On Bethe vectors in \( \mathfrak{g}{\mathfrak{l}}_3 \) invariant integrable models
JHE
Bethe vectors in gl3invariant integrable
A. Liashyk 0 1 3 4 5 6 7 8
N.A. Slavnov 0 1 2 3 7 8
0 6 Usacheva str. , Moscow, 119048 , Russia
1 14b Metrolohichna str. , Kiev, 03143 , Ukraine
2 Theoretical Physics Department, Steklov Mathematical Institute of Russian Academy of Sciences
3 Bogoliubov Institute for Theoretical Physics , NAS of Ukraine
4 Faculty of Mathematics, National Research University Higher School of Economics
5 Center for Advanced Studies, Skolkovo Institute of Science and Technology
6 Department for Theory of Nuclei and Quantum Field Theory
7 8 Gubkina str. , Moscow, 119991 , Russia
8 3 Nobel str. , Moscow, 121205 , Russia
We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl3invariant Rmatrix. We study a new recently proposed approach to construct onshell Bethe vectors of these models. We prove that the vectors constructed by this method are semionshell Bethe vectors for arbitrary values of Bethe parameters. They thus do become onshell vectors provided the system of Bethe equations is fulfilled.
Integrable Field Theories; Lattice Integrable Models

HJEP06(218)
1 Introduction 2 3 4
5
6
Basic notions of NABA
2.1
2.2
Notation
Bethe vectors
Special NABAsolvable models
Main results
4.1
Multiple action of the operator Bg
Proof of proposition 4.1
Action of Bg(z) on Bethe vectors
6.1
6.2
6.3
Action of T23(z2)Tb13(z1)
Action of T13(z2)Tb12(z1)
Action of Bg(z) on semionshell Bethe vectors
7
Proof of proposition 4.2
7.1
Inductive basis
7.2 Inductive step
7.2.1
7.2.2
Contribution W1
Contributions W2 and W3
A Properties of DWPF
B Proof of the connection between two types of Bethe vectors
B.1 First step of induction
B.2 Second step of induction
C Action formulas
C.2.1
C.2.2
Action of Tb13
Action of Tb12
C.1 Actions of the operators Tij on Bethe vectors Ba,b
C.2 Actions of the operators Tbij on Bethe vectors Ba,b
was proposed in [
1
]. In the present paper we study this method by the nested
algebraic Bethe ansatz (NABA) in the case of quantum integrable models with gl3invariant
Rmatrix.
– 1 –
There exist several ways to study quantum integrable models with a high rank of
symmetry. A nested version of the Bethe ansatz [2] was proposed in [3–5]. In the context of
the Quantum Inverse Scattering Method (QISM) [6–9], an algebraic version of this method
(NABA) was developed in [10–12]. One more approach based on the qKZ equation and
Jackson integrals was proposed in [13–15] and generalized to the superalgebras in [16]. We
should also mention a method to construct Bethe vectors via certain projection of Drinfel’d
currents, that was developed in a series of works [17–21]. The Separation of Variables (SoV)
method [22, 23] was applied to the study of gl3invariant quantum spin chains in [24].
The main task of the methods listed above is to construct the eigenfunctions of the
quantum Hamiltonians. Traditionally they are called onshell Bethe vectors. In distinction
of the gl2 based models, a form of these eigenfunctions for the models with higher rank
of symmetry is quite involved. This is due to the fact that these models describe physical
systems with several types of particles. Respectively, one has to consider several creation
operators corresponding to each type of excitations.
For instance, within the framework of QISM, we deal with a quantum monodromy
matrix T (u), whose trace plays the role of generating functional of the integrals of motion.
The uppertriangular entries of the monodromy matrix Tij (u) with i < j are creation
operators, and a physical space of states can be generated by successive action of these
operators on a referent state 0i. In the case of the gl2 based models, there exits only one
creation operator T12(u). Respectively, the eigenvectors of the quantum Hamiltonians have
the form of products of these operators acting onto a referent state 0i. However, already
in the case of the gl3 based models, we deal with three creation operators, and the form of
onshell Bethe vectors immediately becomes much more complex (see e.g. [25] and (2.14)
for explicit formulas).
It was observed in [
1
] that an operator used for constructing the SoV basis of the
gl2invariant spin chain can be also used for generating the basis of the onshell Bethe
vectors. It was conjectured in [
1
] that a similar effect might take place in the spin chains
with higher rank of symmetry. In particular, in the gl3invariant spin chain one should
consider an operator1
(1.1)
(1.2)
Bg(u) = T23(u)T12(u − i)T23(u) − T23(u)T22(u − i)T13(u)
+ T13(u)T11(u − i)T23(u) − T13(u)T21(u − i)T13(u)
for constructing the SoV basis [24]. Here Tij (u) are entries of a twisted monodromy matrix
(see section 3 for more details). Then, in complete analogy with the case of gl2 based
models, onshell Bethe vectors can be presented as a successive action of Bg(ui) onto the
referent state
Bg(u1) . . . Bg(ua)0i.
This conjecture was justified by the computer calculation, however, an analytical proof is
lacking so far. The goal of this paper is to find such the proof.
Our proof of representation (1.2) is given within the framework of NABA. We show
that representation (1.2) for onshell Bethe vectors holds not only for spin chains, but for
1In [
1
] this operator was denoted as Bgood(u). We find this notation too heavy and reduce it to Bg(u).
– 2 –
a more wide class of integrable models possessing gl3invariant Rmatrix. In particular, we
do not use the SoV method.
The paper is organized as follows. We recall basic notions of NABA in section 2.
There we also give a standard description of Bethe vectors within this method. Section 3
is devoted to special NABAsolvable models that usually are applied to the systems of
physical interest. The main results of our paper are gathered in section 4. There we give
explicit representation of the states (1.2) in terms of the monodromy matrix entries acting
on the pseudovacuum vector. We also describe a relationship between the states (1.2) and
the Bethe vectors obtained by the standard NABA approach. In the rest of the paper
we give the proofs of the results of section 4. We identify the state (1.2) with a Bethe
vector in section 5. In section 6 we compute the action of the operator Bg(u) on a generic
Bethe vector. Finally, in section 7 we express the state (1.2) in terms of the monodromy
matrix entries acting on the pseudovacuum vector. Several auxiliary identities for rational
functions are gathered in appendix A. Appendix B contains a proof of connection between
two types of Bethe vectors considered in the paper. Finally, the formulas of the action of
the monodromy matrix entries onto the Bethe vectors are given in appendix C.
2
Basic notions of NABA
We consider quantum integrable models solvable by NABA and possessing the gl3invariant
Rmatrix
constant.2
R(u, v) = I ⊗ I + g(u, v)P,
g(u, v) =
c
u − v
Here I is the identity matrix in C3, P is the permutation matrix in C3 ⊗ C3, and c is a
The monodromy matrix T (u) is a 3 × 3 matrix with operatorvalued entries Tij (u)
acting in a Hilbert space H. Their commutation relations are give by an RT T relation
(2.1)
(2.2)
(2.3)
R(u, v) T (u) ⊗ I I ⊗ T (v) = I ⊗ T (v) T (u) ⊗ I R(u, v).
It follows from (2.2) that an operator
T (u) = tr T (u) =
3
i=1
X Tii(u)
has the following property: [T (u), T (v)] = 0 for arbitrary u and v. This operator is called
a transfer matrix. It plays the role of a generating functional of the integrals of motion of a
quantum model under consideration. One of the main tasks of NABA is to find eigenvectors
of this operator.
If a 3 × 3 cnumber matrix K is such that [R(x, y), K ⊗ K] = 0, then the matrices
KT (u) and T (u)K also satisfy the RT T relation (2.2). A peculiarity of the Rmatrix (2.1)
is that [R(x, y), K ⊗ K] = 0 holds for arbitrary K ∈ gl3. In particular, if K is invertible,
2To compare our presentation with the results of [
1
] one should set c = i.
– 3 –
then one can consider a transformation T (u) → T (K)(u) = KT (u)K−1. Obviously, this
transformation preserves the transfer matrix.
Besides the monodromy matrix T (u), we also will consider a matrix Tb(u) that is closely
associated to a quantum comatrix [26, 27]. First, we introduce quantum minors
tjk11,,jk22 (u) = Tj1,k1 (u)Tj2,k2 (u − c) − Tj2,k1 (u)Tj1,k2 (u − c).
The entries of the quantum comatrix Tejk(u) then are given by
where ¯j = {1, 2, 3} \ j. The quantum comatrix plays the role of the inverse monodromy
HJEP06(218)
where qdet T (u) is a quantum determinant of T (u) [26–29].
The matrix Tb(u) is defined as the transposition of Te(u) with respect to the secondary
It is known [25–27, 30] that a mapping φ : T (u) 7→ Tb(u) is an automorphism of the
RT T algebra (2.2). Thus, the matrix Tb(u) satisfies the RT T relation with the same
R
Using the matrix Tb(u) we can write down the operator Bg(u) (1.1) in a more compact
The following obvious properties of the functions introduced above are useful:
g(u, v) = −g(v, u),
h(u − c, v) =
g(u + c, v) =
f (u − c, v) =
u − v + c
u − v
,
1
g(u, v)
,
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
1
f (v, u)
(2.10)
Similar representation for Bg was used in [24].
