On quantum separation of variables beyond fundamental representations

We describe the extension, beyond fundamental representations of the Yang-Baxter algebra, of our new construction of separation of variables basis for quantum integrable lattice models. The key idea underlying our approach is to use the commuting conserved charges of the quantum integrable models to generate the basis in which their spectral problem is $separated$, i.e. in which the wave functions are factorized in terms of specific solutions of a functional equation. For the so-called"non-fundamental"models we construct two different SoV basis. The first is given from the fundamental quantum Lax operator having isomorphic auxiliary and quantum spaces and that can be obtained by fusion of the original quantum Lax operator. The construction essentially follows the one we used previously for fundamental models and allows us to derive the simplicity and diagonalizability of the transfer matrix spectrum. Then, starting from the original quantum Lax operator and using the full tower of the fused transfer matrices, we introduce a second SoV basis for which the proof of the separation of the transfer matrix spectrum is naturally derived. We show that, under some special choice, this second SoV basis coincides with the one associated to the Sklyanin's approach. Moreover, we derive the finite difference type (quantum spectral curve) functional equation and the set of its solution defining the complete transfer matrix spectrum. This is explicitly implemented for the integrable quantum models associated to the higher spin representations of the general quasi-periodic $Y(gl_{2})$ Yang-Baxter algebra. Our SoV approach also leads to the construction of a $Q$-operator in terms of the fused transfer matrices. Finally, we show that the $Q$-operator family can be equivalently used as the family of commuting conserved charges enabling to construct our SoV basis.


Introduction
In this article we continue the development of our new approach [1][2][3] to generate the separation of variables (SoV) complete characterization of the spectrum of quantum integrable lattice models. We use the framework of the quantum inverse scattering method [4][5][6][7][8][9][10][11][12] and its associated Yang-Baxter algebra. Let us stress that in this context, the quantum version of the separation of variables has been pioneered by E. K. Sklyanin in a series of beautiful seminal works [13][14][15][16][17][18]. The main motivation to develop this new paradigm was to overcome several difficulties in applying the algebraic Bethe ansatz (ABA) in several important cases like the open Toda chain, in particular linked to the absence of an obvious reference state, see e.g. [13]. More conceptually, it was also designed to have a resolution scheme at the quantum level that would be the analog of the standard Hamilton-Jacobi method in classical Hamiltonian mechanics, see e.g. [19]. In particular, the main feature of the SoV method is that it is not an ansatz. As such, it leads to the possibility to find the complete characterization of the spectrum of quantum integrable models, this completeness question being in general a difficult task within the ABA method, see e.g. [20,21].
The key ingredient of the Sklyanin's approach is the construction, from the generators of the Yang-Baxter algebra, of two operator families B(λ) and A(λ), depending on a complex spectral parameter λ ∈ C, and satisfying the following properties. The B-family must be a commuting family of simultaneously diagonalizable operators having simple spectrum. The separate variables are then given by the complete set of commuting operators Y n such that B(Y n ) = 0. Their common eigenbasis defines the SoV basis. The A-family forms also a continuous set of commuting operators, with simple spectrum, which, thanks to their commutation relations with the B-family stemming from the Yang-Baxter algebra, define the shift operators over the spectrum of the separate variables. Moreover, the A-family and the transfer matrices of the model satisfy over the spectrum of the separate variables closed secular equations (the analog of the Hamilton-Jacobi equations), the socalled quantum spectral curve equation that characterizes the spectrum of the given model. This beautiful Sklyanin's picture for the construction of the SoV basis therefore requires the proper identification of the operator families B(λ) and A(λ) and the proof that they indeed satisfy all the outlined required properties. Sklyanin has proposed how to construct these operator families for a large class of models associated to the representation of the 6-vertex Yang-Baxter algebra and even for the higher rank cases. Since then, this method has been successfully implemented, and in some cases partially generalized, to achieve the complete spectrum characterization of several classes of integrable quantum models mainly associated to different representations of the 6-vertex and 8-vertex Yang-Baxter algebras and reflection algebras as well as to their dynamical deformations . Despite its many successes, this construction does not seem however to be completely universal; in particular, some difficulties arise already for the proper identification of the A operator family for the fundamental representations of the higher rank Yang-Baxter algebras, see e.g. [1].
This motivated us to look for a different way for constructing the SoV basis that would not rely on finding such two families of operators A and B. The new idea underlying our approach is to use the action of a well chosen set of commuting conserved charges on some generic co-vector to generate an Hilbert space basis in which their spectral problem is separated. In all the models we considered so far [1][2][3] this set is generated by the transfer matrix itself and provides effectively an SoV basis in which its spectrum can be characterize completely. In particular in such a basis the eigenvectors of the transfer matrix have coordinates given by the products of the corresponding transfer matrices eigenvalues. Hence, the resolution of the spectral problem determines not only the eigenvalues but also gives an algebraic construction of the corresponding eigenvectors in this SoV basis, which is a quite remarkable feature. It is to be emphasized that, in our approach, the SoV basis is directly generated by the quantum symmetries of the considered integrable model. Of course, such a program requires the proof of two main non-trivial steps. First, by using an appropriate set of commuting conserved charges we have to show that we can indeed construct a basis of the given space of the representation of the Yang-Baxter algebra. Second, we need to prove that their spectral problem is indeed separated in this basis. It means that all the wavefunctions should have a factorized form in terms of a well defined class of solutions of an appropriate functional equation (the quantum spectral curve equation). This amounts in fact to get the action of the transfer matrix in this basis which is itself given in terms of the transfer matrix action on the generating co-vector. It turns out that this is equivalent to identify the structure constants of the associative and commutative algebra of the conserved charges generated by the transfer matrices. Again, in all cases we have considered, these structure constants can be computed from the set of fusion relations satisfied by the transfer matrix and the associated quantum determinant evaluated in some specific points that in fact determines the separate variables.
