Helical liquids and Majorana bound states in quantum wires

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Abstract

We show that the combination of spin-orbit coupling with a Zeeman field or strong interactions may lead to the formation of a helical liquid in single-channel quantum wires. In a helical liquid, electrons with opposite velocities have opposite spin precession. We argue that zero-energy Majorana bound states are formed in various situations when the wire is situated in proximity to a conventional s-wave superconductor. This occurs when the external magnetic field, the superconducting gap, or, in particular, the chemical potential vary along the wire. We discuss experimental consequences of the formation of the helical liquid and the Majorana bound states.

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Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect

N Read, Dmitry Green (1999)

We analyze pairing of fermions in two dimensions for fully-gapped cases with broken parity (P) and time-reversal (T), especially cases in which the gap function is an orbital angular momentum (\(l\)) eigenstate, in particular \(l=-1\) (p-wave, spinless or spin-triplet) and \(l=-2\) (d-wave, spin-singlet). For \(l\neq0\), these fall into two phases, weak and strong pairing, which may be distinguished topologically. In the cases with conserved spin, we derive explicitly the Hall conductivity for spin as the corresponding topological invariant. For the spinless p-wave case, the weak-pairing phase has a pair wavefunction that is asympototically the same as that in the Moore-Read (Pfaffian) quantum Hall state, and we argue that its other properties (edge states, quasihole and toroidal ground states) are also the same, indicating that nonabelian statistics is a {\em generic} property of such a paired phase. The strong-pairing phase is an abelian state, and the transition between the two phases involves a bulk Majorana fermion, the mass of which changes sign at the transition. For the d-wave case, we argue that the Haldane-Rezayi state is not the generic behavior of a phase but describes the asymptotics at the critical point between weak and strong pairing, and has gapless fermion excitations in the bulk. In this case the weak-pairing phase is an abelian phase which has been considered previously. In the p-wave case with an unbroken U(1) symmetry, which can be applied to the double layer quantum Hall problem, the weak-pairing phase has the properties of the 331 state, and with nonzero tunneling there is a transition to the Moore-Read phase. The effects of disorder on noninteracting quasiparticles are considered.

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Non-Abelian Anyons and Topological Quantum Computation

Ady Stern, Chetan Nayak, Steven H. Simon … (2007)

Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as {\it Non-Abelian anyons}, meaning that they obey {\it non-Abelian braiding statistics}. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations which are necessary for quantum computation are carried out by braiding quasiparticles, and then measuring the multi-quasiparticle states. The fault-tolerance of a topological quantum computer arises from the non-local encoding of the states of the quasiparticles, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the \nu=5/2 state, although several other prospective candidates have been proposed in systems as disparate as ultra-cold atoms in optical lattices and thin film superconductors. In this review article, we describe current research in this field, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. We address both the mathematical underpinnings of topological quantum computation and the physics of the subject using the \nu=5/2 fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.