Phase diagram of superconductivity in the integer quantum Hall regime.

An interplay between pairing and topological orders has been predicted to give rise to superconducting states supporting exotic emergent particles, such as Majorana particles obeying non-Abelian braid statistics. We consider a system of spin polarized electrons on a Hofstadter lattice with nearest-neighbor attractive interaction and solve the mean-field Bogoliubov-de Gennes equations in a self-consistent fashion, leading to gauge-invariant observables and a rich phase diagram as a function of the chemical potential, the magnetic field, and the interaction. As the strength of the attractive interaction is increased, the system first makes a transition from a quantum Hall phase to a skyrmion lattice phase that is fully gapped in the bulk but has topological chiral edge current, characterizing a topologically nontrivial state. This is followed by a vortex phase in which the vortices carrying Majorana modes form a lattice; the spectrum contains a low-energy Majorana band arising from the coupling between neighboring vortex-core Majorana modes but does not have chiral edge currents. For some parameters, a dimer vortex lattice occurs with no Majorana band. The experimental feasibility and the observable consequences of skyrmions as well as Majorana modes are indicated.

Erratum: Anderson Localization in the Fractional Quantum Hall Effect [Phys. Rev. Lett. 128, 116801 (2022)].

This corrects the article DOI: 10.1103/PhysRevLett.128.116801.

Composite Fermion Pairing Induced by Landau Level Mixing.

Pairing of composite fermions provides a possible mechanism for fractional quantum Hall effect at even denominator fractions and is believed to serve as a platform for realizing quasiparticles with non-Abelian braiding statistics. We present results from fixed-phase diffusion Monte Carlo calculations which predict that substantial Landau level mixing can induce a pairing of composite fermions at filling factors ν=1/2 and ν=1/4 in the l=-3 relative angular momentum channel, thereby destabilizing the composite-fermion Fermi seas to produce non-Abelian fractional quantum Hall states.

Fractional Quantum Hall Effect with Unconventional Pairing in Monolayer Graphene.

Motivated by the observation of even denominator fractional quantum Hall effect in the n=3 Landau level of monolayer graphene [Kim et al., Nat. Phys. 15, 154 (2019)NPAHAX1745-247310.1038/s41567-018-0355-x], we consider a Bardeen-Cooper-Schrieffer variational state for composite fermions and find that the composite-fermion Fermi sea in this Landau level is unstable to an f-wave pairing. Analogous calculation suggests the possibility of a p-wave pairing of composite fermions at half filling in the n=2 graphene Landau level, whereas no pairing instability is found at half filling in the n=0 and n=1 graphene Landau levels. The relevance of these results to experiments is discussed.

Anderson Localization in the Fractional Quantum Hall Effect.

The interplay between interaction and disorder-induced localization is of fundamental interest. This article addresses localization physics in the fractional quantum Hall state, where both interaction and disorder have nonperturbative consequences. We provide compelling theoretical evidence that the localization of a single quasiparticle of the fractional quantum Hall state at filling factor ν=n/(2n+1) has a striking quantitative correspondence to the localization of a single electron in the (n+1)th Landau level. By analogy to the dramatic experimental manifestations of Anderson localization in integer quantum Hall effect, this leads to predictions in the fractional quantum Hall regime regarding the existence of extended states at a critical energy, and the nature of the divergence of the localization length as this energy is approached. Within a mean field approximation, these results can be extended to situations where a finite density of quasiparticles is present.

Exactly Solvable Model for Strongly Interacting Electrons in a Magnetic Field.

States of strongly interacting particles are of fundamental interest in physics and can produce exotic emergent phenomena and topological structures. We consider here two-dimensional electrons in a magnetic field, and, departing from the standard practice of restricting to the lowest LL, introduce a model short-range interaction that is infinitely strong compared to the cyclotron energy. We demonstrate that this model lends itself to an exact solution for the ground as well as excited states at arbitrary filling factors ν<1/2p and produces a fractional quantum Hall effect at fractions of the form ν=n/(2pn+1), where n and p are integers. The fractional quantum Hall states of our model share many topological properties with the corresponding Coulomb ground states in the lowest Landau level, such as the edge physics and the fractional charge of the excitations.

Kohn-Sham Theory of the Fractional Quantum Hall Effect.

We formulate the Kohn-Sham (KS) equations for the fractional quantum Hall effect by mapping the original electron problem into an auxiliary problem of composite fermions that experience a density dependent effective magnetic field. Self-consistent solutions of the KS equations demonstrate that our formulation captures not only configurations with nonuniform densities but also topological properties such as fractional charge and fractional braid statistics for the quasiparticles excitations. This method should enable a realistic modeling of the edge structure, the effect of disorder, spin physics, screening, and of fractional quantum Hall effect in mesoscopic devices.