In our previous articles [1][2][3], we have considered quantum integrable lattice models associated to fundamental representations of the Yang-Baxter algebra Y (gl n ) and U q (gl n ) for arbitrary integer n ≥ 2. In particular, in our first paper [1] we have presented how our new approach to construct the SoV basis for integrable quantum models associated to the fundamental representations of Y (gl n ) with the most general quasi-periodic boundary conditions, and for some simple generalizations of them. Then, we have used it to obtain the explicit and complete characterization of the transfer matrix spectrum first for the cases of Y (gl 2 ) and U q (sl 2 ) and then for Y (gl 3 ). In our second paper these results on the complete transfer matrix spectrum have been extended to the case Y (gl n ), for any integer n ≥ 2 while in our third article, we have obtained similar results for the U q (gl n ) case. In our two first papers for the fundamental representations of Y (gl n≥2 ), we have identified a natural choice of the set of the commuting conserved charges and as well characterized the generating covector to be used as starting point to generate our SoV basis. This has been motivated by the consequent simplicity of the proof that this system of covectors forms indeed a basis of the Hilbert space and that the spectrum of the transfer matrix is indeed separated in such a basis. The results are the introduction of the so-called quantum spectral curve and the exact characterization of the set of its solutions which generates the complete transfer matrix spectrum associating to any solution exactly one nonzero eigenvector up to trivial normalisation. These results allow also to point out how the SoV basis in our construction can be equivalently obtained by the action of the Baxter's Q-operator family  satisfying with the transfer matrices the quantum spectral curve equation. In our first paper [1] we have also shown that, under some specific choice of the co-vector, our SoV basis coincides with the Sklyanin's SoV basis, when Sklyanin's approach applies, for integrable quantum models associated to rank one Yang-Baxter algebra. Based on our analysis of the Y (gl 3 ) case, we also conjectured (and verified on small size chains) that the same should hold for the higher rank cases as well, the recent analysis [57] confirming such a statement for Y (gl n ).
The aim of the present article is to show how our method works in cases going beyond the fundamental representations. Our interest in this situation, besides broadening the application of our method, is to understand and explain how our SoV construction works when we have at our disposal a richer structure of commuting conserved charges. As the current paper is mainly addressed to explain these features, we have chosen to consider the simplest example in this class, namely quantum integrable models associated to the higher spin representations of the rational 6-vertex Yang-Baxter algebra, i.e. Y (gl 2 ). In doing so we also solve the case associated to the most general quasi-periodic boundary conditions 1 . Completely similar results can be derived for others compact non-fundamental representations, as for example the higher spin and cyclic representations of the trigonometric 6-vertex Yang-Baxter algebra or their higher rank cases as it will be described in future publications.
Here, we first show that the SoV construction presented in the fundamental representation can be indeed naturally extended to the compact non-fundamental representations. This is first done by substituting in the SoV basis construction the transfer matrix, i.e. the one associated to the trace over the bi-dimensional auxiliary space, by the fundamental transfer matrix, i.e. the one associated to the trace over the auxiliary space isomorphic to the local quantum space. We give the proof that the SoV basis construction can in this framework be derived following a method very similar to the one used for fundamental representations. One direct consequence of this SoV basis construction is then the simplicity and diagonalizability of the transfer matrix.
Then a second SoV basis construction is presented using the full tower of higher fused transfer matrices. This construction appears to be very natural as the action of the transfer matrix in this basis is easily computed just using the fusion relations. In particular, it allows to prove that the transfer matrix spectral problem is indeed separated in this basis. In fact, we then derive the quantum spectral curve equation and uniquely determine the set of its solutions that characterize the complete spectrum of the transfer matrices.
In this paper we also implement, for the same model, the Sklyanin's construction for the SoV basis for the most general quasi-periodic integrable boundary conditions. This construction leads to generate the eigenbasis of the twisted B-family of commuting operators, or some simple generalization of it, and to prove that it is diagonalizable and simple spectrum for these higher spin representations. We further show that our second SoV basis indeed coincides with Sklyanin's one once we chose the generating co-vector in an appropriate way.
Finally, we show that the quantum spectral curve equation together with the diagonalizability and the simple spectrum character of the transfer matrix family allow us to characterize the Qoperator family in terms of the elements of the monodromy matrix, and in particular in terms of the set of fused transfer matrices themselves. This result also allows us to rewrite our SoV basis as the action of the Q-operator family on some new generating covector, i.e. to use the Q-operator family as the set of commuting conserved charges generating our SoV basis.