2.1
Notation
Besides the function g(u, v) we also introduce two new functions
f (u, v) = 1 + g(u, v) =
Tejk(u) = (−1)j+kt¯k¯(u),
j
Te(u − c)T (u) = qdet T (u) I,
Tbjk(u) = Te4−k,4−j (u).
Before giving a description of the Bethe vectors we formulate a convention on the
notation. We denote sets of variables by a bar: u¯, v¯, and so on. Individual elements of the
sets are denoted by subscripts: uj , vk, and so on. Notation u¯ + c means that the constant c
is added to all the elements of the set u¯. Subsets of variables are denoted by roman indices:
u¯I, u¯II, v¯ii, and so on. In particular, we consider partitions of sets into subsets. Then the
notation u¯ ⇒ {u¯I, u¯II} means that the set u¯ is divided into two disjoint subsets u¯I and u¯II.
The order of the elements in each subset is not essential. A special notation u¯j is used for
subsets complementary to the element uj , that is, u¯j = u¯ \ uj , v¯k = v¯ \ vk and so on.
– 4 –
In order to avoid too cumbersome formulas we use shorthand notation for products of
functions depending on one or two variables. Namely, if the functions g, f , and h depend
on sets of variables, this means that one should take the product over the corresponding
set. For example,
h(u¯, v) =
Y h(uj , v);
uj∈u¯
Y
zj∈z¯
zj6=zi
g(zi, z¯i) =
g(zi, zj );
f (u¯II, u¯I) =
f (uj , uk).
Y
Y
uj∈u¯II uk∈u¯I
In the last equation of (2.11) the set u¯ is divided into two subsets u¯I, u¯II, and the double
product is taken with respect to all uk belonging to u¯I and all uj belonging to u¯II. We use the
1 if at least one of the sets is empty.
the framework of NABA, it is assumed that successive action of these operators onto 0i
generates vectors of the space H. Bethe vectors are special polynomials in Tij (u) with i < j
acting on 0i. Their explicit form will be given later. Here we would like to mention that
in the models with gl3invariant Rmatrix Bethe vectors depend on two sets of complex
parameters u¯ = {u1, . . . , ua} and v¯ = {v1, . . . , vb} called Bethe parameters. We denote
these vectors by Ba,b(u¯; v¯), where a and b respectively are the cardinalities of the sets u¯
and v¯. A characteristic property of the Bethe vectors is that they become eigenvectors of
the transfer matrix T (z) = tr T (z) provided u¯ and v¯ enjoy ceratin constraint. In this case
they are called onshell Bethe vectors. Otherwise, if u¯ and v¯ are generic complex numbers,
then the corresponding vector is called offshell Bethe vector.
In physical models, vectors of the space H describe states with quasiparticles
(excitations) of two different types (colors). We say that a state has coloring {a, b}, if it contains
a quasiparticles of the color 1 and b quasiparticles of the color 2. The vector 0i has zero
coloring. The operator T12 is the creation operator of quasiparticles of the first color, while
the operator T23 creates quasiparticles of the second color. The operator T13 creates one
quasiparticle of the first color and one quasiparticle of the second color. The diagonal
– 5 –
(2.11)
(2.12)
(2.13)
operators Tii are neutral, the matrix elements Tij with i > j play the role of annihilation
operators. Generally, there are no restrictions on the coloring {a, b}, thus, the parameters a
and b are arbitrary nonnegative integers. In specific models, some restrictions may appear.
Different methods to construct Bethe vectors were developed in [12, 15–17]. Several
equivalent explicit representations were found in [25]. One of this representations reads
Recall that here we use the shorthand notation (2.11), (2.12) for the products of the
operators Tij and the functions λ2, f , and g. The sum in (2.14) is taken over partitions of the
sets u¯ ⇒ {u¯I, u¯II} and v¯ ⇒ {v¯I, v¯II} such that #u¯I = #v¯I = n, where n = 0, 1, . . . , min(a, b).
It is easy to see that each term of this sum has a fixed coloring {a, b}, and thus, Bethe
vector Ba,b(u¯; v¯) has coloring that coincides with the cardinalities of the Bethe parameters.
We would like to stress that generically there is no any restriction on the cardinalities of
the Bethe parameters u¯ and v¯. In particular, one might have a < b, that is #u¯ < #v¯.
The function Kn(v¯Iu¯I) in (2.14) is a partition function of the sixvertex model with
domain wall boundary condition (DWPF) [31, 32]. It depends on two sets of variables
v¯ and u¯; the subscript shows that #v¯ = #u¯ = n. The function Kn has the following
determinant representation [32]:
Kn(v¯u¯) = h(v¯, u¯)
n
j<k
Y g(vj, vk)g(uk, uj) dnet
g(vj, uk)
h(vj, uk)
.
(2.15)
Some properties of Kn are gathered in appendix A.
Observe that the normalization in (2.14) differs from the normalization of Bethe vectors
used in [25]. The present normalization is chosen so that the Bethe vector does not have
singularities for vj = uk and vj − c = uk.
We also consider Bethe vectors Bba,b(u¯; v¯) which correspond to the monodromy matrix
HJEP06(218)
λˆ2(v¯II)λˆ2(u¯)g(v¯, u¯)
Kn(v¯Iu¯I)f (u¯I, u¯II)f (v¯II, v¯I) Tb13(u¯I)Tb12(u¯II)Tb23(v¯II)0i, (2.16)
Tb(u). They have the form
Bba,b(u¯; v¯) =
where λˆ2(z) = λ1(z)λ3(z − c).
and Bb:
The automorphism T (u) 7→ Tb(u) generates a connection between the Bethe vectors B
Bbb,a(v¯ + c; u¯) = (−1)a+b+ab λ2(u¯)λ2(v¯) Ba,b(u¯; v¯).
a system of Bethe equations and has the following form:
If the system (2.18) is fulfilled, then
where
,
Below we will need the action formulas of the operators Tij (z) and Tbij (z) on the
generic Bethe vectors. They were obtained in [25]. We give the list of necessary formulas
in appendix C.
3
Special NABAsolvable models
At the first sight, a method to construct onshell Bethe vectors by means of the operator
Bg(u) (1.1) contradicts to the content of the previous section. Indeed, according to the
general scheme, the onshell Bethe vector depends on two sets of variables subject to the
equations (2.18). At the same time, vector (1.2) depends on only one set of variables.
The solution of this contradiction lies in the fact that in some models there is a kind of
hierarchy between the variables u¯ and v¯: the set u¯ plays a basic role, while the variables
v¯ are auxiliary. In particular, the system of Bethe equations can be reformulated as a
constraint on the Bethe parameters u¯ only (see (3.4) below).
This class of models includes the XXX SU(3)invariant Heisenberg chain, for which
the operator Bg(u) was originally constructed in [
1
]. A characteristic property of these
models is that only the operators T12(u) and T13(u) are true creation operators, while
T23(u)0i = 0. In spite of these models are a particular case of the models considered
above, they find a wide application in physics.3
Consider a monodromy matrix T 0(u) such that T203(u)0i = 0. This condition
immediately implies a restriction on the vacuum eigenvalues λj (u). Indeed, it follows from the
RT T relation that
[T302(u), T203(v)] = g(u, v) T202(v)T303(u) − T202(u)T303(v) .
Acting with this equation onto 0i we obtain
0 = λ2(v)λ3(u) − λ2(u)λ3(v) 0i,
3One can also consider models, in which T12(u)0i = 0, while T23(u) and T13(u) are true creation
operators. This case is equivalent to the one considered in this paper, due to an automorphism of the
RT T algebra (2.2) with respect to the replacement Tij(u) → T4−j,4−i(−u).
– 7 –
(2.18)
(2.19)
(2.20)
HJEP06(218)
(3.1)
(3.2)
leading to λ2(u) = κλ3(u), where κ is a constant. Without loss of generality we can set
λ2(z) = κ and λ3(z) = 1. At the same time, the vacuum eigenvalue λ1(z) still remains a
free functional parameter. Below we omit the subscript and denote it λ1(z) = λ(z).
Bethe equations (2.18) take the form
One can show (see e.g. [33]) that this system implies
λ(uj ) = κ
this parameter. As both sides of (3.4) are polynomials in α of degree a, this condition is
equivalent to a set of a equations for a variables u¯ = {u1, . . . , ua} (the free terms in both
sides obviously are equal to 1). We see that the set of auxiliary variables v¯ is eliminated.
According to the coloring prescriptions, quasiparticles of the second color now can
be created by the action of the operator T103(u) only. Since this operator simultaneously
creates a quasiparticle of the first color, we conclude that the coloring of any state in
these models has a property b ≤ a. In particular, Bethe vectors B0a,b(u¯; v¯) for such the
monodromy matrix possess this property. Their explicit form also simplifies:
B0a,b(u¯; v¯) =
X
#u¯I=b
κag(v¯, u¯)
Kb(v¯u¯I)f (u¯I, u¯II) T103(u¯I)T102(u¯II)0i.
these models.
appropriate twist transformation
In distinction of (2.14), here the sum is taken over partitions of the set u¯ ⇒ {u¯I, u¯II} such
that #u¯I = b, while the set v¯ is not divided into subsets. We see that a generic offshell
Bethe vector B0a,b(u¯; v¯) still depends on the set of auxiliary Bethe parameters v¯. We will
show, however, that the auxiliary parameters can be eliminated from onshell Bethe vectors,
as it was done for the system of Bethe equations.
