- [126] F.J. Wegner

**Michael Fisher and Localization**

A. Aharony, O. Entin, D. Huse, and L. Radzihovsky (eds.),

50 years of the renormalization group / dedicated to the memory of Michael E. Fisher

prepared for a volume by World Scientific

First version: Fisher_Localization.pdf

Note added: Fisher_Localizatiom_note23.pdf

Michael Fisher presented many important contributions to critical phenomena and distributed it widely in review articles. His paper with Kenneth Wilson on the expansion of critical exponents in $4-\epsilon$ dimensions gave the field of critical phenomena an enormous boost.

A few years later Alexander Polyakov presented his paper on the expansion around the lower critical dimension 2, which brought an important impetus for non-linear sigma-models. The Anderson localization transition due to disorder is described by such a model of supersymmetric matrices. It is applied to various dimensions and to ensembles with and without time-reversal invariance. - [125] F.J. Wegner

**Fifty years of Ising lattice gauge theory**

Invited talk given at the MCQST workshop on

Discrete lattice gauge theories - emergence and quantum simulations

at the Max-Planck Institute of Quantum Optics (Garching near Munich) May 2022

Ising_gauge22n.pdf

indico.physik.uni-muenchen.de/event/171/contributions/809/attachments/400/787/Ising_gauge22.pdf

I review the introduction of the gauge invariant Ising model and my motivation based on my 1971 paper "Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters". I start with the exact determination of the critical temperature of the two-dimensional Ising model on the square lattice by Kramers and Wannier (1941) before Onsager gave the exact solution for this model in 1944. The basic idea was that this model was self-dual. Therefore I thought whether something similar could be done for the three dimensional model. I realized that the dual model is a gauge-invariant model, but it is not self-dual. Increasing the number of dimensions from three to four the gauge-invariant model is self-dual and its critical temperature is the same as for the two-dimensional conventional Ising model. The gauge-invariant model has the property that it has no local order parameter. Non-vanishing correlations are given by products of spins along a loop, called Wilson-loop. The expectation value obeys at high temperatures an area law and at low temperatures a perimeter law. I mention the dual of the correlations and if time permits some further related work. - [124] F. Wegner

**The Collatz Problem generalized to 3x+k**(13 pages)

arxiv: 2101.08060

The Collatz problem with 3x+k is revisited. Positive and negative limit cycles are given up to k=9997 starting with x0=-2.10^{7}...+2.10^{7}. A simple relation between the probability distribution for the Syracuse iterates for various k (not divisible by 2 and 3) is obtained. From this it follows that the oscillation considered by Tao 2019 (arXiv:1909.03562v2 ) does not depend on k. Thus this piece of the proof of his theorem 1.3 "Almost all Collatz orbits attain almost bounded values" holds for all k not divisible by 2 and 3.

[124S] F. Wegner

**Supplement: List of limit cycles for the Collatz problem 3x+k**(278 pages)

In this supplement to "The Collatz problem generalized to 3x+k" a list of limit cycles is given for k not divisible by 2 and 3. Both positive and negative limit cycles are presented up to k = 9997 starting with x_{0}= -2.10^{7}...+2.10^{7}. - [123] F. Wegner

**From Elastica to Floating Bodies of Equilibrium**

arxiv: 1909.12596

A short historical account of the curves related to the two-dimensional floating bodies of equilibrium and the bicycle problem is given. Bor, Levi, Perline and Tabachnikov found, quite a number had already been described as*Elastica*by Bernoulli and Euler and as*Elastica under Pressure*or*Buckled Rings*by Levy and Halphen. Auerbach already realized that Zindler had described curves for the floating bodies problem. An even larger class of curves solves the bicycle problem.

The subsequent sections deal with some supplemental details: Several derivations of the equations for the elastica and elastica under pressure are given. Properties of Zindler curves and some work on the problem of floating bodies of equilibrium by other mathematicians are considered. Special cases of elastica under pressure reduce to algebraic curves, as shown by Greenhill. Since most of the curves considered here are bicycle curves, a few remarks concerning them are added. - [122] F.J. Wegner and Yury M. Pismak

**Wave packets of bound states of a Dirac field at a material plane**

Theoretical and Mathematical Physics 200 (2019) 1401-1412

Teoreticheskaya I Matematicheskaya Fizika 200 (2019) 553-565

Recently Yury Pismak and Daria Shukhobodskaia introduced a model of a Dirac particle interacting with a material plane.

Depending on the interaction constants the Dirac particle may have bound states, which decay exponentially perpendicular to the plane and move parallel to the plane. However, the waves on both sides of the plane move with different velocities.

Although the states with sharp momenta parallel to the plane do not have currents perpendicular to it, it turns out that wave-packets do have a weak current-component in this direction, which keeps the wave packets above and below the plane together. - [121] Yu. Pismak and F. Wegner

**Dispersion relations and dynamic characteristics of bound states in the model of a Dirac field interacting with a material plane**

EPJ Web of Conferences 191 (2018) 06015

Symanzik's approach for construction of quantum field model in inhomogeneous space-time is used as a basis for modeling the interaction of a macroscopic material body with quantum fields. In quantum electrodynamics it enables one to establish the most general form of the action functional describing the interaction of 2-dimensional material objects with photon and fermion fields. Results obtained within this approach for description of the interaction of the spinor field with a material plane are presented. - [120] F. Wegner

**Phasenübergänge, Renormierung und Flussgleichung**

in I. Appenzeller, D. Dubbers, H.-G. Siebig, A. Winnacker (Hrsg.): Heidelberger Physiker berichten - Rückblicke auf Forschung in der Physik und Astronomie. Band 3: Mikrokosmos und Makrokosmos, (2017) 151-171, heiBooks (Heidelberg)

Inhalt:

Einführung (Herausgeber)

Studium

Kritische Phänomene und Renormierung

Anderson Lokalisierung und nichtlineares σ-Modell

Fluss-Gleichungen

Körper, die in allen Richtungen schwimmen - [119] F.J. Wegner

**In memory of Leo P. Kadanoff**

J. of Stat. Phys. 167 (2017) 420

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 historical account is given. - [118] F. Wegner

**Supermathematics and its Applications in Statistical Physics**

Grassmann Variables and the Method of Supersymmetry

Lecture Notes in Physics 920, Springer (2016)

The book is available at Springer via DOI 10.1007/978-3-662-49170-6

Contents

- Part I Grassmann Variables and Applications

1 Introduction

2 Grassmann Algebra

3 Grassmann Analysis

4 Disordered Systems

5 Substitution of Variables

6 The Complex Conjugate

7 Path Integrals for Fermions and Bosons

8 Dimers in Two Dimensions

9 Two-Dimensional Ising Model

- Part II Supermathematics

10 Supermatrices

11 Functions of Matrices

12 Supersymmetric Matrices

13 Adjoint, Scalar Product, Superunitary Groups

14 Superreal Matrices, Unitary-Orthosymplectic Groups

15 Integral Theorems for the Unitary Group

16 Integral Theorems for the(Unitary)-Orthosymplectic Group

17 More on Matrices

- Part III Supersymmetry in Statistical Physics

18 Supersymmetric Models

19 Supersymmetry in Stochastic Field Equations and in High Energy Physics

20 Dimensional Reduction

21 Random Matrix Theory

22 Diffusive Model

23 More on the Non-linear σ-Model

24 Summary and Additional Remarks

Solutions

References

Index

- Part I Grassmann Variables and Applications
- [117] F.J. Wegner

**Duality in generalized Ising models**

Chapter 5 of C. Chamon, M.O. Goerbig, R. Moessner, and L.F. Cugliandolo (eds.), Topological aspects of condensed matter physics, Lecture Notes of the Les Houches Summer School August 2014

vol. 103 (2017) 219, Oxford University Press

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*[12]. It deals with Ising models with interactions containing products of more than two spins. In contrast to this 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. - [116] D.Yu. Pis'mak, Yu.M. Pis'mak, and F.J. Wegner

**Electromagnetic Waves in a Model with Chern-Simons Potential**

in: Phys. Rev. E 92 (2015) 013204

We investigate the appearance of Chern-Simons terms in electrodynamics at the surface/interface of materials. The requirement of locality, gauge invariance and renormalizability in this model is imposed. Scattering and reflection of electromagnetic waves in three different homogeneous layers of media is determined. Snell's law is preserved. However, the transmission and reflection coefficient depend on the strength of the Chern-Simons interaction, and parallel and perpendicular components are mixed. - [115] F.J. Wegner

**Kenneth Wilson - Renormalization and QCD**

in: Memorial Volume for Professor Kenneth Wilson, World Scientific and International Journal of Modern Physics A

Kenneth Wilson had an enormous impact on field theory, in particular on the renormalization group and critical phenomena, and on QCD. I had the great pleasure to work in three fields to which he contributed essentially: Critical phenomena, gauge-invariance in duality and QCD, and flow equations and similarity renormalization. - [114] F.J. Wegner

**In memory of Kenneth G. Wilson**

in: Journal of Statistical Physics 157 (2014) 628

DOI: 10.1007/s10955-014-0988-9

Kenneth Wilson had an enormous impact on the renormalization group and field theories in general. I had the great pleasure to work in three fields to which he contributed essentially: Critical phenomena, gauge-invariance in duality and confinement, and flow equations and similarity renormalization. - [113] F.J. Wegner

**Inhomogeneous Fixed Point Ensembles Revisited**

arxiv: 1003.0787

in: E. Abrahams (ed)., 50 years of Anderson Localization, World Scientific 2010

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~|E|^{μ}and of the localization length ξ~|E|^{-ν}. This prediction from 1976 is checked against explicit results obtained meanwhile. - [112] F. Wegner

**Floating Bodies of Equilibrium at Density 1/2 in Arbitrary Dimensions**

arxiv.org: 0902.3538

Bodies of density one half (of the fluid in which they are immersed) that can float in all orientations are investigated. It is shown that expansions starting from and deforming the (hyper)sphere are possible in arbitrary dimensions and allow for a large manifold of solutions: One may either (i) expand r(n)+r(-n) in powers of a given difference r(u)-r(-u), (r(n) denoting the distance from the origin in direction n). Or (ii) the envelope of the water planes (for fixed body and varying direction of gravitation) may be given. Equivalently r(n) can be expanded in powers of the distance h(u) of the water planes from the origin perpendicular to u. - [111] Y.M. Pismak, A. Weber, and F.J. Wegner

