Detection of composite fermions by magnetic focusing

Observation of fractionally quantized anomalous Hall effect

The integer quantum anomalous Hall (QAH) effect is a lattice analogue of the quantum Hall effect at zero magnetic field1-3. This phenomenon occurs in systems with topologically non-trivial bands and spontaneous time-reversal symmetry breaking. Discovery of its fractional counterpart in the presence of strong electron correlations, that is, the fractional QAH effect4-7, would open a new chapter in condensed matter physics. Here we report the direct observation of both integer and fractional QAH effects in electrical measurements on twisted bilayer MoTe2. At zero magnetic field, near filling factor ν = -1 (one hole per moiré unit cell), we see an integer QAH plateau in the Hall resistance Rxy quantized to h/e2 ± 0.1%, whereas the longitudinal resistance Rxx vanishes. Remarkably, at ν  =  -2/3 and -3/5, we see plateau features in Rxy at [Formula: see text] and [Formula: see text], respectively, whereas Rxx remains small. All features shift linearly versus applied magnetic field with slopes matching the corresponding Chern numbers -1, -2/3 and -3/5, precisely as expected for integer and fractional QAH states. Additionally, at zero magnetic field, Rxy is approximately 2h/e2 near half-filling (ν  = -1/2) and varies linearly as ν  is tuned. This behaviour resembles that of the composite Fermi liquid in the half-filled lowest Landau level of a two-dimensional electron gas at high magnetic field8-14. Direct observation of the fractional QAH and associated effects enables research in charge fractionalization and anyonic statistics at zero magnetic field.

Composite Fermi Liquid at Zero Magnetic Field in Twisted MoTe_{2}

The pursuit of exotic phases of matter outside of the extreme conditions of a quantizing magnetic field is a long-standing quest of solid state physics. Recent experiments have observed spontaneous valley polarization and fractional Chern insulators in zero magnetic field in twisted bilayers of MoTe_{2}, at partial filling of the topological valence band (ν=-2/3 and -3/5). We study the topological valence band at half filling, using exact diagonalization and density matrix renormalization group calculations. We discover a composite Fermi liquid (CFL) phase even at zero magnetic field that covers a large portion of the phase diagram near twist angle ∼3.6°. The CFL is a non-Fermi liquid phase with metallic behavior despite the absence of Landau quasiparticles. We discuss experimental implications including the competition between the CFL and a Fermi liquid, which can be tuned with a displacement field. The topological valence band has excellent quantum geometry over a wide range of twist angles and a small bandwidth that is, remarkably, reduced by interactions. These key properties stabilize the exotic zero field quantum Hall phases. Finally, we present an optical signature involving "extinguished" optical responses that detects Chern bands with ideal quantum geometry.

Fractional Quantum Hall Effect from Frustration-Free Hamiltonians

We show that there is an emergent lattice description for the continuous fractional quantum Hall (FQH) systems, with a generalized set of few-body coherent states. In particular, model Hamiltonians of the FQH effect (FQHE) are equivalent to the real-space von Neumann lattice of local projection operators imposed on a continuous system in the thermodynamic limit. It can be analytically derived that tuning local one-body potentials in such lattices amounts to the tuning of individual two- or few-body pseudopotentials. For some cases, we can realize pure few-body pseudopotentials important for stabilizing exotic non-Abelian topological phases. Thus, this new approach can potentially lead to the experimental realization of coveted non-Abelian quantum fluids including the Moore-Read state and the Fibonacci state. The reformulation of the FQHE as a sum of local projections opens up new paths for rigorously proving the incompressibility of microscopic Hamiltonians in the thermodynamic limit.

Particle-Hole Symmetry and the Fractional Quantum Hall Effect in the Lowest Landau Level

We report on detailed experimental studies of a high-quality heterojunction insulated-gate field-effect transistor (HIGFET) to probe the particle-hole symmetry of the fractional quantum Hall effect (FQHE) states about half-filling in the lowest Landau level. The HIGFET is specially designed to vary the density of a two-dimensional electronic system under constant magnetic fields. We find in our constant magnetic field, variable density measurements that the sequence of FQHE states at filling factors ν=1/3,2/5,3/7… and its particle-hole conjugate states at filling factors 1-ν=2/3,3/5,4/7… have a very similar energy gap. Moreover, a reflection symmetry can be established in the magnetoconductivities between the ν and 1-ν states about half-filling. Our results demonstrate that the FQHE states in the lowest Landau level are manifestly particle-hole symmetric.