Thus, for the models with the monodromy matrix T 0(u), one can actually restrict
himself with a one set of the Bethe parameters only. However, if we substitute the operators
Ti0j (u) into equation (1.1) for Bg(u), then we see that Bg(u)0i = 0. This is due to the
fact that T203(u)0i = 0. Thus, the operator (1.1) cannot be used as a creation operator in
A nontrivial action of Bg(u) onto the pseudovacuum vector can be provided by an
In paper [
1
], a generic twist matrix K was considered. We restrict ourselves with a ‘minimal’
twist, which provides a condition T23(u)0i 6= 0, but does not change the action of other
operators Tij onto 0i. Let
T (u) = KT 0(u)K−1.
K = I +
β
1 − κ
E23,
– 8 –
(3.3a)
(3.3b)
(3.4)
where β 6= 0 is a complex number and E23 is an elementary unit matrix (E23)ij = δi2δj3.
It is easy to see that the matrix T (u) has the same vacuum eigenvalues λ1(z) = λ(z),
λ2(z) = κ, and λ3(z) = 1. However, now we have T23(u)0i = β0i provided κ 6= 1.
Of course, the twist matrix (3.7) is not the only matrix, ensuring the condition
T23(u)0i 6= 0. We discuss more general twists in Conclusion.
4
Main results
We are now in position to formulate our main results.
Proposition 4.1. Let the vacuum eigenvalues of the monodromy matrix T (u) be given be
and T23(u)0i = β0i. Let u¯ and v¯ be two sets of complex numbers such that #u¯ = a,
#v¯ = b, and the constraint (3.3a) is fulfilled. Then Bethe vector Ba,b(u¯, v¯) has the following
a
X
n=0
βb−n
n
X
κa+bg(v¯, u¯) s=0 #u¯I=s
X
(−κ)n−sλ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III)
× T13(u¯I)T13(u¯II)T12(u¯III)0i.
(4.2)
Here the sum is taken over partitions of the set u¯ into three subsets u¯ ⇒ {u¯I, u¯II, u¯III}. The
cardinalities of the subsets are shown explicitly by the subscripts of the sum symbol in (4.2).
The proof of proposition 4.1 is based on the explicit representation for the Bethe
vectors (2.14). This is done in section 5. Here we give several comments on this proposition.
The condition (3.3a) is a part of Bethe equations, therefore, the corresponding Bethe
vector can be called a semionshell Bethe vector [34]. The constraint (3.3a) is a system
of a equations for a + b variables. In particular, if4 b ≥ a, then we can consider (3.3a)
as the system of equations for the parameters vk, k = 1, . . . , b. At the same time the
parameters u¯ remain generic complex numbers, and one can easily show that the system
is solvable. Furthermore, it follows from representation (4.2) that if vk, k = 1, . . . , b and
vk′, k = 1, . . . , b′ are two different solutions to the system (3.3a), then
κb′−bg(v¯′, u¯)Ba,b′ (u¯, v¯′) = βb′−bg(v¯, u¯)Ba,b(u¯, v¯).
(4.3)
This property is due to the very specific action of the operator T23(z) onto the
pseudovacuum vector: T23(z)0i = β0i. Thus, two semionshell Bethe vectors with different sets of
the Bethe parameters v¯ and v¯′ actually are proportional to each other. In fact, for an
appropriate normalization, semionshell Bethe vectors (4.2) do not depend on the parameters
of the set v¯.
Proposition 4.1 implies that onshell Bethe vectors also have representation (4.2). In
this case the parameters u¯ and v¯ enjoy the additional set of equations (3.3b).
We see,
however, that the condition (3.3a) is already sufficient to eliminate the parameters v¯ from
the representation for the Bethe vector. They are only included in the normalization factor.
4Recall that due to T23(z)0i 6= 0 we have no restriction b ≤ a.
– 9 –
Now we give an explicit representation for the multiple action of the operator Bg onto
pseudovacuum vector 0i. It was shown in [
1
] that [Bg(u), Bg(v)] = 0 for arbitrary u and
v. Thus, given a set u¯ = {u1, . . . , ua}, then the notation
Bg(u¯) = Y Bg(uj )
a
j=1
is well defined.
#u¯ = a. Then
Proposition 4.2. Let the vacuum eigenvalues of the monodromy matrix T (u) be as in
proposition 4.1 and T23(u)0i = β0i. Let a set u¯ consist of generic complex numbers and
a
n=0
n
(−κ)n−sλ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III)
(4.4)
HJEP06(218)
× T13(u¯I)T13(u¯II)T12(u¯III)0i.
(4.5)
Here the sum over partitions of u¯ is taken as in proposition 4.1.
This proposition gives the result of multiple action of the operator Bg onto 0i in terms
of multiple actions of the creation operators T12 and T13. The proof of proposition 4.2 is
given in section 7.
Comparing (4.5) and (4.2) we immediately arrive at
Corollary 4.1. Under the conditions of propositions 4.1 and 4.2
Bg(u¯)0i = β2a−bκa+bg(v¯, u¯)Ba,b(u¯, v¯).
(4.6)
Thus, if the Bethe parameters u¯ and v¯ satisfy Bethe equations (3.3a), (3.3b), then the
vector Bg(u¯)0i is onshell Bethe vector, as it is proportional to the onshell Bethe vector
Ba,b(u¯, v¯). One can also consider the vector Bg(u¯)0i for generic complex u¯. Equation (4.6)
remains true in this case, if the set v¯ satisfies the system (3.3a). Due to the property (4.3)
one can always provide the solvability of this system for generic complex u¯.
5
Proof of proposition 4.1
We begin with an explicit form of Bethe vectors corresponding to the twisted monodromy
matrix T (u) (3.6). This form follows from the general representation (2.14), where one
should take into account the condition T23(u)0i = β0i. Then
Ba,b(u¯; v¯) =
βb−nKn(v¯Iu¯I)f (u¯I, u¯II)f (v¯II, v¯I) T13(u¯I)T12(u¯II)0i.
Here, like in (2.14), the sum is taken over partitions of the sets u¯ ⇒ {u¯I, u¯II} and v¯ ⇒
{v¯I, v¯II}. The subscripts of the sums show that the partitions satisfy restrictions #u¯I =
#v¯I = n, where n = 0, 1, . . . , min(a, b).
The sum over partitions v¯ ⇒ {v¯I, v¯II} can be transformed into a sum over additional
partitions of the subset u¯I via (A.2), in which one should set x¯ = v¯ and y¯ = u¯I. Then
X
#v¯I=n
n
s=0 #u¯i=s
Kn(v¯Iu¯I)f (v¯II, v¯I) = X
X (−1)n−sf (u¯i, u¯ii)f (v¯, u¯i).
(5.2)
Here in the lhs, the sum is taken over partitions v¯ ⇒ {v¯I, v¯II} so that #v¯I = n. In the rhs, the
sum is taken over all possible partitions u¯I ⇒ {u¯i, u¯ii}. Substituting this into (5.1) we find
Ba,b(u¯, v¯) = X
a
βb−n
n
X
n=0 κa+b−ng(v¯, u¯) s=0 #u¯i=s
X
#u¯ii=n−s
(−1)n−sf (u¯i, u¯ii)f (v¯, u¯i)f (u¯i, u¯II)f (u¯ii, u¯II)
× T13(u¯i)T13(u¯ii)T12(u¯II)0i.
(5.3)
In (5.3), the sum is taken over partitions of the set u¯ into three subsets u¯ ⇒ {u¯i, u¯ii, u¯II}.
The cardinalities of subsets are shown explicitly by the subscripts of the sum.
Note that we have replaced the upper summation limit min(a, b) with a in the sum
over n. If a ≤ b, then this replacement certainly is possible. If a > b, then all the terms in
the sum over n with n > b vanish due to proposition A.2. Indeed, due to this proposition
the sum in the r.h.s. of (5.2) gives a determinant (A.14). The latter vanishes for n > b.
Thus, if a > b, then the sum in (5.3) actually breaks at n = b.
Suppose that Ba,b(u¯, v¯) is a semionshell Bethe vector whose Bethe parameters satisfy
the condition (3.3a). Then taking the product of equations (3.3a) over subset u¯i we find
f (v¯, u¯i) = κ−sλ(u¯i) f (u¯i, u¯ii)f (u¯i, u¯II)
f (u¯II, u¯i)f (u¯ii, u¯i) .
(5.4)
Substituting this into (5.3) we arrive at
Ba,b(u¯, v¯) = X
a
βb−n
n
X
n=0 κa+bg(v¯, u¯) s=0 #u¯i=s
X
#u¯ii=n−s
(−κ)n−sλ(u¯i)f (u¯ii, u¯i)f (u¯II, u¯i)f (u¯ii, u¯II)
× T13(u¯i)T13(u¯ii)T12(u¯II)0i.
