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Read-Green resonances in a $p_x+ip_y$ superfluid interacting with a bath

Pieter W. Claeys, Stijn De Baerdemacker, Dimitri Van Neck

Published Jan 15, 2016• 0 citations
OpenAlex

Abstract

We study the weak-pairing phase in a finite-size two-dimensional $p_x+ip_y$ superfluid interacting with an environment. This interaction breaks particle-number symmetry and can be modelled by means of an integrable Hamiltonian. We present the exact wave function and solve the resulting Bethe ansatz equations, from which it is shown how resonances are introduced at fixed interaction strengths. These cannot be observed within mean-field theory and are argued to be caused by zero-energy excitations at topologically protected Read-Green points.

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Read-Green resonances in a px+ipyp_x+ip_ypx​+ipy​ superfluid interacting with a bath
We study the weak-pairing phase in a finite-size two-dimensional px+ipyp_x+ip_ypx​+ipy​ superfluid interacting with an environment. This interaction breaks particle-number symmetry and can be modelled by means of an integrable Hamiltonian. We present the exact wave function and solve the resulting Bethe ansatz equations, from which it is shown how resonances are introduced at fixed interaction strengths. These cannot be observed within mean-field theory and are argued to be caused by zero-energy excitations at topologically protected Read-Green points.
We study a topological superconductor capable of exchanging particles with an environment. This additional interaction breaks particle-number symmetry and can be modeled by means of an integrable Hamiltonian, building on the class of Richardson-Gaudin pairing models. The isolated system supports zero-energy modes at a topological phase transition, which disappear when allowing for particle exchange with an environment. However, it is shown from the exact solution that these still play an important role in system-environment particle exchanges, which can be observed through resonances in low-energy and low-momentum level occupations. These fluctuations signal topologically protected Read-Green points and cannot be observed within traditional mean-field theory.
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.
We analyze a family of two-dimensional BCS Hamiltonians with general lll-wave pairing interactions, classifying the models in this family that are Bethe-ansatz solvable in the finite-size regime. We show that these solutions are characterized by nontrivial winding numbers, associated with topological phases, in some part of the corresponding phase diagrams. By means of a comparative study, we demonstrate benefits and limitations of the mean-field approximation, which is the standard approach in the limit of a large number of particles. The mean-field analysis also allows us to extend part of the results beyond integrability, clarifying the peculiarities associable with the integrability itself.
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.
In this work, we study the full set of eigenstates of a px+ipy{p}_{x}+i{p}_{y}px​+ipy​ topological superconductor coupled to a particle bath, which can be described in terms of an integrable Hamiltonian of the Richardson-Gaudin class. The results derived in this work also characterize the behavior of an anisotropic XXZ central spin model in an external magnetic field since both types of Hamiltonian are known to share the exact same conserved quantities, making them formally equivalent. We show how by ramping the coupling strength (or equivalently the magnetic field acting in the zzz-direction on the central spin), each individual eigenstate undergoes a sequence of gain/loss of excitations when crossing the specific values known as Read-Green points. These features are shown to be completely predictable for every one of the 2N{2}^{N}2N eigenstates, using only two integers easily obtainable from the zero-coupling configuration, which defines the eigenstate in question. These results provide a complete map of the particle-number sectors (superconductor) or magnetization sectors (central spin) involved in the large number of level crossings that occur in these systems at the Read-Green points. It further allows us to define quenching protocols that could create states with remarkably large excitation-number fluctuations.