Density of states for high landau levels and random potential

The density of states of a two-dimensional, non-interacting electron gas under the influence of a strong perpendicular magnetic field and random impurities is examined. In the strong magnetic field the ensemble averaged Green's function can be restricted to a single Landau leveln. Using 1/n as expansion parameter we calculate first order corrections. The results for two models of disorder are compared: white-noise potential and uniformly distributed zero-range scatterers with a Lorentzian distribution of strength.     

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     

The fate of lifshits tails in magnetic fields

We investigate the integrated density of states of the Schrödinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a nonnegative, algebraically decaying, single-impurity potential we prove that the leading asymptotic behavior for small energies is always given by the corresponding classical result, in contrast to the case of vanishing magnetic field. We also show that the integrated density of states of the operator restricted to the eingenspace of any Landau level exhibits the same behavior. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.     

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.     

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.     

Axioms for Euclidean Green's functions

We establish necessary and sufficient conditions for Euclidean Green's functions to define a unique Wightman field theory.Supported by the National Science Foundation under grant GP 31239X.Supported in part by the Air Force Office of Scientific Research, contract AF 44620-70-C-0030.     

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.     

Widths of the Hall Conductance Plateaus

We study the charge transport of the noninteracting electron gas in a two-dimensional quantum Hall system with Anderson-type impurities at zero temperature. We prove that there exist localized states of the bulk order in the disordered-broadened Landau bands whose energies are smaller than a certain value determined by the strength of the uniform magnetic field. We also prove that, when the Fermi level lies in the localization regime, the Hall conductance is quantized to the desired integer and shows the plateau of the bulk order for varying the filling factor of the electrons rather than the Fermi level.     

Vacuum birefringence in strong magnetic fields: (I) Photon polarization tensor with all the Landau levels

Photons propagating in strong magnetic fields are subject to a phenomenon called the “vacuum birefringence” where refractive indices of two physical modes both deviate from unity and are different from each other. We compute the vacuum polarization tensor of a photon in a static and homogeneous magnetic field by utilizing Schwinger’s proper-time method, and obtain a series representation as a result of double integrals analytically performed with respect to proper-time variables. The outcome is expressed in terms of an infinite sum of known functions which is plausibly interpreted as summation over all the Landau levels of fermions. Each contribution from infinitely many Landau levels yields a kinematical condition above which the contribution has an imaginary part. This indicates decay of a sufficiently energetic photon into a fermion–antifermion pair with corresponding Landau level indices. Since we do not resort to any approximation, our result is applicable to the calculation of refractive indices in the whole kinematical region of a photon momentum and in any magnitude of the external magnetic field.     

Levitation of Extended States in a Random Magnetic Field with a Finite Mean

We study the localization properties of electrons in a two-dimensional system in a random magnetic field B(r)=B₀+δB(r) with the average B₀ and the amplitude of the magnetic field fluctuations δB. The localization length of the system is calculated by using the finite-size scaling method combined with the transfer-matrix technique. In the case of weak δB, we find that the random magnetic field system is equivalent to the integer quantum Hall effect system, namely, the energy band splits into a series of disorder broadened Landau bands, at the centers of which states are extended with the localization length exponent ν=2.34±0.02. With increasing δB, the extended states float up in energy, which is similar to the levitation scenario proposed for the integer quantum Hall effect.     

Magnetic field effects on the quantum wire energy spectrum and Green's function

We analyze the energy spectrum and propagation of electrons in a quantum wire on a 2D host medium in a normal magnetic field, representing the wire by a 1D Dirac delta function potential which would support just a single subband state in the absence of the magnetic field. The associated Schrödinger Green's function for the quantum wire is derived in closed form in terms of known functions and the Landau quantized subband energy spectrum is examined.     

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     

A Note on Exponential Decay in the Random Field Ising Model

For the two-dimensional random field Ising model (RFIM) with bimodal (i.e., two-valued) external field, we prove exponential decay of correlations either (i) when the temperature is larger than the critical temperature of the Ising model without external field and the magnetic field strength is small or (ii) at any temperature when the magnetic field strength is sufficiently large. Unlike previous work on exponential decay, our approach is not based on cluster expansions but rather on arguably simpler methods; these combine an analysis of the Kertész line and a coupling of Ising measures (and also their random cluster representations) with different boundary conditions. We also show similar but weaker results for the RFIM with a general field distribution and in any dimension.     

The symmetry of three-electron states in a quantized magnetic field

The states of itinerant electrons on the finite square lattice in a quantized magnetic field are discussed. The single-electron states are classified by irreducible representations (IRs) of the magnetic translation group.The representation correspond to the Landau level, whereas its basis functions are analogs of the cyclotronic orbits. The filling factor of the Landau level is considered in the frame of multi-electron functions. Corresponding states are classified by the IR of MTG as well as by the irreducible representation of a symmetric group (permuting the electrons) and a unitary group (permuting electron states). The discussion is restricted to the case when the magnetic flux per unit cell of the two-dimensional square lattice is a rational number in terms of magnetic flux quanta. The additional parameter of the model is the Hubbard interaction between electrons.     

Localization and the integer-quantized Hall effect: Diffusion constant for high Landau levels and white noise potential

The diffusion constant and the diagonal conductivity for non-interacting electrons in a two-dimensional, disordered system are studied. A homogeneous magnetic field perpendicular to the electron system is assumed. For weak short-range random potentials and high fields the Landau quantum numbern can be used as expansion parameter. In the limit of high Landau levels the system shows metallic behaviour. Corrections for finiten decrease the conductivity and indicate localized states in the whole energy band. A breakdown of the expansion and stronger localization are observed only for the lowest Landau levels if the typical experimental length scale of the quantized Hall effect is used.     

Some properties of random Ising models

We consider an Ising model with random magnetic fieldh _i and random nearest-neighbor couplingsJ _ij . The random variablesh _i andJ _ij are independent and identically distributed with a nice enough distribution, e.g., Gaussian. We will prove that (i) at high temperature, infinite volume correlation functions are independent on the boundary conditions and decay exponentially fast with probability 1 and (ii) for any temperature with sufficiently strong magnetic field the correlation functions are again independent on the boundary conditions and decay exponentially fast with probability 1. We also prove that the averaged magnetization of the ground state configuration of the one-dimensional Ising model with random magnetic field is zero, no matter how small is the variance of theh _i .     

Local Thermodynamical Stability of Fermion Lattice Systems

We study the magnetic moment of neutral atoms in large magnetic fields. Asymptotic formulas for the energy exist with high precision in different regions depending on how large the nuclear charge Z is compared to the magnetic field strength B. All these formulas take the splitting of the kinetic energy into Landau levels as the principal feature and then treat the electric potential as a perturbation. We prove that these approximate formulas predict correctly'(to highest order) the total magnetic moment of the atom. The proof of this fact relies on a 'virial theorem' for Coulomb systems in a constant magnetic field.     

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.     

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.