Class of Exactly Solvable Pairing Models

Abstract

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 fermion interacting with repulsive pairing forces in a two dimensional square lattice. Inspite of the repulsive pairing force the exact results show attractive pair correlations. PACS number: 71.10. Li, 74.20.Fg Exactly solvable models have played an important role in understanding the physics of the quantum many body problem, especially in cases where the system is strongly correlated. Such situations arises e.g. in one dimensional (1D) systems of interest for condensed matter physics and also in strongly correlated finite fermion systems as atomic nuclei. In both branches of physics the study of exactly solvable models has been pursued since long with enormous success.In 1D quantum physics, the exactly solvable models can be classified into three families. The first family begun with Bethe's exact solution of the Heisenberg model. Since then a wide variety 1D models has been solved us- Several exactly solvable models have been developed in the field of nuclear physics from a different perspective [2]. In these models the hamiltonian is written as a linear combination of the Casimir operators of a group decomposition chain ideally representing the properties of a particular nuclear phase. Typical examples are the Elliot SU(3) model describing nuclear deformations and rotations and the U(6) Interacting Boson Model [3] with its three dynamical symmetry limits describing rotational nuclei (SU(3)), vibrational nuclei (U(5)) and gamma unstable nuclei (O(6)). These models were extremely useful in providing a simple understanding of some prototypical nuclei.The impact of the exactly solvable models in condensed matter physics and in nuclear physics is so enormous that one hardly can believe that the exact solution of the Pairing Model (PM), of great interest for both fields, passed almost unnoticed till very recently [4]. The PM, was solved exactly by Richardson in a series of papers in the sixties [5][6][7].Independently of Richardson's exact solution, it was recently demonstrated [8] that the PM is an integrable model. The pairing model may turn out particularly interesting, since recent work [9] has shown that the pure repulsive pairing Hamiltonian in a 2D lattice can be solved exactly in the thermodynamic limit revealing strong superconducting fluctuations. The importance of this finding stems, of course, from the fact that high T c superconductors apparently acquire their superconducting properties through the repulsive Coulomb interaction.We will derive in this letter three families of exactly solvable models based on the pairing interaction for fermion systems as well as for boson systems. The most important feature of the new set models we will present is that they are exactly solvable in any dimension. In [10] we have advanced a numerical solution for a three dimensional confined boson systems, here we wi...

Eigenvalue-based method and form-factor determinant representations for integrable XXZ Richardson-Gaudin models

“…Following Gaudin [6] and Dukelsky et al [12], it is possible to obtain a set of conditions for which the set of operators R i commute mutually. These are given by…”

“…For most of these systems, computationally inexpensive expressions are also known for several form factors and overlaps, which can be used to investigate observables in systems where the Hilbert space is too large for traditional exact methods. Unfortunately, the Bethe ansatz or Richardson-Gaudin equations [6,11,12] that need to be solved are highly nonlinear and give rise to singularities, making a straightforward numerical solution challenging [13,14]. Several methods have been introduced as a way to resolve this difficulty, such as a change in variables [14,15], a (pseudo)deformation of the algebra [16,17], or a Heine-Stieltjes connection, reducing the problem to a differential equation [18].…”

Superconductivity in Ultrasmall Grains: Introduction to Richardson’s Exact Solution

“…To access larger systems and fully recover the bulk limit, fixed-n projected variational BCS wavefunctions (PBCS) were used in [9] (for n ≤ 600); significant improvements over the latter results, in particular in the crossover regime, were subsequently achieved in [10] using the density matrix renormalization group (DMRG) (with n ≤ 400). Finally, Dukelsky and Schuck [11] showed that a self-consistent RPA approach, that in principle can be extended to finite temperatures, describes the f.d. regime rather well (though not as well as the DMRG).…”

“…BCS approach [2,3,4] was applied to a reduced BCS Hamiltonian for uniformly spaced, spin-degenerate levels; it suggested that pairing correlations, as measured by the condensation energy E C , vanish abruptly once d exceeds a critical level spacing d c that depends on the parity (0 or 1) of the number of electrons on the grain, being smaller for odd grains (d c 1 ≃ 0.89∆) than even grains (d c 0 ≃ 3.6∆). A series of more sophisticated canonical approaches (summarized in Section 3 below) confirmed the parity dependence of pairing correlations, but established [6]- [11] that the abrupt vanishing of pairing correlations at d c is an artifact of g.c.…”

“…Recent experiments by Ralph, Black and Tinkham, involving the observation of a spectroscopic gap indicative of pairing correlations in ultrasmall Al grains [1], have inspired a number of theoretical [2]- [11] studies of how superconducting pairing correlations in such grains are affected by reducing the grains' size, or equivalently by increasing its mean level spacing d ∝ Vol −1 until it exceeds the bulk gap ∆. In the earliest of these, a grand-canonical (g.c.)…”