Numerical calculation of the neutral fermion gap at the ν=5/2 fractional quantum Hall state

Signatures of Supersymmetry in the ν=5/2 Fractional Quantum Hall Effect

The Moore-Read state, one of the leading candidates for describing the fractional quantum Hall effect at filling factor ν=5/2, is a paradigmatic p-wave superconductor with non-Abelian topological order. Among its many exotic properties, the state hosts two collective modes: a bosonic density wave and a neutral fermion mode that arises from an unpaired electron in the condensate. It has recently been proposed that the descriptions of the two modes can be unified by postulating supersymmetry (SUSY) that relates them in the long-wavelength limit. Here we extend the SUSY description to construct wave functions of the two modes on closed surfaces, such as the sphere and torus, and we test the resulting states in large-scale numerical simulations. We demonstrate the equivalence in the long-wavelength limit between SUSY wave functions and previous descriptions of collective modes based on the Girvin-MacDonald-Platzman ansatz, Jack polynomials, and bipartite composite fermions. Leveraging the first-quantized form of the SUSY wave functions, we study their energies using the Monte Carlo method and show that realistic ν=5/2 systems are close to the putative SUSY point, where the two collective modes become degenerate in energy.

Collective Excitations at Filling Factor 5/2: The View from Superspace

We present a microscopic theory of the neutral collective modes supported by the non-Abelian fractional quantum Hall states at filling factor 5/2. The theory is formulated in terms of the trial states describing the Girvin-MacDonald-Platzman mode and its fermionic counterpart. These modes are superpartners of each other in a concrete sense, which we elucidate.

Topological Exciton Fermi Surfaces in Two-Component Fractional Quantized Hall Insulators

A wide variety of two-dimensional electron systems allow for independent control of the total and relative charge density of two-component fractional quantum Hall (FQH) states. In particular, a recent experiment on bilayer graphene (BLG) observed a continuous transition between a compressible and incompressible phase at total filling ν_{T}=1/2 as charge is transferred between the layers, with the remarkable property that the incompressible phase has a finite interlayer polarizability. We argue that this occurs because the topological order of ν_{T}=1/2 systems supports a novel type of interlayer exciton that carries Fermi statistics. If the fermionic excitons are lower in energy than the conventional bosonic excitons (i.e., electron-hole pairs), they can form an emergent neutral Fermi surface, providing a possible explanation of an incompressible yet polarizable state at ν_{T}=1/2. We perform exact diagonalization studies that demonstrate that fermionic excitons are indeed lower in energy than bosonic excitons. This suggests that a "topological exciton metal" hidden inside a FQH insulator may have been realized experimentally in BLG. We discuss several detection schemes by which the topological exciton metal can be experimentally probed.

Recent experimental progress of fractional quantum Hall effect: 5/2 filling state and graphene

The phenomenon of fractional quantum Hall effect (FQHE) was first experimentally observed 33 years ago. FQHE involves strong Coulomb interactions and correlations among the electrons, which leads to quasiparticles with fractional elementary charge. Three decades later, the field of FQHE is still active with new discoveries and new technical developments. A significant portion of attention in FQHE has been dedicated to filling factor 5/2 state, for its unusual even denominator and possible application in topological quantum computation. Traditionally, FQHE has been observed in high-mobility GaAs heterostructure, but new materials such as graphene also open up a new area for FQHE. This review focuses on recent progress of FQHE at 5/2 state and FQHE in graphene.

The quantum Hall effect at 5/2 filling factor

Experimental discovery of a quantized Hall state at 5/2 filling factor presented an enigmatic finding in an established field of study that has remained an open issue for more than twenty years. In this review we first examine the experimental requirements for observing this state and outline the initial theoretical implications and predictions. We will then follow the chronology of experimental studies over the years and present the theoretical developments as they pertain to experiments, directed at sets of issues. These topics will include theoretical and experimental examination of the spin properties at 5/2; is the state spin polarized? What properties of the higher Landau levels promote development of the 5/2 state, what other correlation effects are observed there, and what are their interactions with the 5/2 state? The 5/2 state is not a robust example of the fractional quantum Hall effect: what experimental and material developments have allowed enhancement of the effect? Theoretical developments from initial pictures have promoted the possibility that 5/2 excitations are exceptional; do they obey non-abelian statistics? The proposed experiments to determine this and their executions in various forms will be presented: this is the heart of this review. Experimental examination of the 5/2 excitations through interference measurements will be reviewed in some detail, focusing on recent results that demonstrate consistency with the picture of non-abelian charges. The implications of this in the more general physics picture is that the 5/2 excitations, shown to be non-abelian, should exhibit the properties of Majorana operators. This will be the topic of the last review section.

Topological blockade and measurement of topological charge

The fractionally charged quasiparticles appearing in the 5/2 fractional quantum Hall plateau are predicted to have an extra nonlocal degree of freedom, known as topological charge. We show how this topological charge can block the tunneling of these particles, and how such topological blockade can be used to read out their topological charge. We argue that the short time scale required for this measurement is favorable for the detection of the non-Abelian anyonic statistics of the quasiparticles. We also show how topological blockade can be used to measure braiding statistics, and to couple a topological qubit with a conventional one.

