Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary
We solve the XXZ Gaudin model with generic boundary using the modified algebraic Bethe ansatz. The diagonal and triangular cases have been recovered in this general framework. We show that the model for odd or even lengths has two different behaviors. The corresponding Bethe equations are computed f...
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irk-123456789-1492762019-02-20T01:24:36Z Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary Crampe, N. We solve the XXZ Gaudin model with generic boundary using the modified algebraic Bethe ansatz. The diagonal and triangular cases have been recovered in this general framework. We show that the model for odd or even lengths has two different behaviors. The corresponding Bethe equations are computed for all the cases. For the chain with even length, inhomogeneous Bethe equations are necessary. The higher spin Gaudin models with generic boundary is also treated. 2017 Article Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary / N. Crampe // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 44 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R12; 17B80; 37J35 DOI:10.3842/SIGMA.2017.094 http://dspace.nbuv.gov.ua/handle/123456789/149276 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We solve the XXZ Gaudin model with generic boundary using the modified algebraic Bethe ansatz. The diagonal and triangular cases have been recovered in this general framework. We show that the model for odd or even lengths has two different behaviors. The corresponding Bethe equations are computed for all the cases. For the chain with even length, inhomogeneous Bethe equations are necessary. The higher spin Gaudin models with generic boundary is also treated. |
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Crampe, N. |
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Crampe, N. Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary Symmetry, Integrability and Geometry: Methods and Applications |
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Crampe, N. |
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Crampe, N. |
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Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary |
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Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary |
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Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary |
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Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary |
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Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary |
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algebraic bethe ansatz for the xxz gaudin models with generic boundary |
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Інститут математики НАН України |
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Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary / N. Crampe // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 44 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT crampen algebraicbetheansatzforthexxzgaudinmodelswithgenericboundary |
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2025-07-12T21:13:56Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 094, 13 pages
Algebraic Bethe Ansatz for the XXZ Gaudin Models
with Generic Boundary
Nicolas CRAMPE
Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier,
Montpellier, France
E-mail: nicolas.crampe@umontpellier.fr
Received November 01, 2017, in final form December 06, 2017; Published online December 13, 2017
https://doi.org/10.3842/SIGMA.2017.094
Abstract. We solve the XXZ Gaudin model with generic boundary using the modified
algebraic Bethe ansatz. The diagonal and triangular cases have been recovered in this
general framework. We show that the model for odd or even lengths has two different
behaviors. The corresponding Bethe equations are computed for all the cases. For the chain
with even length, inhomogeneous Bethe equations are necessary. The higher spin Gaudin
models with generic boundary is also treated.
Key words: integrability; algebraic Bethe ansatz; Gaudin models; Bethe equations
2010 Mathematics Subject Classification: 81R12; 17B80; 37J35
1 Introduction
The algebraic Bethe ansatz is a powerful method to solve analytically numerous integrable
models [41]. Then, the introduction of integrable boundaries in this framework was initiated
in [39]. However, the problem to solve models with generic integrable boundaries thanks to
the algebraic Bethe ansatz has been overcome only recently [1, 5, 6, 8, 15]. The problem lies
in the fact that there is no simple particular eigenvector, which is normally the starting point
of the method. Therefore, other approaches have been used to deal with this problem: the
functional Bethe ansatz [21, 23, 34, 35], the coordinate Bethe ansatz [16, 17], the separation of
variables [19, 22, 37], the q-Onsager approach [2, 4] and the matrix ansatz [18, 25, 28, 29, 30].
The algebraic Bethe ansatz has been also used in this context [7, 10, 33, 38, 42] but constraints
on the parameters defining the boundaries were necessary. Then, inhomogeneous T-Q relations
were introduced in [11, 12, 13, 36] where the authors obtained the eigenvalues and the Bethe
equations for the XXZ model with generic boundaries. Finally, the modified algebraic Bethe
ansatz allows us to compute the associated eigenvectors [1, 5, 6, 8, 44]. Let us mention also
that this method has been also exploited to study twisted XXX spin chain in [9] or the rational
Gaudin magnets in arbitrarily oriented magnetic fields in [20].
In this letter, we solve the Gaudin model (introduced in [24]) with generic boundaries using
the modified algebraic Bethe ansatz. The Gaudin model is one of the simplest quantum inte-
grable system and we hope that its resolution will shed more light on the method. Let us also
mention that the algebraic Bethe ansatz was already applied to solve open Gaudin model using
the vertex-IRF correspondence of the XXZ spin chain [26, 43]. However, as pointed in [14], the
model considered in [26, 43] is obtained as a limit from the XXZ spin chain and is different of
the one studied here which is constructed directly from a classical r-matrix.
