Moore–Read state

Moore–Read state
Name	Moore–Read state
Other names	Pfaffian state
Proposed by	Gregory Moore, Nicholas Read
Year	1991
Context	Fractional quantum Hall effect
Filling fraction	5/2
Particle statistics	Non-Abelian

Moore–Read state

The Moore–Read state is a proposed non-Abelian quantum Hall wavefunction introduced by Gregory Moore and Nicholas Read for the fractional quantum Hall effect at filling fraction 5/2. It combines concepts from conformal field theory, BCS pairing, and Pfaffian constructions to produce paired composite-fermion correlations believed relevant to two-dimensional electron systems in high magnetic fields. The proposal has motivated extensive theoretical work linking Ising conformal field theory, Majorana fermion, and topological quantum field theory approaches.

Introduction

The Moore–Read construction originated from a correspondence between correlators in conformal field theory and trial wavefunctions for quantum Hall states, a strategy previously used by Robert Laughlin for the Laughlin wavefunction and by Juan Maldacena in related high-energy contexts. Moore and Read applied concepts from Ising CFT, Virasoro algebra, and chiral algebra to generate a paired state of composite fermions resembling a chiral p-wave superconductor inspired by Alexei Kitaev's later work. The state is often called the Pfaffian state due to the Pfaffian factor appearing in the wavefunction, a mathematical construct linked to André Weil and algebraic combinatorics.

Theoretical construction

The Moore–Read wavefunction is built by multiplying a Pfaffian of pairwise functions with a Laughlin wavefunction-like Jastrow factor, combining ideas from Robert Laughlin, B. I. Halperin, and composite-particle theories developed by Jainendra Jain. The Pfaffian term encodes pairing of composite fermions analogous to BCS pairing, while the Jastrow factor enforces the 1/q power-law correlations characteristic of fractionalized fluids. Moore and Read derived the state using correlators of primary fields in Ising conformal field theory and a chiral boson, tying the construction to the representation theory of the Virasoro algebra and modular invariants studied by G. Segal and E. Verlinde. Subsequent formulations connected the Moore–Read state to effective descriptions in terms of Chern–Simons theory, chiral topological quantum field theory, and the Read–Rezayi series. Numerical studies by groups including F. D. M. Haldane, N. Read, and S. Das Sarma employed exact diagonalization and density-matrix renormalization group techniques to test energetic stability on the sphere and torus geometries introduced by Haldane pseudopotentials and F. D. M. Haldane.

Topological properties and quasiparticles

The Moore–Read phase supports excitations with non-Abelian braiding statistics described by the Ising anyon model and the fusion rules of the Ising conformal field theory. Quasiparticles include charge-e/4 vortices that bind zero-energy Majorana fermion modes similar to those predicted in Kitaev chain models and in proposals by Sarma, Freedman, and Nayak. The topological order is characterized by degeneracies on higher-genus surfaces linked to modular tensor category data and topological entanglement entropy measures related to work by Xiao-Gang Wen and Alexei Kitaev. Edge excitations combine a chiral Majorana mode and a charged boson mode, with predictions stemming from Wen's chiral Luttinger liquid theory and boundary conformal field theory analyses by John Cardy.

Experimental signatures and evidence

Experimental searches for the Moore–Read state focus on observations at the second Landau level in high-mobility GaAs/AlGaAs heterostructures studied by groups led by Horst Stormer, Daniel Tsui, and Arthur Gossard. Key signatures include an incompressible plateau at filling fraction 5/2 in magnetotransport measurements, activated energy gaps measured via temperature-dependent conductivity and shot-noise experiments by teams including Raphael de Picciotto and Michael Heiblum, and tunneling spectroscopy consistent with chiral edge structure probed in interferometry experiments by Clarke, Nayak, and Willett. Interferometric proposals by P. Bonderson and C. Nayak seek to detect non-Abelian braiding through Fabry–Pérot and Mach–Zehnder geometries, while thermal conductance measurements by Kane and Fisher and experiments reporting half-integer thermal Hall conductance provide evidence for Majorana edge modes as predicted for the Moore–Read phase. Competing candidate states, including the anti-Pfaffian proposed by Levin and Stern and numerical alternatives discussed by Rezayi and Haldane, complicate interpretation and motivate continued transport, tunneling, and thermal experiments.

Applications in quantum computation

The non-Abelian anyons of the Moore–Read phase implement topologically protected operations relevant to fault-tolerant quantum computation frameworks advanced by Kitaev, Nayak, and Freedman. Braiding of Ising anyons realizes a subset of quantum gates equivalent to the Clifford group; universal quantum computation requires supplementation via magic-state distillation or coupling to ancillary systems as developed in proposals by Bravyi and Kitaev and Bravyi. Experimental platforms seeking to harness Moore–Read physics include engineered semiconductor-superconductor heterostructures inspired by Alicea and Lutchyn proposals, and hybrid devices leveraging proximity-induced superconductivity studied by Mourik and Das Sarma. The Moore–Read state's simpler non-Abelian algebra compared with Fibonacci anyon schemes makes it an attractive intermediate step toward scalable topological quantum computing architectures investigated by Microsoft Station Q and academic consortia.

Extensions and related states

The Moore–Read state sits within the broader Read–Rezayi hierarchy and inspired generalizations including the series associated with parafermion conformal field theories and Z_k-Read–Rezayi states analyzed by Read and Rezayi. The anti-Pfaffian, particle-hole conjugate proposed by Levin and Stern, competes energetically under realistic interactions and disorders considered by Peterson and Simon. Related constructions appear in studies of chiral p-wave superconductors examined by Volovik, spinless fermion models by Kitaev, and lattice fractional Chern insulators explored by Sheng and Neupert. Connections to higher-dimensional topological phases and symmetry-protected topological order have been discussed in work by Chen, Gu, and Wen.

Category:Fractional quantum Hall effect