(5.5)
This representation coincides with (4.2) up to the labels of the subsets. Thus,
proposition 4.1 is proved.
6
Action of Bg(z) on Bethe vectors
We use induction over a in order to prove proposition 4.2. However, before doing this, we
find the action of the operator Bg(z) on an arbitrary Bethe vector Ba,b(u¯, v¯). This will give
us a necessary tool for the proof.
Below, for some time, we do not use restrictions T23(z)0i = β0i, λ2(z) = κ, and
λ3(z) = 1. Instead, we consider the most general case of the monodromy matrix. In order
to avoid new notation, we still denote this monodromy matrix by T (z). However, we do not
assume that the action of T23(z) has some peculiarity, nor do we impose any restrictions
on the eigenvalues λj(z). We simply consider the action of the operator Bg(z) (2.8) on
an arbitrary Bethe vector Ba,b(u¯; v¯) using (2.17) and action formulas (C.2)–(C.4). We also
replace the expression for Bg(z) (2.8) by T23(z2)Tb13(z1) − T13(z2)Tb12(z1) and consider the
limit zk → z (k = 1, 2) in the end of the calculations. Then we specify the obtained result
to the semionshell Bethe vectors described in section 4.
In this section we study the action of T23(z2)Tb13(z1):
Then due to (C.2) we obtain
Λ1 = T23(z2)Tb13(z1)Ba,b(u¯; v¯).
Λ1 = (−1)a+b+ab λ1(u¯)λ3(v¯)
λ2(u¯)λ2(v¯) T23(z2)Tb13(z1)Bbb,a(v¯ + c; u¯).
Λ1 =
(−1)a+b+abλ1(u¯)λ3(v¯)λˆ2(z1)
h(z1, v¯ + c)h(u¯, z1)λ2(u¯)λ2(v¯) T23(z2)Bbb+1,a+1({v¯ + c, z1}; {u¯, z1}).
Turning back from Bb to B and using λˆ2(z) = λ1(z)λ3(z − c) we arrive at
Λ1 =
(−1)a+b+1λ2(z1)λ2(z1 − c)g(z1, v¯)
T23(z2)Ba+1,b+1({u¯, z1}; {v¯, z1 − c}).
It remains to act with T23(z2) onto the obtained vector via (C.4):
Λ1 =
×
X
f (η¯I, η¯II)h(v¯, η¯I)h(z2, η¯I) Ba+1,b+2(η¯II; {v¯, z1 − c, z2}),
where η¯ = {u¯, z1, z2} and the sum is taken over partitions η¯ ⇒ {η¯I, η¯II} so that #η¯I = 1.
We see that η¯I 6= z1 due to the function g(z1, η¯I) in the denominator of (6.5). Thus,
either η¯I = z2 or η¯I = uj, where j = 1, . . . , a. Respectively, we can present Λ1 in the
following form
Λ1 = Λ(10) + X Λ(1j).
a
j=1
Here Λ(10) corresponds to the case η¯I = z2:
(6.1)
(6.2)
(6.3)
(6.4)
(6.5)
(6.6)
(−1)a+b+1λ2(z1)λ2(z2)λ2(z1 − c)g(z1, v¯)g(z1, z2)
× f (uj, u¯j)h(uj, z1)h(v¯, uj)f (z2, uj)Ba+1,b+2({u¯j, z1, z2}; {v¯, z1 − c, z2}).
(6.8)
Due to (C.2) we can present the vector Ba+1,b+2({u¯j, z1, z2}; {v¯, z1 − c, z2}) as a result
Ba+1,b+2({u¯j, z1, z2}; {v¯, z1 − c, z2}) =
h(z2, u¯j)h(z2, z1)h(v¯, z2)
λ2(z2)g(z1, z2)
× T13(z2)Ba,b+1({u¯j, z1}; {v¯, z1 − c}).
(6.9)
Observe that here we can take the limit z1 = z2 = z:
h(uj, z1)g(z2, uj)f (uj, u¯j)h(v¯, uj)
Λ(j)
1 z1=z2=z
=
(−1)aλ2(z)λ2(z − c)g(v¯, z)
Now we study the action of T13(z2)Tb12(z1):
Using again (2.17) we have
and due to (C.3) we obtain
Λ2 = T13(z2)Tb12(z1)Ba,b(u¯; v¯).
Λ2 = (−1)a+b+ab λ1(u¯)λ3(v¯)
λ2(u¯)λ2(v¯) T13(z2)Tb12(z1)Bbb,a(v¯ + c; u¯),
Λ2 =
(−1)a+b+abλˆ2(z1)λ1(u¯)λ3(v¯)g(z1, v¯)
λ2(u¯)λ2(v¯)h(u¯, z1)
T13(z2)
×
X
fg((ξ¯ξ¯III,, vξ¯¯I))hh((zξ¯1I,, ξz¯I1)) Bbb+1,a({v¯ + c, z1}; ξ¯II),
where ξ¯ = {u¯, z1} and the sum is taken over partitions ξ¯ ⇒ {ξ¯I, ξ¯II} so that #ξ¯I = 1.
Turning back to the vector B we find
Λ2 =
(−1)a+1λ2(z1)λ2(z1 − c)g(z1, v¯)
λ1(ξ¯I)f (ξ¯II, ξ¯I)h(ξ¯I, z1) Ba,b+1(ξ¯II; {v¯, z1 − c}).
(6.10)
(6.11)
(6.12)
(6.13)
(6.14)
(6.15)
There is no problem to compute the action of T13(z2), however, we do not do this.
Instead we present the obtained result in the form similar to (6.6)
Λ2 = Λ(20) + X Λ(2j).
a
j=1
Here Λ(20) corresponds to the partition ξ¯I = z1:
Observe that here we can take the limit z1 = z2 = z:
Λ(20) = (−1)a+1λ1(z1)λ2(z1 − c)g(u¯, z1)T13(z2)Ba,b+1(u¯; {v¯, z1 − c}).
Λ(0)
2 z1=z2=z
= (−1)a+1λ1(z)λ2(z − c)g(u¯, z)T13(z)Ba,b+1(u¯; {v¯, z − c}).
The contributions Λ(2j) correspond to the partitions ξ¯I = uj and have the following form:
Λ(2j) =
(−1)aλ2(z1)λ2(z1 − c)g(z1, v¯)
λ1(uj)f (u¯j, uj)f (uj, z1) Ba,b+1({u¯j, z1}; {v¯, z1 − c}).
Here we also can take the limit
Λ(j)
2 z1=z2=z
=
(−1)aλ2(z)λ2(z − c)g(z, v¯)
λ1(uj)f (u¯j, uj)f (uj, z) Ba,b+1({u¯j, z}; {v¯, z − c}).
Action of Bg(z) on semionshell Bethe vectors
Consider the difference of the contributions Λ(1j) and Λ(2j) at z1 = z2 = z. Using (6.11)
and (6.20) we find
Λ(1j) − Λ(2j)
z1=z2=z
= (−1)aλ2(z)λ2(z − c)
f (uj, z)g(v¯, z)
g(v¯, uj)h(u¯, z)
× f (uj, u¯j)f (v¯, uj) −
λ2(uj)
λ1(uj) f (u¯j, uj) T13(z)Ba,b+1({u¯j, z}; {v¯, z − c}).
If Ba,b(u¯; v¯) is a semionshell Bethe vector such that
λ2(uj)
λ1(uj) f (u¯j, uj) = f (uj, u¯j)f (v¯, uj),
then this difference vanishes. In particular, if we impose the constraint (3.3a) (setting
λ1(uj) = λ(uj) and λ2(uj) = κ), then the contributions Λ(j) and Λ(j) cancel each other.
1 2
It is remarkable, however, that the cancellation of these terms takes place in the most
general case of the semionshell Bethe vectors, for which λ1(z) and λ2(z) are free functional
parameters.
(6.16)
(6.17)
(6.18)
(6.19)
(6.20)
(6.21)
(6.22)
HJEP06(218)
Now we are able to prove proposition 4.2 via induction over a. For this, we specify the
action formulas of section 6 to the case λ1(z) = λ(z) and λ2(z) = κ.
Consider the action of Bg(z) onto 0i = B0,0(∅; ∅). Then due to (6.7), (6.18) we have
,
Using (5.3) we easily find B0,1(∅; z − c) = βκ−10i. In the case a = 1, b = 2, equation (5.3)
gives
B1,2(u; {v1, v2}) =
β2
κ3g(v¯, u)
κ
β
T12(u)0i +
(f (v¯, u) − 1)T13(u)0i .
Setting here u = z1, v1 = z1 − c, and v2 = z2 we obtain
leading to
Thus, we arrive at
B1,2(z1; {z1 − c, z2}) =
β2
κ3g(z1, z2)
T12(z1)0i −
κ
β T13(z1)0i ,
Λ(0) = β2T12(z)0i − κβT13(z)0i.
1
Bg(z)0i = Λ(0) − Λ(0) = β2T12(z)0i + β(λ(z) − κ)T13(z)0i.
1 2
It is easy to see that representation (4.5) gives the same result for a = 1:
1
n=0
n
s=0 #z¯I=s
#z¯II=n−s
Bg(z)0i =
X β2−n X
X
(−κ)n−sλ(z¯I)T13(z¯I)T13(z¯II)T12(z¯III)0i.
