Even-denominator Fractional Quantum Hall States Research Articles

Study of polarization for even-denominator fractional quantum Hall states in SU(4) Graphene

Abstract We have focused on studying the even-denominator fractional quantum Hall (EDFQH) states observed in monolayer graphene. In this article, we have studied polarization mainly for the two EDFQH states at filling fractions ν = 1/2, 1/4, which are observed in an experimental study [Nat. Phys. 14, 930 (2018)] a few years ago. We have applied Chern-Simon’s gauge field theory to explain the possible variational wave functions for different polarized states and calculated their ground state energies using the Coulomb potential. We have chosen the lowest energy states using suitable combinations of attached flux quanta to the electrons for different polarized states of those EDFQH states.

Energy Gap of the Even-Denominator Fractional Quantum Hall State in Bilayer Graphene

Energy Gap of the Even-Denominator Fractional Quantum Hall State in Bilayer Graphene

Next-generation even-denominator fractional quantum Hall states of interacting composite fermions.

The discovery of the fractional quantum Hall state (FQHS) in 1982 ushered a new era of research in many-body condensed matter physics. Among the numerous FQHSs, those observed at even-denominator Landau level filling factors are of particular interest as they may host quasiparticles obeying non-Abelian statistics and be of potential use in topological quantum computing. The even-denominator FQHSs, however, are scarce and have been observed predominantly in low-disorder two-dimensional (2D) systems when an excited electron Landau level is half filled. An example is the well-studied FQHS at filling factor [Formula: see text] 5/2 which is believed to be a Bardeen-Cooper-Schrieffer-type, paired state of flux-particle composite fermions (CFs). Here, we report the observation of even-denominator FQHSs at [Formula: see text] 3/10, 3/8, and 3/4 in the lowest Landau level of an ultrahigh-quality GaAs 2D hole system, evinced by deep minima in longitudinal resistance and developing quantized Hall plateaus. Quite remarkably, these states can be interpreted as even-denominator FQHSs of CFs, emerging from pairing of higher-order CFs when a CF Landau level, rather than an electron or a hole Landau level, is half-filled. Our results affirm enhanced interaction between CFs in a hole system with significant Landau level mixing and, more generally, the pairing of CFs as a valid mechanism for even-denominator FQHSs, and suggest the realization of FQHSs with non-Abelian anyons.

Highly Anisotropic Even-Denominator Fractional Quantum Hall State in an Orbitally Coupled Half-Filled Landau Level.

The even-denominator fractional quantum Hall states (FQHSs) in half-filled Landau levels are generally believed to host non-Abelian quasiparticles and be of potential use in topological quantum computing. Of particular interest is the competition and interplay between the even-denominator FQHSs and other ground states, such as anisotropic phases and composite fermion Fermi seas. Here, we report the observation of an even-denominator fractional quantum Hall state with highly anisotropic in-plane transport coefficients at Landau level filling factor ν=3/2. We observe this state in an ultra-high-quality GaAs two-dimensional hole system when a large in-plane magnetic field is applied. By increasing the in-plane field, we observe a sharp transition from an isotropic composite fermion Fermi sea to an anisotropic even-denominator FQHS. Our data and calculations suggest that a unique feature of two-dimensional holes, namely the coupling between heavy-hole and light-hole states, combines different orbital components in the wave function of one Landau level, and leads to the emergence of a highly anisotropic even-denominator fractional quantum Hall state. Our results demonstrate that the GaAs two-dimensional hole system is a unique platform for the exploration of exotic, many-body ground states.

Valley-Tunable Even-Denominator Fractional Quantum Hall State in the Lowest Landau Level of an Anisotropic System.

Fractional quantum Hall states (FQHSs) at even-denominator Landau level filling factors (ν) are of prime interest as they are predicted to host exotic, topological states of matter. We report here the observation of a FQHS at ν=1/2 in a two-dimensional electron system of exceptionally high quality, confined to a wide AlAs quantum well, where the electrons can occupy multiple conduction-band valleys with an anisotropic effective mass. The anisotropy and multivalley degree of freedom offer an unprecedented tunability of the ν=1/2 FQHS as we can control both the valley occupancy via the application of in-plane strain, and the ratio between the strengths of the short- and long-range Coulomb interaction by tilting the sample in the magnetic field to change the electron charge distribution. Thanks to this tunability, we observe phase transitions from a compressible Fermi liquid to an incompressible FQHS and then to an insulating phase as a function of tilt angle. We find that this evolution and the energy gap of the ν=1/2 FQHS depend strongly on valley occupancy.