Prediction of a Non-Abelian Fractional Quantum Hall State with f-Wave Pairing of Composite Fermions in Wide Quantum Wells.

We theoretically investigate the nature of the state at the quarter filled lowest Landau level and predict that, as the quantum well width is increased, a transition occurs from the composite fermion Fermi sea into a novel non-Abelian fractional quantum Hall state that is topologically equivalent to f-wave pairing of composite fermions. This state is topologically distinct from the familiar p-wave paired Pfaffian state. We compare our calculated phase diagram with experiments and make predictions for many observable quantities.

Crystallization in the Fractional Quantum Hall Regime Induced by Landau-Level Mixing.

The interplay between strongly correlated liquid and crystal phases for two-dimensional electrons exposed to a high transverse magnetic field is of fundamental interest. Through the nonperturbative fixed-phase diffusion Monte Carlo method, we determine the phase diagram of the Wigner crystal in the ν-κ plane, where ν is the filling factor and κ is the strength of Landau-level (LL) mixing. The phase boundary is seen to exhibit a striking ν dependence, with the states away from the magic filling factors ν=n/(2pn+1) being much more susceptible to crystallization due to Landau-level mixing than those at ν=n/(2pn+1). Our results explain the qualitative difference between the experimental behaviors observed in n- and p-doped gallium arsenide quantum wells and, in particular, the existence of an insulating state for ν<1/3 and also for 1/3<ν<2/5 in low-density p-doped systems. We predict that, in the vicinity of ν=1/5 and ν=2/9, increasing LL mixing causes a transition not into an ordinary electron Wigner crystal, but rather into a strongly correlated crystal of composite fermions carrying two vortices.

Fractional Quantum Hall Effect at ν=2+6/13: The Parton Paradigm for the Second Landau Level.

The unexpected appearance of a fractional quantum Hall effect (FQHE) plateau at ν=2+6/13 [A. Kumar et al., Phys. Rev. Lett. 105, 246808 (2010)PRLTAO0031-900710.1103/PhysRevLett.105.246808] offers a clue into the physical mechanism of the FQHE in the second Landau level (SLL). Here we propose a "3[over ¯]2[over ¯]111" parton wave function, which is topologically distinct from the 6/13 state in the lowest Landau level. We demonstrate the 3[over ¯]2[over ¯]111 state to be a good candidate for the ν=2+6/13 FQHE, and make predictions for experimentally measurable properties that can reveal the nature of this state. Furthermore, we propose that the "n[over ¯]2[over ¯]111" family of parton states naturally describes many observed SLL FQHE plateaus.

Inter-Landau-level Andreev Reflection at the Dirac Point in a Graphene Quantum Hall State Coupled to a NbSe_{2} Superconductor.

Superconductivity and the quantum Hall effect are distinct states of matter occurring in apparently incompatible physical conditions. Recent theoretical developments suggest that the coupling of the quantum Hall effect with a superconductor can provide fertile ground for realizing exotic topological excitations such as non-Abelian Majorana fermions or Fibonacci particles. As a step toward that goal, we report observation of Andreev reflection at the junction of a quantum Hall edge state in a single layer graphene and a quasi-two-dimensional niobium diselenide (NbSe_{2}) superconductor. Our principal finding is the observation of an anomalous finite-temperature conductance peak located precisely at the Dirac point, providing a definitive evidence for inter-Landau-level Andreev reflection in a quantum Hall system. Our observations are well supported by detailed numerical simulations, which offer additional insight into the role of the edge states in Andreev physics. This study paves the way for investigating analogous Andreev reflection in a fractional quantum Hall system coupled to a superconductor to realize exotic quasiparticles.

Density-Functional Theory of the Fractional Quantum Hall Effect.

A conceptual difficulty in formulating the density-functional theory of the fractional quantum Hall effect is that while in the standard approach the Kohn-Sham orbitals are either fully occupied or unoccupied, the physics of the fractional quantum Hall effect calls for fractionally occupied Kohn-Sham orbitals. This has necessitated averaging over an ensemble of Slater determinants to obtain meaningful results. We develop an alternative approach in which we express and minimize the grand canonical potential in terms of the composite fermion variables. This provides a natural resolution of the fractional-occupation problem because the fully occupied orbitals of composite fermions automatically correspond to fractionally occupied orbitals of electrons. We demonstrate the quantitative validity of our approach by evaluating the density profile of fractional Hall edge as a function of temperature and the distance from the delta dopant layer and showing that it reproduces edge reconstruction in the expected parameter region.

Landau-Level Mixing and Particle-Hole Symmetry Breaking for Spin Transitions in the Fractional Quantum Hall Effect.

