Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables

We apply our new approach of quantum Separation of Variables (SoV) to the complete characterization of the transfer matrix spectrum of quantum integrable lattice models associated to gl(n)-invariant R-matrices in the fundamental representations. We consider lattices with N sites and quasi-periodic boundary conditions associated to an arbitrary twist K having simple spectrum (but not necessarily diagonalizable). In our approach the SoV basis is constructed in an universal manner starting from the direct use of the conserved charges of the models, i.e., from the commuting family of transfer matrices. Using the integrable structure of the models, incarnated in the hierarchy of transfer matrices fusion relations, we prove that our SoV basis indeed separates the spectrum of the corresponding transfer matrices. Moreover, the combined use of the fusion rules, of the known analytic properties of the transfer matrices and of the SoV basis allows us to obtain the complete characterization of the transfer matrix spectrum and to prove its simplicity. Any transfer matrix eigenvalue is completely characterized as a solution of a so-called quantum spectral curve equation that we obtain as a difference functional equation of order n. Namely, any eigenvalue satisfies this equation and any solution of this equation having prescribed properties leads to an eigenvalue. We construct the associated eigenvector, unique up to normalization, by computing its decomposition on the SoV basis that is of a factorized form written in terms of the powers of the corresponding eigenvalues. If the twist matrix K is diagonalizable with simple spectrum then the transfer matrix is also diagonalizable with simple spectrum. In that case, we give a construction of the Baxter Q-operator satisfying a T-Q equation of order n, the quantum spectral curve equation, involving the hierarchy of the fused transfer matrices.


Introduction
In this paper we continue our study of quantum integrable lattice models using the new approach to quantum separation of variables that we recently developed [1]. We use the framework of the quantum inverse scattering method [2][3][4][5][6][7][8][9][10]. In the present article we consider the class of quantum integrable lattice models associated to irreducible representations of the Yang-Baxter algebra obtained from tensor products of the fundamental representation corresponding to the n 2 × n 2 rational gl n -invariant R-matrices for any n ≥ 2 [11][12][13][14][15][16][17][18]. The gl n symmetry of these Rmatrices implies that the monodromy matrix multiplied by an arbitrary n × n twist matrix K still satisfies the same Yang-Baxter algebra governed by the given R-matrix. Let us remark that for any choice of an invertible K-matrix, one gets a quantum integrable model associated in the homogeneous limit to the same bulk Hamiltonian as for K = 1 but having different K-dependent quasi-periodic boundary conditions.
Integrable quantum models in this class have been analyzed in the literature and exact results on the spectrum have been obtained e.g. by the use of generalizations of the standard algebraic Bethe ansatz like nested Bethe ansatz [11,14] and analytic Bethe ansatz [15][16][17][18], with important recent progress towards their dynamics [19][20][21][22][23][24][25][26][27][28]. Here, we investigate them in the framework of the quantum Separation of Variable (SoV) approach pioneered by Sklyanin [29][30][31][32][33][34] along our new method of constructing an SoV basis presented in [1]. The SoV approach has in general the clear advantage to give a simple proof of the completeness of the spectrum description as it has been demonstrated in several important examples, mainly associated to the 6-vertex and 8-vertex representations of the Yang-Baxter and Reflexion algebras, see e.g. , which in Bethe ansatz framework is not an easy task in general. The SoV approach has been also shown to work for several integrable quantum models for which the Algebraic Bethe ansatz fails, in particular due to the absence of a so-called reference state. It also allows to get some universal and straightforward simultaneous characterization of the transfer matrix eigenvalues and associated eigenvectors.
Let us recall that in the Sklyanin's SoV approach 1 , the first step is to identify a one parameter commuting family of operators, the so-called B-operator, which must be diagonalizable and with simple spectrum, the corresponding eigenbasis being labeled by the eigenvalues of the set of the commuting operator zeroes of B(λ), let us denote them by Y n . Second, this operator family should have a "canonical conjugate" operator family, the so-called A-operator, also depending on a spectral parameter λ, which, thanks to the Yang-Baxter commutation relations, when carrefully evaluated at λ = Y n acts as a shift operator over the spectrum of the Y n . Third, the operator families B(λ), A(λ) and the transfer matrices of the model have to satisfy appropriate commutation relations implying that the operator families A(λ) and the transfer matrices over the spectrum of the Y n satisfy a quantum spectral curve equation associated to the monodromy matrix M (λ) satisfying a Yang-Baxter or Reflexion algebra. The fused transfer matrices appear there as operator coefficients and play the role of the quantum spectral invariants of the monodromy matrix M (λ). As shown by Sklyanin, see e.g. [33], they are defined as quantum deformations of the corresponding classical spectral invariants. When these three steps are realized, the Y n are the so-called quantum separate variables for the transfer matrix spectral problem, the associated eigenbasis is the SoV basis and the separate relations are given by the quantum spectral curve equations that are finite difference equations over the spectrum of the separate variables.