This letter is organised as follows. In Section 2, we recall well-known results about the Gaudin
algebra and model to fix the notations used in the following. Then, we provide technical results
in the Sections 3.1 and 3.2 to implement the algebraic Bethe ansatz. In Section 3.3, we recover
the results for the diagonal and triangular boundaries [27, 31, 32]. Section 3.4 contains new
mailto:nicolas.crampe@umontpellier.fr
https://doi.org/10.3842/SIGMA.2017.094
2 N. Crampe
results for the generic boundaries when the length of the chain is odd. Finally, we deal with the
even chain in Section 3.5 where an additional computation is necessary. We explain also why
the computation is different depending on the parity of the length of the chain. In Section 4,
we generalize these results to solve the higher spin Gaudin models with generic boundary.
2 Gaudin models and algebras
In this section, we recall different well-known notions to construct Gaudin models and prove
their integrability.
For a matrix r(x, y) ∈ End(CN ⊗ CN ) depending on 2 parameters x, y, one defines the
fundamental equation, called classical Yang–Baxter equation, given by
[r13(x1, x3), r23(x2, x3)] = [r21(x2, x1), r13(x1, x3)] + [r23(x2, x3), r12(x1, x2)].
In the previous relation, we have used the usual notations: r12(x) = r(x)⊗ I, r23(x) = I⊗ r(x),
. . . , where I is the identity matrix. A solution of the classical Yang–Baxter equation is called
a r-matrix. In addition, if an r-matrix r(x, y) depends only on the quotient x/y and satisfies
the supplementary relations r12(x, y) = r12(x/y) = −r21(y/x) = −r12(y/x)t1t2 then it is called
a skew-symmetric r-matrix. For a given skew-symmetric r-matrix r(x/y), one defines also the
reflection equation
r12(x/y)k1(x)k2(y)− k1(x)k2(y)r21(x/y) = k2(y)r12(xy)k1(x)− k1(x)r21(xy)k2(y).
A solution k(x) of the reflection equation is called k-matrix. From a skew-symmetric r-matrix
and an associated k-matrix, we can construct a new r-matrix [40]
r12(x, y) = r12(x/y)− k1(x)r12(1/(xy))k1(x)−1, (2.1)
which is not skew-symmetric.
To each r-matrix r(x, y), on can associate a Lie algebra R. Indeed, if K (x) ∈ End
(
CN
)
⊗R,
then the following relation
[K1(x),K2(y)] = [r21(y, x),K1(x)] + [K2(y), r12(x, y)] (2.2)
defines a Lie commutator for R. Indeed, the antisymmetry of the product is obvious and the
Jacobi identity is satisfied due to the classical Yang–Baxter equation. For any invertible matrix
M(x) ∈ End
(
CN
)
, there is a Lie algebra isomorphism given by
ΦM : R → RM ,
K0(x) 7→M0(x)−1K0(x)M0(x),
where RM is the Lie algebra defined by (2.2) with the r-matrix
M1(x)−1M2(y)−1r12(x, y)M1(x)M2(y).
From the defining relations (2.2), one can show that the transfer matrix
t(x) =
1
4
tr0
(
K0(x)2
)
(2.3)
satisfies [t(x), t(y)] = 0. Then, when a representation for R is chosen, the coefficients of t(x)
provide an integrable hierarchy. Let us remark that, for a given matrix M(x), R and RM give
the same hierarchy since it is easy to see that tr
(
K0(x)2
)
= tr
(
ΦM (K0(x))2
)
.
Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary 3
In this letter, we focus on the skew-symmetric r-matrix associated to the affine Kac–Moody
algebra ŝl2:
r(x) =
1
x− 1
−1
2(x+ 1) 0 0 0
0 1
2(x+ 1) −2 0
0 −2x 1
2(x+ 1) 0
0 0 0 −1
2(x+ 1)
. (2.4)
The most general k-matrix associated to this r-matrix is given by
k(x) =
(
β + γ/x −α(β+ρ)
2 (x− 1/x)
β−ρ
2α (x− 1/x) β + γx
)
, (2.5)
where α, β, γ and ρ are scalar parameters. As explained previously, the r-matrix and the k-
matrix allow us to construct a new r-matrix (see relation (2.1)). Without loss of generality, we
can use the Lie algebra isomorphism ΦM to transform this r-matrix. Therefore, we use in this
paper the following r-matrix
r̃12(x, y) = M1(x)−1M2(y)−1
(
r12(x/y)− k1(x)r12(1/(xy))k1(x)−1
)
M1(x)M2(y), (2.6)
where r(x) is given by (2.4), k(x) is given by (2.5) and
M(x) =
(
β+ρ
2ρ
α
x
(β−ρ)x
2αρ 1
)
.