Here the sum is taken over partitions of the set z¯ (consisting on one element z) into three
subsets z¯I, z¯II, and z¯III. Clearly, two of these subsets are empty. Because of this reason
we did not write the product of the f functions in (7.6) (see (4.5)), as these products are
taken at least over one empty set. Setting successively in (7.6) z¯I = z, z¯II = z, and z¯III = z
we obtain three contributions coinciding with (7.5). Thus, the induction basis is checked.
It is interesting to write down this result in terms of the entries of the original
monodromy matrix T 0(u). Using (3.6) and (3.7) we find
T13(u) = T103(u) −
β
1 − κ T102(u),
T12(u) = T102(u).
Then replacing z with u in representation (7.5) we obtain
Bg(u)0i = β(λ(u) − κ)T103(u)0i + β
2
1 − λ(u)
1 − κ
T102(u)0i.
(7.1)
HJEP06(218)
(7.2)
(7.3)
(7.4)
(7.5)
(7.6)
(7.7)
(7.8)
The monodromy matrix T 0(u) has two onshell Bethe vectors in the case a = 1:
B10,0(u, ∅) and B10,1(u, v). In the first case, there is only one Bethe equation λ(u) = κ, and
hence, (7.8) yields
Bg(u)0i = β2T102(u)0i.
In the second case we have a system of two Bethe equations
what implies λ(u) = 1. Then (7.8) yields
Both vectors T102(u)0i and T103(u)0i indeed are onshell Bethe vectors respectively for
λ(u) = κ and λ(u) = 1. Thus, the action of Bg(u) onto the pseudovacuum vector does give
the onshell Bethe vectors, if u is a root of Bethe equations.
We assume that (4.5) holds for some a ≥ 1. Then due to corollary 4.1 the action Bg(u¯)0i
with u¯ = {u1, . . . , ua} is proportional to the semionshell Bethe vector Ba,b(u¯, v¯) (4.2),
where the set v¯ enjoys the constraint (3.3a). Hence, we have
Bg(z)Bg(u¯)0i = β2a−bκa+bg(v¯, u¯)Bg(z)Ba,b(u¯, v¯).
The action of Bg(z) onto Ba,b(u¯, v¯) is given by the terms Λ(20) (6.18) and Λ(10) (6.7) (in the
limit z1 = z2 = z). Thus,
Bg(z)Bg(u¯)0i = β2a−bκa+bg(v¯, u¯) Λ(10) − Λ(20)
z1=z2=z
.
Now we should set λ1(z) = λ(z) and λ2(z) = κ in (6.7) and (6.18) for Λ(k0) and substitute
these expressions into (7.13). We obtain
(7.9)
(7.10)
(7.11)
(7.12)
(7.13)
(7.14)
(7.15)
z1=z2=z
(7.16)
where
and
Bg(z)Bg(u¯)0i = M1 + M2,
M1 = β2a−bκa+b+1g(v¯, u¯)λ(z)g(z, u¯)T13(z)Ba,b+1(u¯; {v¯, z − c}),
M2 = β2a−bκa+b+3g(v¯, u¯)
g(v¯, z1)g(u¯, z2)g(z1, z2) Ba+1,b+2({u¯, z1}; {v¯, z1 − c, z2})
It remains to substitute explicit expression (5.3) for the Bethe vectors Ba,b+1(u¯; {v¯, z − c})
and Ba+1,b+2({u¯, z1}; {v¯, z1 − c, z2}) into (7.15) and (7.16). However, before doing this, it
is convenient to describe an expected form of the result.
We expect that multiple action Bg(z)Bg(u¯)0i is given by (4.5), in which one should
replace u¯ with η¯ = {u¯, z} and a → a + 1. That is,
Bg(z)Bg(u¯)0i = X β2a+2−n X
X
(−κ)n−sλ(η¯I)f (η¯II, η¯I)f (η¯III, η¯I)f (η¯II, η¯III)
Let us give more details on the expected form of the result (7.17).
z ∈ η¯II; z ∈ η¯III. Respectively, there are three contributions
There are three possibilities in the sum over partitions in the r.h.s. of (7.17): z ∈ η¯I;
HJEP06(218)
Bg(z)Bg(u¯)0i = W1 + W2 + W3.
(7.18)
× T13(η¯I)T13(η¯II)T12(η¯III)0i.
(7.17)
a+1
n=0
a
n=0
a
n=0
n
In the first case we have
W1 = λ(z)T13(z) X β2a+1−n X
X
Indeed, we can set η¯I = {z, u¯I}, η¯II = u¯II, and η¯III = u¯III. The set η¯I is not empty, thus,
s ∈ [1, . . . , n]. This implies n ∈ [1, . . . , a + 1]. Shifting n → n + 1 and s → s + 1 in (7.17)
we arrive at (7.19).
In the second case η¯II = {z, u¯II}, η¯I = u¯I, and η¯III = u¯III. The set η¯II is not empty, thus,
s ∈ [0, . . . , n − 1]. We also have n ∈ [1, . . . , a + 1], because the union {η¯I, η¯II} is not empty.
Shifting n → n + 1 in (7.17) we arrive at
W2 = −κT13(z) X β2a+1−n X
X
(−κ)n−sλ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III)
× f (z, u¯I)f (z, u¯III)T13(u¯I)T13(u¯II)T12(u¯III)0i.
(7.20)
Finally, in the third case η¯I = u¯I, η¯II = u¯II, and η¯III = {z, u¯III}. The set η¯III is not empty,
thus, n ∈ [0, . . . , a]. We obtain
a
n=0
n
W3 = X β2a+2−n X
X
(−κ)n−sλ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III)
× f (z, u¯I)f (u¯II, z)T13(u¯I)T13(u¯II)T12(u¯III)T12(z)0i.
(7.21)
Thus, our goal is to check that equations (7.14)–(7.16) give all three contributions
Wj, j = 1, 2, 3.
7.2.1
Consider the term M1. Using (5.3) for Ba,b+1(u¯; {v¯, z − c}) we obtain
M1 = λ(z)f (u¯, z)T13(z) X κnβ2a+1−n X
X
(−1)n−s
a
n=0
n
× f (u¯I, u¯II) f (u¯I, z)
f (v¯, u¯I) f (u¯I, u¯III)f (u¯II, u¯III) T13(u¯I)T13(u¯II)T12(u¯III)0i.
(7.22)
Taking into account (3.3a) we arrive at
HJEP06(218)
M1 = λ(z)T13(z) X β2a+1−n X
X
(−κ)n−sf (u¯II, z)f (u¯III, z)
Using (3.3a) we arrive at
M2(1) = −κT13(z) X β2a+1−n X
X
(−κ)n−sf (z, u¯I)f (z, u¯III)
and we see that this is exactly W2 (7.20).
× f (u¯I, u¯II)f (v¯, u¯I)f (u¯I, u¯III)f (u¯II, u¯III) T13(u¯I)T13(u¯II)T12(u¯III)0i.
(7.25)
× λ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III) T13(u¯I)T13(u¯II)T12(u¯III)0i,
(7.26)
a+1
n=0
We see that this is exactly W1 (7.19).
Contributions W2 and W3
Consider the term M2.
{z1, u¯} = η¯ we obtain
M2 = X κnβ2a+2−n X
X
n
n
n
× λ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III) T13(u¯I)T13(u¯II)T12(u¯III)0i.
(7.23)
Using (5.3) for Ba+1,b+2({u¯, z1}; {v¯, z1 − c, z2}) and setting
(−1)n−s f (v¯, η¯I)f (z2, η¯I)
f (η¯I, z1)
× f (η¯I, η¯II)f (η¯I, η¯III)f (η¯II, η¯III) T13(η¯I)T13(η¯II)T12(η¯III)0i.
(7.24)
We see that z1 ∈/ η¯I, otherwise 1/f (η¯I, z1) = 0. Thus, either z1 ∈ η¯II or z1 ∈ η¯III.
Respectively, M2 consists of two contributions: M2(1) corresponding to the case z1 ∈ η¯II and M2(2)
corresponding to the case z1 ∈ η¯III.
Let z1 ∈ η¯II. Then we can set η¯I = u¯I, η¯III = u¯III, and η¯II = {z1, u¯II}. We also have
n − s > 0, and thus, s ∈ [0, . . . , n − 1] and n ∈ [1, . . . , a + 1]. Shifting n → n + 1 and setting
z1 = z2 = z we obtain
M2(1) = −κT13(z) X κnβ2a+1−n X
X
(−1)n−sf (z, u¯I)f (z, u¯III)
a
n=0
a
n=0
× f (u¯I, u¯II)f (v¯, u¯I)f (u¯I, u¯III)f (u¯II, u¯III) T13(u¯I)T13(u¯II)T12(u¯III)T12(z)0i.
(7.27)
HJEP06(218)
M2(2) =
X κnβ2a+2−n X
X
(−1)n−sf (z, u¯I)f (u¯II, z)
n
Let now z1 ∈ η¯III. Then we can set η¯I = u¯I, η¯II = u¯II, and η¯III = {z1, u¯III}. We also have
n ∈ [0, . . . , a]. Setting z1 = z2 = z we obtain
× λ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III) T13(u¯I)T13(u¯II)T12(u¯III)T12(z)0i.