It is worth to comment that on the basis of all our current results for both fundamental and non-fundamental compact representations of the Yang-Baxter algebra our SoV construction based on the use of the Q-operator family as the generating set of commuting conserved charges always lead to the same natural choice of the SoV basis induced by the fusion of transfer matrices.
Clearly in order to use directly the Q-operator family to generate SoV for others integrable quantum lattice models all the following fundamental elements have to be accessible: first we need to have an SoV independent characterization of the Q-operator family; second we have to design some criteria to identify appropriate generating co-vectors (as starting point of our SoV construction) as well as the exact subset of commuting conserved charges in the Q-operator family (i.e. the spectrum of the separate variables); third a proof that the set of co-vectors generated is indeed a basis; fourth that the transfer matrix spectrum is indeed separated in this basis.
In fact, it is important to stress that in our current construction it is indeed the structure of the transfer matrix fusion relations and the fact that they simplify for special choices of the spectral parameters that allows us to naturally select the subset of commuting conserved charges to be used to generate the SoV basis. Furthermore, these fusion relations determine the structure constants of the associative and commutative algebra of conserved charges generated by the transfer matrices.
The present paper is organized in five sections. In section 2, we recall the higher spin representations of the rational rank one Yang-Baxter algebra and the fused transfer matrix general properties. In section 3, we present the Sklyanin's type SoV basis construction giving an explicit representation of its co-vectors. In section 4, we present our SoV basis construction and the consequent complete characterization of the transfer matrix spectrum. In subsection 4.1, this is done producing an SoV basis which is the natural generalization of those generated in the case of the fundamental representations in [1][2][3]. In subsection 4.2, we present a second SoV basis on which the action of the transfer matrix is easily computed by using the fusion relations. This new basis is shown there to coincide with the Sklyanin's one under a proper choice of the generating co-vector. Finally, in subsection 5.1 we prove the reformulation of the discrete SoV complete spectrum characterization in terms the so-called quantum spectral curve equation. This last result allows us to determine the Q-operator in subsection 5.2 while we use it to reconstruct our second SoV basis in subsection 5.3.
2 The quasi-periodic Y (gl 2 ) higher spin representations Let us recall that the first studies of the integrable higher spin quantum Heisenberg chains have been developed in [81][82][83][84][85][86][87][88][89][90][91][92][93]. The next two subsections are used to recall the higher spin representations of the rank one rational Yang-Baxter algebra and the properties of the fused transfer matrices which will be used in the next sections to develop our analysis in the framework of the separation of variables.

Higher spin representations
The generators of the sl(2) algebra: admit the following spin-s representation: where x n (j) ≡ j(2s n + 1 − j), in a spin-s n representation associated the linear space V (2sn) ≃ C 2sn+1 with 2s n ∈ Z >0 . Then, the following Lax operator: associated to each local quantum space V (2sn) n , satisfies the following Yang-Baxter algebra: associated to the rational 6-vertex R-matrix: The scalar Yang-Baxter equation: is satisfied by any K ∈ End(C 2 ), which defines the gl 2 invariance of rational 6-vertex R-matrix. We can then introduce the monodromy matrix associated to a quantum lattice model with N sites and carrying at each site n ∈ {1, . . . , N} a representation V where we have define H = ⊗ N n=1 V (2sn) n and the ξ n are the inhomogeneity parameters. In the following we will assume these parameters to be in generic positions, namely ξ i = ξ j (modη) whenever i = j. This monodromy matrix satisfies also the same rational 6-vertex Yang-Baxter algebra: which implies that the transfer matrix: is a one-parameter family of commuting operators and that the quantum determinant: is a central elements of the Yang-Baxter algebra of the following form: where M (I|1) 0 (λ) is the monodromy matrix associated to the 2 × 2 identity twist matrix K = I 2×2 , and and we have used the notation λ ± ≡ λ ± η/2.