Hence, this beautiful Sklyanin's picture for the construction of the SoV requires the identification of the operator families B(λ) and A(λ) and the proof that they satisfy all the outlined required properties. Sklyanin has proposed a way to identify these operator families 2 for a large class of models associated to the representation of the 6-vertex Yang-Baxter algebra and has implemented the procedure for some important models. As already mentioned, using his identification or simple generalization of it (see for example the idea of pseudo-diagonalizability of the B-operator family [59,60]) it has been possible to widely implement the Sklyanin's SoV approach for integrable quantum model. Nevertheless, the Sklyanin's identification of the operator families B(λ) and A(λ) does not seem to be universal. In particular, for the higher rank cases, it appeared [1] that for the fundamental representation of the rational Yang-Baxter model associated to gl 3 it does not apply, as the proposed A(λ) does not seem to act as a shift operator over the full B-spectrum.
Hence, until now, despite several important progress in their understanding [34,37,66,71,72], a systematic SoV description of higher rank quantum integrable models for a generic K-matrix has represented a longstanding open problem. Here, we solve it for the class of models associated to the fundamental representations of the Yangian Y (gl n ) for general quasi-periodic boundary conditions associated to a matrix K having simple spectrum (but not necessarily diagonalizable). This is done by implementing our new construction of the Separation of Variables (SoV) basis according to the general lines described in [1] where it was already applied to the cases n = 2 and n = 3.
The key point is that our SoV construction allows us to overcome the above mentioned problem of the identification of the operator families B(λ) and A(λ) and the proof of their required properties, e.g. the characterization of the B-spectrum and the proof of its diagonalizability and simplicity. In the case in which the Sklyanin's SoV approach works our SoV construction can be made coinciding with the Sklyanin's one by choosing appropriately our SoV-basis while our SoV construction applies for larger classes of models, as the higher rank cases that we are going to describe in this article.
In our approach [1] the SoV basis is constructed in an universal manner starting from the direct use of the conserved charges of the models, namely from the repeated action of the transfer matrices on a generic co-vector of the Hilbert space. The integrable structure of the model, incarnated in the transfer matrix fusion rules, are the basic tools used to prove the separation of the transfer matrix spectrum in our SoV basis. The complete characterization of the transfer matrix spectrum (eigenvalues and eigenvectors) is then obtained and its simplicity is proven. For any fixed eigenvalue the associated (unique up to normalization) eigenvector has coefficients in the SoV basis of factorized form written in terms of powers of the given eigenvalues. The eigenvalues admit both a discrete and a functional equation characterization. Our SoV approach naturally leads to a characterization of the eigenvalues as the set of solutions to a system of N (number of sites of the lattice) equations of order n for N unknowns. Then, in a second characterization, that we prove to be equivalent to the first one, they are obtained as the set of solutions of a functional equation which is an order n finite difference functional equation. The coefficients of this quantum spectral curve equation are related to the quantum spectral invariant eigenvalues of the model, i.e. the eigenvalues of the fused transfer matrices. Finally, under the further condition that the twist matrix K is diagonalizable with simple spectrum, we prove that the transfer matrices are also diagonalizable with simple spectrum. It allows us in this case to define a single higher rank analog of the Baxter Q-operator [73][74][75][76][77][78][79][80][81][82][83][84][85][86][87][88][89][90] satisfying with the transfer matrices the same n order finite difference functional quantum spectral curve equation.
It is interesting here to make some further comments on the role played by the integrability in our SoV construction and on the subsequent characterization of the transfer matrix spectrum. We have 2 On the basis of pure algebraic properties inferred by the Yang-Baxter commutation relations. recalled, in our previous paper [1], that the concept of independence of the charges, which is natural in classical integrability, is hard to define in the quantum case. Instead we have used there the requirement of w-simplicity (non-degeneracy) of the transfer matrix spectrum as an independence condition for generating the SoV basis. One has however to point out that from the w-simplicity of the spectrum of the commuting family of the transfer matrices it follows that we can always fix some value of the spectral parameter for which the corresponding transfer matrix, let us say T (λ 0 ) is itself w-simple. Then, remarking that any operator commuting with a w-simple operator can be written as a polynomial of it of maximal degree the dimension of the Hilbert space [91,92], one natural question that can emerge concerns the role of the other conserved charges and of the integrable structure. What we have shown in [1] is their role in the solution of the common spectral problem. Indeed, using only one w-simple operator T (λ 0 ) we can in fact construct our basis according to formula (2.17) of our previous paper, so that the factorized characterization of the eigenvectors in terms of the eigenvalues of the Lemma 2.1 works. However this can be seen only as a pre-SoV characterization. Indeed, it leads to a characterization of the eigenvalues through a polynomial equation (given by the characteristic polynomial) of degree n N (the dimension of the Hilbert space) for N sites. Instead, exploiting the full integrable structure of the model, in particular using the hierarchy of fused transfer matrices and their fusion relations, allows for the introduction of different type of spectrum characterization (discrete and functional) giving rise to equations of the quantum spectral curve of degree n.
