Fractional Quantum Hall Effect States Research Articles

Prediction of non-Abelian fractional quantum Hall effect at ν=2+411

The fractional quantum Hall effect (FQHE) in the second Landau level (SLL) likely stabilizes non-Abelian topological orders. Recently, a parton sequence has been proposed to capture many of the fractions observed in the SLL [Ajit C. Balram, SciPost Phys. {\bf 10}, 083 (2021)]. We consider the first member of this sequence which has not yet been studied, which is a non-Abelian state that occurs at $4/11$. As yet FQHE in the SLL at this fraction has not been observed in experiments. Nevertheless, by studying its competition with other candidate FQHE states in the SLL we show that this parton state might be viable. We also make predictions for experimentally measurable properties of the parton state which can distinguish it from other topological orders.

Unconventional Fractional Quantum Hall States in a Wide Quantum Well

A bilayer electron system that is formed in a 60-nm-wide GaAs quantum well and has a large difference of the electron densities in the layers has been studied. It has been found that, when a magnetic field is tilted from the normal to the plane of the system, integer quantum Hall effect states at the filling factors of Landau levels of 1 and 2 disappear; instead, fractional quantum Hall effect states in the interval between these filling factors appear at the filling factors νF = 4/3, 10/7, and 6/5 with odd denominators and at the filling factor νF = 5/4. Several different states can be observed under the variation of the magnetic field. The detected fractional quantum Hall effect states are interpreted as combined states with the same filling factor 1 in the layer with the higher density and with the filling factors νF – 1 in the layer with the lower density. These states are formed because of the redistribution of electrons between the layers, which occurs under the variation of the magnetic field. The appearance of the state with the filling factor νF = 5/4 with the even denominator is presumably attributed to the dominance of the interlayer electron–electron interaction over the intralayer one for electrons in the layer with the lower density.

The model for non-Abelian field topology for the multilayer fractional quantum anomalous Hall device

From the recent empirical discovery of the quantum anomalous Hall effect (QAHE), the interaction of the particle with spin–orbit coupling (SOC) plays an essential role in the cause of the QAHE, which includes three terms: external, internal, and chiral symmetric terms. Then, the non-Abelian quantum field theory was adopted to analyze and prove the conjecture on the causes that can lead to the fractional quantum Hall effect (FQHE). The spontaneously topological chiral symmetry breaking is the main contribution to the FQHE, which also includes two terms: the hopping of sublattice and Coulomb energy by the interaction of many-body particles. More generally, this exciton possesses an intermediate characteristic between the Wannier regimes and displays a peculiar two-dimensional wavefunction in the three-dimensional FQHE states. Finally, a bilayer three-dimensional model is proposed to implement the FQHE on the lattice by incorporating ferromagnetic dopants into three-dimensional topological insulative thin films. This study theoretically predicts the FQHE on the basis of other reports that have experimentally verified the rationality of the proposed model in magnetic topological insulators.

Birotons and “Dark” Quantum Hall Hierarchies

A computational scheme is suggested to estimate neutral excitation energies in the fractional quantum Hall effect (FQHE) states. The FQHE states are systematized according to the Farey-number hierarchy structure. We show that besides the widely known Laughlin–Jain hierarchy of fractional states, there exist other “dark” hierarchies. Although hardly observed in the highest mobility samples, they can significantly affect the thermodynamics and spectral characteristics of the FQHE states. The known problems in the interpretation of the FQHE’s experimental results are explained in terms of the coexistence of two fundamentally different transformations of the electron system, one of which is a neutral excitation in the FQHE state, whereas the other is a transition between two FQHE ground states, one of which represents the Laughlin–Jain FQHE hierarchy and the other a state of “dark” hierarchies.

Majorana Anyon Composites in Magneto-Photoluminescence Spectra of Natural Quantum Hall Puddles.

In magneto-photoluminescence (magneto-PL) spectra of quasi two-dimensional islands (quantum dots) having seven electrons and Wigner–Seitz radius rs~1.5, we revealed a suppression of magnetic field (B) dispersion, paramagnetic shifts, and jumps of the energy of the emission components for filling factors ν > 1 (B < 10 T). Additionally, we observed B-hysteresis of the jumps and a dependence of all these anomalous features on rs. Using a theoretical description of the magneto-PL spectra and an analysis of the electronic structure of these dots based on the single-particle Fock–Darwin spectrum and many-particle configuration-interaction calculations, we show that these observations can be described by the rs-dependent formation of the anyon (magneto-electron) composites (ACs) involving single-particle states having non-zero angular momentum and that the anyon states observed involve Majorana modes (MMs), including zero-B modes having an equal number of vortexes and anti-vortexes, which can be considered as Majorana anyons. We show that the paramagnetic shift corresponds to a destruction of the equilibrium self-formed ν~5/2 AC by the external magnetic field and that the jumps and their hysteresis can be described in terms of Majorana qubit states controlled by B and rs. Our results show a critical role of quantum confinement in the formation of magneto-electrons and implies the liquid-crystal nature of fractional quantum Hall effect states, the Majorana anyon origin of the states having even ν, i.e., composite fermions, which provide new opportunities for topological quantum computing.