**Critical Behavior of a General O(n)-symmetric Model of two n-Vector Fields in D=4-2 ε**

J. Phys. A: Math. Theor. 42 (2009) 095003

arxiv.org: 0809.1568

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 ε expansion. Depending on the coupling constants the β-functions, fixed points and critical exponents are calculated up to the one- and two-loop order, resp. (η 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. - [110] F. Wegner

**Floating Bodies of Equilibrium in Three Dimensions. The central symmetric case**

arXiv.org: 0803.1043

Three-dimensional central symmetric bodies different from spheres that can float in all orientations are considered. For relative density ρ=1/2 there are solutions, if holes in the body are allowed. For ρ≠1/2 the body is deformed from a sphere. A set of nonlinear shape-equations determines the shape in lowest order in the deformation. It is shown that a large number of solutions exists. An expansion scheme is given, which allows a formal expansion in the deformation to arbitrary order under the assumption that apart from x=0,±1 there is no x, which obeys P_{p,2}(x)=0 for two different integers p, where P are Legendre functions. - [109] F. Wegner

**Rigid unit modes in tetrahedral crystals**

J. Phys. C: Condens. Matter 19 (2007) 406218

cond-mat/0703486

The 'rigid unit mode' (RUM) model requires unit blocks, in our case tetrahedra of SiO_{4}groups, to be rigid within first order of the displacements of the O-ions. The wave-vectors of the lattice vibrations, which obey this rigidity, are determined analytically. Lattices with inversion symmetry yield generically surfaces of RUMs in reciprocal space, whereas lattices without this symmetry yield generically lines of RUMs. Only in exceptional cases as in beta-quartz a surface of RUMs appears, if inversion symmetry is lacking. The occurence of planes and bending surfaces, straight and bent lines is discussed. Explicit calculations are performed for five modifications of SiO_{2}crystals. - [108] F.J. Wegner

**Floating Bodies of Equilibrium in 2D and the Tire Track Problem**

physics/0701241

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. - [107] F. Wegner

**Floating Bodies of Equilibrium. Explicit Solution**

physics/0603160

Explicit solutions of the two-dimensional floating body problem (bodies that can float in all positions) for relative density ρ 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 in [102]. - [106] F. Wegner
**Flow Equations and Normal Ordering. A Survey**

J. Phys. A: Math. Gen. 39 (2006) 8221-8230

cond-mat/0511660

First we give an introduction to the method of diagonalizing or block-diagonalizing continuously a Hamiltonian and explain how this procedure can be used to analyze the two-dimensional Hubbard model. Then we give a short survey on applications of this flow equation on other models. Finally we outline, how symmetry breaking can be introduced by means of a symmetry breaking of the normal ordering, not of the Hamiltonian. - [105] E. Körding ang F. Wegner

**Flow Equations and Normal Ordering**

J. Phys. A: Math. Gen. 39 (2006) 1231-1237

cond-mat/0509801

In this paper we consider flow-equations where we allow a normal ordering which is adjusted to the one-particle energy of the Hamiltonian. We show that this flow converges nearly always to the stable phase. Starting out from the symmetric Hamiltonian and symmetry-broken normal ordering nearly always yields symmetry breaking below the critical temperature. - [104] V. Hankevych and F. Wegner

**Possible Phases of the Two-Dimensional t-t' Hubbard Model**

Eur. Phys. Journal B 31 (2003) 497

We present a stability analysis of the 2D t-t' Hubbard model on a square lattice for various values of the next-nearest-neighbor hopping t' and electron concentration. Using the free energy expression, derived by means of the flow equations method, we have performed numerical calculation for the various representations under the point group C_{4mm}in order to determine the phase diagram. A surprising large number of phases has been observed. Some of them have an order parameter with many nodes in k-space. Commonly discussed types of order found by us are antiferromagnetism, d_{x2-y2}-wave singlet superconductivity, d-wave Pomeranchuk instability and flux phase. A few instabilities newly observed are a triplet analog of the flux phase, a particle-hole instability of p-type symmetry in the triplet channel which gives rise to a phase of magnetic currents, an s*-magnetic phase, a g-wave Pomeranchuk instability and the band splitting phase with p-wave character. Other weaker instabilities are found also. We study the interplay of these phases and favorable situations of their occurrences. A comparison with experiments is made. - [103] F. Wegner and V. Hankevych

**Superconductivity and Instabilities in the t-t' Hubbard Model**

Acta Phys. Pol. B 34 (2003) 497, Erratum 34 (2003) 1591

Contributed paper to the International Conference on Strongly Correlated Electron Systems SCES'02 in Cracow

We present a stability analysis of the 2D t-t' Hubbard model on a square lattice for t' = -t/6. We find possible phases of the model (d-wave Pomeranchuk and superconducting states, band splitting, singlet and triplet flux phases), and study the interplay of them. - [102] F. Wegner

**Floating Bodies of Equilibrium**

Studies in Applied Mathematics 111 (2003) 167-183

A long cylindrical body of circular cross-section and homogeneous density may float in all orientations around the cylinder axis. It is shown that there are also bodies of non-circular cross-sections which may float in any direction. Apart from those found by Auerbach for ρ = 1/2 there are one-parameter families of cross-sections for ρ ≠ 1/2 which have a p-fold rotation symmetry. For given p they have this property for p-2 different densities ρ. The differential equation governing the non-circular boundary curves is derived. Its solution is expressed in terms of an elliptic integral. - [101] V. Hankevych, I. Grote, and F. Wegner

**Pomeranchuk and other Instabilities in the t-t' Hubbard model at the Van Hove Filling**

Phys. Rev. B66 (2002) 094516

We present a stability analysis of the two-dimensional t-t' Hubbard model for various values of the next-nearest-neighbor hopping t', and electron concentrations close to the Van Hove filling by means of the flow equation method. For t' > -t/3 a d_{x2-y2}-wave Pomeranchuk instability dominates (apart from antiferromagnetism at small t'). At t' < -t/3 the leading instabilities are a g-wave Pomeranchuk instability and p-wave particle-hole instability in the triplet channel at temperatures T < 0.15t, and an s^{*}-magnetic phase for T > 0.15t; upon increasing the electron concentration the triplet analog of the flux phase occurs at low temperatures. Other weaker instabilities are found also. - [100] I. Grote, E. Körding, and F. Wegner

**Stability Analysis of the Hubbard-Model**

Journal of Low Temperature Physics 126 (2002) 1385

An effective Hartree-Fock-Bogoliubov-type interaction is calculated for the Hubbard model in second order in the coupling by means of flow equations. A stability analysis is performed in order to obtain the transition into various possible phases.

We find, that the second order contribution weakens the tendency for the antiferromagnetic transition. Apart from a possible antiferromagnetic transition the d-wave Pomeranchuk instability recently reported by Halboth and Metzner is usually the strongest instability. A newly found instability is of p-wave character and yields band-splitting. In the BCS-channel one obtains the strongest contribution for d_{x2-y2}-waves. Other types of instabilities of comparable strength are also reported. - [99] F.J. Wegner

**Flow Equations for Hamiltonians**

in: S. Arnone, Y.A. Kubyshin, T.R. Morris, K. Yoshida, Proceedings of the 2nd Conference on the Exact Renormalization Group, Rome 2000

Int. J. Mod. Phys. A 16 (2001) 1941

A method to diagonalize or block-diagonalize Hamiltonians by means of an appropriate continuous unitary transformation is reviewed. We will outline (i) the procedure for the elimination of the electron-phonon interaction and the construction of the effective attractive electron-electron interaction, and (ii) the application to some systems with electron-electron interaction (n-orbital model and Hubbard model). - [98] F. Wegner

**Flow Equations for Hamiltonians**

in B. Kramer (ed.)

Advances in Solid State Physics 40 (2000) 133

A method to diagonalize or block-diagonalize Hamiltonians by means of an appropriate continuous unitary transformation is reviewed. Main advantages among others are: (i) In perturbation theory one obtains new results for effective interactions which are less singular than those obtained by conventional perturbation theory, eg. for the effective pair interaction by eliminating the electron-phonon interaction. (P. Lenz and F.W.) (ii) In systems with impurities as for example in the spin-boson problem large parameter regions can be treated in a consistent way (S. Kehrein and A. Mielke). - [97] F.J. Wegner

**Flow Equations for Hamiltonians**

in H.C. Pauli and L.C.L. Hollenberg (eds.), Non-Perturbative QCD and Hadron Phenomenology

Nucl. Phys. B (Proc. Suppl.) 90 (2000) 141

A method to diagonalize or block-diagonalize Hamiltonians by means of an appropriate continuous unitary transformation is reviewed. - [96] F.J. Wegner

**Flow Equations for Hamiltonians**

in: D.O.Connor, C.R. Stephens, Renormalization Group Theory in the new Millenium. II (Proceedings of RG 2000 in Taxco, Mexico)

Physics Reports 348 (2001) 77

A recently developed method to diagonalize or block-diagonalize Hamiltonians by means of an appropriate continuous unitary transformation is reviewed. The main aspects will be discussed: (i) Elimination of off-diagonal matrix elements at different energy scales and (ii) problems and advantageous of this method. Two applications in condensed matter physics are given as examples: The interaction of an n-orbital model of fermions in the limit of large n is brought to block-diagonal form, and the generation of the effective attractive two-electron interaction due to the elimination of electron-phonon interaction is given. The advantage of this method in particular in comparison with conventional perturbation theory is pointed out. - [95] F.J. Wegner

**Orthogonality constraints and entropy in the SO(5)-Theory of High T**_{c}Superconductivity

Eur. Phys. J. B14 (2000) 11-17

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. - [94] E.L. Gubankova, H-C. Pauli, F.J. Wegner, and G. Papp

**Light-cone Hamiltonian Flow for the Positronium**

MPI-H-V33-1998

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, such that the approach can ultimately be used to address to the same problem for quantum chromodynamics. - [93] M. Ragwitz and F. Wegner

