The complete exact solution of the T 1 neutron-proton pairing Hamiltonian is presented in the context of the SO(5) Richardson-Gaudin model with nondegenerate single-particle levels and including isospin symmetry-breaking terms. The power of the method is illustrated with a numerical calculation for 64 Ge for a pf g 9=2 model space which is out of reach of modern shell-model codes. DOI: 10.1103/PhysRevLett.96.072503 PACS numbers: 21.60.Fw, 03.65.ÿw, 27.50.+e, 74.20.Rp Exactly solvable models (ESM) provide important insights into the structure of many-body quantum systems. The two main advantages of ESMs are: (1) they can describe in an analytical or exact numerical way a wide variety of elementary phenomena. (2) They can be and have been used as a testing ground for various many-body approaches.A particular class of ESMs, extensively used in nuclear physics, are the dynamical-symmetry models. In this case the Hamiltonian can be expressed in terms of Casimir operators of a chain of nested algebras. An example often used to introduce nuclear superconductivity [see, e.g., Ref.[1] ] is the rank-1 (Lie) algebra SU(2). Examples of dynamical-symmetry models associated with a rank-2 algebra are Elliott's SU(3) model of nuclear deformation [2] and the SO(5) model of T 1 isovector pairing between neutrons and protons [3] which has found many applications in nuclei [see, e.g., Ref. [4] ].The concept of quantum integrability, closely linked with exact solvability, goes beyond the limits of the dynamical-symmetry approach. A quantum system is integrable if there exist as many commuting Hermitian operators (integrals of motion) as quantum degrees of freedom [5]. The set of Casimir operators of a chain of nested algebras satisfies this condition.Dynamical-symmetry models are usually defined for degenerate single-particle levels. Lifting this degeneracy breaks the dynamical symmetry but may still preserve integrability. The pairing model with nondegenerate single-particle levels, of which an exact solution was found by Richardson in the 1960s [6], represents an example of an ESM with such characteristics. Recently, more general exactly solvable pairing models, both for fermions and for bosons, called Richardson-Gaudin (RG) models, have been proposed [7,8].The RG pairing models are based on rank-1 algebras: SU(2) for fermions and SU(1,1) for bosons. In this Letter we carry out the first step in extending the RG models to higher-rank algebras by considering a RG model based on the rank-2 algebra SO(5). The model Hamiltonian describes a two-component system consisting of neutrons and protons interacting through an isovector (T 1) pairing force and distributed over nondegenerate orbits. This neutron-proton (np) pairing Hamiltonian with nondegenerate orbits has been studied by Richardson [9] who proposed an exact solution. However, it was shown subsequently that Richardson's solution is incorrect for more than two nucleon pairs [10] by explicitly solving the case of three-nucleon pairs. Independently, Links et al. derived an ex...
The particle number projected BCS (PBCS) approximation is tested against the exact solution of the SO(5) Richardson-Gaudin model for isovector pairing in a system of nondegenerate single-particle orbits. Two isovector PBCS wave functions are considered. One is constructed as a single proton-neutron pair condensate; the other corresponds to a product of a neutron pair condensate and a proton pair condensate. The PBCS equations are solved using a recurrence method and the analysis is performed for systems with an equal number of neutrons and protons distributed in a sequence of equally spaced fourfold (spin-isospin) degenerate levels. The results show that although PBCS offers significant improvement over BCS, the agreement of PBCS with the exact solution is less satisfactory than in the case of the SU(2) Richardson model for pairing between like particles.