Possible fractional quantum Hall effect and Wigner crystal states in two-dimensional Coulomb gases as a result of anyon correlations

Recently, we have derived an approximate analytic formula for the ground state energy of (2D) charged anyon gas. This result allows us in the present paper to estimate the exchangecorrelation energy, which is as it is well known from a literature the difference of the ground state energy and the Hartree-Fock energy, for fermions with the anyon parameter ν=1, and to formulate the anyon correlation energy for other ν&lt;1. The minima of the last one for all fixed values of ν as well as the minima of the ground state energy at the same ν are only the ν dependent, which means they are only the anyon statistics dependent with no effect of the Coulomb interaction. From this, we conclude that there should appear the localized real space macroscopic structure for any fixed ν from ν&lt;1. For instance, since the Landau level fillingfactor (LLFF) of the fractional quantum Hall effect (FQHE) quasi-particles is equal to the anyon parameter then these quasi-particles should condense in the some real space structure at this LLFF, forming the FQHE plateau and showing that the FQHE is a result of anyoncorrelations. On the other hand, expressing the above mentioned minima function of ν through the density parameter rs of the Coulomb gas, we obtain the values of rs, for which the 2D Wigner crystal (WC) can be observed. It is interesting that the simultaneous observation of the FQHE and WC states in accord with our consideration.

Resonant Photoluminescence of a Two-Dimensional Electron System upon the Formation of a Bulk 1/3 State of the Fractional Hall Effect

Resonant and nonresonant photoluminescence spectra of a two-dimensional electron system have been studied under the conditions of the formation of the 1/3 fractional quantum Hall effect state. It has been shown that resonant photoluminescence, in contrast to nonresonant photoluminescence, is a universal marker of the formation of the 1/3 state in the bulk of the two-dimensional system. It has been found that the probabilities of optical transitions from the zeroth Landau level of electrons in the 1/3 state vary so strongly that these variations cannot be explained within the existence theoretical concepts.

Explanation of an unexpected occurrence of fractional quantum Hall effect states in monolayer graphene

Recent experiment reveals the appearance of incompressible fractional quantum Hall effect states in monolayer graphene at and , substituting the compressible Hall metal states at these fillings in the lowest Landau level in a narrow magnetic field window depending on the sample parameters. Simultaneously, none such behavior has been observed either for or , . We propose an explanation of these observations in terms of homotopy of monolayer graphene in consistence with a general theory of correlated states in planar Hall systems.

High-Quality Magnetotransport in Graphene Using the Edge-Free Corbino Geometry.

We report fabrication of graphene devices in a Corbino geometry consisting of concentric circular electrodes with no physical edge connecting the inner and outer electrodes. High device mobility is realized using boron nitride encapsulation together with a dual-graphite gate structure. Bulk conductance measurement in the quantum Hall effect (QHE) regime outperforms previously reported Hall bar measurements, with improved resolution observed for both the integer and fractional QHE states. We identify apparent phase transitions in the fractional sequence in both the lowest and first excited Landau levels (LLs) and observe features consistent with electron solid phases in higher LLs.

Nuclear spin dynamics in [001] AlAs quantum well in the regime of integer and fractional quantum Hall effect

Relaxation of nuclear spins located in the vicinity of 2D electron system confined in a 16 nm [001] AlAs/AlGaAs quantum well was studied with the aid of electron spin resonance (ESR) in the regime of integer and fractional quantum Hall effect. Nuclear spin-lattice relaxation time τ was measured from the time decay of the Overhauser shift near different filling factors of the electron system. The resultant dependence of τ on filling factor turned out to be nontrivial. At the temperature 1.5 K τ reached its maximal value at the exact filling ν = 1 and decreased when ν was altered, yet this maximum vanished when the system was cooled down to 0.5 K. The fractional quantum Hall effect state at the filling of 2∕3 was formed at the temperature of 0.5 K, and the development of this state was accompanied by the slowing of the nuclear spin relaxation. This observation suggests the enhancement of energy gap in the spin excitation spectrum of two-dimensional electrons at 2/3 state.