(7.28)
We see that this is exactly W3 (7.21). Thus, the multiple action Bg(z)Bg(u¯)0i is given by
the formulas (7.19)–(7.21), leading to (7.17). Hence, proposition 4.2 is proved.
Conclusion. In this paper we have proved one of conjectures of [
1
]. Namely we have
shown that the successive action of the operator Bg (1.1) on the pseudovacuum vector
generates onshell Bethe vectors in gl3invariant models, provided the arguments of these
operators satisfy Bethe equations. Furthermore, if the arguments of the Bg operators are
generic complex numbers, then the successive action of Bg gives a semionshell Bethe
vector. This property holds not only for gl3invariant spin chains, but for a wider class of
models, for instance, for the twocomponent generalization of the LiebLiniger model [3, 35–
37]. At the same time, we would like to emphasize that the operator Bg can not be used
to construct onshell Bethe vectors in generic NABAsolvable models. The restriction
T203(u)0i = 0 is crucial. On the other hand, the existence of this restriction was clear from
the outset, since within the framework of the new approach Bethe vectors depend only
on one set of variables by construction, rather than two sets, as is the case of the Bethe
vectors of the general form.
In this paper, we considered the minimal twist (3.7). A general twist Kgen can be
treated as further twisting of the matrix T (u). It is quite natural to expect that the effect
of the general twist must be similar to what one has in the case of gl2 based models [38].
Namely, we saw that for the minimal twist, the multiple action of Bg was equivalent to
the one semionshell Bethe vector Ba,b(u¯, v¯). Most probably, that the multiple action
of Bg in the case of the general twist is equivalent to a linear combination of
semionshell Bethe vectors with different sets of the Bethe parameters. However, as soon as we
impose Bethe equations, only one term in this linear combination should survive. The
proof of this property in the case of gl2invariant models is very simple (see [38]). However,
a generalization of this proof to the models with gl3invariant Rmatrix meats certain
technical difficulties. Therefore, we did not consider the case of the general twist.
Despite the fact that we have proved the property of Bg(u) to generate onshell Bethe
vectors, we still do not have a clear understanding of why this is happening. In this
context, the most intrigues looks the cancellation of ‘unwanted’ terms (6.21). Recall that
this cancellation takes place for a general semionshell Bethe vector. We do not need to
assume any specific form of λj(u) and specific action of T23(u) onto 0i. Perhaps this is
due to some hidden structure of the operator Bg(u), which is not yet clear. It would be
very interesting to find this structure.
Finally it is worth mentioning that a generalization of the operator Bg(u) to the glN
invariant spin chains (N > 3) was also proposed in [
1
]. It was conjectured that this operator
also generates onshell Bethe vectors, similarly to the gl3 case. Basing on the results of this
paper we can assume that the successive action of Bg(u) is equivalent to a semionshell
Bethe vector of a certain glN invariant integrable model. However, the method that we
used in this paper hardly can be applied to the case N > 3, as it becomes very bulky.
Acknowledgments
We would like to thank I. Kostov, F. LevkovichMaslyuk, S. Pakuliak, and E. Ragoucy for
numerous and fruitful discussions. The work of A.L. has been funded by Russian Academic
Excellence Project 5100 and by Young Russian Mathematics award.
A part of this work, section 6, was performed in Steklov Mathematical Institute of
Russian Academy of Sciences by N.A.S. and he was supported by the Russian Science
Foundation under grant 145000005.
A
Properties of DWPF
The DWPF Kn(x¯y¯) defined by (2.15) is a rational function of x¯ and y¯. It is symmetric
over x¯ and symmetric over y¯. If xj → ∞ (or yj → ∞) and all other variables are fixed,
then Kn(x¯y¯) → 0. This function has simple poles at xj = yk, j, k = 1, . . . , n. The residues
in these poles can be expressed in terms of Kn−1. Due to the symmetry of Kn over x¯ and
over y¯, it is enough to consider the residue at xn = yn:
xn→yn
Kn(x¯y¯)
= g(xn, yn)f (x¯n, xn)f (yn, y¯n)Kn−1(x¯ny¯n) + reg,
(A.1)
where reg means regular part.
the function Kn(x¯y¯) unambiguously [31, 32].
Proposition A.1. Let #x¯ = m and #y¯ = n so that m ≥ n. Then
The properties listed above, together with the initial condition K1(xy) = g(x, y) fix
X
Kn(x¯Iy¯)f (x¯II, x¯I) = X
X (−1)n−kf (y¯I, y¯II)f (x¯, y¯I).
(A.2)
Here in the lhs, the sum is taken over partitions x¯ ⇒ {x¯I, x¯II} so that #x¯I = n. In the rhs,
the sum is taken over all possible partitions y¯ ⇒ {y¯I, y¯II}.
n
Proof. We use induction over n. For n = 1, equation (A.2) takes the form
m
j=1
X g(xj, y)f (x¯j, xj) = f (x¯, y) − 1.
Obviously, the l.h.s. of (A.3) is partial fraction decomposition of the r.h.s. Thus,
identity (A.2) is valid for n = 1 and arbitrary m ≥ 1.
Assume that (A.2) holds for some n − 1 and arbitrary m ≥ n − 1. Let
X
X (−1)n−kf (y¯I, y¯II)f (x¯, y¯I).
Consider properties of Hnℓ,m and Hnr,m as functions of yn at other variables fixed. Both
functions are rational functions of yn. Due to the properties of Kn(x¯Iy¯), the function
Hnℓ,m(x¯; y¯) vanishes as yn → ∞. Let us show that Hnr,m(x¯; y¯) has the same property. We
use the fact that for arbitrary finite z the functions f (z, yn) and f (yn, z) go to 1 as yn → ∞.
Clearly, we have either yn ∈ y¯I or yn ∈ y¯II in the sum over partitions over y¯. Consider
the first case. Then k > 0 and we can set y¯I = {yn, y¯i}. We obtain
X (−1)n−kf (yn, y¯II)f (y¯i, y¯II)f (x¯, yn)f (x¯, y¯i) = X
X (−1)n−k−1f (y¯i, y¯II)f (x¯, y¯i).
ynli→m∞
n
X
k=1 #y¯i=k−1
lim
yn→∞
n−1
X
In the second case k < n and we can set y¯II = {yn, y¯ii}. We obtain
X (−1)n−kf (y¯I, yn)f (y¯I, y¯ii)f (x¯, y¯I) = X
X (−1)n−kf (y¯I, y¯ii)f (x¯, y¯I).
n−1
k=0 #y¯i=k
n−1
The remaining sum over partitions gives Hnℓ−1,m−1(x¯j; y¯n), and we finally arrive at
yn→xj
Hnℓ,m(x¯; y¯)
= g(xj, yn)f (x¯j, xj)f (yn, y¯n)Hnℓ−1,m−1(x¯j; y¯n) + reg.
Relabeling y¯i → y¯I in (A.5) and y¯ii → y¯II in (A.6) we see that the obtained sums over
partitions cancel each other. Thus Hnr,m(x¯; y¯) → 0 as yn → ∞.
It remains to compare the residues of two rational functions in the poles yn = xj,
j = 1, . . . , m. Let yn → xj in the function Hnℓ,m(x¯; y¯). The pole occurs if and only if
xj ∈ x¯I. Setting x¯I = {xj, x¯I′} and using (A.1) we find
Hnℓ,m(x¯; y¯)
yn→xj
=
X
#x¯I′=n−1
g(xj, yn)f (x¯I′, xj)f (yn, y¯n)Kn−1(x¯I′y¯n)f (x¯II, x¯I′)f (x¯II, xj)+reg,
where reg means regular part. Obviously f (x¯I′, xj)f (x¯II, xj) = f (x¯j, xj). Hence,
Hnℓ,m(x¯; y¯)
yn→xj
= g(xj, yn)f (x¯j, xj)f (yn, y¯n)
Kn−1(x¯I′y¯n)f (x¯II, x¯I′) + reg.
X
#x¯I′=n−1
(A.3)
(A.4)
Consider now the behavior of Hnr,m(x¯; y¯) at yn → xj. The pole occurs if and only if
yn ∈ y¯I. Setting y¯I = {yn, y¯I′} we obtain
Hnr,m(x¯; y¯)
yn→xj
= X
X
k=1 #y¯I′=k−1
(−1)n−kf (yn, y¯II)f (y¯I′, y¯II)
Using f (yn, y¯I′)f (yn, y¯II) = f (yn, y¯n) and changing k → k + 1 we find
× f (x¯j, y¯I′)g(xj, yn)f (x¯j, xj)f (yn, y¯I′) + reg.
(A.10)
Hnr,m(x¯; y¯)
yn→xj
= g(xj, yn)f (x¯j, xj)f (yn, y¯n) X
X (−1)n−1−kf (y¯I′, y¯II)f (x¯j, y¯I′)+reg.
The remaining sum over partitions gives Hnr−1,m−1(x¯j; y¯n), and we finally arrive at
yn→xj
Hnr,m(x¯; y¯)
= g(xj, yn)f (x¯j, xj)f (yn, y¯n)Hnr−1,m−1(x¯j; y¯n) + reg.
