Scalar Products in Twisted XXX Spin Chain. Determinant Representation
We consider the XXX spin-1/2 Heisenberg chain with non-diagonal boundary conditions. We obtain a compact determinant representation for the scalar product of on-shell and off-shell Bethe vectors. In the particular case when both Bethe vectors are on shell, we obtain a determinant representation for...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2019 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2019
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210229 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Scalar Products in Twisted XXX Spin Chain. Determinant Representation / S. Belliard, N.A. Slavnov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 33 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859505312829014016 |
|---|---|
| author | Belliard, S. Slavnov, N.A. |
| author_facet | Belliard, S. Slavnov, N.A. |
| citation_txt | Scalar Products in Twisted XXX Spin Chain. Determinant Representation / S. Belliard, N.A. Slavnov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 33 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We consider the XXX spin-1/2 Heisenberg chain with non-diagonal boundary conditions. We obtain a compact determinant representation for the scalar product of on-shell and off-shell Bethe vectors. In the particular case when both Bethe vectors are on shell, we obtain a determinant representation for the norm of the on-shell Bethe vector and prove orthogonality of the on-shell vectors corresponding to the different eigenvalues of the transfer matrix.
|
| first_indexed | 2025-12-07T21:24:50Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 066, 30 pages
Scalar Products in Twisted XXX Spin Chain.
Determinant Representation
Samuel BELLIARD † and Nikita A. SLAVNOV ‡
† Institut Denis-Poisson, Université de Tours, Université d’Orléans,
Parc de Grammont, 37200 Tours, France
E-mail: samuel.belliard@gmail.com
‡ Steklov Mathematical Institute of Russian Academy of Sciences,
8 Gubkina Str., Moscow, 119991, Russia
E-mail: nslavnov@mi-ras.ru
Received June 19, 2019, in final form August 27, 2019; Published online September 03, 2019
https://doi.org/10.3842/SIGMA.2019.066
Abstract. We consider XXX spin-1/2 Heisenberg chain with non-diagonal boundary con-
ditions. We obtain a compact determinant representation for the scalar product of on-shell
and off-shell Bethe vectors. In the particular case when both Bethe vectors are on shell,
we obtain a determinant representation for the norm of on-shell Bethe vector and prove or-
thogonality of the on-shell vectors corresponding to the different eigenvalues of the transfer
matrix.
Key words: XXX chain; non-diagonal boundary conditions; scalar product; determinant
2010 Mathematics Subject Classification: 82B23; 81R50
1 Introduction
The algebraic Bethe ansatz (ABA) [14, 15, 32] is a powerful method to study quantum integrable
systems. Besides the studying the spectra of quantum Hamiltonians, this method is also used to
calculate the correlation functions [18, 21, 25, 27]. The main tool for solving this problem within
the framework of the ABA is the calculation of scalar products of Bethe vectors. In this context,
it should be noted works [16, 17, 26] in which the norm of the Hamiltonian eigenstate (on-shell
Bethe vector) was computed, and also paper [31], where a compact determinant formula was
obtained for the scalar product of off-shell and on-shell Bethe vectors (OFS-ONS scalar product).
The results listed above concern models possessing U(1) symmetry. However, in real physical
systems this symmetry can often be violated, for example, because of non-trivial boundary
conditions. The study of quantum integrable models without U(1) symmetry has led to the
development of new techniques to perform the ABA [29]. Among the proposed methods such
as the off-diagonal Bethe ansatz [10, 11, 28, 33] or the separation of the variables [13, 22, 30],
the modified algebraic Bethe ansatz (MABA) allows one to understand this new method from
the ABA point of view [1, 3, 4, 5, 12].
This paper is a continuation of the series of works [6, 8, 9] devoted to the study of the MABA.
We consider closed XXX spin-1
2 chain with the Hamiltonian
H =
N∑
k=1
(
σxk ⊗ σxk+1 + σyk ⊗ σ
y
k+1 + σzk ⊗ σzk+1
)
, (1.1)
subject to the following non-diagonal boundary conditions
γσxN+1 =
κ̃2 + κ2 − κ2
+ − κ2
−
2
σx1 + i
κ2 − κ̃2 − κ2
+ + κ2
−
2
σy1 + (κκ− − κ̃κ+)σz1 , (1.2)
mailto:email@address
mailto:email@address
https://doi.org/10.3842/SIGMA.2019.066
2 S. Belliard and N.A. Slavnov
γσyN+1 = i
κ̃2 − κ2 − κ2
+ + κ2
−
2
σx1 +
κ̃2 + κ2 + κ2
+ + κ2
−
2
σy1 − i(κ̃κ+ + κκ−)σz1 , (1.3)
γσzN+1 = (κκ+ − κ̃κ−)σx1 + i(κ̃κ− + κκ+)σy1 + (κ̃κ+ κ+κ−)σz1 . (1.4)
The twist parameters {κ, κ̃, κ+, κ−} are generic complex numbers and γ = κ̃κ − κ+κ−. The
Pauli matrices1 σαk with α = x, y, z act non-trivially on the kth component of the quantum
space H = ⊗Nk=1Vk with Vk = C2.
The main object of our study is the OFS-ONS scalar product. It was conjectured in [6] that
this scalar product admits a determinant representation similar to the one obtained in [24, 31]
in the case of the usual ABA. In this paper we prove this conjecture. The main tool of our prove
is an analog of Izergin–Korepin formula for the scalar product of off-shell Bethe vectors [20, 26]
(see (3.6)). This formula was generalized for the case of the MABA in [9]. In this paper we
specify it to the particular case when one of the Bethe vectors is on-shell. This allows us to
compute the sum over partitions of the Bethe parameters in the form of a single determinant.
In the particular case, the obtained determinant representation describes the norm of on-shell
Bethe vector. This representation also allows us to prove orthogonality of the on-shell vectors
corresponding to the different eigenvalues of the transfer matrix.
The paper is organised as follows. In Section 2 we recall basic notions of the MABA and
define modified Bethe vectors. We also introduce notation used in the paper. In Section 3 we
give definition of a modified Izergin determinant, which is one of the most important tools for
studying scalar products of the modified Bethe vectors. Section 4 contains the main results
of the paper. Here we give different determinant representations for OFS-ONS scalar product,
a determinant formula for the norm of the modified on-shell Bethe vector, and different forms of
the inhomogeneous Bethe equations. In the rest of the paper we present the proofs of the results
of Section 4. In Section 5 we prove alternative forms of the inhomogeneous Bethe equations.
In Section 6 we consider some properties of the on-shell modified Izergin determinant. Finally,
in Section 7 we present the proof of determinant formula for OFS-ONS scalar product. In
particular, we prove the equivalence of two determinant representations for OFS-ONS scalar
product in Section 7.1. Several useful formulas and auxiliary lemmas are gathered in appendices.
Appendix A contains a list of properties of the modified Izergin determinant. In Appendix B,
we give some identities for rational functions. Appendices C and D are devoted to the properties
of the on-shell modified Izergin determinants.
2 Basic notions
To describe the Hamiltonian (1.1) within the framework of the quantum inverse scattering
method (QISM), we first introduce a gl2-invariant R-matrix acting in C2 ⊗ C2:
R(u, v) =
u− v
c
I + P. (2.1)
Here c is a constant, I is the identity operator, and P is the permutation operator. The R-
matrix (2.1) is called gl2-invariant, because
[R(u, v),K ⊗K] = 0,
for any matrix K ∈ gl2.
The key object of the QISM is a quantum monodromy matrix
T (u) =
(
t11(u) t12(u)
t21(u) t22(u)
)
. (2.2)
1σz =
(
1 0
0 −1
)
, σ+ = ( 0 1
0 0 ), σ− = ( 0 0
1 0 ), σx = σ+ + σ−, σy = i(σ− − σ+).
Scalar products in twisted XXX spin chain. Determinant Representation 3
This matrix acts as a 2 × 2 matrix in auxiliary space C2. The entries tij(u) are operators
depending on a complex parameter u and acting in the Hilbert space H of the Hamiltonian (1.1).
The commutation relations of these operators are given by an RTT -relation
R(u, v)
(
T (u)⊗ I
)(
I ⊗ T (v)
)
=
(
I ⊗ T (v)
)(
T (u)⊗ I
)
R(u, v).
Here I is the identity operator in C2. Equivalently, these commutation relations can be written
in the form
[tij(u), tkl(v)] = g(u, v)
(
tkj(v)til(u)− tkj(u)til(v)
)
.
To obtain the Hamiltonian (1.1) we first introduce an inhomogeneous monodromy matrix
T (u) = R0N (u, θN ) · · ·R01(u, θ1), (2.3)
where θj are inhomogeneity parameters. Each R-matrix R0k(u, θk) in (2.3) acts non-trivially
in the space V0 ⊗ Vk, where V0 is the auxiliary space of the monodromy matrix and Vk is the
quantum space associated with the kth site of the chain. The Hamiltonian of the XXX chain
with periodic boundary conditions can be obtained from the monodromy matrix (2.3) in the
homogeneous limit θj = 0, j = 1, . . . , N (see [2]). In order to obtain the Hamiltonian (1.1) with
the boundary condition (1.2)–(1.4) we introduce a twisted monodromy matrix
TK(u) = KT (u), K =
(
κ̃ κ+
κ− κ
)
.
Then, defining a twisted transfer matrix T (u) by
T (u) = trTK(u),
we obtain
H = 2c
d
du
(
log
(
T (u)
))∣∣
u→0, θi→0
−N.
It is convenient to present the twist matrix K in the form
K = BDA,
where
A =
√
µ
1
ρ2
κ−ρ1
κ+
1
, B =
√
µ
1
ρ1
κ−ρ2
κ+
1
, D =
(
κ̃− ρ1 0
0 κ− ρ2
)
,
and the parameters ρi and µ enjoy the following constraints:
ρ1ρ2 − ρ2κ̃− ρ1κ+ κ+κ− = 0, µ =
1
1− ρ1ρ2
κ+κ−
. (2.4)
Then we have
T (u) = tr
(
DT (u)
)
, T (u) = AT (u)B =
(
ν11(u) ν12(u)
ν21(u) ν22(u)
)
. (2.5)
Thus, instead of the twisted monodromy matrix TK(u) we can consider a modified monodromy
matrix T (u) (2.5). Respectively, the transfer matrix T (u) now can be understood as the trace
of the twisted modified monodromy matrix T (u) with the diagonal twist D.
4 S. Belliard and N.A. Slavnov
2.1 Highest weight representation of the Yangian
Recall that the Hilbert space of the Hamiltonian (1.1) is H = ⊗Nk=1Vk, where Vk = C2. We
define a highest weight vector |0〉 ∈ H as the state with all spins up
|0〉 = ( 1
0 )1 ⊗ · · · ⊗ ( 1
0 )N .
Then the action of the monodromy matrix entries tij(u) (2.2) on |0〉 is
tii(u)|0〉 = λi(u)|0〉, i = 1, 2,
t21(u)|0〉 = 0, (2.6)
where
λ1(u) =
1
cN
N∏
k=1
(u− θk + c), λ2(u) =
1
cN
N∏
k=1
(u− θk). (2.7)
The representation space of the Yangian is then spanned by the Bethe vectors Bm
0 (v̄)
Bm
0 (v̄) =
m∏
i=1
t12(vi)|0〉, m = 0, 1, . . . , N,
which provide a formal basis depending on the parameters v̄ = {v1, . . . , vm}.
To study scalar products of Bethe vectors we also consider the dual highest weight vector
〈0| = |0〉T belonging to the dual space H∗. This vector possesses the following properties
〈0|tii(u) = λi(u)〈0|, i = 1, 2, 〈0|t12(u) = 0, 〈0|0〉 = 1,
where the functions λi(u) are given by (2.7). Dual Bethe vectors are constructed by successive
application of the operator t21 to the vector 〈0|
Cm
0 (v̄) = 〈0|
m∏
i=1
t21(vi), m = 0, 1, . . . , N.
Within the framework of the MABA the states of the space H (resp. the dual space H∗) are
generated by the successive application of the operators ν12 (resp. ν21) to the state |0〉 (resp. 〈0|).
The modified Bethe vectors are given by
Bm(v̄) =
m∏
i=1
ν12(vi)|0〉, Cm(v̄) = 〈0|
m∏
i=1
ν21(vi). (2.8)
In these formulas, v̄ = {v1, . . . , vm} are generic complex numbers. They are called Bethe para-
meters.
