Fractional Hall Effect Research Articles

The Principle and State-of-Art Application of Quantum Hall Effect

Contemporarily, quantum hall effect (QHE), as the quantum version of classic Hall effect, represented a turning point in the understanding of material physics, metrology, high energy physics, astrophysics. In the category of QHE, there are plenty of sub-types including fractional QHE (FQHE), anomalous QHE (QAHE), spin QHE (SPHE) etc. This paper will mainly focus on the latest progress and the state-of-art applications regarding to QHE. To be specific, a brief introduction of the principle of Hall effect and the differences among different types will be introduced. Afterwards, the applications will be discussed and demonstrated. Overall, these results would provide the audience a general view on QHE, and some recent applications regards it, which shed light on guiding further exploration for QHE.

A highly correlated topological bubble phase of composite fermions

Strong interactions and topology drive a wide variety of correlated ground states. Some of the most interesting of these ground states, such as fractional quantum Hall states and fractional Chern insulators, have fractionally charged quasiparticles. Correlations in these phases are captured by the binding of electrons and vortices into emergent particles called composite fermions. Composite fermion quasiparticles are randomly localized at high levels of disorder and may exhibit charge order when there is not too much disorder in the system. However, more complex correlations are predicted when composite fermion quasiparticles cluster into a bubble, and then these bubbles order on a lattice. Such a highly correlated ground state is termed the bubble phase of composite fermions. Here we report the observation of such a bubble phase of composite fermions, evidenced by the re-entrance of the fractional quantum Hall effect. We associate this re-entrance with a bubble phase with two composite fermion quasiparticles per bubble. Our results demonstrate the existence of a new class of strongly correlated topological phases driven by clustering and charge ordering of emergent quasiparticles. Composite fermions emerge in the fractional quantum Hall effect. Now, it has been shown that these objects can group into bubbles and that these can order into a lattice.

Birotons and Dark Hierarchies in the Fractional Quantum Hall Effect

Birotons and Dark Hierarchies in the Fractional Quantum Hall Effect

Birotons and ‘dark hierarchies’ in fractional quantum Hall effect

A computational scheme has been developed to determine the energies of neutral excitations in fractional states of the quantum Hall effect. The fractional states are brought into a general hierarchical system (Farey-number hierarchy structure). It is shown that in addition to the well-known hierarchy of fractional states (Laughlin–Jane), there are other ‘dark hierarchies’ of fractional quantum Hall effect that are not observed experimentally in the most highly mobile samples but can have a significant impact on the thermodynamics and spectral characteristics of fractional quantum Hall effect states. The connection of the ‘dark hierarchies’ of fractional quantum Hall effect with the existing experimental data is considered.

Coexistence of two hole phases in high-quality p−GaAs/AlGaAs in the vicinity of Landau-level filling factors ν=1 and ν=(1/3)

We focused on the transverse AC magnetoconductance of a high-mobility $p$-GaAs/AlGaAs quantum well $(p=1.2\ifmmode\times\else\texttimes\fi{}{10}^{11}{\mathrm{cm}}^{\ensuremath{-}2})$ in the vicinity of two values of the Landau-level filling factor $\ensuremath{\nu}$: $\ensuremath{\nu}=1$ (integer quantum Hall effect) and $\ensuremath{\nu}=1/3$ (fractional quantum Hall effect). The complex transverse AC conductance ${\ensuremath{\sigma}}_{xx}^{\mathrm{AC}}(\ensuremath{\omega})$ was found from simultaneous measurements of attenuation and velocity of surface acoustic waves propagating along the interface between a piezoelectric crystal and the two-dimensional hole system under investigation. We analyzed both the real and imaginary parts of the hole conductance and compared the similarities and differences between the results for filling factor 1 and filling factor 1/3. Both to the left and to the right of these values maxima of a specific shape, ``wings,'' arose in the $\ensuremath{\sigma}(\ensuremath{\nu})$ dependences at those two $\ensuremath{\nu}$. Analysis of the results of our acoustic measurements at different temperatures and surface acoustic wave frequencies allowed us to attribute these wings to the formation of collective localized states, namely, the domains of a pinned Wigner crystal, i.e., a Wigner solid. While the Wigner solid has been observed in 2D hole systems previously, we were able to detect it at the highest hole density and, therefore, the lowest hole-hole interaction reported.

