Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models
We study gl(2|1) symmetric integrable models solvable by the nested algebraic Bethe ansatz. Using explicit formulas for the Bethe vectors we derive the actions of the monodromy matrix entries onto these vectors. We show that the result of these actions is a finite linear combination of Bethe vectors...
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irk-123456789-1478642019-02-17T01:23:22Z Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models Hutsalyuk, A. Liashyk, A. Pakuliak, S.Z. Ragoucy, E. Slavnov, N.A. We study gl(2|1) symmetric integrable models solvable by the nested algebraic Bethe ansatz. Using explicit formulas for the Bethe vectors we derive the actions of the monodromy matrix entries onto these vectors. We show that the result of these actions is a finite linear combination of Bethe vectors. The obtained formulas open a way for studying scalar products of Bethe vectors. 2016 Article Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models / A. Hutsalyuk. A. Liashyk, S.Z. Pakuliak, E. Ragoucy, N.A. Slavnov // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 54 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 82B23; 81R12; 81R50; 17B80 DOI:10.3842/SIGMA.2016.099 http://dspace.nbuv.gov.ua/handle/123456789/147864 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We study gl(2|1) symmetric integrable models solvable by the nested algebraic Bethe ansatz. Using explicit formulas for the Bethe vectors we derive the actions of the monodromy matrix entries onto these vectors. We show that the result of these actions is a finite linear combination of Bethe vectors. The obtained formulas open a way for studying scalar products of Bethe vectors. |
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Article |
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Hutsalyuk, A. Liashyk, A. Pakuliak, S.Z. Ragoucy, E. Slavnov, N.A. |
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Hutsalyuk, A. Liashyk, A. Pakuliak, S.Z. Ragoucy, E. Slavnov, N.A. Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models Symmetry, Integrability and Geometry: Methods and Applications |
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Hutsalyuk, A. Liashyk, A. Pakuliak, S.Z. Ragoucy, E. Slavnov, N.A. |
| author_sort |
Hutsalyuk, A. |
| title |
Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models |
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Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models |
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Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models |
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Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models |
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Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models |
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multiple actions of the monodromy matrix in gl(2|1)-invariant integrable models |
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Інститут математики НАН України |
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2016 |
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Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models / A. Hutsalyuk. A. Liashyk, S.Z. Pakuliak, E. Ragoucy, N.A. Slavnov // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 54 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT hutsalyuka multipleactionsofthemonodromymatrixingl21invariantintegrablemodels AT liashyka multipleactionsofthemonodromymatrixingl21invariantintegrablemodels AT pakuliaksz multipleactionsofthemonodromymatrixingl21invariantintegrablemodels AT ragoucye multipleactionsofthemonodromymatrixingl21invariantintegrablemodels AT slavnovna multipleactionsofthemonodromymatrixingl21invariantintegrablemodels |
| first_indexed |
2025-07-11T02:59:21Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 099, 22 pages
Multiple Actions of the Monodromy Matrix
in gl(2|1)-Invariant Integrable Models?
Arthur HUTSALYUK †1, Andrii LIASHYK †2†3, Stanislav Z. PAKULIAK †4†1,
Eric RAGOUCY †5 and Nikita A. SLAVNOV †6
†1 Moscow Institute of Physics and Technology, Dolgoprudny, Moscow region, Russia
E-mail: hutsalyuk@gmail.com
†2 Bogoliubov Institute for Theoretical Physics, NAS of Ukraine, Kyiv, Ukraine
E-mail: a.liashyk@gmail.com
†3 National Research University Higher School of Economics, Russia
†4 Laboratory of Theoretical Physics, JINR, Dubna, Moscow region, Russia
E-mail: stanislav.pakuliak@jinr.ru
†5 Laboratoire de Physique Théorique LAPTh, CNRS and USMB, Annecy-le-Vieux, France
E-mail: eric.ragoucy@lapth.cnrs.fr
†6 Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
E-mail: nslavnov@mi.ras.ru
Received June 24, 2016, in final form October 03, 2016; Published online October 08, 2016
http://dx.doi.org/10.3842/SIGMA.2016.099
Abstract. We study gl(2|1) symmetric integrable models solvable by the nested algebraic
Bethe ansatz. Using explicit formulas for the Bethe vectors we derive the actions of the
monodromy matrix entries onto these vectors. We show that the result of these actions is
a finite linear combination of Bethe vectors. The obtained formulas open a way for studying
scalar products of Bethe vectors.
Key words: algebraic Bethe ansatz; superalgebras; scalar product of Bethe vectors
2010 Mathematics Subject Classification: 82B23; 81R12; 81R50; 17B80
1 Introduction
Algebraic Bethe ansatz is one of the most famous applications of the quantum inverse scat-
tering method. It was developed at the end of the 70s of the last century by the Leningrad
school [15, 16]. From a mathematical point of view the method consists of the study of the
highest weight representations of some Hopf algebra that depends on the model under considera-
tion [12, 22, 23]. This method allows one to find eigenvectors of the transfer matrix (a generating
function of the commuting Hamiltonians of the quantum models), leading to a diagonalization
of the physical Hamiltonians. The obtained determinant representations for scalar products of
the Bethe vectors [29, 45] allow then to calculate the form factors in various integrable sys-
tems [25, 27, 30]. Using the form factor representations, many interesting physical results were
obtained during the last few years [10, 11, 20, 21, 24, 26, 28, 41, 42, 44].
The results listed above mostly concerned the models with gl(2) symmetry or its q-defor-
mation. An important problem remains open: the case of models based on the gl(N) algebras
(N > 2). These models are quite more involved, therefore they are less studied. The models
?This paper is a contribution to the Special Issue on Recent Advances in Quantum Integrable Systems. The
full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2016.html
mailto:hutsalyuk@gmail.com
mailto:a.liashyk@gmail.com
mailto:stanislav.pakuliak@jinr.ru
mailto:eric.ragoucy@lapth.cnrs.fr
mailto:nslavnov@mi.ras.ru
http://dx.doi.org/10.3842/SIGMA.2016.099
http://www.emis.de/journals/SIGMA/RAQIS2016.html
2 A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, E. Ragoucy and N.A. Slavnov
with higher rank symmetries were considered within the framework of a nested Bethe ansatz
approach [48, 49, 53]. The algebraic version of this method was developed in the works [31,
32, 33, 34]. Later on, new methods to construct Bethe vectors in the models of higher rank
symmetries were proposed in [23, 50]. A generalization to the case of superalgebras gl(m|n) was
given in [8].
Recently, the models with gl(3) symmetries were studied more deeply by the nested algebraic
Bethe ansatz in the works [3, 4, 5, 7, 37, 38, 39, 51, 52]. There, determinant representations
for scalar products of Bethe vectors and form factors of local operators were obtained. Some of
these results were generalized to the models with q-deformed gl(3) symmetry [35, 36, 46].
In this paper, we calculate multiple actions of the monodromy matrix entries onto Bethe
vectors in the models with gl(2|1) symmetry. This is a part of a larger program devoted to
the study of quantum integrable models based on gl(m|n) superalgebras. This program was
initiated in [8] and continued in [40], where the Bethe vectors for these models were constructed.
The calculation of the monodromy matrix entries multiple actions onto the Bethe vectors is the
next step of our investigation. Further steps that include the calculation of the scalar products
and form factors of the monodromy matrix entries will be reported elsewhere. They will lead
eventually to determinant representations for form factors of local operators in the models of
physical interest. In particular, we expect to obtain compact explicit formulas for form factors of
local operators in the supersymmetric t-J model [1, 14, 17, 18, 43, 54] and the U -model of strongly
correlated electrons [2, 9] since these models are based on the gl(1|2) superalgebra. Knowing
form factors of local operators, one can then study correlation functions via a summation of their
form factor series. We expect to come back on these physical models in further works. We also
hope that our results will be of some interest in the context of super-Yang–Mills theories, when
studied in the integrable systems framework. Indeed in these theories, the general approach
relies on a spin chain based on the psl(2, 2|4) superalgebra, but there are closed subsectors
based on the gl(1|2) or gl(2|1) superalgebras. We believe that the present results will be a first
step to understand these subsectors, and then the entire theory.
The paper is organized as follows. Section 2 is devoted to the definitions and the notation. In
Section 3 we present the main results of the paper: multiple actions of the monodromy matrix
entries onto Bethe vectors. In Section 4 we consider the action of the transfer matrix onto on-
shell Bethe vectors. Sections 5–7 contain proofs of the formulas presented in Section 3. Finally,
the appendix gathers some identities for rational functions that we use in our proofs.
2 Definitions
We consider a normalized rational R-matrix
R(x, y) = I + g(x, y)P, g(x, y) =
c
x− y
, (2.1)
where c is a constant and P is a graded permutation operator [34]. The R-matrix of gl(2|1)-
based models acts in the tensor product C2|1 ⊗ C2|1, where C2|1 is the Z2-graded vector space
with the grading [1] = [2] = 0, [3] = 1. The R-matrix (2.1) satisfies the graded Yang–Baxter
equation
R12(x, y)R13(x, z)R23(y, z) = R23(y, z)R13(x, z)R12(x, y) (2.2)
written in the tensor product of graded spaces C2|1 ⊗ C2|1 ⊗ C2|1. The subscripts in (2.2) show
in which copies of the C2|1 space the R-matrix acts non-trivially. The monodromy matrix T (u)
is also graded according to the rule [Tij(u)] = [i] + [j]. It satisfies an intertwining relation (RTT
relation):
R(u, v)
(
T (u)⊗ I
)(
I⊗ T (v)
)
=
(
I⊗ T (v)
)(
T (u)⊗ I
)
R(u, v). (2.3)
Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models 3
Equation (2.3) holds in the tensor product of graded spaces C2|1⊗C2|1⊗H, where H is a Hilbert
space of a Hamiltonian. It implies commutation relations between the monodromy matrix
entries:
[Tij(u), Tkl(v)} = (−1)[i]([k]+[l])+[k][l]g(u, v)
[
Tkj(v)Til(u)− Tkj(u)Til(v)
]
, (2.4)
where we introduced a graded commutator
[Tij(u), Tkl(v)} = Tij(u)Tkl(v)− (−1)([i]+[j])([k]+[l])Tkl(v)Tij(u).
The graded transfer matrix is defined as the supertrace of the monodromy matrix
T (u) = strT (u) =
3∑
i=1
(−1)[i]Tii(u).
It defines an integrable system, due to the relation [T (u), T (v)] = 0 which is implied by the
relation (2.3).
In order to make our formulas more compact we use several auxiliary functions and conven-
tions on the notation. In addition to the functions g(x, y) we also introduce the functions
f(x, y) = 1 + g(x, y) =
x− y + c
x− y
, h(x, y) =
f(x, y)
g(x, y)
=
x− y + c
c
,
t(x, y) =
g(x, y)
h(x, y)
=
c2
(x− y + c)(x− y)
.
The following obvious properties of these functions are useful
g(x, y) = −g(y, x), h(x, y) =
1
g(x, y − c)
, f(x− c, y) =
1
f(y, x)
.
Below, we will permanently have to deal with sets of variables which will be denoted by
a bar: ū, v̄, η̄ etc. Individual elements of the sets are denoted by subscripts and without a bar:
uk, v`, ηj etc. The notation ū ± c means that all the elements of the set ū are shifted by ±c:
ū±c = {u1±c, . . . , un±c}. As a rule, the number of elements in the sets is not shown explicitly;
however we give these cardinalities in special comments to the formulas. Subsets of variables
are denoted by roman or other subscripts, easily distinguishable from Latin indices used for
individual element: ūI, v̄II, η̄ii, ξ̄0 etc. For example, the notation ū⇒ {ūI, ūII} means that the set
ū is divided into two disjoint subsets ūI and ūII. We assume that the elements in every subset are
ordered in such a way that the sequence of their subscripts is strictly increasing. For the union
of two sets into another one we use the notation {v̄, z̄} = ξ̄. Finally we use a special notation
ūj , v̄k and so on for the sets ū \ {uj}, v̄ \ {vk} etc.
In order to avoid excessively cumbersome formulas we use shorthand notation for products
of functions depending on one or two variables. Namely, whenever such a function depends on
a set of variables, this means that we deal with the product of this function with respect to the
corresponding set, as follows
h(ū, v) =
∏
uj∈ū
h(uj , v), g(xk, ξ̄`) =
∏
ξj∈ξ̄
ξj 6=ξ`
g(xk, ξj), f(ūII, ūI) =
∏
uj∈ūII
∏
uk∈ūI
f(uj , uk).
This notation is also used for the product of commuting operators,
Tij(ū) =
∏
uk∈ū
Tij(uk), if [i] + [j] = 0, mod (2). (2.5)
4 A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, E. Ragoucy and N.A. Slavnov
One can easily see from the commutation relations (2.4) that in this case [Tij(u), Tij(v)] = 0, and
hence, the operator product (2.5) is well defined. However, if [i]+ [j] = 1, then [Tij(u), Tij(v)] 6=
0, but we can introduce symmetric operator products
Tj3(v̄) =
Tj3(v1) · · ·Tj3(vn)∏
n≥`>m≥1
h(v`, vm)
, T3j(v̄) =
T3j(v1) · · ·T3j(vn)∏
n≥`>m≥1
h(vm, v`)
, j = 1, 2.
In various formulas the Izergin determinant Kn(x̄|ȳ) appears1. It is defined for two sets x̄
and ȳ with common cardinality #x̄ = #ȳ = n,
Kn(x̄|ȳ) = h(x̄, ȳ)
n∏
`<m
g(x`, xm)g(ym, y`) det
n
[t(xi, yj)]. (2.6)
We draw the readers attention that according to the convention on the shorthand notation h(x̄, ȳ)
in (2.6) means the double product of the h-functions over all parameters x̄ and ȳ. It is easy to
see from definition (2.6) that K1(x|y) = g(x, y) and
Kn(x̄|ȳ + c) = (−1)n
Kn(ȳ|x̄)
f(ȳ, x̄)
. (2.7)
2.1 Bethe vectors
For further calculation we need explicit formulas for gl(2|1) Bethe vectors in terms of the mon-
odromy matrix entries Tij(u). We recall that generically Bethe vectors are special polynomials
in operators Tij(u) with i ≤ j applied to the pseudovacuum vector Ω. This vector possesses the
following properties:
Tii(u)Ω = λi(u)Ω, i = 1, 2, 3, Tij(u)Ω = 0, 3 ≥ i > j ≥ 1. (2.8)
Here λi(u) are some scalar functions depending on a specific model. Below we will also use the
ratios of these functions
r1(u) =
λ1(u)
λ2(u)
, r3(u) =
λ3(u)
λ2(u)
.
We extend the convention on the shorthand notation to the products of the functions λi and rk.
For example,
λ2(z̄) =
∏
zj∈z̄
λ2(zj), r1(η̄II) =
∏
ηj∈η̄II
r1(ηj).
We denote the Bethe vectors as Ba,b(ū; v̄). They depend on two sets of variables (Bethe pa-
rameters) ū = {u1, . . . , ua} and v̄ = {v1, . . . , vb}, where a, b = 0, 1, . . . . Explicit representations
for gl(2|1) Bethe vectors were obtained in2 [40].
Definition 2.1. For #ū = a and #v̄ = b define a Bethe vector
Ba,b(ū; v̄) =
∑
g(v̄I, ūI)
f(ūI, ūII)g(v̄II, v̄I)h(ūI, ūI)
λ2(ū)λ2(v̄II)f(v̄, ū)
T13(ūI)T12(ūII)T23(v̄II)Ω. (2.9)
1Note that by definition this function depends on two sets of variables. Therefore, the convention on shorthand
notations for the products is not applicable in this case.
2The formulas for the Bethe vectors obtained in [40] differ from (2.9) and (2.10) by a normalization factor
λ2(v̄)λ2(ū)f(v̄, ū).
Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models 5
Here the sum is taken over partitions v̄ ⇒ {v̄I, v̄II} and ū⇒ {ūI, ūII} with the restriction #ūI =
#v̄I = n, where n = 0, 1, . . . ,min(a, b). We recall also that we use the shorthand notation for
the products of all the functions and the operators in (2.9).
An alternative formula for the Bethe vector is
Ba,b(ū; v̄) =
∑
Kn(v̄I|ūI)
f(ūI, ūII)g(v̄II, v̄I)
λ2(ūII)λ2(v̄)f(v̄, ū)
T13(v̄I)T23(v̄II)T12(ūII)Ω, (2.10)
where Kn is the Izergin determinant (2.6) and the sum is the same as in (2.9).
A distinctive feature of the Bethe vectors is that under certain conditions on ū and v̄ (Bethe
equations), they become eigenvectors of the transfer matrix. In this case we call them on-shell
Bethe vectors. We will show in Section 4 that the vectors (2.9), (2.10) do possess this property.
It is obvious within the framework of the current approach [23] that the action of any mono-
dromy matrix entry Tij(u) on a Bethe vector produces a linear combination of a finite number of
Bethe vectors. This follows from the presentations of the monodromy matrix elements and Bethe
vectors in terms of the Cartan–Weyl or current generators of the Yangian double DY(gl(2|1)) and
normal ordering of these generators according to certain cyclic ordering [13, 19]. In summary,
we can rewrite the action formulas as normal ordering problem for the current generators and
then translate the result of this ordering back into finite sum of the Bethe vectors. However, it
is not so obvious if we deal with the explicit representations (2.9), (2.10). Furthermore, in spite
of the action of Tij(z) onto Ba,b(ū; v̄) formally can be derived via (2.4) and (2.8), actually it is
a pretty nontrivial problem.
Fortunately, similarly to the gl(3) case [6] the gl(2|1) Bethe vectors obey recursion relations
over the number of the Bethe parameters [40]. The first recursion has the form
T12(z)Ba,b(ū; v̄) = λ2(z)f(v̄, z)Ba+1,b({ū; z}; v̄)
+
b∑
j=1
g(z, vj)g(v̄j , vj)T13(z)Ba,b−1(ū; v̄j). (2.11)
The second recursion reads
T23(z)Ba,b(ū; v̄) = λ2(z)h(v̄, z)f(z, ū)Ba,b+1(ū; {v̄, z})
+
a∑
j=1
g(uj , z)f(uj , ūj)T13(z)Ba−1,b(ūj ; v̄). (2.12)
We recall that in these formulas v̄j and ūj respectively mean v̄\{vj} and ū\{uj}. The shorthand
notation for the products of the functions g and f is also used.
Equations (2.11) and (2.12) allow us to built recursively Bethe vectors starting with the
simplest cases
Ba,0(ū;∅) =
T12(ū)
λ2(ū)
Ω, B0,b(∅; v̄) =
T23(v̄)
λ2(v̄)
Ω. (2.13)
One can also easily derive the actions of Tij onto either Ba,0(ū;∅) or B0,b(∅; v̄), and then, using
induction over a or b obtain the action rule in the general case. This will be our main strategy
in the derivation of the action formulas.
3 Multiple actions of the operators Tij onto Bethe vectors
The main result of this paper consists of explicit formulas of the multiple actions of the mono-
dromy matrix entries onto Bethe vectors. We show that these actions always reduce to finite
linear combinations of Bethe vectors.
6 A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, E. Ragoucy and N.A. Slavnov
Everywhere in this section we assume that ū, v̄, and z̄ are three sets of generic complex
numbers with cardinalities #ū = a, #v̄ = b, and #z̄ = n, a, b, n = 0, 1, . . . . We also set
η̄ = {ū, z̄} and ξ̄ = {v̄, z̄}.
3.1 Actions of Tij(z̄) with i < j
• Multiple action of T13(z):
T13(z̄)Ba,b(ū; v̄) = λ2(z̄)h(v̄, z̄)Ba+n,b+n(η̄; ξ̄). (3.1)
• Multiple action of T12(z):
T12(z̄)Ba,b(ū; v̄) = λ2(z̄)h(ξ̄, z̄)
∑ g(ξ̄II, ξ̄I)
h(ξ̄I, z̄)
Ba+n,b(η̄; ξ̄II). (3.2)
Here the sum is taken over partitions ξ̄ ⇒ {ξ̄I, ξ̄II} with #ξ̄I = n.