Generalization of Laughlin's theory for the fractional quantum Hall effect

Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor [Formula: see text] (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelling an electron into a different Hilbert subspace from the original Hilbert space of the Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions [Formula: see text], respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their 'conjugates', formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for [Formula: see text] sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transition between two topologically distinct phases each corresponding to a family of states: one consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.

Cyclotron Orbits of Composite Fermions in the Fractional Quantum Hall Regime

We study a bilayer GaAs hole system that hosts two distinct many-body phases at low temperatures and high perpendicular magnetic fields. The higher-density (top) layer develops a Fermi sea of composite fermions (CFs) in its half-filled lowest Landau level, while the lower-density (bottom) layer forms a Wigner crystal (WC) as its filling becomes very small. Owing to the interlayer interaction, the CFs in the top layer feel the periodic Coulomb potential of the WC in the bottom layer. We measure the magnetoresistance of the top layer while changing the bottom-layer density. As the WC layer density increases, the resistance peaks separating the adjacent fractional quantum Hall states in the top layer change nonmonotonically and attain maximum values when the cyclotron orbit of the CFs encloses one WC lattice point. These features disappear at T=275  mK when the WC melts. The observation of such geometric resonance features is unprecedented and surprising as it implies that the CFs retain a well-defined cyclotron orbit and Fermi wave vector even deep in the fractional quantum Hall regime, far from half-filling.

Nonlinear Conductivity and Collective Charge Excitations in the Lowest Landau Level

For weakly disordered fractional quantum Hall phases, the nonlinear photoconductivity is related to the charge susceptibility of the clean system by a Floquet boost. Thus, it may be possible to probe collective charge modes at finite wave vectors by electrical transport. Incompressible phases, irradiated at slightly above the magnetoroton gap, are predicted to exhibit negative photoconductivity and zero resistance states with spontaneous internal electric fields. Nonlinear conductivity can probe composite fermions' charge excitations in compressible filling factors.

Geometric Resonance of Composite Fermions near Bilayer Quantum Hall States

Via the application of a parallel magnetic field, we induce a single-layer to bilayer transition in two-dimensional electron systems confined to wide GaAs quantum wells and study the geometric resonance of composite fermions (CFs) with a periodic density modulation in our samples. The measurements reveal that CFs exist close to bilayer quantum Hall states, formed at Landau level filling factors ν=1 and 1/2. Near ν=1, the geometric resonance features are consistent with half the total electron density in the bilayer system, implying that CFs prefer to stay in separate layers and exhibit a two-component behavior. In contrast, close to ν=1/2, CFs appear single-layer-like (single component) as their resonance features correspond to the total density.

Bosonic Analogue of Dirac Composite Fermi Liquid

We introduce a particle-hole-symmetric metallic state of bosons in a magnetic field at odd-integer filling. This state hosts composite fermions whose energy dispersion features a quadratic band touching and corresponding 2π Berry flux protected by particle-hole and discrete rotation symmetries. We also construct an alternative particle-hole symmetric state-distinct in the presence of inversion symmetry-without Berry flux. As in the Dirac composite Fermi liquid introduced by Son [Phys. Rev. X 5, 031027 (2015)], breaking particle-hole symmetry recovers the familiar Chern-Simons theory. We discuss realizations of this phase both in 2D and on bosonic topological insulator surfaces, as well as signatures in experiments and simulations.

Explicit Derivation of Duality between a Free Dirac Cone and Quantum Electrodynamics in (2+1) Dimensions

We explicitly derive the duality between a free electronic Dirac cone and quantum electrodynamics in (2+1) dimensions (QED_{3}) with N=1 fermion flavors. The duality proceeds via an exact, nonlocal mapping from electrons to dual fermions with long-range interactions encoded by an emergent gauge field. This mapping allows us to construct parent Hamiltonians for exotic topological-insulator surface phases, derive the particle-hole-symmetric field theory of a half-filled Landau level, and nontrivially constrain QED_{3} scaling dimensions. We similarly establish duality between bosonic topological insulator surfaces and N=2 QED_{3}.