**Flow Equations for Electron-Phonon Interactions: Phonon Damping**

Eur. Phys. J. B8 (1999) 9-17

A recently proposed method of a continuous sequence of unitary transformations will be used to investigate the dynamics of phonons, which are coupled to an electronic system. This transformation decouples the interaction between electrons and phonons. Damping of the phonons enters through the observation, that the phonon creation and annihilation operators decay under this transformation into a superposition of electronic particle-hole excitations with a pronounced peak, where these excitations are degenerate in energy with the renormalized phonon frequency. This procedure allows the determination of the phonon correlation function and the spectral function. The width of this function is proportional to the square of the electron-phonon coupling and agrees with the conventional results for electron-phonon damping. The function itself is non-Lorentzian, but apart from these scales independent of the electron-phonon coupling. - [92] E.L. Gubankova and F. Wegner

**Flow Equations for QED in Light Front Dynamics**

Phys. Rev. D 58 (1998) 025012

The method of flow equations is applied to QED on the light front. Requiring that the particle number conserving terms in the Hamiltonian are considered to be diagonal and the other terms off-diagonal an effective Hamiltonian is obtained which reduces the positronium problem to a two-particle problem, since the particle number violating contributions are eliminated. No infrared divergencies appear. The ultraviolet renormalization can be performed simultaneously. - [91] R. Pietig and F.J. Wegner

**Low temperature expansion of the gonihedric Ising model**

Nucl. Phys. B525 (1998) 549

We investigate a model of closed (d-1)-dimensional self-avoiding random surfaces on a d-dimensional cubic lattice. The energy of a surface configuration is given by E=J(n_{2}+4kn_{4}), where n_{2}is the number of edges, where two plaquettes meet at a right angle and n_{4}is the number of edges, where 4 plaquettes meet. This model can be represented as a Z_{2}-spin system with ferromagnetic nearest-neighbour-, antiferromagnetic next-nearest-neighbour- and plaquette interaction. It corresponds to a special case of a general class of spin systems introduced by Wegner and Savvidy. Since there is no term proportional to the surface area, the bare surface tension of the model vanishes, in contrast to the ordinary Ising model. By a suitable adaption of Peierls' argument, we prove the existence of infinitely many ordered low temperature phases for the case k=0. A low temperature expansion of the free energy in 3 dimensions up to order x^{38}(x=e^{-βJ}) shows that for k>0 only the ferromagnetic low temperature phases remain stable. An analysis of low temperature expansions up to order x^{44}for the magnetization, susceptibility and specific heat in 3 dimensions yield critical exponents, which are in agreement with previous results. - [90] F. Wegner

**Hamiltonian Flow in Condensed Matter Physics**

in M. Grange et al (eds.), New Non-Perturbative Methods and Quantization on the Light Cone. Les Houches School 1997, Editions de Physique/Springer 8 (1998) 33

A recently developed method to diagonalize or block-diagonalize Hamiltonians by means of an appropriate continuous transformation is reviewed. Two applications in condensed matter physics are given as examples: (i) the interaction of an n-orbital model of fermions in the limit of large n is brought to block-diagonal form, and (ii) the generation of the effective attractive two-electron interaction due to the elimination of electron-phonon interaction is given. The advantage of this method in particular in comparison to conventional perturbation theory is pointed out. - [89] F. Wegner

**Flow Equations of Hamiltonians**

Proceedings of the Bar-Ilan 1997 Minerva Workshop on Mesoscopics, Fractals, and Neural Networks, Eilat

Phil. Mag. B77 (1998) 1249

A recently developed method to diagonalize or block-diagonalize Hamiltonians is reviewed. As an example it is applied to the elimination of the electron-phonon interaction. A discussion of the advantage of this method is given. - [88] A. Kabel and F. Wegner

**Flow Equations for Hamiltonians: Crossover from Luttinger to Landau-Liquid Behaviour in the n-Orbital Model**

Z. Physik B103 (1997) 555

Flow equations for Hamiltonians are a novel method for diagonalizing Hamiton operators. They were applied by one of the authors to a one-dimensional SU(n)-symmetric fermionic system, solving the occuring equations to first order of a 1/n-expansion. In this paper we generalize the procedure to an arbitrary number of spatial dimensions. Although the resulting equations cannot be solved analytically, some information can be extracted about the particle number near the Fermi surface. The results suggest a nonuniversal behaviour for d=1 which breaks down in favour of that of a Landau liquid in any dimension >1. - [87] P. Lenz and F. Wegner

**Flow Equations for Electron-Phonon Interactions**

Nucl. Phys. B482 (1996) 693

A recently proposed method of continuous unitary transformations is used to decouple the interaction between electrons and phonons. The differential equations for the couplings represent an infinitesimal formulation of a sequence of Fröhlich transformations. The two approaches are compared. Our result will turn out to be less singular than Fröhlich's. Furthermore the interaction between electrons belonging to a Cooper pair will always be attractive in our approach. Even in the case where Fröhlich's transformation is not defined (Fröhlich actually excluded these regions from the transformation), we obtain an elimination of the electron-phonon interaction. This is due to a sufficiently slow change of the phonon energies as a function of the flow parameter. - [86] R. Pietig and F.J. Wegner

**Phase Transition in Lattice Surface Systems with Gonihedric Action**

Nucl. Phys. B466 (1996) 513

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 S.K. Savvidy as a lattice version of the gonihedric string, which is an action for triangulated random surfaces. - [85] G.K. Savvidy, K.G. Savvidy, and F.J. Wegner

**Geometrical String and Dual Spin Systems**

Nucl. Phys. B443 (1995) 565

We are able to perform the duality transformation of the spin system which was found before as a lattice realization of the string with linear action. In four and higher dimensions this spin system can be described in terms of a two-plaquette gauge hamiltonian. The duality transformation is constructed in geometrical and algebraic language. The dual hamiltonian represents a new type of spin system with local gauge invariance. At each vertex ξ there are d(d-1)/2 Ising spins Λ_{ν,μ}= Λ_{μ,ν}, μ≠&nu =1, ...d and one Ising spin Γ on every link (ξ,ξ+e_{μ}). For the frozen spin Γ=1 the dual hamiltonian factorizes into d(d-1)/2 two-dimensional Ising ferromagnets and into antiferromagnets in the case Γ=-1. For fluctuating Γ it is a sort of spin-glass system with local gauge invariance. The generalization to p-membranes is given. - [84] Z. Pluhar, H.A. Weidenmüller, J.A. Zuk, C.H. Lewenkopf, and F.J. Wegner

**Crossover from Orthogonal to Unitary Symmetry for Ballistic Electron Transport in Chaotic Microstructures**

Annals of Physics (New York) 243 (1995) 1

We study the ensemble-averaged conductance as a function of applied magnetic field for ballistic electron transport across few-channel microstructures constructed in the shape of classically chaotic billiards. We analyze the results of recent experiments, which show suppression of weak localization due to magnetic field, in the framework of random-matrix theory. By analyzing a random-matrix Hamiltonian for the billiard-lead system with the aid of Landauer`s formula and Efetov`s supersymmetry technique, we derive a universal expression for the weak-localization contribution to the mean conductance that depends only on the number of channels and the magnetic flux. We consequently gain a theoretical understanding of the continuous crossover from orthogonal symmetry to unitary arising from the violation of time-reversal invariance for generic chaotic systems. - [83] S.K. Kehrein and F. Wegner

**The structure of the spectrum of anomalous dimensions in the N-Vector model in 4-ε dimensions**

Nucl. Phys. B424 (1994) 521

In a recent publication we have investigated the spectrum of anomalous dimensions for arbitrary composite operators in the critical N-vector model in 4-ε dimensions. We could establish properties like upper and lower bounds for the anomalous dimensions in one-loop order. In this paper we extend these results and explicitely derive parts of the one-loop spectrum of anomalous dimensions. This analysis becomes possible by an explicit representation of the conformal symmetry group on the operator algebra. Still the structure of the spectrum of anomalous dimensions is quite complicated and does generally not resemble the algebraic structure familiar from two-dimensional conformal field theories. - [82] F. Wegner

**Flow Equations for Hamiltonians**

Annalen der Physik (Berlin) 3 (1994) 77

Flow-equations are introduced in order to bring Hamiltonians closer to diagonalization. It is characteristic for these equations that matrix-elements between degenerate or almost degenerate states do not decay or decay very slowly. In order to understand different types of physical systems in this framework it is probably necessary to classify various types of these degeneracies and to investigate the corresponding physical behavior. In general these equations generate many-particle interactions. However, for an n-orbital model the equations for the two-particle interaction are closed in the limit of large n. Solutions of these equatiuons for a one-dimensional model are considered. There appear convergency problems, which are removed, if instead of diagonalization only a block-diagonalization into blocks with the same number of quasiparticles is performed. - [81] G.K. Savvidy and F.J. Wegner

**Geometrical String and Spin Systems**

Nucl. Phys. B413 (1994) 605

We formulate the geometrical string which has been proposed in earlier articles on the euclidean lattice. There are two essentially distinct cases which correspond to non-self-avoiding surfaces and to soft-self-avoiding ones. For the last case it is possible to find such spin systems with local interaction which reproduce the same surface dynamics. In the three-dimensional case this spin system is a usual Ising ferromagnet with additional diagonal antiferromagnetic interaction and with specially adjusted coupling constants. In the four-dimensional case the spin-system coincides with the gauge Ising system with an additional double-plaquette interaction and also with specially tuned coupling constants. We extend this construction to random walks and random hypersurfaces (membrane and p-branes) of high dimensionality. We compare these spin systems with the eight-vertex model and BNNNI models. - [80] S. Hikami, M. Shirai, and F. Wegner

**Anderson Localization in the lowest Landau level for a two-subband model**

Nucl. Phys. B408 (1993) 415

The quantum Hall effect is related to the extended states in the Landau level. A model with a special scattering process of the two-dimensional electrons in a strong magnetic field is introduced which allows the electrons to be scattered between two different states. For this model a nonvanishing conductivity is obtained at the band center of the lowest Landau level and the density of states becomes singular at this band center. The exact value of the diagonal conductivity is evaluated for a gaussian white-noise potential. The singularity of the density of states is studied in a 1/N expansion. - [79] S.K. Kehrein, F. Wegner, and Yu. Pismak

