Edge states and tunneling of non-Abelian quasiparticles in the nu=5/2 quantum Hall state and p+ip superconductors

TL;DR

The paper develops a rigorous framework for describing tunneling of non-Abelian quasiparticles at edges of topological states, focusing on the p+ip superconductor and the Moore–Read Pfaffian state. It shows that edge quasiparticle correlators live in a multi-dimensional conformal-block space, resolved by specifying fusion channels via the Moore–Seiberg braiding rules, and constructs a bosonized representation of the tunneling operators. This bosonization maps the problem onto Kondo-type Hamiltonians, revealing a deep connection between edge tunneling and impurity physics, including entropy loss and infrared fixed points. The authors analyze the strong-coupling limits and instanton expansions for multiple parameter regimes, uncovering universal infrared behavior such as perfect backscattering in certain cases and non-generic fixed points in others, with clear predictions for transport quantities like $R_{xx}$ and their temperature dependences. The framework also sets the stage for extensions to other non-Abelian states and potential experimental tests via tunneling and interferometry.

Abstract

We study quasiparticle tunneling between the edges of a non-Abelian topological state. The simplest examples are a p+ip superconductor and the Moore-Read Pfaffian non-Abelian fractional quantum Hall state; the latter state may have been observed at Landau-level filling fraction nu=5/2. Formulating the problem is conceptually and technically non-trivial: edge quasiparticle correlation functions are elements of a vector space, and transform into each other as the quasiparticle coordinates are braided. We show in general how to resolve this difficulty and uniquely define the quasiparticle tunneling Hamiltonian. The tunneling operators in the simplest examples can then be rewritten in terms of a free boson. One key consequence of this bosonization is an emergent spin-1/2 degree of freedom. We show that vortex tunneling across a p+ip superconductor is equivalent to the single-channel Kondo problem, while quasiparticle tunneling across the Moore-Read state is analogous to the two-channel Kondo effect. Temperature and voltage dependences of the tunneling conductivity are given in the low- and high-temperature limits.

Figures (7)

• Figure 1: A voltage $V_G$ applied to gates on either side of a Hall bar forms a constriction, causing tunneling between the edges. For a weak constriction in a $\nu=5/2$ state, quasiparticles can tunnel between the edges of the half-filled first excited Landau level. Tunneling between the integer quantum Hall edges of the filled lowest Landau level can be neglected because these edges are further apart.
• Figure 2: When a large gate voltage $V_G$ is applied, the quantum Hall droplet is broken in two. Electrons can tunnel between the two droplets. At $\nu=5/2$, it is the $\nu=1/2$ droplet in the first excited Landau level which is broken in two. The $\nu=2$ integer quantum Hall droplet remains unbroken.
• Figure 3: We redraw the point contact with the bottom edge flipped so that the description is completely chiral. In the strong constriction limit, an incoming $a$ mode becomes an outgoing $b$ mode and vice versa. The dotted line with arrows on both ends in the top two pictures represents the tunneling path between the top and bottom edges in the weak constriction limit. The dashed line in the bottom two pictures represents a tunneling path from the left droplet to the right one. This dashed line is transposed to the top two pictures to illustrate how these tunneling paths cross in real space but are parallel in the flipped representation.
• Figure 4: We can fold the $x>0$ half-plane onto the $x<0$ half-plane so that right-moving modes in the $x>0$ half-plane now become left-moving modes in the $x<0$ half-plane. The resulting non-chiral modes are coupled only at the origin.
• Figure 5: Spacetime history of quasiparticle-quasihole pair creation processes contributing to transport across the point contact. A pair is created at the middle of the point contact; the quasiparticle moves to one edge and the quasihole moves to the other. When two such processes happen in succession, we can ask how the two quasiparticles which end up on the same edge fuse. The amplitude for such fusion into the $1$ and $\psi$ channels is given by the Jones-Kauffman bracket evaluation of the knot diagrams at the bottom left and right of the figure.