Franz Wegner

Bounds on the density of states in disordered systems

For a class of tight-binding models governed by short-range one-particle Hamiltonians with site-diagonal and/or off-diagonal disorder and continuous distribution of the matrix elements it is proven that the averaged density of states does neither vanish nor diverge inside the band. This refutes for these models conjectures that the density of states vanishes or diverges at the mobility edge.

The mobility edge problem: Continuous symmetry and a conjecture

An apparently overlooked symmetry of the disordered electron problem is derived. It yields the well-known Ward-identity connecting the one- and two-particle Greens function. This symmetry and the apparent shortrange behaviour of the averaged one-particle Greens function are used to conjecture that the critical behaviour near the mobility edge coincides with that of interacting matrices which have two different eigenvalues of multiplicity zero (due to replicas). As a consequence the exponents of the d.c. conductivity is expected to approach 1 for real matrices and 1/2 for complex matrices as the dimensionality of the system approaches two from above. In two dimensions no metallic conductivity is expected.

Disordered system withn orbitals per site: Lagrange formulation, hyperbolic symmetry, and goldstone modes

We give a Lagrange formulation of the gauge invariantn-orbital model for disordered electronic systems recently introduced by Wegner. The derivation proceeds analytically without use of diagrams, and it identifies the previously discussedn→∞ limit as the saddle-point approximation of the Lagrangian formulation. We discover that the Lagrangian model crucially depends on the position with respect to the real axis of the energies involved. If the energies occur on both sides of the real axis as is the case in the calculation of the conductivity, then the order parameter field takes values in a set of complex non-hermitean matrices. If all energies are on the same side of the real axis then a hermitean matrix model emerges. This difference reflects a difference in the symmetries. Whereas in the latter case normal unitary symmetry holds, the symmetry in the former case is of hyperbolic nature. The corresponding symmetry group is not compact and this might be a source of singularities also in the region of localized states. Eliminating massive modes in tree approximation we derive an effective Lagrangian for the Goldstone modes. The structure of this Lagrangian resembles the non-linear σ-model and is a very general consequence of broken isotropic symmetry. We also consider the first correction to the tree approximation which is related to the invariant measure of the generalized non-linear σ-model.

Electrons in disordered systems. Scaling near the mobility edge

Renormalization group arguments are applied to an ensemble of disordered electronic systems (without electron-electron interaction). The renormalization group procedure consists of a sequence of transformations of the length and the energy scales, and of orthogonal transformations of the electronic states. Homogeneity and power laws are obtained for various one and two-particle correlations and for the low-temperature conductivity in the vicinity of the mobility edge. Two types of fixed point ensembles are proposed, a homogeneous ensemble which is roughly approximated by a cell model, and an inhomogeneous ensemble.

Inverse participation ratio in 2+ε dimensions

The averaged moments of the eigenfunctions (including the inverse participation ratio) of a particle in a random potential are considered near the mobility edge. The exponents of the power laws are given in anε-expansion in one-loop order for ad=2+ε dimensional system. The calculation is based on a recent formulation of the mobility edge problem which maps it onto a model of interacting matrices.

Scaling approach to anisotropic magnetic systems statics

Scaling laws are stated for anisotropic magnetic systems, where the anisotropy parameters are either scaled or held fixed. Combining the two ways of scaling, the critical behavior of thermodynamic quantities in anisotropic systems is determined. Particular attention is drawn to the temperature range where the anisotropy becomes important, and to the dependence there of the different quantities on the anisotropy parameters. In a transverse magnetic field the phase transition of an anisotropic magnet takes place along aλ-line. Assuming the singular part of the free enthalpy to depend on the distance from theλ-line, anomalous corrections to the transverse susceptibility and magnetization are calculated. For an experimental verification of many of the results, experiments including a variation of the anisotropy parameters or a finite transverse field are necessary.

Four-loop-order β-function of nonlinear σ-models in symmetric spaces

Abstract The beta-function for the grassmannian nonlinear σ-model of symmetry U (N) U (p) ∗ U (N−p) has been calculated directly in four-loop order in d=2+e dimensions. Using isomorphisms and information from 1 N expansions I obtain the four-loop β-function for a large class of manifolds. Consequences are: (i) the degeneracy of the exponent ν for chiral models on the group manifolds SU(N) and SO(N) in three-loop order is lifted in four-loop order; (ii) the conductivity exponent at the mobility edge for the orthogonal case acquires a negative correction; (iii) the β-function bends over in the symplectic (i.e. spin-orbit coupling) case which suggests a nontrivial mobility edge fixed-point in d = 2 dimensions.

Exact density of states for lowest Landau level in white noise potential superfield representation for interacting systems

The density of states of two-dimensional electrons in a strong perpendicular magnetic field and white-noise potential is calculated exactly under the provision that only the states of the free electrons in the lowest Landau level are taken into account. It is used that the integral over the coordinates in the plane perpendicular to the magnetic field in a Feynman graph yields the inverse of the number λ of Euler trails through the graph, whereas the weight by which a Feynman graph contributes in this disordered system is λ times that of the corresponding interacting system. Thus the factors λ cancel which allows the reduction of thed dimensional disordered problem to a (d-2) dimensional φ4 interaction problem. The inverse procedure and the equivalence of disordered harmonic systems with interacting systems of superfields is used to give a mapping of interacting systems withU(1) invariance ind dimensions to interacting systems with UPL(1,1) invariance in (d+2) dimensions. The partition function of the new systems is unity so that systems with quenched disorder can be treated by averaging exp(−H) without recourse to the replica trick.

Anomaly in the band centre of the one-dimensional Anderson model

We calculate the density of states and various characteristic lengths of the one-dimensional Anderson model in the limit of weak disorder. All these quantities show anomalous fluctuations near the band centre. This has already been observed for the density of states in a different model by Gorkov and Dorokhov, and is in close agreement with a Monte-Carlo calculation for the localization length by Czycholl, Kramer and Mac-Kinnon.

Spin-ordering in a planar classical Heisenberg model

AbstractWe consider aD-dimensional system of classical spins rotating in a plane and interacting via a Heisenberg coupling. The spin-correlation functiongD(r) is calculated for large distancesr in a low-temperature approximation (taking into account shortrange order):