Multiple Actions of the Monodromy Matrix in $\mathfrak{gl}(2|1)$-Invariant Integrable Models

We study $\mathfrak{gl}(2|1)$ 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.


Introduction
Algebraic Bethe ansatz is one of the most famous applications of the quantum inverse scattering 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 consideration [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 systems [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-deformation. 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 Equation (2.3) holds in the tensor product of graded spaces C 2|1 ⊗ C 2|1 ⊗ H, where H is a Hilbert space of a Hamiltonian. It implies commutation relations between the monodromy matrix entries: [T ij (u), T kl (v)} = (−1) [

i]([k]+[l])+[k]
[l] g(u, v) T kj (v)T il (u) − T kj (u)T il (v) , (2.4) where we introduced a graded commutator The graded transfer matrix is defined as the supertrace of the monodromy matrix 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 conventions on the notation. In addition to the functions g(x, y) we also introduce the functions .
Below, we will permanently have to deal with sets of variables which will be denoted by a bar:ū,v,η etc. Individual elements of the sets are denoted by subscripts and without a bar: u k , v , η j etc. The notationū ± c means that all the elements of the setū are shifted by ±c: u ± c = {u 1 ± c, . . . , u n ± 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:ū I ,v II ,η ii ,ξ 0 etc. For example, the notationū ⇒ {ū I ,ū II } means that the set u is divided into two disjoint subsetsū I andū II . 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 {v,z} =ξ. Finally we use a special notation u j ,v k and so on for the setsū \ {u j },v \ {v k } 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 This notation is also used for the product of commuting operators, In various formulas the Izergin determinant K n (x|ȳ) appears 1 . It is defined for two setsx andȳ with common cardinality #x = #ȳ = n, (2.6) We draw the readers attention that according to the convention on the shorthand notation h(x,ȳ) in (2.6) means the double product of the h-functions over all parametersx andȳ. It is easy to see from definition (2.6) that K 1 (x|y) = g(x, y) and

Bethe vectors
For further calculation we need explicit formulas for gl(2|1) Bethe vectors in terms of the monodromy matrix entries T ij (u). We recall that generically Bethe vectors are special polynomials in operators T ij (u) with i ≤ j applied to the pseudovacuum vector Ω. This vector possesses the following properties: Here λ i (u) are some scalar functions depending on a specific model. Below we will also use the ratios of these functions We extend the convention on the shorthand notation to the products of the functions λ i and r k . For example, We denote the Bethe vectors as B a,b (ū;v). They depend on two sets of variables (Bethe parameters)ū = {u 1 , . . . , u a } andv = {v 1 , . . . , v b }, where a, b = 0, 1, . . . . Explicit representations for gl(2|1) Bethe vectors were obtained in 2 [40].
Here the sum is taken over partitionsv ⇒ {v I ,v II } andū ⇒ {ū I ,ū II } with the restriction #ū I = #v I = 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 where K n 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 onū andv (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 monodromy matrix entry T ij (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 T ij (z) onto B a,b (ū;v) 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 (2.11) The second recursion reads We recall that in these formulasv j andū j respectively meanv \{v j } andū\{u j }. 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 One can also easily derive the actions of T ij onto either B a,0 (ū; ∅) or B 0,b (∅;v), 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.

Multiple actions of the operators T ij onto Bethe vectors
The main result of this paper consists of explicit formulas of the multiple actions of the monodromy matrix entries onto Bethe vectors. We show that these actions always reduce to finite linear combinations of Bethe vectors.

Actions of T ij (z) with i < j
• Multiple action of T 13 (z): (3.1) • Multiple action of T 12 (z): Here the sum is taken over partitionsξ ⇒ {ξ I ,ξ II } with #ξ I = n.
• Multiple action of T 23 (z): Here the sum is taken over partitionsη ⇒ {η I ,η II } with #η I = n.
• Multiple action of T 31 (z): Here the sum is taken over partitionsξ The proofs of the multiple action formulas will be given in Sections 5-7.

On-shell Bethe vectors
The action formulas (3.1)-(3.9) are valid for generic complex numbersz,ū, andv. In this section we consider them for on-shell Bethe vector B a,b (ū;v), that is when the parametersū andv satisfy a system of Bethe equations (see (4.6)).
In order to find explicitly the result of the transfer matrix action onto B a,b (ū;v) one should set n = 1 in (3.4)-(3.6). Then the subsetsη I andξ I consist of one element only. Obviously, there are two essentially different types of partitions of the setη = {z,ū}: Similarly, there are two different types of partitions of the setξ = {z,v}: Thus, the action of T (z) onto B a,b (ū;v) can be written in the form where τ , Λ j ,Λ k , and M jk are numerical coefficients. In order to find τ (z|ū;v) we substitute the partitions (4.1) and (4.3) into (3.4)-(3.6). We obtain where we have used h(z, z) = 1 and In order to find Λ j we substitute the partitions (4.2) and (4.3) into (3.4)-(3.6). We find Similarly, in order to findΛ k we substitute the partitions (4.1) and (4.4) into (3.4)-(3.6). This gives us is an eigenvector of T (z), then the coefficients Λ j andΛ k must vanish for arbitrary z. Setting Λ j = 0 for j = 1, . . . , a andΛ k = 0 for k = 1, . . . , b we arrive at a system of equations Let us check that M jk = 0 provided the system (4.6) is fulfilled. Substituting the partitions (4.2) and (4.4) into (3.4)-(3.6) we obtain Substituting here r 1 (u j ) and r 3 (v k ) from equations (4.6), we immediately find that M jk = 0 due to the identity Thus, the system (4.6) can be treated as the system of Bethe equations for the parametersū andv. If (4.6) holds, then the corresponding Bethe vector B a,b (ū;v) 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,v) has no poles in the points z = u j , j = 1, . . . , a, and z = v k , k = 1, . . . , b due to the system (4.6).