• Multiple action of T23(z):
T23(z̄)Ba,b(ū; v̄) = (−1)nλ2(z̄)h(v̄, z̄)
∑
Kn(z̄|η̄I + c)f(η̄I, η̄II)Ba,b+n(η̄II; ξ̄). (3.3)
Here the sum is taken over partitions η̄ ⇒ {η̄I, η̄II} with #η̄I = n.
3.2 Actions of Tii(z̄)
In formulas (3.4)–(3.6) the sums are taken over partitions ξ̄ ⇒ {ξ̄I, ξ̄II} and η̄ ⇒ {η̄I, η̄II} with
#ξ̄I = #η̄I = n.
• Multiple action of T11(z):
T11(z̄)Ba,b(ū; v̄) = (−1)nλ2(z̄)h(ξ̄, z̄)
×
∑
r1(η̄I)
f(η̄II, η̄I)g(ξ̄II, ξ̄I)
h(ξ̄I, z̄)f(ξ̄II, η̄I)
Kn(η̄I|ξ̄I + c)Ba,b(η̄II; ξ̄II). (3.4)
• Multiple action of T22(z):
T22(z̄)Ba,b(ū; v̄) = (−1)nλ2(z̄)h(ξ̄, z̄)
×
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)
h(ξ̄I, z̄)
Kn(z̄|η̄I + c)Ba,b(η̄II; ξ̄II). (3.5)
• Multiple action of T33(z):
T33(z̄)Ba,b(ū; v̄) = λ2(z̄)h(ξ̄, z̄)
∑
r3(ξ̄I)
f(η̄I, η̄II)g(ξ̄II, ξ̄I)h(η̄I, η̄I)
h(ξ̄I, η̄I)h(η̄I, z̄)f(ξ̄I, η̄II)
Ba,b(η̄II; ξ̄II). (3.6)
3.3 Actions of Tij(z̄) with i > j
• Multiple action of T21(z):
T21(z̄)Ba,b(ū; v̄) = λ2(z̄)h(ξ̄, z̄)
∑
r1(η̄I)
f(η̄II, η̄I)f(η̄II, η̄III)f(η̄III, η̄I)g(ξ̄II, ξ̄I)
h(ξ̄I, z̄)f(ξ̄II, η̄I)
×Kn(z̄|η̄II + c)Kn(η̄I|ξ̄I + c)Ba−n,b(η̄III; ξ̄II). (3.7)
Here the sum is taken over partitions ξ̄ ⇒ {ξ̄I, ξ̄II} and η̄ ⇒ {η̄I, η̄II, η̄III} with #ξ̄I = #η̄I =
#η̄II = n.
Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models 7
• Multiple action of T32(z):
T32(z̄)Ba,b(ū; v̄) = (−1)
n(n−1)
2 λ2(z̄)h(ξ̄, z̄)
×
∑
r3(ξ̄I)
f(η̄I, η̄II)g(ξ̄II, ξ̄I)g(ξ̄III, ξ̄II)g(ξ̄III, ξ̄I)
h(η̄I, z̄)h(ξ̄I, η̄I)h(ξ̄II, z̄)f(ξ̄I, η̄II)
h(η̄I, η̄I)Ba,b−n(η̄II; ξ̄III). (3.8)
Here the sum is taken over partitions ξ̄ ⇒ {ξ̄I, ξ̄II, ξ̄III} and η̄ ⇒ {η̄I, η̄II} with #ξ̄I = #ξ̄II =
#η̄I = n.
• Multiple action of T31(z):
T31(z̄)Ba,b(ū; v̄) = (−1)
n(n+1)
2 λ2(z̄)h(ξ̄, z̄)
∑
r3(ξ̄I)r1(η̄II)
g(ξ̄II, ξ̄I)g(ξ̄III, ξ̄II)g(ξ̄III, ξ̄I)
h(η̄I, z̄)h(ξ̄I, η̄I)h(ξ̄II, z̄)
× f(η̄I, η̄II)f(η̄I, η̄III)f(η̄III, η̄II)h(η̄I, η̄I)
f(ξ̄I, η̄II)f(ξ̄I, η̄III)f(ξ̄III, η̄II)
Kn(η̄II|ξ̄II + c)Ba−n,b−n(η̄III; ξ̄III). (3.9)
Here the sum is taken over partitions ξ̄ ⇒ {ξ̄I, ξ̄II, ξ̄III} and η̄ ⇒ {η̄I, η̄II, η̄III} with #ξ̄I =
#ξ̄II = #η̄I = #η̄II = n.
The proofs of the multiple action formulas will be given in Sections 5–7.
4 On-shell Bethe vectors
The action formulas (3.1)–(3.9) are valid for generic complex numbers z̄, ū, and v̄. In this
section we consider them for on-shell Bethe vector Ba,b(ū; v̄), that is when the parameters ū
and v̄ satisfy a system of Bethe equations (see (4.6)).
In order to find explicitly the result of the transfer matrix action onto Ba,b(ū; v̄) one should
set n = 1 in (3.4)–(3.6). Then the subsets η̄I and ξ̄I consist of one element only. Obviously, there
are two essentially different types of partitions of the set η̄ = {z, ū}:
η̄I = z, η̄II = ū, (4.1)
η̄I = uj , η̄II = {z, ūj}, j = 1, . . . , a. (4.2)
Similarly, there are two different types of partitions of the set ξ̄ = {z, v̄}:
ξ̄I = z, ξ̄II = v̄, (4.3)
ξ̄I = vk, ξ̄II = {z, v̄k}, k = 1, . . . , b. (4.4)
Thus, the action of T (z) onto Ba,b(ū; v̄) can be written in the form
T (z)Ba,b(ū; v̄) = τ(z|ū; v̄)Ba,b(ū; v̄) +
a∑
j=1
ΛjBa,b({z, ūj}; v̄)
+
b∑
k=1
Λ̃kBa,b(ū; {z, v̄k}) +
a∑
j=1
b∑
k=1
MjkBa,b({z, ūj}; {z, v̄k}),
where τ , Λj , Λ̃k, and Mjk are numerical coefficients. In order to find τ(z|ū; v̄) we substitute
the partitions (4.1) and (4.3) into (3.4)–(3.6). We obtain
τ(z|ū, v̄) = λ1(z)f(ū, z) + λ2(z)f(z, ū)f(v̄, z)− λ3(z)f(v̄, z), (4.5)
where we have used h(z, z) = 1 and K1(z|z + c) = g(z, z + c) = −1.
8 A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, E. Ragoucy and N.A. Slavnov
In order to find Λj we substitute the partitions (4.2) and (4.3) into (3.4)–(3.6). We find
Λj = λ2(z)h(v̄, z)g(v̄, z)g(z, uj)
(
r1(uj)
f(ūj , uj)
f(v̄, uj)
− f(uj , ūj)
)
.
Similarly, in order to find Λ̃k we substitute the partitions (4.1) and (4.4) into (3.4)–(3.6). This
gives us
Λ̃k = λ2(z)f(z, ū)g(v̄k, vk)h(v̄k, z)g(z, vk)
(
1− r3(vk)
f(vk, ū)
)
.
If Ba,b(ū; v̄) is an eigenvector of T (z), then the coefficients Λj and Λ̃k must vanish for arbitrary z.
Setting Λj = 0 for j = 1, . . . , a and Λ̃k = 0 for k = 1, . . . , b we arrive at a system of equations
r1(uj) =
f(uj , ūj)
f(ūj , uj)
f(v̄, uj), j = 1, . . . , a,
r3(vk) = f(vk, ū), k = 1, . . . , b. (4.6)
Let us check that Mjk = 0 provided the system (4.6) is fulfilled. Substituting the parti-
tions (4.2) and (4.4) into (3.4)–(3.6) we obtain
Mjk = λ2(z)h(v̄k, z)g(v̄k, vk)
(
r1(uj)f(ūj , uj)
f(v̄, uj)
g(vk, uj)g(z, vk)
+ f(uj , ūj)g(uj , z)
[
g(z, vk) +
r3(vk)
f(vk, ū)
g(vk, uj)
])
.
Substituting here r1(uj) and r3(vk) from equations (4.6), we immediately find that Mjk = 0 due
to the identity
g(vk, uj)g(z, vk) + g(uj , z)g(z, vk) + g(uj , z)g(vk, uj) = 0.
Thus, the system (4.6) can be treated as the system of Bethe equations for the parameters ū
and v̄. If (4.6) holds, then the corresponding Bethe vector Ba,b(ū; v̄) is on-shell, i.e., it is an
eigenvector of the transfer matrix T (z). The eigenvalue of this on-shell vector is given by (4.5).
At the same time, it is easy to see that the function τ(z|ū, v̄) has no poles in the points z = uj ,
j = 1, . . . , a, and z = vk, k = 1, . . . , b due to the system (4.6).
5 Proofs of multiple actions for Tij with i < j
Bethe vectors consist of the elements from the upper triangular part of the monodromy matrix
applied to pseudovacuum Ω (2.9), (2.10). Then, it is intuitively clear that actions of the ele-
ments Tij with i < j are the simplest. We begin our consideration from the right-upper corner of
monodromy matrix and will move along anti-diagonal direction successively proving the action
relations.
5.1 Proof for T13
For n = 1 equation (3.1) follows directly from the definitions of the Bethe vectors. Let us
take, for instance, (2.9) and set there ū = {z, ū′} and v̄ = {z, v̄′}. Then the product 1/f(v̄, ū)
vanishes, as it contains 1/f(z, z). This zero, however, can be compensated if and only if z ∈ ūI
and z ∈ v̄I. Indeed, in this case the product g(v̄I, ūI) contains a singular factor g(z, z). Thus,
Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models 9
we should consider only such partitions, for which z ∈ ūI and z ∈ v̄I. Therefore we should set:
ūI = {z, ū′I} and v̄I = {z, v̄′I}; ūII = ū′II and v̄II = v̄′II. Then we obtain
Ba,b({z, ū′}; {z, v̄′}) =
∑
g(v̄′I, ū
′
I)
f(ū′I, ū
′
II)g(v̄′II, v̄
′
I)h(ū′I, ū
′
I)
λ2(ū′)λ2(v̄′II)f(v̄′, ū′)
g(v̄′I, z)g(z, ū′I)
× f(z, ū′II)g(v̄′II, z)h(z, ū′I)h(ū′I, z)
λ2(z)f(v̄′, z)f(z, ū′)
T13(z)
h(ū′I, z)
T13(ū′I)T12(ū′II)T23(v̄′II)Ω.
