Bethe Ansatz analysis of the open and closed p + ip-pairing BCS models

The p+ip-wave pairing BCS model has attracted much attention due to its potential applications inexperimental physics, and the developing theory of topological superconductors. In this thesis, weinvestigate the p + ip BCS model as an integrable model with exact Bethe Ansatz solutions. Theground-state energies of this model will be studied with various methods.  First a brief introduction on quantum integrability is given, followed by an introduction to theQuantum Inverse Scattering Method, algebraic Bethe Ansatz and BCS theory of superconductivity. Wethen introduce a two-dimensional BCS pairing Hamiltonian with px +ipy symmetry. In the frameworkof the Quantum Inverse Scattering Method, the p+ip Hamiltonian can be constructed from a specificsolution of the non-standard classical Yang-Baxter equation. This solution produces a family ofconserved operators satisfying quadratic relations. Remarks on topological superconductivity for thep+ip-wave pairing model are provided.  We move on to investigate methods for calculating the ground-state energy for the p+ip Hamil-tonian. We first consider the Hamiltonian isolated from its environment (closed model) through twoforms of Bethe Ansatz solution, which generally have complex-valued Bethe roots. A continuum limitapproximation, leading to an integral equation, is applied to compute the ground-state energy. Theevolution of the root distribution curve with respect to a range of parameters is presented, followed bya discussion of the limitations of this method. Despite the limitations, the ground-state energy andthe ‘gap’ equation derived from the integral approximation are consistent with mean-field results fromprevious study.  We then consider the Hamiltonian interacting with its environment (open model) and begin witha mean-field analysis. We discuss the relation between the Bethe roots and the conserved operatoreigenvalues. The Bethe Ansatz solution of the open model is studied numerically, revealing complicatedbehaviour of the Bethe roots without a clear indication for an integral approximation.  Next we consider an alternative approach based on the quadratic relations among the real-valuedconserved operator eigenvalues. An integral equation is first established for the closed model. Thisequation is shown to admit an exact solution associated with the ground state.  Finally we extend the aforementioned alternative approach to the open model. Combined withresults from mean-field analysis, we are able to establish an integral equation with an exact solutionthat corresponds to the ground state for the open case.