Due to the induction assumption Hnr−1,m−1(x¯j; y¯n) = Hnℓ−1,m−1(x¯j; y¯n). Hence, the residues
of Hnr,m(x¯; y¯) and Hnℓ,m(x¯; y¯) in the poles at yn = xj coincide. Since both functions vanish
at yn → ∞ we conclude that Hnr,m(x¯; y¯) = Hnℓ,m(x¯; y¯).
Proposition A.2. Let #x¯ = m and #y¯ = n. Then
X
Here the sum is taken over all possible partitions y¯ ⇒ {y¯I, y¯II}. If m < n, then
det
n
f (yj, y¯j)f (x¯, yj) − δjk
= 0.
Proof. Expanding the determinant in the r.h.s. of (A.13) over diagonal minors we find
s
s
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
det
n
h(yj, yk)
f (yj, y¯j)f (x¯, yj) − δjk
= (−1)n + X(−1)n−s
X
Thus, we obtain
det
n
h(yj, yk)
f (yj, y¯j)f (x¯, yj) − δjk
= (−1)n + X(−1)n−s
X
n
s=1
n
s=1
The determinant in the r.h.s. is the Cauchy determinant, hence,
1≤j1<···<js≤n p=1
1
det
s h(yji, yjk )
=
s
Y
p6=q
p,q=1 f (yjp, yjq )
1
.
Y f (yjp, y¯jp)f (x¯, yjp) dset h(yji, yjk )
.
1
Y f (yjp, y¯jp)f (x¯, yjp)
1≤j1<···<js≤n p=1
s
Y
p6=q
1
p,q=1 f (yjp, yjq )
. (A.17)
This is exactly the sum over partitions in the l.h.s. of (A.13).
Let now m < n. Obviously,
det
f (yj, y¯j)f (x¯, yk) − δjk ,
because both matrices are related by a similarity transformation. It is easy to see that the
matrix in the r.h.s. of (A.18) has an eigenvector with zero eigenvalue:
where
Indeed, consider a function
.
Due to the condition m < n this function vanishes as z → ∞. Hence, it has the following
partial fraction decomposition
cn−m Qpm=1(xp −z)
Qqn=1(z −yq +c)
= X
cn−m Qpm=1(xp −yk +c)
k=1 (z −yk +c) Qqn=1,q6=k(yk −yq)
k=1
= Xn g(yk, y¯k)h(x¯, yk) .
h(z, yk)
Setting here z = yj we arrive at
k=1
Xn g(yk, y¯k)h(x¯, yk) −
= 0.
On the other hand, substituting νj from (A.20) into (A.19) we immediately obtain the
l.h.s. of (A.23).
B
Proof of the connection between two types of Bethe vectors
The proof of (2.17) is based on the double induction, first on a, and then on b.
B.1
First step of induction
We first assume that b = 0. Then (2.17) takes the form
Bb0,a(∅; u¯) = (−1)a λλ12((uu¯¯)) Ba,0(u¯; ∅).
For a = 0, (B.1) turns into a trivial identity: 0i = 0i. It is easy to see that (B.1) also
holds for a = 1:
Bb0,1(∅; u) = Tb23(u)0i
λˆ2(u)
= − T12(u)0i
λ1(u)
= − λλ21((uu)) B1,0(u; ∅),
where we used (C.8) for Tb23(u).
(A.18)
(A.19)
(A.20)
(A.21)
(A.22)
(A.23)
(B.1)
(B.2)
HJEP06(218)
Assume now that (B.1) holds for some a ≥ 1. Then we have for #u¯ = a
Bb0,a+1(∅; {u¯, z}) = Tb23(z) λˆTb22(3z()u¯λˆ)2(0u¯i)
= (−1)aTb23(z) λˆ2(z)λ1(u¯)
T12(u¯)0i
.
Substituting here Tb23(z) from (C.8) we find
Bb0,a+1(∅; {u¯, z}) = (−1)a T13(z)T32(z − c) − T12(z)T33(z − c) λˆ2(z)λ1(u¯)
.
T12(u¯)0i
To calculate the obtained action we use commutation relations of the monodromy matrix
[Tij(u), Tkl(v)] = g(u, v) Tkj(v)Til(u) − Tkj(u)Til(v) .
In particular, we have
T32(u)T12(v) = T12(v)T32(u)f (u, v) − T12(u)T32(v)g(u, v).
We see that permuting the operators T32 and T12 we obtain the annihilation operator T32
on the right. Eventually, this operator approaches the vector 0i and annihilates it. Thus,
the contribution from the term T13(z)T32(z − c) vanishes.
The commutation relations (B.5) also imply
T33(u)T12(v) = T12(v)T33(u) + g(u, v) T13(v)T32(u) − T13(u)T32(v) .
(B.7)
We see that when the operator T33 is permuted with the operator T12, it either commutes
or generates the operator T32. As we have already seen, the latter annihilates the state
T12(u¯)0i. Thus, the operator T33(z − c) acts on the state T12(u¯)0i as
T33(z − c)T12(u¯)0i = λ3(z − c)T12(u¯)0i.
Substituting this into (B.4), we arrive at
Bb0,a+1(∅; {u¯, z}) = (−1)a+1λ3(z −c) T12(z)T12(u¯)0i
λ1(u¯)λˆ2(z)
= (−1)a+1 λλ21((uu¯¯))λλ21((zz)) Ba+1,0({u¯, z}; ∅),
what completes the first step of the induction. Thus, equation (2.17) holds for b = 0 and
a arbitrary nonnegative.
B.2
Second step of induction
We pass to the second step of induction. This time we use a recursion for the Bethe vectors
Bb [25]
λˆ2(z)g(u¯, z)Bbb+1,a({v¯ + c, z}; u¯) = Tb12(z)Bbb,a(v¯ + c; u¯)
(B.3)
(B.4)
(B.5)
(B.6)
(B.8)
(B.9)
a
j=1
+ X g(uj, z)
fg((u¯ujj,,uv¯j)) Tb13(z)Bbb,a−1(v¯ + c; u¯j). (B.10)
This recursion allows us to uniquely construct the Bethe vector Bbb+1,a, knowing the Bethe
Assume that (2.17) holds for some b ≥ 0 and a arbitrary. Then we can replace the
Bethe vectors Bb by B in the r.h.s. of (B.10). We obtain
λˆ2(z)g(u¯, z)Bbb+1,a({v¯+c, z}; u¯) = (−1)a+b+ab λ2(u¯)λ2(v¯) (
Tb12(z) Ba,b(u¯; v¯)
j=1
We should compute the action of the operator Tb12(z) on Ba,b(u¯; v¯) and the action of the
operator Tb13(z) on Ba−1,b(u¯j; v¯). This is done in sections C.2.2 and C.2.1 respectively. The
results have the following form:
Tb13(z)Ba−1,b(u¯j; v¯) = (−1)a+bλ2(z)λ2(z − c)
hg((u¯zj,,v¯z)) Ba,b+1({u¯j, z}; {v¯, z − c}), (B.12)
and
(
Tb12(z)Ba,b(u¯; v¯) = (−1)a+1λ2(z − c) λ1(z)g(u¯, z)Ba,b+1(u¯; {v¯, z − c})
j=1 λ2(uj)g(v¯, uj)h(u¯j, z)
)
+ λ2(z)g(v¯, z) Xa λ1(uj)g(z, uj)f (u¯j, uj) Ba,b+1({u¯j, z}; {v¯, z − c}) .
Λ(z) =
λ2(z)
h(v¯, z)h(z, u¯)
Substituting these formulas into (B.11) we immediately arrive at
λˆ2(z)Bbb+1,a({v¯ + c, z}; u¯) = (−1)1+b+ab λ2(u¯)λ2(v¯)
λ1(u¯)λ3(v¯) λ1(z)λ2(z − c)Ba,b+1(u¯; {v¯, z − c}).
Finally, using λˆ2(z) = λ1(z)λ3(z − c) we obtain
This completes the second step of the induction.
Bbb+1,a({v¯+c, z}; u¯) = (−1)a+(b+1)+a(b+1) λλ21((uu¯¯))λλ23((vv¯¯))λλ23((zz −−cc)) Ba,b+1(u¯; {v¯, z −c}). (B.15)
C
C.1
also set
5We set by definition Bbb,−1 = 0.
Action formulas
Actions of the operators Tij on Bethe vectors Ba,b
In this section we give a list of formulas for the actions of the operators Tij(z) on the Bethe
vectors Ba,b(u¯; v¯). These formulas were obtained in [25]. Here they are adopted to the new
normalization of the Bethe vectors. In all action formulas η¯ = {z, u¯} and ξ¯ = {z, v¯}. We
(B.13)
(B.14)
(C.1)
• Action of T12(z):
• Action of T23(z):
T13(z)Ba,b(u¯; v¯) = Λ(z)Ba+1,b+1(η¯; ξ¯).
T12(z)Ba,b(u¯; v¯) = Λ(z) X
h(z, ξ¯I)
f (ξ¯II, ξ¯I)h(ξ¯I, η¯) Ba+1,b(η¯; ξ¯II).
