Lifshitz Tails in Constant Magnetic Fields

We consider the 2D Landau Hamiltonian H perturbed by a random alloy-type potential, and investigate the Lifshitz tails, i.e. the asymptotic behavior of the corresponding integrated density of states (IDS) near the edges in the spectrum of H. If a given edge coincides with a Landau level, we obtain different asymptotic formulae for power-like, exponential sub-Gaussian, and super-Gaussian decay of the one-site potential. If the edge is away from the Landau levels, we impose a rational-flux assumption on the magnetic field, consider compactly supported one-site potentials, and formulate a theorem which is analogous to a result obtained by the first author and T. Wolff in [25] for the case of a vanishing magnetic field.     

Density of states of random Schrödinger operators with a uniform magnetic field

We consider the density of states of Schrödinger operators with a uniform magnetic field and a random potential with a Gaussian distribution. We show that the restriction to the states of the first Landau level is equivalent to a scaling limit where one looks at the density of states near to the energy of the first Landau level and simultaneously lets the strength of the coupling to the random potential go to zero. We also consider a different limit where we look at the suitably normalised density of states near to the energy of the first Landau level when the intensity of the magnetic field goes to infinity.     

Landau Hamiltonians with random potentials: Localization and the density of states

We prove the existence of localized states at the edges of the bands for the two-dimensional Landau Hamiltonian with a random potential, of arbitrary disorder, provided that the magnetic field is sufficiently large. The corresponding eigenfunctions decay exponentially with the magnetic field and distance. We also prove that the integrated density of states is Lipschitz continuous away from the Landau energies. The proof relies on a Wegner estimate for the finite-area magnetic Hamiltonians with random potentials and exponential decay estimates for the finitearea Green's functions. The proof of the decay estimates for the Green's functions uses fundamental results from two-dimensional bond percolation theory.Supported in part by CNRS.Supported in part by NSF grants INT 90-15895 and DMS 93-07438.Unité Propre de Recherche 7061.     

The Absolute Continuity of the Integrated Density¶of States for Magnetic Schrödinger Operators¶with Certain Unbounded Random Potentials

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr?dinger operator with magnetic field and a random potential which may be unbounded from above and below. In case that the magnetic field is constant and the random potential is ergodic and admits a so-called one-parameter decomposition, we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative, the density of states. This local Lipschitz continuity of the integrated density of states is derived by establishing a Wegner estimate for finite-volume Schr?dinger operators which holds for rather general magnetic fields and different boundary conditions. Examples of random potentials to which the results apply are certain alloy-type and Gaussian random potentials. Besides we show a diamagnetic inequality for Schr?dinger operators with Neumann boundary conditions. Received: 20 October 2000 / Accepted: 8 March 2001     

The effect of exchange interaction on quasiparticle Landau levels in narrow-gap quantum well heterostructures

Using the 'screened' Hartree-Fock approximation based on the eight-band k·p Hamiltonian, we have extended our previous work (Krishtopenko et al 2011 J. Phys.: Condens. Matter 23 385601) on exchange enhancement of the g-factor in narrow-gap quantum well heterostructures by calculating the exchange renormalization of quasiparticle energies, the density of states at the Fermi level and the quasiparticle g-factor for different Landau levels overlapping. We demonstrate that exchange interaction yields more pronounced Zeeman splitting of the density of states at the Fermi level and leads to the appearance of peak-shaped features in the dependence of the Landau level energies on the magnetic field at integer filling factors. We also find that the quasiparticle g-factor does not reach the maximum value at odd filling factors in the presence of large overlapping of spin-split Landau levels. We advance an argument that the behavior of the quasiparticle g-factor in weak magnetic fields is defined by a random potential of impurities in narrow-gap heterostructures.     

Broadening of the lowest Landau level by a Gaussian random potential with an arbitrary correlation length: an efficient continued-fraction approach

For an electron in the Euclidean plane subjected to a perpendicular constant magnetic field and a homogeneous Gaussian random potential with a Gaussian covariance function we approximate the averaged density of states restricted to the lowest Landau level. To this end, we extrapolate the first nine coefficients of the underlying continued fraction consistently with the coefficients’ high-order asymptotics. The latter derives from the known asymptotic decay of the density of states in the tails. We thus achieve on the one hand a reliable extension of Wegner’s exact result [Z. Phys. B 51, 279 (1983)] for the delta-correlated case to the physically more relevant case of a non-zero correlation length. On the other hand, we have thereby found a paragon for the power of continued-fraction expansions for designing approximations to spectral densities.     