Observe that despite of the new operators νij satisfy the same commutation relations as
the tij(z),
[νij(u), νkl(v)] = g(u, v)
(
νkj(v)νil(u)− νkj(u)νil(v)
)
, (2.9)
their actions on the highest weight vector (2.6) change. It is easy to see that now they are given by
ν11(u)|0〉 = λ1(u)|0〉+ β2ν12(u)|0〉,
ν22(u)|0〉 = λ2(u)|0〉+ β1ν12(u)|0〉,
ν21(u)|0〉 =
(
β1λ1(u) + β2λ2(u)
)
|0〉+ β1β2ν12(u)|0〉,
where βi = ρi/κ
+.
Concluding this section we would like to mention that the operators tii(u) are polynomials
in u of degree N , while tij(u) with i 6= j are polynomials in u of degree N − 1. This statement
follows from representation (2.3) and (2.1). Since for generic twist parameters, any νij(u) is
a linear combination of all tkl(u), we conclude that the operators νij(u) are polynomials in u of
degree N for all i and j.
Scalar products in twisted XXX spin chain. Determinant Representation 5
2.2 Shorthand notation
Before moving on, we introduce a notation and several important conventions that will be used
throughout the paper. First of all, we introduce the following rational functions:
g(u, v) =
c
u− v
, f(u, v) = 1 + g(u, v) =
u− v + c
u− v
,
h(u, v) =
f(u, v)
g(u, v)
=
u− v + c
c
. (2.10)
Actually, all these functions depend on the difference of their arguments. However, we do not
stress this dependence, which will allow us to introduce special shorthand notation for their
products. It is easy to see that the functions introduced above possess the following properties:
χ(u, v)
∣∣∣
c→−c
= χ(v, u), χ(−u,−v) = χ(v, u), χ(u− c, v) = χ(u, v + c),
where χ is any of the three functions. One can also convince himself that
g(u, v − c) =
1
h(u, v)
, h(u, v + c) =
1
g(u, v)
, f(u, v + c) =
1
f(v, u)
. (2.11)
Let us formulate now a convention on the notation. We denote sets of variables by a bar, for
example, ū = {u1, . . . , un}. Notation ū± c means that ±c is added to all the arguments of the
set ū. Individual elements of the sets or subsets are denoted by Latin subscripts, for instance,
uj is an element of ū. As a rule, the number of elements in the sets is not shown explicitly in the
equations, however we give these cardinalities in special comments to the formulas, if necessary.
We also consider subsets of variables. We agree upon that the notation ūk refers to a subset
that is complementary to the element uk, that is, ūk = ū \ uk. In all other cases, we denote
subsets by subscripts so that they can be easily distinguished from the elements of sets. In
particular, when dealing with partitions of the sets into subsets we mostly denote the latter by
the Roman numbers ūI, ūII, and so on. Notation {ūI, ūII} ` ū means that the set ū is divided
into two disjoint subsets ūI and ūII. The order of the elements in each subset is not essential.
To make the formulas more compact, we use a shorthand notation for the products of the
rational functions (2.10), the operators νkl(u) (2.2), and the vacuum eigenvalues λi(u) (2.6).
Namely, if a function (an operator) depends on a (sub)set of variables, then one should take
a product with respect to the corresponding (sub)set. For example,
νkl(ū) =
∏
uj∈ū
νkl(uj), f(z, ūI) =
∏
uj∈ūI
f(z, uj), f(ūk, uk) =
∏
uj∈ū
uj 6=uk
f(uj , uk). (2.12)
Note that due to commutativity of the νkl-operators (which follows from (2.9)) the first product
in (2.12) is well defined. Notation f(ū, v̄) means a double product over the sets ū and v̄. By
definition any product over the empty set is equal to 1. A double product is equal to 1 if at
least one of the sets is empty.
In particular, using this convention we can write down the modified Bethe vectors and their
dual ones (2.8) in the form
Bm(v̄) = ν12(v̄)|0〉, Cm(v̄) = 〈0|ν21(v̄),
where v̄ = {v1, . . . , vm}. The eigenvalues (2.7) take the form
λ1(u) = h(u, θ̄), λ2(u) = g(u, θ̄)−1,
λ1(u)
λ2(u)
= f(u, θ̄), (2.13)
where θ̄ = {θ1, . . . , θN}.
6 S. Belliard and N.A. Slavnov
2.3 On-shell Bethe vectors
Within the framework of the MABA the operator (2.5)
T (z) = tr
(
DT (z)
)
= (κ̃− ρ1)ν11(z) + (κ− ρ2)ν22(z)
appears to be a generating function of the integrals of motion. Thus, the eigenstates of this
operator also are the eigenstates of the Hamiltonian (1.1). They are commonly called modified
on-shell Bethe vectors.
The on-shell Bethe vectors in the MABA solvable models were found in [6, 10, 12]. Here we
briefly recall this construction.
Let ū = {u1, . . . , uN}, where N is the number of sites of the chain. Then a modified Bethe
vector BN (ū) (2.8) becomes on-shell, if the Bethe parameters ū satisfy a system of modified
Bethe equations
(κ− ρ2)λ2(uj)f(uj , ūj)− (κ̃− ρ1)λ1(uj)f(ūj , uj)
+ (ρ1 + ρ2)g(uj , ūj)λ1(uj)λ2(uj) = 0, (2.14)
for j = 1, . . . , N . Then
T (z)BN (ū) = Λ(z|ū)BN (ū),
where the eigenvalue Λ(z|ū) is
Λ(z|ū) = (κ̃− ρ1)λ1(z)f(ū, z) + (κ− ρ2)λ2(z)f(z, ū) + (ρ1 + ρ2)λ1(z)λ2(z)g(z, ū). (2.15)
For further application, it is convenient to introduce a function (cf. [23])
Y(z|ū) =
Λ(z|ū)
λ2(z)g(z, ū)
. (2.16)
The explicit expression for this function reads
Y(z|ū) = (−1)N (κ̃− ρ1)
λ1(z)
λ2(z)
h(ū, z) + (κ− ρ2)h(z, ū) + (ρ1 + ρ2)λ1(z).
Then Bethe equations (2.14) take the form
Y(uj |ū) = 0, j = 1, . . . , N.
Similarly, a modified dual Bethe vector CN (ū) (2.8) becomes on-shell, if the Bethe parameters
ū satisfy the system (2.14). Then
CN (ū)T (z) = Λ(z|ū)CN (ū),
with the eigenvalue (2.15).
The main goal of this paper is to study the scalar products of the modified Bethe vectors,
in which at least one of the vectors is on-shell. In fact, it is enough to consider the case when
the vector BN (ū) is on-shell, while the dual state CN (v̄) is a generic modified dual Bethe vector
(see (3.7) below).
3 Modified Izergin determinant and scalar products
Within the framework of the ABA, the Izergin determinant allows us to obtain a formula for
the scalar product of Bethe vectors as a sum over partitions of the Bethe parameters [20, 26].
For the MABA, we should introduce the modified Izergin determinant. Then one can obtain an
analogous formula for the scalar product of the modified Bethe vectors [9].
Scalar products in twisted XXX spin chain. Determinant Representation 7
3.1 Modified Izergin determinant
Definition 3.1. Let ū = {u1, . . . , un}, v̄ = {v1, . . . , vm}, and z be complex numbers. Then the
modified Izergin determinant K
(z)
n,m(ū|v̄) is defined by
K(z)
n,m(ū|v̄) = det
m
(
−zδjk +
f(ū, vj)f(vj , v̄j)
h(vj , vk)
)
. (3.1)
Alternatively the modified Izergin determinant can be presented as
K(z)
n,m(ū|v̄) = (1− z)m−n det
n
(
δjkf(uj , v̄)− z f(uj , ūj)
h(uj , uk)
)
. (3.2)
Recall that we use the shorthand notation for the products (2.12) in representations (3.1)
and (3.2). The proof of the equivalence of these representations can be found in [19]. It is based
on the recursive property (A.6).
It is also convenient to introduce a conjugated modified Izergin determinant as
K
(z)
n,m(ū|v̄) = K(z)
n,m(ū|v̄)
∣∣∣
c→−c
= det
m
(
−zδjk +
f(vj , ū)f(v̄j , vj)
h(vk, vj)
)
, (3.3)
or equivalently,
K
(z)
n,m(ū|v̄) = (1− z)m−n det
n
(
δjkf(v̄, uj)− z
f(ūj , uj)
h(uk, uj)
)
.
In the particular case z = 1 and #ū = #v̄ = n the modified Izergin determinant turns into
the ordinary Izergin determinant, that we traditionally denote by Kn(ū|v̄):
K(1)
n,n(ū|v̄) = Kn(ū|v̄).
This property can be seen from the recursion (A.6) and the initial condition (A.2). Let us recall
one more representation for the ordinary Izergin determinant [20]
Kn(ū|v̄) = h(ū, v̄)∆′(ū)∆(v̄) det
n
(
g(uj , vk)
h(uj , vk)
)
, (3.4)
where
∆(v̄) =
∏
1≤j<k≤n
g(vk, vj), ∆′(ū) =
∏
1≤j<k≤n
g(uj , uk). (3.5)
It is easy to see that the modified Izergin determinants K
(z)
n,m(ū|v̄) and K
(z)
n,m(ū|v̄) are rational
functions of ū and v̄. They are symmetric over ū and symmetric over v̄. Other properties of the
modified Izergin determinant are collected in Appendix A.
3.2 Scalar product
Definition 3.2. Let ū = {u1, . . . , un}, v̄ = {v1, . . . , vm}. The scalar product of two modified
Bethe vectors is defined as
Sm,nν (v̄, ū) = Cm(v̄)Bn(ū) = 〈0|ν21(v̄)ν12(ū)|0〉.
The function Sm,nν (v̄, ū) has several important properties. First, it follows from the commuta-
tion relations [ν12(u), ν12(v)] = 0 and [ν21(u), ν21(v)] = 0 that the scalar product is a symmetric
function of v̄ and a symmetric function of ū. Furthermore, since any νij(z) is a polynomial in z
of degree N , the scalar product is a polynomial in any vj and in any uj of degree N .
8 S. Belliard and N.A. Slavnov
Proposition 3.3 ([9]). Let #ū = n and #v̄ = m. Then
Sm,nν (v̄, ū) = µ2m(µ− 1)n−m
∑
{v̄I,v̄II}`v̄
{ūI,ūII}`ū
βpII−qII1 βpI−qI2 λ2(v̄I)λ1(v̄II)λ2(ūII)λ1(ūI)
× f(v̄I, v̄II)f(ūII, ūI)K
(1/µ)
qII,pII
(ūII|v̄II)K
(1/µ)
qI,pI
(ūI|v̄I). (3.6)
Here pI = #v̄I, pII = #v̄II, qI = #ūI, and qII = #ūII. The sum is taken over all partitions
{v̄I, v̄II} ` v̄ and {ūI, ūII} ` ū. There is no restriction on the cardinalities of the subsets. The
functions K
(1/µ)
qII,pII and K
(1/µ)
qI,pI
respectively are the modified Izergin determinants (3.1) and (3.3)
at z = 1/µ.
Equation (3.6) was derived in [9].
Proposition 3.4. The scalar product of generic modified Bethe vectors satisfies a condition
Sm,nν (v̄, ū) =
(
κ−
κ+
)m−n
Sn,mν (ū, v̄). (3.7)
Proof. Replacing in (3.6) ū↔ v̄, n↔ m, qI ↔ pI, and qII ↔ pII we obtain
Sn,mν (ū, v̄) = µ2n(µ− 1)m−n
∑
{v̄I,v̄II}`v̄
{ūI,ūII}`ū
βqII−pII1 βqI−pI2 λ2(ūI)λ1(ūII)λ2(v̄II)λ1(v̄I)
× f(ūI, ūII)f(v̄II, v̄I)K
(1/µ)
pII,qII
(v̄II|ūII)K
(1/µ)
pI,qI
(v̄I|ūI). (3.8)
Using (A.1) we find
K(1/µ)
pII,qII
(v̄II|ūII)K
(1/µ)
pI,qI
(v̄I|ūI) =
(
1− 1
µ
)n−m
K
(1/µ)
qII,pII
(ūII|v̄II)K(1/µ)
qI,pI
(ūI|v̄I),
and hence,
Sn,mν (ū, v̄) = µn+m
∑
{v̄I,v̄II}`v̄
{ūI,ūII}`ū
βqII−pII1 βqI−pI2 λ2(ūI)λ1(ūII)λ2(v̄II)λ1(v̄I)
× f(ūI, ūII)f(v̄II, v̄I)K
(1/µ)
qII,pII
(ūII|v̄II)K(1/µ)
qI,pI
(ūI|v̄I).