ν=2/3 fractional quantum Hall state in an AlAs quantum well probed by electron spin resonance

The electron spin resonance (ESR) of two-dimensional electrons confined in a high-quality, 16-nm AlAs quantum well was investigated near the filling factor $\ensuremath{\nu}=2/3$ of the fractional quantum Hall effect (FQHE). The spin resonance was robust in the vicinity of the fractional filling $\ensuremath{\nu}=2/3$, indicating that the $\ensuremath{\nu}=2/3$ state is at least partially spin polarized. The formation of the $2/3$ FQHE state did not result in any modifications of the ESR linewidth and, hence, of the electron spin relaxation rate. Yet the nuclear spin-lattice relaxation rate extracted from the time decay of the ESR Overhauser shift demonstrated a strong nonmonotonic dependence on the electron filling factor with a minimum near $\ensuremath{\nu}=2/3$. This observation suggests the enhancement of the energy gap in the spin excitation spectrum of two-dimensional electrons at the $\ensuremath{\nu}=2/3$ state.

Edge reconstruction in fractional quantum Hall states

Questions on the nature of edge reconstruction and "where does the current flow" in the quantum Hall effect (QHE) have been debated for years. Moreover, the recent observation of proliferation of "upstream" neutral modes in the fractional QHE raised doubts about the present models of edge channels. In this article we focus on hole-conjugate states, nu=2/3 and nu=3/5, and present a new picture of their edge reconstruction. For example, while the present model for nu=2/3 consists of a single downstream charge channel with conductance 2/3 e^2/h and an upstream neutral mode, we show that the current is carried by two separate downstream edge channels, each with conductance 1/3 e^2/h, accompanied by upstream neutral mode(s). We find that if the two downstream channels are not equilibrated, inter-mode equilibration (via particle exchange) takes place over a distance of microns, with the two channels effectively behaving as a single channel. Moreover, the inter-channel equilibration is accompanied by an excitation of upstream neutral modes. In turn, the counter-propagating neutral modes, moving in close proximity to the charge modes, fragment into propagating charges, inducing thus downstream current fluctuations with zero net current - a novel mechanism for non-equilibrium noise. This unexpected edge reconstruction underlines the need for better understanding of edge reconstruction and energy transport in all fractional QHE states.

Microwave spectroscopic observation of a Wigner solid within the ν=1/2 fractional quantum Hall effect

The fractional quantum Hall effect (FQHE) states at half integer Landau fillings ($\nu$) have long been of great interest, since they have correlations that differ from those of the fundamental Laughlin states found at odd denominators. At $\nu=1/2$ the FQHE has been observed in wide or double quantum wells, and is ascribed to the two-component Halperin-Laughlin $\Psi_{331}$ state. $\Psi_{331}$ excitations carry charge $\pm e/4$, like the carriers of $\nu=5/2$ states which are of interest in quantum computation. Further, such an excitation (quasiparticle or -hole) of $\Psi_{331}$ has unequal, opposite charge in the top and bottom layers, and hence an up or down dipole moment. Here we report evidence for a Wigner solid (WS) of such dipolar quasiholes from a quantitative study of the microwave spectra of a wide quantum well (WQW) at $\nu$ close to 1/2.

Gapped excitations of unconventional fractional quantum Hall effect states in the second Landau level

We report the observation of low-lying collective charge and spin excitations in the second Landau level at $\ensuremath{\nu}=2+1/3$ and also for the very fragile states at $\ensuremath{\nu}=2+2/5$ and $2+3/8$ in inelastic light scattering experiments. These modes exhibit a clear dependence on filling factor and temperature substantiating the unique access to the characteristic neutral excitation spectra of the incompressible fractional quantum Hall effect (FQHE) states. A detailed mode analysis reveals low-energy modes at around 70 $\ensuremath{\mu}\mathrm{eV}$ and a sharp mode slightly below the Zeeman energy interpreted as gap and spin-wave excitation, respectively. The lowest-energy collective charge excitation spectrum at $\ensuremath{\nu}=2+1/3$ exhibits significant qualitative similarities with its cousin state in the lowest Landau level at $\ensuremath{\nu}=1/3$ suggesting similar magnetoroton minima in the neutral excitations. The mode energies differ by a scaling of 0.15 indicating different interaction physics in the $N=0$ and $N=1$ Landau levels. The striking polarization dependence in elastic and inelastic light scattering is discussed in the framework of anisotropic electron phases that allow for the stabilization of nematic FQHE states. The observed excitation spectra provide new insights by accessing quantum phases in the bulk of electron systems and facilitate comparison with different theoretical descriptions of those enigmatic FQHE states.

Evidence for a fractional fractal quantum Hall effect in graphene superlattices.

The Hofstadter energy spectrum provides a uniquely tunable system to study emergent topological order in the regime of strong interactions. Previous experiments, however, have been limited to low Bloch band fillings where only the Landau level index plays a role. We report measurements of high-mobility graphene superlattices where the complete unit cell of the Hofstadter spectrum is accessible. We observed coexistence of conventional fractional quantum Hall effect (QHE) states together with the integer QHE states associated with the fractal Hofstadter spectrum. At large magnetic field, we observed signatures of another series of states, which appeared at fractional Bloch filling index. These fractional Bloch band QHE states are not anticipated by existing theoretical pictures and point toward a distinct type of many-body state.