Explicitly, this r-matrix reads as
r̃(x, y) =
(
−1
2ω(x, y)σz b(x)σz + f(y, x)σ−
c(x)σz − f(1/y, 1/x)σ+ 1
2ω(x, y)σz
)
,
where σz =
(
1 0
0 −1
)
, σ+ =
(
0 1
0 0
)
and σ− =
(
0 0
1 0
)
and
δ(x) =
((β − ρ)x2 + 2γx+ ρ+ β)
2ρ(x− 1/x)
, b(x) = − α
δ(x)
, c(x) =
ρ2 − β2
4αρ2δ(1/x)
, (2.7)
ω(x, y) =
x+ y
x− y
+
xy + 1
xy − 1
and f(x, y) = ω(x, y)
δ(x)
δ(y)
. (2.8)
Let us emphasize that the r-matrix r̃(x, y) has a simpler form than the one without the isomor-
phism (i.e., r12(x/y) − k1(x)r12(1/(xy))k1(x)−1). Without the isomorphism, the 16 entries of
the r-matrix should be non-zero whereas r̃(x, y) has 6 vanishing entries and each two by two
block submatrices are diagonal or triangular.
As explained previously, the r-matrix r̃(x, y) allows one to define a Lie algebra by using
relation (2.2) with
K̃0(x) =
(
A(x) B̃(x)
C̃(x) −A(x)
)
. (2.9)
We denote by R̃ this Lie algebra. Let us mention that the Lie algebra R̃ for special choice of
the parameters in the k-matrix has been identified as Onsager algebras [3]1.
1To compare this letter with [3], we must use the additional property r12(x)t1 = σy
1r12(x)σy
1 satisfied by the
r-matrix (2.4).
4 N. Crampe
Finally, let us introduce the following representation2
K̃0(x) =
L∑
j=1
r̃0j(x, vj). (2.10)
By using the classical Yang–Baxter equation, it is obvious to show that (2.10) satisfies the
commutation relations of R̃. The parameters vi are called inhomogeneous parameters. The
Hamiltonians are usually defined by
H̃j = Resx=vj t(x) =
1
4
Resx=vj tr0
(
K̃0(x)2
)
= −vj
L∑
p=1
p 6=j
r̃jp(vj , vp) + Lvj
(
3v2j
v2j − 1
− ν(vj)
)
,
where
ν(z) =
z2(ρ2 − β2)− 2βγz + ρ2 − β2
2ρ2(z2 − 1)δ(z)δ(1/z)
. (2.11)
These Hamiltonians are integrable due to the commutation of the transfer matrix and [H̃j , H̃k]
= 0. They are related to the Gaudin models with boundary (or also called BC Gaudin models).
Indeed, by conjugating the previous Hamilonians by M1(v1) · · ·ML(vL), one gets the Hamilto-
nians
Hj = −vj
L∑
p=1
p 6=j
(
rjp
(
vj
vp
)
+ kj(vj)rjp
(
1
vjvp
)
kj(vj)
−1
)
+ Lvj
(
3v2j
v2j − 1
− ν(vj)
)
,
which are the usual form for the XXZ Gaudin model with boundary.
3 Modified algebraic Bethe ansatz
In this section, we want to diagonalize the matrix 1
4 tr0
(
K̃0(x)2
)
when the 4 parameters in the
k-matrix are generic. The diagonalisation when the k-matrix is diagonal was obtained in [27]
and for the triangular case in [31, 32]. For the generic case, the situation is more complicated.
Here, we use the modified algebraic Bethe ansatz, introduced and used in different contexts
[1, 5, 6, 8, 15], to compute the eigenvalues and the eigenvectors for the generic case. The first
two subsections are technical. Then, Section 3.3 contains the known results about the diagonal
or triangular boundaries recovered here and Sections 3.4 and 3.5 contains the new results for
the chain of length odd or even.
