Abstract
We describe the extension, beyond fundamental representations of the Yang-Baxter algebra, of our new construction of separation of variables bases 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 bases 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 types of SoV bases. 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 type of SoV bases for which the proof of the separation of the transfer matrix spectrum is naturally derived. We show that, under some special choice, this second type of SoV bases coincides with the one associated to Sklyanin’s approach. Moreover, we derive the finite difference type (quantum spectral curve) functional equation and the set of its solutions 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})Y(gl2) Yang-Baxter algebra. Our SoV approach also leads to the construction of a QQ-operator in terms of the fused transfer matrices. Finally, we show that the QQ-operator family can be equivalently used as the family of commuting conserved charges enabling to construct our SoV bases.
Highlights
In our two first papers for the fundamental representations of Y (g ln), n ≥ 2, we have identified a natural choice of the set of the commuting conserved charges and as well characterized the generating co-vector to be used as starting point to generate our separation of variables (SoV) basis
In subsection 4.2, we introduce another SoV basis constructed from the full tower of fused transfer matrices, which we argue to be the most natural with respect to the action of the transfer matrix that becomes explicitly linear in that basis thanks to the fusion rules satisfied by the quantum spectral invariants
In the non-fundamental representations we considered in this article, we can make the above description of the SoV basis construction using the Q-operator
Summary
In this article we continue the development of our approach [1,2,3] to generate the separation of variables (SoV) complete characterization of the spectrum of quantum integrable lattice models. In order to use directly the Q-operator family to generate SoV bases 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 a basis; fourth that the transfer matrix spectrum is separated in this basis. The 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 to develop our analysis in the framework of the separation of variables
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