J. Dukelsky

Colloquium: Exactly solvable Richardson-Gaudin models for many-body quantum systems

The use of exactly solvable Richardson-Gaudin models to describe the physics of systems with strong pair correlations is reviewed. The article begins with a brief discussion of Richardsons early work, which demonstrated the exact solvability of the pure pairing model, and then shows how that work has evolved recently into a much richer class of exactly solvable models. The Richardson solution leads naturally to an exact analogy between these quantum models and classical electrostatic problems in two dimensions. This analogy is then used to demonstrate formally how BCS theory emerges as the large-

Class of Exactly Solvable Pairing Models

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EQUIVALENCE OF THE VARIATIONAL MATRIX PRODUCT METHOD AND THE DENSITY MATRIX RENORMALIZATION GROUP APPLIED TO SPIN CHAINS

limit of the pure pairing Hamiltonian. Several applications to problems of relevance to condensed-matter physics, nuclear physics, and the physics of confined systems are considered. Some of the interesting effects that are discussed in the context of these exactly solvable models include (i) the crossover from superconductivity to a fluctuation-dominated regime in small metallic grains; (ii) the role of the nucleon Pauli principle in suppressing the effects of high-spin bosons in interacting boson models of nuclei, and (iii) the possibility of fragmentation in confined boson systems. Interesting insight is also provided into the origin of the superconducting phase transition both in two-dimensional electronic systems and in atomic nuclei, based on the electrostatic image of the corresponding exactly solvable quantum pairing models.

Neutron-proton correlations in an exactly solvable model

We present three classes of exactly solvable models for fermion and boson systems, based on the pairing interaction. These models are solvable in any dimension. As an example we show the first results for fermions interacting with repulsive pairing forces in a two-dimensional square lattice. In spite of the repulsive pairing force the exact results show attractive pair correlations.

Exactly-solvable models derived from a generalized Gaudin algebra

We study the relationship between the Density Matrix Renormalization Group (DMRG) and the variational matrix product method (MPM). In the latter method one can also define a density matrix whose eigenvalues turn out to be numerically close to those of the DMRG. We illustrate our ideas with the spin-1 Heisenberg chain, where we compute the ground-state energy and the spin correlation length. We also give a rotational invariant formulation of the MPM.

Density Matrix Renormalization Group Study of Ultrasmall Superconducting Grains

We examine isovector and isoscalar neutron-proton correlations in an exactly solvable model based on the algebra SO(8). We look particularly closely at Gamow-Teller strength and double β decay, both to isolate the effects of the two kinds of pairing and to test two approximation schemes: the renormalized neutron-proton quasiparticle random phase approximation (QRPA) and generalized BCS theory. When isoscalar pairing correlations become strong enough a phase transition occurs and the dependence of the Gamow-Teller β+ strength on isospin changes in a dramatic and unfamiliar way, actually increasing as neutrons are added to an N=Z core. Renormalization eliminates the well-known instabilities that plague the QRPA as the phase transition is approached, but only by unnaturally suppressing the isoscalar correlations. Generalized BCS theory, on the other hand, reproduces the Gamow-Teller strength more accurately in the isoscalar phase than in the usual isovector phase, even though its predictions for energies are equally good everywhere. It also mixes T=0 and T=1 pairing, but only on the isoscalar side of the phase transition.

Large-N limit of the exactly solvable BCS model: analytics versus numerics

Abstract We introduce a generalized Gaudin Lie algebra and a complete set of mutually commuting quantum invariants allowing the derivation of several families of exactly solvable Hamiltonians. Different Hamiltonians correspond to different representations of the generators of the algebra. The derived exactly-solvable generalized Gaudin models include the Hamiltonians of Bardeen–Cooper–Schrieffer, Suhl–Matthias–Walker, Lipkin–Meshkov–Glick, the generalized Dicke and atom–molecule, the nuclear interacting boson model, a new exactly-solvable Kondo-like impurity model, and many more that have not been exploited in the physics literature yet.

Exactly Solvable Models for Atom-Molecule Hamiltonians

This work was supported by the DGES Spanish Grants No. PB95-01123 (J. D.) and No. PB97-1190 (G. S.).

Generalized Brückner-Hartree-Fock theory and self-consistent RPA

Abstract We have studied the numerical solutions of Richardson equations of the BCS model in the limit of large number of energy levels at half-filling, and compare them with the analytic results derived by Gaudin and Richardson, which in turn leads to the standard BCS solution. We focus on the location and density of the roots, the eigenvalues of the conserved quantities, and the scaling properties of the total energy for the equally spaced and the two-level models.

A self-consistent description of systems with many interacting bosons

We present a family of exactly solvable generalizations of the Jaynes-Cummings model involving the interaction of an ensemble of SU(2) or SU(1,1) quasispins with a single boson field. They are obtained from the trigonometric Richardson-Gaudin models by replacing one of the SU(2) or SU(1,1) degrees of freedom by an ideal boson. The application to a system of bosonic atoms and molecules is reported.