We present a new two-channel integrable model describing a system of spinless fermions interacting through a p-wave Feshbach resonance. Unlike the BCS-BEC crossover of the s-wave case, the pwave model has a third order quantum phase transition. The critical point coincides with the deconfinement of a single molecule within a BEC of bound dipolar molecules. The exact manybody wavefunction provides a unique perspective of the quantum critical region suggesting that the size of the condensate wavefunction, that diverges logarithmically with the chemical potential, could be used as an experimental indicator of the phase transition. PACS numbers:In recent years p-wave paired superfluids have attracted a lot of attention, in part due to their exotic properties [1]. Of particular interest is the chiral twodimensional (2D) p x + ip y superfluid of spinless fermions, that supports a topological phase with zero energy Majorana modes [2]. The latter are theorized to serve as a basic element for a topological quantum computer [3]. [7] with the potential to manipulate the system from the weak (BCS) to the strong (BEC) pairing regime. However, these gases revealed to be unstable due to atom-molecule relaxation processes in which the molecule decays to a deep bound state while the atom escapes with excess energy [8]. Other atomic and molecular gases are now in consideration, as well as different mechanisms to suppress relaxation.From a theoretical standpoint, despite great efforts to describe these systems, a complete understanding of the BCS-BEC transition and the corresponding phase diagram is still missing. Recently, by means of an exactly solvable p x + ip y pairing model [9,10] it was shown that the quantum phase transition (QPT) taking place from weak pairing to strong pairing can be understood as the deconfinement of bound Cooper pairs [11]. In this Letter we introduce an exactly solvable two-channel p x + ip y pairing model with a Feshbach resonance. Our model is exactly solvable in arbitrary dimensions, although we will concentrate on its 2D realization. We propose a way to experimentally detect the so-called topological QPT [12] from weak to strong pairing, by measuring a densitydensity correlation function. This together with the analysis of the size of a Cooper pair in terms of the exact solution allows the characterization of the transition as one of a confinement-deconfinement type without Landau order parameter. Moreover, the transition is shown to be third order in the Ehrenfest classification. Another way to theoretically detect that QPT is by analyzing the behavior of the quantum fidelity z of the ground state wavefunction. Interestingly, the second order derivative of ln |z| displays a logarithmic singularity at the transition point confirming its third order character.Consider the 2D two-channel p x + ip y -wave model† is a bosonic creation operator, andOur next goal is to show that this model is a particular realization of a family of exactly-solvable atom-molecule Hamiltonians of physical relevance in th...

The Lipkin-Meshkov-Glick (LMG) model has a Schwinger boson realization in terms of a two-level boson pairing Hamiltonian. Through this realization, it has been shown that the LMG model is a particular case of the SU(1, 1) Richardson-Gaudin (RG) integrable models. We exploit the exact solvability of the model to study the behavior of the spectral parameters (pairons) that completely determine the wave function in the different phases, and across the phase transitions. Based on the relation between the Richardson equations and the Lamé ordinary differential equation we develop a method to obtain numerically the pairons. The dynamics of pairons in the ground and excited states provide new insights into the first, second and third order phase transitions, as well as into the crossings taking place in the LMG spectrum.

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