**Conformal Symmetry and the Spectrum of Anomalous Dimensions in the N-Vector Model in 4-ε Dimensions**

Nucl. Phys. B402 (1993) 669

The subject of this paper is to study the critical N-vector model in 4-ε dimensions in one-loop order. We analyze the spectrum of anomalous dimensions of composite operators with an arbitrary number of fields and gradients. For composite operators with three elementary fields and gradients we work out the complete spectrum of anomalous dimensions, thus extending the old solution of Wilson and Kogut for two fields and gradients. In the general case we prove some properties of the spectrum, in particular a lower limit 0+O(ε^{2}). Thus one-loop contributions generally improve the stability of the nontrivial fixed point in contrast to some 2+ε expansions. Furthermore we explicitely find conformal invariance at the nontrivial fixed point. - [78] H. Mall and F. Wegner

**Anomalous Dimensions of High Gradient Operators in the Orthogonal Matrix Model**

Nucl. Phys. B393 (1993) 495

A complete classification of all polynomial eigenoperators under the renormalization group and their critical exponents are given for operators with an arbitrary number of gradients which do not vanish in two dimensions, in a 2+ε expansion in one-loop order for the orthogonal matrix model of symmetry O(m_{+}+m_{-})/O(m_{+})*O(m_{-}). Similarly as in the unitary case the correction in one-loop order increases with the square of the number of gradients. In contrast to the unitary case the eigenoperators are characterized by five Young tableaux. - [77] F.J. Wegner

**Heisenberg-Antiferromagnet and Loop-Soup**

Z. Phys. B85 (1991) 259

An analytic approximation to the loop-soup approach of Liang et al. to the spin ½ Heisenberg antiferromagnet is introduced. It allows for a staggered long-range correlation of the spins in d>1 dimensions. The wave-vector dependence for the static spin-correlation function and for the averaged spin-wave energy agrees qualitatively with that obtained in spinwave approximation. Since my approximation does not exclude the intersection of loops, the expectation value of the spin-spin correlation at short distances is larger by a factor of approximately 3/2, similarly as in the Boson mean-field approximation. The elementary bosonic excitations of this theory correspond in my case to single unpaired spins moving on one sublattice through the system with (apart from a different overall prefactor) the same dispersion. Within the present approach the amplitudes h of the singlets in the wave function fall off like r^{-d-1}for pairs of spins a distance r apart, if long-range order is present. This suggests that the loop-soup picture may be a good starting point for further investigations. - [76] R. Gade and F. Wegner

**The n=0 Replica Limit of U(n) and U(n)/SO(n) Models**

Nucl. Phys. B360 (1991) 213

Recently the replica limit n=0 of the U(n) and U(n)/SO(n) models have attracted interest since they describe the Anderson localization behaviour in the band-centre of a two-sublattice model. For n≠0 the theories can be decomposed into one with symmetry U(1) and one for SU(n) and SU(n)/SO(n) resp. This does no longer hold for n=0. We show how the β-functions and zeta-functions for operators without derivatives can be obtained in the limit from those of SU(n) and SU(n)/SO(n) and draw consequences for these functions in this limit. - [75] F. Wegner

**Anomalous Dimensions of High-Gradient Operators in the Unitary Matrix-Model**

Nucl. Phys. B354 (1991) 441

A complete classification of all polynomial eigenoperators with an arbitrary number of gradients and which do not vanish in two dimensions and their critical exponents are given in a 2+ε expansion in one-loop order for the unitary matrix model of symmetry U(m_{+}+m_{-})/U(m_{+})*U(m_{-}). The calculation is performed by means of a formulation manifestly invariant under the full symmetry group. - [74] I.V. Lerner and F. Wegner

**High-Gradient Operators in the Unitary Matrix Model**

Z. Phys. B81 (1990) 95

A complete classification of all rotationally invariant operators of the two-dimensional unitary matrix model composed of gradients of the field Q and their anomalous dimensions are given in one-loop order. Similarly as in the orthogonal case and for the n-vector model the leading correction of operators with 2n factors ∂Q grows with n(n-1). - [73] F. Wegner

**Anomalous Dimensions of High-Gradient Operators in the n-Vector Model in 2+ε Dimensions**

Z. Phys. B78 (1990) 33

The anomalous dimensions of operators with an arbitrary number of gradients are determined for the n-vector model in d=2+ε dimensions in one-loop order. For those operators which do not vanish in d=2 dimensions all anomalous dimensions can be given explicitly. Among the scalar operators (under O(n) and O(d)) with 2s derivatives there is an operator with the full dimension y=2(1-s)+ε(1+s(s-1)/(n-2)) + O(ε^{2}). Thus similarly as for the Q-matrix model investigated by Kravtsov, Lerner, and Yudson, large positive corrections in one-loop order are obtained for the n-vector model. Possible consequences of the corrections are discussed. - [72] F. Wegner

**Four-Loop Order β-Function of Nonlinear σ-Models in Symmetric Spaces**

Nucl. Phys. B316 (1989) 663

The β-function of 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+ε 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. - [71] F. Wegner

**Berry's Phase and the Quantized Hall Effect**

Publication de l'Institut de Recherche Mathématique Avancée Université Louis Pasteur, Strasbourg, R.C.P. 25, vol. 39 (1988) 21

The connection between Berry's phase and the quantized Hall effect is reviewed. In the first section an introduction to the quantized Hall effect is given, in the second Berry's phase is introduced and determined. In the third section Avron's and Seiler's proof of quantized transport is given in an elementary way and the connection to Berry's phase is made. Remarks are added on the quantization of the Hall conductance in periodic potentials and on a generalization to the fractional quantized Hall effect. Since this seminar was given nearly two years ago I take the liberty to include a few remarks and references. - [70] F. Wegner

**Electrons in a Random Potential and Strong Magnetic Field: Lowest Landau Level**

in G. Landwehr (ed.) High Magnetic Fields in Semiconductor Physics, Springer Heidelberg (1987) 28

In this review the properties of two-dimensional independent electrons in a strong perpendicular magnetic field and a random potential are considered in the lowest Landau level. For the density of states of point scatterers an exact expression exists. For the d.c. conductivity and the inverse participation ratio series expansions in the Green's functions are available. They yield an estimate for the d.c. conductivity in the band centre and for the exponent which describes the vanishing of the inverse participation ratio in the band centre. - [69] W. Bernreuther and F.J. Wegner

**Four-Loop Order β-Function for two dimensional non-linear σ models**

Phys. Rev. Lett. 57 (1986) 1383

We determine the β function of the O(n) nonlinear σ model in 2+ε dimensions to four-loop order using the recently calculated ζ function and the critical exponent η through order 1/n^{3}. This β function determines completely, according to Hikami, the four-loop order β function for a large class of nonlinear σ models. As an application the conductivity exponent of the Anderson metal-insulator transition is calculated for the unitary case to order ε. This exponent turns out to be smaller than ½. - [68] F. Wegner

**Anomalous Dimensions for the Nonlinear σ-Model in 2+ε Dimensions II**

Nucl. Phys. B280 [FS18] (1987) 210

The anomalous dimensions of the composite scaling operators for the nonlinear σ-model of symmetry G(m_{1}+m_{2})/G(m_{1})*G(m_{2}) calculated in the previous paper are expressed in terms of the group characters of the Young tableaux which classify these operators. As an application the exponents for the averaged moments of the wave functions and the crossover exponents for the Anderson localization are determined. Comparison with numerical calculations show that the four-loop order contribution to the exponent of the participation ratio yields an overestimate, but of the expected sign. - [67] F. Wegner

**Anomalous Dimensions for the Nonlinear σ-Model in 2+ε Dimensions I**

Nucl. Phys. B280 [FS18] (1987) 193

The anomalous dimensions of composite scaling operators without spatial derivations are given for the G(m_{1}+m_{2})/G(m_{1})*G(m_{2}) matrix models (G=O, U, Sp) in a 2+ε expansion in four-loop order. They are expressed in the expansion coefficients in the transversal components of the operators invariant under G(m_{1})*G(m_{2}). In contrast to the three-loop order result the dimensions are no longer proportional to the first expansion coefficient. Special cases discussed are the operators of the O(n)-vector model, the CP^{n-1}model, and the HP^{n-1}model. The ζ-function of the O(n) model contributes a piece to the comparison of the 1/n and ε-expansions of the exponent η which allows us the determination of the unknown coefficient B_{1}of Hikami's four-loop β-function (Bernreuther and Wegner). - [66] F. Wegner

**Phasenübergänge und Renormierung**

Physikalische Blätter 42 (1986) 185

Kritische Phänomene wie das Verhalten einer Flüssigkeit in der Umgebung des kritischen Punktes oder eines Ferromagneten nahe der Curie-Temperatur beschäftigen Physiker seit über 100 Jahren. Die Geschichte und die Grundideen der Theorie des kritischen Verhaltens werden skizziert. Aus den vielen Systemen, die mit dieser Theorie behandelt werden können, wird im zweiten Teil der Anderson-Übergang, ein Metall-Isolator-Übergang, vorgestellt. - [65] D. Höf and F. Wegner

**Calculation of Anomalous Dimensions for the Nonlinear σ Model**

Nucl. Phys. B275 [FS17] (1986) 561

The relevant scaling operators without derivatives for the orthogonal, unitary and symplectic nonlinear σ-model are classified and their anomalous dimensions are calculated up to three-loop order. The exponents for the participation ratio and for higher averaged moments of the wave functions for Anderson localization are obtained. For nonmagnetic scattering the two-loop and three-loop terms vanish. - [64] F.J. Wegner

**Crossover of the Mobility Edge Behaviour**

Nucl. Phys. B270 [FS16] (1986) 1

The mobility edge behaviour of a particle in a random one-particle potential which conserves (i) spin and (ii) time-reversal invariance is governed by the orthogonal fixed point. Addition of a random potential which violates one of these symmetries yields a crossover to the unitary and symplectic fixed points, respectively, with crossover exponent φ_{a}=2ν+O(ε^{3}) in d=2+ε dimensions. If both symmetries are broken simultaneously, then the (leading) crossover exponent is in general φ_{s}=2ν+3+O(ε^{3}). This holds in particular for local spin-scattering potentials. - [63] F. Wegner

**Metal Insulator Transition in Disordered Solids**

Interdisciplinary Science Reviews 11 (1986) 164

The beauty and rarity of crystals has fascinated mankind for thousands of years. Most solids, however, are either disordered by imperfections or they lack a periodic lattice of atoms at all: they are amorphous. The periodicity of a crystal simplifies the theoretical understanding of its physical properties. At low concentrations the effect of imperfections can often be thought of as a superposition of single imperfections. At high concentrations these imperfections may interfere and lead to new cooperative phenomena. Statistical mechanics and solid state physics have gained many insights into strongly disordered systems during the last decade. New structures and fascinating symmetries appear on a deeper theoretical level. One of the phenomena where progress has been made is the metal-insulator transition induced by disorder. - [62] A. Mielke and F, Wegner