Fusion relations for higher spin transfer matrices
The fusion procedure was first developed in [82] for the case of the rational 6-vertex representations of the type analyzed here and later in [90] for the trigonometric ones. Let us define the following symmetric and antisymmetric projectors: where P π is the permutation operator: with P − 1 = I. Note that in our current representations, we have that V a ≃ C 2 and the P + 1...m is a rank m + 1 projector, so that, it is an (m + 1)-dimensional vector space. Then, we can define the following higher transfer matrices: where we have defined the higher spin monodromy matrices by: for which the following identity holds: These transfer matrices define commuting families of operators: satisfying the fusion relations: where for simplicity we use the notation: These fusion relations define uniquely any higher spin transfer matrix T (K|l) (λ) in terms of the original transfer matrix T (K) (λ). Moreover, it is possible to write an explicit determinant formula that solves the hierarchy of fusion relations as follows. Let D l (T (K) (λ)) be the following tridiagonal l × l matrix: then: The proof can be done by an elementary induction. Indeed the formula trivially holds for l = 1 and for l = 2 it reduces to the fusion relation defining T (K|2) (λ): Let us now suppose the formula is true up to some integer l ≥ 2. Then by expanding det l+1 D l+1 (T (K) (λ)) w.r.t. its first column we get: where the l × l-matrix ∆ l (λ) is given by: Then expanding det l ∆ l (λ) by its first row and applying the induction hypothesis for l and l − 1 we get: which completes the proof. Finally, let us note that the polynomial T (K|l) (λ) admits the following central divisor polynomial: where we have used the Heaviside step function θ defined by θ(x) = 0 for x ≤ 0 and θ(x) = 1 for x > 0. Let us comment that if the original 2 × 2 twist matrix K is diagonalizable, simple and invertible then the same is true for all the fused twist matrices. In particular, given the distinct nonzero eigenvalues k 1 and k 2 of such 2 × 2 twist matrix K then the fused twist matrix K (a) has the following simple spectrum for all h ∈ {1, ..., a + 1}, for any fixed a ∈ Z >0 . Finally, let us remark that the following commutation relations hold: and so we have also: Note that here and in the following we use a shorter notation to represent the spaces in the fused twist matrices, i.e. we can write K

Sklyanin's type construction of the SoV basis
Following the Sklyanin's approach a separation of variable (SoV) representation [13][14][15][16][17][18] for the T (K) -spectral problem can be defined for representations for which the commutative family of operators B (K) (λ) (or C (K) (λ)) is diagonalizable and with simple spectrum. As already explained in our previous paper for the case of fundamental representation such a statement can be extended by using the gl 2 invariance. In order to do so we have to use the following remark that given a either it satisfies the condition b = 0 (or c = 0) directly or it exists a W (K) ∈ End(C 2 ) such that: so that we can state the following:

3)
and the twist matrix for any α ∈ C, i.e. K is not proportional to the identity, then the T (K) -spectral problem admits Sklyanin's like separate variable representations. More in detail: ) is diagonalizable and with simple spectrum and the quantum separate variables are generated by the B (K) -operator (or C (K) -operator) zeros.
is diagonalizable and with simple spectrum and the quantum separate variables are generated by thẽ is diagonalizable and with simple spectrum and the quantum separate variables are generated by thẽ C (K) -operator zeros.
In the next subsection we construct explicitly the B (K) -eigenbasis andB (K) -eigenbasis, the construction for C (K) -eigenbasis andC (K) -eigenbasis can be similarly derived, in this way giving a constructive proof of the above theorem.

Construction of the SoV representation in
where we have defined for any value opf n the following 2s n +1-covector (local left references states): and let us denote by k 1 and k 2 the eigenvalues of K ∈ End(C 2 ), then: 3) are verified and K ∈ End(C 2 ) is any 2 × 2 matrix non proportional to the identity, invertible 2 and satisfying the condition b = 0, then the set of covectors h n ∈ {0, ..., 2s n } for all the n ∈ {1, ..., N} and: defines a covector B (K) -eigenbasis of H: with the distinct eigenvalues: Moreover it holds: 3) are verified and K ∈ End(C 2 ) is any 2 × 2 matrix non proportional to the identity, invertible and such that b = 0, then the covectors h| Sk ≡ h 1 , ..., h N | Sk , defined by: define a covectorB (K) -eigenbasis of H: where: Moreover it holds: Proof. The proof that the covectors (3.12) are eigencovectors of B (K) (λ) with the above defined eigenvalues is standard, see e.g. [44], it uses just the Yang-Baxter commutation relations and the fact that the left reference covector is B (K) -eigencovector. Indeed, it holds and Then the proof that the operators A (K) (λ) and D (K) (λ) have the given representation in the B (K)eigencovectors is once again a direct consequence of the Yang-Baxter commutation relations. Finally, note that the above construction generates i.e. the dimension of the representation, B (K) -eigencovectors which are independent and so form a basis as soon as they are all nonzero as they are associated to different eigenvalues of B (K) (λ). This last statement can be for example shown by constructing the B (K) -eigenvectors and proving that the action of a B (K) -eigencovector on the B (K) -eigenvector associated to the same eigenvalue is nonzero. We omit this steps as they can be done following exactly the same lines described in the case of the antiperiodic boundary conditions [44]. If the condition b = 0 is satisfied, then the above results allow similarly to show that the one parameter operator family B (K) (λ) is diagonalizable with simple spectrum and they allow to derive its left eigenbasis. Then the same statements hold for the one parameter operator familyB (K) (λ) defining the SoV basis for b = 0, being it similar to B (K) (λ):

New SoV basis and complete spectrum characterization
In this section we construct two different SoV basis from two natural sets of conserved charges of the considered models using the method developed in [1]. Moreover, we show how these SoV basis indeed separate the quantum spectral problem for the transfer matrix. We have already presented such a procedure [1,2] in the case of models associated to fundamental representations of the rational Yang-Baxter algebra. Here we explain how this procedure can be developed for non-fundamental representations, using as an example higher spin representations of the rational gl 2 Yang-Baxter algebra. In subsection 4.1, we introduce a first set of SoV covectors generated from the transfer matrices obtained from the fundamental Lax operators, i.e., the Lax operators for which the auxiliary space is isomorphic to its local quantum space at some site n. They are obtained by fusion from the original Lax operator having an auxiliary space in the spin-1/2 representation. In this case the proof that these conserved charges generate a basis of the Hilbert space is given following the same main steps used for the fundamental representations in [1]. This SoV basis is quite natural in this respect and it allows to prove also the simplicity of the transfer matrix spectrum and derive its complete characterization. In subsection 4.2, we then introduce another SoV basis constructed from the full tower of fused transfer matrices, which we argue to be the most natural w.r.t. the fusion rules satisfied by the quantum spectral invariants. There, we prove also that under some special choice of the generating covector this SoV basis coincides with the Sklyanin's SoV basis presented in the previous section.