Fractional quantum Hall effect in optical lattices

In this research, we study the bosonic fractional quantum Hall (FQH) states in a system of ultracold bosons in a two-dimensional optical lattice in the presence of a synthetic magnetic field, described by the bosonic Harper–Hofstadter Hamiltonian. We use the cluster Gutzwiller mean-field and exact diagonalization techniques in our work. We obtain incompressible states as ground states at various filling factors similar to those of the FQH states. We focus in particular on the ν = 1/2 FQH state, and it is characterized by the two-point correlation function and the many-body Chern number. We further investigate the effect of dipolar interaction on the ν = 1/2 FQH state. We find that the dipolar interaction stabilizes the FQH state against the competing superfluid state.

Quantized fractional Thouless pumping of solitons

In many contexts, interaction between particles gives rise to emergent phenomena. An example is the fractional quantum Hall effect, where the interaction between electrons leads to fractionally quantized Hall conductance. In photonic systems, the nonlinear response of an ambient medium mediates the interaction between photons, and, in the mean-field limit, these dynamics are described by the nonlinear Schrödinger (also called Gross–Pitaevskii) equation. It was recently shown that at weak nonlinearity, soliton motion in nonlinear Thouless pumps—a dimensionally reduced implementation of a Chern insulator—could be quantized to the Chern number, because solitons track the single-band Wannier function throughout the pumping cycle. Here using arrays of coupled optical waveguides, we show that a sufficiently strong nonlinearity fractionally quantizes the motion of solitons. Specifically, we find that the soliton follows maximally localized multi-band Wannier functions and therefore returns to itself only after multiple cycles of the Thouless pump—but displaced by an integer number of unit cells—leading to a rich fractional plateau structure describing soliton motion. Our results represent an example of emergent behaviour in topologically non-trivial systems in the presence of interactions. Interactions between photons arise due to the presence of optical nonlinearities. In topological Thouless pumps, a sufficiently strong nonlinearity leads to soliton transport with a fractionally quantized plateau structure—reminiscent of transport in the fractional quantum Hall effect.

Emergent XY* transition driven by symmetry fractionalization and anyon condensation

Anyons in a topologically ordered phase can carry fractional quantum numbers with respect to the symmetry group of the considered system, one example being the fractional charge of the quasiparticles and quasiholes in the fractional quantum Hall effect. When such symmetry-fractionalized anyons condense, the resulting phase must spontaneously break the symmetry and display a local order parameter. In this paper, we study the phase diagram and anyon condensation transitions of a \mathbb{Z}_2ℤ2 topological order perturbed by Ising interactions in the Toric Code. The interplay between the global (``onsite’’) Ising (\mathbb{Z}_2ℤ2) symmetry and the lattice space group symmetries results in a non-trivial symmetry fractionalization class for the anyons, and is shown to lead to two characteristically different confined, symmetry-broken phases. To understand the anyon condensation transitions, we use the recently introduced critical torus energy spectrum technique to identify a line of emergent 2+1D XY* transitions ending at a fine-tuned (Ising^22)* critical point. We provide numerical evidence for the occurrence of two symmetry breaking patterns predicted by the specific symmetry fractionalization class of the condensed anyons in the explored phase diagram. In combination with large-scale quantum Monte Carlo simulations we measure unusually large critical exponents \etaη for the scaling of the correlation function at the continuous anyon condensation transitions, and we further identify lines of (weakly) first order transitions in the phase diagram. As an important additional result, we discuss the phase diagram of a resulting 2+1D Ashkin-Teller model, where we demonstrate that torus spectroscopy is capable of identifying emergent XY/O(2) critical behaviour, thereby solving some longstanding open questions in the domain of the 3D Ashkin-Teller models. To establish the generality of our results, we propose a field theoretical description capturing the transition from a \mathbb{Z}_2ℤ2 topological order to either \mathbb{Z}_2ℤ2 symmetry broken phase, which is in excellent agreement with the numerical results.

Photonic flatband resonances for free-electron radiation.

Flatbands have become a cornerstone of contemporary condensed-matter physics and photonics. In electronics, flatbands entail comparable energy bandwidth and Coulomb interaction, leading to correlated phenomena such as the fractional quantum Hall effect and recently thosein magic-angle systems. In photonics, they enable properties including slow light1 and lasing2. Notably, flatbands support supercollimation-diffractionless wavepacket propagation-in both systems3,4. Despite these intense parallel efforts, flatbands have never been shown to affect the core interaction between free electrons and photons. Their interaction, pivotal for free-electron lasers5, microscopy and spectroscopy6,7, and particle accelerators8,9, is, in fact, limited by a dimensionality mismatch between localized electrons and extended photons. Here we reveal theoretically that photonic flatbands can overcome this mismatch and thus remarkably boost their interaction. We design flatband resonances in a silicon-on-insulator photonic crystal slab to control and enhance the associatedfree-electron radiation by tuning their trajectory and velocity. We observe signatures of flatband enhancement, recording a two-order increase from the conventional diffraction-enabled Smith-Purcell radiation. The enhancement enables polarization shaping of free-electron radiation and characterizationof photonic bands through electron-beam measurements. Our results support the use of flatbands as test beds for strong light-electron interaction, particularly relevant for efficient and compact free-electron light sources and accelerators.

