Noncompact SL(2,R) spin chain

Abstract

We consider the integrable spin chain model - the noncompact SL(2,R) spin magnet. The spin operators are realized as the generators of the unitary principal series representation of the SL(2,R) group. In an explicit form, we construct R-matrix, the Baxter Q-operator and the transition kernel to the representation of the Separated Variables (SoV). The expressions for the energy and quasimomentum of the eigenstates in terms of the Baxter Q-operator are derived. The analytic properties of the eigenvalues of the Baxter operator as a function of the spectral parameter are established. Applying the diagrammatic approach, we calculate Sklyanin's integration measure in the separated variables and obtain the solution to the spectral problem for the model in terms of the eigenvalues of the Q-operator. We show that the transition kernel to the SoV representation is factorized into a product of certain operators each depending on a single separated variable.

29 pages, 12 figures

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• Baxter operators with deformed symmetry
• On exact overlaps in integrable spin chains
• Baxter Q-operators of the XXZ chain and R-matrix factorization
• Baxter operators for arbitrary spin II
• Baxter operators for arbitrary spin
• Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma models
• A new representation for two- and three-point correlators of operators from sl(2) sector
• Intertwining operators for Sklyanin algebra and elliptic hypergeometric series
• Baxter operators for the quantum sl(3) invariant spin chain
• Analytic continuation of Bethe energies and application to the thermodynamic limit of the $SL(2,\mathbb{C})$ non-compact spin chains
• Jordanian spin chains for twisted strings in $Ad{S}_5\times S^5$
• On Derkachov--Manashov $R$-matrices for the principal series of unitary representations
• Hypergeometry from $\hat{\mathrm{P}}$-Symmetry: Feynman Integrals in One and Two Dimensions