On quantum separation of variables

Abstract

We present a new approach to construct the separate variables basis leading to the full characterization of the transfer matrix spectrum of quantum integrable lattice models. The basis is generated by the repeated action of the transfer matrix itself on a generically chosen state of the Hilbert space. The fusion relations for the transfer matrix, stemming from the Yang-Baxter algebra properties, provide the necessary closure relations to define the action of the transfer matrix on such a basis in terms of elementary local shifts, leading to a separate transfer matrix spectral problem. Hence our scheme extends to the quantum case a key feature of Liouville-Arnold classical integrability framework where the complete set of conserved charges defines both the level manifold and the flows on it leading to the construction of action-angle variables. We work in the framework of the quantum inverse scattering method. As a first example of our approach, we give the construction of such a basis for models associated to Y(gln) and argue how it extends to their trigonometric and elliptic versions. Then we show how our general scheme applies concretely to fundamental models associated to the Y(gl2) and Y(gl3) R-matrices leading to the full characterization of their spectrum. For Y(gl2) and its trigonometric deformation a particular case of our method reproduces Sklyanin's construction of separate variables. For Y(gl3) it gives new results, in particular through the proper identification of the shifts acting on the separate basis. We stress that our method also leads to the full characterization of the spectrum of other known quantum integrable lattice models, including in particular trigonometric and elliptic spin chains, open chains with general integrable boundaries, and further higher rank cases that we will describe in forthcoming publications.

61 pages, v2: small typos corrected and some references added

References in corpus (12)

• Quenching the Anisotropic Heisenberg Chain: Exact Solution and Generalized Gibbs Ensemble Predictions
• Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
• Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors
• Separation of Variables in the open XXX chain
• Antiperiodic dynamical 6-vertex model I: Complete spectrum by SOV, matrix elements of the identity on separate states and connections to the periodic 8-vertex model
• Functional Bethe ansatz methods for the open XXX chain
• Completeness of Bethe Ansatz by Sklyanin SOV for Cyclic Representations of Integrable Quantum Models
• Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I
• Baxter-Bazhanov-Stroganov model: Separation of Variables and Baxter Equation
• On Bethe vectors in ${\mathfrak{gl}}_3$-invariant integrable models
• The tau_2-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin's SOV method
• Separation of variables for the quantum SL(3,C) spin magnet: eigenfunctions of Sklyanin B-operator

Cited by in corpus (36)

• Correlation functions and transport coefficients in generalised hydrodynamics
• Algebraic construction of current operators in integrable spin chains
• Basso-Dixon Correlators in Two-Dimensional Fishnet CFT
• Dual Separated Variables and Scalar Products
• Exactly solvable magnet of conformal spins in four dimensions
• Separation of variables and scalar products at any rank
• Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables
• Gaudin Models and Multipoint Conformal Blocks II: Comb channel vertices in 3D and 4D
• Separation of variables for rational gl(n) spin chains in any compact representation, via fusion, embedding morphism and Backlund flow
• Exactly solvable single-trace four point correlators in $\chi$CFT${}_4$
• On exact overlaps for $\mathfrak{gl}(N)$ symmetric spin chains
• Open Fishchain in N=4 Supersymmetric Yang-Mills Theory
• 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
• Introduction to the nested algebraic Bethe ansatz
• On Separation of Variables for Reflection Algebras
• Correlation functions of integrable $O(n)$ spin chains
• Wrapping corrections for long range spin chains
• 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
• Boundary dissipative spin chains with partial solvability inherited from system Hamiltonians
• Determinant Form of Correlators in High Rank Integrable Spin Chains via Separation of Variables
• Reduction of divisors for classical superintegrable $GL(3)$ magnetic chain
• Separation of variables for higher rank integrable models: review of recent progress
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• Bethe ansatz inside Calogero-Sutherland models
• Canonical formulation for the thermodynamics of $s{l}_n$-invariant integrable spin chains
• Twisted and Non-Twisted Deformed Virasoro Algebra via Vertex Operators of $U_q({\widehat{\mathfrak{sl}}}_2)$
• The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint
• Orthogonality of Q-Functions up to Wrapping in Planar N=4 Super Yang-Mills Theory
• BC-type open $SL(2,\mathbb{C})$ spin chain