**Scaling Behaviour of One-Dimensional Weakly Disordered Models**

Z. Phys. B62 (1985) 1

The density of states and various characteristic lengths of one-dimensional tight-binding models and disordered harmonic chains are calculated in the limit of weak disorder at the band edge of the ordered system. The density of states and a localization length of the one-dimensional Anderson model were already calculated by Derrida and Gardner; we recover their results. For the tight-binding models with off-diagonal disorder our results are in agreement with numerical calculations by Krey. - [61] K.H. Hoffmann, S. Großmann, and F. Wegner

**Random Walk on a Fractal: Eigenvalue Analysis**

Z. Phys. B60 (1985) 401

The eigenvalues of the master equation describing the motion on a nested hierarchy of d-dimensional intervals with selfsimilar scaling of spatial extension as well as of the level dependent transition rates are derived. Based on this spectrum the diffusion behaviour is obtained, which is anomalous, either exponential or obeying a power law with various exponents. Emphasis is put on the insight into the mechanism of the anomalous diffusion, in particular the geometrical structure of the decay rate spectrum. - [60] F. Wegner

**Density Correlations near the Mobility Edge**

in H. Fritzsche and D. Adler (eds.), Localization and Metal-Insulator Transitions, Plenum Press New York (1985) 337

The correlations of the eigenfunctions of a particle in a spinindependent time-reversal invariant random potential near the mobility edge in d=2+ε dimensions are detzermined. The formulation in terms of the nonlinear σ model is used and the previously employed technique+to derive the participation ratio near criticality is extended to correlations by means of the operator-product expansion. - [59] S. Großmann, F. Wegner, and K.H. Hoffmann

**Anomalous Diffusion on a Selfsimilar Hierarchical Structure**

J. de Physique Lett. 46 (1985) L575

**Résumé.**- Nous étudions la croissance temporelle des moments de la distribution de particules diffusant sur un fractal à portée de saut variable avec une coupure inférieure. Les paramètres essentiels sont: le taux de croissance, le facteur d'échelle de la longueur et celui du temps le long de la hiérarchie; ce dernier critère est nouveau. Nous trouvons des lois de croissance algébriques et exponentielles et des corrections logarithmiques, ou un piégeage si la coupure est éliminée. Une augmentation anormale du taux de croissance de la variance σ∝t^{θ}, θ étant supérieure à 2, comme cela a déjà été observé pour la turbulence, est obtenue pur la première fois.

**Abstract.**- The temporal increase of the moments in diffusion on a fractal with large hopping range and lower cut-off is given. The essential parameters are the growth ratio, the length scaling and, as a new feature, the time scaling along the hierarchy. We find algebraical and exponential increase, logarithmic corrections, or trapping if the cut-off is removed. For the first time anomalous enhancement of the variance increase σ∝t^{θ}, θ larger tan 2, is obtained as observed in turbulence. - [58] F. Wegner and S. Großmann

**Diffusion and Trapping on a Nested Fractal Structure**

Z. Phys. B59 (1985) 197

We consider the spreading of an ensemble of phase points on a nested hierarchy of levels whose spatial extension scales self-similar. In order to model turbulent pair separation, a deterministic dynamical law is defined that maps a given scale to all (infinitely many) smaller scales and also to the next larger scale. The model can be solved analytically. We find anomalous diffusion (exponential increase of the variance) or trapping (finite limiting value of low order moments) depending on the dominance of level-up or level-down mapping. - [57] F. Wegner

**Anderson Transition and Nonlinear σ-Model**

in B. Kramer, G. Bergmann, Y. Bruinseraede (eds.), Localization, Interaction, and Transport Phenomena, Springer Series in Solid-State Sciences 61 (1985) 99

A particle (e.g. an electron) moving in a random one-particle potential may have localized and extended eigenstates depending on the energy of the particle. The energy E_{c}which separates the localized states from the extended states is called the mobility edge. Extended states can carry a direct current whereas localized states are bound to a certain region and can move only with the assistance of other mechanisms (e.g. phonon-assisted hopping). Thus the residual conductivity is expected to vanish for Fermi energies E in the region of localized states, and to be nonzero for E in the region of extended states. This transition from an insulating behaviour to a metallic one is called Anderson transition.

This problem can be mapped onto a field theory of interacting matrices. The critical behaviour near the mobility edge will be discussed. The theory has a G(m,m) symmetry which, for finite frequency, breaks to a G(m)*G(m) symmetry. Depending on the potential, G stands for the unitary, orthogonal and symplectic group. Due to the replica trick m equals 0. The replica trick can be circumvented by using fields composed of commuting and anticommuting components. Then one deals with unitary graded and unitary orthosymplectic symmetries.

I refer to lectures given in Les Houches [45], Sanda-Shi [51], Trieste[54], and Sitges [55]. Most of the material presented here can be found in ref. [] and in the original papers []. A few remarks concerning developments for interacting systems on similar lines are added. - [56] F. Wegner

**Disorder, Dimensional Reduction and Supersymmetry**

in K. Dietz, R. Flume, G.v. Gehlen and V. Rittenberg (eds.), Supersymmetry, NATO ASI Series B125 (1985) 697

During the last five years supersymmetric theories have become of interest for the explanation of dimensional reduction in disordered systems. In at least two cases the disordered system is closely related to a pure system in two fewer dimensions . In both cases the this dimensional reduction can be explained as the consequence of an underlying hidden supersymmetry [] which adds a pair of anticommuting coordinates to the d conventional real space ones. In both cases the Lagrangian is invariant under rotations in superspace and thus the expectation values are reduced to those of the (d-2)-dimensional system. These two systems are

(i) Ferromagnets in a random magnetic field.

(ii) Electrons in a strong magnetic field and random potential in the lowest Landau level. The calculation of averaged Green's functions is considerably simplified if one restricts oneself to the lowest Landau level. In this case the calculation of the density of states is reduced from a two-dimensional to a zero-dimensional problem.

In the following these two systems and their supersymmetric field theories will be reviewed. - [55] F. Wegner

**Anderson Transition and the Nonlinear σ-Model**

Lecture Notes in Physics 216 (1985) 141

A particle (e.g. an electron) moving in a random one-particle potential may have localized and extended eigenstates depending on the energy of the particle. The enrgy E_{c}which separates the localized states from the extended states is called the mobility edge. Extended states can carry a direct current whereas localized states are bound to a certain region and can move only with the assistance of other mechanisms (e.g. phonon-assisted hopping). Thus the residual conductivity is expected to vanish for Fermi energies E in the region of localized states, and to be nonzero for E in the region of extended states. This transition from an insulating behaviour to a metallic one is called Anderson transition.

This problem can be mapped onto a field theory of interacting matrices. The critical behaviour near the mobility edge will be discussed. The theory has a G(m,m) symmetry which for finite frequency breaks to a G(m)*G(m) symmetry. Depending on the potential G stands for the unitary, orthogonal and symplectic group. Due to the replica trick m equals 0. The replica trick can be circumvented by using fields composed of commuting and anticommuting components. Then one deals with unitary graded and unitary orthosymplectic symmetries.

I refer to lectures given in Les Houches [45], Sanda-Shi [51], and Trieste [54]. Most of the material presented here can be found in the original papers []. Some references to recent applications to electrons in strong magnetic fields, to interacting systems, and to applications in nuclear physics are given. - [54] F. Wegner

**Anderson Transition and the Nonlinear σ-Model**

Lecture Notes in Physics 201 (1984) 454

A particle (e.g. an electron) moving in a random one-particle potential may have localized and extended eigenstates depending on the energy of the particle. The energy E_{c}which separates the localized states from the extended states is called the mobility edge. Extended states can carry a direct current whereas localized states are bound to a certain region and can move only with the assistance of other mechanisms (e.g. phonon-assisted hopping). Thus the residual conductivity is expected to vanish for Fermi energies E in the region of localized states, and to be nonzero for E in the region of extended states. This transition from an insulating behaviour to a metallic one is called Anderson transition.

It will be shown that this problem can be mapped onto a field theory of interacting matrices. The critical behaviour near the mobility edge will be discussed. The theory has a G(m,m) symmetry which for finite frequency breaks to a G(m)*G(m) symmetry. Depending on the potential G stands for the unitary, orthogonal and symplectic group. Due to the replica trick m equals 0. The replica trick can be circumvented by using fields composed of commuting and anticommuting components. Then one deals with unitary graded and unitary orthosymplectic symmetries.

I refer to lectures given in Les Houches [45] and in Sanda-Shi [51], where, however, the graded groups have not yet been used. Most of the material presented here can be found in the original papers []. - [53] F. Wegner

**Exact Density of States for lowest Landau Level in White Noise Potential. Superfield Representation for Interacting Systems**

Z. Phys. B51 (1983) 279

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 contibutes in this disordered system is λ times that of the corresponding interacting system. Thus the factors λ cancel which allows the reduction of the d dimensional disordered problem to a (d-2) dimensional Φ^{4}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 with U(1) invariance in d 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. - [52] F. Wegner

**Algebraic Derivation of Symmetry Relations for Disordered Electronic Systems**

Z. Phys. B49 (1983) 297

By means of "superfields" two time-reversal invariant disordered electronic n-orbital models one without, the other with a spin-dependent random potential can be described by the same Lagrangian except for the sign of an overall prefactor. Similarly two different treatments of a system which breaks time-reversal invariance yields the same Lagrangian but with opposite sign of the prefactor. Since this prefactor is proportional to n, identical saddle point expansions in powers of ± n^{-1}for the averaged Green's functions are obtained, relations first found diagrammatically by Oppermann and Jüngling. The invariance of the Lagrangian under unitary graded and unitary ortho-symplectic transformations of the fields for systems without and with time-reversal invariance, respectively, is pointed out. - [51] F.J. Wegner

**The Anderson Transition and the Nonlinear σ-Model**

in Y. Nagaoka and H. Fukuyama (eds.), Anderson Localization, Springer Series in Solid-State Sciences 39 (1982) 8

A particle (e.g. an electron) moving in a random one-particle tight-binding potential

H=Σ_{r,r'}v_{r,r'}|r> <r'|

may have localized and extended eigenstates depending on the energy of the particle. The energy E_{c}which separates the localized from the extended states is called the mobility edge. Extended states can carry a direct current whereas localized states are bound to a certain region and can move only with the assistance of other mechanisms (e.g. by phonon-assisted hopping).Thus the residual conductivity is expected to vanish for Fermi energies E in the region of localized states, and to be nonzero for E in the region of extended states.