Luttinger Theorem for the Strongly Correlated Fermi Liquid of Composite Fermions

While an ordinary Fermi sea is perturbatively robust to interactions, the paradigmatic composite-fermion (CF) Fermi sea arises as a nonperturbative consequence of emergent gauge fields in a system where there was no Fermi sea to begin with. A mean-field picture suggests two Fermi seas, of composite fermions made from electrons or holes in the lowest Landau level, which occupy different areas away from half filling and thus appear to represent distinct states. Using the microscopic theory of composite fermions, which satisfies particle-hole symmetry in the lowest Landau level to an excellent approximation, we show that the Fermi wave vectors at filling factors ν and 1-ν are equal when expressed in units of the inverse magnetic length, and are generally consistent with the experimental findings of Kamburov et al. [Phys. Rev. Lett. 113, 196801 (2014)]. Our calculations suggest that the area of the CF Fermi sea may slightly violate the Luttinger area rule.

Composite fermions with a warped Fermi contour

Via measurements of commensurability features near the Landau filling factor ν=1/2, we probe the shape of the Fermi contour for hole-flux composite fermions confined to a wide GaAs quantum well. The data reveal that the composite fermions are strongly influenced by the characteristics of the Landau level in which they are formed. In particular, their Fermi contour is warped when their Landau level originates from a hole band with significant warping.

What determines the Fermi wave vector of composite fermions?

Composite fermions (CFs), exotic particles formed by pairing an even number of flux quanta to each electron, provide a fascinating description of phenomena exhibited by interacting two-dimensional electrons at high magnetic fields. At and near Landau level filling ν=1/2, CFs occupy a Fermi sea and exhibit commensurability effects when subjected to a periodic potential modulation. We observe a pronounced asymmetry in the magnetic field positions of the commensurability resistance minima of CFs with respect to the field at ν=1/2. This unexpected asymmetry is consistent with the CFs' Fermi wave vector being determined by the minority carriers in the lowest Landau level, and suggests a breaking of the particle-hole symmetry for CFs near ν=1/2.

Quantum anomalous Hall insulator of composite fermions

We show that a weak hexagonal periodic potential can transform a two-dimensional electron gas with an even-denominator magnetic filling factor into an quantum anomalous hall insulator of composite fermions, giving rise to the fractionally quantized Hall effect. The system provides a realization of the Haldane honeycomb-net model, albeit in a composite fermion system. We further propose a trial wave function for the state, and numerically evaluate its relative stability against the competing Hofstadter state. Possible sets of experimental parameters are proposed.

Composite fermions with tunable Fermi contour anisotropy

The composite fermion formalism elegantly describes some of the most fascinating behaviors of interacting two-dimensional carriers at low temperatures and in strong perpendicular magnetic fields. In this framework, carriers minimize their energy by attaching two flux quanta and forming new quasiparticles, the so-called composite fermions. Thanks to the flux attachment, when a Landau level is half-filled, the composite fermions feel a vanishing effective magnetic field and possess a Fermi surface with a well-defined Fermi contour. Our measurements in a high-quality two-dimensional hole system confined to a GaAs quantum well demonstrate that a parallel magnetic field can significantly distort the hole-flux composite fermion Fermi contour.

Commensurability oscillations of hole-flux composite fermions

We report the observation of commensurability oscillations of hole-flux composite fermions near a filling factor of ν=1/2 in a high-mobility two-dimensional hole system confined to a GaAs quantum well and subjected to a weak, strain-induced, unidirectional periodic potential modulation. The oscillations, which are consistent with ballistic transport of fully spin-polarized composite fermions in a weak periodic effective magnetic field, are surprisingly strong and exhibit up to third-order minima. We extract a ballistic mean-free path of about 0.1 μm for the hole-flux composite fermions.

Global phase diagram of the fractional quantum Hall effect arising from repulsive three-body interactions

The model of fermions in a magnetic field interacting via a purely three-body repulsive interaction has attracted interest because it produces, in the limit of short range interaction, the Pfaffian state with non-Abelian excitations. We show that this is part of a rich phase diagram containing a host of fractional quantum Hall states, a composite fermion Fermi sea, and a pairing transition. This is entirely unexpected, because the appearance of composite fermions and fractional quantum Hall effect is ordinarily thought to be a result of strong two-body repulsion. Recent breakthroughs in ultracold atoms have facilitated the realization of such a system, where this physics can be tested.

Dynamic nuclear polarization in the fractional quantum Hall regime

We investigate dynamic nuclear polarization in quantum point contacts (QPCs) in the integer and fractional quantum Hall regimes. Following the application of a dc bias, fractional plateaus in the QPC shift symmetrically about half filling of the lowest Landau level, ν=1/2, suggesting an interpretation in terms of composite fermions. Polarizing and detecting at different filling factors indicates that Zeeman energy is reduced by the induced nuclear polarization. Mapping effects from integer to fractional regimes extends the composite fermion picture to include hyperfine coupling.

