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AIP conference proceedings

New Generalizations of the Richardson-Gaudin Models

J. Dukelsky

Published Jan 1, 2004• 0 citations
Publisher

Abstract

We discuss the extension of the Richardson‐Gaudin models along two lines. We £rst describe a new family of exactly solvable atom‐molecule Hamiltonians obtained by coupling a bosonic mode to the rational family. We then present the generalization of the Richardson‐Gaudin models to any simple Lie algebra. As an example of relevance to nuclear structure, we show the exact solution of the T=1 pairing Hamiltonian with non‐degenerate single‐particle energies.

DOI
10.1063/1.1805906

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We discuss the extension of the Richardson‐Gaudin models along two lines. We £rst describe a new family of exactly solvable atom‐molecule Hamiltonians obtained by coupling a bosonic mode to the rational family. We then present the generalization of the Richardson‐Gaudin models to any simple Lie algebra. As an example of relevance to nuclear structure, we show the exact solution of the T=1 pairing Hamiltonian with non‐degenerate single‐particle energies.
The exact solvability of several nuclear models with non-degenerate single-particle energies is outlined and leads to a generalization of integrable Richardson-Gaudin models, like the su(2)su(2)su(2)-based fermion pairing, to any simple Lie algebra. As an example, the so(5)∼sp(4)so(5)\sim sp(4)so(5)∼sp(4) model of T=1 pairing is discussed and illustrated for the case of 64^{64}64Ge with non-degenerate single-particle energies.
The exact solution of the SU(2) pairing Hamiltonian with non-degenerate single particle orbits was introduced by Richardson in the early sixties. Since then it has been extended to several families of integrable models, called the Richardson-Gaudin (RG) models. In particular, the rational family of integrable RG models has been widely applied to mesoscopic systems like small grains, quantum dots and nuclear systems where finite size effects play an important role. We introduce these families of integrable models and then we describe a new separable pairing Hamiltonian derived from the hyperbolic family that provides an excellent reproduction of Gogny self-consistent mean-field calculations for heavy nuclei.
We present the exact solution of the Richardson–Gaudin models associated with the SU (3) Lie algebra. The derivation is based on a Gaudin algebra valid for any simple Lie algebra in the rational, trigonometric and hyperbolic cases. For the rational case additional cubic integrals of motion are obtained, whose number is added to that of the quadratic ones to match, as required from the integrability condition, the number of quantum degrees of freedom of the model. We discuss different SU (3) physical representations and elucidate the meaning of the parameters entering in the formalism. By considering a bosonic mapping limit of one of the SU (3) copies, we derive new integrable models for three level systems interacting with two bosons. These models include a generalized Tavis–Cummings model for three level atoms interacting with two modes of the quantized electric field.