A first SoV basis construction and the associated spectrum characterization
The following proposition holds: Proof. This is a corollary of the general Propositions 2.4 and 2.5 of our previous paper [1]. Indeed, the proof follows the same lines that in these propositions once we observe that the fused R-matrices satisfies the following identity: and so: 5) and moreover that the leading asymptotic of any R-matrices R (2sn|2sm) nm (λ) is proportional to the identity. Moreover, the existence of S, n| is implied by the fact that the K (2sn) n are diagonalizable with simple spectrum.
Remark: It is worth to point out that we have proven our proposition only in the case in which the twist matrix K ∈ End(C 2 ) has not only simple spectrum but is in addition diagonalizable and invertible. These requirements in the case of fundamental representations are not needed, here they are imposed to get that the fused twist matrices keep the simple spectrum nature which allows us to use the general Propositions 2.4 of our first paper [1]. It is then interesting to point out that in the next subsection 4.2 the SoV basis will be constructed in our approach using the commuting conserved charges without imposing these additional constraints, but just asking the simplicity of its spectrum which for a 2 × 2 matrix is equivalent to ask that it isn't proportional to the identity.
As explained in our previous papers once the SoV basis is constructed, by using the action of the quantum spectral invariants on some given generating covector, then the transfer matrix fusion equations allow for the full characterization of the transfer matrix spectrum. Indeed, the transfer matrix satisfies the following:
Proof. The computation of the asymptotics is easily derived from the known asymptotics of the elements of the R-matrix R (1|2sn) an ). Then the system of equations satisfied by the transfer matrix is just a direct consequence of the fusion relations for the transfer matrices resolved in (2.24) and of the emerging central zeroes (2.30).
Remark: It is interesting to point out that the Proposition 4.2 can be also derived as consequence of the SoV characterization of the transfer matrix spectrum. Indeed, in the Sklyanin's framework, one can prove that any transfer matrix eigenvalue has to satisfy the discrete system of equations (4.10) written bellow by computing the action of the transfer matrix on the generic eigenvector in the SoV representation, as it was done in the antiperiodic case in [44]. Then, the Proposition 4.2 holds for any diagonalizable and simple spectrum twist matrix; indeed by Proposition 4.1 the transfer matrix share the same properties and then it satisfies the system of equations (4.7). Now it is enough to observe that the determinants on the l.h.s. of (4.7) are polynomials in the elements of the twist matrix to derive that (4.7) has to hold for any twist matrix as it holds for almost any value of its elements.
Here, instead, we use the above proposition to prove that any solution to this system of equations indeed generates an eigenvalue and eigenvector in our SoV basis. Indeed, let us define the function: then we can state the following Theorem 4.1. Under the same conditions allowing to define the SoV basis (4.1), the spectrum of T (K) (λ) coincides with the set of polynomials:  characterizes the associated unique eigenvector |t , up-to the overall normalization fixed by S|t = 1.
Proof. From the previous proposition it follows that any eigenvalue t(λ) ∈ Σ T (K) is indeed a degree N polynomial in λ with central asymptotics (4.6) and solutions of the system (4.7) as a consequence of the fusion equations.
We have now to prove the reverse statement that any solution of this system of equations {x 1 , ..., x N } ∈ D T (K) indeed define through the polynomial interpolation formula (4.9) a transfer matrix eigenvalue. The very existence of the SoV basis, for almost any value of the inhomogeneities, has as a direct consequence that the spectrum of all transfer matrices T (K|2sn) is simple. All these fused transfer matrices being polynomials in the transfer matrix it implies (otherwise we get an immediate contradiction) that T (K) (λ) has simple spectrum and it is diagonalizable. Indeed, the Proposition 2.5 of our first paper [1] applies here too. So that there exists d {sn} different eigenvalues t(λ) ∈ Σ T (K) , i.e. they are in the same number of the dimension of the representation.
It is easy now to remark that, as the theorem states, (4.10) is a system of N polynomial equations in the N unknowns {x a≤N } of degree 2s n + 1 for any n ∈ {1, ..., N}. Then the Theorem of Bezout 5 states that the system admits a finite number d {sn} of solutions {x 1 , ..., x N }, if the N polynomials of degree 2s n + 1 in the N variables {x 1 , ..., x N }, defining the system, have no common components. As we have already shown that to any t(λ) ∈ Σ T (K) it is uniquely associated a solution {x a≤N ≡ t(ξ (0) a≤N )} ∈ D T (K) , then we can state that the system (4.10) admits at least d {sn} distinct solutions. Then, once the condition of no common components is satisfied, we derive that the system admits exactly d {sn} distinct solutions and each one is associated to a transfer matrix eigenvalue, which complete the equivalence proof.