Dirac fermion spectrum of the fractional quantum Hall states

Applying a unified approach, we study the integer quantum Hall effect (IQHE) and fractional quantum Hall effect (FQHE) in the Hofstadter model with short range interactions between fermions. An effective field, that takes into account the interaction between fermions, is determined by both amplitude and phase. Its amplitude is proportional to the interaction strength, the phase corresponds to the minimum energy. In fact, the problem is reduced to the Harper equation with two different scales: the first is a magnetic scale with the cell size corresponding to a unit quantum magnetic flux, the second scale determines the inhomogeneity of the effective field, forms the steady fine structure of the Hofstadter spectrum and leads to the realization of fractional quantum Hall states. In a sample of finite size with open boundary conditions, the fine structure of the Hofstadter spectrum consists of the Dirac branches of the fermion excitations and includes the fine structure of the edge chiral modes. The Chern numbers of the topological Hofstadter bands are conserved during the formation of their fine structure. The edge modes are formed into the Hofstadter bands. They connect the nearest-neighbor subbands and determine the conductance for the fractional filling.

Fractional quantum Hall effect and $M$-theory

Fractional quantum Hall effect and $M$-theory

Quantum transport and fractional hall effect in Moiré correlated/anticorrelated interface channels

The energy-resolved transmission of twisted circular (anticorrelated) nanoribbons depends on twisted angles, where the hybridization of local fermion momenta as f-orbitals and itinerant conduction electrons as c-orbitals play an important role.

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.

Unconventional exciton evolution from the pseudogap to superconducting phases in cuprates

Electron quasiparticles play a crucial role in simplifying the description of many-body physics in solids with surprising success. Conventional Landau’s Fermi-liquid and quasiparticle theories for high-temperature superconducting cuprates have, however, received skepticism from various angles. A path-breaking framework of electron fractionalization has been established to replace the Fermi-liquid theory for systems that show the fractional quantum Hall effect and the Mott insulating phenomena; whether it captures the essential physics of the pseudogap and superconducting phases of cuprates is still an open issue. Here, we show that excitonic excitation of optimally doped Bi2Sr2CaCu2O8+δ with energy far above the superconducting-gap energy scale, about 1 eV or even higher, is unusually enhanced by the onset of superconductivity. Our finding proves the involvement of such high-energy excitons in superconductivity. Therefore, the observed enhancement in the spectral weight of excitons imposes a crucial constraint on theories for the pseudogap and superconducting mechanisms. A simple two-component fermion model which embodies electron fractionalization in the pseudogap state provides a possible mechanism of this enhancement, pointing toward a novel route for understanding the electronic structure of superconducting cuprates.

Lattice Chern-Simons model for FQHE

Motivated by [60][61] and using topological invariance and cycle homology, we construct an effective gauge invariant lattice Chern-Simons (CS) model for the fractional quantum Hall effect (FQHE) with applications to (i) the Laughlin state on 3D crystals, (ii) Haldane hierarchy, and (iii) composite fermions. First, we give a new derivation of the lattice CS action and demonstrate its gauge symmetry and invariance under lattice duality. Then, we use this lattice construction to study the filling factors of the above mentioned QH states. We show amongst others that the νlattL of Laughlin state on lattice is given by the ratio Φ0/ΦWL with ΦWL designating the flux of CS link variables through Wilson loops WL. For composite fermions, we use the adb formalism to show that the factor νlattCF is given by the charge Qb of the dumbbell field under the dual gauge link variables. Our model gives explicitly the effective filling fraction of composite fermions on the lattice as νlatt⁎=νlattCF/(1−2nνlattCF) with νlattCF being the bare lattice filling factor in agreement with the continuum limit result.

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.

