector with the
highest spin. In the present case, it is the unique vector in the sector Sz = Ls:
|∅i = |si⊗|si···⊗|si.
Using the explicit forms for the β’s, we get h |si⊗|si = 0. Thus, |∅i is a H-eigenvector with
12
vanishing eigenvalue.
The second step consists in adding pseudo-excitations. These pseudo-excitations are not
physical excitations (hence the name pseudo-excitation). They are obtained by acting with a
creation operator e−, conjugated to s−, on the pseudo-vacuum |∅i. Let us remark that this
j j
operator in this finite representation does not satisfy (e−)2 = 0 but rather (e−)2s+1 = 0. This
explains the supplementary difficulties to deal with s > 1. Indeed, in the case s = 1, no more
2 2
than one pseudo-excitation can beat the same site: we have strict exclusion. In the general case
of spin s, we have a weaker exclusion. More precisely, we can have up to 2s pseudo-excitations
at the same site. This behavior appears already for s = 1.
4 N. Cramp´e, E. Ragoucy and L. Alonzi
3 Coordinate Bethe ansatz for general spin s
We define a state in the sector Sz = Ls−m, for 1 ≤ x ≤ x ≤ ··· ≤ x ≤ L
1 2 m
|x ,x ,...,x i = e−e− ···e− |∅i, (3.1)
1 2 m x1 x2 xm
where e− is conjugated to s−:
s s
s+n (2s)!
e− = |n−1ihn| = gs−g−1 with g = |nihn|.
s−n+1 (s−n)!
n=−s+1r n=−s
X X
This choice for e− is for later convenience. Theset of such non-vanishingvectors (i.e. x > x
j+2s j
for 1 ≤ j ≤ L−2s) provides a basis for this sector. As already noticed, contrarily to the usual
case (s = 1), several (up to 2s) pseudo-excitations at the same site are allowed, that is to say,
2
some x ’s can be equal. The restriction that no more than 2s particles are on the same site
j
is implemented by the fact that (e−)2s = 0. Using the explicit form of e−, we can rewrite the
excited states as follows:
|x ,x ,...,x i = α ···α |si⊗···⊗|si⊗|s−m i⊗|si⊗···⊗|si⊗|s−m i⊗··· ,
1 2 m m1 mk 1 2
x1−1 xm1+1−xm1−1
| {z } | {z }
where m is the number of times x appears and
j j
2s
α = ,
m m
s
(cid:18) (cid:19)
z
where = z(z−1)···(z−k+1) is the binomial coefficient. Let us remark that, if m > 2s the
k k! j
(cid:18) (cid:19)
vector |s−m i has no meaning but the normalization α vanishes.
j mj
Any eigenvector in the sector Sz = Ls−m is a linear combination of the vectors (3.1). Then,
let us introduce the vector
Ψ = a(x ,x ,...,x )|x ,x ,...,x i,
m 1 2 m 1 2 m
x1≤x2X≤···≤xm
where a(x ,x ,...,x ) are complex-valued functions to be determined. As in the case of s = 1,
1 2 m 2
we assume a plane wave decomposition for these functions (Bethe ansatz)
a(x ,...,x )= A (k)exp i(k x +···+k x ) ,
1 m P P1 1 Pm m
PX∈Sm (cid:8) (cid:9)
where S is the permutation group of m elements and A (k) are functions on the symmetric
m P
group algebra depending on some parameters k which will be specified below2. Using the fact
that the states |x ,x ,...,x i form a basis, we can project the eigenvalue equation
1 2 m
HΨ = EΨ (3.2)
m m
on these different basis vectors to determine the A (k) parameters.
P
Since H is a sum of operators acting on two neighbouring sites only, one has to single out
the cases where the x’s obey the following constraints:
2In the following, to lighten thepresentation, thek-dependencewill not be written explicitly.
Coordinate Bethe Ansatz for Spin s XXX Model 5
• all the x ’s are far away one from each other and are not on the boundary sites 1 and L
j
(this case will be called generic),
• x +1 = x for some j,
j j+1
• x = x for some j,
j j+1
• x = x = ··· = x and x +1 = x = ··· = x for some positive
j j+1 j+m1 j j+m1+1 j+m1+1+m2
integers m and m ,
1 2
• x = 1, or x = L.
1 m
As the eigenvalue problem is a linear problem, it is enough to treat the cases where at most one
of the particular cases appears: more complicated cases just appear as superposition of these
‘simple’ cases.