3.1 Algebraic relations
In this subsection, by using the commutation relations obtained from (2.2), we provide necessary
propositions to perform the algebraic Bethe ansatz. Firstly, let us mention that the commutation
relations between A(x), B̃(x) and C̃(x) given by (2.2) with the r-matrix (2.6) and K̃ (x) given
by (2.9) are quite complicated in comparison to the usual ones. This is why we introduce the
two following shifted generators
B(x, n) = B̃(x)− (2n− 1)b(x) and C(x, n) = C̃(x)− (2n− 1)c(x),
2We keep the same notation K̃ (x) for the algebraic element or its representation.
Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary 5
where b(x) and c(x) are defined by (2.7). For these generators and the generators A(x), the
commutation relations become
A(x)A(y) = A(y)A(x), (3.1)
B(x, n)B(y, n+ 1) = B(y, n)B(x, n+ 1), (3.2)
C(x, n)C(y, n− 1) = C(y, n)C(x, n− 1), (3.3)
A(x)B(y,m) = B(y,m)A(x) + ω(x, y)B(y,m)− f(x, y)B(x,m), (3.4)
A(x)C(y,m) = C(y,m)A(x)− ω(x, y)C(y,m)− f(1/x, 1/y)C(x,m), (3.5)
C(x, n)B(y, n) = B(y, n+ 1)C(x, n+ 1)
+ 2f(x, y)A(x)− 2f(1/y, 1/x)A(y)− 8nb(y)c(x), (3.6)
where ω(x, y) and f(x, y) are defined by (2.8).
We introduce also a shifted transfer matrix
t(x, j) =
1
2
A(x)2 +
1
4
B(x, j + 1)C(x, j + 1) +
1
4
C(x, j)B(x, j)− 1
2
b(x)c(x). (3.7)
For j = 0 in the previous relation, the transfer matrix (2.3) is recovered: t(x) = 1
4 tr0 K̃0(x)2 =
t(x, 0).
Secondly, we present a lemma permitting to commute the shifted transfer matrix with the
generator B(x, p). This lemma is standard in the context of the algebraic Bethe ansatz for the
Gaudin model except for the modification of the shift for the transfer matrix.
Lemma 3.1. If the algebraic relations (3.1)–(3.6) are satisfied, the following relation holds
t(x, p− 1)B(z, p)−B(z, p)t(x, p) = B(z, p)
(
ω(x, z)(A(x) + ν(x)) + 2b(x)c(x)
)
−B(x, p)
(
f(1/z, 1/x)(A(z) + ν(z)) + 4(p− 1)c(x)b(z)
)
, (3.8)
where ν(z) is defined by (2.11).
Proof. By straightforward computations using the commutation relations (3.1)–(3.6), one gets
t(x, p− 1)B(z, p)−B(z, p)t(x, p) = ω(x, z)B(z, p)A(x)− f(1/z, 1/x)B(x, p)A(z)
+
1
2
(
ω(x, z)2 + f(z, x)f(1/z, 1/x)
)
B(z, p)
− 1
2
(
ω(z, x)f(1/z, 1/x) + ω(x, z)f(x, z) + 4(2p− 1)c(x)b(z)
)
B(x, p).
Then, by using the following functional relations
ω(x, z)2 + f(z, x)f(1/z, 1/x) = 2ν(x)ω(x, z) + 4b(x)c(x),
ω(z, x)f(1/z, 1/x) + ω(x, z)f(x, z) = 2ν(z)f(1/z, 1/x)− 4c(x)b(z)
one gets relation (3.8). �
Let us define the following product
B(z) = B(z1, 1)B(z2, 2) · · ·B(zM ,M),
where z = {z1, z2, . . . , zM}. Let us remark that, due to relation (3.2), B(z) is invariant under
any permutation of the zi which justifies the definition of z as a set. One introduces also, for
k = 1, 2, . . . ,M
B(zk, x) = B(z1, 1) · · ·B(zk−1, k − 1)B(x, k)B(zk+1, k + 1) · · ·B(zM ,M).
Finally, we are in position to compute the action of the transfer matrix with the product B(z):
6 N. Crampe
Proposition 3.2. If the algebraic relations (3.1)–(3.6) are satisfied, the following relation holds
t(x)B(z) =
1
2
B(z)B(x,M + 1)C(x,M + 1)
+ B(z)
Λ(x) +
M∑
p=1
ω(x, zp)
A(x) + ν(x) +
∑
q 6=p
(x− 1/x)ω(zp, zq)
zp − 1/zp
−
M∑
p=1
B(zp, x)f(1/zp, 1/x)
A(zp) + ν(zp) +
∑
q 6=p
ω(zp, zq)
,
where
Λ(x) =
1
2
A(x)2 + xA′(x) + ν(x)A(x)− x2 + 1
x2 − 1
A(x)− 1
2
b(x)c(x).