Proofs of multiple actions for T ij 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 elements T ij 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.

Proof for T 13
For n = 1 equation (3.1) follows directly from the definitions of the Bethe vectors. Let us take, for instance, (2.9) and set thereū = {z,ū } andv = {z,v }. Then the product 1/f (v,ū) vanishes, as it contains 1/f (z, z). This zero, however, can be compensated if and only if z ∈ū I and z ∈v I . Indeed, in this case the product g(v I ,ū I ) contains a singular factor g(z, z). Thus, we should consider only such partitions, for which z ∈ū I and z ∈v I . Therefore we should set: After evident cancellations we arrive at 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 and thus, (3.1) is proved. Using (3.1) one can recast recursions (2.11) and (2.12) as follows and One can easily recognize in these equations the actions (3.2) and (3.3) for n = 1. Then one should use induction over n.

Proof for T 12
Assume that (3.2) holds for some n − 1. Then Hereη = {z n ,ū},ξ = {z n ,v}, and the sum runs through the partitionsξ ⇒ {ξ I ,ξ II } with #ξ I = n − 1. Acting with T 12 (z n ) we obtain Here alreadyη = {z,ū} andξ = {z,v}. The sum first is taken over partitions {z n ,v} ⇒ {ξ I ,ξ II } with #ξ I = n − 1, and then over partitions {z n ,ξ II } ⇒ {ξ i ,ξ ii } with #ξ i = 1. One can say that the sum is taken over partitions {z,v} =ξ ⇒ {ξ I ,ξ i ,ξ ii } with restrictions z n / ∈ξ I , #ξ I = n − 1, and #ξ i = 1. Presentingξ II asξ II = {ξ i ,ξ ii } \ {z n } we obtain and hence, Observe that the condition z n / ∈ξ I is ensured by the product g(z n ,ξ I ) in the denominator. Hence, we can say that the sum is taken over partitionsξ ⇒ {ξ I ,ξ i ,ξ ii } with the restrictions on the cardinalities of the subsets only. Setting {ξ I ,ξ i } =ξ 0 we recast (5.2) as follows The sum over partitionsξ 0 ⇒ {ξ I ,ξ i } can be computed via Lemma A.1 where we took into account that #ξ 0 = n. Thus, we arrive at which coincides with (3.2) up to a relabeling of the subsets.
6 Proof of the multiple action of the operator T 22 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 T 22 (u). The strategy of the proof is the following. First, we prove equation (3.5) for a = #ū = 0 and n = #z = 1. This can be done either via the standard consideration of the algebraic Bethe ansatz or using induction over b = #v. In both cases we use (2.13) and the relation 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) Finally, when equation (3.5) is proved for n = 1 and arbitrary a and b we use induction over n. 6.1 Action of T 22 (z) at a = 0 and z = 1 In the particular case a = 0 and n = 1 equation (3.5) turns into The sum is taken over partitions {z,v} =ξ ⇒ {ξ I ,ξ II } with #ξ I = 1. We prove this action using the standard scheme of the algebraic Bethe ansatz. The vector B 0,b (∅;v) is given by the second equation (2.13). Thus, we should move the operator T 22 (z) to the right through the product of the operators T 23 (v j ). Using (6.1) we easily find where Λ and Λ j are some coefficients to be determined. Obviously, in order to obtain the coefficient of B 0,b (∅;v) one should use only the first term in the r.h.s. of (6.1). From this we immediately find Then, due to the symmetry of T 23 (v) overv it is enough to find Λ 1 only. Permuting T 22 (z) with T 23 (v 1 ) we should use the second term in the r.h.s. of (6.1). We have where U W T means unwanted terms, i.e., the terms that cannot give a contribution to the coefficient Λ 1 . Now, moving T 22 (v 1 ) through the product T 23 (v) we should use only the first term in the r.h.s. of (6.1), which gives us . It remains to combine T 23 (z) and T 23 (v 1 ) into T 23 ({z,v 1 }) and we arrive at Thus, we eventually obtain It is easy to see that this formula coincides with (6.4). Indeed the first term in (6.5) corresponds to the partitionξ I = z andξ II =v in (6.4). The other terms arise in the case of the partitions ξ I = v j , j = 1, . . . , b, andξ II = {z,v j }. Thus, action (6.4) is proved.

