Classical tau-function for quantum spin chains

Abstract

For an arbitrary generalized quantum integrable spin chain we introduce a "master T -operator" which represents a generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space. We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bilinear equations of the Hirota form for the master T -operator, which allows one to identify it with τ-function of an integrable hierarchy of classical soliton equations. In this paper we consider spin chains with rational GL(N)-invariant R-matrices but the result is independent of a particular functional form of the transfer matrices and directly applies to quantum integrable models with more general (trigonometric and elliptic) R-matrices and to supersymmetric spin chains.

67 pages, 2 figures; one figure and several explanations added in version 2

References in corpus (22)

• Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
• Hidden Grassmann Structure in the XXZ Model II: Creation Operators
• Supersymmetric Bethe Ansatz and Baxter Equations from Discrete Hirota Dynamics
• Hidden Grassmann structure in the XXZ model
• Baxter Q-Operators and Representations of Yangians
• Baxter's Q-operators for supersymmetric spin chains
• Solving the AdS/CFT Y-system
• Wronskian Solution for AdS/CFT Y-system
• Quantization of the Gaudin System
• From Characters to Quantum (Super)Spin Chains via Fusion
• The Baxter's Q-operator for the W-algebra $W_N$
• Backlund transformations for difference Hirota equation and supersymmetric Bethe ansatz
• Wronskian solutions of the T, Q and Y-systems related to infinite dimensional unitarizable modules of the general linear superalgebra gl(M|N)
• KZ Characteristic Variety as the Zero Set of Classical Calogero-Moser Hamiltonians
• Gaudin Hamiltonians generate the Bethe algebra of a tensor power of vector representation of gl_N
• Factorization of R-matrix and Baxter Q-operators for generic sl(N) spin chains
• XXZ scalar products and KP
• Coupled Modified KP Hierarchy and Its Dispersionless Limit
• Rational Ruijsenaars-Schneider hierarchy and bispectral difference operators
• Fermionic basis for space of operators in the XXZ model
• New Developments in the Eight Vertex Model
• Jacobi-Trudy formula for generalised Schur polynomials

Cited by in corpus (43)

• T-system on T-hook: Grassmannian Solution and Twisted Quantum Spectral Curve
• Free fermions and tau-functions
• Conformal algebra: R-matrix and star-triangle relation
• New Construction of Eigenstates and Separation of Variables for SU(N) Quantum Spin Chains
• Spectrum of Quantum Transfer Matrices via Classical Many-Body Systems
• QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains
• The master T-operator for the Gaudin model and the KP hierarchy
• Trigonometric version of quantum-classical duality in integrable systems
• Quantum groups and functional relations for higher rank
• Quantum Spectral Curve of $\gamma$-twisted $\mathcal{N}=4$ SYM theory and fishnet CFT
• Asymptotic representations and q-oscillator solutions of the graded Yang-Baxter equation related to Baxter Q-operators
• Supersymmetric quantum spin chains and classical integrable systems
• Vertex operators arising from Jacobi-Trudi identities
• Dualities in quantum integrable many-body systems and integrable probabilities -- I
• ABJM Matrix Model and 2D Toda Lattice Hierarchy
• Negative flows of generalized KdV and mKdV hierarchies and their gauge-Miura transformations
• The Master T-Operator for Inhomogeneous XXX Spin Chain and mKP Hierarchy
• Two-Point Functions in ABJM Matrix Model
• QKZ-Ruijsenaars correspondence revisited
• Universal Baxter TQ-relations for open boundary quantum integrable systems
• Symmetric polynomials, generalized Jacobi-Trudi identities and τ-functions
• Supersymmetric extension of qKZ-Ruijsenaars correspondence
• Quantum groups, Verma modules and $q$-oscillators: General linear case
• Off-shell scalar products for the $XXZ$ spin chain with open boundaries
• KZ-Calogero correspondence revisited
• On Plücker Equations Characterizing Grassmann Cones
• Jacobi-Trudi Identity in Super Chern-Simons Matrix Model
• Conformal Four-Point Integrals: Recursive Structure, Toda Equations and Double Copy
• Bethe ansatz and Hirota equation in integrable models
• On the Expansion Coefficients of Tau-function of the BKP Hierarchy
• Folding QQ-relations and transfer matrix eigenvalues: towards a unified approach to Bethe ansatz for super spin chains
• Quantum-classical duality for Gaudin magnets with boundary
• Integrable systems and AdS/CFT duality
• Quantum spin chains and integrable many-body systems of classical mechanics
• Asymmetric 6-vertex model and classical Ruijsenaars-Schneider system of particles
• $Q$-Boson model and relations with integrable hierarchies
• Quantum-classical correspondence for gl(1|1) supersymmetric Gaudin magnet with boundary
• Comments on Slavnov products, Temperley-Lieb open spin chains, and KP tau functions
• Decomplexification of the Capelli identities and holomorphic factorization
• On the Integrable Structure of Super Yang-Mills Scattering Amplitudes
• Giambelli type formulae in the BKP hierarchy
• Character expansion of matrix integrals
• Boson-Fermion correspondence, QQ-relations and Wronskian solutions of the T-system