Let us comment that the use of the fusion relations [12,14] in the framework of the quantum inverse scattering method to investigate the transfer matrix spectrum has already found several applications in the literature of quantum integrable models. A first systematic use of them has been introduced by Reshetikhin in his analytic Bethe ansatz method [15][16][17][18], see also [93]. There, these fusion relations are used to introduce an ansatz on the form of the transfer matrix eigenvalues which eventually leads to a nested system of Bethe equations by the requirement of analyticity.
It is also interesting to remark that the connection with the fusion relations of the transfer matrices is evident already in the Sklyanin's SoV approach. This is intrinsically contained in the SoV as the condition of existence of transfer matrix eigenvectors [54][55][56] is just equivalent to the fusion relations of the transfer matrices computed in the spectrum of the separate variables and their shifted values. In [56] indeed it was explicitly stated, in the case of the 8-vertex model, that these fusion relations together with the known analyticity properties of the transfer matrix can be used to characterize the transfer matrix eigenvalues (as solutions to a system of quadratic equations of N equations in N unknowns). There it was however pointed out that such a purely functional approach does not allow to identify the solutions to the system of equations which correspond to true eigenvalues. This is the case for the purely functional methods which do not allow for the construction of eigenvectors. Such a problem is indeed solved by our construction of the SoV basis in [1]. Indeed, our SoV basis with the combined use of transfer matrix fusion relations and their known analytic properties allows to identify the eigenvalues as the solutions to the system for which a unique (up to normalization) corresponding eigenvector can be constructed.
Finally, let us mention that while this work was already completed, an interesting paper [94] appeared, using the ideas of [1], also dealing with quantum integrable models associated to gl ninvariant R-matrices. In this article, the conjecture discussed in [1] (and proven there for the gl 2 case) that our SoV basis can be constructed in such a way that it is an eigenbasis for the Sklyanin B-operator is proven for the gl n case. Let us comment however that the strategy we use in the present article (and also in [1]) to completely characterize the transfer matrix spectrum does not use the properties of the Sklyanin B-operator. Rather, as will be shown in the following, we use directly the properties of our SoV basis constructed from repeated actions of transfer matrices on a generic co-vector of the Hilbert space. From there, we will show that the structure constants of the commutative and associative Bethe algebra of conserved charges generated by the transfer matrices can be completely determined from the transfer matrix fusion relations and their asymptotic behavior properties (see eq.(3.20)). It leads in a direct way to the complete characterization of the spectrum of these transfer matrices (eigenvalues and corresponding eigenvectors). Notice that for Gaudin gl n models the interest of such Bethe algebra was pointed out in [95]. Let us finally remark that it would be quite interesting to investigate the possible role of these structure constants in the approach of [96] to quantum integrable models.
This article is organized as follows. In section 2 we fix the basic definitions and the essential fusion and asymptotic behavior properties of the transfer matrices that we need for our purposes. In section 3 we completely characterize the spectrum of the transfer matrix for gl n models in the fundamental representations in terms of a discrete set of equations. In section 4 we prove that this characterization is equivalent to a functional quantum spectral curve equation. Further, in the case where the twist matrix is diagonalizable with simple spectrum we give a reconstruction of the Baxter Q-operator satisfying the corresponding T -Q equation together with the fused transfer matrices. Finally in section 5 we give an Algebraic Bethe ansatz like rewriting of the transfer matrix complete spectrum. Important properties of the transfer matrices commutative algebra used in section 3 are proven in Appendix A. Similarly, technical proofs needed in section 4 are gathered in Appendix B.