So we are left with the proof of this condition of no common components. In order to prove this statement it is enough to show it for some special value of the twist eigenvalues to imply its validity for almost any value of these parameters, as the polynomials defining the system are polynomial in the twist parameters too. Let us here consider the case k 1 = 0 and k 2 = 0, then the system of equations reads: Finally, by definition of the SoV basis for any t(λ) ∈ Σ T (K) the uniquely associated eigenvector has the factorized wavefunctions: where the polynomial t (2sn) (λ) is the eigenvalue of the fused transfer matrix T (K|2sn) (λ) uniquely defined in terms of the t(λ) ∈ Σ T (K) by the use of the fusion equations. Then the statement of the theorem follows observing that the following identities:

The natural SoV basis and the associated spectrum characterization
Here we show that there exists a different choice of the SoV basis for which the action of the transfer matrix becomes very simple as a consequence of the fusion relations and in particular of the Proposition 4.2.
Theorem 4.2. The set Proof. Let us remark that the determinant of the d {sn} × d {sn} matrix whose columns are the elements of the covectors (4.19) in the natural basis is a polynomial of order at most d {sn} in the components of the covector S|. So that it is enough to prove that this determinant is nonzero for a special choice of S| to show that it is nonzero for almost any choice of S|. So to prove the theorem it is enough to prove that the statements a) and b) hold. Let us first observe that by using the quantum determinant condition we have the following rewriting of the Sklyanin's type SoV basis: and In order to do so, we expand the following covector: (4.26) where for any i ∈ {1, ..., N}\{n} we have fixed h i ≥h i ≥ 1, which by definition implies: Then the induction assumption implies that it holds: Now by definition of the transfer matrix and the Sklyanin's type SoV basis we have that it holds: .

(4.29)
So that by using the quantum determinant identity: it holds: by definition of the Sklyanin's type SoV basis. So that replacing these results in (4.28) we get our identity: for any h i ≥h i ≥ 1 with i ∈ {1, ..., N}\{n}, which proves the induction. In the case b), following the same proof of case a) for the transfer matrices we show the identities: for any h n ∈ {0, ..., 2s n } and n ∈ {1, ..., N} from which our statement easily follows.
As we have anticipated before this is the natural SoV basis as the action of the transfer matrix on it is easily derived by the fusion identities. Indeed, we have the following: now we can use the fusion relations to rewrite the following operator product: (4.39) so that using the l.h.s. we get: from which our statement follows by the definition of the SoV basis.
From the above proposition, we get the following characterization of the transfer matrix spectrum in our new SoV basis. Clearly the discrete characterization of the transfer matrix eigenvalues is invariant w.r.t. the chosen SoV basis so it coincides with the one we have given in the theorem of the previous section. Of course what changes is the characterization of the transfer matrix eigenvectors in the new SoV basis; indeed, it holds: Theorem 4.3. Let us assume K ∈ End(C 2 ) non proportional to the identity and invertible, then for almost any value of the inhomogeneities T (K) (λ) has simple spectrum, is diagonalizable and the set of its eigenvalues Σ T (K) coincides with the set of degree N polynomials (4.9). For any t(λ) ∈ Σ T (K) the associated unique (up-to normalization fixed by S|t = 1) eigenvector |t has the following factorized wavefunction in the SoV covector basis: . (4.41) Proof. The proof can be done along the same lines as in the theorem of the previous section. It is however interesting to point out that in the current SoV basis the direct action of the transfer matrix allows to provide a simple and alternative proof of the fact that any solution {x 1 , ..., x N } ∈ D T (K) defines a transfer matrix eigenvalue through the polynomial interpolation formula (4.9). Indeed, the following identity: is trivially deduced from the definition (4.41) of the state |t and the previous proposition on the action of the transfer matrix on the SoV basis. Then the above results together with the asymptotics of the polynomials (4.9) imply our statement.

The quantum spectral curve
A functional equation which provides an equivalent characterization of the SoV discrete characterization of the spectrum we derived above, the so-called quantum spectral curve, is given here. In the case at hand it is a second order Baxter difference equation.
Theorem 5.1. Let the twist matrix K ∈ End(C 2 ) be such that 7 k 1 = k 2 , k 1 = 0, k 2 = 0 and the inhomogeneities {ξ 1 , ..., ξ N } ∈ C N satisfy the condition (3.3). Then an entire function t(λ) is an element of Σ T (K) iff there exists a unique polynomial: for any (a, b) ∈ {1, ..., M} × {1, ..., N}, satisfying the following quantum spectral curve functional equation: where we have defined: Up to an overall normalization the associated transfer matrix eigenvector |t admits the following rewriting in the left SoV basis: Proof. Let us start proving that the quantum spectral curve equation admits at most one polynomial solution Q t (λ) for a given function t(λ). Indeed, if we assume the existence of two such polynomial solutions P (λ) and Q(λ), it holds: which can be rewritten k 1 a(λ) W P,Q (λ) = k 2 d(λ) W P,Q (λ + η). (5.6) W P,Q (λ) is the quantum Wronskian of these two solutions: Taking into account that the zeros of d(λ) coincides with those of a(λ) shifted by 2s n η for any n ∈ {1, ..., N} it follows: where w P,Q (λ) is a polynomial in λ which moreover has to satisfy the following quasi-periodicity condition: k 1 w P,Q (λ + η) = k 2 w P,Q (λ). (5.9) Being k 1 = k 2 this implies 8 w P,Q (λ) = 0. Let us now assume the existence of Q t (λ) satisfying with t(λ) the functional equation (5.2), then it follows that t(λ) is a polynomial of degree N with leading coefficient t N+1 satisfying the equation: which imposes t N+1 =trK = k 1 + k 2 . It is now easy to verify that particularizing the functional equation in the points λ = ξ (hn) n + η, for any h n ∈ {0, ..., 2s n }, then the condition Q t (ξ (2sn) n ) = 0 implies that t(λ) satisfies the equation (4.10) for any fixed n ∈ {1, ..., N} which together with its asymptotic behavior implies that t(λ) is a transfer matrix eigenvalue.