After evident cancellations we arrive at
Ba,b({z, ū′}; {z, v̄′}) =
T13(z)
λ2(z)h(v̄′, z)
Ba−1,b−1(ū′; v̄′),
which coincides with (3.1) at n = 1. The same result arises from the analysis of equation (2.10).
Now we use induction over n. Assume that (3.1) holds for some n− 1. Then
T13(z̄)Ba,b(ū; v̄) =
T13(zn)T13(z̄n)
h(z̄n, zn)
Ba,b(ū; v̄)
= λ2(z̄n)
h(v̄, z̄n)
h(z̄n, zn)
T13(zn)Ba+n−1,b+n−1({ū, z̄n}; {v̄, z̄n})
= λ2(z̄)h(v̄, z̄)Ba+n,b+n({ū, z̄}; {v̄, z̄}),
and thus, (3.1) is proved.
Using (3.1) one can recast recursions (2.11) and (2.12) as follows
T12(z)Ba,b(ū; v̄) = λ2(z)f(v̄, z)Ba+1,b({ū, z}; v̄)
+ λ2(z)
b∑
j=1
g(z, vj)g(v̄j , vj)h(v̄j , z)Ba+1,b({ū, z}; {v̄j , z}), (5.1)
and
T23(z)Ba,b(ū; v̄) = λ2(z)h(v̄, z)
(
f(z, ū)Ba,b+1(ū; {v̄, z})
+
a∑
j=1
g(uj , z)f(uj , ūj)Ba,b+1({ūj , z}; {v̄, z})
)
.
One can easily recognize in these equations the actions (3.2) and (3.3) for n = 1. Then one
should use induction over n.
5.2 Proof for T12
Assume that (3.2) holds for some n− 1. Then
T12(z̄)Ba,b(ū; v̄) = T12(zn)λ2(z̄n)h(ξ̄, z̄n)
∑ g(ξ̄II, ξ̄I)
h(ξ̄I, z̄n)
Ba+n−1,b(η̄; ξ̄II).
Here η̄ = {z̄n, ū}, ξ̄ = {z̄n, v̄}, and the sum runs through the partitions ξ̄ ⇒ {ξ̄I, ξ̄II} with
#ξ̄I = n− 1. Acting with T12(zn) we obtain
T12(z̄)Ba,b(ū; v̄) = λ2(z̄)
h(ξ̄, z̄n)
h(zn, z̄n)
∑ g(ξ̄II, ξ̄I)
h(ξ̄I, z̄n)
h(ξ̄II, zn)
g(ξ̄ii, ξ̄i)
h(ξ̄i, zn)
Ba+n,b(η̄; ξ̄ii).
10 A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, E. Ragoucy and N.A. Slavnov
Here already η̄ = {z̄, ū} and ξ̄ = {z̄, v̄}. The sum first is taken over partitions {z̄n, v̄} ⇒ {ξ̄I, ξ̄II}
with #ξ̄I = n− 1, and then over partitions {zn, ξ̄II} ⇒ {ξ̄i, ξ̄ii} with #ξ̄i = 1. One can say that
the sum is taken over partitions {z̄, v̄} = ξ̄ ⇒ {ξ̄I, ξ̄i, ξ̄ii} with restrictions zn /∈ ξ̄I, #ξ̄I = n− 1,
and #ξ̄i = 1. Presenting ξ̄II as ξ̄II = {ξ̄i, ξ̄ii} \ {zn} we obtain
g(ξ̄II, ξ̄I) =
g(ξ̄i, ξ̄I)g(ξ̄ii, ξ̄I)
g(zn, ξ̄I)
, h(ξ̄II, zn) = h(ξ̄i, zn)h(ξ̄ii, zn),
and hence,
T12(z̄)Ba,b(ū; v̄) = λ2(z̄)
h(ξ̄, z̄n)
h(zn, z̄n)
∑ g(ξ̄i, ξ̄I)g(ξ̄ii, ξ̄I)g(ξ̄ii, ξ̄i)
g(zn, ξ̄I)h(ξ̄I, z̄n)
h(ξ̄ii, zn)Ba+n,b(η̄; ξ̄ii). (5.2)
Observe that the condition zn /∈ ξ̄I is ensured by the product g(zn, ξ̄I) in the denominator. Hen-
ce, we can say that the sum is taken over partitions ξ̄ ⇒ {ξ̄I, ξ̄i, ξ̄ii} with the restrictions on the
cardinalities of the subsets only. Setting {ξ̄I, ξ̄i} = ξ̄0 we recast (5.2) as follows
T12(z̄)Ba,b(ū; v̄) = λ2(z̄)
h(ξ̄, z̄)
h(zn, z̄n)
∑ g(ξ̄i, ξ̄I)g(zn, ξ̄i)h(ξ̄i, z̄n)
g(zn, ξ̄0)h(ξ̄0, z̄)
g(ξ̄ii, ξ̄0)Ba+n,b(η̄; ξ̄ii).
The sum over partitions ξ̄0 ⇒ {ξ̄I, ξ̄i} can be computed via Lemma A.1∑
ξ̄0⇒{ξ̄I,ξ̄i}
g(ξ̄i, ξ̄I)g(zn, ξ̄i)h(ξ̄i, z̄n) = g(zn, ξ̄0)h(zn, z̄n), (5.3)
where we took into account that #ξ̄0 = n. Thus, we arrive at
T12(z̄)Ba,b(ū; v̄) = λ2(z̄)h(ξ̄, z̄)
∑ g(ξ̄ii, ξ̄0)
h(ξ̄0, z̄)
Ba+n,b(η̄; ξ̄ii),
which coincides with (3.2) up to a relabeling of the subsets.
5.3 Proof for T23
Assume that (3.3) holds for some n− 1. Let #z̄ = n. Then we have
T23(z̄)Ba,b(ū; v̄) =
T23(zn)T23(z̄n)
h(z̄n, zn)
Ba,b(ū; v̄)
= (−1)n−1λ2(z̄n)
h(v̄, z̄n)
h(z̄n, zn)
∑
Kn−1(z̄n|η̄I + c)f(η̄I, η̄II)T23(zn)Ba,b+n−1(η̄II; ξ̄).
Here η̄ = {z̄n, ū}, ξ̄ = {z̄n, v̄}, and the sum runs through the partitions η̄ ⇒ {η̄I, η̄II} with
#η̄I = n− 1. Acting with T23(zn) we obtain
T23(z̄)Ba,b(ū; v̄) = (−1)nλ2(z̄)
h(v̄, z̄n)
h(z̄n, zn)
×
∑
Kn−1(z̄n|η̄I + c)f(η̄I, η̄II)h(ξ̄, zn)K1(zn|η̄i + c)f(η̄i, η̄ii)Ba,b+n(η̄ii; ξ̄).
Here already η̄ = {z̄, ū}, ξ̄ = {z̄, v̄}, and the sum is taken first over partitions {z̄n, ū} ⇒ {η̄I, η̄II}
with #η̄I = n− 1, and then over partitions {zn, η̄II} ⇒ {η̄i, η̄ii} with #η̄i = 1. Substituting here
η̄II = {η̄i, η̄ii} \ {zn} we find
T23(z̄)Ba,b(ū; v̄) = (−1)nλ2(z̄)h(v̄, z̄)
×
∑
Kn−1(z̄n|η̄I + c)K1(zn|η̄i + c)
f(η̄I, η̄i)f(η̄I, η̄ii)f(η̄i, η̄ii)
f(η̄I, zn)
Ba,b+n(η̄ii; ξ̄).
Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models 11
Setting η̄0 = {η̄I, η̄i} and using K1(zn|η̄i + c) = −K1(η̄i|zn)/f(η̄i, zn) we obtain
T23(z̄)Ba,b(ū; v̄) = (−1)n−1λ2(z̄)h(v̄, z̄)
×
∑
Kn−1(z̄n|η̄I + c)K1(η̄i|zn)f(η̄I, η̄i)
f(η̄0, η̄ii)
f(η̄0, zn)
Ba,b+n(η̄ii; ξ̄).
Now we can compute the sum over partitions η̄0 ⇒ {η̄I, η̄i} via (A.2)∑
η̄0⇒{η̄I,η̄i}
Kn−1(z̄n|η̄I + c)K1(η̄i|zn)f(η̄I, η̄i) = −f(η̄0, zn)Kn(z̄|η̄0 + c), (5.4)
which gives us
T23(z̄)Ba,b(ū; v̄) = (−1)nλ2(z̄)h(v̄, z̄)
∑
Kn(z̄|η̄0 + c)f(η̄0, η̄ii)Ba,b+n(η̄ii; ξ̄).
This is exactly (3.3) up to the labeling of the subsets.
6 Proof of the multiple action of the operator T22
The proofs for the actions (3.4)–(3.9) are much more involved than the ones considered in the
previous section. Fortunately, they all are quite similar. Therefore, we only detail one as a typical
example, the other actions being proven in the same manner. We focus on the operator T22(u).
The strategy of the proof is the following. First, we prove equation (3.5) for a = #ū = 0
and n = #z̄ = 1. This can be done either via the standard consideration of the algebraic Bethe
ansatz or using induction over b = #v̄. In both cases we use (2.13) and the relation
T22(u)T23(v) = f(v, u)T23(v)T22(u) + g(u, v)T23(u)T22(v), (6.1)
that follows from (2.4).
The next step of the proof is an induction over a. We assume that (3.5) is valid for n = 1
and some a and use recursion (2.11). Hereby, we use some of commutation relations (2.4)
T22(u)T12(v) = f(u, v)T12(v)T22(u) + g(v, u)T12(u)T22(v), (6.2)
[T22(u), T13(v)] = g(u, v)
(
T12(v)T23(u)− T12(u)T23(v)
)
. (6.3)
Finally, when equation (3.5) is proved for n = 1 and arbitrary a and b we use induction over n.