The exact solution of proton-neutron isoscalar-isovector (T =0,1) pairing Hamiltonian with nondegenerate single-particle orbits and equal pairing strengths (gT =1=gT =0) is presented for the first time. The Hamiltonian is a particular case of a family of integrable SO(8) Richardson-Gaudin (RG) models. The exact solution of the T =0,1 pairing Hamiltonian is reduced to a problem of 4 sets of coupled non linear equations that determine the spectral parameters of the complete set of eigenstates. The microscopic structure of individual eigenstates is analyzed in terms of evolution of the spectral parameters in the complex plane for system of A=80 nucleons. The spectroscopic trends of the exact solutions are discussed in terms of generalized rotations in isospace.PACS numbers: 02.30. Ik, 21.60.Fw, 74.20.Rp The exactly solvable models introduced by Richardson [1] and by Gaudin [2] belong nowadays to classic theoretical tools in mesoscopic physics. Indeed, these models based on the rank 1 SU(2) algebra for fermions or the SU(1,1) algebra for bosons were applied to a large variety of quantum many-body systems including the atomic nucleus, superconducting grains, cold atomic gases, etc., see review article In this letter we will derive for the first time the exact solution for the rank 4 SO(8) RG integrable model with non-degenerate single-particle (sp) spectrum and arbitrary degeneracies. As a particular realization of the rank 4 SO(8) RG model we will consider the nuclear isoscalarisovector (T =0,1) pairing Hamiltonian introduced for a single degenerate shell in Ref. [7] and further developed in [8,9]. We will solve the model for a realistic case of A=80 nucleons moving in fourfold degenerated equidistant sp spectrum working in T =0,1 pair representation of the SO(8) algebra. It should be mentioned that other representations like the Ginnocchio model [10] can lead to interesting exactly solvable models in nuclear structure as well as to models of spin 3/2 cold atoms [11].The study of proton-neutron (p-n) pairing has gained a renewed interest due to the new generation of radioactive-beam facilities that will open the access to proton-rich nuclei close to the N =Z line. In spite of vigorous activity in this field, see [12] and refs. therein, the fundamental questions concerning the basic building blocks and experimental fingerprints of the p-n pairing are still a matter of debate. So are the theoretical problems concerning generalization of well established nuclear pairing models to include p-n pairing, proper treatment of isospin degree of freedom or α-like clustering. All these problems set clear motivation for realistic exact-model studies of the p-n pairing undertaken in this work.Let us begin our derivation by introducing the 28 generators of the SO(8) algebra [7]: three (T =1,S=0) and three (T =0, S=1) pair creators, together with their respective annihilation operators:† , where the triads in the couplings represent, respectively, angular momentum, isospin and spin. The fermionic operators a † limτ σ create a pa...
We introduce an exactly solvable model for trapped three-color atom gases. Application to a system of cigar-shaped trapped cold fermions reveals a complex structure of breached pairing phases. We find two competing superfluid phases at weak and intermediate couplings, each with two-color pair condensates that can be distinguished with density profile measurements. DOI: 10.1103/PhysRevA.79.051603 PACS number͑s͒: 03.75.Ss, 03.75.Mn, 05.30.Fk, 02.30.Ik Color superconductivity is predicted to occur in quark matter at sufficiently high density and low temperatures ͓1͔. Quarks, having three different colors ͓red ͑R͒, green ͑G͒, and blue ͑B͔͒ and a strong attractive interaction, allow for more diverse pairing patterns compared to the SU͑2͒ Cooper pairing in classical metallic superconductors. Such diversity, likewise, makes it hard to establish the particular pairing symmetry favored by nature. With the advent of ultracold trapped Fermi gases a window of opportunities has opened to address some of these fundamental questions, at least, in a qualitative fashion ͑see, for example, ͓2͔͒. One can certainly manipulate different atomic species and hyperfine states to effectively generate multicolor Fermi gases with attractive interactions.The goal of this Rapid Communication is to investigate the superfluid behavior of an imbalanced three-color Fermi gas by means of an exactly solvable pairing model. We will present an exactly solvable color-pairing Hamiltonian derived from the SO͑6͒ quadratic invariants of the generalized Richardson-Gaudin ͑RG͒ models ͓3,4͔. Previous studies using standard mean-field ͓5͔, density-matrix renormalization-group ͓6͔, or Bethe ansatz ͓7͔ technique concentrated on the competition between a trionic or baryonic phase and a color superfluid phase. A main result of our work is the competition between breached-pair ͑BP͒ and unbreached-pair ͑UP͒ superfluid phases in a polarized multicolor Fermi gas. As in the two-color case, where density profiles have been recently investigated experimentally ͓8͔ and theoretically ͓9͔, an analog of the BP or Sarma phase ͓10,11͔ appears. We find a complex structure of breached pairing as well as the coexistence of two pair condensates. While the possibility of coexistence of several superfluid phases has been suggested in ͓12͔ using a localdensity approximation ͑LDA͒ theory, we predict the existence of two distinct color fermionic condensates and propose ways to detect them. Within our model this is a genuine effect, although care must be exercised when contrasted to experiments since interactions not included in our model could make this phase unstable against the formation of a fraction of bound trions in the strong-coupling limit. However, population imbalance as well as the experimental realization of a stable three-color atomic gas with different atomic masses and/or different Feshbach resonances ͓13,14͔ could stabilize it.Consider the SU͑3͒ color-symmetric Hamiltonianfor L levels i of energy i , whereare the pair creators, and g Ͼ 0 is the pairing strength. H...
We first review the development of the Richardson-Gaudin exactly-solvable pairing models and then discuss several new models based on rank-two algebras and their applications to problems in nuclear structure.
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