Projection on |x ,x ,...,x i with x +1 < x , ∀j, x > 1 and x < L. As
1 2 m j j+1 1 m
usual, we start by projecting (3.2) on a generic vector |x ,x ,...,x i. This leads to
1 2 m
m
A (β0 +β0 +β−1eikPj +β1 e−ikPj)−E exp(i(k x +···+k x )) = 0
P 1,0 0,1 1,0 0,1 P1 1 Pm m
PX∈Sm Xj=1 !
which must be true for any choice of generic x’s. Therefore, we get for the energy (using the
explicit forms of the β’s given in Section 2)
m
1
E = − 2−eikj −e−ikj .
2s
j=1
X(cid:0) (cid:1)
After the change of variable
λ +is
eikj = j , (3.3)
λ −is
j
the energy becomes
m
2s
E = − .
λ +s2
j
j=1
X
This form for the energy is the one obtained by algebraic Bethe ansatz [10].
Projection on |x ,x ,...,x i with x +1 = x for some j. Let us consider now
1 2 m j j+1
the projection of (3.2) when two pseudo-excitations are nearest neighbours. Using the form of
the energy previously found, we get
AP β10,1−β10,0−β00,1+(β1−,11α2−β1−,01)eikPj +(β11,1α2−β01,1)e−ikP(j+1)
PX∈Sm (cid:0) (cid:1)
×ei(...kPjxj+kP(j+1)(1+xj)...) = 0.
Thisequationistriviallysatisfiedfors > 1 sinceusingexplicitvalueswefindβ0 −β0 −β0 = 0,
2 1,1 1,0 0,1
β−1α −β−1 = 0 and β1 α −β1 = 0. For the case s = 1, we find a constraint between A
1,1 2 1,0 1,1 2 0,1 2 P
and A (where T is the transposition of j and j +1). Explicitly, it is given by (3.5) with
PTj j
s = 1.
2
Projection on |x ,x ,...,x i with x = x for some j. For s > 1, we must also
1 2 m j j+1 2
consider the case when several particles are on the same site. Defining S , the shift operator
j
adding 1 to the jth variable, we get the following relation, when two particles are on the same
site
1
(1+S−1S−1) S S +(2s−1)S −(2s+1)S +1 a(...,x ,x ,...)= 0
2s(1−2s) i i+1 i i+1 i i+1 i i
(cid:0) (cid:1)
6 N. Cramp´e, E. Ragoucy and L. Alonzi
⇒ S S +(2s−1)S −(2s+1)S +1 a(...,x ,x ,...)= 0. (3.4)
i i+1 i i+1 i i
(cid:0) (cid:1)
Using the plane waves decomposition, we get the following constraint
APTj = σ eikPj,eikP(j+1) AP, (3.5)
where T is the(cid:0)transposition o(cid:1)f j and j +1, and we have introduced the scattering matrix
j
uv+(2s−1)u−(2s+1)v +1
σ(u,v) = − . (3.6)
uv+(2s−1)v−(2s+1)u+1
As in the case s = 1, relation (3.5) allows us to express all the A ’s in terms of only one,
2 P
for instance A (where Id is the identity of S ). More precisely, one expresses P ∈ S as
Id m m
a product of T , and then uses (3.5) recursively to express A in terms of A . At this point,
i P Id
one must take into account that the expression of P in terms of T is not unique, because of the
i
relations
T2 = Id, [T ,T ]= 0 (|j −i| > 1) and T T T = T T T .
i j i i i+1 i i+1 i i+1
Therefore, for the construction to be consistent, the function σ has to satisfy the relations
σ(u,v)σ(v,u) = 1, [σ(u,v),σ(w,z)] = 0,
σ(u ,u )σ(u ,u )σ(u ,u )= σ(u ,u )σ(u ,u )σ(u ,u ).
1 2 1 3 2 3 2 3 1 3 1 2
By direct computation, we can show that (3.6) indeed satisfies these relations. We can solve
the recursive defining relations for A and we find, with a particular choice of normalisation
P
for A , the following explicit form, for any P ∈ S
Id m
1 (eikPj −1)(eikPk −1)
A = 1+ .
P 2s eikPj −eikPk
j<k(cid:18) (cid:19)
Y
This scattering matrix becomes after the change of variables u= λ+is and v = µ+is
λ−is µ−is
λ−µ−i
σ(λ,µ) = .