Resonant inelastic light scattering investigation of low-lying gapped excitations in the quantum fluid at ν=5/2

The low-lying neutral excitation spectrum of the incompressible quantum Hall fluid at ν=5/2 is investigated by inelastic light scattering. Gapped modes are observable only in a very narrow filling factor range centered at 5/2 at energies that overlap estimates from transport activation gaps. The modes are interpreted as critical points in the wave-vector dispersion of excitations that preserve spin orientation. For very small changes |δν|≲0.01 the gapped modes disappear and a continuum of low-lying excitations takes over indicating the transition from an incompressible fluid at 5/2 to a compressible state. Observations of spin wave modes indicate spin polarization of the 5/2 and 2+1/3 quantum Hall fluids.

Quantum phase transitions and the ν=5/2 fractional Hall state in wide quantum wells

We study the nature of the ν=5/2 quantum Hall state in wide quantum wells under the mixing of electronic subbands and Landau levels. A general method is introduced to analyze the Moore-Read pfaffian state and its particle-hole conjugate, the anti-pfaffian state, under periodic boundary conditions in a "quartered" Brillouin zone scheme containing both even and odd numbers of electrons. By examining the rotational quantum numbers on the torus, we show spontaneous breaking of the particle-hole symmetry can be observed in finite-size systems. In the presence of electronic-subband and Landau-level mixing, the particle-hole symmetry is broken in such a way that the anti-pfaffian state is unambiguously favored, and becomes more robust in the vicinity of a transition to the compressible phase, in agreement with recent experiments.

Microscopic model of quasiparticle wave packets in superfluids, superconductors, and paired Hall states

We study the structure of Bogoliubov quasiparticles, bogolons, the fermionic excitations of paired superfluids that arise from fermion (BCS) pairing, including neutral superfluids, superconductors, and paired quantum Hall states. The naive construction of a stationary quasiparticle in which the deformation of the pair field is neglected leads to a contradiction: it carries a net electrical current even though it does not move. However, treating the pair field self-consistently resolves this problem: in a neutral superfluid, a dipolar current pattern is associated with the quasiparticle for which the total current vanishes. When Maxwell electrodynamics is included, as appropriate to a superconductor, this pattern is confined over a penetration depth. For paired quantum Hall states of composite fermions, the Maxwell term is replaced by a Chern-Simons term, which leads to a dipolar charge distribution and consequently to a dipolar current pattern.

Model wave functions for the collective modes and the magnetoroton theory of the fractional quantum Hall effect

We construct model wave functions for the collective modes of fractional quantum Hall systems. The wave functions are expressed in terms of symmetric polynomials characterized by a root partition that defines a "squeezed" basis, and show excellent agreement with exact diagonalization results for finite systems. In the long wavelength limit, we prove that the model wave functions are identical to those predicted by the single-mode approximation, leading to intriguing interpretations of the collective modes from the perspective of the ground-state guiding-center metric.

Fractionalization of itinerant anyons in one-dimensional chains

We construct models of interacting itinerant non-Abelian anyons moving along one-dimensional chains, focusing, in particular, on itinerant Ising anyon chains, and derive effective anyonic t-J models for the low-energy sectors. Solving these models by exact diagonalization, we find a fractionalization of the anyons into charge and (non-Abelian) anyonic degrees of freedom--a generalization of spin-charge separation of electrons which occurs in Luttinger liquids. A detailed description of the excitation spectrum by combining spectra for charge and anyonic sectors requires a subtle coupling between charge and anyonic excitations at the microscopic level (which we also find to be present in Luttinger liquids), despite the macroscopic fractionalization.

Unpaired composite fermion, topological exciton, and zero mode

The paired state of composite fermions is expected to support two kinds of excitations: vortices and unpaired composite fermions. We construct an explicit microscopic description of the unpaired composite fermions, which we demonstrate to be accurate for a 3-body model interaction and, possibly, adiabatically connected to the Coulomb solution. This understanding reveals that an unpaired composite fermion carries with it a charge-neutral "topological" exciton, which, in turn, helps provide microscopic insight into the origin of zero modes, fusion rules, and energetics.

Neutral fermion excitations in the Moore-Read state at filling factor ν = 5/2

We present evidence supporting the weakly paired Moore-Read phase in the half-filled second Landau level, focusing on some of the qualitative features of its excitations. Based on numerical studies, we show that systems with odd particle number at the flux N(ϕ)=2N-3 can be interpreted as a neutral fermion mode of one unpaired fermion, which is gapped. The mode is found to have two distinct minima, providing a signature that could be observed by photoluminescence. In the presence of two quasiparticles the same neutral fermion excitation is shown to be gapless, confirming expectations for non-abelian statistics of the Ising model with degenerate fusion channels 1 and ψ.

Weakly coupled Pfaffian as a type I quantum Hall liquid

The Pfaffian phase in the proximity of a half-filled Landau level is understood to be a p+ip superconductor of composite fermions. We consider the properties of this paired quantum Hall phase when the pairing energy is small, i.e., in the weak-coupling, BCS limit, where the coherence length is much larger than the charge screening length. We find that, as in a type I superconductor, vortices attract so that, upon varying the magnetic field from its magic value at ν=5/2, the system exhibits Coulomb frustrated phase separation. We propose that the weakly and strongly coupled Pfaffians exemplify a general dichotomy between type I and type II quantum Hall fluids.