| Moore–Read state | |
|---|---|
| Name | Moore–Read state |
| Introduced | 1991 |
| Authors | Gregory Moore and Nicholas Read |
| Field | Condensed matter physics |
| Related | Fractional quantum Hall effect, Non-Abelian anyon |
Moore–Read state
The Moore–Read state is a proposed quantum many-body state of two-dimensional electron systems that exhibits paired fermionic correlations and supports non-Abelian quasiparticle excitations. Introduced in 1991 by Gregory Moore and Nicholas Read as a trial wavefunction for the Fractional quantum Hall effect at filling fraction 5/2, it is central to studies of topological phases, topological order, and potential fault-tolerant quantum computation.
The Moore–Read state arises in low-temperature, high-magnetic-field two-dimensional electron gases hosted by semiconductor heterostructures such as GaAs/AlGaAs quantum wells and in engineered systems like graphene and ZnO heterostructures. It is relevant when Landau level filling fractions produce even-denominator fractions, most notably the experimentally observed 5/2 state first reported by Willett et al. The state occupies a distinguished role among proposed states of the Fractional quantum Hall effect because it breaks the simple composite-fermion hierarchy by incorporating pairing analogous to a p-wave superconductor, connecting ideas from BCS theory and topological superconductivity.
The Moore–Read wavefunction is constructed as a product of a pairing factor and a Laughlin-like Jastrow factor. In second-quantized form and on the plane it can be written using a Pfaffian: Pf(1/(z_i - z_j)) ∏_{i
A defining feature of the Moore–Read state is the emergence of quasiparticles with non-Abelian braid statistics. These excitations, often described as Ising anyons, carry a topological degeneracy when multiple quasiparticles are present; braiding acts by noncommuting unitary transformations on the degenerate ground state manifold. The non-Abelian statistics are captured by the fusion rules and modular data of the Ising CFT, connecting to mathematical structures studied in tensor category theory and modular tensor categories. The topological order of the Moore–Read phase is distinct from Abelian Laughlin states and supports protected edge modes described by a chiral Majorana fermion plus a chiral boson, with consequences for thermal transport quantization as explored in Kane and Fisher and subsequent literature.
Within the hierarchy of fractional quantum Hall states, the Moore–Read state provides a candidate explanation for the even-denominator 5/2 plateau observed in high-mobility samples. It competes with alternative candidate states such as the anti-Pfaffian (the particle-hole conjugate), Abelian composite fermion paired states, and stripe or nematic phases. Theoretical work using exact diagonalization, density-matrix renormalization group (DMRG), and matrix-product-state methods—pioneered in part by groups at Princeton University, Columbia University, University of California, Santa Barbara, and Microsoft Station Q collaborations—has evaluated overlaps and energetics under realistic Hamiltonians including Coulomb interactions, finite width, and Landau level mixing.
Experimental probes relevant to the Moore–Read state include measurements of quantized Hall conductance, thermal Hall conductance, shot noise, tunneling spectroscopy, interferometry, and scanning probe microscopy. Observations by groups led by researchers such as R. R. Du, J. P. Eisenstein, and Jainendra K. Jain have been influential in characterizing the 5/2 plateau. Thermal Hall measurements by the Heiblum group and others have aimed to detect the expected half-integer central charge contribution from the chiral Majorana mode. Interferometry experiments at Weizmann Institute of Science, Bell Labs, and IBM Research attempt to directly measure braiding statistics, while experiments in engineered platforms—such as proximitized semiconductor nanowires (inspired by proposals of Roman Lutchyn and Sau-Das Sarma) and heterostructures supporting topological superconductivity—seek to realize similar non-Abelian excitations under more controlled conditions.
The Moore–Read state is tightly connected to the Ising conformal field theory and a U(1) chiral boson. The Pfaffian wavefunction is generated as a correlator of Majorana fermion fields and vertex operators; its quasiholes correspond to spin-field insertions with nontrivial braiding encoded by the monodromy of conformal blocks. Mathematical analysis employs braid group representations, fusion algebra, and topological quantum field theory (TQFT) frameworks such as SU(2)_2 Chern–Simons theory. Rigorous advances relate the edge CFT to bulk TQFT via bulk–boundary correspondence and employ techniques from complex analysis, representation theory, and numerical spectral studies.
Because non-Abelian Ising anyons implement a limited set of topologically protected operations, Moore–Read-like platforms have been proposed for fault-tolerant quantum gates that are intrinsically robust against local perturbations. While braiding Ising anyons yields Clifford gates, which are not universal alone, proposals combine non-topological operations or alternative non-Abelian phases to achieve universality. Stability of the Moore–Read phase against disorder, Landau level mixing, and competing phases remains a practical concern; achieving robust, reproducible realizations is an active area connecting condensed matter experiment, materials growth at institutions like Bell Labs and Sandia National Laboratories, and theoretical methods developed by Kitaev, Nayak, and collaborators. The state thus sits at the nexus of tradition and innovation in quantum condensed matter: preserving topological coherence while offering routes toward transformative technologies such as topological quantum computation.
Category:Quantum Hall effect Category:Topological phases of matter