Study of Wigner crystal in n-GaAs/AlGaAs by surface acoustic waves

The surface acoustic wave (SAW) technique was used for studying a 65 nm wide GaAs quantum well containing a two dimensional electron system (2DES) with the density of 5·1010 cm-2 and the mobility of 8·106 cm2/V·s. A SAW propagated in proximity of the quantum well. Both SAW attenuation α and SAW velocity V were measured in perpendicular magnetic field of up to 18 T in the frequency f range 28.5–306 MHz at temperatures 40–380 mK. Then, both real and imaginary parts of the ac conductance σ(f) = σ1 (f) + i·σ2 (f) of the 2DES were calculated from α and V. Dependencies of α, V, σ1, and σ2 on magnetic fields showed a rich oscillation pattern, in particular, the fractional quantum Hall effect (FQHE). Thus, at low temperature, the FQHE state was observed at the filling factor ν = 0.20, whereas, in vicinity of this value, at ν = 0.19 and 0.21, the 2DES behaved as a 2D Wigner crystal weakly pinned by a disorder random potential. The calculated pinning frequencies at high and zero magnetic fields were 193 MHz and 12....

Fractional quantum Hall effect in Hofstadter butterflies of Dirac fermions

We report on the influence of a periodic potential on the fractional quantum Hall effect (FQHE) states in monolayer graphene. We have shown that for two values of the magnetic flux per unit cell (one-half and one-third flux quantum) an increase of the periodic potential strength results in a closure of the FQHE gap and appearance of gaps due to the periodic potential. In the case of one-half flux quantum this causes a change of the ground state and consequently the change of the momentum of the system in the ground state. While there is also crossing between low-lying energy levels for one-third flux quantum, the ground state does not change with the increase of the periodic potential strength and is always characterized by the same momentum. Finally, it is shown that for one-half flux quantum the emergent gaps are due entirely to the electron–electron interaction, whereas for the one-third flux quantum per unit cell these are due to both non-interacting electrons (Hofstadter butterfly pattern) and the electron–electron interaction.

Anomalously low magnetoroton energies of the unconventional fractional quantum Hall States of composite fermions.

We show a generic formation of the primary magnetorotons in the collective modes of the observed "unconventional" fractional quantum Hall effect states of the composite fermions at the filling factors 4/11, 4/13, 5/13, 5/17, and 3/8 at very low wave vectors with anomalously low energies which do not have any analog to the conventional fractional quantum Hall states. Rather slow decay of the oscillations of the pair-correlation functions in these states is responsible for the low-energy magnetorotons. This is a manifestation of the distinct topology predicted previously for these fractional quantum Hall effect states. Experimental consequences of our theory are also discussed.

Composite Fermion Theory of Exotic Fractional Quantum Hall Effect

The fractional quantum Hall effect (FQHE) arises from strong correlations between electrons when they are confined to two dimensions and exposed to a strong magnetic field. The underlying physics is the formation of topological particles called composite fermions (CFs), electron-vortex bound states whose integer quantum Hall effect explains a large majority of the observed FQHE states. In recent years, the focus has shifted to the more exotic states that originate from a weak residual interaction between composite fermions. These include chiral p-wave paired states of composite fermions at certain even denominator fractions, unconventional FQHE of composite fermions, and a series of CF crystals at low fillings. Aside from these states, we also review the FQHE in multicomponent systems, which has attracted renewed attention because of the observation of well-developed FQHE in several multivalley systems, such as graphene and AlAs quantum wells.

Emergent chiral spin liquid: fractional quantum Hall effect in a kagome Heisenberg model.

The fractional quantum Hall effect (FQHE) realized in two-dimensional electron systems under a magnetic field is one of the most remarkable discoveries in condensed matter physics. Interestingly, it has been proposed that FQHE can also emerge in time-reversal invariant spin systems, known as the chiral spin liquid (CSL) characterized by the topological order and the emerging of the fractionalized quasiparticles. A CSL can naturally lead to the exotic superconductivity originating from the condense of anyonic quasiparticles. Although CSL was highly sought after for more than twenty years, it had never been found in a spin isotropic Heisenberg model or related materials. By developing a density-matrix renormalization group based method for adiabatically inserting flux, we discover a FQHE in a isotropic kagome Heisenberg model. We identify this FQHE state as the long-sought CSL with a uniform chiral order spontaneously breaking time reversal symmetry, which is uniquely characterized by the half-integer quantized topological Chern number protected by a robust excitation gap. The CSL is found to be at the neighbor of the previously identified Z2 spin liquid, which may lead to an exotic quantum phase transition between two gapped topological spin liquids.