Remark 6.1. We begin the proof with the case n = 1, a = 0, and arbitrary b. However, one
could also begin with the case n = 1, b = 0, and arbitrary a. For the action of the operator
T22(z) this is a matter of choice. For other operators these two starting cases could be essentially
different. For instance, one can easily see that T21(z)B0,b(∅, v̄) = 0 for arbitrary b. On the other
hand, the action T21(z)Ba,0(ū,∅) is highly nontrivial, although it is clear that it should coincide
with the analogous action in the models with gl(3)-invariant R-matrix. Obviously, in this case
it is better to begin the proof with the vector B0,b(∅, v̄).
6.1 Action of T22(z) at a = 0 and z = 1
In the particular case a = 0 and n = 1 equation (3.5) turns into
T22(z)B0,b(∅; v̄) = λ2(z)h(v̄, z)
∑ g(ξ̄II, ξ̄I)
h(ξ̄I, z)
B0,b(∅; ξ̄II). (6.4)
12 A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, E. Ragoucy and N.A. Slavnov
The sum is taken over partitions {z, v̄} = ξ̄ ⇒ {ξ̄I, ξ̄II} with #ξ̄I = 1. We prove this action using
the standard scheme of the algebraic Bethe ansatz. The vector B0,b(∅; v̄) is given by the second
equation (2.13). Thus, we should move the operator T22(z) to the right through the product of
the operators T23(vj). Using (6.1) we easily find
T22(z)B0,b(∅; v̄) = ΛB0,b(∅; v̄) +
b∑
j=1
ΛjB0,b(∅; {v̄j , z)),
where Λ and Λj are some coefficients to be determined. Obviously, in order to obtain the
coefficient of B0,b(∅; v̄) one should use only the first term in the r.h.s. of (6.1). From this we
immediately find
Λ = λ2(z)f(v̄, z).
Then, due to the symmetry of T23(v̄) over v̄ it is enough to find Λ1 only. Permuting T22(z)
with T23(v1) we should use the second term in the r.h.s. of (6.1). We have
T22(z)
T23(v̄)
λ2(v̄)
Ω = T22(z)
T23(v1)T23(v̄1)
λ2(v̄)h(v̄1, v1)
Ω = g(z, v1)T23(z)
T22(v1)T23(v̄1)
λ2(v̄)h(v̄1, v1)
Ω + UWT,
where UWT means unwanted terms, i.e., the terms that cannot give a contribution to the
coefficient Λ1. Now, moving T22(v1) through the product T23(v̄) we should use only the first
term in the r.h.s. of (6.1), which gives us
T22(z)
T23(v̄)
λ2(v̄)
Ω = g(z, v1)g(v̄1, v1)T23(z)
T23(v̄1)
λ2(v̄)
λ2(v1)Ω + UWT,
where we used g(v̄1, v1) = f(v̄1, v1)/h(v̄1, v1). It remains to combine T23(z) and T23(v̄1) into
T23({z, v̄1}) and we arrive at
T22(z)
T23(v̄)
λ2(v̄)
Ω = λ2(z)g(z, v1)g(v̄1, v1)h(v̄1, z)
T23({z, v̄1})
λ2(v̄1)λ2(z)
Ω + UWT,
leading to
Λ1 = λ2(z)g(z, v1)g(v̄1, v1)h(v̄1, z).
Thus, we eventually obtain
T22(z)B0,b(∅; v̄) = λ2(z)f(v̄, z)B0,b(∅; v̄)
+ λ2(z)
b∑
j=1
g(z, vj)g(v̄j , vj)h(v̄j , z)B0,b(∅; {v̄j , z}). (6.5)
It is easy to see that this formula coincides with (6.4). Indeed the first term in (6.5) corresponds
to the partition ξ̄I = z and ξ̄II = v̄ in (6.4). The other terms arise in the case of the partitions
ξ̄I = vj , j = 1, . . . , b, and ξ̄II = {z, v̄j}. Thus, action (6.4) is proved.
6.2 Induction over a
For n = 1 equation (3.5) takes the form
T22(z)Ba,b(ū; v̄) = λ2(z)h(v̄, z)
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)
h(η̄I, z)h(ξ̄I, z)
Ba,b(η̄II; ξ̄II). (6.6)
Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models 13
The sum is taken over partitions ξ̄ ⇒ {ξ̄I, ξ̄II} and η̄ ⇒ {η̄I, η̄II} with #ξ̄I = #η̄I = 1. We assume
that (6.6) is valid for some a ≥ 0 and b arbitrary. Then, due to recursion (2.11) we have
T22(z1)Ba+1,b({ū; z2}; v̄) = T22(z1)
T12(z2)Ba,b(ū; v̄)
λ2(z2)f(v̄, z2)
− T22(z1)
b∑
j=1
g(z2, vj)g(v̄j , vj)T13(z2)Ba,b−1(ū; v̄j)
λ2(z2)f(v̄, z2)
. (6.7)
We see that in order to compute the action of T22(z1) onto Ba+1,b({z2, ū}; v̄) we should calculate
the successive actions of the operators T22(z1)T12(z2) and T22(z1)T13(z2). This can be done
via (6.2) and (6.3)
T22(z1)T12(z2)Ba,b(ū; v̄)
=
(
f(z1, z2)T12(z2)T22(z1) + g(z2, z1)T12(z1)T22(z2)
)
Ba,b(ū; v̄), (6.8)
T22(z1)T13(z2)Ba,b−1(ū; v̄j) = T13(z2)T22(z1)Ba,b−1(ū; v̄j)
+ g(z1, z2)
(
T12(z2)T23(z1)− T12(z1)T23(z2)
)
Ba,b−1(ū; v̄j). (6.9)
Thus, we have reduced the action T22(z1)Ba+1,b({ū; z2}; v̄) to the calculation of several successive
actions. In all of them the operator T22 acts either on Ba,b(ū; v̄) or on Ba,b−1(ū; v̄j), which are
known due to the induction assumption. The actions of other operators Tij with i < j are
already known for a and b arbitrary.
6.2.1 Successive action of T12 and T23
We begin our calculation with the successive action of the operators T12 and T23. Using (3.3)
we have
T12(z2)T23(z1)Ba,b(ū; v̄) = λ2(z1)h(v̄, z1)
∑ f(η̄I, η̄II)
h(η̄I, z1)
T12(z2)Ba,b+1(η̄II; ξ̄).
Here η̄ = {z1, ū} and ξ̄ = {z1, v̄}. The sum is taken over partitions η̄ ⇒ {η̄I, η̄II} with #η̄I = 1.
Then we use (3.2) and find
T12(z2)T23(z1)Ba,b(ū; v̄)
= λ2(z̄)h(v̄, z1)h(ξ̄, z2)
∑ f(η̄I, η̄II)
h(η̄I, z1)
g(ξ̄II, ξ̄I)
h(ξ̄I, z2)
Ba+1,b+1({η̄II, z2}; ξ̄II). (6.10)
Here already ξ̄ = {z̄, v̄}, however we still have η̄ = {z1, ū}. Replacing {η̄II, z2} by η̄II we re-
cast (6.10) in the form
T12(z2)T23(z1)Ba,b(ū; v̄)
= λ2(z̄)h(v̄, z̄)h(z1, z2)
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)
f(η̄I, z2)h(η̄I, z1)h(ξ̄I, z2)
Ba+1,b+1(η̄II; ξ̄II).
Here η̄ = {z̄, ū}, and the sum is taken over partitions η̄ ⇒ {η̄I, η̄II} and ξ̄ ⇒ {ξ̄I, ξ̄II} with
#η̄I = #ξ̄I = 1. Note that the condition z2 /∈ η̄I is ensured automatically. Indeed, if z2 ∈ η̄I,
then 1/f(η̄I, z2) = 0.
Replacing here z1 ↔ z2 we obtain
T12(z1)T23(z2)Ba,b(ū; v̄)
= λ2(z̄)h(v̄, z̄)h(z2, z1)
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)
f(η̄I, z1)h(η̄I, z2)h(ξ̄I, z1)
Ba+1,b+1(η̄II; ξ̄II).
14 A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, E. Ragoucy and N.A. Slavnov
Thus, we find
g(z1, z2)
(
T12(z2)T23(z1)− T12(z1)T23(z2)
)
Ba,b(ū; v̄)
= λ2(z̄)h(v̄, z̄)
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)
h(η̄I, z̄)
Ba+1,b+1(η̄II; ξ̄II)
×
{
f(z1, z2)
g(η̄I, z2)h(ξ̄I, z2)
+
f(z2, z1)
g(η̄I, z1)h(ξ̄I, z1)
}
. (6.11)
6.2.2 Successive action of T13 and T22
Combining the actions (6.6) and (3.1) we obtain
T13(z2)T22(z1)Ba,b(ū; v̄)
= λ2(z̄)h(v̄, z1)
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)h(ξ̄II, z2)
h(η̄I, z1)h(ξ̄I, z1)
Ba+1,b+1({η̄II, z2}; {ξ̄II, z2}).
Here η̄ = {ū, z1} and ξ̄ = {v̄, z1}. Replacing {η̄II, z2} with η̄II and {ξ̄II, z2} with ξ̄II we arrive at
T13(z2)T22(z1)Ba,b(ū; v̄) = λ2(z̄)h(v̄, z̄)h(z1, z2)
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)Ba+1,b+1(η̄II; ξ̄II)
f(η̄I, z2)g(z2, ξ̄I)h(η̄I, z1)h(ξ̄I, z̄)
. (6.12)
Here already η̄ = {ū, z̄} and ξ̄ = {v̄, z̄}. The sum is taken over partitions η̄ ⇒ {η̄I, η̄II} and
ξ̄ ⇒ {ξ̄I, ξ̄II} with #η̄I = #ξ̄I = 1.
Now we are able to calculate the successive action T22(z1)T13(z2) on Ba,b(ū; v̄). Indeed, due
to (6.9) this successive action is given by a combination of (6.11) and (6.12). A straightforward
calculation leads us to the following representation:
T22(z1)T13(z2)Ba,b(ū; v̄) = λ2(z̄)h(v̄, z̄)h(z2, z1)
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)
h(η̄I, z1)h(ξ̄I, z1)
Ba+1,b+1(η̄II; ξ̄II).
Remark 6.2. Taking into account (3.1) we conclude that if the action (6.6) is valid on the
vector Ba,b(ū; v̄), then it is also valid on vectors of the special type Ba+1,b+1(ū′; v̄′), if ū′∩ v̄′ 6= ∅.