Size dependence- and induced transformations- of fractional quantum Hall effects under tilted magnetic fields

Two-dimensional electron systems subjected to high transverse magnetic fields can exhibit Fractional Quantum Hall Effects (FQHE). In the GaAs/AlGaAs 2D electron system, a double degeneracy of Landau levels due to electron-spin, is removed by a small Zeeman spin splitting, g mu _B B, comparable to the correlation energy. Then, a change of the Zeeman splitting relative to the correlation energy can lead to a re-ordering between spin polarized, partially polarized, and unpolarized many body ground states at a constant filling factor. We show here that tuning the spin energy can produce fractionally quantized Hall effect transitions that include both a change in nu for the R_{xx} minimum, e.g., from nu = 11/7 to nu = 8/5, and a corresponding change in the R_{xy}, e.g., from R_{xy}/R_{K} = (11/7)^{-1} to R_{xy}/R_{K} = (8/5)^{-1}, with increasing tilt angle. Further, we exhibit a striking size dependence in the tilt angle interval for the vanishing of the nu = 4/3 and nu = 7/5 resistance minima, including “avoided crossing” type lineshape characteristics, and observable shifts of R_{xy} at the R_{xx} minima- the latter occurring for nu = 4/3, 7/5 and the 10/7. The results demonstrate both size dependence and the possibility, not just of competition between different spin polarized states at the same nu and R_{xy}, but also the tilt- or Zeeman-energy-dependent- crossover between distinct FQHE associated with different Hall resistances.

High-Temperature Quantum Hall Effect in Graphite-Gated Graphene Heterostructure Devices with High Carrier Mobility.

Since the discovery of the quantum Hall effect in 1980, it has attracted intense interest in condensed matter physics and has led to a new type of metrological standard by utilizing the resistance quantum. Graphene, a true two-dimensional electron gas material, has demonstrated the half-integer quantum Hall effect and composite-fermion fractional quantum Hall effect due to its unique massless Dirac fermions and ultra-high carrier mobility. Here, we use a monolayer graphene encapsulated with hexagonal boron nitride and few-layer graphite to fabricate micrometer-scale graphene Hall devices. The application of a graphite gate electrode significantly screens the phonon scattering from a conventional SiO2/Si substrate, and thus enhances the carrier mobility of graphene. At a low temperature, the carrier mobility of graphene devices can reach 3 × 105 cm2/V·s, and at room temperature, the carrier mobility can still exceed 1 × 105 cm2/V·s, which is very helpful for the development of high-temperature quantum Hall effects under moderate magnetic fields. At a low temperature of 1.6 K, a series of half-integer quantum Hall plateaus are well-observed in graphene with a magnetic field of 1 T. More importantly, the ν = ±2 quantum Hall plateau clearly persists up to 150 K with only a few-tesla magnetic field. These findings show that graphite-gated high-mobility graphene devices hold great potential for high-sensitivity Hall sensors and resistance metrology standards for the new Système International d'unités.

Two-particle time-domain interferometry in the fractional quantum Hall effect regime

Quasi-particles are elementary excitations of condensed matter quantum phases. Demonstrating that they keep quantum coherence while propagating is a fundamental issue for their manipulation for quantum information tasks. Here, we consider anyons, the fractionally charged quasi-particles of the Fractional Quantum Hall Effect occurring in two-dimensional electronic conductors in high magnetic fields. They obey anyonic statistics, intermediate between fermionic and bosonic. Surprisingly, anyons show large quantum coherence when transmitted through the localized states of electronic Fabry-Pérot interferometers, but almost no quantum interference when transmitted via the propagating states of Mach-Zehnder interferometers. Here, using a novel interferometric approach, we demonstrate that anyons do keep quantum coherence while propagating. Performing two-particle time-domain interference measurements sensitive to the two-particle Hanbury Brown Twiss phase, we find 53 and 60% visibilities for anyons with charges e/5 and e/3. Our results give a positive message for the challenge of performing controlled quantum coherent braiding of anyons.

Nonrelativistic supergeometry in the Moore-Read fractional quantum Hall state

The Moore-Read state is one the most well known non-Abelian fractional quantum Hall states. It supports non-Abelian Ising anyons in the bulk and a chiral bosonic and chiral Majorana modes on the boundary. It has been recently conjectured that these modes are superpartners of each other and described by a supersymmetric conformal field theory [1]. We propose a non-relativistic supergeometric theory that is compatible with this picture and gives rise to an effective spin-3/2 field in the bulk. After breaking supersymmetry through a Goldstino, the spin-3/2 field becomes massive and can be seen as the neutral collective mode that characterizes the Moore-Read state. By integrating out this fermion field, we obtain a purely bosonic topological action that properly encodes the Hall conductivity, Hall viscosity and gravitational anomaly. Our work paves the way to the exploration of the fractional quantum Hall effect through non-relativistic supergeometry.