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

Abstract

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.

43 pages, v2: several references added

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Cited by in corpus (22)

• Basso-Dixon Correlators in Two-Dimensional Fishnet CFT
• Separation of variables and scalar products at any rank
• Separation of variables for rational gl(n) spin chains in any compact representation, via fusion, embedding morphism and Backlund flow
• A review of the AdS/CFT Quantum Spectral Curve
• Mirror channel eigenvectors of the $d$-dimensional fishnets
• On Scalar Products in Higher Rank Quantum Separation of Variables
• Completeness of Wronskian Bethe equations for rational gl(m|n) spin chains
• On quantum separation of variables beyond fundamental representations
• Complete spectrum of quantum integrable lattice models associated to ${\mathcal{U}}_q(\widehat{g{l}_n})$ by separation of variables
• Separation of variables bases for integrable $g{l}_{\mathcal{M}|\mathcal{N}}$ and Hubbard models
• Form-factors and complete basis of observables via separation of variables for higher rank spin chains
• On Separation of Variables for Reflection Algebras
• Correlation functions by Separation of Variables: the XXX spin chain
• Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields
• Extended systems of Baxter Q-functions and fused flags I: simply-laced case
• All correlation functions of the open XXX spin 1/2 quantum chains for unparallel boundary magnetic fields with one constraint
• Determinant Form of Correlators in High Rank Integrable Spin Chains via Separation of Variables
• Bethe ansatz inside Calogero-Sutherland models
• Separation of variables for higher rank integrable models: review of recent progress
• On scalar products and form factors by Separation of Variables: the antiperiodic XXZ model
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint