Optical probing of the spin polarization of the ν=5/2 quantum Hall state

A platform for far-infrared spectroscopy of quantum materials at millikelvin temperatures

Optical spectroscopy of quantum materials at ultralow temperatures is rarely explored, yet it may provide critical characterizations of quantum phases not possible using other approaches. We describe the development of a novel experimental platform that enables optical spectroscopic studies, together with standard electronic transport, of materials at millikelvin temperatures inside a dilution refrigerator. The instrument is capable of measuring both bulk crystals and micrometer-sized two-dimensional van der Waals materials and devices. We demonstrate its performance by implementing photocurrent-based Fourier transform infrared spectroscopy on a monolayer WTe2 device and a multilayer 1T-TaS2 crystal, with a spectral range available from the near-infrared to the terahertz regime and in magnetic fields up to 5 T. In the far-infrared regime, we achieve spectroscopic measurements at a base temperature as low as ∼43 mK and a sample electron temperature of ∼450 mK. Possible experiments and potential future upgrades of this versatile instrumental platform are envisioned.

Topological Interface between Pfaffian and Anti-Pfaffian Order in ν=5/2 Quantum Hall Effect

A recent thermal Hall experiment triggered renewed interest in the problem of ν=5/2 quantum Hall effect, which motivated novel interpretations based on the formation of mesoscopic puddles made of Pfaffian and anti-Pfaffian topological orders. Here, we study an interface between the Pfaffian and anti-Pfaffian states, which may play crucial roles in thermal transport, by means of state-of-the-art, density-matrix renormalization group simulations. We demonstrate that an intrinsic electric dipole moment emerges at the interface, similar to the "p-n" junction sandwiched between N-type and P-type semiconductor. Importantly, we elucidate the topological origin of this dipole moment, whose formation is to counterbalance the mismatch of guiding-center Hall viscosity of bulk Pfaffian and anti-Pfaffian states. In addition, these results imply that the formation of a dipole moment could be helpful to stabilize the puddles made of Pfaffian and anti-Pfaffian states in experimental conditions.

Direct Observation of Composite Fermions and Their Fully-Spin-Polarized Fermi Sea near ν=5/2

The enigmatic even-denominator fractional quantum Hall state at Landau level filling factor ν=5/2 is arguably the most promising candidate for harboring Majorana quasiparticles with non-Abelian statistics and, thus, of potential use for topological quantum computing. The theoretical description of the ν=5/2 state is generally believed to involve a topological p-wave pairing of fully-spin-polarized composite fermions through their condensation into a non-Abelian Moore-Read Pfaffian state. There is, however, no direct and conclusive experimental evidence for the existence of composite fermions near ν=5/2 or for an underlying fully-spin-polarized Fermi sea. Here, we report the observation of composite fermions very near ν=5/2 through geometric resonance measurements and find that the measured Fermi wave vector provides direct demonstration of a Fermi sea with full spin polarization. This lends crucial credence to the model of 5/2 fractional quantum Hall effect as a topological p-wave paired state of composite fermions.

Competing ν = 5/2 fractional quantum Hall states in confined geometry

Some theories predict that the filling factor 5/2 fractional quantum Hall state can exhibit non-Abelian statistics, which makes it a candidate for fault-tolerant topological quantum computation. Although the non-Abelian Pfaffian state and its particle-hole conjugate, the anti-Pfaffian state, are the most plausible wave functions for the 5/2 state, there are a number of alternatives with either Abelian or non-Abelian statistics. Recent experiments suggest that the tunneling exponents are more consistent with an Abelian state rather than a non-Abelian state. Here, we present edge-current-tunneling experiments in geometrically confined quantum point contacts, which indicate that Abelian and non-Abelian states compete at filling factor 5/2. Our results are consistent with a transition from an Abelian state to a non-Abelian state in a single quantum point contact when the confinement is tuned. Our observation suggests that there is an intrinsic non-Abelian 5/2 ground state but that the appropriate confinement is necessary to maintain it. This observation is important not only for understanding the physics of the 5/2 state but also for the design of future topological quantum computation devices.

Optical Emission Spectroscopy Study of Competing Phases of Electrons in the Second Landau Level

Quantum phases of electrons in the filling factor range 2≤ν≤3 are probed by the weak optical emission from the partially populated second Landau level and spin wave measurements. Observations of optical emission include a multiplet of sharp peaks that exhibit a strong filling factor dependence. Spin wave measurements by resonant inelastic light scattering probe breaking of spin rotational invariance and are used to link this optical emission with collective phases of electrons. A remarkably rapid interplay between emission peak intensities manifests phase competition in the second Landau level.

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.

Electron spin resonance and cyclotron resonance for fractional quantum Hall states in narrow-gap QW heterostructures

We report a theoretical study of the energies of cyclotron resonance (CR) and electron spin resonance (ESR) for fractional quantum Hall states (FQHS) in n-type narrow-gap quantum well (QW) heterostructures. Using the generalized single-mode approximation (GSMA) based on the 8-band k·p Hamiltonian, we calculate the many-body corrections to the CR and ESR energies for FQHS, providing theoretical evidence of the Kohn and Larmor theorem violation in narrow-gap QWs. We predict the correlation-induced reduction of CR energies and the correlation-induced enhancement of ESR energies as compared with the values obtained within the Hartree-Fock approximation. We demonstrate a nonlinear dependence of the CR and ESR energies on a Landau level filling factor.

Enhancement of the ν = 5/2 fractional quantum Hall state in a small in-plane magnetic field

Using a 50-nm-width ultraclean GaAs/AlGaAs quantum well, we have studied the Landau level filling factor ν=5/2 fractional quantum Hall effect in a perpendicular magnetic field B∼1.7  T and determined its dependence on tilted magnetic fields. Contrary to all previous results, the 5/2 resistance minimum and the Hall plateau are found to strengthen continuously under an increasing tilt angle 0<θ<25° (corresponding to an in-plane magnetic field 0<B(∥)<0.8  T). In the same range of θ, the activation gaps of both the 7/3 and the 8/3 states are found to increase with tilt. The 5/2 state transforms into a compressible Fermi liquid upon tilt angle θ>60°, and the composite fermion series [2+p/(2p±1), p=1,2] can be identified. Based on our results, we discuss the relevance of a Skyrmion spin texture at ν=5/2 associated with small Zeeman energy in wide quantum wells, as proposed by Wójs et al. [Phys. Rev. Lett. 104, 086801 (2010)].

NMR probing of the spin polarization of the ν=5/2 quantum Hall state

Resistively detected nuclear magnetic resonance is used to measure the Knight shift of the 75As nuclei and determine the electron spin polarization of the fractional quantum Hall states of the second Landau level. We show that the 5/2 state is fully polarized within experimental error, thus confirming a fundamental assumption of the Moore-Read theory. We measure the electron heating under radio frequency excitation and show that we are able to detect NMR at electron temperatures down to 30 mK.

Stable Pfaffian state in bilayer graphene

Here, we show that the incompressible Pfaffian state originally proposed for the 5/2 fractional quantum Hall states in conventional two-dimensional electron systems can actually be found in a bilayer graphene at one of the Landau levels. The properties and stability of the Pfaffian state at this special Landau level strongly depend on the magnetic field strength. The graphene system shows a transition from the incompressible to a compressible state with increasing magnetic field. At a finite magnetic field of ~10 T, the Pfaffian state in bilayer graphene becomes more stable than its counterpart in conventional electron systems.

Characterizing neutral modes of fractional states in the second Landau level

Fractionally charged quasiparticles, which obey non-abelian statistics, were predicted to exist in the ν=8/3, ν=5/2, and ν=7/3 fractional quantum Hall states (in the second Landau level). Here we present measurements of charge and neutral modes in these states. For both ν=7/3 and ν=8/3 states, we found a quasiparticle charge e=1/3 and an upstream neutral mode only in ν=8/3-excluding the possibility of non-abelian Read-Rezayi states and supporting Laughlin-like states. The absence of an upstream neutral mode in the ν=7/3 state also proves that edge reconstruction was not present in the ν=7/3 state, suggesting its absence also in ν=5/2 state, and thus may provide further support for the non-abelian anti-pfaffian nature of the ν=5/2 state.

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.

Rapid collapse of spin waves in nonuniform phases of the second Landau level

The spin degree of freedom in quantum phases of the second Landau level is probed by resonant light scattering. The long wavelength spin wave, which monitors the degree of spin polarization, is at the Zeeman energy in the fully spin-polarized state at ν = 3. At lower filling factors, the intensity of the Zeeman mode collapses, indicating loss of polarization. A novel continuum of low-lying excitations emerges that dominates near ν = 8/3 and ν = 5/2. Resonant Rayleigh scattering reveals that quantum fluids for ν < 3 break up into robust domain structures. While the state at ν = 5/2 is considered to be fully polarized, these results reveal unprecedented roles for spin degrees of freedom.

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

We present the first numerical computation of the neutral fermion gap, Δ(F), in the ν=5/2 quantum Hall state, which is analogous to the energy gap for a Bogoliubov-de Gennes quasiparticle in a superconductor. We find Δ(F)≈0.027e(2)/εℓ(0), comparable to the charge gap. We also deduce an effective Fermi velocity v(F) for neutral fermions from the low-energy spectra for odd numbers of electrons, and thereby obtain a correlation length ξ(F)=v(F)/Δ(F)≈1.3ℓ(0). We comment on implications for experiments, topological quantum information processing, and electronic mechanisms of superconductivity.