Relabeling the subsets as ūI ↔ ūII and v̄I ↔ v̄II (and respectively relabeling their cardinalities)
we finally arrive at
Sn,mν (ū, v̄) = µn+m(β1β2)n−m
∑
{v̄I,v̄II}`v̄
{ūI,ūII}`ū
βpII−qII1 βpI−qI2 λ2(ūII)λ1(ūI)λ2(v̄I)λ1(v̄II)
× f(ūII, ūI)f(v̄I, v̄II)K
(1/µ)
qI,pI
(ūI|v̄I)K(1/µ)
qII,pII
(ūII|v̄II). (3.9)
Comparing equations (3.9) and (3.8) we see that
Sm,nν (v̄, ū)
Sn,mν (ū, v̄)
=
(
µβ1β2
µ− 1
)n−m
,
and substituting here explicit expressions for βi = ρi/κ+ and µ (2.4) we immediately arrive
at (3.7). �
Scalar products in twisted XXX spin chain. Determinant Representation 9
4 Main results
In this section we give a list of the main results obtained in this paper. Most of the proofs are
given in the remaining part of the text.
4.1 Determinant representations for the scalar product
Theorem 4.1. Let #ū = #v̄ = N . Let the set ū solve Bethe equations (2.14), while the set v̄
consist of arbitrary complex numbers. Then the scalar product SNν (v̄, ū) of the modified on-shell
Bethe vector BN (ū) and generic dual modified Bethe vector CN (v̄) has the following determinant
representation:
SNν (v̄, ū) =
(µβ)N∆(v̄)∆′(ū)λ2(ū)λ2(v̄)
(βκ+ κ̃− 2ρ1)N
K
(−1/β)
N,N (ū|θ̄) det
N
(
c
g(vk, ū)λ2(vk)
∂Λ(vk|ū)
∂uj
)
. (4.1)
Here Λ(v|ū) is the eigenvalue (2.15), β = ρ1/ρ2, and ∆(v̄), ∆′(ū) are given by (3.5) with n = N .
Representation (4.1) was conjectured in [6]. We also would like to point out that similarly
to the scalar products of on-shell and off-shell Bethe vectors in ABA [24, 31] this representation
involves Jacobian of the transfer matrix eigenvalue:
c
g(vk, ū)λ2(vk)
∂Λ(vk|ū)
∂uj
= (−1)N−1(κ̃− ρ1)
λ1(vk)
λ2(vk)
g(uj , vk)
h(uj , vk)
h(ū, vk)
+ (κ− ρ2)
g(vk, uj)
h(vk, uj)
h(vk, ū) + (ρ1 + ρ2)λ1(vk)g(vk, uj). (4.2)
Theorem 4.2. Under the condition of Theorem 4.1 the scalar product has the following deter-
minant representation:
SNν (v̄, ū) = λ2(ū)λ2(v̄)
(µ
α
)N
K
(µ+(µ−1)/β)
N,N (ū|θ̄)K(α)
2N,N ({ū, v̄}|θ̄), (4.3)
where
α =
κ− ρ2
κ̃− ρ1
. (4.4)
The equivalence of representations (4.1) and (4.3) is proved in Section 7.1.
Corollary 4.3. Dual and ordinary modified on-shell Bethe vectors corresponding to different
eigenvalues are orthogonal.
Proof. Let
γj =
g(uj , ūj)
g(uj , v̄)
.
Observe that if the vectors BN (ū) and CN (v̄) correspond to different transfer matrix eigenvalues
(i.e., ū 6= v̄), then there exists at least one γj 6= 0.
Using formulas of Appendix B.1 we obtain
N∑
j=1
cγj
g(vk, ū)λ2(vk)
∂Λ(vk|ū)
∂uj
= − 1
g(vk, v̄k)
(
(κ̃− ρ1)
λ1(vk)
λ2(vk)
f(v̄k, vk)
− (κ− ρ2)f(vk, v̄k) + (ρ1 + ρ2)g(vk, v̄k)λ1(vk)
)
. (4.5)
If the set v̄ satisfies the system of modified Bethe equations (2.14), then the r.h.s. of (4.5)
vanishes. Thus, the rows of the matrix (4.2) are linearly dependent, and hence, the Jacobian of
the transfer matrix eigenvalue in (4.1) vanishes. �
10 S. Belliard and N.A. Slavnov
Setting v̄ = ū in (4.3) we obtain an expression for square of the norm of on-shell modified
Bethe vector2. Another possibility is to set v̄ = ū in (4.1). Then one should resolve singularities
in the diagonal elements of (4.2). The result is described by the following theorem.
Theorem 4.4. The square of the norm of on-shell modified Bethe vector has the following
determinant representation:
SNν (ū, ū) =
(µβ)N∆(ū)∆′(ū)λ2
2(ū)
(βκ+ κ̃− 2ρ1)N
K
(−1/β)
N,N (ū|θ̄) det
N
(
c
∂Y(uk|ū)
∂uj
)
,
where Y(z|ū) is given by (2.16).
Proof. It suffices to substitute vj = uj in (4.1) and compare the result with the partial deriva-
tives c∂Y(uk|ū)/∂uj . �
4.2 Alternative form of the Bethe equations
The initial form of Bethe equations (2.14) is not always convenient for applications. There
exists, however, various alternative forms of these equations. Two of them are described by the
following proposition.
Proposition 4.5. Inhomogeneous Bethe equations (2.14) can be written in the forms
κ− ρ2
f(ū, θj)
+ (κ̃− ρ1)
N∑
k=1
f(θk, θ̄k)
h(θk, θj)
f(ū, θk) = κ+ κ̃, j = 1, . . . , N, (4.6)
or
(κ̃− ρ1)f(ū, θj) + (κ− ρ2)
N∑
k=1
f(θ̄k, θk)
h(θj , θk)
1
f(ū, θk)
= κ+ κ̃, j = 1, . . . , N. (4.7)
The proof of this theorem and other forms of Bethe equations are given in Section 5.
4.3 On-shell modified Izergin determinant
Determinant representations for the scalar product (4.1) contains the modified Izergin deter-
minant K
(z)
N,N (ū|θ̄), where the set ū solves Bethe equations (2.14). We call it on-shell modified
Izergin determinant. This determinant enjoys an identity described by the following proposition.
Proposition 4.6. Let a set ū consist of the roots of Bethe equations (2.14). Then
K
(z)
N,N (ū|θ̄) =
N∏
i=1
(di − z), (4.8)
for arbitrary complex z. Here di = d± for i = 1, . . . , N , and
d± =
κ+ κ̃±
√
(κ+ κ̃)2 − 4(κ− ρ2)(κ̃− ρ1)
2(κ̃− ρ1)
. (4.9)
The proof of this proposition is given in Section 6.
Remark 4.7. Similar property of the on-shell modified Izergin determinant holds for the XXX
spin chain with diagonal boundary condition (see (6.7)).
2Traditionally SNν (ū, ū) = CN (ū)BN (ū) is called the square of the norm even if CN (ū) 6=
(
BN (ū)
)†
.
Scalar products in twisted XXX spin chain. Determinant Representation 11
5 Proof of the alternative forms of Bethe equations
In this section we prove Proposition 4.5.
Proof. Let us divide Bethe equations (2.14) by the product λ1(uj)λ2(uj):
(κ− ρ2)
f(uj , ūj)
λ1(uj)
− (κ̃− ρ1)
f(ūj , uj)
λ2(uj)
+ (ρ1 + ρ2)g(uj , ūj) = 0, j = 1, . . . , N.
Multiplying each of equations by a monic polynomial Pn(uj) in uj of degree n < N and taking
the sum over j we obtain
N∑
j=1
(
(κ− ρ2)f(uj , ūj)
λ1(uj)
− (κ̃− ρ1)f(ūj , uj)
λ2(uj)
+ (ρ1 + ρ2)g(uj , ūj)
)
Pn(uj) = 0.
The sums over j can be written is a contour integral
1
2πic
∮
Γ(ū)
(
(κ− ρ2)f(z, ū)
λ1(z)
+
(κ̃− ρ1)f(ū, z)
λ2(z)
+ (ρ1 + ρ2)g(z, ū)
)
Pn(z) dz = 0.
Here anticlockwise oriented contour Γ(ū) surrounds the roots of Bethe equations ū and does not
contain any other singularities of the integrand. Substituting here explicit expressions for λ1(z)
and λ2(z) (2.13) we obtain
1
2πi
∮
Γ(ū)
(
(κ− ρ2)f(z, ū)
h(z, θ̄)
+ (κ̃− ρ1)f(ū, z)g(z, θ̄) + (ρ1 + ρ2)g(z, ū)
)
Pn(z) dz = 0.
The integral can now be taken by the residues outside the integration contour, that is in the
points z = θk and z = θk − c, k = 1, . . . , N . Taking into account that n < N we find
N∑
k=1
(
(κ− ρ2)
Pn(θk − c)
f(ū, θk)
+ (κ̃− ρ1)f(ū, θk)Pn(θk)
)
g(θk, θ̄k) = cN−1(κ+ κ̃)δn,N−1, (5.1)
where we used (2.11).
This is the most general form of Bethe equations. It involves an arbitrary polynomial Pn(θk)
of degree n < N . Now we can consider several particular cases.
Let us take a system of polynomials
P
(j)
N−1(z) = cN−1h(z, θ̄j) =
N∏
k=1
k 6=j
(z − θk + c), j = 1, . . . , N. (5.2)
Then equation (5.1) yields
N∑
k=1
(
(κ− ρ2)
h(θk − c, θ̄j)
f(ū, θk)
+ (κ̃− ρ1)f(ū, θk)h(θk, θ̄j)
)
g(θk, θ̄k) = κ+ κ̃, j = 1, . . . , N.
Using h(θk − c, θ̄j) = δjk/g(θk, θ̄k) we arrive at (4.6).
Similarly, setting in (5.1)
P
(j)
N−1(z) =
cN−1
g(z, θ̄j)
=
N∏
k=1
k 6=j
(z − θk), j = 1, . . . , N, (5.3)
we arrive at (4.7). �
12 S. Belliard and N.A. Slavnov
One can also consider a choice of P
(j)
N−1(z) that interpolates between (5.2) and (5.3). Let θ̄A
and θ̄B be two fixed disjoint subsets of the inhomogeneities such that {θ̄A, θ̄B} = θ̄. Assume
that θ̄B 6= ∅. Let
P
(j)
N−1(z) = cN−1g(z, θj)
h(z, θ̄A)
g(z, θ̄B)
, θj ∈ θ̄B. (5.4)
Then equation (5.1) takes the form
(κ− ρ2)
∑
{θ̄B1
,θ̄B2
}`θ̄B
#θ̄B1
=1
f(θ̄B2 , θ̄B1)
h(θj , θ̄B1)
1
f(ū, θ̄B1)
= κ+ κ̃− (κ̃− ρ1)f(ū, θj)f(θj , θ̄A)
+ (κ̃− ρ1)
∑
{θ̄A1
,θ̄A2
}`θ̄A
#θ̄A1
=1
g(θj , θ̄A1)f(ū, θ̄A1)f(θ̄A1 , θ̄A2).
Here in the l.h.s. the sum is taken over partitions {θ̄B1 , θ̄B2} ` θ̄B such that #θ̄B1 = 1. In the
r.h.s. the sum is taken over partitions {θ̄A1 , θ̄A2} ` θ̄A such that #θ̄A1 = 1.
Remark 5.1. The examples given above do not exhaust all possible forms of the Bethe equa-
tions. In particular, on the basis of the polynomial (5.4) we can construct
P
(j)
N−1(z) = cN−1
∑
{θ̄A,θ̄B}`θ̄
g(z, θj)
h(z, θ̄A)
g(z, θ̄B)
G(θ̄A, θ̄B), θj ∈ θ̄B, (5.5)
where G(θ̄A, θ̄B) is some function that depends on the subsets θ̄A and θ̄B. An example of the
polynomial (5.5) is considered in Appendix D.2.
6 Proof of the on-shell modified Izergin determinant properties
In this section we prove Proposition 4.6.
Proof. Let us introduce two N ×N matrices Ω(θ̄) and Z:
Ωjk(θ̄) =
f(θk, θ̄k)
h(θk, θj)
, Zjk = Ωjk(θ̄)f(ū, θk), j, k = 1, . . . , N.