λ−µ+i
Let us remark that after this change of variables the scattering matrix σ(λ,µ) does not depend
on the value of spin and is similar to the one obtained for s = 1.
2
Projection on |x ,...,x ,...,x ,x +1,...,x +1,...,x i. One can compute
1 i i i i m
m1 m2
| {z } | {z }
P (S ,...,S ;S ,...,S )
m1,m2 i i+m1−1 i+m1 i+m1+m2−1
h
+P S−1 ,...,S−1 ;S−1 ,...,S−1
m2,m1 i+m1+m2−1 i+m1 i+m1−1 i
×a(...,x(cid:0),...,x ,x +1,...,x +1,...) = 0, (cid:1)i (3.7)
i i i i
where
m2
P (y;z) = α α α βn z ···z
m1,m2 n m2−n m1 0,m2 m2−n+1 m2
n=1
X
M1
+ α α β−n y ···y
m1−n m2+n m1,m2 m1−n+1 m1
n=1
X
Coordinate Bethe Ansatz for Spin s XXX Model 7
m1 β0 +β0 m2 β0 +β0
−α α 1,0 0,1 +β−1y −α α 1,0 0,1 +β−1z
m1 m2 2 1,0 j m1 m2 2 1,0 j
! !
j=1 j=1
X X
1
+ α α (β0 +β0 +β0 ),
2 m1 m2 m1,m2 0,m1 0,m2
and we have used the notation (2.3) and
y = (y ,y ,...,y ) with y = S , 1≤ k ≤ m ,
1 2 m1 k i+k−1 1
z = (z ,z ,...,z ) with z = S , 1 ≤ k ≤ m .
1 2 m2 k i+m1+k−1 2
Relation (3.7) is implied by (3.4). The sketch of the proof goes as follows.
First, one can check directly that3 P (S ,S ) =0, ∀j, since P (S ,S ) corresponds to
0,2 j j+1 0,2 j j+1
relation (3.4). The same is true for P (S−1 ,S−1) (after multiplication by S−1 S−1).
2,0 j+1 j j+1 j
Next, we rewrite (3.7) as
P (y;z)+P (z;y)= 0, (3.8)
m1,m2 m2,m1
where we have defined
y = y−1,...,y−1,y−1 , i.e. y = S−1 , 1 ≤ k ≤ m ,
m1 2 1 k i+m1−k 1
z = (cid:0)zm−12,...,z2−1,z1−1(cid:1), i.e. zk = Si−+1m1+m2−k, 1 ≤ k ≤ m2.
(cid:0) (cid:1)
These variables are such that if P (z ,z ) = 0 ∀j, then we have also P (z ,z ) = 0.
0,2 j j+1 0,2 j j+1
Hence, a property valid for P (y;z) will be also valid for P (z;y).
m1,m2 m2,m1
We first focus on
m m
P (z) = α α βn z ···z −α β−1 z +α β0 −mβ0 .
0,m n m−n 0,m m−n+1 m m 1,0 j m 0,m 1,0
n=1 j=1
X X (cid:0) (cid:1)
If we suppose that we have variables z such that P (z ,z ) = 0, ∀j, then from expres-
j 0,2 j j+1
sion (2.2), and after some calculation, one can show that
m
(m)
z z ···z = 1−m+ χ z ,
1 2 m j j
j=1
X
m−j j−1
2s 2s
χ(m) = (−1)m+j −1 +1 .
j k ℓ
k=1 (cid:18) (cid:19)ℓ=1(cid:18) (cid:19)
Y Y
Thus, the polynomial P (z) becomes a linear function of the z ’s. Looking at the coefficient
0,m j
of z and at the constant term, one checks that they identically vanish, so that P (z) = 0.
j 0,m
Looking at the general polynomial P (y,z) and using the relation
m1,m2
α α β−n = α α α βn ,
m1−n m2+n m1,m2 M2 M1−n n 0,M1
one can rewrite it as
P (y,z) = α P (z)+α P (y ,...,y )+α α R (y),
m1,m2 m1 0,m2 M2 0,M1 m1−M1+1 m1 M1 M2 m1,m2
m1−M1 β0 +β0 1
R (y) = − 1,0 0,1 +β−1y + β0 −β0 +β0 −β0 .
m1,m2 2 1,0 j 2 0,M2 M1,0 0,m1 0,m2
!
j=1
X (cid:0) (cid:1)
3Multiplication on the right by a(...,xi,...,xi,xi+1,...,xi+1,...) will beunderstood duringtheproof.
