arXiv subjects
Publications and source records attributed to Franz J. Wegner.
Leo Kadanoff has worked in many fields of statistical mechanics. His contributions had an enormous impact. This holds in particular for critical phenomena, where he explained Widom's homogeneity laws by means of block-spin transformations and laid the basis for Wilson's renormalization group equation. I had the pleasure to work in his group for one year. A short historically account is given.
This paper rests to a large extend on a paper I wrote some time ago on 'Duality in generalized Ising models and phase transitions without local order parameter'. It deals with Ising models with interactions containing products of more than two spins. In contrast to the old paper I will first give examples before I come to the general statements. Of particular interest is a gauge invariant Ising model in four dimensions. It has important properties in common with models for quantum chromodynamics as developed by Ken Wilson. One phase yields an area law for the Wilson-loop yielding an interaction increasing proportional to the distance and thus corresponding to quark-confinement. The other phase yields a perimeter law allowing for a quark-gluon plasma.
The density of states of disordered systems in the Wigner-Dyson classes approaches some finite non-zero value at the mobility edge, whereas the density of states in systems of the chiral and Bogolubov-de Gennes classes shows a divergent or vanishing behavior in the band centre. Such types of behavior were classified as homogeneous and inhomogeneous fixed point ensembles within a real-space renormalization group approach. For the latter ensembles the scaling law $μ=dν-1$ was derived for the power laws of the density of states $ρ\propto|E|^μ$ and of the localization length $ξ\propto|E|^{-ν}$. This prediction from 1976 is checked against explicit results obtained meanwhile.
The critical behaviour of the O(n)-symmetric model with two n-vector fields is studied within the field-theoretical renormalization group approach in a D=4-2 epsilon expansion. Depending on the coupling constants the beta-functions, fixed points and critical exponents are calculated up to the one- and two-loop order, resp. (eta in two- and three-loop order). Continuous lines of fixed points and O(n)*O(2) invariant discrete solutions were found. Apart from already known fixed points two new ones were found. One agrees in one-loop order with a known fixed point, but differs from it in two-loop order.
Explicit solutions of the two-dimensional floating body problem (bodies that can float in all positions) for relative density different from 1/2 and of the tire track problem (tire tracks of a bicycle, which do not allow to determine, which way the bicycle went) are given, which differ from circles. Starting point is the differential equation given by the author in archive physics/0205059 and Studies in Appl. Math. 111 (2003) 167-183. The curves are also trajectories of charges in a perpendicular magnetic field.
S.C. Zhang has put forward the idea that high-temperature-superconductors can be described in the framework of an SO(5)-symmetric theory in which the three components of the antiferromagnetic order-parameter and the two components of the two-particle condensate form a five-component order-parameter with SO(5) symmetry. Interactions small in comparison to this strong interaction introduce anisotropies into the SO(5)-space and determine whether it is favorable for the system to be superconducting or antiferromagnetic. Here the view is expressed that Zhang's derivation of the effective interaction V_{eff} based on his Hamiltonian H_a is not correct. However, the orthogonality constraints introduced several pages after this 'derivation' give the key to an effective interaction very similar to that given by Zhang. It is shown that the orthogonality constraints are not rigorous constraints, but they maximize the entropy at finite temperature. If the interaction drives the ground-state to the largest possible eigenvalues of the operators under consideration (antiferromagnetic ordering, superconducting condensate, etc.), then the orthogonality constraints are obeyed by the ground-state, too.
The technique of Hamiltonian flow equations is applied to the canonical Hamiltonian of quantum electrodynamics in the front form and 3+1 dimensions. The aim is to generate a bound state equation in a quantum field theory, particularly to derive an effective Hamiltonian which is practically solvable in Fock-spaces with reduced particle number. The effective Hamiltonian, obtained as a solution of flow eqautions to the second order, is solved numerically for positronium spectrum. The impact of different similarity functions is explicitly studied. The approach discussed can ultimately be used to address to the same problem for quantum chromodynamics.
We prove the existence of an ordered low temperature phase in a model of soft-self-avoiding closed random surfaces on a cubic lattice by a suitable extension of Peierls contour method. The statistical weight of each surface configuration depends only on the mean extrinsic curvature and on an interaction term arising when two surfaces touch each other along some contour. The model was introduced by F.J. Wegner and G.K. Savvidy as a lattice version of the gonihedric string, which is an action for triangulated random surfaces.