Then the system of equations (4.6) takes the form
κ− ρ2
f(ū, θj)
+ (κ̃− ρ1)
N∑
k=1
Zjk = κ+ κ̃, j = 1, . . . , N.
Multiplying each equation by Zij we obtain
(κ− ρ2)Ωij(θ̄) + (κ̃− ρ1)
N∑
k=1
ZijZjk = (κ+ κ̃)Zij , i, j = 1, . . . , N.
Taking the sum over j and using3
N∑
j=1
Ωij(θ̄) = 1, (6.1)
3Equation (6.1) can be derived by the contour integral method described in Appendix B.1.
Scalar products in twisted XXX spin chain. Determinant Representation 13
we arrive at
N∑
k=1
(
(κ− ρ2)δik + (κ̃− ρ1)
(
Z2
)
ik
− (κ+ κ̃)Zik
)
= 0, i = 1, . . . , N.
Setting here Z = S−1DS, where D = diag(d1, . . . , dN ), we find
N∑
k=1
(
(κ− ρ2)δik + (κ̃− ρ1)
(
S−1D2S
)
ik
− (κ+ κ̃)
(
S−1DS
)
ik
)
= 0, i = 1, . . . , N.
Multiplying this equation from the left by S we arrive at
N∑
k=1
(
(κ− ρ2) + (κ̃− ρ1)d2
i − (κ+ κ̃)di
)
Sik = 0, i = 1, . . . , N. (6.2)
Since S is invertible, we conclude that linear combination (6.2) vanishes if and only if
(κ̃− ρ1)d2
i − (κ+ κ̃)di + κ− ρ2 = 0, i = 1, . . . , N. (6.3)
Thus, the matrix Z has only two eigenvalues d+ and d− (4.9). It also follows from the
above consideration that the on-shell modified Izergin determinant K
(z)
N,N (ū|θ̄) has the following
presentation:
K
(z)
N,N (ū|θ̄) = det
N
(Zjk − zδjk) =
N∏
i=1
(di − z), (6.4)
for arbitrary complex z, provided the parameters ū are on-shell. Thus, (4.8) is proved. �
In particular, setting z = 0 in (6.4) we obtain
f(ū, θ̄) =
N∏
i=1
di. (6.5)
Using equation (6.3) one can find identities for the on-shell modified Izergin determinants
with different z. For example, (6.3) yields
β(κ− ρ2)
βκ+ κ̃− 2ρ1
(di − 1)
(
di +
1
β
)
= di
(
di − µ−
µ− 1
β
)
, i = 1, . . . , N.
Taking the product over i we obtain(
β(κ− ρ2)
βκ+ κ̃− 2ρ1
)N N∏
i=1
(di − 1)
(
di +
1
β
)
=
N∏
i=1
di
(
di − µ−
µ− 1
β
)
.
Then, due to equations (6.4), (6.5) we find(
β(κ− ρ2)
βκ+ κ̃− 2ρ1
)N
K
(1)
N,N (ū|θ̄)K(−1/β)
N,N (ū|θ̄) = f(ū, θ̄)K
(µ+(µ−1)/β)
N,N (ū|θ̄). (6.6)
Concluding this section we would like to mention that exactly the same method can be used
to transform Bethe equations of the XXX spin chain with diagonal boundary condition
κ
f(uj , ūj)
λ1(uj)
− κ̃f(ūj , uj)
λ2(uj)
= 0, j = 1, . . . ,M,
14 S. Belliard and N.A. Slavnov
where #ū = M ≤ N . Similarly to (4.8) we then obtain
K
(z)
M,N (ū|θ̄) =
N∏
i=1
(di − z), (6.7)
where di ∈ {1, κ/κ̃}. Using definition of the modified Izergin determinant (3.2) we can write
down (6.7) as
(1− z)N−M det
M
(
δjkf(uj , θ̄)− z
f(uj , ūj)
h(uj , uk)
)
=
N∏
i=1
(di − z).
7 Calculation of the scalar product
In this section we prove Theorem 4.1. First, we show the equivalence of representations (4.3)
and (4.1). Then we prove (4.3) starting with equation (3.6) and specifying it for the case
#ū = #v̄ = N
SNν (v̄, ū) = µ2N
∑
{v̄I,v̄II}`v̄
{ūI,ūII}`ū
βpII−qII1 βpI−qI2 λ2(v̄I)λ1(v̄II)λ2(ūII)λ1(ūI)f(v̄I, v̄II)f(ūII, ūI)
×K(1/µ)
qII,pII
(ūII|v̄II)K
(1/µ)
qI,pI
(ūI|v̄I). (7.1)
Recall that here the sum is taken over all possible partitions {v̄I, v̄II} ` v̄ and {ūI, ūII} ` ū without
any restriction on the cardinalities of the subsets.
We have pointed out in Section 3.2 that the function SN (v̄|ū) is a polynomial in v̄ of degree N
in each vj . Since this polynomial is symmetric over v̄, it has
(
2N
N
)
independent coefficients. Hence,
in order to find this polynomial, it is enough to compute it in
(
2N
N
)
points.
Let us consider an arbitrary partition of the inhomogeneities θ̄ = {θ̄A, θ̄B} with #θ̄A = nA
and #θ̄B = nB. Suppose that v̄ = {θ̄A, θ̄B − c}. Clearly, there exists exactly
(
2N
N
)
choices of
this type. Hence, if we compute the scalar product for all possible v̄ = {θ̄A, θ̄B − c}, then the
polynomial is completely determined for an arbitrary set v̄. This will be done in Sections 7.2
and 7.3.
7.1 Transformation of Jacobian
In this subsection we reduce the Jacobian of the transfer matrix eigenvalue to the modified
Izergin determinant. We thus prove Theorem 4.2.
Proof. Assume that representation (4.1) holds. Let
M(uj , vk) =
c
g(vk, ū)λ2(vk)
∂Λ(vk|ū)
∂uj
,
where Λ(vk|ū) is given by (2.15). Then
M(uj , vk) = (−1)N−1(κ̃− ρ1)f(vk, θ̄)h(ū, vk)
g(uj , vk)
h(uj , vk)
+ (κ− ρ2)h(vk, ū)
g(vk, uj)
h(vk, uj)
+ (ρ1 + ρ2)h(vk, θ̄)g(vk, uj). (7.2)
Scalar products in twisted XXX spin chain. Determinant Representation 15
Recall that here the set v̄ consists of arbitrary complex numbers, while the set ū solves Bethe
equations. Therefore, in particular, the matrix elements (7.2) have no poles at vk = uj :
ResM(uj , vk)
∣∣∣
vk=uj
= 0.
Indeed, it is easy to see that this residue is proportional to the Bethe equations (2.14).
Obviously, for any non-degenerated N ×N matrix A, one has
det
N
M =
detN H
detN A
, where H = MA.
Let
A(vj , θk) =
g(vj , v̄j)
h(vj , θk)
.
Then the determinant of this matrix is proportional to the Cauchy determinant
det
N
A(vj , θk) =
∆(v̄)
∆(θ̄)h(v̄, θ̄)
,
and we obtain
det
N
M =
∆(θ̄)h(v̄, θ̄)
∆(v̄)
det
N
H.
Let us compute explicitly the entries Hjl =
N∑
k=1
M(uj , vk)A(vk, θl). Consider an auxiliary
contour integral
I =
1
2πic
∮
|z|=R→∞
g(z, v̄)
h(z, θl)
M(uj , z) dz. (7.3)
It is easy to see that the integrand behaves as z−2 as z →∞. Thus, I = 0. On the other hand,
taking the sum of the residues within the integration contour we obtain
I = Hjl + (−1)N
M(uj , θl − c)
h(v̄, θl)
+
1
c
N∑
k=1
g(θk, v̄)
h(θk, θl)
ResM(uj , z)
∣∣∣
z=θk
.
Substituting here explicit representation (7.2) for M(uj , vk) we find
(−1)N−1Hjl =
(κ− ρ2)g(uj , θl)
g(θl, ū)h(v̄, θl)h(uj , θl)
− (κ̃− ρ1)
N∑
k=1
g(θk, v̄)
h(θk, θl)
f(θk, θ̄k)h(ū, θk)
g(uj , θk)
h(uj , θk)
=
N∑
k=1
(κ− ρ2)g(uj , θk)
g(θk, ū)h(v̄, θk)h(uj , θk)
(
δkl −
κ̃− ρ1
κ− ρ2
f(θk, θ̄k)
h(θk, θl)
f(ū, θk)f(v̄, θk)
)
.
Thus
det
N
H = det
N
H(1) det
N
H(2),
where
H
(1)
jk =
(κ− ρ2)
g(θk, ū)h(v̄, θk)
g(uj , θk)
h(uj , θk)
,
16 S. Belliard and N.A. Slavnov
and
H
(2)
jk = δjk −
κ̃− ρ1
κ− ρ2
f(θj , θ̄j)
h(θj , θk)
f(ū, θj)f(v̄, θj).
It is easy to see that det
N
H(1) is proportional to the ordinary Izergin determinant (3.4)
det
N
H(1) =
(κ− ρ2)N
g(θ̄, ū)h(v̄, θ̄)
det
N
(
g(uj , θk)
h(uj , θk)
)
=
(−1)N (κ− ρ2)N
h(v̄, θ̄)f(ū, θ̄)∆′(ū)∆(θ̄)
K
(1)
N,N (ū|θ̄),
while det
N
H(2) is proportional to the modified Izergin determinant
det
N
H(2) = (−1)N
(
κ̃− ρ1
κ− ρ2
)N
K
(α)
2N,N ({ū, v̄}|θ̄), where α =
κ− ρ2
κ̃− ρ1
.
Thus, we finally obtain
det
N
M =
(κ̃− ρ1)N
f(ū, θ̄)∆′(ū)∆(v̄)
K
(1)
N,N (ū|θ̄)K(α)
2N,N ({ū, v̄}|θ̄). (7.4)
We would like to stress that we essentially used the fact that ū satisfies Bethe equations (2.14).
Therefore, M(uj , vk) has no pole at vk = uj . Otherwise the contour integral (7.3) would have
an additional contribution. Thus, equation (7.4) holds only on-shell, that is ū satisfies Bethe
equations (2.14).
Substituting (7.4) into (4.1) we obtain
SNν (v̄, ū) =
(
µβ(κ̃− ρ1)
βκ+ κ̃− 2ρ1
)N λ2(ū)λ2(v̄)
f(ū, θ̄)
K
(−1/β)
N,N (ū|θ̄)K(1)
N,N (ū|θ̄)K(α)
2N,N ({ū, v̄}|θ̄).
Using identity (6.6) we immediately arrive at (4.3). Thus, representations (4.1) and (4.3) are
equivalent. �
7.2 Transformation of the scalar product for generic ū
We now turn back to equation (7.1). Using equation (A.1) and λ1(uj)/λ2(uj) = f(uj , θ̄) we
obtain
SNν (v̄, ū) = µ2Nλ2(ū)
∑
{v̄I,v̄II}`v̄
{ūI,ūII}`ū
βpII−qII1 βpI−qI2
(
1− 1
µ
)pI−qI
λ2(v̄I)λ1(v̄II)f(ūI, θ̄)
× f(v̄I, v̄II)f(ūII, ūI)K
(1/µ)
qII,pII
(ūII|v̄II)K(1/µ)
pI,qI
(v̄I|ūI).
Now we set v̄ = {θ̄A, θ̄B − c} and denote the corresponding scalar product by SAB. Since
λ2(θj) = 0 and λ1(θj − c) = 0 for all j = 1, . . . , N , we see that the sum over partitions
{v̄I, v̄II} ` v̄ is frozen. The only non-vanishing contribution occurs for v̄I = θ̄B − c and v̄II = θ̄A.
Then we have
SAB = µ2Nλ2(ū)GAB
∑
{ūI,ūII}`ū
βnA−qII
(
1− 1
µ
)nB−qI
f(ūI, θ̄)f(ūII, ūI)
×K(1/µ)
qII,nA
(ūII|θ̄A)K(1/µ)
nB ,qI
(θ̄B − c|ūI), (7.5)
Scalar products in twisted XXX spin chain. Determinant Representation 17
where we used nA + nB = qI + qII = N , and
GAB =
λ2(θ̄B − c)λ1(θ̄A)
f(θ̄A, θ̄B)
= (−1)NnB
h(θ̄, θ̄B)h(θ̄A, θ̄)
f(θ̄A, θ̄B)
. (7.6)
Recall also that β = β1/β2. Our goal now is to calculate the sum over partitions of the set ū.
