Wave function multifractality and dephasing at metal–insulator and quantum Hall transitions

We analyze the critical behavior of the dephasing rate induced by short-range electron–electron interaction near an Anderson transition of metal–insulator or quantum Hall type. The corresponding exponent characterizes the scaling of the transition width with temperature. Assuming no spin degeneracy, the critical behavior can be studied by performing the scaling analysis in the vicinity of the non-interacting fixed point, since the latter is stable with respect to the interaction. We combine an analytical treatment (that includes the identification of operators responsible for dephasing in the formalism of the non-linear sigma-model and the corresponding renormalization-group analysis in 2 + ? dimensions) with numerical simulations on the Chalker–Coddington network model of the quantum Hall transition. Finally, we discuss the current understanding of the Coulomb interaction case and the available experimental data.     

Localization length at the conductivity minima of the quantum Hall effect

The quantum localization is known to be responsible for the deep conductivity minima of the quantum Hall effect. In this paper we calculate the localization length as a function of magnetic field at such minima for several models of disorder (“white-noise”, short-range, and long-range random potentials). We find that with the exponent between one and , depending on the model. In particular, for the “white-noise” random potential roughly coincides with the classical cyclotron radius. Our results are in agreement with available experimental data.     

Towards a field theory of the plateau transitions in the integer quantum Hall effect

We suggest a procedure for calculating correlation functions of the local densities of states (DOS) at the plateau transitions in the integer quantum Hall effect (IQHE). We argue that their correlation functions are appropriately described in terms of the SL( )/SU(2) WZNW model (at the usual Ka –Moody point and with the level 6≤k≤8 ). In this model we have identified the operators corresponding to the local DOS, and derived the partial differential equation determining their correlation functions. The OPEs for powers of the local DOS obtained from this equation are in agreement with available results.     

A new network model for the integer quantum Hall effect

A Landauer–Büttiker-type formulation of backscattering between pairs of opposite directed channels is used to describe the coupling at the nodes of a network. Physically, these nodes correspond to saddle points of a slowly varying lateral potential modulation in a 2D electron system in the high magnetic field regime. We show that the network can be solved without needing a transfer matrix as used by Chalker and Coddington. We use an exponential dependence of the coupling on the filling factor of the associated Landau level. We demonstrate that our network representation allows a quantitative modeling of almost every realistic sample geometry in the quantum Hall regime, including the effect of gate electrodes across a Hall bar.     

Integer quantum Hall effect in a PbTe quantum well

Transport measurements have been carried out on a 10 nm n-type PbTe/Pb_0.9Eu_0.1Te quantum well at millikelvin temperatures. The Hall and longitudinal resistances are measured in a Van der Pauw geometry under high magnetic fields up to 23 T. A robust signature of the integer quantum Hall effect is observed without any sign of parasitic parallel conduction. The unconventional sequence of filling factors associated with the integer quantum Hall effect is discussed in terms of the occupancy of multiple valleys.     

Particle–hole symmetric Luttinger liquids in a quantum Hall circuit

We report finite-bias differential conductance measurements through a split-gate constriction in the integer quantum Hall regime at ν=1. Both enhanced and suppressed zero-bias inter-edge backscattering can be obtained in a controllable way by changing the split-gate voltage. This behavior is interpreted in terms of local charge depletion and particle–hole symmetry. We discuss the relevance of particle–hole symmetry in connection with the chiral Luttinger model of edge states.     

Detection of hidden localized states by the quantum Hall effect in graphene

We fabricated a monolayer graphene transistor device in the shape of the Hall-bar structure, which produced an exactly symmetric signal following the sample geometry. During electrical characterization, the device showed the standard integer quantum Hall effect of monolayer graphene except for a broader range of several quantum Hall plateaus corresponding to small filling factors in the electron region. We investigated this anomaly on the basis of localized states owing to the presence of possible electron traps, whose energy levels were estimated to be near the Dirac point. In particular, the inequality between the filling of electrons and holes was ascribed to the requirement of excess electrons to fill the trap levels. The relations between the quantum Hall plateau, Landau level, and filling factor were carefully analyzed to reveal the details of the localized states in this graphene device.     

