Quasi-classical descendants of disordered vertex models with boundaries

Abstract

We study descendants of inhomogeneous vertex models with boundary reflections when the spin-spin scattering is assumed to be quasi--classical. This corresponds to consider certain power expansion of the boundary-Yang-Baxter equation (or reflection equation). As final product, integrable $su(2)$-spin chains interacting with a long range with $XXZ$ anisotropy are obtained. The spin-spin couplings are non uniform, and a non uniform tunable external magnetic field is applied; the latter can be obtained when the boundary conditions are assumed to be quasi-classical as well. The exact spectrum is achieved by algebraic Bethe ansatz. Having realized the $su(2)$ operators in terms of fermions, the class of models we found turns out to describe confined fermions with pairing force interactions. The class of models presented in this paper is a one-parameter extension of certain Hamiltonians constructed previously. Extensions to $su(n)$-spin open chains are discussed.

27 pages; 2 eps figures; elsart. Revised version, appendix C added

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• Exact solution of the p+ip pairing Hamiltonian and a hierarchy of integrable models
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• Inner products in integrable Richardson-Gaudin models
• $Z_n$ elliptic Gaudin model with open boundaries
• Integrable spin-boson interaction in the Tavis-Cummings model from a generic boundary twist
• Exact solution of the XXX Gaudin model with the generic open boundaries
• Determinant representations for scalar products of the XXZ Gaudin model with general boundary terms
• A Bethe ansatz solvable model for superpositions of Cooper pairs and condensed molecular bosons
• Quantum-classical duality for Gaudin magnets with boundary
• Mesoscopic BCS pairing in the repulsive 1d-Hubbard model
• Relaxation of nonlinear oscillations in BCS superconductivity
• Quantum-classical correspondence for gl(1|1) supersymmetric Gaudin magnet with boundary
• $osp(1|2)$ off-shell Bethe ansatz equation with boundary terms
• Exact solution of the Izergin-Korepin Gaudin model with periodic and open boundaries
• Coherent oscillations in superconducting cold Fermi atoms and their applications