Using (A.5) we rewrite (7.5) as follows:
SAB = µ2Nλ2(ū)GAB
∑
{ūI,ūII}`ū
βnA−qII
(
− 1
µ
)nB
f(ūI, θ̄A)f(ūII, ūI)
×K(1/µ)
qII,nA
(ūII|θ̄A)K(µ)
qI,nB
(ūI|θ̄B). (7.7)
The next step is to use the second equation (A.4) for both modified Izergin determinants in (7.7):
K(1/µ)
qII,nA
(ūII|θ̄A) =
(
1− 1
µ
)nA−q2−q4 ∑
{ū2,ū4}`ūII
(
− 1
µ
)q2
f(ū4, θ̄A)f(ū2, ū4),
K(µ)
qI,nB
(ūI|θ̄B) = (1− µ)nB−q1−q3
∑
{ū1,ū3}`ūI
(−µ)q1f(ū3, θ̄B)f(ū1, ū3).
Here we use Arabic numeration of the subsets, in order to avoid too cumbersome Roman nu-
merals. We also set qj = #ūj . Substituting these equations into (7.7) we obtain
SAB = µ2Nλ2(ū)GAB
∑
{ū1,ū2,ū3,ū4}`ū
βnA−q2−q4
(
− 1
µ
)N−q1−q4
f(ū3, θ̄B)f(ū1, θ̄A)
× f(ū3, θ̄A)f(ū4, θ̄A)f(ū2, ū1)f(ū2, ū3)f(ū4, ū1)f(ū4, ū3)f(ū2, ū4)f(ū1, ū3). (7.8)
Here the sum is taken over partitions of the set ū into four subsets {ū1, ū2, ū3, ū4} ` ū.
Let ū0 = {ū1, ū4}. Then (7.8) transforms as follows:
SAB = µ2Nλ2(ū)GAB
∑
{ū0,ū2,ū3}`ū
βnA−q2
(
− 1
µ
)N−q0
f(ū0, θ̄A)
× f(ū3, θ̄)f(ū2, ū0)f(ū2, ū3)f(ū0, ū3)
∑
{ū1,ū4}`ū0
β−q4f(ū4, ū1),
where q0 = #ū0. The sum over partitions is now organized into two steps. First, the set ū
is divided into three subsets {ū0, ū2, ū3} ` ū, and then the subset ū0 is divided once more as
{ū1, ū4} ` ū0. Obviously, the last sum over partitions of the subset ū0 gives (1+β−1)q0 (see [7]),
and we arrive at
SAB = (−µ)Nλ2(ū)GAB
∑
{ū0,ū2,ū3}`ū
(−µ)q0(1 + β)q0βnA−q2−q0f(ū0, θ̄A)f(ū3, θ̄)
× f(ū2, ū0)f(ū2, ū3)f(ū0, ū3). (7.9)
Now we combine the subsets ū0 and ū2 into one subset. Let ū1 = {ū0, ū2} with #ū1 = q1.
Then equation (7.9) takes the form
SAB = (−µ)Nλ2(ū)GAB
∑
{ū1,ū3}`ū
βnA−q1f(ū3, θ̄)f(ū1, ū3)
×
∑
{ū0,ū2}`ū1
(−µ)q0(1 + β)q0f(ū0, θ̄A)f(ū2, ū0).
18 S. Belliard and N.A. Slavnov
The sum over partitions {ū0, ū2} ` ū1 gives a new modified Izergin determinant due to the
second equation (A.4), and we find
SAB = (−µ)Nλ2(ū)GAB
∑
{ū1,ū3}`ū
f(ū3, θ̄)f(ū1, ū3)
× βnA−q1(−µ− µβ)nA
(1− µ− µβ)nA−q1
K(1/(µ+µβ))
q1,nA
(ū1|θ̄A).
We now develop the modified Izergin determinant K
(1/(µ+µβ))
q1,nA (ū1|θ̄A) via the first equa-
tion (A.4), as a sum over partitions {θ̄A1 , θ̄A2} ` θ̄A:
SAB = (−µ)Nλ2(ū)GAB
(µ+ µβ)nAβnA
(µ+ µβ − 1)nA
∑
{θ̄A1
,θ̄A2
}`θ̄A
∑
{ū1,ū3}`ū
(
1− µ
β
− µ
)q1
f(ū3, θ̄)
×
(
− 1
µ+ µβ
)nA2
f(ū1, ū3)f(ū1, θ̄A1)f(θ̄A1 , θ̄A2), (7.10)
where nA1 = #θ̄A1 , nA2 = #θ̄A2 = nA − nA1 . Let {θ̄B, θ̄A2} = θ̄C . Then (7.10) can be written
as follows:
SAB = (−µ)Nλ2(ū)GAB
(µ+ µβ)nAβnA
(µ+ µβ − 1)nA
∑
{θ̄A1
,θ̄A2
}`θ̄A
(
− 1
µ+ µβ
)nA2
f(ū, θ̄A1)f(θ̄A1 , θ̄A2)
×
∑
{ū1,ū3}`ū
(
1− µ
β
− µ
)q1
f(ū3, θ̄C)f(ū1, ū3). (7.11)
The sum over partitions {ū1, ū3} ` ū gives again the modified Izergin determinant due to the
second equation (A.4):
∑
{ū1,ū3}`ū
(
1− µ
β
− µ
)q1
f(ū3, θ̄C)f(ū1, ū3)
=
(
1− µ
β
− µ
)N−nB−nA2
K
(µ+(µ−1)/β)
N,N−nA1
(ū|{θ̄ \ θ̄A1}).
Substituting this into (7.11) we finally arrive at
SAB =
(−µ)Nλ2(ū)GABβ
nA
(1− µ− µβ)nA
∑
{θ̄A1
,θ̄A2
}`θ̄A
ξnA1f(ū, θ̄A1)f(θ̄A1 , θ̄A2)
×K(µ+(µ−1)/β)
N,N−nA1
(ū|{θ̄ \ θ̄A1}), (7.12)
where
ξ =
µ
β
(β + 1)2(µ− 1). (7.13)
Thus, we computed the sum over partitions of the set ū. However, instead of the original
partitions, we obtained a new sum over partitions of the inhomogeneities θ̄.
Scalar products in twisted XXX spin chain. Determinant Representation 19
7.3 Further transformation of the scalar product for ū on-shell
Observe that up to now the parameters ū were arbitrary complex numbers. Now we require
them to be on-shell. Then due to Theorem C.1 we have
ξnA1f(ū, θ̄A1)K
(µ+(µ−1)/β)
N,N−nA1
(ū|{θ̄ \ θ̄A1}) = ηnA1K
(µ+(µ−1)/β)
N,N (ū|θ̄)K(1/η)
N,nA1
(ū|θ̄A1),
where η =
(µ−1
β + µ
) κ̃−ρ1
κ−ρ2 (see (C.1)). Then (7.12) takes the form
SAB =
(−µ)Nλ2(ū)GABβ
nA
(1− µ− µβ)nA
K
(µ+(µ−1)/β)
N,N (ū|θ̄)
×
∑
{θ̄A1
,θ̄A2
}`θ̄A
ηnA1f(θ̄A1 , θ̄A2)K
(1/η)
N,nA1
(ū|θ̄A1). (7.14)
The remaining sum over partitions now can be computed via (A.8):∑
{θ̄A1
,θ̄A2
}`θ̄A
ηnA1f(θ̄A1 , θ̄A2)K
(1/η)
N,nA1
(ū|θ̄A1) = ηnAK
(0)
N,nA
(ū|θ̄A) = ηnAf(ū, θ̄A).
Substituting this into (7.14) we immediately arrive at
SAB = (−1)nBµNα−nAλ2(ū)GABf(ū, θ̄A)K
(µ+(µ−1)/β)
N,N (ū|θ̄), (7.15)
where α is given by (4.4).
It remains to compare this result with representation (4.3) at v̄ = {θ̄A, θ̄B − c}. For this we
should calculate λ2(v̄)K
(α)
2N,N ({ū, v̄}|θ̄) for given values of v̄. Replacing θ̄A with θ̄A + ε we obtain
lim
ε→0
λ2(θ̄A + ε)λ2(θ̄B − c)K(α)
2N,N ({ū, θ̄A + ε, θ̄B − c}|θ̄)
= (−α)nB lim
ε→0
(−1)NnBh(θ̄, θ̄B)
g(θ̄A, θ̄A + ε)g(θ̄A, θ̄B)
K
(α)
2N,N ({ū, θ̄A + ε}|θ̄A),
where we used representation (2.13) for the function λ2(u) and reduction property (A.3). Using
now (A.7) we find
lim
ε→0
λ2(θ̄A + ε)λ2(θ̄B − c)K(α)
2N,N ({ū, θ̄A + ε, θ̄B − c}|θ̄)
= (−α)nB
(−1)NnBh(θ̄, θ̄B)h(θ̄A, θ̄A)
g(θ̄A, θ̄B)
f(ū, θ̄A) = (−α)nBGABf(ū, θ̄A),
where GAB is given by (7.6). Substituting this expression for λ2(v̄)K
(α)
2N,N ({ū, v̄}|θ̄) at v̄ =
{θ̄A, θ̄B − c} into (4.3) we arrive at (7.15).
Thus, for an arbitrary partition θ̄ = {θ̄A, θ̄B}, the values of the polynomial SNν (v̄, ū) at
v̄ = {θ̄A, θ̄B − c} are given by equation (4.3). As we discussed above, this means that SNν (v̄, ū)
is given by (4.3) for arbitrary complex v̄.
8 Conclusion
In this paper we proved the determinant representation for the OFS-ONS scalar product conjec-
tured in [6] within the framework of the MABA. Similarly to the known determinant formulas,
this representation contains the Jacobian of the transfer matrix eigenvalue. However, due to the
peculiarities of the MABA this Jacobian can be reduced to the modified Izergin determinant.
20 S. Belliard and N.A. Slavnov
This possibility arises due to the fact that the number of the Bethe parameters in the modified
on-shell Bethe vector coincides with the number of sites of the chain.
In spite of representation (4.1) formally looks very similar to the one obtained in [24, 31] for
the XXX chain with periodic boundary conditions, the proof is very different. Recall that the
determinant formula [31] for the ABA solvable models holds for the most general case when
the Hilbert space of the monodromy matrix entries is not specified. In particular, it can be
infinite-dimensional. On the contrary, the proof given in this paper is essentially based on the
fact that we work with a finite-dimensional representation of the RTT algebra. In its turn, this
yields a set of very specific properties of the on-shell Izergin determinant. The latter also is an
essential part of our proof.
All this does not mean, however, that there is no other proof of the representation (4.1),
which would be more general and would not rely on the properties of a particular model. The
are planning to publish such a proof in our forthcoming publication.
A Properties of the modified Izergin determinant
In this section we give a list of properties of the modified Izergin determinant introduced in
Section 3. The proofs can be found in [9]. In all the formulas listed below ū, v̄, and z are
arbitrary complex numbers such that #ū = n and #v̄ = m.
We begin with a direct relation between modified Izergin determinant and its conjugated:
K
(z)
n,m(ū|v̄) = (1− z)m−nK(z)
m,n(v̄|ū), K(z)
n,m(−ū| − v̄) = K
(z)
n,m(ū|v̄). (A.1)
Due to this relation all other properties are given for K
(z)
n,m(ū|v̄) only.
• For arbitrary complex w
K(z)
n,m(ū− w|v̄) = K(z)
n,m(ū|v̄ + w).
• The modified Izergin determinant has the following initial conditions:
K
(z)
n,0(ū|∅) = 1, K
(z)
0,n(∅|v̄) = (1− z)n. (A.2)
• If one of the arguments goes to infinity, then
K(z)
n,m(ū|v̄)
∣∣∣
uk→∞
= K
(z)
n−1,m(ūk|v̄),
K(z)
n,m(ū|v̄)
∣∣∣
vk→∞
= (1− z)K(z)
n,m−1(ū|v̄k).
• For arbitrary complex w
K
(z)
n+1,m+1({ū, w − c}|{v̄, w}) = −zK(z)
n,m(ū|v̄). (A.3)
• The modified Izergin determinant has the following representations as sums over partitions
K(z)
n,m(ū|v̄) =
∑
{v̄I,v̄II}`v̄
(−z)#v̄IIf(ū, v̄I)f(v̄I, v̄II),
K(z)
n,m(ū|v̄) = (1− z)m−n
∑
{ūI,ūII}`ū
(−z)#ūIf(ūII, v̄)f(ūI, ūII). (A.4)
Here the sum is taken over all partitions {v̄I, v̄II} ` v̄ in the first equation, while in the
second equation, the sum is taken over all partitions {ūI, ūII} ` ū.