Aharonov–Bohm Oscillations in Bilayer Graphene Quantum Hall Edge State Fabry–Pérot Interferometers

Bernal-stacked bilayer graphene exhibits a wealth of interaction-driven phenomena, including robust even-denominator fractional quantum Hall states. We construct Fabry-Pérot interferometers using a split-gate design and present measurements of the Aharonov-Bohm oscillations. The edge state velocity is found to be approximately 6 × 104 m/s at filling factor ν = 2 and decreases with increasing filling factor. The dc bias and temperature dependence of the interference point to electron-electron interaction induced decoherence mechanisms. These results pave the way for the quest of fractional and non-Abelian braiding statistics in this promising device platform.

Even-Denominator Fractional Quantum Hall State at Filling Factor ν=3/4.

Fractional quantum Hall states (FQHSs) exemplify exotic phases of low-disorder two-dimensional (2D) electron systems when electron-electron interaction dominates over the thermal and kinetic energies. Particularly intriguing among the FQHSs are those observed at even-denominator Landau level filling factors, as their quasiparticles are generally believed to obey non-Abelian statistics and be of potential use in topological quantum computing. Such states, however, are very rare and fragile, and are typically observed in the excited Landau level of 2D electron systems with the lowest amount of disorder. Here we report the observation of a new and unexpected even-denominator FQHS at filling factor ν=3/4 in a GaAs 2D hole system with an exceptionally high quality (mobility). Our magnetotransport measurements reveal a strong minimum in the longitudinal resistance at ν=3/4, accompanied by a developing Hall plateau centered at (h/e^{2})/(3/4). This even-denominator FQHS is very unusual as it is observed in the lowest Landau level and in a 2D hole system. While its origin is unclear, it is likely a non-Abelian state, emerging from the residual interaction between composite fermions.

Even-denominator fractional quantum Hall state in conventional triple-gated quantum point contact

The even-denominator states have attracted considerable attention owing to their possible applications in future quantum technologies. In this letter, we first report a 3/2 diagonal resistance, indicating the existence of a 3/2 state in a nanometer-sized triple-gated quantum point contact (QPC) fabricated on a high-mobility (not ultra-high-mobility) single-layer two-dimensional (2D) GaAs wafer. The center gate plays a crucial role in realizing the QPC’s 3/2 state. Our observation of the 3/2 state using a conventional QPC device, which is a suitable building block for semiconductor quantum devices, paves a new path for the development of semiconductor-based quantum technologies.

Even-denominator fractional quantum Hall state in bilayer graphene

At a half-filled Landau level, composite fermions with chiral <i>p</i>-wave pairing will form a Moore-Read state which hosts charge-<i>e</i>/4 fractional excitation. This excitation supports non-Abelian statistics and has potential to enable topological quantum computation. Owing to the <i>SU</i>(4) symmetry of electron and electric-field tunability, the bilayer graphene becomes an ideal platform for exploring physics of multi-component quantum Hall state and is candidate for realizing non-Abelian statistics. In this work, high-quality bilayer graphene/hBN heterostructure is fabricated by using dry-transfer technique, and electric transport measurement is performed to study quantum Hall state behavior in bilayer graphene under electric field and magnetic field. Under strong magnetic field, the sequences of incompressible state with quantized Hall conductivity are revealed at –5/2, –1/2, 3/2 filling of Landau level. The feature of even-denominator quantum Hall state is more visible then weaker with increasing magnetic field, and this corresponds to the polarization of Landau level wave function. The experimental results indicate that the observed even-denominator fractional quantum Hall state belongs to the topological phase described by Pfaffian wavefunction.

Stability of even-denominator fractional quantum Hall states in systems with strong Landau-level mixing

Mixing of Landau levels has been understood to be essential in governing the nature of the ground state for the even-denominator fractional quantum Hall effect. The incompressibility of the ground state at filling factor $5/2$ in the strong Landau-level-mixed systems, such as the ZnO quantum well, is not always stable. Here we present an approach to generally deal with this kind of system and satisfactorily explain the recent experiments [Falson et al., Sci. Adv. 4, eaat8742 (2018)] by implementing the screening plus the thickness effect. Further, the phase diagrams of the incompressibility of the ground state indicate that the phase transitions can be explicitly extracted by observing the lowest gap of the collective modes when the magnetic field and the width of the quantum well are tuned. We also predict the incompressibility of the two-dimensional electron gas in higher Landau levels in another strong Landau-level-mixed system, viz., the black phosphorene, by considering the screening effect where the relevant even-denominator fractional quantum Hall effects can possibly be observed.