The enigma of the quantum Hall effect in graphene

We apply Laughlin’s gauge argument to analyze the ν=0 quantum Hall effect observed in graphene when the Fermi energy lies near the Dirac point, and conclude that this necessarily leads to divergent bulk longitudinal resistivity in the zero temperature thermodynamic limit. We further predict that in a Corbino geometry measurement, where edge transport and other mesoscopic effects are unimportant, one should find the longitudinal conductivity vanishing in all graphene samples which have an underlying ν=0 quantized Hall effect. We argue that this ν=0 graphene quantum Hall state is qualitatively similar to the high field insulating phase (also known as the Hall insulator) in the lowest Landau level of ordinary semiconductor two-dimensional electron systems. We establish the necessity of having a high magnetic field and high mobility samples for the observation of the divergent resistivity as arising from the existence of disorder-induced density inhomogeneity at the graphene Dirac point.     

Theory of the integer quantum Hall effect in graphene

A Hall resistivity formula for the 2DES in graphene is derived from the zero-mass Dirac field model adopting the electron reservoir hypothesis. The formula reproduces perfectly the experimental resistivity data [K.S. Novoselov, et al., Nature 438 (2005) 201]. This perfect agreement cannot be achieved by any other existing models. The electron reservoir is shown to be the 2DES itself.     

Observation of the quantum Hall effect in cleaved InAs surfaces

We have performed the in-plane magnetotransport measurements on the two-dimensional electron gas at the cleaved p-InAs (1 1 0) surface by deposition of Ag. The surface electron density N_s is determined from the Hall coefficient at . The coverage dependence of N_s is well explained by the assumption that each adsorbed Ag atom denotes one electron into InAs until the surface Fermi level reaches the adsorbate-induced donor level. The electron mobility μ is about and does not show a clear dependence on the coverage over . In the high-magnetic field regime of B>1/μ, Shubnikov–de Hass oscillations were observed. A beating pattern due to the strong spin–orbit interaction appears for high N_s. For lower N_s of , an apparent quantum Hall plateau for ν=4 and vanishing of the longitudinal resistivity were observed around .     

An alternative formulation of Hall effect and quantum phases in noncommutative space

A recent method of constructing quantum mechanics in noncommutative coordinates, alternative to implying noncommutativity by means of star product is discussed. Within this approach we study Hall effect as well as quantum phases in noncommutative coordinates. The θ-deformed phases which we obtain are velocity independent.     

Anyons and the quantum Hall effect—A pedagogical review

The dichotomy between fermions and bosons is at the root of many physical phenomena, from metallic conduction of electricity to super-fluidity, and from the periodic table to coherent propagation of light. The dichotomy originates from the symmetry of the quantum mechanical wave function to the interchange of two identical particles. In systems that are confined to two spatial dimensions particles that are neither fermions nor bosons, coined “anyons”, may exist. The fractional quantum Hall effect offers an experimental system where this possibility is realized. In this paper we present the concept of anyons, we explain why the observation of the fractional quantum Hall effect almost forces the notion of anyons upon us, and we review several possible ways for a direct observation of the physics of anyons. Furthermore, we devote a large part of the paper to non-abelian anyons, motivating their existence from the point of view of trial wave functions, giving a simple exposition of their relation to conformal field theories, and reviewing several proposals for their direct observation.     

Three-dimensional quantum Hall effect in Weyl and double-Weyl semimetals

The well-known quantum Hall effect (QHE) was usually studied in 2D systems. In this work, we investigate the integer QHE in 3D Weyl and double-Weyl semimetals. Based on the lattice models of Weyl and double-Weyl semimetals subjected to a uniform magnetic field, we derive the generalized 3D spinfull Hofstadter Hamiltonians and Harper equations for the two systems, and obtain their corresponding energy spectra. Furthermore, we show that for proper hopping parameters and rational magnetic fluxes, both systems exhibit the 3D QHE when the Fermi level lies in some band gaps. The 3D QHE is topologically characterized by three Chern numbers with one or two nonzero Chern values which are respectively defined for three crystal planes. The possible experimental realization and detection of the 3D QHE are also discussed.     