Scalar products in twisted XXX spin chain. Determinant Representation 21
• The order of the sets ū and v̄ can be changed via
K(z)
n,m(ū|v̄ + c) =
(−z)n(1− z)m−n
f(v̄, ū)
K(1/z)
m,n (v̄|ū). (A.5)
• The modified Izergin determinant has simple poles at uj = vk. The residue at un = vm is
given by
K(z)
n,m(ū|v̄)
∣∣∣
un→vm
= g(un, vm)f(ūn, un)f(vm, v̄m)K
(z)
n−1,m−1(ūn|v̄m) + reg, (A.6)
where reg means regular part. It follows from (A.6) that if #w̄ = #w̄′ = l, then
lim
w̄′→w̄
K
(z)
n+l,m+l(ū|v̄)
f(w̄′, w̄)
= f(ū, w̄)f(w̄, v̄)K(z)
n,m(ū|v̄). (A.7)
• Let ū and v̄ be sets of arbitrary complex numbers such that #ū = n and #v̄ = m. Then∑
{v̄I,v̄II}`v̄
zlII1 K
(z2)
n,lI
(ū|v̄I)f(v̄I, v̄II) = K(z2−z1)
n,m (ū|v̄). (A.8)
Here lII = #v̄II. The sum is taken with respect to all partitions {v̄I, v̄II} ` v̄. There is no
any restriction on the cardinalities of the subsets.
B Identities for rational functions
B.1 Sums of rational functions
To prove (4.5) we use the following summation formulas:
N∑
j=1
g(uj , vk)
h(uj , vk)
γj =
h(v̄, vk)
h(ū, vk)
,
N∑
j=1
g(vk, uj)
h(vk, uj)
γj = −h(vk, v̄)
h(vk, ū)
,
N∑
j=1
g(vk, uj)γj = −1. (B.1)
All these formulas can be obtained by calculation of special contour integrals. For example,
consider an integral
I =
1
2πic
∮
|z|=R→∞
g(z, vk)
h(z, vk)
g(z, ū)
g(z, v̄)
dz
=
1
2πi
∮
|z|=R→∞
cdz
(z − vk)(z − vk + c)
N∏
i=1
z − vi
z − ui
.
The integration is taken over anticlockwise oriented contour around infinity. Clearly, I = 0, as
the integrand behaves as z−2 at z → ∞. On the other hand, the integral is equal to the sum
of residues within the contour. The residues at z = uj give the sum in the l.h.s. of the first
equation (B.1). One more contribution comes from the residue at z = vk − c. Thus, we obtain
I = 0 =
N∑
j=1
g(uj , vk)
h(uj , vk)
γj −
N∏
i=1
vk − vi − c
vk − ui − c
.
22 S. Belliard and N.A. Slavnov
This immediately implies the first identity (B.1). Other identities can be proved exactly in the
same manner.
Using now representation (4.2) we obtain
N∑
j=1
cγj
g(vk, ū)λ2(vk)
∂Λ(vk|ū)
∂uj
= (−1)N−1(κ̃− ρ1)
λ1(vk)
λ2(vk)
h(v̄, vk)
− (κ− ρ2)h(vk, v̄)− (ρ1 + ρ2)λ1(vk),
leading to (4.5).
B.2 Matrices with rational elements
Let x̄ = {x1, . . . , xn} and let Ω(x) be an n× n matrix with the elements
Ωjk(x̄) =
f(x̄k, xk)
h(xj , xk)
.
Let us prove that the inverse matrix has the following entries:(
Ω−1(x̄)
)
jk
=
f(xk, x̄k)
h(xk, xj)
= Ωjk(x̄)
∣∣∣
c→−c
.
For this we consider the product of these two matrices. Let
Gjk =
n∑
l=1
f(x̄l, xl)
h(xj , xl)
f(xk, x̄k)
h(xk, xl)
.
Consider an auxiliary contour integral
I =
1
2πic
∮
|z|=R→∞
f(x̄, z)
h(xj , z)
f(xk, x̄k)
h(xk, z)
dz,
where the anticlockwise oriented integration contour is around infinite point. Since the integrand
behaves as z−2 at z →∞, we have I = 0. On the other hand, this integral is equal to the sum of
the residues within the integration contour. The contribution of the poles at z = xl, l = 1, . . . , n
gives −Gjk. One more contribution comes from the pole at z = xk+c at j = k. Thus, we obtain
I = 0 = −Gjk + δjkf(xk, x̄k)f(x̄k, xk + c) = −Gjk + δjk,
where we used (2.11). Thus, Gjk = δjk, what ends the proof.
C Relations between on-shell modified Izergin determinants
Theorem C.1. Consider an arbitrary partition of inhomogeneities {θ̄A, θ̄B} ` θ̄ such that
#θ̄A = nA and θ̄B = N−nA. Let ū be on-shell, that is the set ū satisfies Bethe equations (2.14).
Then
f(ū, θ̄A)K
(µ+(µ−1)/β)
N,N−nA (ū|θ̄B) =
(
η
ξ
)nA
K
(µ+(µ−1)/β)
N,N (ū|θ̄)K(1/η)
N,nA
(ū|θ̄A),
where
η =
(
µ− 1
β
+ µ
)
κ̃− ρ1
κ− ρ2
. (C.1)
and ξ is given by (7.13).
Scalar products in twisted XXX spin chain. Determinant Representation 23
The proof of Theorem C.1 is based on two preparatory lemmas.
Lemma C.2. Let θ̄A and θ̄B be two fixed partitions of the set θ̄ such that #θ̄B = nB > 0. Let
θj ∈ θ̄B. Then for arbitrary complex γ the following identity holds:
∑
{θ̄B1
,θ̄B2
}`θ̄B
#θ̄B1
=1
f(θ̄B2 , θ̄B1)
h(θj , θ̄B1)
K
(γ)
N,nB−1(ū|θ̄B2)
=
1
γ
(
f(ū, θj)K
(γ)
N,nB−1(ū|{θ̄B \ θj})−K(γ)
N,nB
(ū|θ̄B)
)
.
Here ū and θ̄ are arbitrary complex numbers. In particular, we do not require the set ū to be
on-shell.
Lemma C.3. Let θ̄A and θ̄B be two fixed partitions of the set θ̄ such that #θ̄A = nA < N . Let
θj ∈ θ̄B and ū be on-shell. Then
η
∑
{θ̄B1
,θ̄B2
}`θ̄B
#θ̄B1
=1
f(θ̄B2 , θ̄B1)
h(θs, θ̄B1)
K
(1/η)
N,nA+1(ū|{θ̄A, θ̄B1})
f(ū, θ̄B1)
=
β
µβ + µ− 1
(
ηK
(1/η)
N,nA+1(ū|{θ̄A, θs})− ξK(1/η)
N,nA
(ū|θ̄A)
)
. (C.2)
The proofs of these lemmas respectively are given in Appendices D.1 and D.2.
Proof of Theorem C.1. The proof partly uses induction over nA. For nA = 0 the statement
of the theorem is obvious. Assume that it is valid for some nA ≥ 0:
ξnAf(ū, θ̄A)K
(µ+(µ−1)/β)
N,nB
(ū|θ̄B) = ηnAK
(µ+(µ−1)/β)
N,N (ū|θ̄)K(1/η)
N,nA
(ū|θ̄A),
Here θ̄A and θ̄B are arbitrary subsets of θ̄ with cardinalities #θ̄A = nA and #θ̄B = nB = N−nA
respectively. We assume that K
(µ+(µ−1)/β)
N,nB
(ū|θ̄B) 6= 0 for generic inhomogeneities θ̄ and generic
twist parameters.
Now we make new subsets. We fix some θ` ∈ θ̄B and consider subsets {θ̄A, θ`} and {θ̄B \ θ`}.
Let us introduce Y (θ`) as
Y (θ`) = ξnA+1f(ū, θ̄A)K
(µ+(µ−1)/β)
N,nB−1 (ū|{θ̄B \ θ`})
− ηnA+1K
(µ+(µ−1)/β)
N,N (ū|θ̄)
K
(1/η)
N,nA
(ū|{θ̄A, θ`})
f(ū, θ`)
. (C.3)
Obviously, if we prove that Y (θ`) = 0, then we prove Theorem C.1 for n = nA + 1.
Let Ω be an nB × nB matrix with the entries
Ωjk =
f(θ̄Bk , θ
B
k )
h(θBj , θ
B
k )
.
Here we temporary used a superscript for the subset θ̄B = θ̄B. Respectively, θBk is k-th element
of the subset θ̄B. We also used θ̄Bk = θ̄B \ θBk . It can be easily checked (see Appendix B.2) that
(
Ω−1
)
jk
=
f(θBk , θ̄
B
k )
h(θBk , θj)
.
24 S. Belliard and N.A. Slavnov
Let us compute the action of Ω on the vector Y (C.3). Obviously
nB∑
`=1
Ωj`Y (θ`) =
∑
{θ̄B1
,θ̄B2
}`θ̄B
#θ̄B1
=1
f(θ̄B2 , θ̄B1)
h(θj , θ̄B1)
Y (θ̄B1).
Thus, we can use the results of Lemmas C.2 and C.3. Simple straightforward calculation gives
nB∑
`=1
Ωj`Y (θ`) =
βf(ū, θj)
µβ + µ− 1
Y (θj)−
βξ
µβ + µ− 1
×
{
ξnAf(ū, θ̄A)K
(µ+(µ−1)/β)
N,nB
(ū|θ̄B)− ηnAK(µ+(µ−1)/β)
N,N (ū|θ̄)K(1/η)
N,nA
(ū|θ̄A)
}
.
The term in the second line vanishes due to the induction assumption. Hence,
nB∑
`=1
Ωj`Y (θ`) =
βf(ū, θj)
µβ + µ− 1
Y (θj).
or equivalently,
nB∑
`=1
((
µ+
µ− 1
β
)
Ωj` − δj`f(ū, θj)
)
Y (θ`) = 0.
Multiplying this equation from the left by
(
Ω−1
)
ij
and taking the sum over θj ∈ θ̄B we arrive at
nB∑
`=1
(
f(ū, θi)
(
Ω−1
)
i`
−
(
µ+
µ− 1
β
)
δi`
)
Y (θ`) = 0.
It is easy to see that
det
(
f(ū, θi)
(
Ω−1
)
i`
−
(
µ+
µ− 1
β
)
δi`
)
= K
(µ+(µ−1)/β)
N,nB
(ū|θ̄B).
Due to the induction assumption K
(µ+(µ−1)/β)
N,nB
(ū|θ̄B) 6= 0. Then Y (θ`) = 0, and hence, Theo-
rem C.1 is proved for n = nA + 1. �
D Proof of preparatory lemmas
D.1 Proof of Lemma C.2
Let
F =
∑
{θ̄B1
,θ̄B2
}`θ̄B
#θ̄B1
=1
f(θ̄B2 , θ̄B1)
h(θj , θ̄B1)
K
(γ)
N,nB−1(ū|θ̄B2).
Consider an auxiliary contour integral
I =
1
2πic
∮
|z|=R→∞
f(θ̄B, z)
h(θj , z)
K
(γ)
N+1,nB
({ū, z − c}|θ̄B).
Scalar products in twisted XXX spin chain. Determinant Representation 25
On the one hand, this integral is equal to the residue at infinity:
I = −K(γ)
N,N−1(ū|θ̄B). (D.1)
On the other hand, this integral is equal to the sum of residues within the contour. The sum in
the points z = θk ∈ θ̄B gives γF (due to (A.3)). One more pole occurs at z = θj +c. Using (A.6)
we find
I = γF − f(ū, θj)K
(γ)
N,nB−1(ū|{θ̄B \ θj}). (D.2)
Comparing (D.2) and (D.1) we arrive at the statement of the lemma.
D.2 Proof of Lemma C.3
Let θ̄B1 = θl in (C.2). Consider K
(1/η)
N,nA+1(ū|{θ̄A, θl}) as a function of θl. Using the first equa-
tion (A.4) we have
K
(1/η)
N,nA+1(ū|{θ̄A, θl}) =
∑
{θ̄I,θ̄II}`{θ̄A,θl}
(
−1
η
)#θ̄II
f(ū, θ̄I)f(θ̄I, θ̄II).
