The use of exactly-solvable Richardson-Gaudin (R-G) models to describe the physics of systems with strong pair correlations is reviewed. We begin with a brief discussion of Richardson's early work, which demonstrated the exact solvability of the pure pairing model, and then show how that work has evolved recently into a much richer class of exactly-solvable models. We then show how the Richardson solution leads naturally to an exact analogy between such quantum models and classical electrostatic problems in two dimensions. This is then used to demonstrate formally how BCS theory emerges as the large-N limit of the pure pairing Hamiltonian and is followed by several applications to problems of relevance to condensed matter physics, nuclear physics and the physics of confined systems. Some of the interesting effects that are discussed in the context of these exactly-solvable models include: (1) the crossover from superconductivity to a fluctuation-dominated regime in small metallic grains, (2) the role of the nucleon Pauli principle in suppressing the effects of high spin bosons in interacting boson models of nuclei, and (3) 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.2

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...

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.

We determine the zero temperature quantum phase diagram of a 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 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 confinementdeconfinement type without local order parameter. We show that this phase transition is not limited to the px + ipy pairing model, but can be found in any representation of the hyperbolic RichardsonGaudin model and is related to a symmetry that is absent in the rational Richardson-Gaudin model.

Several variants of the recently proposed density matrix embedding theory (DMET) [G. Knizia and G. K-L. Chan, Phys. Rev. Lett. 109, 186404 (2012)] are formulated and tested. We show that spin symmetry breaking of the lattice mean-field allows precise control of the lattice and fragment filling while providing very good agreement between predicted properties and exact results. We present a rigorous proof that at convergence this method is guaranteed to preserve lattice and fragment filling. Differences arising from fitting the fragment one-particle density matrix alone versus fitting fragment plus bath are scrutinized. We argue that it is important to restrict the density matrix fitting to solely the fragment. Furthermore, in the proposed broken symmetry formalism, it is possible to substantially simplify the embedding procedure without sacrificing its accuracy by resorting to density instead of density matrix fitting. This simplified density embedding theory (DET) greatly improves the convergence properties of the algorithm.