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Using the exact Bethe Ansatz solution, we investigate methods for calculatingthe ground-state energy for the $p + ip$-pairing Hamiltonian. We first considerthe Hamiltonian isolated from its environment (closed model) through two formsof Bethe Ansatz solutions, which generally have complex-valued Bethe roots. Acontinuum limit approximation, leading to an integral equation, is applied tocompute the ground-state energy. We discuss the evolution of the rootdistribution curve with respect to a range of parameters, and the limitationsof this method. We then consider an alternative approach that transforms theBethe Ansatz equations to an equivalent form, but in terms of the real-valuedconserved operator eigenvalues. An integral equation is established for thetransformed solution. This equation is shown to admit an exact solutionassociated with the ground state. Next we discuss results for a recentlyderived Bethe Ansatz solution of the open model. With the aforementionedalternative approach based on real-valued roots, combined with mean-fieldanalysis, we are able to establish an integral equation with an exact solutionthat corresponds to the ground-state for this case.

Ground-state energies of the open and closed $p+ip$-pairing models from the Bethe Ansatz