2 Transfer matrices for quasi-periodic Y (gl n ) fundamental model Here and in the following we denote by N the number of lattice sites of the model. Using the same notations as in [1], let us consider the Yangian gl n R-matrix where P a,b is the permutation operator on the tensor product V a ⊗ V b and η is an arbitrary complex number. It is solution of the Yang-Baxter equation written in End(V a ⊗ V b ⊗ V c ): and any matrix K ∈ End(C n ) satisfies: i.e. it realizes the gl n invariance of the considered R-matrix. Then we can define the general (twisted) monodromy matrix, which satisfies the Yang-Baxter equation, , with H ≡ ⊗ N l=1 V l and its dimension d = n N . Hence it defines a representation of the Yang-Baxter algebra associated to this R-matrix and the following one parameter family of commuting transfer matrices: In the above formulae, and in all this article, the complex parameters {ξ 1 , ..., ξ N } are called inhomogeneity parameters, and we will assume in the following that they are in generic position such that the above Yang-Baxter algebra representation is irreducible. Let us define the following antisymmetric projectors: where: with P − 1 = I. Then the following proposition holds: 12,13,93]). The fused transfer matrices (quantum spectral invariants): generate n one parameter families of commuting operators: The last quantum spectral invariant, the so-called quantum determinant: is moreover a central element of the algebra: m (λ) has degree mN in λ and central asymptotic behavior given by: b) the quantum determinant reads: The next fusion identities hold: together with the following central zeroes structure: Proof. The general fusion identity [12,13,93] reduce to the above ones if computed in the inhomogeneities and their shifted values and moreover they imply the central zeroes structure stated in the proposition.
Let us introduce the functions and ), (2.17) then the following corollary holds: (λ) allows to completely characterize all the higher transfer matrices T (K) m (λ) by the fusion equations and central zeros structure in terms of the following interpolation formulae: Proof. The known central zeroes and asymptotic behavior imply the above interpolation formula once we use the fusion equations to write T 3 Complete transfer matrix spectrum in our SoV approach 3.1 SoV covector basis for the quasi-periodic Y (gl n ) fundamental model The fundamental Proposition 2.4 proven in [1] for the construction of the SoV covector basis applies to the fundamental representation of the gl n rational Yang-Baxter algebra, i.e. the Yangian gl n . Let us introduce the following notations with K J the Jordan form of K with the following block form: where any K  ). Let K be a n × n w-simple matrix, then form a covector basis of H = N a=1 V a , with V a ≃ C n , for almost any choice of S| and of the inhomogeneities satisfying the irreducibility condition. In particular, the state S| can take the next tensor product form: where: 1 , ..., x

5)
as soon as we take: M a=1 x (a)

Transfer matrix spectrum characterization
Let us define the following n − 1 polynomials in λ and functions of a n × n matrix K and of a point and from it: for any m ∈ {1, ..., n − 2}. Then, the following characterization of the transfer matrix spectrum holds: Theorem 3.1. Let us assume that the same conditions implying that (3.3) is a covector basis are satisfied, then we have the following characterization of the spectrum of T Σ T being the set of solutions to the following system of N equations of order n: (λ) is non-degenerate and for any t 1 (λ) ∈ Σ T (K) the associated unique eigenvector |t has the following wave-function in the left SoV basis, up-to an overall normalization: Finally, if the n × n twist matrix K has simple spectrum and it is diagonalizable then T (λ) is diagonalizable and with simple spectrum, for almost any choice of the inhomogeneity parameters. Proof. The system of N equations of order n in the N unknown {x 1 , ..., x N } coincides with the rewriting of the transfer matrix fusion equations: for the eigenvalues of the transfer matrices T n−1 (λ). Moreover the recursion relations (3.8) just follow from the corresponding recursion relations for the fused transfer matrices in (2.18). So any eigenvalue of the transfer matrix has to satisfy this system of equations and the associated eigenvector |t has the given characterization in the covector SoV basis.
The reverse statement has to be proven now. That is we have to prove that any polynomial t 1 (λ), of the above given form, which is solution of this system is an eigenvalue of the transfer matrix. We prove this showing that the vector |t characterized by (3.11) is a transfer matrix eigenvector, i.e. that for any h 1 , ..., h N | it holds: This proof being quite lengthy, the main technical part of it is presented in the Appendix A. Let us just here give the main steps. The first remark is that the common spectrum of the transfer matrices evaluated in the ξ i being simple, any operator commuting with the transfer matrix can be obtained as a polynomial of maximal degree n N (the dimension of the Hilbert space) in the transfer matrix itself, see e.g., [91,92]. It means that the vector space B T (the so-called Bethe algebra) spanned by the operators commuting with the transfer matrices T (K) m (λ) evaluated at an arbitrary spectral parameter λ is of maximal dimension n N . Now, let us denote by T .., n − 1} ⊗N the following products: (3.14) The fact that (3.3) defines a basis of our Hilbert space immediately implies that the system of operators given in (3.14) forms a free family of n N operators. Moreover any of these operators obviously commutes with the transfer matrix. Hence, the system of operators given in (3.14) forms a basis of the vector space B T of operators commuting with the transfer matrix evaluated at arbitrary values of the spectral parameter. As already noted above, to obtain the proof that the polynomial t 1 (λ) is an eigenvalue, we need to consider the action of T (K) 1 (ξ a ), for a = 1, . . . , N on an arbitrary vector h 1 , ..., h N | of our SoV basis and to give its decomposition again on this basis. Then, the vector |t being completely defined by its components on the SoV basis we can compute the necessary objects entering (3.13). Obviously, this amounts to be able to write the product T (ξ a ) for any given h as a linear combination of T is an operator commuting with any transfer matrix, it belongs to the vector space B T , and can be decomposed linearly on the basis given by the set (3.14). Namely, there exist sets of complex numbers C (a) hh ′ depending on the two sets of parameters h and h ′ and on the index (a) such that: T Then, using the interpolation formula for T Moreover, the results of Appendix A show that the set of complex numbers C (a) hh ′ and hence of polynomials C hh ′ (λ) are completely determined by the hierarchy of fusion relations for the transfer matrices supplemented by their asymptotic behavior. Consequently, the above relation is also satisfied by the quantities t 1 (ξ a ), namely we have, and, Hence we get for any choice of the co-vector h 1 , ..., h N |, This relation being true on the SoV basis, it proves that |t is an eigenvector of the transfer matrix T The final statement of the theorem about simplicity and diagonalizability of the transfer matrix has been already shown in Proposition 2.5 of [1].
