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arXiv:1507.04068 · Journal of Physics A Mathematical and Theoretical

An integrable case of the p + i p pairing Hamiltonian interacting with its environment

Inna Lukyanenko, Phillip S. Isaac, Jon Links

Published Jan 20, 2016• 36 citations
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Abstract

Abstract We consider a generalization of the p + i p pairing Hamiltonian, with external interaction terms of a particular form. These terms allow for the exchange of particles between the system and its environment. As a result the symmetry associated with conservation of particle number, present in the p + i p Hamiltonian, is broken. Nonetheless the generalized model is integrable. We establish integrability using the boundary quantum inverse scattering method, with one of the reflection matrices chosen to be non-diagonal. We also derive the corresponding Bethe ansatz equations, the roots of which parametrize the exact solution for the energy spectrum.

DOI
10.1088/1751-8113/49/8/084001

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Abstract We consider a generalization of the p + i p pairing Hamiltonian, with external interaction terms of a particular form. These terms allow for the exchange of particles between the system and its environment. As a result the symmetry associated with conservation of particle number, present in the p + i p Hamiltonian, is broken. Nonetheless the generalized model is integrable. We establish integrability using the boundary quantum inverse scattering method, with one of the reflection matrices chosen to be non-diagonal. We also derive the corresponding Bethe ansatz equations, the roots of which parametrize the exact solution for the energy spectrum.
Using the exact Bethe Ansatz solution, we investigate methods for calculating the ground-state energy for the p + i p -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.
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.
Abstract The Bose–Hubbard model is a paradigm for the study of strongly correlated bosonic systems. We study the two-site Bose–Hubbard model with generic integrable open boundaries specified by the most general non-diagonal reflecting matrices. Besides the inhomogeneous parameters, the model itself has three free boundary parameters, which break the U(1)-symmetry, in other words, break the particle number conservation. The Hamiltonian H under these circumstances is constructed. With the help of the off-diagonal Bethe Ansatz method, we successfully obtain the corresponding Bethe Ansatz equations as well as the eigenvalues.
A pairing model for nucleons, introduced by Richardson in 1966, which describes proton–neutron pairing as well as proton–proton and neutron–neutron pairing, is re-examined in the context of the quantum inverse scattering method. Specifically, this shows that the model is integrable by enabling the explicit construction of the conserved operators. We determine the eigenvalues of these operators in terms of the Bethe ansatz, which in turn leads to an expression for the energy eigenvalues of the Hamiltonian.