8 N. Cramp´e, E. Ragoucy and L. Alonzi
Thus, to prove relation (3.8), it is enough to show that
R (y)+R (z) = 0. (3.9)
m1,m2 m2,m1
Using the expression of M and M , see (2.3), it is easy to see that
1 2
m −M = m −M ≡ m ≥ 0.
1 1 2 2 12
Two cases have to be distinguished: m = 0 or m > 0. In the first case, relation (3.9) is
12 12
trivially satisfied. In the second case, equation (3.9) can be rewritten as
m
12
y +z = 2m ,
j j 12
j=1
X(cid:0) (cid:1)
which is obeyed if
S +S−1 = 2.
i+j−1 i+2s+j−1
To prove this last relation, we use recursively (3.4) to show
2s 2s
S S +1+ −1 S − +1 S = 0 ∀j.
k k+ℓ k k+ℓ
ℓ ℓ
(cid:18) (cid:19) (cid:18) (cid:19)
Taking k = i+j −1, ℓ = 2s and using m +m = 2s+m gives the result.
1 2 12
Hence, relation (3.8) is satisfied if the variables are such that P (z ,z ) = P (y ,y ) =
0,2 j j+1 0,2 j j+1
0, ∀j. This ends the proof.
This step concludes the bulk part of the problem, the other possible equations being fulfilled
by linearity. It remains to take into account the periodicboundarycondition. It is donethrough
the following projection.
Projection on |1,x ,...,x i. As usual, this leads to a constraint on the parameters k .
2 m j
It is not surprising since these parameters can be interpreted as momenta: we are quantizing
them since we are on a line with (periodic) boundary conditions. Namely, this leads to
A exp(i(k x +···+k x ))
P P2 2 Pm m
PX∈Sm (cid:0)
−exp(i(k x +k x +···+k x +k L)) = 0.
P1 2 P2 3 P(m−1) m Pm
Now, we first perform the change of variable in the summation P(cid:1) → PT ···T in the second
1 m−1
term of the previous relation. Then, using recursively relation (3.5) and projecting on inde-
pendent exponential functions, we get the quantization of the momenta via the so-called Bethe
equations
eiLkj = σ eikℓ,eikj for j = 1,2,...,m. (3.10)
ℓ6=j
Y (cid:0) (cid:1)
Since these equations express the periodicity of the chain, they are equivalent to the ones ob-
tained through projection on |x ,...,x ,Li (as it can be checked explicitly). Thus, we do
1 m−1
not have any new independent equations through projections, and the eigenvalue problem has
been solved (up to the resolution of the Bethe equations).
Note that using the change of variables (3.3) and the expression (3.6) for the scattering
matrix, equations (3.10) can be rewritten as
m
λ +is λ −λ −i
j ℓ j
= − .
λ −is λ −λ +i
j ℓ j
ℓ=1
Y
One recognizes the usual Bethe equations of the spin s chain [10, 11].
Coordinate Bethe Ansatz for Spin s XXX Model 9
Action of su(2) generators
Since the su(2) generators commute with the Hamiltonian, from any eigenfunction Ψ , one
m
can construct (possibly) new eigenfunctions by application of Sα, α = z,± on Ψ . As already
m
mentioned, it is a straightforward calculation to check that
SzΨ = (Ls−m)Ψ .
m m
Moreover, it is part of the ansatz to suppose that the eigenvector Ψ is a highest weight vector
m
of the su(2) symmetry algebra,
S+Ψ = 0.
m
Let us stress that for Ψ to be an eigenvector, one has to assume that the rapidities λ have to
m j
obey the Bethe equations. In the same way, Ψ is a highest weight vector only when the Bethe
m
equations are fulfilled. In the context of coordinate Bethe ansatz, there exists no general proof
(for generic spin s) of it (at least to our knowledge). Note however that for spin 1, the proof
2
was given in [12]. Nevertheless, one can check the highest weight property on different cases,
and we illustrate it below by the calculation of S+Ψ , S+Ψ and S+Ψ . We also show on the
1 2 3
last example where the proof used by Gaudin does not work anymore for spin s > 1.