The nature of the spectrum for a Landau Hamiltonian with delta impurities

We consider a single-band approximation to the random Schrödinger operator in an external magnetic field. The random potential consists of delta functions of random strengths situated on the sites of a regular two-dimensional lattice. We characterize the entire spectrum of this Hamiltonian when the magnetic field is sufficiently high. We show that the whole spectrum is pure point, the energy coinciding with the first Landau level in the absence of a random potential being infinitely degenerate, while the eigenfunctions corresponding to energies in the rest of the spectrum are localized.     

Quantum mechanical Hamiltonians with large ground-state degeneracy

Nonrelativistic Hamiltonians with large, even infinite, ground-state degeneracy are studied by connecting the degeneracy to the property of a Dirac operator. We then identify a special class of Hamiltonians, for which the full space of degenerate ground states in any spatial dimension can be exhibited explicitly. The two-dimensional version of the latter coincides with the Pauli Hamiltonian, and recently-discussed models leading to higher-dimensional Landau levels are obtained as special cases of the higher-dimensional version of this Hamiltonian. But, in our framework, it is only the asymptotic behavior of the background ‘potential’ that matters for the ground-state degeneracy. We work out in detail the ground states of the three-dimensional model in the presence of a uniform magnetic field and such potential. In the latter case one can see degenerate stacking of all 2d Landau levels along the magnetic field axis.     

Magneto-oscillations in a two-dimensional electron gas: A simplified analytic formulation

We derive a simplified thermodynamical formulation for a two-dimensional electron gas (2DEG) in a uniform magnetic field, with a large Landau level width and at low temperatures. Our analytic results clearly bring out dependences of magneto-oscillations on the Landau level broadening.1. In contrast with Gaussian broadening, different Landau levels do not overlap in the case of semi-elliptic density of states (Ando and Uemura, 1974), were such a normalization constant needs not to be introduced.     

Landau levels in a simple hexagonal bilayer graphene

A new approach, which makes the Hamiltonian of the Peierls tight-binding model change into a band matrix, is used to investigate the Landau levels in a AA-stacked bilayer graphene. The interlayer atomic hoppings could induce an energy gap, the asymmetry of the Landau levels about the chemical potential, the random variation in the level spacing, more fourfold degenerate Landau levels at low energy, and the oscillatory Landau levels and the complicated state degeneracies at moderate energy. For the low-energy Landau levels, their dependence on the quantum number and the field strength cannot be well characterized by a simple power law. They exhibit a anomalous oscillation during the variation of the magnetic field. The main features of the magnetoelectronic states are directly reflected in density of states.     

On the thermodynamics of a degenerate two-dimensional electron gas in a strong magnetic field: density of states

Systematic expansions, in powers ofB ^–1, for the free energy and the density of states, are derived for a two-dimensional degenerate electron gas in the presence of a strong magnetic field and an arbitrary potential. They are then applied to a system involving random impurities. Landau levels are shown to be broadened, with level widths related to the impurity concentration and potential. We show that level broadenings, induced by long range electron-impurity ineractions, do not depend on the magnetic field in the strong field limit, confirming the existing theories. But broadened Landau levels can have a large variety of shapes as one changes the impurity potential, distribution and concentration. Our theory, with a Gaussian potential, leads to a good agreement with the recent experiment on the de Haas-van Alphen effect in Br₂-graphite intercalation compounds     

Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities

We consider a random Schr?dinger operator in an external magnetic field. The random potential consists of delta functions of random strengths situated on the sites of a regular two-dimensional lattice. We characterize the spectrum in the lowest N Landau bands of this random Hamiltonian when the magnetic field is sufficiently strong, depending on N. We show that the spectrum in these bands is entirely pure point, that the energies coinciding with the Landau levels are infinitely degenerate and that the eigenfunctions corresponding to energies in the remainder of the spectrum are localized with a uniformly bounded localization length. By relating the Hamiltonian to a lattice operator we are able to use the Aizenman–Molchanov method to prove localization. Received: 1 June 1998 / Accepted: 29 January 1999     

Non-Abelian Chern-Simons particles in an external magnetic field

The quantum mechanics and thermodynamics of SU(2) non-Abelian Chern-Simons particles (non-Abelian anyons) in an external magnetic field are addressed. We derive the N-body Hamiltonian in the (anti-) holomorphic gauge when the Hilbert space is projected onto the lowest Landau level of the magnetic field. In the presence of an additional harmonic potential, the N-body spectrum depends linearly on the coupling (statistics) parameter. We calculate the second virial coefficient and find that in the strong magnetic field limit it develops a step-wise behavior as a function of the statistics parameter, in contrast to the linear dependence in the case of Abelian anyons. For small enough values of the statistics parameter we relate the N-body partition functions in the lowest Landau level to these of SU(2) bosons and find that the cluster (and virial) coefficients dependence on the statistics parameter cancels.     