Clearly, either θl ∈ θ̄I or θl ∈ θ̄II. In the first case we set θ̄I = {θl, θ̄A1} and θ̄II = θ̄A2 . In the
second case we set θ̄I = θ̄A1 and θ̄II = {θl, θ̄A2}. Thus, we obtain
K
(1/η)
N,nA+1(ū|{θ̄A, θl})
f(ū, θl)
=
∑
{θ̄A1
,θ̄A2
}`θ̄A
(
−1
η
)#θ̄A2
f(ū, θ̄A1)f(θ̄A1 , θ̄A2)
×
(
f(θl, θ̄A2)− 1
η
f(θ̄A1 , θl)
f(ū, θl)
)
.
Substituting this to the l.h.s. of (C.2) we have
η
∑
{θ̄B1
,θ̄B2
}`θ̄B
#θ̄B1
=1
f(θ̄B2 , θ̄B1)
h(θs, θ̄B1)
K
(1/η)
N,nA+1(ū|{θ̄A, θ̄B1})
f(ū, θ̄B1)
= Λ1 + Λ2,
where
Λ1 = η
∑
{θ̄A1
,θ̄A2
}`θ̄A
(
−1
η
)#θ̄A2
f(ū, θ̄A1)f(θ̄A1 , θ̄A2)
×
∑
{θ̄B1
,θ̄B2
}`θ̄B
#θ̄B1
=1
f(θ̄B2 , θ̄B1)f(θ̄B1 , θ̄A2)
h(θs, θ̄B1)
, (D.3)
and
Λ2 = −
∑
{θ̄A1
,θ̄A2
}`θ̄A
(
−1
η
)#θ̄A2
f(ū, θ̄A1)f(θ̄A1 , θ̄A2)
×
∑
{θ̄B1
,θ̄B2
}`θ̄B
#θ̄B1
=1
f(θ̄B2 , θ̄B1)f(θ̄A1 , θ̄B1)
h(θs, θ̄B1)f(ū, θ̄B1)
. (D.4)
26 S. Belliard and N.A. Slavnov
D.2.1 First contribution
Consider the sum Λ1 (D.3). Let
G =
∑
{θ̄B1
,θ̄B2
}`θ̄B
#θ̄B1
=1
f(θ̄B2 , θ̄B1)f(θ̄B1 , θ̄A2)
h(θs, θ̄B1)
.
Consider an auxiliary contour integral
J =
1
2πic
∮
|z|=R→∞
f(θ̄B, z)
h(θs, z)
f(z, θ̄A2).
Taking the residue at infinity we obtain J = −1. On the other hand, this integral is equal to
the sum of residues within the contour. The sum in the points z = θk ∈ θ̄B gives −G. One more
series of poles occurs at z = θk ∈ θ̄A2 . Thus, we find
G = 1 +
∑
{θ̄A3
,θ̄A′2
}`θ̄A2
#θ̄A3
=1
f(θ̄B, θ̄A3)
h(θs, θ̄A3)
f(θ̄A3 , θ̄A′2). (D.5)
Substituting this into (D.3) we arrive at
Λ1 = ηK
(1/η)
N,nA
(ū|θ̄A)− Λ̃1, (D.6)
where
Λ̃1 =
∑
{θ̄A1
,θ̄A2
,θ̄A3
}`θ̄A
#θ̄A3
=1
(
−1
η
)#θ̄A2 f(ū, θ̄A1)
h(θs, θ̄A3)
f(θ̄A1 , θ̄A2)f(θ̄A1 , θ̄A3)
× f(θ̄A3 , θ̄A2)f(θ̄B, θ̄A3). (D.7)
Here, when substituting (D.5) into (D.3) we first set θ̄A2 = {θ̄A3 , θ̄A′2} and then relabeled θ̄A′2 →
θ̄A2 .
D.2.2 Second contribution
In order to compute the contribution Λ2 we should transform the sum over partitions of θ̄B
in (D.4). Let {θ̄B, θ̄A1} = θ̄C . Then
∑
{θ̄B1
,θ̄B2
}`θ̄B
#θ̄B1
=1
f(θ̄B2 , θ̄B1)f(θ̄A1 , θ̄B1)
h(θs, θ̄B1)f(ū, θ̄B1)
=
∑
{θ̄C1
,θ̄C2
}`θ̄C
#θ̄C1
=1
f(θ̄C2 , θ̄C1)
h(θs, θ̄C1)f(ū, θ̄C1)
−
∑
{θ̄A′1
,θ̄A3
}`θ̄A1
#θ̄A3
=1
f(θ̄B, θ̄A3)f(θ̄A′1 , θ̄A3)
h(θs, θ̄A3)f(ū, θ̄A3)
. (D.8)
In other words, instead of taking the sum over the subset θ̄B, we take the sum over the subset
θ̄C = {θ̄B, θ̄A1} and then subtract the sum over the subset θ̄A1 . In this case we can compute the
sum over the subset θ̄C = {θ̄B, θ̄A1} via Bethe equations (4.7).
Scalar products in twisted XXX spin chain. Determinant Representation 27
However, we first consider contribution coming from the second term in the r.h.s. of (D.8).
We have
Λ
(2)
2 =
∑
{θ̄A1
,θ̄A2
}`θ̄A
(
−1
η
)#θ̄A2
f(ū, θ̄A1)f(θ̄A1 , θ̄A2)
∑
{θ̄A′1
,θ̄A3
}`θ̄A1
#θ̄A3
=1
f(θ̄B, θ̄A3)f(θ̄A′1 , θ̄A3)
h(θs, θ̄A3)f(ū, θ̄A3)
.
Substituting here θ̄A1 = {θ̄A′1 , θ̄A3} and relabeling θ̄A′1 → θ̄A1 we find
Λ
(2)
2 =
∑
{θ̄A1
,θ̄A2
,θ̄A3
}`θ̄A
#θ̄A3
=1
(
−1
η
)#θ̄A2 f(ū, θ̄A1)
h(θs, θ̄A3)
f(θ̄A1 , θ̄A2)f(θ̄A3 , θ̄A2)
× f(θ̄A1 , θ̄A3)f(θ̄B, θ̄A3). (D.9)
We see that this sum is equal to the term Λ̃1 (D.7). Thus, common contribution of these terms
vanishes.
Consider now contribution coming from the first term in the r.h.s. of (D.8). We have
Λ
(1)
2 = −
∑
{θ̄A1
,θ̄A2
}`θ̄A
(
−1
η
)#θ̄A2
f(ū, θ̄A1)f(θ̄A1 , θ̄A2)
×
∑
{θ̄C1
,θ̄C2
}`θ̄C
#θ̄C1
=1
f(θ̄C2 , θ̄C1)
h(θs, θ̄C1)f(ū, θ̄C1)
. (D.10)
Using Bethe equations (4.7) we find
∑
{θ̄C1
,θ̄C2
}`θ̄C
#θ̄C1
=1
f(θ̄C2 , θ̄C1)
h(θs, θ̄C1)f(ū, θ̄C1)
=
κ̃+ κ
κ− ρ2
− 1
α
f(ū, θs)f(θs, θ̄A2)
+
1
α
∑
{θ̄A3
,θ̄A′2
}`θ̄A2
#θ̄A3
=1
g(θs, θ̄A3)f(ū, θ̄A3)f(θ̄A3 , θ̄A′2).
Substituting this into (D.10) we arrive at
Λ
(1)
2 = − κ̃+ κ
κ− ρ2
K
(1/η)
N,nA
(ū|θ̄A) +
1
α
(M1 −M2), (D.11)
where
M1 =
∑
{θ̄A1
,θ̄A2
}`{θ̄A,θs}
θs∈θ̄A1
(
−1
η
)#θ̄A2
f(ū, θ̄A1)f(θ̄A1 , θ̄A2),
and
M2 =
∑
{θ̄A1
,θ̄A2
}`θ̄A
(
−1
η
)#θ̄A2
f(ū, θ̄A1)f(θ̄A1 , θ̄A2)
28 S. Belliard and N.A. Slavnov
×
∑
{θ̄A3
,θ̄A′2
}`θ̄A2
#θ̄A3
=1
g(θs, θ̄A3)f(ū, θ̄A3)f(θ̄A3 , θ̄A′2). (D.12)
Presenting M1 as
M1 =
∑
{θ̄A1
,θ̄A2
}`{θ̄A,θs}
(
−1
η
)#θ̄A2
f(ū, θ̄A1)f(θ̄A1 , θ̄A2)
−
∑
{θ̄A1
,θ̄A2
}`{θ̄A,θs}
θs∈θ̄A2
(
−1
η
)#θ̄A2
f(ū, θ̄A1)f(θ̄A1 , θ̄A2),
we obtain
M1 = K
(1/η)
N,nA+1(ū|{θ̄A, θs})−
∑
{θ̄A1
,θ̄A2
}`{θ̄A,θs}
θs∈θ̄A2
(
−1
η
)#θ̄A2
f(ū, θ̄A1)f(θ̄A1 , θ̄A2).
Setting here θ̄A2 = {θ̄A′2 , θs} and relabeling θ̄A′2 → θ̄A2 we arrive at
M1 = K
(1/η)
N,nA+1(ū|{θ̄A, θs})−
∑
{θ̄A1
,θ̄A2
}`θ̄A
(
−1
η
)#θ̄A2
+1
f(ū, θ̄A1)f(θ̄A1 , θ̄A2)f(θ̄A1 , θs).
Consider now contribution M2. Setting θ̄A2 = {θ̄A′2 , θ̄A3} in (D.12) and relabeling θ̄A′2 → θ̄A2
we arrive at
M2 =
∑
{θ̄A1
,θ̄A2
,θ̄A3
}`θ̄A
#θ̄A3
=1
(
−1
η
)#θ̄A2+1
f(ū, θ̄A1)f(θ̄A1 , θ̄A2)f(θ̄A1 , θ̄A3)g(θs, θ̄A3)
× f(ū, θ̄A3)f(θ̄A3 , θ̄A2). (D.13)
Let {θ̄A1 , θ̄A3} = θ̄A0 . Then (D.13) takes the form
M2 =
∑
{θ̄A0
,θ̄A2
}`θ̄A
(
−1
η
)#θ̄A2+1
f(ū, θ̄A0)f(θ̄A0 , θ̄A2)
×
∑
{θ̄A1
,θ̄A3
}`θ̄A0
#θ̄A3
=1
g(θs, θ̄A3)f(θ̄A1 , θ̄A3). (D.14)
The sum over partitions {θ̄A1 , θ̄A3} ` θ̄A0 can be easily computed by the contour integral∑
{θ̄A1
,θ̄A3
}`θ̄A0
#θ̄A3
=1
g(θs, θ̄A3)f(θ̄A1 , θ̄A3) = 1− f(θ̄A0 , θs).
Substituting this into (D.14) we obtain
M2 =
∑
{θ̄A0
,θ̄A2
}`θ̄A
(
−1
η
)#θ̄A2+1
f(ū, θ̄A0)f(θ̄A0 , θ̄A2)
(
1− f(θ̄A0 , θs)
)
Scalar products in twisted XXX spin chain. Determinant Representation 29
= −1
η
K
(1/η)
N,nA
(ū|θ̄A)−
∑
{θ̄A1
,θ̄A2
}`θ̄A
(
−1
η
)#θ̄A2+1
f(ū, θ̄A1)f(θ̄A1 , θ̄A2)f(θ̄A1 , θs).
Thus,
M1 −M2 = K
(1/η)
N,nA+1(ū|{θ̄A, θs}) +
1
η
K
(1/η)
N,nA
(ū|θ̄A),
and due to (D.11) we arrive at
Λ
(1)
2 =
(
1
ηα
− κ̃+ κ
κ− ρ2
)
K
(1/η)
N,nA
(ū|θ̄A) +
1
α
K
(1/η)
N,nA+1(ū|{θ̄A, θs}),
Finally, taking into account (D.6), (D.7), and (D.9) we obtain after simple algebra
Λ1 + Λ2 =
β
βµ+ µ− 1
(
ηK
(1/η)
N,nA+1(ū|{θ̄A, θs})− ξK(1/η)
N,nA
(ū|θ̄A)
)
.
Thus, Lemma C.3 is proved.
Acknowledgements
The work was performed at the Steklov Mathematical Institute of Russian Academy of Sciences,
Moscow. This work is supported by the Russian Science Foundation under grant 19-11-00062.