Let us further remark that the above results have an interesting consequence. Indeed, using recursively the algebraic relation (3.15), one can obtain the following algebraic structure of the space B T : T for a set of complex numbers coefficients C h ′′ hh ′ that are completely determined (and computable) from the fusion relations satisfied by the transfer matrices. These coefficients are the structure constants of the associative and commutative algebra B T . 4 Transfer matrix spectrum by quantum spectral curve 4

.1 The quantum spectral curve equation
The transfer matrix spectrum in our SoV basis is equivalent to the quantum spectral curve functional reformulation as stated in the next theorem.
for any m ∈ {1, ..., n−2}, and ϕ t (λ) are solutions of the following quantum spectral curve functional equation: where we have defined and andᾱ is solution of the characteristic equation: i.e.ᾱ is an eigenvalue of the matrix K. Moreover, up to a normalization the common transfer matrix eigenvector |t admits the following separate representation: Proof. Let the entire function t 1 (λ) satisfies with the polynomial t m (λ) and ϕ t (λ) the functional equation then it is a degree N polynomial in λ with leading coefficient t 1,N solution of the equation: which beingᾱ an eigenvalue of K implies: Now for λ = ξ a it holds: 11) and the functional equation reduces to: .
While, for any fixed s such that 1 ≤ s ≤ n − 1, for λ = ξ a + sη it holds: so that the functional equation reduces to: .
Now from the identities: the previous identities imply that the following equations are satisfied: so that by our previous theorem we have that t m (λ) are eigenvalues of the transfer matrices T (K) m (λ), associated to the same eigenvector |t .
We derive now the reverse statement. That is, let t 1 (λ) be eigenvalue of the transfer matrix T (K) 1 (λ) then we show that there exists a polynomial ϕ t (λ) satisfying with the t m (λ) the functional equation. The polynomial ϕ t (λ) is here characterized by imposing the next set of conditions: The fact that this characterizes uniquely a polynomial of the form (4.1) is an essential point in the derivation of the functional equation starting from the SoV characterization of the spectrum and we have detailed it in Appendix B. Here, we prove that this characterization of ϕ t (λ) implies the validity of the functional equation. The l.h.s. of the functional equation is a polynomial in λ of maximal degree (n + 1)N so to show that it is identically zero we have to prove it in (n + 1)N distinct points, being zero the leading coefficient by the choice ofᾱ to be an eigenvalue of K. We use the following (n+1)N points ξ a +k a η, for any a ∈ {1, ..., N } and k a ∈ {−1, 0, ..., n − 1}. Indeed, for λ = ξ a − η it holds: from which the functional equation is satisfied for any a ∈ {1, ..., N } and in the remaining nN points the functional equation reduces to the nN equations (4.12)-(4.15). Now, being the fusion equations satisfied by the transfer matrix eigenvalues, these equations are all equivalent to the discrete characterization (4.18) implying our statement. Finally, we show that the SoV characterization of the transfer matrix eigenvectors is equivalent to that presented in this theorem, up to an overall normalization. Indeed, multiplying the eigenvector |t by the non-zero product of the ϕ n−1 t (ξ a ) over all the a ∈ {1, ..., N } it holds: (4.20)

Reconstruction of the Q-operator and the Baxter T -Q equation
The previous characterization of the transfer matrix spectrum indeed allows to reconstruct the Q-operator in terms of the elements of the monodromy matrix and more precisely in terms of the fundamental transfer matrix, as it is stated in the following: Corollary 4.1. Let us assume that K is a n × n diagonalizable matrix with simple spectrum. Let us take 3 ξ N +1 = ξ i≤N and let exist an i ∈ {1, . . . ,N } such that k i = 0, then, for almost any values of {ξ i≤N } and of {k j≤n }, a Q-operator is given by the following polynomial family of commuting operators of maximal degree N : where we have defined: 22) and the central matrix of rank one: Indeed, it satisfies with the transfer matrices the quantum spectral curve at operator level: where we have defined T Proof. This result is a direct consequence of the SoV characterization of the spectrum and of the proof of the previous theorem. As shown in Appendix B, for any t 1 (λ) eigenvalue of the transfer matrix T (K) 1 (λ) we can associate the polynomial ϕ t (λ) of the form (4.1) solution of the quantum spectral curve equation with the transfer matrix eigenvalues. In that proof, we have shown that, up to an irrelevant overall