The ket |r> stands for an atomic orbital at site r. We assume these orbitals to be orthonormal. The matrix elements v_{r,r'}are random variables (with independent distributions or short-range correlations). In amorphous materials r itself will be the random position of an ion. Here we consider only a Bravais lattice of sites r. Here the emphasis will be on the underlying symmetries of the problem, the mapping on a field theoretic model of interacting matrices and the consequences for the behaviour in two and 2+ε dimensions. Besides the lectures on disordered systems given in this volume and the original literature I also refere to lectures given in Brasov 1979 [41] and Les Houches 1980 [45]. - [50] M. Kappus and F. Wegner

**Anomaly in the Band Centre of the One-Dimensional Anderson Model**

Z. Phys. B45 (1981) 15

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. - [49] F. Wegner

**Bounds on the Density of States in Disordered Systems**

Z. Phys. B44 (1981) 9

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. - [48] F.J. Wegner

**Relations between Nonlinear σ-Models of Various Symmetries**

Nucl. Phys. B180 [FS2] (1981) 77

In a formal expansion in powers of T, m_{1}and m_{2}it is shown that the correlation functions of the non-linear σ-model with unitary symplectic symmetry Sp(m_{1}+m_{2})/Sp(m_{1})*Sp(m_{2}) at temperature T equal (apart from an overall factor) the correlations of the model with orthogonal symmetry O(-2m_{1}-2m_{2})/O(-2m_{1})*O(-2m_{2}) at temperature -½T. Similarly the non-linear σ-model with unitary symmetry U(m_{1}+m_{2})/U(m_{1})*U(m_{2}) yields correlations which are invariant under a simultaneous change of sign in T, m_{1}, and m_{2}. - [47] L. Schäfer and F. Wegner

**Lattice Instantons, A Basis for a Treatment of Localized States?**

Z. Phys. B39 (1980) 281

We consider instanton-type solutions for a lattice model of the disordered electronic system where both the diagonal and the offdiagonal matrix elements are taken as Gaussian distributed random variables. For a large range of distributions we show that the dominant instanton solution is localized at a single site. This solution is taken as the starting point for a perturbation expansion in powers of 1/E^{2}in the region of localized states. This expansion has many features in common with the well-known high-temperature expansions, and we suggest to use it for an estimate of critical exponents. We evaluate the first few terms in the expansions of the density of states, the participation ratio, and the localization length for the case of nearest-neighbour hopping on a simple d-dimensional lattice. The tendency of the numerical results is promising. - [46] A. Houghton, L. Schäfer, and F.J. Wegner

**The Two-Particle Spectral Function and a.c. Conductivity of an Amorphous System far below the Mobility Edge: A Problem of Interacting Instantons**

Phys. Rev. B22 (1980) 3598

We use a variational approach to calculate the two-particle spectral function S_{2}(x_{1},x'_{1},x_{2},x'_{2},E_{1},E_{2}) of a Gaussian-disordered electron system in the limit of deeply localized states and small energy differences ω=E_{1}-E_{2}. The solution of the variational equations yields a two-center potential, each center in lowest order being determined by the square of an instanton function. The two instantons interact via the constraint that the Hamiltonian has to have lowest eigenvalues E_{1}, E_{2}. As the two centers approach the minimum distance allowed for given ω by the tunnel effect, we are confronted with a problem of confluent saddle points, which forces us to introduce an additional constraint. Our method is rigorous in the limit of weak disorder |E_{1}+E_{2}|→∞, ω/|E_{1}+E_{2}|→=const«1. We also apply it to the hydrodynamic limit ω/|E_{1}+E_{2}|→0, |E_{1}+E_{2}| large. It is found that these limits cannot be interchanged. In both limits we evaluate the ac conductivity. The result σ(ω)∼ω²(lnω)^{d+1}is found in the hydrodynamic limit. - [45] F. Wegner

**Disordered Electronic System as a Model of Interacting Matrices**

Phys. Repts. 67 (1980) 15

The behaviour of a quantum mechanic particle moving in a random potential is considered with special emphasis on the aspect of local gauge-invariance. Symmetry arguments are reviewed which allow the mapping of such a system onto a field theoretic model of interacting matrices. The model yields an expansion of the critical exponents at the mobility edge around the lower critical dimensionality two. - [44] L. Schäfer and F. Wegner

**Disordered System with n Orbitals per Site: Lagrange Formulation, Hyperbolic Symmetry, and Goldstone Modes**

Z. Phys. B38 (1980) 113

We give a Lagrangian formulation of the gauge invariant n-orbital model for disordered electronic systems recently introduced by Wegner. The derivation proceeds analytically without use of diagrams, and it identifies the previously discussed n→infinity 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 nonlinear 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. - [43] F. Wegner

**Inverse Participation Ratio in 2+ε Dimensions**

Z. Phys. B36 (1980) 209

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 an ε-expansion in one-loop order for a d=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. - [42] F. Wegner

**Inequality for the Mobility Edge Behaviour**

J. Physics C13 (1980) L45

It is shown that the averaged Green's function G of a particle in a random potential cannot diverge like (E-E_{c})^{-γ}as a function of the energy E with γ>1. Thus the conventional critical behaviour of the n=0 vector model cannot yield a description of the mobility edge behaviour. - [41] F.J. Wegner

**Renormalization Group and the Anderson Model of Disordered Systems**

in International Summer School "Recent Advances in Statistical Mechanics", Brasov (1979) 63

(without abstract) - [40] F. Wegner

**The Mobility Edge Problem: Continuous Symmetry and a Conjecture**

Z. Phys. B35 (1979) 207

An apparently overlooked symmetry of the disordered electron problem is derived. It yields the well-known Ward-identity connecting the one- and two-particle Green's function. This symmetry and the apparent shortrange behaviour of the averaged one-particle Green's 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 exponent s 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. - [39] R. Oppermann and F. Wegner

**Disordered System with n Orbitals per Site: 1/n Expansion**

Z. Physik B34 (1979) 327

Averaged Green's functions for a disordered electronic system with n orbitals per site are expanded in powers of 1/n. These expansions should be valid in the region of extended states. The expansion coefficients for the d.c. conductivity are finite for dimensionality d>2 and diverge as d approaches 2. Similarities of two types of two-particle Green's functions with the transverse and longitudinal susceptibilities of a ferromagnet with broken continuous symmetry are pointed out. Arguments for two being the lower critical dimensionality for the hydrodynamics and the mobility edge are given. Provided our series can be exponentiated we find that no metallic conductivity exists for finite n and d=2 in one of our models. Critical exponents for d infinitesimal above two are given. In this limit ν diverges like 1/(d-2) and the conductivity vanishes linearly at the mobility edge. The diagrams of the Green's functions are given in terms of vertices of short-range order and of the two-particle propagators of the n=∞ limit. Diagrams with s loops contribute in order n^{-s}. The diagrams can be rearranged so that a number of vertices vanishes like the square of the wavevector. This feature prevents infrared divergencies for the d.c. conductivity for d>2. - [38] F.J. Wegner

**Disordered System with n orbitals per site: n=∞ Limit**

Phys. Rev. B19 (1979) 783

A model of randomly disordered system with n electronic states at each site of a d-dimensional lattice is introduced. It is a generalization of a model by Wigner to d dimensions and an extension of the usually considered model for disordered systems to n states per site. In the limit n=∞, which is the limit of a dense system of weak scatterers, the one- and two-particle Green's function can be calculated exactly. The eigenstates are extended and the residual conductivity is finite, provided the Fermi energy is inside the band. Two special cases are considered more closely: (i) In the case of mere site-diagonal disorder the n=∞ solution agrees with the n=1 coherent-potential approximation for a semicircle distribution of the site-diagonal elements. (ii) In a "local gauge invariant model," where the phases at different sites are completely uncorrelated, the Green's functions vanish unless the points coincide pairwise in local space. Except for a special case of the gauge-invariant model, the systems (i) and (ii) show the same long-range correlation between eigenstates over a length L which diverges like |ω|^{-½}as the energy difference ω vanishes. - [37] F.J. Wegner

**Electrons in Disordered Systems. Scaling near the Mobility Edge**

in W.E. Spear (ed.), Proceedings of the Seventh International Conference on Amourphous and Liquid Semiconductors, Edinburgh (1977) 301

Electronic states near the mobility edge of disordered systems (without two-electron interaction) are investigated by means of renormalization group arguments. 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 residual conductivity according to Kubo and Greenwood. Two types of fixed point ensembles are proposed. - [36] F.J. Wegner

**Electrons in Disordered Systems. Scaling near the Mobility Edge**

Z. Phys. B25 (1976) 327

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. - [35] F.J. Wegner

**Critical Phenomena and Scale Invariance**

Lecture Notes in Physics 54 (1976) 1-29

The concept of scale invariance has turned out to be a very fruitful idea to explain critical phenomena. This idea gives a very intuitive picture of the behaviour in the critical region. It is based on the idea of a fixed point hamiltonian which is invariant under change of the length scale. This theory has confirmed most of the phenomenological assumptions and heuristic observations on critical systems, and has reproduced the features of exact model solutions. Moreover, the theory gave a deeper insight into the complicated nonanalytic behaviour at the critical point.