Disorder broadening of even-denominator fractional quantum Hall states in the presence of a short-range alloy potential

We study energy gaps of the $\nu=7/2$ and $\nu=5/2$ fractional quantum Hall states in a series of two-dimensional electron gases containing alloy disorder. We found that gaps at these two filling factors have the same suppression rate with alloy disorder. The dimensionless intrinsic gaps in our alloy samples obtained from the model proposed by Morf and d'Ambrumenil are consistent with numerical results, but are larger than those obtained from experiments on pristine samples published in the literature. The disorder broadening parameter has large uncertainties. However, a modified analysis relying on shared intrinsic gaps yields consistent results for both the $\nu=5/2$ and $7/2$ fractional quantum Hall states and establishes a linear relationship between the disorder broadening parameter and alloy concentration. Furthermore, we find that we can separate contributions to the disorder broadening of the long-range and short-range scattering.

Odd- and even-denominator fractional quantum Hall states in monolayer WSe2.

Monolayer semiconducting transition-metal dichalcogenides (TMDs) represent a unique class of two-dimensional (2D) electron systems. Their atomically thin structure facilitates gate tunability just like graphene does, but unlike graphene, TMDs have the advantage of a sizable band gap and strong spin-orbit coupling. Measurements under large magnetic fields have revealed an unusual Landau level (LL) structure1-3, distinct from other 2D electron systems. However, owing to the limited sample quality and poor electrical contact, probing the lowest LLs has been challenging, and observation of electron correlations within the fractionally filled LL regime has not been possible. Here, through bulk electronic compressibility measurements, we investigate the LL structure of monolayer WSe2 in the extreme quantum limit, and observe fractional quantum Hall states in the lowest three LLs. The odd-denominator fractional quantum Hall sequences demonstrate a systematic evolution with the LL orbital index, consistent with generic theoretical expectations. In addition, we observe an even-denominator state in the second LL that is expected to host non-Abelian statistics. Our results suggest that the 2D semiconductors can provide an experimental platform that closely resembles idealized theoretical models in the quantum Hall regime.

Finite-thickness effect and spin polarization of the even-denominator fractional quantum Hall states

The spin-polarized even-denominator fractional quantum Hall (FQH) states in the second Landau level (LL), i.e. 5/2 and 7/2, may possess novel quasi-particle excitations obeying non-Abelian statistics. However, the spin polarization of the 7/2 FQH state has not been investigated experimentally and the spin polarization of the 5/2 FQH state from tilted field experiments remains controversial. Using a piezo-driven sample rotator with the lowest electron temperature down to 25 mK, we studied the energy gap of the even-denominator FQH states in the second LL by precise control of the tilted angles with a resolution less than 0.1{\deg}. We observed two different energy gap dependences on the in-plane magnetic field for 5/2, 7/2, other FQH states (7/3 and 8/3) in the second LL and reentrant integer quantum Hall (RIQH) states in the third LL. Though the transition fields vary from states, their corresponding in-plane magnetic lengths are comparable to the quantum well thickness of the sample, which may result from the influence of the finite-thickness effect. At low in-plane magnetic fields, before the conjectured finite-thickness effect starts to dominate, the energy gaps of both 5/2 and 7/2 states show a non-decreasing behavior, supporting spin-polarized ground states. Our results also suggest that the 7/3, 8/3 FQH states, and the RIQH states in the third LL are spin-polarized or partially spin-polarized.

3/2 fractional quantum Hall plateau in confined two-dimensional electron gas

Even-denominator fractional quantum Hall (FQH) states, such as 5/2 and 7/2, have been well known in a two-dimensional electron gas (2DEG) for decades and are still investigated as candidates of non-Abelian statistics. In this paper, we present the observation of a 3/2 FQH plateau in a single-layer 2DEG with lateral confinement at a bulk filling factor of 5/3. The 3/2 FQH plateau is quantized at left( {frac{h}{{e^2}}} right)/left( {frac{3}{2}} right) within 0.02%, and can survive up to 300 mK. This even-denominator FQH plateau may imply intriguing edge structure and excitation in FQH system with lateral confinement. The observations in this work demonstrate that understanding the effect of the lateral confinement on the many-body system is critical in the pursuit of important theoretical proposals involving edge physics, such as the demonstration of non-Abelian statistics and the realization of braiding for fault-tolerant quantum computation.