Bulk and edge properties of the Chern-Simons Ginzburg-Landau theory for the fractional quantum Hall effect

The Chern-Simons Ginzburg-Landau theory for the fractional quantum Hall effect is studied in the presence of a confining potential We review the bulk properties of the model and discuss how the plateau formation emerges without any impurity potential. The effect is related to changes, by accumulation of charge, at the edge when the chemical potential is changed. Fluctuations about the ground state are examined and an expression is found for the velocity of the massless edge mode in terms of the confining potential. The effect of including spin is examined for the case when the system is fully polarized in the bulk. In general a spin texture may appear at the edge, and we examine this effect in the case of a small spin-down component. The low-frequency edge modes are examined and a third-order equation is found for velocities which indicates the presence of three different modes. The discussions are illustrated by numerical studies of the ground states, both for the one- and two-component cases.     

Universal transverse conductance between quantum Hall regions and (2+1) D bosonization

Using bosonization techniques for (2+1) D systems, we show that the transverse conductance for a system with general current interactions, when measured between perfect Hall regions is not renormalized at low temperatures. Our method extends two results we have recently obtained on low dimensional fermionic systems: on the one hand, the relationship between universality of Landauer conductance and universality of bosonization rules for (1+1) D systems, and on the other hand, the universal character of the bosonized topological current associated to a (2+1) D fermionic system with current interactions.     

Electrostatic theory for imaging experiments on local charges in quantum Hall systems

We use a simple electrostatic treatment to model recent experiments on quantum Hall systems, in which charging of localised states by addition of integer or fractionally charged quasiparticles is observed. Treating the localised state as a compressible quantum dot or antidot embedded in an incompressible background, we calculate the electrostatic potential in its vicinity as a function of its charge, and the chemical potential values at which its charge changes. The results offer a quantitative framework for analysis of the observations.     

Experimental probes of emergent symmetries in the quantum Hall system

Experiments studying renormalization group flows in the quantum Hall system provide significant evidence for the existence of an emergent holomorphic modular symmetry Γ₀(2)${\Gamma }_0(2)$. We briefly review this evidence and show that, for the lowest temperatures, the experimental determination of the position of the quantum critical points agrees to the parts per mille   level with the prediction from Γ₀(2)${\Gamma }_0(2)$. We present evidence that experiments giving results that deviate substantially from the symmetry predictions are not cold enough to be in the quantum critical domain. We show how the modular symmetry extended by a non-holomorphic particle–hole duality leads to an extensive web of dualities related to those in plateau–insulator transitions, and we derive a formula relating dual pairs (B,Bd)$(B,B_d)$ of magnetic field strengths across any transition. The experimental data obtained for the transition studied so far is in excellent agreement with the duality relations following from this emergent symmetry, and rule out the duality rule derived from the “law of corresponding states”. Comparing these generalized duality predictions with future experiments on other transitions should provide stringent tests of modular duality deep in the non-linear domain far from the quantum critical points.     

Elliptic theta functions and the fractional quantum Hall effect

Algebro-geometric methods are applied to the theoretical understanding of the fractionary quantum Hall effect on a periodic lattice. The fermionic Fock space of the many-electron system is precisely identified, and as a consequence, the variational Haldane-Rezayi ground state is decomposed in terms of one-particle wave functions at the first Landau level; the filling factor is thus analytically computed. Quasi-hole and quasi-particle excitations are also analyzed. The center of mass dynamics is described in terms of a section in a very subtle stable vector bundle. The Hall conductance arises as a topological invariant; namely, the slope of the vector bundle previously mentioned.     

Current breakdown of the integer and fractional quantum Hall effects detected by torque magnetometry

We have developed a novel technique that enables measurements of the breakdown of both the integer and fractional quantum Hall effects in a two-dimensional electron system without the need to contact the sample. The critical Hall electric fields that we measure are significantly higher than those reported by other workers, and support the quasi-elastic inter-Landau-level tunnelling model of breakdown. Comparison of the fractional quantum Hall effect results with those obtained on the integer quantum Hall effect allows the fractional quantum Hall effect energy gap to be determined and provides a test of the composite-fermion theory. The temperature dependence of the critical current gives an insight into the mechanism by which momentum may be conserved during the breakdown process.     

Electron–hole states in the fractional quantum Hall regime probed by photoluminescence

The electron–hole states in the fractional quantum Hall regime is investigated with a back-gated undoped quantum well by photoluminesccence in magnetic fields. The evolution of the photoluminescence spectra is discussed depending on the electron density. We find anomalies of the photoluminescence at the integer as well as the fractional filling factors.