The sum is taken over partitions ξ¯ ⇒ {ξ¯I, ξ¯II} so that #ξ¯I = 1.
T23(z)Ba,b(u¯; v¯) = Λ(z)
f (η¯I, η¯II)h(ξ¯, η¯I) Ba,b+1(η¯II; ξ¯).
The sum is taken over partitions η¯ ⇒ {η¯I, η¯II} so that #η¯I = 1.
(C.2)
(C.3)
(C.4)
• Action of T22(z):
T22(z)Ba,b(u¯; v¯) = Λ(z)
• Action of T11(z):
T11(z)Ba,b(u¯; v¯) = Λ(z)
X
#ξ¯I=#η¯I=1
X
#ξ¯I=#η¯I=1
h(z, ξ¯I)h(η¯I, z)
f (η¯I, η¯II)f (ξ¯II, ξ¯I)h(ξ¯I, η¯)h(ξ¯II, η¯I) Ba,b(η¯II; ξ¯II). (C.5)
λ2(η¯I)g(ξ¯II, η¯I)h(z, ξ¯I)
λ1(η¯I)f (η¯II, η¯I)f (ξ¯II, ξ¯I)h(ξ¯I, η¯II) Ba,b(η¯II; ξ¯II). (C.6)
The sum is taken over partitions ξ¯ ⇒ {ξ¯I, ξ¯II} and η¯ ⇒ {η¯I, η¯II} so that #ξ¯I = #η¯I = 1.
The sum is taken over partitions ξ¯ ⇒ {ξ¯I, ξ¯II} and η¯ ⇒ {η¯I, η¯II} so that #ξ¯I = #η¯I = 1.
• Action of T21(z):
T21(z)Ba−1,b(u¯; v¯) = Λ(z)
X
× h(ξ¯I, η¯II)h(ξ¯I, η¯III)h(ξ¯II, η¯II)Ba−1,b(η¯III; ξ¯II). (C.7)
The sum is taken over partitions ξ¯ ⇒ {ξ¯I, ξ¯II} and η¯ ⇒ {η¯I, η¯II, η¯III} so that
#ξ¯I = #η¯I = #η¯II = 1.
The actions of Tbij onto Bba,b(u¯; v¯) are given by the same formulas, where we should put
hats for the operators, the vacuum eigenvalues λk(z), and the Bethe vectors.
The action formulas (C.2)–(C.7) allow us to derive the actions of the operators Tbij onto
the Bethe vectors Ba,b(u¯; v¯). For this we should express Tbij in terms of the original entries
Tij via (2.5)–(2.7). In particular, we have
Tb12(z) = T21(z)T13(z − c) − T23(z)T11(z − c),
Tb23(z) = T13(z)T32(z − c) − T12(z)T33(z − c),
Tb13(z) = T12(z)T23(z − c) − T13(z)T22(z − c).
(C.8)
HJEP06(218)
Then the actions of Tbij onto Ba,b(u¯; v¯) can be obtained via successive application of the
formulas (C.2)–(C.7). Below we give some details of this derivation for the action of Tb13
and Tb12.
C.2.1
Action of Tb13
The operator Tˆ13(z) is given by the last equation (C.8). It is convenient to consider the
following combination
T12(x)T23(y) − T13(x)T22(y)
(C.9)
and set x = z, y = z − c in the end. Such the replacement of Tb13(z) allows us to avoid
singular expressions in the intermediate computations.
Applying successively, first (C.5) and (C.2), and then (C.4) and (C.3) we obtain
T13(x)T22(y)Ba,b(u¯; v¯) =
and
T12(x)T23(y)Ba,b(u¯; v¯) =
Λ(x, y)
h(y, x)
Λ(x, y)
h(y, x)
X
#ξ¯I=1
X
f (η¯I, η¯II)f (ξ¯II, ξ¯I) h(ξ¯I, η¯)h(ξ¯II, η¯I) Ba+1,b+1(η¯II; ξ¯II),
h(η¯I, y)h(x, ξ¯I)f (η¯I, x)
f (η¯I, η¯II)f (ξ¯II, ξ¯I)h(ξ¯I, η¯)h(ξ¯II, η¯I) Ba+1,b+1(η¯II; ξ¯II).
Here η¯ = {u¯, x, y} and ξ¯ = {v¯, x, y}. The sum is taken over partitions η¯ ⇒ {η¯I, η¯II} and
ξ¯ ⇒ {ξ¯I, ξ¯II} so that #η¯I = #ξ¯I = 1. Here we also introduced
Λ(x, y) =
λ2(x)λ2(y)
h(x, y)h(v¯, x)h(v¯, y)h(x, u¯)h(y, u¯)
.
Taking the difference of (C.10) and (C.11) we arrive at
Tb13(z)Ba,b(u¯; v¯) = Λ(x, y) X
h(η¯I, y)h(x, ξ¯I)h(y, ξ¯I)f (η¯I, x)
f (η¯I, η¯II)f (ξ¯II, ξ¯I)h(ξ¯I, η¯)h(ξ¯II, η¯I) Ba+1,b+1(η¯II; ξ¯II)
(C.10)
(C.11)
(C.12)
x=z .
y=z−c
(C.13)
Now we should consider several cases. First of all, we see that η¯I 6= x, because otherwise
the factor 1/f (η¯I, x) in (C.13) is equal to zero. Thus, either η¯I = y or η¯I = uj, j = 1, . . . , a.
Consider the first case η¯I = y and denote this contribution by G. Then
G = Λ(x, y)f (y, u¯)h(v¯, y)h(x, y) X
f (ξ¯II, ξ¯I)h(ξ¯I, x)h(ξ¯I, u¯) Ba+1,b+1({u¯, x}; ξ¯II)
This case respectively should be divided into subcases.
• ξ¯I = x, hence, ξ¯II = {v¯, y}. Then, substituting (C.12) in (C.14) we find
x=z .
y=z−c
(C.14)
(C.15)
(C.16)
(C.17)
G1 = (−1)a+b+1λ2(z)λ2(z − c)
hg((zu¯,, v¯z)) Ba+1,b+1({u¯, z}; {v¯, z − c}).
• ξ¯I = y, hence, ξ¯II = {v¯, x}. Then h(ξ¯I, x) = h(y, x) → 0, as x → z and y → z − c.
Thus, this contribution vanishes.
• ξ¯I = vj, j = 1, . . . , b, hence, ξ¯II = {v¯j, x, y}. Then
G(12) = λ2(x)λ2(y)g(y, u¯)f (v¯j, vj)g(x, vj)g(y, vj)h(vj, u¯) Ba+1,b+1({u¯, x}; {v¯j, x, y}) y=x=z−zc .
h(v¯j, x)h(x, u¯)
In this case the Bethe vector Ba,b({u¯, x}; {v¯j, x, y}) vanishes in the limit x = z and
y = z − c. Indeed, we have due to (C.2)
Ba+1,b+1({u¯, x}; {v¯j, x, y}) =
h(y, x)h(v¯j, x)h(x, u¯)T13(x)Ba,b(u¯; {v¯j, y}),
1
and the r.h.s. of (C.17) vanishes, because h(z − c, z) = 0.
Similarly, one should consider the case η¯I = uj, j = 1, . . . , a. The analysis of this case
Bethe vector Ba,b(u¯; v¯) is given by (C.15):
shows that all the corresponding contributions vanish. Thus, the action of Tb13(z) on the
Tb13(z)Ba,b(u¯; v¯) = (−1)a+b+1λ2(z)λ2(z − c)
hg((zu¯,, v¯z)) Ba+1,b+1({u¯, z}; {v¯, z − c}).
(C.18)
C.2.2
The action of Tb12(z) can be considered exactly in the same manner. Using (C.8) and the
action formulas (C.2)–(C.7) we obtain
Tb12(z)Ba,b(u¯; v¯) = Λ(x, y)
X
× h(ξ¯, η¯II)h(ξ¯I, η¯III)Ba,b+1(η¯III; ξ¯II)
x=z .
y=z−c
(C.19)
Here η¯ = {u¯, x, y} and ξ¯ = {v¯, x, y}. The sum is taken over partitions η¯ ⇒ {η¯I, η¯II, η¯III} and
ξ¯ ⇒ {ξ¯I, ξ¯II} so that #η¯I = #η¯II = #ξ¯I = 1.
Again one should consider several cases. The analysis shows that nonvanishing
contributions arise if and only if ξ¯I = x and η¯II = y. Then
Tb12(z)Ba,b(u¯; v¯) =
λ2(x)λ2(y)g(y, u¯)g(v¯, x)
h(x, u¯)
×
X
λ1(η¯I)f (η¯III, η¯I)h(x, η¯III) Ba,b+1(η¯III; {v¯, y}) x=z ,
y=z−c
(C.20)
HJEP06(218)
where η¯ = {u¯, x} and the sum is taken over partitions η¯ ⇒ {η¯I, η¯III} so that #η¯I = 1. Then
we should consider two cases. First, we can set η¯I = x and η¯III = u¯. Then we obtain the
first term in (B.13). The second case is η¯I = uj and η¯III = {u¯j , x}, j = 1, . . . , a. Then we
obtain the second term in (B.13).
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