Unconventional Anisotropic Even-Denominator Fractional Quantum Hall State in a System with Mass Anisotropy.

The fractional quantum Hall state (FQHS) observed at a half-filled Landau level in an interacting two-dimensional electron system (2DES) is among the most exotic states of matter as its quasiparticles are expected to be Majorana excitations with non-Abelian statistics. We demonstrate here the unexpected presence of such a state in a novel 2DES with a strong band-mass anisotropy. The FQHS we observe has unusual characteristics. While its Hall resistance is well quantized at low temperatures, it exhibits highly anisotropic in-plane transport resembling compressible stripe or nematic charge-density-wave phases. More striking, the anisotropy sets in suddenly below a critical temperature, suggesting a finite-temperature phase transition. Our observations highlight how anisotropy modifies the many-body phases of a 2DES, and should further fuel the discussion surrounding the enigmatic even-denominator FQHS.

Author Correction: Even-denominator fractional quantum Hall states at an isospin transition in monolayer graphene

In the version of this Article originally published, the sketch of ‘PSP’ in Fig. 1a was incorrect; it has now been replaced. Please see the correction note to compare the original and corrected figure.

Even-denominator fractional quantum Hall states at an isospin transition in monolayer graphene

Magnetic fields quench the kinetic energy of two dimensional electrons, confining them to highly degenerate Landau levels. In the absence of disorder, the ground state at partial Landau level filling is determined only by Coulomb interactions, leading to a variety of correlation-driven phenomena. Here, we realize a quantum Hall analog of the Ne\'el-to-valence bond solid transition within the spin- and sublattice- degenerate monolayer graphene zero energy Landau level by experimentally controlling substrate-induced sublattice symmetry breaking. The transition is marked by unusual isospin transitions in odd denominator fractional quantum Hall states for filling factors $\nu$ near charge neutrality, and the unexpected appearance of incompressible even denominator fractional quantum Hall states at $\nu=\pm1/2$ and $\pm1/4$ associated with pairing between composite fermions on different carbon sublattices.

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.

Even-denominator fractional quantum Hall states in bilayer graphene.

The distinct Landau level spectrum of bilayer graphene (BLG) is predicted to support a non-abelian even-denominator fractional quantum Hall (FQH) state similar to the [Formula: see text] state first identified in GaAs. However, the nature of this state has remained difficult to characterize. Here, we report transport measurements of a robust sequence of even-denominator FQH in dual-gated BLG devices. Parallel field measurement confirms the spin-polarized nature of the ground state, which is consistent with the Pfaffian/anti-Pfaffian description. The sensitivity of the even-denominator states to both filling fraction and transverse displacement field provides new opportunities for tunability. Our results suggest that BLG is a platform in which topological ground states with possible non-abelian excitations can be manipulated and controlled.

Search for composite fermions at filling factor 5/2: Role of Landau level and subband index

The pairing of composite fermions (CFs), electron-flux quasi-particles, is commonly proposed to explain the even-denominator fractional quantum Hall state observed at $\nu=5/2$ in the first excited ($N=1$) Landau level (LL) of a two-dimensional electron system (2DES). While well-established to exist in the lowest ($N=0$) LL, much is unknown about CFs in the $N=1$ LL. Here we carry out geometric resonance measurements to detect CFs at $\nu=5/2$ by subjecting the 2DES to a one-dimensional density modulation. Our data, taken at a temperature of 0.3 K, reveal no geometric resonances for CFs in the $N=1$ LL. In stark contrast, we observe clear signatures of such resonances when $\nu=5/2$ is placed in the $N=0$ LL of the anti-symmetric subband by varying the 2DES width. This finding implies that the CFs' mean-free-path is significantly smaller in the $N=1$ LL compared to the $N=0$ LL. Our additional data as a function of in-plane magnetic field highlight the role of subband index and establish that CFs at $\nu=5/2$ in the $N=0$ LL are more anisotropic in the symmetric subband than in the anti-symmetric subband.