6.2.3 Successive action of T12 and T22
Using (6.6) we obtain
T12(z2)T22(z1)Ba,b(ū; v̄) = λ2(z1)h(v̄, z1)
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)
h(η̄I, z1)h(ξ̄I, z1)
T12(z2)Ba,b(η̄II; ξ̄II).
Here η̄ = {ū, z1} and ξ̄ = {v̄, z1}. Applying (3.2) to this formula we find
T12(z2)T22(z1)Ba,b(ū; v̄)
= λ2(z̄)h(v̄, z1)
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)
h(η̄I, z1)h(ξ̄I, z1)
h(ξ̄II, z2)
g(ξ̄ii, ξ̄i)
h(ξ̄i, z2)
Ba+1,b({η̄II, z2}; ξ̄ii).
Here we first have partitions η̄ = {ū, z1} ⇒ {η̄I, η̄II} and ξ̄ = {v̄, z1} ⇒ {ξ̄I, ξ̄II}. Then we
combine ξ̄II with z2 and divide this set into new subsets {ξ̄II, z2} ⇒ {ξ̄i, ξ̄ii}. The restrictions are:
#ξ̄i = #ξ̄I = #η̄I = 1, z2 /∈ η̄I, and z2 /∈ ξ̄I. As we already did before, we replace {η̄II, z2} with
η̄II and use ξ̄II = {ξ̄i, ξ̄ii} \ {z2}. Then
T12(z2)T22(z1)Ba,b(ū; v̄) = λ2(z̄)h(v̄, z̄)h(z1, z2)
×
∑ f(η̄I, η̄II)g(ξ̄i, ξ̄I)g(ξ̄ii, ξ̄I)g(ξ̄ii, ξ̄i)
f(η̄I, z2)h(η̄I, z1)h(ξ̄I, z1)g(z2, ξ̄I)h(ξ̄I, z2)h(ξ̄i, z2)
Ba+1,b(η̄II; ξ̄ii). (6.13)
Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models 15
Setting ξ̄0 = {ξ̄i, ξ̄I} we recast (6.13) as follows
T12(z2)T22(z1)Ba,b(ū; v̄) = λ2(z̄)h(v̄, z̄)h(z1, z2)
×
∑ f(η̄I, η̄II)g(ξ̄i, ξ̄I)g(ξ̄ii, ξ̄0)
g(η̄I, z2)h(η̄I, z̄)h(ξ̄I, z1)g(z2, ξ̄I)h(ξ̄0, z2)
Ba+1,b(η̄II; ξ̄ii).
The sum over partitions of the set ξ̄ = {z1, z2, v̄} is organized as follows: first, we have partitions
ξ̄ ⇒ {ξ̄ii, ξ̄0}; second we divide ξ̄0 ⇒ {ξ̄i, ξ̄I}. The latter sum consists of two terms and can be
computed straightforwardly. This leads us to
T12(z2)T22(z1)Ba,b(ū; v̄) = λ2(z̄)h(v̄, z̄)h(z̄, z̄)
∑ f(η̄I, η̄II)g(ξ̄ii, ξ̄0)
g(η̄I, z2)h(η̄I, z̄)h(ξ̄0, z̄)
Ba+1,b(η̄II; ξ̄ii),
and relabeling ξ̄0 → ξ̄I, ξ̄ii → ξ̄II we finally obtain
T12(z2)T22(z1)Ba,b(ū; v̄)
= λ2(z̄)h(v̄, z̄)h(z̄, z̄)
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)
g(η̄I, z2)h(η̄I, z̄)h(ξ̄I, z̄)
Ba+1,b(η̄II; ξ̄II). (6.14)
Here the sum is taken over partitions η̄ ⇒ {η̄I, η̄II}, ξ̄ ⇒ {ξ̄I, ξ̄II}. The cardinalities of the subsets
are #η̄I = 1, #ξ̄I = 2.
6.2.4 Successive action of T22 and T12
Using (6.8) and (6.14) we are able to calculate the action of T22(z1)T12(z2) onto Ba,b(ū; v̄). It
is clear that for this we should take the following combination: equation (6.14) multiplied with
f(z1, z2) and the same equation with z1 ↔ z2 multiplied with g(z2, z1). This straightforward
calculation gives
T22(z1)T12(z2)Ba,b(ū; v̄) = λ2(z̄)h(v̄, z̄)h(z̄, z̄)
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)
h(η̄I, z1)h(ξ̄I, z̄)
Ba+1,b(η̄II; ξ̄II). (6.15)
Here the sum is taken over partitions η̄ ⇒ {η̄I, η̄II}, ξ̄ ⇒ {ξ̄I, ξ̄II}. The cardinalities of the subsets
are #η̄I = 1, #ξ̄I = 2.
6.3 Recursion formula
Now everything is ready for the use of recursion (6.7). Due to (5.1) we can write it as follows
Ba+1,b({ū, z2}; v̄) =
1
λ2(z2)f(v̄, z2)
(
T12(z2)Ba,b(ū; v̄)−Ψ
)
, (6.16)
where
Ψ = λ2(z2)h(v̄, z2)
∑
z2 /∈ξ̄I
g(ξ̄II, ξ̄I)
h(ξ̄I, z2)
Ba+1,b(η̄; ξ̄II). (6.17)
Here η̄ = {z2, ū} and ξ̄ = {z2, v̄}, and we used h(z2, z2) = 1. The sum is taken over partitions
ξ̄ ⇒ {ξ̄I, ξ̄II} with #ξ̄I = 1. One more restriction z2 /∈ ξ̄I is shown explicitly by the subscript of
the sum.
Recall that we assume that the action of T22(z1) on the vectors Ba,b(ū; v̄) is given by (6.6)
at some value of a ≥ 0 and arbitrary b. All the vectors in the linear combination (6.17) have
16 A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, E. Ragoucy and N.A. Slavnov
the form Ba+1,b({ū, z2}; {v̄j , z2}), that is {ū, z2} ∩ {v̄j , z2} 6= ∅. Hence, taking into account
Remark 6.2, the action of T22(z1) on these vectors is known and it is given by (6.6):
T22(z1)Ψ = λ2(z̄)h(v̄, z2)
∑
z2 /∈ξ̄I
g(ξ̄II, ξ̄I)
h(ξ̄I, z2)
h(ξ̄II, z1)
f(η̄I, η̄II)g(ξ̄ii, ξ̄i)
h(η̄I, z1)h(ξ̄i, z1)
Ba+1,b(η̄II; ξ̄ii). (6.18)
In this formula η̄ = {z̄, ū} and ξ̄ = {z̄, v̄}. At the first step we have partitions {z2, v̄} ⇒ {ξ̄I, ξ̄II}.
Then we obtain additional partitions {z̄, v̄} ⇒ {ξ̄i, ξ̄ii} and η̄ ⇒ {η̄I, η̄II}. Hereby #η̄I = #ξ̄I =
#ξ̄i = 1. Thus, one can say that the set ξ̄ = {z̄, v̄} is divided into subsets {ξ̄I, ξ̄i, ξ̄ii} with the
restrictions z1 /∈ ξ̄I and z2 /∈ ξ̄I. Substituting ξ̄II = {ξ̄i, ξ̄ii} \ {z1} into (6.18) we obtain
T22(z1)Ψ = λ2(z̄)h(v̄, z2)
∑
z2 /∈ξ̄I
g(ξ̄ii, ξ̄I)g(ξ̄i, ξ̄I)g(ξ̄ii, ξ̄i)f(η̄I, η̄II)h(ξ̄ii, z1)
g(z1, ξ̄I)h(ξ̄I, z2)h(η̄I, z1)
Ba+1,b(η̄II; ξ̄ii). (6.19)
Observe that the restriction z1 /∈ ξ̄I holds automatically due to the factor g(z1, ξ̄I)
−1. In order
to get rid of the restriction z2 /∈ ξ̄I we present T22(z1)Ψ as a difference of two terms. The first
term is just the sum over partitions in (6.19), where no restrictions on the partitions of the set ξ̄
are imposed. In the second term we simply set ξ̄I = z2. Thus,
T22(z1)Ψ = Ψ′ −Ψ′′,
where
Ψ′ = λ2(z̄)h(v̄, z2)
∑ g(ξ̄ii, ξ̄I)g(ξ̄i, ξ̄I)g(ξ̄ii, ξ̄i)f(η̄I, η̄II)h(ξ̄ii, z1)
g(z1, ξ̄I)h(ξ̄I, z2)h(η̄I, z1)
Ba+1,b(η̄II; ξ̄ii), (6.20)
and
Ψ′′ = λ2(z̄)h(v̄, z2)
∑ g(ξ̄ii, z2)g(ξ̄i, z2)g(ξ̄ii, ξ̄i)f(η̄I, η̄II)h(ξ̄ii, z1)
g(z1, z2)h(η̄I, z1)
Ba+1,b(η̄II; ξ̄ii). (6.21)
In (6.21) we have {ξ̄i, ξ̄ii} = {v̄, z1}, therefore
Ψ′′ = λ2(z̄)h(v̄, z1)f(v̄, z2)
∑ g(ξ̄ii, ξ̄i)f(η̄I, η̄II)
h(ξ̄i, z1)h(η̄I, z1)
Ba+1,b(η̄II; ξ̄ii). (6.22)
In (6.20) we can take the sum over partitions into subsets ξ̄i and ξ̄I, because it consists of two
terms only
g(ξ̄i, ξ̄I)
g(z1, ξ̄I)h(ξ̄I, z2)
+
g(ξ̄I, ξ̄i)
g(z1, ξ̄i)h(ξ̄i, z2)
=
h(z1, z2)
h(ξ̄0, z2)
,
where ξ̄0 = {ξ̄i, ξ̄I}. Thus,
Ψ′ = λ2(z̄)h(v̄, z2)h(z1, z2)
∑ g(ξ̄ii, ξ̄0)f(η̄I, η̄II)h(ξ̄ii, z1)
h(ξ̄0, z2)h(η̄I, z1)
Ba+1,b(η̄II; ξ̄ii), (6.23)
and extracting the product h(ξ̄, z1) we recast (6.23) as follows
Ψ′ = λ2(z̄)h(v̄, z̄)h(z̄, z̄)
∑ g(ξ̄ii, ξ̄0)f(η̄I, η̄II)
h(ξ̄0, z̄)h(η̄I, z1)
Ba+1,b(η̄II; ξ̄ii).