In view of the numerous papers on this subject the reader is in many cases referred to the review articles [] and the extensive literature cited therein. Whereas the references [] appeared before Wilson's formulation of the renormalization group the references [] report on this concept and its consequences. - [34] F.J. Wegner

**Phase Transitions and Critical Behaviour**

in J. Treusch (ed.), Festkörperprobleme (Advances in Solid State Physics), Vieweg, Braunschweig XVI (1976) 1

An introduction to the theory of critical phenomena and the renormalization group as promoted by Wilson is given. The main emphasis is on the idea of the fixed point hamiltonian (asymptotic invariance of the critical hamiltonian under change of the length scale) and the resulting homogeneity laws. - [33] F.J. Wegner

**Critical Phenomena and the Renormalization Group**

in H. Odabasi and Ö. Akyüz (eds.), Topics in Mathematical Physics, Proceedings of the Bogazici International Symposium 1975, Colorado Associated University Press, Boulder (1975) 75

The recent theory of critical phenomena and the renormalization group as promoted by Wilson is considered on an introductory level. The main emphasis is on the idea of a fixed point Hamiltonian (asymptotic invariance of the critical Hamiltonian under change of the length scale) and the resulting homogeneity laws. - [32] F.J. Wegner

**Statistics of Disordered Chains**

Z. Physik B22 (1975) 273

The statistical weights of the stationary states in various ensembles of isotopically disordered harmonic chains are compared. It is proven that the ensemble defined by the boundary conditions at both ends of the chain is constructed with correct statistical weights as follows: One matches at site i the stationary states of the ensemble defined by the boundary condition at one end and frequency ω with those of the ensemble defined by the boundary condition at the other end and ω. Finally one integrates over ω and sums over all sites i. This confirms and substantiates the conjecture on the exponential localization of the eigenstates in one dimension. The matching procedure yields an equation for the density of states. - [31] F.J. Wegner

**Exponents for Critical Points of Higher Order**

Phys. Lett. 54A (1975) 1

Critical exponents for nontrivial fixed points of order σ branching off from the trivial fixed point at dimensionality d_{σ}=2σ/(σ-1) and the exponent η are reported in order ε and ε², respectively. - [30] F.J. Wegner

**The Critical State, General Aspects**

in C. Domb and M.S. Green (eds.) Phase Transitons and Critical Phenomena, Academic Press, 6 (1976) 7

- I. Introduction
- II. The Renormalization Group

A. Order parameter, critical exponents

B. From discrete to continuous models

C. Scale invariance and basic properties of the renormalization group

D. Definitions and Notations

E. Renormalization group equations with smooth momentum cut-off

F. Other renormalization group transformations

- III. Linearized Theory

A. Fixed point, linearized renprmalization group equations

B. Redundant operators

C. Scaling of the free energy

D. Correlation functions in momentum space

E. Correltion functions in coordinate space

F. Trivial (Gaussian) fixed point

G. Comments

- IV. Nonlinear Theory: The Nontrivial Fixed Point

A. The nonlinear term

B. The nontrivial fixed point in order ε

C. Exponent η

D. Isotropic n-component model

- V. Nonlinear Theory: Homogeneous Systems

A. Scaling fields

B. Invariance properties

C. Universality

D. Coexistence curve

E. Logarithmic anomalies

F. The limit case y_{E}=0: Phase transitions of infinite order

- VI. Nonlinear Theory: Correlations

A. Order parameter correlations

B. Recursion equation for correlation functions

C. Correlation functions for finite wave-lengths near T_{c}

D. Scaling fields for inhomogeneous perturbations, universality of scaling

- [29] F.J. Wegner

**Correlation Functions near the Critical Point**

J. Physics A8 (1975) 1

Using renormalization group arguments we expand n-point correlation functions (for non-exceptional wavevectors) in expectation values of translational invariant short-range operators O_{i}. We use the fact that the Fourier components of our operators become negligible for wavevectors q large in comparison to the momentum cut-off.

The correltion functions show the same non-analyticities at the critical point as the expectation values <O_{i}>. The expansion coefficients are regular in the thermodynamic variables for q≠0. They can be expressed in terms of (a) functions which become singular at q=0 and yield the scaling behaviour, and (b) functions which are regular at q=0. The expansion coefficients of the two-point correlation function are sums of both types of functions. - [28] F.J. Wegner

**Critical Phenomena and the Renormalization Group**

Lecture Notes in Physics 27 (1975) 171

The recent theory of critical phenomena and the renormalization group as promoted by Wilson is considered on an introductory level. The main emphasis is on the idea of the fixed point Hamiltonian (asymptotic invariance of the critical Hamiltonian under change of the length scale) and the resulting homogeneity laws. - [27] A. Houghton and F.J. Wegner

**Feynman-Graph Calculation of the (0,l) Critical Exponents to Order ε²**

Phys. Rev. A10 (1974) 435

It is shown that the critical indices corresponding to (0,l) perturbations can be calculated by a Feynman-diagram method. At order ε², the results are consistent with the special cases L=1, 2, and 4 obtained previously. There is also general agreement to order ε with the exponents calculated from Wilson's recurrence relation. - [26] F.J. Wegner

**Some Invariance Properties of the Renormalization Group**

J. Physics C7 (1974) 2098

A generalization of Wilson's renormalization group equation is discussed. A large class of equivalent fixed points exists for every fixed point. The eigenperturbations fall into two classes: (i) scaling operators whose scaling exponents are independent of the fixed point within the class of equivalent fixed points, and (ii) redundant operators whose exponents depend on the choice of the renormalization group equation. The exponents of the redundant operators are spurious since the free energy depends only on the scaling fields of the scaling operators but not on the scaling fields of the redundant operators. - [25] F.J. Wegner

**Magnetic Phase Transitions on Elastic Isotropic Lattices**

J. Physics C7 (1974) 2109

The elimination of the elastic degrees of freedom of a harmonic isotropic lattice transforms the magnetoelastic coupling into an effective magnetic interaction which consists of a short-range interaction recently discussed by Aharony and a long-range interaction due to the free surface of the systems. Within the renormalization group approach to critical phenomena it is shown that the critical behaviour of the system is described appropriately by the magnetothermomechanics provided that the additional short-range interaction does not change the critical behaviour. - [24] E.K. Riedel and F.J. Wegner

**Effective Critical and Tricritical Exponents**

Phys. Rev. B9 (1974) 294

A semi-microscopic scaling-field is developed for crossover phenomena near critical and tricritical points. The theory is based on a renormalization-group description of a model with two competing fixed points (such as a critical and a tricritical fixed point) in terms of scaling fields. The coupled nonlinear differential equations for scaling fields are truncated such as to preserve the physics essential for crossover phenomena. The approach allows the explicit calculation of thermodynamic functions for (i) tricritical systems and (ii) critical systems with an irrelevant scaling field. We obtain, for example, an explicit expression for the scaling function of the susceptibility, which describes the crossover from the tricritical to the critical region. The idea of "flow diagrams" in the scaling-field space is used to characterize crossover phenomena globally in the whole critical region. The concept of asymptotical critical exponents is generalized and effective critical exponents are introduced as logarithmic derivatives of thermodynamic quantities with respect to experimental fields and scaling fields, respectively. By using the method of effective exponents the size of the crossover region between regions of different asymptotic critical behavior is estimated. For the susceptibility, the width of the crossover region in decades of the effective temperature variable is roughly equal to the inverse of the crossover exponent. In the case of a critical system with a slow transient the asymptotic critical exponent is only reached extremely close to the critical point (unless the amplitude of the transient vanishes). It might then be impossible to determine the asymptotic critical exponent experimentally or by conventional series-expansion techniques, and an analysis of the data in terms of effective exponents is the alternative. The scaling-field approach is applied to three systems with crossover phenomena: (i) the model for tricritical systems with molecular-field critical exponents, (ii) the Ashkin-Teller model in three dimensions, and (iii) a model for phase transitions with Fisher exponent renormalization due to a constraint. - [23a] F.J. Wegner

**Differential Form of the Renormalization Group**

in J.D. Gunton and M.S. Green (eds.), Renormalization Group in Critical Phenomena and Quantum Field Theory: Proceedings of a Conference, Temple University, Philadelphia, Pa., USA (1974) p. 46

Several forms of exact renormalization group differential equations (RGDE) have been derived:

(a) Starting from a Hamiltonian with a sharp momentum cut-off one obtains [] an exact RGDE by integrating over the Fourier components in an infinitesimal small shell. The sharp momentum cut-off RGEs have the disadvantage that they lead to nonanalyticities in k-space and therefore long range interactions in r space. Nevertheless they can be used to expand critical exponents around dimensionality 4.

A RGDE with smooth momentum cut-off has been derived by Wilson [} by means of an "incomplete integration". From a generalization one finds: To each fixed point Hamiltonian there exists a large class of equivalent fixed points. The eigenperturbations fall into two classes: (i) scaling operators whose scaling exponents are independent of the representation of the fixed point within the class of equivalent fixed points and (ii) spurious scaling operators whose exponents depend on the choice of the RGE. The spurious scaling operators do not contribute to the critical behavior. - [23b] F.J. Wegner

**Corrections to Thermodynamic Scaling Behaviour**

in J.D. Gunton and M.S. Green (eds.), Renormalization Group in Critical Phenomena and Quantum Field Theory: Proceedings of a Conference, Temple University, Philadelphia, Pa., USA (1974) p. 73

The effects of higher-order contributions to the linearized renormalization group equations are discussed.[] The analysis divides into four parts:

(i) An exact scaling law for redefined fields g is obtained.

(ii) The theory explains why logarithmic terms can exist in the free energy.

(iii) In the case of a marginal operator, one may obtain a power law times a fractional power of a logarithm.

(iv) Finally, the case where the energy itself is a marginal operator leads to an asymptotic series for the free energy in the temperature variable.

