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arXiv:1001.1591 · Journal of Statistical Mechanics Theory and Experiment

Exact solution of the p + ip pairing Hamiltonian and a hierarchy of integrable models

Clare Dunning, Miguel Ibáñez-Berganza, Jon Links, Germán Sierra, Shao-you Zhao A

Published Aug 26, 2010• 57 citations
arXivHTMLPDF

Abstract

Using the well-known trigonometric six-vertex solution of the Yang-Baxter equation we derive an integrable pairing Hamiltonian with anyonic degrees of freedom. The exact algebraic Bethe ansatz solution is obtained using standard techniques. From this model we obtain several limiting models, including the pairing Hamiltonian with p+ip-wave symmetry. An in-depth study of the p+ip model is then undertaken, including a mean-field analysis, analytic and numerical solution of the Bethe ansatz equations, and an investigation of the topological properties of the ground-state wavefunction. Our main result is that the ground-state phase diagram of the p+ip model consists of three phases. There is the known boundary line with gapless excitations that occurs for vanishing chemical potential, separating the topologically trivial strong pairing phase and the topologically non-trivial weak pairing phase. We argue that a second boundary line exists separating the weak pairing phase from a topologically trivial weak coupling BCS phase, which includes the Fermi sea in the limit of zero coupling. The ground state on this second boundary line is the Moore-Read state.

DOI
10.1088/1742-5468/2010/08/p08025

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Using the well-known trigonometric six-vertex solution of the Yang-Baxter equation we derive an integrable pairing Hamiltonian with anyonic degrees of freedom. The exact algebraic Bethe ansatz solution is obtained using standard techniques. From this model we obtain several limiting models, including the pairing Hamiltonian with p+ip-wave symmetry. An in-depth study of the p+ip model is then undertaken, including a mean-field analysis, analytic and numerical solution of the Bethe ansatz equations, and an investigation of the topological properties of the ground-state wavefunction. Our main result is that the ground-state phase diagram of the p+ip model consists of three phases. There is the known boundary line with gapless excitations that occurs for vanishing chemical potential, separating the topologically trivial strong pairing phase and the topologically non-trivial weak pairing phase. We argue that a second boundary line exists separating the weak pairing phase from a topologically trivial weak coupling BCS phase, which includes the Fermi sea in the limit of zero coupling. The ground state on this second boundary line is the Moore-Read state.
We present the exact Bethe ansatz solution for the two-dimensional BCS pairing Hamiltonian with p_x + i p_y symmetry. Using both mean-field theory and the exact solution we obtain the ground-state phase diagram parameterized by the filling fraction and the coupling constant. It consists of three phases denoted weak coupling BCS, weak pairing, and strong pairing. The first two phases are separated by a topologically protected line where the exact ground state is given by the Moore-Read pfaffian state. In the thermodynamic limit the ground-state energy is discontinuous on this line. The other two phases are separated by the critical line, also topologically protected, previously found by Read and Green. We establish a duality relation between the weak and strong pairing phases, whereby ground states of the weak phase are dressed versions of the ground states of the strong phase by zero energy (Moore-Read) pairs and characterized by a topological order parameter.
We present the exact Bethe ansatz solution for the two-dimensional BCS pairing Hamiltonian with px+ipy{p}_{x}+i{p}_{y}px​+ipy​ symmetry. Using both mean-field theory and the exact solution we obtain the ground-state phase diagram parametrized by the filling fraction and the coupling constant. It consists of three phases that are denoted weak-coupling BCS, weak pairing, and strong pairing. The first two phases are separated by a topologically protected line where the exact ground state is given by the Moore-Read pfaffian state. In the thermodynamic limit the ground-state energy is discontinuous on this line. The other two phases are separated by the critical line, also topologically protected, previously found by Read and Green. We establish a duality relation between the weak and strong pairing phases, whereby ground states of the weak phase are ``dressed'' versions of the ground states of the strong phase by zero energy (Moore-Read) pairs and characterized by a topological order parameter.
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.
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.
Motivated by the work of Kitaev, we construct an exactly soluble spin-½ model on a honeycomb lattice whose ground states are identical to Δ 1 x p x +Δ 1 y p y + i (Δ 2 x p x +Δ 2 y p y )-wave paired fermions on a square lattice, with tunable paring order parameters. We derive a universal phase diagram for this general p -wave theory which contains a gapped A-phase and a topologically non-trivial B-phase. We show that the gapless condition in the B-phase is governed by a generalized inversion (G-inversion) symmetry under p x ↔ (Δ 1 y / Δ 1 x ) p y . The G-inversion symmetric gapless B-phase near the phase boundaries is described by (1+1)-dimensional gapless Majorana fermions in the asymptotic long-wavelength limit, i.e. the c =1/2 conformal field theory. The gapped B-phase has G-inversion symmetry breaking and is the weak pairing phase described by the Moore-Read Pfaffian. We show that in the gapped B-phase, vortex pair excitations are separated from the ground state by a finite energy gap.
We study the Bethe ansatz/ordinary differential equation (BA/ODE) correspondence for Bethe ansatz equations that belong to a certain class of coupled, nonlinear, algebraic equations. Through this approach we numerically obtain the generalized Heine–Stieltjes and Van Vleck polynomials in the degenerate, two-level limit for four cases of integrable Bardeen–Cooper–Schrieffer (BCS) pairing models. These are the s-wave pairing model, the p + ip-wave pairing model, the p + ip pairing model coupled to a bosonic molecular pair degree of freedom, and a newly introduced extended d + id-wave pairing model with additional interactions. The zeros of the generalized Heine–Stieltjes polynomials provide solutions of the corresponding Bethe ansatz equations. We compare the roots of the ground states with curves obtained from the solution of a singular integral equation approximation, which allows for a characterization of ground-state phases in these systems. Our techniques also permit the computation of the roots of the excited states. These results illustrate how the BA/ODE correspondence can be used to provide new numerical methods to study a variety of integrable systems.