Beyond the Fermi liquid paradigm: hidden Fermi liquids

An intense investigation of possible non-Fermi liquid states of matter has been inspired by two of the most intriguing phenomena discovered in the past quarter century, namely, high-temperature superconductivity and the fractional quantum Hall effect. Despite enormous conceptual strides, these two fields have developed largely along separate paths. Two widely employed theories are the resonating valence bond theory for high-temperature superconductivity and the composite fermion theory for the fractional quantum Hall effect. The goal of this perspective article is to note that they subscribe to a common underlying paradigm: They both connect these exotic quantum liquids to certain ordinary Fermi liquids residing in unphysical Hilbert spaces. Such a relation yields numerous nontrivial experimental consequences, exposing these theories to rigorous and definitive tests.

Nature of the spin transition in the half-filled Landau level

We report transport and nuclear spin relaxation studies of a density tunable two-dimensional electron system at filling nu=1/2 in tilted magnetic fields. The transition from partial to full spin polarization with an in-plane field leaves a clear signature in the resistance. Nuclear spin relaxation studies suggest that puddles of minority spins are responsible for an observed non-Korringa temperature dependence. This inhomogeneous spin polarization, similarly encountered in manganites where it strongly affects resistance, may help with understanding the spin dependent transport at nu=1/2.

Dispersion of the composite-fermion cyclotron-resonance mode

The dispersion of the composite-fermion cyclotron-resonance mode is measured with an optical detection scheme under the combined excitation of microwave radiation and a surface acoustic wave from an interdigital transducer. The slowly traveling surface wave defines the transferred wave vector. Momenta up to 10;{8} m;{-1} are accessible. The cyclotron-resonance mode exhibits strong negative dispersion, which suggests predominant short range residual interaction among composite fermions. From an extrapolation, the cyclotron mass at k=0 is obtained and investigated as a function of electron density.

Composite-fermion spin excitations as nu approaches 1/2: interactions in the Fermi sea

Measurements of low-lying spin excitations by inelastic light scattering unveil a delicate balance between spin reversal and Fermi energies in the Fermi sea of composite fermions that emerges in the limit of nu --> 1/2. The interplays between these two fundamental quasiparticle interactions are uncovered in lowest spin-flip excitations in which the spin orientation and Landau level index of composite fermions change simultaneously. A collapse of the spin-flip excitation gap as nu --> 1/2 is linked to vanishing quasiparticle energy level spacings and loss of full spin polarization.

Half-filled lowest Landau level on a thin torus

We solve a model that describes an interacting electron gas in the half-filled lowest Landau level on a thin torus, with radius of the order of the magnetic length. The low-energy sector consists of noninteracting, one-dimensional, neutral fermions. The ground state, which is homogeneous, is the Fermi sea obtained by filling the negative energy states, and the excited states are gapless neutral excitations out of this one-dimensional sea. Although the limit considered is extreme, the solution has a striking resemblance to the composite fermion description of the bulk nu=1/2 state-the ground state is homogeneous and the excitations are neutral and gapless. This suggests a one-dimensional Luttinger liquid description, with possible observable effects in transport experiments, of the bulk state where it develops continuously from the state on a thin torus as the radius increases.

Detecting spin-polarized currents in ballistic nanostructures

We demonstrate a mesoscopic spin polarizer/analyzer system that allows the spin polarization of current from a quantum point contact in a large in-plane magnetic field to be measured. A transverse electron focusing geometry is used to couple current from an emitter point contact into a collector point contact. At large in-plane fields, with the point contacts biased to transmit only a single spin (g<e(2)/h), the voltage across the collector depends on the spin polarization of the current incident on it. Spin polarizations of >70% are found for both emitter and collector at 300 mK and 7 T in-plane field.

Theory of quantum Hall nematics

Transport measurements on two-dimensional electron systems in moderate magnetic fields suggest the existence of a spontaneously orientationally ordered, compressible liquid state. We develop and analyze a microscopic theory of such a "quantum Hall nematic" (QHN) phase, predict the existence of a novel, highly anisotropic q(3) density-director mode, find that the T = 0 long-range orientational order is unstable to weak disorder, and compute the tunneling into such a strongly correlated state. This microscopic approach is supported and complemented by a hydrodynamic model of the QHN, which, in the dissipationless limit, reproduces the modes of the microscopic model.

Possible pairing-induced even-denominator fractional quantum Hall effect in the lowest Landau level

We report on our theoretical investigations that point to the possibility of a fractional quantum Hall effect with partial spin polarization at nu = 3/8. The physics of the incompressible state proposed here involves p-wave pairing of composite fermions in the spin reversed sector. The temperature and magnetic field regimes for the realization of this state are estimated.

Cyclotron resonance of composite fermions

It is occasionally possible to interpret strongly interacting many-body systems within a single-particle framework by introducing suitable fictitious entities, or 'quasi-particles'. A notable recent example of the successful application of such an approach is for a two-dimensional electron system that is exposed to a strong perpendicular magnetic field. The conduction properties of the system are governed by electron-electron interactions, which cause the fractional quantum Hall effect. Composite fermions, electrons that are dressed with magnetic flux quanta pointing opposite to the applied magnetic field, were identified as apposite quasi-particles that simplify our understanding of the fractional quantum Hall effect. They precess, like electrons, along circular cyclotron orbits, but with a diameter determined by a reduced effective magnetic field. The frequency of their cyclotron motion has hitherto remained enigmatic, as the effective mass is no longer related to the band mass of the original electrons and is entirely generated from electron-electron interactions. Here we demonstrate enhanced absorption of a microwave field in the composite fermion regime, and interpret it as a resonance with the frequency of their circular motion. From this inferred cyclotron resonance, we derive a composite fermion effective mass that varies from 0.7 to 1.2 times that of the electron mass in vacuum as their density is tuned from 0.6 x 10(11) cm(-2) to 1.2 x 10(11) cm(-2).

Stripe formation in the fractional quantum Hall regime

An understanding of the physics of the half-filled lowest Landau level has been achieved in terms of a Fermi sea of composite fermions, but the nature of the state at other even-denominator fractions has remained unclear. We investigate theoretically Landau level fillings of the form nu = (2n + 1)/(4n + 4), which correspond to composite-fermion fillings nu* = n + 1/2. By considering various plausible candidate states with complete spin polarization, we rule out the composite-fermion Fermi sea and paired composite-fermion state at these filling factors, and predict that the system phase separates into stripes of n and n + 1 filled Landau levels of composite fermions.

Optical investigation of spin-wave excitations in fractional quantum hall states and of interaction between composite fermions

We demonstrate that the temperature dependence of the electron spin polarization for the fractional states nu = 1/3 and nu = 2/3 displays activated behavior. This study enables the first measurement of the fractional quantum Hall spin-flip gaps. They are found to be systematically larger in comparison with the gaps simultaneously measured in transport. For nu = 1/3 and nu = 1/2, these spin-flip gaps allow the determination of the composite fermion interaction energy. This energy is investigated as a function of the finite width of the 2D channel.

Missing 2k(F) response for composite fermions in phonon drag

The response of composite fermions to large wave vector scattering has been studied through phonon drag measurements. While the response retains qualitative features of the electron system at zero magnetic field, notable discrepancies develop as the system is varied from a half-filled Landau level by changing density or field. These deviations, which appear to be inconsistent with the current picture of composite fermions, are absent if half filling is maintained while changing density. There remains, however, a clear deviation from the temperature dependence anticipated for 2k(F) scattering.

Cooper instability of composite fermions

When confined to two dimensions and exposed to a strong magnetic field, electrons screen the Coulomb interaction in a topological fashion; they capture an even number of quantum vortices and transform into particles called 'composite fermions' (refs 1-3). The fractional quantum Hall effect occurs in such a system when the ratio (or 'filling factor, nu) of the number of electrons and the degeneracy of their spin-split energy states (the Landau levels) takes on particular values. The Landau level filling nu = 1/2 corresponds to a metallic state in which the composite fermions form a gapless Fermi sea. But for nu = 5/2, a fractional quantum Hall effect is observed instead; this unexpected result is the subject of considerable debate and controversy. Here we investigate the difference between these states by considering the theoretical problem of two composite fermions on top of a fully polarized Fermi sea of composite fermions. We find that they undergo Cooper pairing to form a p-wave bound state at nu = 5/2, but not at nu = 1/2. In effect, the repulsive Coulomb interaction between electrons is overscreened in the nu = 5/2 state by the formation of composite fermions, resulting in a weak, attractive interaction.