The reverse statement is proven now. Let us assume that t(λ) is a transfer matrix eigenvalue then we can show the existence of the polynomial Q t (λ) of the form (5.1) satisfying the required functional equation. On the l.h.s. of the equation there is a polynomial in λ of maximal degree 2N + M, with M ≤ N s , so that in order to prove that the functional equation is satisfied we have to show that it is zero in 2N + N s + 1 different points. First, at infinity, thanks to (5.10), the leading coefficient of this polynomial is zero as t(λ) is a transfer matrix eigenvalue. It is easy to remark that in the N points ξ ∀n ∈ {1, ..., N}. (5.11) is satisfied. As a consequence of the fact that t(λ) is an eigenvalue we have: det 2sn+1 D t,n = 0 ∀n ∈ {1, ..., N}, (5.12) so that the previous system is equivalent (for example) to the system of the last 2s n equations which can be resolved in terms of Q t (ξ (2sn) n ) as it follows: where we define Q (2sn) t,n = 1 and the others, due to the trigonal form of the matrix D t,n , are defined in a unique way recursively by: , ∀h n ∈ {1, ..., 2s n − 1}, (5.14) . (5.15) Due to the tridiagonal form of the matrix D t,n these recursion relations can be solved explicitly in terms of the transfer matrix eigenvalues in a very simple way as: hence leading to the result (5.4) by direct comparison with (4.41) with the convention that t (1) (λ) = t(λ). The proof can be done by an elementary induction. Indeed, the formula holds for h n = 1 from the relation (5.15). Suppose the formula is true up to a value l ∈ {1, ..., 2s n − 2} then let us prove it for l + 1. We have using (5.14) and then applying the induction hypothesis for l and l − 1: , (5.18) hence leading to the formula: which proves the induction hypothesis and hence (5.16) for any h n ∈ {1, ..., 2s n − 1}.
Therefore, the existence of the polynomial Q t (λ) of the form (5.1) which satisfies with t(λ) the quantum spectral curve equation is reduced to the proof of the existence of a polynomial of maximal degree N s that interpolates the values (5.13) in the Q t (ξ  (5.20) where ζ is an arbitrary value different of ξ (hn) n for any h n ∈ {1, ..., 2s n } and n ∈ {1, ..., N}. Imposing now that Q t (λ) satisfies (5.13) for any h n ∈ {1, ..., 2s n } and n ∈ {1, ..., N}, we get: where we have introduced the notations: n that were not used in the interpolation formula (5.21). So imposing these conditions indeed constitutes N constraints on the possible values of the q t,a , for a ∈ {1, ..., N}. So that we are left with the system of N equations obtained by imposing that the interpolation formula (5.21) indeed satisfies the equations (5.13) in the points h n = 0 for n ∈ {1, ..., N}. This is an homogeneous linear system of N equations in N + 1 unknowns, the q t,a for any a ∈ {0, ..., N}, or equivalently an inhomogeneous system of N equations in the N unknowns, the q t,a for any a ∈ {1, ..., N} in terms of the normalization q t,0 : ζ ] ab has the following elements where the coefficients Q . . , N}, are given from the transfer matrix eigenvalues by (5.16) as the solution to the linear system (5.11). The linear system (5.23) admits always one nonzero solution which produces one polynomial Q t (λ) satisfying the functional equation (5.2) with t(λ). This and the unicity of the polynomial solution implies that det N [C (t) ζ ] is nonzero and finite for almost any choice of ζ. Then, for any given choice of q t,0 = 0, there exists one and only one nontrivial solution (q t,1 , . . . , q t,N ) of the system (5.23), which is given by Cramer's rule: for almost any values of the parameters. Let us consider the case k 1 = 0 then the transfer matrix reduces to k 2 D(λ), which coincides with B (K) (λ) for K = k 2 σ 1 , whose diagonalizability and spectrum simplicity follows for example by Theorem 3.2. The spectrum explicitly reads: {0, . . . , 2s n }. (5.28) In this special case we can solve explicitly in the class of the polynomial solutions the quantum spectral curve equation which reads: Indeed, the polynomial it is easily checked to satisfy the above functional equation (5.29) with t (k 1 =0,k 2 =0) h (λ) for any fixed h ∈ N n=1 {0, . . . , 2s n }. Moreover, it is the only solution to the above equation for any fixed h. This follows by the general proof of unicity, given above by the quantum Wronskian argument, or by rederiving it directly in this special case. Indeed, any polynomial solution of (5.29) has to have the factorized form:Q for some polynomial P (λ) which has to satisfy then the equation: which is only possible for P (λ) constant. Let us observe that all our polynomials are indeed of degree M ≤ N s ≡ 2 N n=1 s n , so that for any fixed h we can interpolate them in N s + 1 points according to the interpolation formula (5.20). Moreover, as the couple t h (λ) satisfies the functional equation (5.29) then they satisfy also the homogeneous system of equations (5.11) or equivalently the (5.13) with (5.14) and (5.15), for k 1 = 0. This means that the interpolation formula (5.21) also holds for our polynomial Q t (k 1 =0,k 2 =0) h (λ) and so imposing the condition (5.13) for any h a = 0, we get the linear system (5.23), which is satisfied for:  [2], we can argue that being these determinant nonzero at k 1 = 0 for any transfer matrix eigenvalue this implies that this must be true for almost any values of the parameters.