Asymptotic expansion for the density of states of the magnetic Schrödinger operator with a random potential

We study the asymptotics for the density of states of the magnetic Schrödinger operator with a random potential. By using the methods of effective Hamiltonian, complex dilation and complex translation, we obtain in the large magnetic field limit, the asymptotic expansion for the density of states measure considered as a distribution.     

Spatial fluctuations of the density of states in magnetic fields observed with scanning tunneling spectroscopy

Scanning tunneling spectroscopy images on n-InAs(110) exhibit a strong magnetic field dependent contrast on the 50 nm length scale, indicating fluctuations in the density of states of the sample. The contrast is correlated to previously observed Landau oscillations in dI/dV curves. Its origin is a spatial fluctuation of the Landau level energy of 3-4 meV caused by the inhomogeneous distribution of dopant atoms. Besides inducing large-scale fluctuations in the density of states, dopants preserve their ability to scatter electron waves. The resulting wave pattern is found to depend on the magnetic field. It is suggested that the dependence is guided by the condensation of the electronic states on Landau tubes.     

Lifshitz Tails for a Class of Schrödinger Operators with Random Breather-Type Potential

We derive bounds on the integrated density of states for a class of Schrödinger operators with a random potential. The potential depends on a sequence of random variables, not necessarily in a linear way. An example of such a random Schrödinger operator is the breather model, as introduced by Combes, Hislop and Mourre. For these models, we show that the integrated density of states near the bottom of the spectrum behaves according to the so called Lifshitz asymptotics. This result can be used to prove Anderson localization in certain energy/disorder regimes.     

Density of states of a two-dimensional electron in a strong magnetic field and a random potential

The density of states of a two-dimensional electron in a strong magnetic field moving in a periodic and a random potential is calculated. The results are compared with the density of states of the Landau model with disorder as obtained in the single band approximation. The limitations of the single band model are discussed.     

Semi‐Classical Model of Strongly Correlated Coulomb Systems in Weak Magnetic Field

The integer and fractional quantum Hall effects are two remarkable macroscopic quantum phenomena occurring in two‐dimensional strongly correlated electronic systems at high magnetic fields and low temperatures. Quantization of Hall resistivity in the very high magnetic field regime at partial filling of the lowest Landau level indicates the stabilization of an electronic liquid quantum Hall phase of matter. Other interesting phases that differ from the quantum Hall phases take prominence in weaker magnetic fields when many more Landau levels are filled. These states manifest anisotropic magneto‐transport properties and, under certain conditions, appear to mimic charge density waves and/or liquid crystalline phases. One way to understand such a behavior has been in terms of effective interaction potentials confined to the highest Landau level partially filled with electrons. In this work we show that, for weak magnetic fields, such a quantum treatment of these strongly correlated Coulomb systems resembles a semi‐classical model of rotating electrons in which the time‐averaged interaction potential can be expressed solely in terms of guiding center coordinates. We discuss how the features of this semi‐classical effective potential may affect the stability of various strongly correlated electronic phases in the weak magnetic field regime (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantum mechanics on the noncommutative torus

We analyze the algebra of observables of a charged particle on a noncommutative torus in a constant magnetic field. We present a set of generators of this algebra which coincide with the generators for a commutative torus but at a different value of the magnetic field, and demonstrate the existence of a critical value of the magnetic field for which the algebra reduces. We then obtain the irreducible representations of the algebra and relate them to noncommutative bundles. Finally we comment on Landau levels, density of states and the critical case.     

Lifshitz Tails for Random Schrödinger Operators¶with Negative Singular Poisson Potential

We develop a method of asymptotic study of the integrated density of states (IDS) N(E) of a random Schr?dinger operator with a non-positive (attractive) Poisson potential. The method is based on the periodic approximations of the potential instead of the Dirichlet-Neumann bracketing used before. This allows us to derive more precise bounds for the rate of approximations of the IDS by the IDS of respective periodic operators and to obtain rigorously for the first time the leading term of log N(E) as E→−∞ for the Poisson random potential with a singular single-site (impurity) potential, in particular, for the screened Coulomb impurities, dislocations, etc. Received: 18 November 1998 / Accepted: 9 March 1999