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...

What distinguishes trivial superfluids from topological superfluids in interacting many-body systems where the number of particles is conserved? Building on a class of integrable pairing Hamiltonians, we present a number-conserving, interacting variation of the Kitaev model, the Richardson-Gaudin-Kitaev chain, that remains exactly solvable for periodic and antiperiodic boundary conditions. Our model allows us to identify fermion parity switches that distinctively characterize topological superconductivity (fermion superfluidity) in generic interacting many-body systems. Although the Majorana zero modes in this model have only a power-law confinement, we may still define many-body Majorana operators by tuning the flux to a fermion parity switch. We derive a closed-form expression for an interacting topological invariant and show that the transition away from the topological phase is of third order. DOI: 10.1103/PhysRevLett.113.267002 PACS numbers: 74.20.-z, 03.65.Vf, 74.45.+c, 74.90.+n In recent years, the physics of Majorana zero-energy modes has become a key subfield of condensed matter physics [1][2][3][4]. On the theory side, a central result is the bulkboundary correspondence [5] that associates Majorana zero modes to the boundary of (or defects in) a topologically nontrivial superconductor, with the Kitaev chain as a prototypical example [6]. The mathematical formalism underlying this correspondence relies on the symmetries and topological invariants of the Bogoliubov-de Gennes equation [7], a mean-field description of the superconducting state in which the conservation of the number of fermions (a continuous symmetry) is broken down to a discrete symmetry, the conservation of fermion-number parity. Majorana zero modes are directly linked to the spontaneous breaking of this residual discrete symmetry [8].As the experimental side of Majorana physics continues to develop [9][10][11][12][13][14], it becomes crucial to unveil how much of the mean-field picture survives beyond its natural limits. This has motivated recent studies [15][16][17][18][19][20][21], with the focus on the anomalous 2Φ 0 ¼ h=e flux periodicity of the Josephson effect-the hallmark of a topological superconductor [6].A main thrust of this Letter is the characterization of interacting many-body, number-conserving, topological superconductors, or superfluids, leading to a subsequent analysis on the meaning of Majorana zero modes beyond mean field. The theoretical study of any interacting quantum system is hampered by the exponential growth of the Hilbert space with the number of particles. An additional complication of superconducting systems is the lack of simple principles to guide the design of particlenumber conserving models, in which the phase of the order parameter is not a good quantum number. To overcome both obstacles, we have constructed an exactly solvable, number-conserving variation of the Kitaev chain. Because our model belongs to a class of integrable pairing models [22-24] based on the s-wave reduced BCS Hamiltonian firs...

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.

The traditional nuclear pairing problem is shown to be in one-to-one correspondence with a classical electrostatic problem in two dimensions. We make use of this analogy in a series of calculations in the tin region, showing that the extremely rich phenomenology that appears in this classical problem can provide interesting new insights into nuclear superconductivity.

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We study the Richardson equations close to the critical values of the paring strength gc where the occurrence of divergencies preclude numerical solutions. We derive a set of equations for determining the critical g values and the non-collapsing pair energies. Studying the behavior of the solutions close to the critical points, we develop a procedure to solve numerically the Richardson equations for arbitrary coupling strength.

We study the structure of the number projected BCS (PBCS) wave function in the particle-hole basis, displaying its similarities with coupled clusters theory (CCT). The analysis of PBCS together with several modifications suggested by the CCT wave function is carried out for the exactly solvable Richardson model involving a pure pairing hamiltonian acting in a space of equally-spaced doublydegenerate levels. We point out the limitations of PBCS to describe the non-superconducting regime and suggest possible avenues for improvement.

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