B. Widom's question "Is there a singularity in the coexistence curve at the liquid-vapor critical point?" is answered. - [22] F. Wegner

**On the Magnetic Phase Diagram of (Mn,Fe)WO**_{4}Wolframite

Solid State Comm. 12 (1973) 785

The magnetic phase diagram of mixed crystals of type (Mn,Fe)WO_{4}is discussed within molecular field approximation. The two species of magnetic ions tend to allign in different orderings. Depending on the interaction strengths the two ordered phases are separated either by a phase in which both orderings are superimposed or by a first order transition. - [21] F.J. Wegner

**A Transformation including the Weak-Graph Theorem and the Duality Transformation**

Physica 68 (1973) 570

A transformation in classical lattice statistics which generalizes the weak-graph theorem and includes the duality transformation is described. The energy of the configurations is expressed by a function of "quantum numbers" which are subject to certain constraints. These constraints generate the variables of the transformed system. Examples of transformations in vertex models, and two- and many-component Ising models are given. This method simplifies the derivation of a number of transformations. As an example, the transformation of the Ising model on a triangular lattice into a dual on a triangular lattice can be accomplished without an intermediate step via the honeycomb lattice. - [20] F.J. Wegner and A. Houghton

**Renormalization Group Equation for Critical Phenomena**

Phys. Rev. A8 (1973) 401

An exact renormalization equation is derived by making an infinitesimal change in the cutoff in momentum space. From this equation the expansion for critical exponents around dimensionality 4 and the limit n=∞ of the n-vector model are calculated. We obtain agreement with the results by Wilson and Fisher, and with the spherical model. - [19] F.J. Wegner and E.K. Riedel

**Logarithmic Corrections to the Molecular Field Behaviour of Critical and Tricritical Systems**

Phys. Rev. B7 (1973) 248

The asymptotic critical form of thermodynamic functions is analyzed by means of renormalization-group techniques. If certain exponent relations are satisfied, then the critical behavior is not described by a simple power law, but a power law multiplied by a fractional power of a logarithm. The approach is applied to two special systems whose critical exponents are molecular-field-like. (i) For ordinary critical transitions in four dimensions we find the same logarithmic factors previously computed by Larkin and Khmel'nitskii. (ii) For tricritical transitions in three dimensions we compute the logarithmic corrections to the molecular-field tricritical behavior discussed in an earlier publication. - [18] E.K. Riedel and F.J. Wegner

**Tricritical Exponents and Scaling Fields**

Phys. Rev. Lett. 29 (1972) 349

The tricritical behavior of a classical three-well-potential model for two-component systems (such as He^{3}-He^{4}mixtures) is discussed by using renormalization group techniques. The tricritical exponents and scaling fields are calculated for three dimensions. - [17] F.J. Wegner

**Duality Relation between the Ashkin-Teller and the Eight-Vertex Model**

J. Physics C5 (1972) L131

The Ashkin-Teller model and the eight-vertex model are related by a duality relation. - [16] M.K. Grover, L.P. Kadanoff, and F.J. Wegner

**Critical Exponents for the Heisenberg Model**

Phys. Rev. B6 (1972) 311

A recursion relation obtained by Wilson is used for the numerical calculation of critical indices for the d=3 classical Heisenberg model. Some of the results obtained are γ=1.36 and φ=1.24. - [15] F.J. Wegner

**Corrections to Scaling Laws**

Phys. Rev. B5 (1972) 4529

The effects of higher-order contributions to the linearized renormalization group equations in critical phenomena are discussed. This analysis leads to three quite different results: (i) An exact scaling law for redefined fields is obtained. These redefined fields are normally analytic functions of the physical fields. Corrections to the standard power laws are derived from this scaling law, (ii) The theory explains why logarithmic terms can exist in the free energy. (iii) The case in which the energy scales like the dimensionality is analyzed to show that quite anomalous results may be obtained in this special situation. - [14] F.J. Wegner

**Critical Exponents in Isotropic Spin Systems**

Phys. Rev. B6 (1972) 1891

Critical indices for isotropic systems of n-dimensional spins in (d=4-ε)-dimensional lattices are calculated to order ε. All critical indices corresponding to perturbations of the spin probability distribution are given. Such perturbations might arise from the effects of external or crystal fields on the spin system. - [13] L.P. Kadanoff and F.J. Wegner

**Some Critical Properties of the Eight-Vertex Model**

Phys. Rev. B4 (1971) 3989

The eight-vertex model solved by Baxter is shown to be equivalent to two Ising models with nearest-neighbor coupling interacting with one another via a four-spin coupling term. The critical properties of the model in the weak-coupling limit are in agreement with the scaling hypothesis. In this limit where α→0, the critical indices obey γ/γ_{0}=β/β_{0}=ν/ν_{0}=1-½α, δ/δ_{0}=η/η_{0}=1, with the subscripts zero denoting the index values for the ordinary two-dimensional Ising model. - [12] F.J. Wegner

**Duality in Generalized Ising Models and Phase Transitions without Local Order Parameter**

J. Math. Phys. 12 (1971) 2259-2272

Reprinted in C. Rebbi (ed.), Lattice Gauge Theories and Monte Carlo Simulations, World Scientific, Singapore (1983) p. 60-73

It is shown that any Ising model with positive coupling constants is related to another Ising model by a duality transformation. We define a class of Ising models M_{dn}on d-dimensional lattices characterized by a number n=1, 2, ..., d (n=1 corresponds to the Ising model with two-spin interaction). These models are related by two duality transformations. The models with 1<n$lt;d exhibit a phase transition without local order parameter. A nonanalyticity in the specific heat and a different qualitative behavior of certain spin correlation functions in the low and the high temperature phases indicate tzhe existence of a phase transition. The Hamiltonian of the simple cubic dual model contains products of four Ising spin operators. Applying a star square transformation, one obtains an Ising model with competing interactions exhibiting a singularity in the specific heat but no long-range order of the spins in the low temperature phase. - [11] F. Wegner and E. Riedel

**Critical Phenomena in Anisotropic Magnetic Systems**

J. Physique 32 Suppl. C1 (1971) 519

**Résumé.**- En introduissant l'anisotropie comme une autre variable critique une loi d'échelle universelle existe pour les susceptibilités et les temps de relaxation dans des systèmes anisotropes magnétiques. L'état critique se devise en régimes à compartement critique différent, et des lois de puissance sont valables dans ces régimes. Le comportement dans les régimes intermèdiaires ne peut pas être décrire par des lois de puissance. En utilisant l'approximation des modes couplés nous avons calculé les relaxations critiques. - Nous trouvons un bon accord avec des expériences récentes.

**Abstract.**- Introducing the anisotropy as a further critical variable a universal scaling law for the susceptibilities and relaxation rates in anisotropic magnetic systems exists. The critical state splits into regions of different critical behavior. In these regions power laws apply. The behavior in the cross over regions cannot be desribed by power laws. Using lowest order mode-mode approximation we calculate numerically critical spin relaxations. We find good agreement with recent experiments. - [10] E. Riedel and F. Wegner

**Critical Spin Dynamics, Symmetry and Conservation Laws**

in J.I. Budnick and M.P. Kawatra (eds.), Dynamical Aspects of Critical Phenomena, Gordon and Breach, New York (1972) 19

Dynamic critical phenomena depend strongly on the symmetry and conservation laws of the system under consideration. For small deviations from a certain symmetry point the critical state of the system can split into regions of different critical behavior. Here we review a microscopic theory for the dynamic spin correlation functions in the paramagnetic critical state as functions of the anisotropy of the system.

In the case of weak anisotropy, the system shows effectively isotropic critical behavior in an outer and anisotropic behavior in an inner critical region. The behavior in the crossover region cannot be descibed by power laws. - [9] E. Riedel and F. Wegner

**Crossover Effects in Dynamical Critical Phenomena in MnF**_{2}and FeF_{2}

Phys. Lett. 32A (1970) 273

Crossover effects, due to anisotropy, in the temperature dependent linewidths of critical spin fluctuations are discussed quantitatively for the uniaxial antiferromagnets MnF_{2}and FeF_{2}. - [8] E. Riedel and F. Wegner

**Dynamic Scaling Theory for Anisotropic Magnetic Systems**

Phys. Rev. Lett. 24 (1970) 730, 930E

Dynamic scaling laws for anisotropic magnetic systems are derived where the anisotropy parameters are explicitely treated. The approach is applied to calculate the critical spin relaxation rates for Heisenberg ferromagnets and antiferromagnets with one, two, and three easy axes of magnetization at T≥T_{c}. - [7] F. Wegner

**Slater Integrals for the Wavefunctions of the Harmonic Oscillator**

Nucl. Phys. A141 (1970) 609

Formulae for calculating Slater integrals of local two-particle interactions for the wave-functions of the harmonic oscillator are given. They are derived by means of an addition theorem for Laguerre polynomials connecting the polynomials in the squares of the relative and the single-particle coordinates. Therefore no transformation of single-particle states to relative and c.m. coordinates is necessary. The number of terms to be evaluated is smaller than in formulae of earlier papers. The formulae can also be applied to anisotropic systems and to systems with particles of different oscillator constants involved. - [6] E. Riedel and F. Wegner

**Anomaly of the Ferromagnetic Susceptibility χ**_{q}near T_{c}

Phys. Lett. 29A (1969) 77

Observing that the ferromagnetic phase transition in an inhomogeneous field h_{q}occurs along a λ-line, a correction to the wavenumber-dependent susceptibility χ_{q}, proportional to the entropy of the system, is derived. - [5] E. Riedel and F. Wegner

**Scaling Approach to Anisotropic Magnetic Systems, Statics**

Z. Physik 225 (1969) 195

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 a λ-line. Assuming the singular part of the free enthalpy to depend on the distance from the λ-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. - [4] F. Wegner

**On the Dynamics of the Heisenberg Antiferromagnet at T**_{N}

Z. Physik 218 (1969) 260

The dynamic spin autocorrelation function of the Heisenberg antiferromagnet with isotropic interaction is calculated numerically at the Néel temperature in the hydrodynamic limit. - [3] F. Wegner

**On the Heisenberg Model in the Paramagnetic Region and at the Critical Point**

Z. Physik 216 (1968) 433

An exact diagram technique suitable especially for calculating the time dependent correlation functions in the Heisenberg model is given. We apply it to investigate the equal-time and the dynamic spin-spin correlations of this model in the paramagnetic region and at the critical point. At T_{c}numerical results are given. - [2] F. Wegner

**Spin-Ordering in a Planar Classical Heisenberg Model**

Z. Physik 206 (1967) 465

We consider a D-dimensional systems of classical spins rotating in a plane and interacting via a Heisenberg coupling. The spin-correlation function g_{D}(r) is calculated for large distances r in a low-temperature approximation (taking into account short-range order): g_{1}(r)=exp(-C_{1}Tr), g_{2}(r)=r^{-C2T}, lim_{r→∞}g_{3}(r)=exp(-C_{3}T).

In two dimensions this model exhibits infinite susceptiblity χ=1/2T Σ_{r}g_{2}(r) at low temperatures.

Comparison is made of g_{1}with the exact result and of g_{3}with a spinwave-treatment showing agreement of ln g(r) within order T. Spontaneous magnetization for D=1 and 2 is ruled out exactly. - [1] F. Wegner

**Magnetic Ordering in One and Two Dimensional Systems**

Phys. Lett. 24A (1967) 131

Using the Bogoliubov inequality the absence of spontaneous magnetization in the itinerant electron model in one and two dimensions is proved.

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