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Journal of Physics Conference Series

Richardson-Gaudin models: the hyperbolic family

J. Dukelsky, Stefan Rombouts, Gerardo Ortíz

Published Sep 16, 2011• 0 citations
Publisher

Abstract

We show that the hyperbolic family of Richardson-Gaudin models could describe p-wave superfluids in two dimensional lattices. The analysis of the exact solution reveals an exotic phase diagram with a third order quantum phase transition. Moreover, we propose a separable integrable pairing Hamiltonian within the same family that could be useful to describe exactly superconductivity in heavy nuclei.

DOI
10.1088/1742-6596/321/1/012021

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We show that the hyperbolic family of Richardson-Gaudin models could describe p-wave superfluids in two dimensional lattices. The analysis of the exact solution reveals an exotic phase diagram with a third order quantum phase transition. Moreover, we propose a separable integrable pairing Hamiltonian within the same family that could be useful to describe exactly superconductivity in heavy nuclei.
The exact solution of the SU(2) pairing Hamiltonian with non-degenerate single particle orbits was introduced by Richardson in the early sixties. The exact solution passed almost unnoticed till was recovered in the last decade in an effort to describe the disappearance of superconductivity in ultrasmall superconducting grains. Since then it has been extended to several families 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 will first introduce these families of integrable models and then we will describe the first applications of the hyperbolic family to spinless cold fermionic atonies in two dimensional lattices and to heavy nuclei.
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 determine the zero-temperature quantum phase diagram of a px+ipy{p}_{x}+i{p}_{y}px​+ipy​ pairing model based on the exactly solvable hyperbolic Richardson-Gaudin model. We present analytical and large-scale numerical results for this model. In the continuum limit, the exact solution exhibits a third-order quantum phase transition, separating a strong-pairing from a weak-pairing phase. The mean-field solution allows to connect these results to other models with px+ipy{p}_{x}+i{p}_{y}px​+ipy​ pairing order. We define an experimentally accessible characteristic length scale, associated with the size of the Cooper pairs, that diverges at the transition point, indicating that the phase transition is of a confinement-deconfinement type without local order parameter. We propose an experimental measurement to detect the transition. We show that this phase transition is not limited to the px+ipy{p}_{x}+i{p}_{y}px​+ipy​ pairing model but can be found in any representation of the hyperbolic Richardson-Gaudin model and is related to a symmetry that is absent in the rational Richardson-Gaudin model.
Recently, interest has increased in the hyperbolic family of integrable Richardson-Gaudin (RG) models. It was pointed out that a particular linear combination of the integrals of motion of the hyperbolic RG model leads to a Hamiltonian that describes ppp-wave pairing in a two-dimensional system. Such an interaction is found to be present in fermionic superfluids (3He{}^{3}\text{He}3He), ultracold atomic gases, and ppp-wave superconductivity. Furthermore the phase diagram is intriguing, with the presence of the Moore-Read and Read-Green lines. At the Read-Green line a rare third-order quantum phase transition occurs. The present paper makes a connection between collective bosonic states and the exact solutions of the px+ipy{p}_{x}+i{p}_{y}px​+ipy​ pairing Hamiltonian. This makes it possible to investigate the effects of the Pauli principle on the energy spectrum, by gradually reintroducing the Pauli principle. It also introduces an efficient and stable numerical method to probe all the eigenstates of this class of Hamiltonians. We extend the phase diagram to repulsive interactions, an area that was not previously explored due to the lack of a proper mean-field solution in this region. We found a connection between the point in the phase diagram where the ground state connects to the bosonic state with the highest collectivity, and the Moore-Read line where all the Richardson-Gaudin (RG) variables collapse to zero. In contrast with the reduced BCS case, the overlap between the ground state and the highest collective state at the Moore-Read line is not the largest. In fact it shows a minimum when most other bosonic states show a maximum of the overlap. We found remnants of the Read-Green line for finite systems, by investigating the total spectrum. A symmetry was found between the Hamiltonian with and without single-particle part. When the interaction was repulsive we found four different classes of trajectories of the RG variables.