The spin transitions in the fractional quantum Hall effect provide a direct measure of the tiny energy differences between differently spin-polarized states and thereby serve as an extremely sensitive test of the quantitative accuracy of the theory of the fractional quantum Hall effect, and, in particular, of the role of Landau-level mixing in lifting the particle-hole symmetry. We report on an accurate quantitative study of this physics, evaluating the effect of Landau-level mixing in a nonperturbative manner using a fixed-phase diffusion Monte Carlo method. We find excellent agreement between our calculated critical Zeeman energies and the experimentally measured values. In particular, we find, as also do experiments, that the critical Zeeman energies for fractional quantum Hall states at filling factors ν=2-n/(2n±1) are significantly higher than those for ν=n/(2n±1), a quantitative signature of the lifting of particle-hole symmetry due to Landau-level mixing.

Observation of the Quantum Anomalous Hall Insulator to Anderson Insulator Quantum Phase Transition and its Scaling Behavior.

Fundamental insight into the nature of the quantum phase transition from a superconductor to an insulator in two dimensions, or from one plateau to the next or to an insulator in the quantum Hall effect, has been revealed through the study of its scaling behavior. Here, we report on the experimental observation of a quantum phase transition from a quantum-anomalous-Hall insulator to an Anderson insulator in a magnetic topological insulator by tuning the chemical potential. Our experiment demonstrates the existence of scaling behavior from which we extract the critical exponent for this quantum phase transition. We expect that our work will motivate much further investigation of many properties of quantum phase transition in this new context.

Total hip Replacement in Crowe type IV dysplastic hips - average 5 year follow-up and literature review.

This is a retrospective study of 29 patients (34 hips) of Crowe grade IV dysplastic hips aged between 19 and 75 years who underwent THR for osteoarthritis secondary to DDH. The hips were evaluated radiologically for Sharp's acetabular angle, cup inclination, loosening, and ectopic bone formation. Clinically the results were evaluated by pre and postoperative -Harris hip scoring. The mean acetabular angle was 60.8° (range, 45°-68°) preoperatively. In 18 hips, subtrochanteric femoral osteotomy was performed. Pre-operatively, the mean leg length discrepancy was 5 cm (range, 2-8 cm). -Correction within 1 cm was possible in all patients -except in 4 patients. The mean Harris hip Score was 40.80 (32-45.90) preoperatively and 87.96 (74.78-94.72) at last follow-up. THR is successful in high dislocation dysplastic hips. Although there is no gold standard technique of THR in dysplastic hips and treatment of each patient should be individualized. Level of evidence : IV.

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.

Zero-Field Dissipationless Chiral Edge Transport and the Nature of Dissipation in the Quantum Anomalous Hall State.

The quantum anomalous Hall (QAH) effect is predicted to possess, at a zero magnetic field, chiral edge channels that conduct a spin polarized current without dissipation. While edge channels have been observed in previous experimental studies of the QAH effect, their dissipationless nature at a zero magnetic field has not been convincingly demonstrated. By a comprehensive experimental study of the gate and temperature dependences of local and nonlocal magnetoresistance, we unambiguously establish the dissipationless edge transport. By studying the onset of dissipation, we also identify the origin of dissipative channels and clarify the surprising observation that the critical temperature of the QAH effect is 2 orders of magnitude smaller than the Curie temperature of ferromagnetism.

Long-Range Spin-Triplet Helix in Proximity Induced Superconductivity in Spin-Orbit-Coupled Systems.

We study proximity induced triplet superconductivity in a spin-orbit-coupled system, and show that the d vector of the induced triplet superconductivity undergoes precession that can be controlled by varying the relative strengths of Rashba and Dresselhaus spin-orbit couplings. In particular, a long-range spin-triplet helix is predicted when these two spin-orbit couplings have equal strengths. We also study the Josephson junction geometry and show that a transition between 0 and π junctions can be induced by controlling the spin-orbit coupling with a gate voltage. An experimental setup is proposed to verify these effects. Conversely, the observation of these effects can serve as a direct confirmation of triplet superconductivity.

Fractional angular momentum in cold-atom systems.

The quantum statistics of bosons or fermions are manifest through the even or odd relative angular momentum of a pair. We show theoretically that, under certain conditions, a pair of certain test particles immersed in a fractional quantum Hall state possesses, effectively, a fractional relative angular momentum, which can be interpreted in terms of fractional braid statistics. We propose that the fractionalization of the angular momentum can be detected directly through the measurement of the pair correlation function in rotating ultracold atomic systems in the fractional quantum Hall regime. Such a measurement will also provide direct evidence for the effective magnetic field resulting from Berry phases arising from attached vortices, and of excitations with a fractional particle number, analogous to the fractional charge of the electron fractional quantum Hall effect.

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.