Separation of variables for rational gl(n) spin chains in any compact representation, via fusion, embedding morphism and Backlund flow

Abstract

We propose a way to separate variables in a rational integrable $\mathfrak{gl}(n)$ spin chain with an arbitrary finite-dimensional irreducible representation at each site and with generic twisted periodic boundary conditions. Firstly, we construct a basis that diagonalises a higher-rank version of the Sklyanin B-operator; the construction is based on recursive usage of an embedding of a $\mathfrak{gl}(k)$ spin chain into a $\mathfrak{gl}(k+1)$ spin chain which is induced from a Yangian homomorphism and controlled by dual diagonals of Gelfand-Tsetlin patterns. Then, we show that the same basis can be equivalently constructed by action of Backlund-transformed fused transfer matricies, whence the Bethe wave functions factorise into a product of ascending Slater determinants in Baxter Q-functions. Finally, we construct raising and lowering operators -- the conjugate momenta -- as normal-ordered Wronskian expressions in Baxter Q-operators evaluated at zeros of B -- the separated variables. It is an immediate consequence of the proposed construction that the Bethe algebra comprises the maximal possible number of mutually commuting charges -- a necessary property for Bethe equations to be complete.

References in corpus (37)

• Integrable Structure of Conformal Field Theory II. Q-operator and DDV equation
• Quantum spectral curve for AdS_5/CFT_4
• Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations
• Quantum spectral curve for arbitrary state/operator in AdS${}_5$/CFT${}_4$
• Noncompact Heisenberg spin magnets from high-energy QCD: I. Baxter Q-operator and Separation of Variables
• Supersymmetric Bethe Ansatz and Baxter Equations from Discrete Hirota Dynamics
• Baxter Q-Operators and Representations of Yangians
• Analytic Bethe ansatz and functional equations associated with any simple root systems of the Lie superalgebra sl(r+1|s+1)
• Quantum Spectral Curve and Structure Constants in N=4 SYM: Cusps in the Ladder Limit
• Solutions of the T-system and Baxter equations for supersymmetric spin chains
• Baxter Q-operator and Separation of Variables for the open SL(2,R) spin chain
• Separation of variables in the quantum integrable models related to the Yangian Y[sl(3)]
• Analytic Bethe Ansatz for Fundamental Representations of Yangians
• Separation of variables for the quantum SL(2,R) spin chain
• Bi-scalar integrable CFT at any dimension
• Baxter's Q-operators and operatorial Backlund flow for quantum (super)-spin chains
• New Construction of Eigenstates and Separation of Variables for SU(N) Quantum Spin Chains
• Fast analytic solver of rational Bethe equations
• Basso-Dixon Correlators in Two-Dimensional Fishnet CFT
• Reconstruction of Baxter Q-operator from Sklyanin SOV for cyclic representations of integrable quantum models
• Analytic Bethe ansatz and functional equations for Lie superalgebra sl(r+1|s+1)
• On quantum separation of variables
• Backlund transformations for difference Hirota equation and supersymmetric Bethe ansatz
• The Holographic Fishchain
• Completeness of Bethe Ansatz by Sklyanin SOV for Cyclic Representations of Integrable Quantum Models
• Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame
• Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables
• Discrete Hirota's equation in quantum integrable models
• On determinant representations of scalar products and form factors in the SoV approach: the XXX case
• Classical Functional Bethe Ansatz for $SL(N)$: separation of variables for the magnetic chain
• The full spectrum of AdS5/CFT4 I: Representation theory and one-loop Q-system
• On Bethe vectors in ${\mathfrak{gl}}_3$-invariant integrable models
• Quantum Fishchain in $Ad{S}_5$
• A new integral representation for the scalar products of Bethe states for the XXX spin chain
• New Compact Construction of Eigenstates for Supersymmetric Spin Chains
• Manin matrices and Talalaev's formula
• Complete spectrum of quantum integrable lattice models associated to ${\mathcal{U}}_q(\widehat{g{l}_n})$ by separation of variables

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• Form-factors and complete basis of observables via separation of variables for higher rank spin chains
• Hexagonalization of Fishnet integrals I: mirror excitations
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• Completeness of SoV Representation for $\mathrm{SL}(2,\mathbb{R})$ Spin Chains
• Determinant Form of Correlators in High Rank Integrable Spin Chains via Separation of Variables
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• Separation of variables for higher rank integrable models: review of recent progress
• Unitarity of the SoV Transform for $\mathrm{SL}(2,\mathbb{C})$ Spin Chains
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