nonzero normalization, ϕ t (λ) admits the following representation: in terms of the above defined matrices where we have just replaced the transfer matrix T (ξ a ) with the eigenvalues t 1 (ξ a ). In Proposition 2.5 of [1], we have also shown that for almost any value of the parameters {ξ i≤N } and {k j≤n } such that K is diagonalizable and with simple spectrum then the transfer matrix T (K) 1 (λ) is diagonalizable and with simple spectrum. This implies that we can uniquely define the polynomial operator family Q i (λ) by its action on the eigenbasis of the transfer matrix imposing: for any t 1 (λ) eigenvalue of the transfer matrix T (K) 1 (λ) and |t the uniquely (up to normalization) associated eigenstate. Then, by definition this operator family satisfies with the transfer matrices the quantum spectral curve equation, admits the announced representation in terms of the transfer matrix T (K) 1 (λ) and the spectrum of its zeros never intersect the set of the {ξ i≤N }, which completes our proof.

Algebraic Bethe ansatz like rewriting of the spectrum
We can show that the previous SoV representation of the transfer matrix eigenvectors can be written in an Algebraic Bethe Ansatz form. Let us first observe that we can find one common eigenvector of the transfer matrices T Lemma 5.1. Let K be a n × n w-simple matrix and let us denote with K J its Jordan form with K = W K K J W −1 K , then: is a common eigenvector of the transfer matrices T (K) m (λ): where: and where the t m,0 (λ) satisfy the quantum spectral curve with constant ϕ t (λ): Proof. Note that the vector |0 is eigenvector of the transfer matrix T (K J ) 1 (λ) with eigenvalue t 1,0 (λ). In fact, this is proven by showing that: from which it easily follows that the vector |0 is eigenvector of all the others transfer matrices T (K J ) m (λ) with eigenvalues t m,0 (λ). Then, by the similarity relation: we get our statement about the original transfer matrices. Note that these eigenvalues t m,0 (λ) have to satisfy the quantum spectral curve as a consequence of the previous theorem. We have just to observe now that for the choiceᾱ = k 1 it holds: so that it follows that the associated ϕ t (λ) satisfies the equations: and so ϕ t (λ) is constant. Indeed, defined: this is a degree N − 1 polynomial in λ which is zero in N different points so that it is identically zero.
Let us now define 11) which is diagonal in the SoV basis and it is characterized by: from which it follows: where: then the next corollary follows: for the eigenvector associated to the generic eigenvalue t 1 (λ) ∈ Σ T 1 . Here the λ a are the roots of the polynomial ϕ t (λ) satisfying with the t m (λ) the quantum spectral curve functional equation.
Proof. The following chain of identities holds: where we have used that: which coincides with the last SoV characterization of the same transfer matrix eigenvector.

A Appendix A
This appendix is dedicated to complete the proof of the Theorem 3.1. In the first two subsections, we first introduce some partial results and some tools to compute the action of transfer matrices on the elements of the covector SoV basis then we use them in the third subsection to complete the proof of the Theorem 3.1.

A.1 Transfer matrix action on inner covectors of the SoV basis
Let us start introducing some notations for the SoV covector basis (3.3) for our current aims, we denote: and in the following we suppress the index {h} for brevity. Moreover, we introduce the further notations: for any 1 ≤ m ≤ n, and we use the shorted notation for the generic element of the SoV covector basis when all the information that we need to know are contained in this two numbers. Then, the following lemma holds: Lemma A.1. Under the same assumption of the Theorem 3.1, the following identities hold: Proof. Let us start observing that for h a ≤ n − 2 and h b ∈ {0, ..., n − 1}, for any b ∈ {1, ..., N }\a, it holds: Then, the above identity (A.6) imply the following ones: and so, being t 1 (λ) a polynomial of degree N with known asymptotics which coincides by definition with the central one of T Let us remark now that the interpolation formulae (2.18), for the higher transfer matrices T (K) m (λ), and the formulae (3.8), for the higher functions t m (λ), can be rewritten as it follows: where r a 1 ,...,am (λ) are some computable by induction polynomials of degree one in λ, whose explicit values are not required for the following. From these formulae it follows that:  {N 1 ,...,Nn} (λ) of the above developments are the same, will follow from the fact that we make exactly the same type of operations on the two type of matrix elements, as described in the next.

A.2.1 Interpolation expansions and fusion properties
First, we introduce the following rules to use the interpolation formulae Second, we use the fusion equations to rewrite where: Now denoted with N − m−1 and N + m−1 the integers associated to the above covectors, it holds: whereN − m−1 andN + m−1 are the integers associated to the covector N 1 , ...,N n |. While, if n − m ≤ h a ≤ n − 1, then we have the following rewriting: and t m+1 (ξ a ) N 1 , ..., N n |t , (A. 33) where: and now it holds: The following lemma will be used in the proof of the Theorem 3.1: so that using the standard interpolation formula and fusion identities we prove the above relation (A.39).  24). So that, for all the b such that in the covector N 1 , ..., N n | of (A.27) it holds n + 1 − m ≤ h b ≤ n − 1, we are lead to the matrix elements:

A.2.2 Generation of loops
However, for all the b such that in the covector N 1 , ..., N n | of (A.27) it holds 0 ≤ h b ≤ n − m, we are lead to the matrix elements: and Similarly, ifN − m+1 = 0 then the terms (A.32) and (A.33) have to be developed by using respectively the interpolation formula (A.22) and (A.23) with the choice of the points (A. 24). So that, for all the b such that in the covector N 1 , ..., N n | of (A.32) it holds 0 ≤ h b ≤ n − 2 − m, we are lead to the matrix elements: and We have done in this way one loop (we call it L However, for all the b such that in the covector N 1 , ..., N n | of (A.32) it holds n−1−m ≤ h b ≤ n−1, we are lead to the matrix elements: for any λ ∈ C, we have just to be able to compute matrix elements of the type: n−2 (ξ a − η) and t n−2 (ξ a − η)) are known to coincide otherwise we have to expand them in the usual way by using the interpolation formula and this produces a loop L and While if we develop an x-step loop of up type L (m) +x,−x then we are lead to matrix elements of the form: so that in both the case it holds: Proof. Let us assume that the original state N 1 ,N 2 , ...,N n | is such that starting from the matrix elements ,n−(m+r) ) + δ t (r,+) ,n−(m+r) ∀s ∈ {0, ..., n − (m + x + 2)},  Now from the above formulae it follows for the final state the following rules:  75) and the others of the type: +,− which can be written in a common notation as it follows: +,− with r 3 + s 3 = 1 and the others of the type: +,− which can be written in a common notation as it follows: This process is continued up to x loops, where x is the smaller integer such thatN − 1+2x ≥ 1 and N − 1+2(x+1) = 0 or x = {(n − 1) /2 for n odd, (n − 2) /2 for n even}, in this way producing for each one of our original matrix elements (A.74) the development in the same linear combination of at most N 1+2x matrix elements, of the type: with r 1+2a + s 1+2a = 1 for 1 ≤ a ≤ x − 1 and the remaining one of the type: From this point any further loop development of (A.82) does not generate new type of terms but instead it generates matrix elements of the type: 83) and Starting from the terms (A.82) making at mostN − 1+2(x+1) loops we arrive at matrix elements of the type (A.83) and to  86) and new border terms of the type (A.84) with N − 1+2x ≤N − 1+2x − 1, either this integer is zero or we repeat for these border terms the same procedure explained above ending up with only matrix elements of the type (A.81) for 1 ≤ a ≤ x − 2 while for a = x − 1, we are left with all terms of the type (A.86) so that we have reduced of one unit the value of N T 1+2(x−1) with respect to the value in (A.83) for which it was holding We can repeat this procedure now and after at most a total ofN T 2x−1 − 2x − N , we generate matrix elements with both a = x − 1 and a = x for which the identity between operator and scalar matrix elements is known. So on we are lead to develop matrix elements with a = x − 2 reducing, at any step of the above procedure, at least of one unit the value of N T 1+2(x−2) up to arrive to matrix elements for which the identity of the operator and scalar terms is known. At this point we are left with a = x − 3 and so on up to arrive to be left only with matrix elements of the type (A.81) for a = 1 for which we have N + 1 ≤N + 1 − 1, here if we want we can do induction and prove the theorem.