Here the sum is taken over partitions η̄ ⇒ {η̄I, η̄II}, ξ̄ ⇒ {ξ̄0, ξ̄ii} with #η̄I = 1, #ξ̄0 = 2.
Comparing this expression with (6.15) we see that
Ψ′ = T22(z1)T12(z2)Ba,b(ū; v̄).
Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models 17
Thus, we find from the recursion (6.16)
T22(z1)Ba+1,b({ū, z2}; v̄) =
T22(z1)T12(z2)Ba,b(ū; v̄)−Ψ′ + Ψ′′
λ2(z2)f(v̄, z2)
=
Ψ′′
λ2(z2)f(v̄, z2)
.
Substituting (6.22) in this expression, we arrive at
T22(z1)Ba+1,b({ū, z2}; v̄) = λ2(z1)h(v̄, z1)
∑ g(ξ̄II, ξ̄I)f(η̄I, η̄II)
h(ξ̄I, z1)h(η̄I, z1)
Ba+1,b(η̄II; ξ̄II),
where we have relabeled ξ̄i → ξ̄I and ξ̄ii → ξ̄II. Thus, the induction step is completed.
6.4 Induction over n
Actually, the induction over n for the action of T22(z̄) is a combination of the corresponding
proofs for the actions of T12(z̄) and T23(z̄). Assume that (3.5) is valid for some n− 1. Then
T22(z̄)Ba,b(ū; v̄) = (−1)n−1λ2(z̄n)h(ξ̄, z̄n)
×
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)
h(ξ̄I, z̄n)
Kn−1(z̄n|η̄I + c)T22(zn)Ba,b(η̄II; ξ̄II).
Here the sum is taken over partitions {z̄n, v̄} = ξ̄ ⇒ {ξ̄I, ξ̄II} and {z̄n, ū} = η̄ ⇒ {η̄I, η̄II} with
#ξ̄I = #η̄I = n− 1. Acting with T22(zn) onto Ba,b(η̄II; ξ̄II) we obtain
T22(z̄)Ba,b(ū; v̄) = (−1)nλ2(z̄)
h(ξ̄, z̄n)
h(zn, z̄n)
∑ f(η̄I, η̄II)g(ξ̄II, ξ̄I)
h(ξ̄I, z̄n)
Kn−1(z̄n|η̄I + c)
× h(ξ̄II, zn)
f(η̄i, η̄ii)g(ξ̄ii, ξ̄i)
h(ξ̄i, zn)
K1(zn|η̄i + c)Ba,b(η̄ii; ξ̄ii).
Here already ξ̄ = {z̄, v̄} and η̄ = {z̄, ū}, and we have additional partitions {ξ̄II, zn} ⇒ {ξ̄i, ξ̄ii}
and {η̄II, zn} ⇒ {η̄i, η̄ii} with #ξ̄i = #η̄i = 1. Thus, we can say that we have the sum over
partitions ξ̄ ⇒ {ξ̄I, ξ̄i, ξ̄ii} and η̄ ⇒ {η̄I, η̄i, η̄ii} with restrictions zn /∈ η̄I and zn /∈ ξ̄I.
Substituting ξ̄II = {ξ̄i, ξ̄ii} \ {zn}, η̄II = {η̄i, η̄ii} \ {zn} and denoting ξ̄0 = {ξ̄i, ξ̄I}, η̄0 = {η̄i, η̄I}
we obtain
T22(z̄)Ba,b(ū; v̄) = (−1)nλ2(z̄)
h(ξ̄, z̄)
h(zn, z̄n)
∑ f(η̄I, η̄i)f(η̄0, η̄ii)g(ξ̄i, ξ̄I)g(ξ̄ii, ξ̄0)
f(η̄I, zn)g(zn, ξ̄I)h(ξ̄I, z̄n)h(ξ̄0, zn)
×Kn−1(z̄n|η̄I + c)K1(zn|η̄i + c)Ba,b(η̄ii; ξ̄ii). (6.24)
Observe that the restrictions zn /∈ η̄I and zn /∈ ξ̄I hold automatically due to the factors f(η̄I, zn)
and g(zn, ξ̄I) in the denominator of (6.24). Using K1(zn|η̄i + c) = −K1(η̄i|zn)/f(η̄i, zn) we
recast (6.24) in the form
T22(z̄)Ba,b(ū; v̄) = (−1)n−1λ2(z̄)
h(ξ̄, z̄)
h(zn, z̄n)
∑ f(η̄0, η̄ii)g(ξ̄ii, ξ̄0)
f(η̄0, zn)g(zn, ξ̄0)h(ξ̄0, z̄)
×
{
g(ξ̄i, ξ̄I)g(zn, ξ̄i)h(ξ̄i, z̄n)
}{
Kn−1(z̄n|η̄I + c)K1(η̄i|zn)f(η̄I, η̄i)
}
Ba,b(η̄ii; ξ̄ii).
The sums over partitions ξ̄0 ⇒ {ξ̄i, ξ̄I} and η̄0 ⇒ {η̄i, η̄I} (see the terms in braces) were already
computed (see (5.3) and (5.4)). Thus, we arrive at
T22(z̄)Ba,b(ū; v̄) = (−1)nλ2(z̄)h(ξ̄, z̄)
∑ f(η̄0, η̄ii)g(ξ̄ii, ξ̄0)
h(ξ̄0, z̄)
Kn(z̄|η̄0 + c)Ba,b(η̄ii; ξ̄ii),
which ends the proof.
18 A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, E. Ragoucy and N.A. Slavnov
7 Induction over n for the actions of Tij(z̄) with i > j
The action formulas for all other elements of the monodromy matrix can be proved exactly in
the same manner. However, it is clear that the technical difficulty of the proofs increases when
moving from the right top corner of the monodromy matrix to the left bottom corner. It is due
to the form of the recursion formulas and the commutation relations (2.4). For example, we
have seen that for the derivation of the action of T22(z̄) one should know the actions of T12(z̄)
and T23(z̄) onto Bethe vectors. The latest are relatively simple. However, one can easily convince
oneself that to get the action of Tij(z̄) with i > j, it is necessary to know the actions of the
diagonal elements Tii(z̄), which are more involved. Therefore, we omit the detailed proofs of the
multiple actions of the operators T11(z̄) (3.4), T33(z̄) (3.6), and the operators Tij(z̄) from the
lower-triangular part of the monodromy matrix (3.7)–(3.9). However, as an illustration of the
method, we prove the multiple action of the operator T21(z̄) assuming that the action of a single
operator T21(z) is known.
As previously, the proof goes by induction over n = #z̄. We assume that the action (3.7)
holds for some n− 1. Then acting successively with T21(z̄n) and T21(zn) on Ba,b(ū; v̄) we obtain
T21(z̄)Ba,b(ū; v̄) = λ2(z̄)
h(ξ̄, z̄n)
h(zn, z̄n)
∑
r1(η̄I)
f(η̄II, η̄I)f(η̄II, η̄III)f(η̄III, η̄I)g(ξ̄II, ξ̄I)
h(ξ̄I, z̄n)f(ξ̄II, η̄I)
×Kn−1(z̄n|η̄II + c)Kn−1(η̄I|ξ̄I + c)h(ξ̄II, zn)r1(η̄i)
f(η̄ii, η̄i)f(η̄ii, η̄iii)f(η̄iii, η̄i)g(ξ̄ii, ξ̄i)
h(ξ̄i, zn)f(ξ̄ii, η̄i)
×K1(zn|η̄ii + c)K1(η̄i|ξ̄i + c)Ba−n,b(η̄iii; ξ̄ii). (7.1)
Here #η̄i = #η̄ii = #ξ̄i = 1 and #η̄I = #η̄II = #ξ̄I = n − 1. Originally we have partitions
{z̄n, ū} = η̄ ⇒ {η̄I, η̄II, η̄III} and {z̄n, v̄} = ξ̄ ⇒ {ξ̄I, ξ̄II}. Then we have additional partitions
{zn, η̄II} ⇒ {η̄i, η̄ii, η̄iii} and {zn, ξ̄II} ⇒ {ξ̄i, ξ̄ii}. Thus, in equation (7.1) we have {z̄, ū} = η̄ and
{z̄, v̄} = ξ̄. Setting there η̄III = {η̄i, η̄ii, η̄iii} \ {zn} and ξ̄II = {ξ̄i, ξ̄ii} \ {zn} we arrive at
T21(z̄)Ba,b(ū; v̄) = λ2(z̄)
h(ξ̄, z̄)
h(zn, z̄n)
∑
r1(η̄I)r1(η̄i)
f(η̄ii, η̄i)f(η̄ii, η̄iii)f(η̄iii, η̄i)g(ξ̄ii, ξ̄i)
h(ξ̄i, zn)f(ξ̄ii, η̄i)
× f(η̄II, η̄I)f(η̄II, η̄i)f(η̄II, η̄ii)f(η̄II, η̄iii)f(η̄i, η̄I)f(η̄ii, η̄I)f(η̄iii, η̄I)g(ξ̄i, ξ̄I)g(ξ̄ii, ξ̄I)
h(ξ̄I, zn)h(ξ̄I, z̄n)f(ξ̄i, η̄I)f(ξ̄ii, η̄I)f(η̄II, zn)g(zn, ξ̄I)
×Kn−1(z̄n|η̄II + c)K1(zn|η̄ii + c)Kn−1(η̄I|ξ̄I + c)K1(η̄i|ξ̄i + c)Ba−n,b(η̄iii; ξ̄ii). (7.2)
Now we set {η̄I, η̄i} = η̄0, {η̄II, η̄ii} = η̄0′ , and {ξ̄I, ξ̄i} = ξ̄0. We also transform K1(zn|η̄ii + c) =
−K1(η̄ii|zn)/f(η̄ii, zn) and K1(η̄i|ξ̄i + c) = −K1(ξ̄i|η̄i)/f(ξ̄i, η̄i). Then (7.2) takes the form
T21(z̄)Ba,b(ū; v̄) = λ2(z̄)
h(ξ̄, z̄)
h(zn, z̄n)
∑ r1(η̄0)f(η̄0′ , η̄0)f(η̄0′ , η̄iii)f(η̄iii, η̄0)g(ξ̄ii, ξ̄0)g(ξ̄i, ξ̄I)
h(ξ̄I, z̄)f(ξ̄i, η̄0)f(η̄0′ , zn)f(ξ̄ii, η̄0)g(zn, ξ̄I)h(ξ̄i, zn)
×
{
Kn−1(z̄n|η̄II + c)K1(η̄ii|zn)f(η̄II, η̄ii)
}{
Kn−1(η̄I|ξ̄I + c)K1(ξ̄i|η̄i)f(η̄i, η̄I)
}
× Ba−n,b(η̄iii; ξ̄ii). (7.3)
The sums over partitions in braces can be computed via (A.2):∑
η̄0′⇒{η̄II,η̄ii}
Kn−1(z̄n|η̄II + c)K1(η̄ii|zn)f(η̄II, η̄ii) = −f(η̄0′ , zn)Kn(z̄|η̄0′ + c),
and ∑
η̄0⇒{η̄I,η̄i}
Kn−1(η̄I|ξ̄I + c)K1(ξ̄i|η̄i)f(η̄i, η̄I) = (−1)n−1Kn(ξ̄0|η̄0)
f(ξ̄I, η̄0)
.
Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models 19
Substituting this into (7.3) we arrive at
T21(z̄)Ba,b(ū; v̄) = λ2(z̄)
h(ξ̄, z̄)
h(zn, z̄n)
∑ r1(η̄0)f(η̄0′ , η̄0)f(η̄0′ , η̄iii)f(η̄iii, η̄0)g(ξ̄ii, ξ̄0)
h(ξ̄0, z̄)f(ξ̄ii, η̄0)g(zn, ξ̄0)
×Kn(z̄|η̄0′ + c)Kn(η̄0|ξ̄0 + c)
{
g(ξ̄i, ξ̄I)h(ξ̄i, z̄n)g(zn, ξ̄i)
}
Ba−n,b(η̄iii; ξ̄ii),
where we again replaced Kn(ξ̄0|η̄0) by Kn(η̄0|ξ̄0 + c) via (2.7). The sum over partitions in braces
can be calculated via Lemma A.1∑
ξ̄0⇒{ξ̄I,ξ̄i}
g(ξ̄i, ξ̄I)h(ξ̄i, z̄n)g(zn, ξ̄i) = h(zn, z̄n)g(zn, ξ̄0).
Thus, we finally obtain
T21(z̄)Ba,b(ū; v̄) = λ2(z̄)h(ξ̄, z̄)
∑ r1(η̄0)f(η̄0′ , η̄0)f(η̄0′ , η̄iii)f(η̄iii, η̄0)g(ξ̄ii, ξ̄0)
h(ξ̄0, z̄)f(ξ̄ii, η̄0)
×Kn(z̄|η̄0′ + c)Kn(η̄0|ξ̄0 + c)Ba−n,b(η̄iii; ξ̄ii),
which coincides with the original formula up to the labeling of the subsets.
8 Conclusion
In this paper we obtained compact expressions for the multiple actions of the operators Tij onto
the Bethe vectors in the models with gl(2|1) symmetry. Formally, our method can be extended
to higher rank algebras as well, although the technical complexity of the calculations increases
significantly with the rank and can be rapidly untractable.
Comparing the actions of Tij onto Ba,b(ū; v̄) with the analogous actions in the gl(3)-based
models [6] one can observe that they are quite similar. The main difference is that some
of the Izergin determinants in the gl(3) actions are replaced with the Cauchy determinants
in their gl(2|1) analogs. This is the consequence of Z2 grading that ‘trivializes’ some of the
relations and helps to go further in the calculation. It is well possible that there exist some
general formulas, which valid for both cases. It would be important to find such formulas,
especially for the models with graded algebras of higher rank.
The obtained formulas allow one to tackle the problem of scalar products of the Bethe vectors.
For this we can use explicit expressions for the dual Bethe vectors Ca,b(ū, v̄) obtained in [40].
Actually, they can be obtained by a transposition3 of (2.9), (2.10), for instance,
Ca,b(ū, v̄) = (−1)b(b−1)/2
∑
g(v̄I, ūI)
f(ūI, ūII)g(v̄II, v̄I)h(ūI, ūI)
λ2(v̄II)λ2(ū)f(v̄, ū)
Ω†T32(v̄II)T21(ūII)T31(ūI).
Having this representation we reduce the evaluation of the scalar product to the calculation of
successive multiple actions of the operators T31, T21, and T32 onto Bethe vectors. This will be
the subject of our further publication.
A Identities for rational functions
Lemma A.1. Let w̄, ū and v̄ be sets of complex variables with #ū = m1, #v̄ = m2, and
#w̄ = m1 +m2, where m1 and m2 are fixed arbitrary integers. Then∑
g(w̄I, ū)g(w̄II, v̄)g(w̄II, w̄I) =
g(w̄, ū)g(w̄, v̄)
g(ū, v̄)
. (A.1)
3More precisely, one should use an antimorphism of the algebra gl(2|1) relating Tij and Tji (see [40] for details).
20 A. Hutsalyuk, A. Liashyk, S.Z. Pakuliak, E. Ragoucy and N.A. Slavnov
The sum is taken with respect to all partitions of the set w̄ into subsets w̄I and w̄II with #w̄I = m1
and #w̄II = m2.
The proof of this lemma is given in [47]. Let us show how Lemma A.1 works. In equation (5.3)
we have a sum
I =
∑
ξ̄0⇒{ξ̄I,ξ̄i}
g(ξ̄i, ξ̄I)g(zn, ξ̄i)h(ξ̄i, z̄n),
where #ξ̄i = 1 and #ξ̄I = n − 1. First, we reduce this sum to the form (A.1) using h(u, v) =
1/g(u, v − c). We have
I = −h(ξ̄0, z̄n)
∑
ξ̄0⇒{ξ̄I,ξ̄i}
g(ξ̄i, ξ̄I)
g(ξ̄i, zn)
h(ξ̄I, z̄n)
= −h(ξ̄0, z̄n)
∑
ξ̄0⇒{ξ̄I,ξ̄i}
g(ξ̄i, ξ̄I)g(ξ̄i, zn)g(ξ̄I, z̄n − c).
Now we can directly apply (A.1), and we arrive at
I = −h(ξ̄0, z̄n)
g(ξ̄0, zn)g(ξ̄0, z̄n − c)
g(z̄n − c, zn)
= g(zn, ξ̄0)h(zn, z̄n).
Lemma A.2. Let w̄, ū and v̄ be sets of complex variables with #ū = m1, #v̄ = m2, and
#w̄ = m1 +m2. Then∑
Km1(w̄I|ū)Km2(v̄|w̄II)f(w̄II, w̄I) = (−1)m1f(w̄, ū)Km1+m2({ū− c, v̄}|w̄). (A.2)
The sum is taken with respect to all partitions of the set w̄ into subsets w̄I and w̄II with #w̄I = m1
and #w̄II = m2.
The proof of this lemma is given in [5].
Acknowledgements
The work of A.L. has been funded by the Russian Academic Excellence Project 5-100 and by
joint NASU-CNRS project F14-2016. The work of S.P. was supported in part by the RFBR
grant 16-01-00562-a. N.A.S. was supported by the grants RFBR-15-31-20484-mol-a-ved and
RFBR-14-01-00860-a.
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http://dx.doi.org/10.1007/BF01095104
http://dx.doi.org/10.1007/BF01091463
http://dx.doi.org/10.1007/s11232-014-0180-z
http://arxiv.org/abs/1401.4355
http://dx.doi.org/10.1007/s11232-014-0145-2
http://arxiv.org/abs/1311.3500
http://dx.doi.org/10.1088/1751-8113/48/43/435001
http://arxiv.org/abs/1503.00546
http://dx.doi.org/10.3842/SIGMA.2015.064
http://arxiv.org/abs/1502.01966
http://dx.doi.org/10.1016/j.nuclphysb.2015.02.006
http://arxiv.org/abs/1412.6037
http://arxiv.org/abs/1604.02311
http://dx.doi.org/10.1103/PhysRevLett.96.257202
http://arxiv.org/abs/cond-mat/0603681
http://dx.doi.org/10.1088/1742-5468/2007/08/P08022
http://arxiv.org/abs/0706.4327
http://dx.doi.org/10.1103/PhysRevB.36.5177
http://dx.doi.org/10.1103/PhysRevB.36.5177
http://dx.doi.org/10.1088/1742-5468/2007/08/P08030
http://arxiv.org/abs/0705.3569
http://dx.doi.org/10.1007/BF01016531
http://dx.doi.org/10.1088/1742-5468/2015/03/P03019
http://arxiv.org/abs/1501.06253
http://arxiv.org/abs/1604.05343
http://dx.doi.org/10.1103/PhysRevLett.20.98
http://dx.doi.org/10.1103/PhysRevB.12.3795
http://dx.doi.org/10.3842/SIGMA.2013.048
http://arxiv.org/abs/math.QA/0702277
http://dx.doi.org/10.1016/j.nuclphysb.2013.06.015
http://arxiv.org/abs/1306.0552
http://dx.doi.org/10.1007/s00220-014-2019-8
http://arxiv.org/abs/1204.2089
http://dx.doi.org/10.1103/PhysRevLett.19.1312
http://dx.doi.org/10.1103/PhysRevB.37.3759
1 Introduction
2 Definitions
2.1 Bethe vectors
3 Multiple actions of the operators Tij onto Bethe vectors
3.1 Actions of Tij() with i<j
3.2 Actions of Tii()
3.3 Actions of Tij() with i>j
4 On-shell Bethe vectors
5 Proofs of multiple actions for Tij with i<j
5.1 Proof for T13
5.2 Proof for T12
5.3 Proof for T23
6 Proof of the multiple action of the operator T22
6.1 Action of T22(z) at a=0 and z=1
6.2 Induction over a
6.2.1 Successive action of T12 and T23
6.2.2 Successive action of T13 and T22
6.2.3 Successive action of T12 and T22
6.2.4 Successive action of T22 and T12
6.3 Recursion formula
6.4 Induction over n
7 Induction over n for the actions of Tij() with i>j
8 Conclusion
A Identities for rational functions
References
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