Induction over a
For n = 1 equation (3.5) takes the form The sum is taken over partitionsξ ⇒ {ξ I ,ξ II } andη ⇒ {η I ,η II } with #ξ I = #η I = 1. We assume that (6.6) is valid for some a ≥ 0 and b arbitrary. Then, due to recursion (2.11) we have We see that in order to compute the action of T 22 (z 1 ) onto B a+1,b ({z 2 ,ū};v) we should calculate the successive actions of the operators T 22 (z 1 )T 12 (z 2 ) and T 22 (z 1 )T 13 (z 2 ). This can be done via (6.2) and (6.3)

Successive action of T 13 and T 22
Combining the actions (6.6) and (3.1) we obtain Here alreadyη = {ū,z} andξ = {v,z}. The sum is taken over partitionsη ⇒ {η I ,η II } and ξ ⇒ {ξ I ,ξ II } with #η I = #ξ I = 1. Now we are able to calculate the successive action T 22 (z 1 )T 13 (z 2 ) on B a,b (ū;v). 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: Remark 6.2. Taking into account (3.1) we conclude that if the action (6.6) is valid on the vector B a,b (ū;v), then it is also valid on vectors of the special type B a+1,b+1 (ū ;v ), ifū ∩v = ∅.

Successive action of T 22 and T 12
Using (6.8) and (6.14) we are able to calculate the action of T 22 (z 1 )T 12 (z 2 ) onto B a,b (ū;v). It is clear that for this we should take the following combination: equation (6.14) multiplied with f (z 1 , z 2 ) and the same equation with z 1 ↔ z 2 multiplied with g(z 2 , z 1 ). This straightforward calculation gives Here the sum is taken over partitionsη ⇒ {η I ,η II },ξ ⇒ {ξ I ,ξ II }. The cardinalities of the subsets are #η I = 1, #ξ I = 2.

Recursion formula
Now everything is ready for the use of recursion (6.7). Due to (5.1) we can write it as follows Hereη = {z 2 ,ū} andξ = {z 2 ,v}, and we used h(z 2 , z 2 ) = 1. The sum is taken over partitions ξ ⇒ {ξ I ,ξ II } with #ξ I = 1. One more restriction z 2 / ∈ξ I is shown explicitly by the subscript of the sum.
Recall that we assume that the action of T 22 (z 1 ) on the vectors B a,b (ū;v) is given by (6.6) at some value of a ≥ 0 and arbitrary b. All the vectors in the linear combination (6.17) have the form B a+1,b ({ū, z 2 }; {v j , z 2 }), that is {ū, z 2 } ∩ {v j , z 2 } = ∅. Hence, taking into account Remark 6.2, the action of T 22 (z 1 ) on these vectors is known and it is given by (6.6): Observe that the restriction z 1 / ∈ξ I holds automatically due to the factor g(z 1 ,ξ I ) −1 . In order to get rid of the restriction z 2 / ∈ξ I we present T 22 (z 1 )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 setξ are imposed. In the second term we simply setξ I = z 2 . Thus, (6.20) and In (6.20) we can take the sum over partitions into subsetsξ i andξ I , because it consists of two terms only (6.23) and extracting the product h(ξ, z 1 ) we recast (6.23) as follows Here the sum is taken over partitionsη ⇒ {η I ,η II },ξ ⇒ {ξ 0 ,ξ ii } with #η I = 1, #ξ 0 = 2. Comparing this expression with (6.15) we see that Thus, we find from the recursion (6.16) .
Substituting (6.22) in this expression, we arrive at where we have relabeledξ i →ξ I andξ ii →ξ II . Thus, the induction step is completed.
Observe that the restrictions z n / ∈η I and z n / ∈ξ I hold automatically due to the factors f (η I , z n ) and g(z n ,ξ I ) in the denominator of (6.24). Using K 1 (z n |η i + c) = −K 1 (η i |z n )/f (η i , z n ) we recast (6.24) in the form The sums over partitionsξ 0 ⇒ {ξ i ,ξ I } andη 0 ⇒ {η i ,η I } (see the terms in braces) were already computed (see (5.3) and (5.4)). Thus, we arrive at which ends the proof.
The sum is taken with respect to all partitions of the setw into subsetsw I andw II with #w I = m 1 and #w II = m 2 .
Now we can directly apply (A.1), and we arrive at I = −h(ξ 0 ,z n ) g(ξ 0 , z n )g(ξ 0 ,z n − c) g(z n − c, z n ) = g(z n ,ξ 0 )h(z n ,z n ). The sum is taken with respect to all partitions of the setw into subsetsw I andw II with #w I = m 1 and #w II = m 2 .
The proof of this lemma is given in [5].