Proof. One gets, using a telescopic sum, the following relation
t(x, 0)B(z)− B(z)t(x,M) =
M∑
p=1
B1 · · ·Bp−1 (t(x, p− 1)Bp −Bpt(x, p))Bp+1 · · ·BM ,
where Bp = B(zp, p). Then, using the Lemma 3.1, we express t(x, p − 1)Bp − Bpt(x, p) in the
previous sum. Then, we use repetitively relation (3.4) to put the operators A on the right.
Then, we rearrange the terms using the following relations
ω(x, zp)f(x, zq)− f(1/zp, 1/x)f(zp, zq) = ω(zq, zp)f(1/zq, 1/x)− 4c(x)b(zq),∑
q>p
ω(x, zp)ω(x, zq) =
∑
q 6=p
x− 1/x
zp − 1/zp
ω(zp, zq)ω(x, zp).
Finally, t(x,M) is simplified using relation (3.7) and (3.6) for y → x to prove the proposition. �
Let us emphasize again that this type of computation and result are the same for the periodic
or diagonal Gaudin model. The only difference lies on the shift in the operators.
3.2 Explicit representation
Up to now, we have only used the commutation relations of the algebra to prove the previous
results. Now, we choose the representation for K̃ (x) given by (2.10). Therefore, we get explicitly
A(x) = −1
2
L∑
j=1
ω(x, vj)σ
z
j ,
B(x, n) = (1− 2n)b(x) + b(x)
L∑
j=1
σzj +
L∑
j=1
f(vj , x)σ−j , (3.9)
C(x, n) = (1− 2n)c(x) + c(x)
L∑
j=1
σzj −
L∑
j=1
f(1/vj , 1/x)σ+j .
Let us introduce the following vectors Ω =
(
1
0
)⊗L
, called usually pseudo-vacuum. We deduce
that
A(x)Ω = a(x)Ω and C(x, n)Ω = (L+ 1− 2n)c(x)Ω (3.10)
with a(x) = −1
2
L∑
j=1
ω(x, vj).
Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary 7
The Bethe vectors are defined as follows
V(z) = B(z)Ω. (3.11)
These Bethe vectors are of course invariant under any permutation of the zi since B(z) and C(z)
are invariant. From the previous results (Proposition 3.2 and relations (3.10)), we get easily the
action of the transfer matrix on these Bethe vectors
t(x)V(z) =
L− 2M − 1
2
c(x)B(z)B(x,M + 1)Ω
+
λ(x) +
M∑
p=1
ω(x, zp)
a(x) + ν(x) +
∑
q 6=p
(x− 1/x)ω(zp, zq)
zp − 1/zp
V(z)
−
M∑
p=1
f(1/zp, 1/x)
a(zp) + ν(zp) +
∑
q 6=p
ω(zp, zq)
V(zp, x), (3.12)
where
λ(x) =
1
2
a(x)2 + xa′(x) + ν(x)a(x)− x2 + 1
x2 − 1
a(x)− 1
2
b(x)c(x).
The second term in (3.12) is the wanted term (i.e., is proportional to the Bethe vector) and the
third term is the unwanted term. These terms are already present for the periodic or triangular
case. For the generic boundary, we must deal with the first term in the previous relation (3.12).
There are different possibilities:
• c(x) = 0, i.e., ρ = ±β. This case corresponds to triangular (or diagonal) boundaries and
the Bethe ansatz works as usual. We get different sectors depending on the number M of
excitations (see Section 3.3).
• The length of chain is odd and M = L−1
2 . It corresponds to a case where the number
of excitations is fixed. From relation (3.12), one gets only the half of the spectrum. The
other half is obtained starting from t(x)V(z) (see Section 3.4).
• The length of the chain is even. In this case, we must compute B(z)B(x,M + 1)Ω. It
corresponds to the modified algebraic Bethe ansatz (see Section 3.5).
At first sight, the difference of the behavior for the lengths odd or even seems strange.
