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

TL;DR

This paper uses the Bethe Ansatz to analyze and compute the ground-state energies of both open and closed $p+ip$-pairing models, introducing new integral equations and methods for accurate energy calculations.

Contribution

It presents novel integral equation approaches for calculating ground-state energies of $p+ip$-pairing models using Bethe Ansatz, including for open and closed systems, with exact solutions and analysis of root distributions.

Findings

Derived integral equations for ground-state energies.

Established exact solutions for the transformed Bethe Ansatz equations.

Analyzed root distribution evolution and limitations of continuum approximations.

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.