Finally, let us note that Q t (λ) being a non zero (by construction) polynomial of maximal know degree, its highest coefficient can always be normalized to unity as required.

Reconstruction of the Q-operator
On the basis of the results derived in the SoV framework we can present a reconstruction of the Q-operator in terms of the elements of the monodromy matrix, more precisely it holds: Corollary 5.1. Let us assume that the twist matrix K ∈ End(C 2 ) is such that k 1 = k 2 , k 1 = 0, k 2 = 0. Then the polynomial family of commuting operators of maximal degree N s , defined by: is well defined for any fixed value 9 of ζ and for almost any values of {ξ i≤N } and of {k j≤2 } and it is a Q-operator family. Here, we have introduced the following definitions: . . , N}, (5.35) 9 We can fix for example ζ = ξ where: , ∀ h ∈ {1, . . . , 2s a }, a ∈ {1, . . . , N}, (5.36) and the central matrix of rank one: where we are imposing that 2s N+1 = 1, ξ (1) N+1 = ζ. Indeed, the quantum spectral curve at operator level is satisfied: Proof. We have shown in the previous theorem that a unique polynomial Q t (λ) of the form (5.1) satisfies the quantum spectral curve equation for any fixed t(λ) eigenvalue of the transfer matrix T (K) (λ). Then by using the reconstruction of Q t (λ) in the points ξ (2sa) a for any a ∈ {1, . . . , N} and its interpolation formula (5.21) it is easy to prove that the following determinant representation holds: where we have replaced the transfer matrix T (K) (ξ a ) by its eigenvalue t(ξ a ) in the above defined matrices. Now as a corollary of Proposition 2.5 of our first paper [1], we know that the transfer matrix T (K) (λ) is diagonalizable and with simple spectrum for almost any value of the parameters {ξ (0) i≤N } and {k j≤2 } for the twist matrix K diagonalizable and with simple spectrum. The polynomial operator family Q(λ) can be then uniquely defined by its action on the eigenbasis of the transfer matrix as it follows: Q(λ)|t = |t Q t (λ), (5.40) for any t(λ) eigenvalue and uniquely (up to normalization) associated eigenvector |t of the transfer matrix T (K) (λ). It is then evident that this operator family satisfies by definition the quantum spectral curve equation with the transfer matrices, that it admits the given determinant representation in terms of the transfer matrix T (K) (λ) and that it is invertible in the points {ξ (2sa) a≤N }.

On the general role of the Q-operator as SoV basis generator
The known results on SoV in the literature and, in particular, our general construction of the SoV basis allows to show that for a large class of integrable quantum models associated to finite dimensional representation of the Yang-Baxter, reflection algebra or dynamical generalization of them, the transfer matrix (or some simple extension of it) defines a diagonalizable and simple spectrum one parameter family of commuting operators. Moreover, for the same class of models we know for a fixed normalization of the eigenvector that the corresponding wavefunction admits the following factorized form: clearly the definition of L| is fixed up to the choice of the normalization of all the transfer matrix eigenvectors. These observations naturally lead to the idea that we are able to construct the SoV basis once the Q-operator is known. There is anyhow an important comments we have to make, i.e. some further informations are indeed required beyond the knowledge of the Q-operator. In particular, the right choice of the vector L| which has to satisfy the condition: L|t = 0 ∀t(λ) ∈ Σ T (K) (5.45) and even more importantly we have to have a criterion to chose the spectrum of the separate variables. In our SoV construction there are indeed the fusion relations and the fact that they are simplified for some specific choice of the values of the spectral parameter to guide us to the proper choice of the spectrum of the quantum separate variables.
In the non-fundamental representations we considered in this article, we can now make the above description of the SoV basis construction using the Q-operator. In particular, we have the following: Corollary 5.2. Let us assume that the twist matrix K ∈ End(C 2 ) is such that k 1 = k 2 , k 1 = 0, k 2 = 0, then: a) In the case b = 0, we have that the set of covectors (5.43) coincides with the SoV basis, i.e. the B (K) -eigenbasis, once we fix: A last remark about the construction of the SoV basis starting from the Q-operator should be outlined. For several integrable quantum models it is in fact the construction of the SoV basis and the corresponding characterization of the transfer matrix spectrum that allow for the explicit construction of the Q-operator, as we have explained in this article.