However, we can show that, in the case of odd length, two subspaces of (C2)⊗L are stabilized
by the transfer matrix. More precisely, if we denote by VJ the eigenspace of the total spin
Sz = 1
2
L∑
j=1
σzj with eigenvalue J , one gets
t(x)V1/2 ⊕ V3/2 ⊕ · · · ⊕ VL/2 ⊂ V1/2 ⊕ V3/2 ⊕ · · · ⊕ VL/2,
t(x)V−1/2 ⊕ V−3/2 ⊕ · · · ⊕ V−L/2 ⊂ V−1/2 ⊕ V−3/2 ⊕ · · · ⊕ V−L/2. (3.13)
Then it is natural that the spectrum for odd chain splits into two sectors. For the chain of even
length, this feature disappears: there is no stable subspace.
3.3 Triangular and diagonal boundaries
In this subsection, we deal with the triangular boundary (i.e., ρ = ±β). In this case, we can
easily see that one gets, for M = 0, 1, . . . , L,
t(x)VL/2−M ⊕ VL/2−M+1 ⊕ · · · ⊕ VL/2 ⊂ VL/2−M ⊕ VL/2−M+1 ⊕ · · · ⊕ VL/2.
8 N. Crampe
For M = 0, 1, . . . , L, the Bethe vectors V(z) defined by (3.11) is just in the sector VL/2−M ⊕
· · · ⊕ VL/2 and becomes an eigenvector of the transfer matrix if the Bethe roots zi satisfy the
Bethe equations:
a(zp) + ν(zp) +
M∑
q=1
q 6=p
ω(zp, zq) = 0 for p = 1, 2, . . . ,M. (3.14)
Explicitly, they read, for p = 1, . . . ,M ,
−1
2
L∑
j=1
(
zp + vj
zp − vj
+
zpvj + 1
zpvj − 1
)
+
1
2
(
γz − β
γz + β
+
βz − γ
βz + γ
)
+
M∑
q=1
q 6=p
(
zp + zq
zp − zq
+
zpzq + 1
zpzq − 1
)
= 0. (3.15)
The corresponding eigenvalue of the transfer matrix t(x) = 1
4 tr0
(
K̃0(x)2
)
is
λ(x) +
M∑
p=1
ω(x, zp)
a(x) + ν(x) +
M∑
q=1
q 6=p
(x− 1/x)ω(zp, zq)
zp − 1/zp
. (3.16)
Starting from the triangular boundary, we recover the diagonal one for α = 0. We see that the
eigenvalue (3.16) and the Bethe equations (3.15) do not depend on α. Then, as shown previously
for the Gaudin model [31, 32] (or for other models [7, 38]), the spectrum for triangular boundaries
is similar to the one with diagonal boundaries. Evidently, the Bethe eigenvectors depends on α,
via the function b(x) present in B(x, j) (see (3.9)). In particular, the function b(x) vanishes for
α = 0 and the Bethe vectors V(z) are now in the sector VL/2−M which is consistent with
t(x)VL/2−M ⊂ VL/2−M .
The last equation means that the total spin is conserved by the transfer matrix in the case of
diagonal boundary.
3.4 Generic boundaries for an odd chain
If the length of the chain is odd (i.e., there is an integer ` such that L = 2`+ 1), we can chose
the number of excitations as M = L−1
2 = ` and the coefficient in front of B(z)B(x,M + 1)Ω
vanishes. From the explicit expression (3.9) of B(x, j), we see that
V(z) ∈ V1/2 ⊕ V3/2 ⊕ · · · ⊕ VL/2 for M = `,
which is one of the subspaces stabilized by the transfer matrix, as explained previously. Then,
the eigenvalues for this sector are given by (3.16) with the Bethe equations (3.14) for M = `.
The spectrum and the associated eigenvectors for the other sector (3.13) are obtained from
the Bethe vectors
V(z) = C(z1, 0)C(z2,−1) · · ·C(z`,−`+ 1)
(
0
1
)⊗L
.
It is easy to show that V(z) belongs to the second sector. A relation similar to (3.12) by
replacing V(z) by V(z) can be proved. Then, we show that, starting from V(z), one gets
Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary 9
exactly the expression (3.16) for the eigenvalues and (3.14) for the Bethe equations. Therefore,
the spectrums in both sectors are the same.
Let us also mention that the separation into two sectors is not new in this context. Indeed,
the complete spectrum of the XXZ spin chain with certain constraints between the parameters
of the boundaries is also obtained from two different Bethe vectors [42].
3.5 Generic boundaries for an even chain
For a chain with an even length, we must deal with the term B(z)B(x,M+1)Ω. For a generic M ,
this vector has no particular property but for M = L, special feature appears. This is the crucial
point of the modified algebraic Bethe ansatz.
