Abstract
Using the exact Bethe Ansatz solution, we investigate methods for calculating the ground-state energy for the p+ip-pairing Hamiltonian. We first consider the Hamiltonian isolated from its environment (closed model) through two forms of Bethe Ansatz solutions, which generally have complex-valued Bethe roots. A continuum limit approximation, leading to an integral equation, is applied to compute the ground-state energy. We discuss the evolution of the root distribution curve with respect to a range of parameters, and the limitations of this method. We then consider an alternative approach that transforms the Bethe Ansatz equations to an equivalent form, but in terms of the real-valued conserved operator eigenvalues. An integral equation is established for the transformed solution. This equation is shown to admit an exact solution associated with the ground state. Next we discuss results for a recently derived Bethe Ansatz solution of the open model. With the aforementioned alternative approach based on real-valued roots, combined with mean-field analysis, we are able to establish an integral equation with an exact solution that corresponds to the ground-state for this case.
Highlights
The p + ip-pairing Hamiltonian is an example of a Bardeen-CooperSchrieffer (BCS) model which admits an exact Bethe Ansatz solution
For the open model there is no conservation of total particle number, due to interaction terms which accommodate particle exchange with the system’s environment
Starting with the closed model, we revisited the formulations of [3, 4, 13] which undertake calculations by assuming a form of density function for the Bethe root distribution
Summary
The p + ip-pairing Hamiltonian is an example of a Bardeen-CooperSchrieffer (BCS) model which admits an exact Bethe Ansatz solution. It is worthy of mention that an extended discussion and application of the electrostatic analogy for more general Richardson-Gaudin systems can be found in [13] Following this approach, the continuum limit approximation has been adopted in [3, 4] for the closed p + ip-pairing model. In the continuum limit the ground-state energy is the same as that predicted by mean-field theory, across all values of the model parameters This will be proved by exploiting a completely different approach which does not use the Bethe root distribution at all. The origins of the new approach that will be followed trace back to the work of Babelon and Talalaev [14] who showed that, through a change of variables, the Bethe Ansatz equations for Richardson-Gaudin type systems could be recast into a set of coupled polynomial equations The roots of these equations are related to the eigenvalues of the self-adjoint conserved operators, and as such that are necessarily real-valued.
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