References
[1] Avan J., Belliard S., Grosjean N., Pimenta R.A., Modified algebraic Bethe ansatz for XXZ chain on the
segment – III – Proof, Nuclear Phys. B 899 (2015), 229–246, arXiv:1506.02147.
[2] Baxter R.J., One-dimensional anisotropic Heisenberg chain, Ann. Physics 70 (1972), 323–337.
[3] Belliard S., Modified algebraic Bethe ansatz for XXZ chain on the segment – I: Triangular cases, Nuclear
Phys. B 892 (2015), 1–20, arXiv:1408.4840.
[4] Belliard S., Crampé N., Heisenberg XXX model with general boundaries: eigenvectors from algebraic Bethe
ansatz, SIGMA 9 (2013), 072, 12 pages, arXiv:1309.6165.
[5] Belliard S., Pimenta R.A., Modified algebraic Bethe ansatz for XXZ chain on the segment – II – General
cases, Nuclear Phys. B 894 (2015), 527–552, arXiv:1412.7511.
[6] Belliard S., Pimenta R.A., Slavnov and Gaudin–Korepin formulas for models without U(1) symmetry: the
twisted XXX chain, SIGMA 11 (2015), 099, 12 pages, arXiv:1506.06550.
[7] Belliard S., Slavnov N.A., A note on gl2-invariant Bethe vectors, J. High Energy Phys. 2018 (2018), no. 4,
031, 15 pages, arXiv:1802.07576.
[8] Belliard S., Slavnov N.A., Vallet B., Modified algebraic Bethe ansatz: twisted XXX case, SIGMA 14 (2018),
054, 18 pages, arXiv:1804.00597.
[9] Belliard S., Slavnov N.A., Vallet B., Scalar product of twisted XXX modified Bethe vectors, J. Stat. Mech.
Theory Exp. (2018), 093103, 29 pages, arXiv:1805.11323.
[10] Cao J., Yang W.-L., Shi K., Wang Y., Off-diagonal Bethe ansatz and exact solution a topological spin ring,
Phys. Rev. Lett. 111 (2013), 137201, 5 pages, arXiv:1305.7328.
[11] Cao J., Yang W.-L., Shi K., Wang Y., Off-diagonal Bethe ansatz solution of the XXX spin chain with
arbitrary boundary conditions, Nuclear Phys. B 875 (2013), 152–165, arXiv:1306.1742.
[12] Crampé N., Algebraic Bethe ansatz for the totally asymmetric simple exclusion process with boundaries,
J. Phys. A: Math. Theor. 48 (2015), 08FT01, 12 pages, arXiv:1411.7954.
[13] Derkachev S.E., The R-matrix factorization, Q-operator, and variable separation in the case of the XXX
spin chain with the SL(2,C) symmetry group, Theoret. and Math. Phys. 169 (2011), 1539–1550.
https://doi.org/10.1016/j.nuclphysb.2015.08.006
https://arxiv.org/abs/1506.02147
https://doi.org/10.1016/0003-4916(72)90270-9
https://doi.org/10.1016/j.nuclphysb.2015.01.003
https://doi.org/10.1016/j.nuclphysb.2015.01.003
https://arxiv.org/abs/1408.4840
https://doi.org/10.3842/SIGMA.2013.072
https://arxiv.org/abs/1309.6165
https://doi.org/10.1016/j.nuclphysb.2015.03.016
https://arxiv.org/abs/1412.7511
https://doi.org/10.3842/SIGMA.2015.099
https://arxiv.org/abs/1506.06550
https://doi.org/10.1007/JHEP04(2018)031
https://arxiv.org/abs/1802.07576
https://doi.org/10.3842/SIGMA.2018.054
https://arxiv.org/abs/1804.00597
https://doi.org/10.1088/1742-5468/aaddac
https://doi.org/10.1088/1742-5468/aaddac
https://arxiv.org/abs/1805.11323
https://doi.org/10.1103/PhysRevLett.111.137201
https://arxiv.org/abs/1305.7328
https://doi.org/10.1016/j.nuclphysb.2013.06.022
https://arxiv.org/abs/1306.1742
https://doi.org/10.1088/1751-8113/48/8/08FT01
https://arxiv.org/abs/1411.7954
https://doi.org/10.1007/s11232-011-0131-x
30 S. Belliard and N.A. Slavnov
[14] Faddeev L.D., How the algebraic Bethe ansatz works for integrable models, in Symétries quantiques (Les
Houches, 1995), North-Holland, Amsterdam, 1998, 149–219, arXiv:hep-th/9605187.
[15] Faddeev L.D., Sklyanin E.K., Takhtajan L.A., Quantum inverse problem. I, Theoret. and Math. Phys. 40
(1979), 688–706.
[16] Gaudin M., Modèles exacts en mécanique statistique: la méthode de Bethe et ses généralisations, Preprint,
Centre d’Etudes Nucléaires de Saclay, CEA-N-1559:1, 1972.
[17] Gaudin M., La fonction d’onde de Bethe, Collection du Commissariat à l’Énergie Atomique: Série Scien-
tifique, Masson, Paris, 1983.
[18] Göhmann F., Klümper A., Seel A., Integral representations for correlation functions of the XXZ chain at
finite temperature, J. Phys. A: Math. Gen. 37 (2004), 7625–7651, arXiv:hep-th/0405089.
[19] Gorsky A., Zabrodin A., Zotov A., Spectrum of quantum transfer matrices via classical many-body systems,
J. High Energy Phys. 2014 (2014), no. 1, 070, 28 pages, arXiv:1310.6958.
[20] Izergin A.G., Partition function of the six-vertex model in a finite volume, Sov. Phys. Dokl. 32 (1987),
878–879.
[21] Kitanine N., Kozlowski K.K., Maillet J.M., Slavnov N.A., Terras V., Form factor approach to dynami-
cal correlation functions in critical models, J. Stat. Mech. Theory Exp. 2012 (2012), P09001, 33 pages,
arXiv:1206.2630.
[22] Kitanine N., Maillet J.M., Niccoli G., Terras V., The open XXX spin chain in the SoV framework: scalar
product of separate states, J. Phys. A: Math. Theor. 50 (2017), 224001, 35 pages, arXiv:1606.06917.
[23] Kitanine N., Maillet J.M., Slavnov N.A., Terras V., Master equation for spin-spin correlation functions of
the XXZ chain, Nuclear Phys. B 712 (2005), 600–622, arXiv:hep-th/0406190.
[24] Kitanine N., Maillet J.M., Terras V., Form factors of the XXZ Heisenberg spin- 1
2
finite chain, Nuclear
Phys. B 554 (1999), 647–678, arXiv:math-ph/9807020.
[25] Kitanine N., Maillet J.M., Terras V., Correlation functions of the XXZ Heisenberg spin- 1
2
chain in a mag-
netic field, Nuclear Phys. B 567 (2000), 554–582, arXiv:math-ph/9907019.
[26] Korepin V.E., Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982), 391–418.
[27] Korepin V.E., Bogoliubov N.M., Izergin A.G., Quantum inverse scattering method and correlation functions,
Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1993.
[28] Nepomechie R.I., An inhomogeneous T -Q equation for the open XXX chain with general boundary terms:
completeness and arbitrary spin, J. Phys. A: Math. Theor. 46 (2013), 442002, 7 pages, arXiv:1307.5049.
[29] Sklyanin E.K., Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen. 21 (1988),
2375–2389.
[30] Sklyanin E.K., Quantum inverse scattering method. Selected topics, in Quantum Group and Quantum
Integrable Systems, Nankai Lectures Math. Phys., World Sci. Publ., River Edge, NJ, 1992, 63–97, arXiv:hep-
th/9211111.
[31] Slavnov N.A., Calculation of scalar products of wave functions and form factors in the framework of the
algebraic Bethe ansatz, Theoret. and Math. Phys. 79 (1989), 502–508.
[32] Takhtadzhan L.A., Faddeev L.D., The quantum method for the inverse problem and the Heisenberg XY Z
model, Russian Math. Surveys 34 (1979), no. 5, 11–68.
[33] Wang Y., Yang W.-L., Cao J., Shi K., Off-diagonal Bethe ansatz for exactly solvable models, Springer,
Heidelberg, 2015.
https://arxiv.org/abs/hep-th/9605187
https://doi.org/10.1007/BF01018718
https://doi.org/10.1088/0305-4470/37/31/001
https://arxiv.org/abs/hep-th/0405089
https://doi.org/10.1007/JHEP01(2014)070
https://arxiv.org/abs/1310.6958
https://doi.org/10.1088/1742-5468/2012/09/p09001
https://arxiv.org/abs/1206.2630
https://doi.org/10.1088/1751-8121/aa6cc9
https://arxiv.org/abs/1606.06917
https://doi.org/10.1016/j.nuclphysb.2005.01.050
https://arxiv.org/abs/hep-th/0406190
https://doi.org/10.1016/S0550-3213(99)00295-3
https://doi.org/10.1016/S0550-3213(99)00295-3
https://arxiv.org/abs/math-ph/9807020
https://doi.org/10.1016/S0550-3213(99)00619-7
https://arxiv.org/abs/math-ph/9907019
https://doi.org/10.1007/BF01212176
https://doi.org/10.1017/CBO9780511628832
https://doi.org/10.1088/1751-8113/46/44/442002
https://arxiv.org/abs/1304.7978
https://doi.org/10.1088/0305-4470/21/10/015
https://arxiv.org/abs/hep-th/9211111
https://arxiv.org/abs/hep-th/9211111
https://doi.org/10.1007/BF01016531
https://doi.org/10.1070/RM1979v034n05ABEH003909
https://doi.org/10.1007/978-3-662-46756-5
1 Introduction
2 Basic notions
2.1 Highest weight representation of the Yangian
2.2 Shorthand notation
2.3 On-shell Bethe vectors
3 Modified Izergin determinant and scalar products
3.1 Modified Izergin determinant
3.2 Scalar product
4 Main results
4.1 Determinant representations for the scalar product
4.2 Alternative form of the Bethe equations
4.3 On-shell modified Izergin determinant
5 Proof of the alternative forms of Bethe equations
6 Proof of the on-shell modified Izergin determinant properties
7 Calculation of the scalar product
7.1 Transformation of Jacobian
7.2 Transformation of the scalar product for generic
7.3 Further transformation of the scalar product for on-shell
8 Conclusion
A Properties of the modified Izergin determinant
B Identities for rational functions
B.1 Sums of rational functions
B.2 Matrices with rational elements
C Relations between on-shell modified Izergin determinants
D Proof of preparatory lemmas
D.1 Proof of Lemma C.2
D.2 Proof of Lemma C.3
D.2.1 First contribution
D.2.2 Second contribution
References
|
| id | nasplib_isofts_kiev_ua-123456789-210229 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:50Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Belliard, S. Slavnov, N.A. 2025-12-04T13:03:39Z 2019 Scalar Products in Twisted XXX Spin Chain. Determinant Representation / S. Belliard, N.A. Slavnov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 33 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 82B23; 81R50 arXiv: 1906.06897 https://nasplib.isofts.kiev.ua/handle/123456789/210229 https://doi.org/10.3842/SIGMA.2019.066 We consider the XXX spin-1/2 Heisenberg chain with non-diagonal boundary conditions. We obtain a compact determinant representation for the scalar product of on-shell and off-shell Bethe vectors. In the particular case when both Bethe vectors are on shell, we obtain a determinant representation for the norm of the on-shell Bethe vector and prove orthogonality of the on-shell vectors corresponding to the different eigenvalues of the transfer matrix. The work was performed at the Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow. This work is supported by the Russian Science Foundation under grant 19-11-00062. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Scalar Products in Twisted XXX Spin Chain. Determinant Representation Article published earlier |
| spellingShingle | Scalar Products in Twisted XXX Spin Chain. Determinant Representation Belliard, S. Slavnov, N.A. |
| title | Scalar Products in Twisted XXX Spin Chain. Determinant Representation |
| title_full | Scalar Products in Twisted XXX Spin Chain. Determinant Representation |
| title_fullStr | Scalar Products in Twisted XXX Spin Chain. Determinant Representation |
| title_full_unstemmed | Scalar Products in Twisted XXX Spin Chain. Determinant Representation |
| title_short | Scalar Products in Twisted XXX Spin Chain. Determinant Representation |
| title_sort | scalar products in twisted xxx spin chain. determinant representation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210229 |
| work_keys_str_mv | AT belliards scalarproductsintwistedxxxspinchaindeterminantrepresentation AT slavnovna scalarproductsintwistedxxxspinchaindeterminantrepresentation |