Firstly, let us state the following lemma:
Lemma 3.3. The entries of the matrix
M(z) =
−→
L∏
q=1
(
(z − vq)(z − 1/vq)
(z − zq)(z − 1/zq)
B(zq, q)
)
×
(
(z − 1/z)
(z − x)(z − 1/x)
B(x, L+ 1)
)(
δ(z)B(z, L+ 1)
)−1
have only simple poles at z = 0,∞, z1, z2, . . . , zL, 1/z1, 1/z2, . . . , 1/zL, x, 1/x.
Proof. The proof of the existence of the poles ofM(z) at z = z1, z2, . . . , zL, 1/z1, 1/z2, . . . , 1/zL
and x, 1/x is straightforward. One gets
B̂(z) = δ(z)B(z, L+ 1) = α(2L+ 1)− α
L∑
j=1
σzj +
L∑
j=1
2z(vj − 1/vj)
(z − vj)(z − 1/vj)
δ(vj)σ
−
j .
This matrix B̂(z) is a lower triangular matrix with the diagonal entries in the set {α(L + 1),
α(L + 3), . . . , α(3L + 1)}. In particular, this shows that it is invertible which justifies the
definition of M(z). Due to the Cayley–Hamilton theorem, B̂(z)−1 is a polynomial of B̂(z).
Then, by using the fact that (σ−)2 = 0, we prove that the poles at z = vj and z = 1/vj
of B̂(z)−1 remain simple. Then, M(z) has no pole at these points. In addition, its becomes
a non-singular diagonal matrix for z = 0 which proves the simplicity of the pole of M(z) at
z = 0. The existence of the pole at z =∞ is proven using the property M(z) = −1/z2M(1/z)
which concludes the proof. �
Similar technical lemma has been proven and used in [15]. We are now in position to provide
the following proposition:
Proposition 3.4. The following relation holds
B(z)B(x, L+ 1)Ω = λ̂(x)V(z) +
L∑
p=1
(zp − 1/zp) Resx=zp
(
λ̂(x)
)
(zp − x)(zp − 1/x)
V(zp, x)
with z = {z1, z2, . . . , zL} and
λ̂(x) =
α(L+ 1)
δ(x)
L∏
q=1
(
(x− vq)(x− 1/vq)
(x− zq)(x− 1/zq)
)
.
10 N. Crampe
Proof. The residues of M(z)Ω are obtained by straightforward computations:
Resz=zp(M(z)Ω) =
1
δ(zp)
L∏
q=1
q 6=p
(
(zp − vq)(zp − 1/vq)
(zp − zq)(zp − 1/zq)
)(
(zp − vp)(zp − 1/vp)
(zp − x)(zp − 1/x)
)
V(zp, x),
Resz=x(M(z)Ω) =
1
δ(x)
L∏
q=1
(
(x− vq)(x− 1/vq)
(x− zq)(x− 1/zq)
)
V(z),
Resz=0(M(z)Ω) = − 1
α(L+ 1)
B(z)B(x, L+ 1)Ω.
By using M(z) = −1/z2M(1/z), we deduce that Resz=1/z0(M(z)Ω) = Resz=z0(M(z)Ω) where
z0 = 0, z1, z2, . . . , zL, x. The proof is concluded by using the fact that the sum over all the
residues (with the point at infinity) of a rational function vanishes. �
Therefore, by using relation (3.12) for M = L and the result of Proposition 3.4, we prove
that V(z) (for z = {z1, z2, . . . , zL}) is an eigenvector of the transfer matrix t(x) if the Bethe
roots satisfy the following Bethe equations
a(zp) + ν(zp) +
L∑
q=1
q 6=p
ω(zp, zq) =
(L+ 1)c(zp)
4zp
Resx=zp
(
λ̂(x)
)
, for p = 1, 2, . . . , L. (3.17)
To get the previous relation, we have used the functional relation
− (L+ 1)(z − 1/z)c(x)
2f(1/z, 1/x)(z − x)(z − 1/x)
=
(L+ 1)c(z)
4z
.