2
The Ψ vectors should be also related to the ones obtained through algebraic Bethe ansatz
m
(ABA). Such a correspondence, for the case of spin 1, has been done in [13] using an iteration
2
trick based on the comultiplication [14, 15]. Let us note that in [13] they used the relation
(T )2 = 0 which is not true anymore for s > 1. Their proof must be generalized to apply in
12 2
our case. Let us also notice the other method using the Drinfel’d twist [16]. Moreover, since it
is known that the ABA construction leads to su(2) highest weight vectors, and assuming the
same property for the coordinate Bethe approach, it is clear that the two methods should lead
to the same vectors, up to a normalisation.
For instance, considering Ψ , its ABA “counterpart” takes the form
1
L
Φ = T(1)(λ )···T(x−1)(λ )T(x)(λ )T(x+1)(λ )···T(L)(λ )|∅i,
1 11 1 11 1 12 1 22 1 22 1
x=1
X
where T(j)(λ) is the representation of the monodromy matrix at site j:
1 λ+isz is−
T(j)(λ) = j j
λ−is is+ λ−isz
(cid:18) j j (cid:19)
and λ is the Bethe parameter. This leads to
1
i L λ +is x
Φ = 1 s−|∅i,
1 λ +is λ −is x
1 x=1(cid:18) 1 (cid:19)
X
that has to be compared with
L L
Ψ = eik1x|xi = eik1xe−|∅i.
1 x
x=1 x=1
X X
Using the change of variable (3.3), it is clear that, apart from a normalisation factor, the two
vectors are equal.
Calculation of S+Ψ and S+Ψ . A direct calculation leads to
1 2
y
S+Ψ = 2s 1−yL |∅i with y = eik1,
1
1−y
(cid:0) (cid:1)
10 N. Cramp´e, E. Ragoucy and L. Alonzi
which is identically zero using the Bethe equation yL = 1. Hence, Ψ is indeed a highest weight
1
vector for the su(2) symmetry.
In the same way, one can compute
S+Ψ = A (k ,k )
2 P 1 2
PX∈S2
L
× (2s−1)ei(k1+k2)x|xi+ 2sei(kP1x1+kP2x2) |x i+|x i .
1 2
 
Xx=1 1≤x1X<x2≤L (cid:0) (cid:1)
Using the relation 
A (k ,k )= σ(k ,k )A (k ,k )
T1 1 2 1 2 Id 1 2
and the normalisation A (k ,k ) = 1, one gets
Id 1 2
L yx+1−yL+1
S+Ψ = (2s−1)(y y )x 1+σ(y ,y ) +2s 2 2 yx
2 1 2 1 2 1−y 1
( " 2
x=1
X (cid:0) (cid:1)
yx+1−yL+1 y −yx y −yx
+σ(y ,y ) 1 1 yx+ 1 1yx+σ(y ,y ) 2 2yx |xi,
1 2 1−y 2 1−y 2 1 2 1−y 1
1 1 2 #)
where yj = eikj, j =1,2.
Now, from the Bethe equations
yL = σ(y ,y ) and yL = σ(y ,y ),
1 1 2 2 2 1
one simplifies it as
L
y 1
S+Ψ = (y y )x 2s−1+2s 2 −
2 1 2
1−y 1−y
x=1 ( (cid:18) 2 1(cid:19)
X
y 1
1
+σ(y ,y ) 2s−1+2s − |xi.
1 2
1−y 1−y
" (cid:18) 1 2(cid:19)#)
Finally, the form of the scattering matrix σ ensures that the quantity within brackets {···}
vanishes.
Calculation of S+Ψ . Performing the same kind of calculation on Ψ , we get
3 3
L yx+1−yL+1
S+Ψ = A (k) (y y )x (2s−2)yx+2s 3 3
3 P 1 2 3 1−y
PX∈S3 (Xx=1 " 3
(y −yx)
+2sσ σ 3 3 |x,xi+ yx1yx2 (2s−1)σ yx1 +(2s−1)yx2
23 13 1−y 1 2 23 3 3
3 # "
1≤x1X<x2≤L
yx2+1−yL+1 y −yx1 yx1+1−yx2
+2s 3 3 +2sσ σ 3 3 +2sσ 3 3 |x ,x i ,
23 13 23 1 2
1−y 1−y 1−y
3 3 3 # )
where4
y = eikPj, j = 1,2,3 and σ = σ(y ,y ), 1≤ j 6= ℓ ≤ 3.
j jℓ j ℓ
4Let us stress thedependenceon P ∈S3 in thedefinition of yj: it is used below.