However, it is interesting to get a bound on the maximal number of loops to implement in order to be reduced to linear combinations of matrix elements for which the identity (A.5) applies. From the above analysis, to get the reduction of at least one unit in N + 1 , we have to implement before x loops, to generate all the types of allowed matrix elements, then we have to do less than loops. This means that using the above procedure, we have to do less than loops in order to generate only matrix elements for which the identity (A.5) applies. These means that starting from the original matrix elements (A.17) and (A.18), by implementing all these loops, they will be rewritten as the linear combination of less than N 1+2N T matrix elements for which the identity (A.5) applies with the same coefficients, namely for any i = 1, . . . , N we have: It is interesting to remark here that all the formulae derived in this Appendix could have been obtained exactly in the same manner for the products of operators themselves appearing in any of the considered average value between the co-vector S| and the vector |t . Hence a direct consequence of the above relation is the equation (3.15) with the property that the coefficients C

B Appendix B
This appendix is devoted to complete the proof of the Theorem 4.1. In a first subsection, we first recall for self-consistence some elementary properties of the algebraic functions that we will use. Then we present in the second subsection the proof the Theorem 4.1.

B.1 Some elementary properties of algebraic functions
Let us present a couple of elementary lemmas on algebraic functions. First, let us recall that a function f (x 1 , ..., x N ) of N variables: is by definition algebraic iff there exist M + 1 polynomials a m (x 1 , ..., x N ) such that it holds: where we have defined the polynomial P (y|x 1 , ..., x N ) as it follows: 3) associated to f (x 1 , ..., x N ), then it is an eigenvalue of a family of square matrices P(x 1 , ..., x N ) with characteristic polynomial coinciding with P (y|x 1 , ..., x N ). Note that we have the following representation of P(x 1 , ..., x N ) in terms of the Frobenius companion matrix: where V M is any invertible M × M square matrix and where is itself an algebraic function.

B.2 Proof of Theorem 4.1
The proof of the Theorem 4.1 is now developed using as main ingredient the properties of the algebraic functions. , for all a, b ∈ {1, . . . , N }. Then our statement is a consequence of the following proposition which ensures that all these determinants are nonzero for almost any values of their parameters. Now by using the rank one N × N matrix ∆ ξ N+1 (λ), defined in (4.23), the interpolation formula for ϕ (i) t (λ) can be rewritten in a one determinant form: or equivalently: where we have defined: Let us define the functions: where we have defined: Clearly, it holds: (y − f τ l 1 (ξ 1 ),....,τ l N (ξ N ) (ξ N +1 , η)), (B.40) they are polynomials in all their variables and the f τ l 1 (ξ 1 ),....,τ l N (ξ N ) (ξ N +1 , η) and g τ l 1 (ξ 1 ),....,τ l N (ξ N ),j (ξ N +1 , η) are the respective associated algebraic roots. Let us remark now that the transfer matrix of the fundamental representation of Y (gl n ) associated to a twist matrix satisfying the eigenvalues conditions k i = δ 1,i is similar to the diagonal entry A 1 (λ) of the monodromy matrix. The spectrum of which is known and has the following form over all the values of R ≤ N and π ∈ S N (permutations of {1, . . . ,N }). So that for any fixed τ l≤n N (λ|{k i }), eigenvalue of the transfer matrix, there exist a R l ≤ N and a π l ∈ S N such that: It is then simple to observe that are polynomial of their parameters: ξ N +1 , {ξ j≤N } and η. So that if we prove that they are nonzero in a point they are so almost everywhere. From the equation (B.46) it follows that for any {l 1 , ..., l N } ∈ {1, ..., n N } ⊗N there exist a R {l j } ≤ N and a π {l j } ∈ S N such that: Then to compute f τ l 1 (ξ 1 ),....,τ l N (ξ N ) (ξ N +1 , η) π(i) for i ∈ {1, . . . , R l } are kept free, where we have omitted the {l j } dependence of the permutation π to simplify the notation. Let us remark that it holds:    We can now observe that the following block structure emerges: [c τ l 1 (ξ (π) 1 (ǫ)),....,τ l N (ξ π(a) −ξ This result implies that f τ l 1 (ξ (π) 1 (ǫ)),....,τ l N (ξ π(i) = ξ (π) π(i) for i ∈ {1, . . . , R l } are kept free, where we have omitted the {l j } dependence of the permutation π to simplify the notation. From this point we can proceed as for f τ l 1 (ξ 1 ),....,τ l N (ξ N ) (ξ N +1 , η) k i =δ 1,i and we get: lim ǫ→0 ǫ S {l i } g τ l 1 (ξ (π) 1 (ǫ)),....,τ l N (ξ (π) It is interesting to remark that the rational functions f τ l (ξ N +1 , {ξ (π) j≤N }, {k i = δ 1,i }, η) and g τ l ,j (ξ N +1 , {ξ (π) j≤N }, {k i = δ 1,i }, η) admits some simple explicit expression as a consequence of the calculations developed in the previous proposition.
Corollary B.2. There exist a R l ≤ N and a π l ∈ S N such that: π(i) for i ∈ {1, . . . , R l } are kept free.