Explicitly, they become
−1
2
L∑
j=1
(
zp + vj
zp − vj
+
zpvj + 1
zpvj − 1
)
+ ν(zp) +
L∑
q=1
q 6=p
(
zp + zq
zp − zq
+
zpzq + 1
zpzq − 1
)
=
(ρ2 − β2)(L+ 1)2
16ρ2
(zp − vp)(zp − 1/vp)
δ(zp)δ(1/zp)(z2p − 1)
L∏
q=1
q 6=p
(zp − vq)(zp − 1/vq)
(zp − zq)(zp − 1/zq)
,
where ν(z) is given by (2.11). The corresponding eigenvalue is
λ(x)− L+ 1
2
c(x)λ̂(x) +
M∑
p=1
ω(x, zp)
a(x) + ν(x) +
∑
q 6=p
(x− 1/x)ω(zp, zq)
zp − 1/zp
. (3.18)
Then the l.h.s. of the Bethe equations is similar to the ones for the diagonal or triangular
boundaries. The r.h.s. corresponds to the inhomogeneous part of the TQ relation discovered
in [11, 12] to solve the XXZ spin chain with twist or boundaries.
4 Higher spin Gaudin models
In this section, we generalize the previous construction and consider the higher spin Gaudin
models. Indeed, it is well-known that we can find representations of the algebra R using
Algebraic Bethe Ansatz for the XXZ Gaudin Models with Generic Boundary 11
different representations of sl(2) at each sites. Indeed, the following matrix K̃ (x) satisfies the
defining relation (2.2):
K̃0(x) =
L∑
j=1
L
(sj)
0j (x, vj), (4.1)
where sj = 1/2, 1, 3/2, . . . is the spin of the representation at the sites j and
L (s)(x, y) =
(
−ω(x, y)Sz 2b(x)Sz + f(y, x)S−
2c(x)Sz − f(1/y, 1/x)S+ ω(x, y)Sz
)
. (4.2)
In (4.2), Sz, S+ and S− are the usual matrices representing the sl(2) generators for spin s. The
only necessary knowledge about these matrices for the following constructions is the fact that
there exists a highest-weight vector ws such that
Szws = sws and S+ws = 0 for s = 1/2, 1, 3/2, . . .
For s1, s2, . . . , sL = 1
2 , the representation (4.1) becomes (2.10). Let us emphasize that the
representation of sl(2) may be different at each sites.
Now, we can generalize the results of Section 3. All the results of Section 3.1 remains
unchanged since it is independent of the representation.
The pseudo-vacuum becomes3
Ω = ws1 ⊗ ws2 ⊗ · · · ⊗ wsL .
Then, relation (3.12) is formally unchanged except the definition of the function a(x) which
becomes
a(x) = −
L∑
j=1
ω(x, vj)sj (4.3)
and the L in the first line must be replaced by L = 2
L∑
p=1
sp.
Then, there is again three possibilities to pursue the computation:
• The triangular case corresponding to c(x) = 0. The Bethe equations and the eigenvalues
are given respectively by (3.14) and (3.16) with a(x) given by (4.3). The number M ranges
now from 0 to L.
• The odd case M = L−1
2 . The discussion of Section 3.4 is still valid.
• The even case. To deal with this case, we must generalize the Lemma 3.3. This lemma is
still valid if we define M(z) as follows
M(z) =
L∏
q=1
(z − vq)2sq(z − 1/vq)
2sq
−→
L∏
q=1
B(zq, q)
(z − zq)(z − 1/zq)
× (z − 1/z)
(z − x)(z − 1/x)
B(x, L+ 1)
(
δ(z)B(z, L+ 1)
)−1
.
Let us emphasize that the first product is up to L, the number of sites, and the second up
to L. The demonstration is very similar to the one given previously except we must use
3In this section, we keep the same notations for all the objects which generalize the objects of the previous
sections.
12 N. Crampe
(S−)2s+1 = 0 for a spin s representation. Then, the Bethe equations and the eigenvalues
are given respectively by (3.17) and (3.18) with L replace by L, a(z) given by (4.3) and
λ̂(x) =
α(L+ 1)
δ(x)
L∏
q=1
(x− vq)2sq(x− 1/vq)
2sq
L∏
q=1
1
(x− zq)(x− 1/zq)
.
Acknowledgements
I thank P. Baseilhac, S. Belliard and V. Caudrelier for their interest. This work has been done
during the stay of the author at the “Laboratoire de Mathématiques et Physique Théorique
CNRS/UMR 7350, Université de Tours”. I thank warmly the LMPT for hospitality.
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1 Introduction
2 Gaudin models and algebras
3 Modified algebraic Bethe ansatz
3.1 Algebraic relations
3.2 Explicit representation
3.3 Triangular and diagonal boundaries
3.4 Generic boundaries for an odd chain
3.5 Generic boundaries for an even chain
4 Higher spin Gaudin models
References
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