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.
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
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.
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
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
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
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
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.
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.
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.
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.
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.
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.
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.
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
Pp,2(x)=0 for two different integers p, where P are Legendre
functions.
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 SiO4 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 SiO2
crystals.
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.
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].
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.
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.
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 C4mm 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,
dx2-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.
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.
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.
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 dx2-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.
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
dx2-y2-waves. Other types of instabilities of
comparable strength are also reported.
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).
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).
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.
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.
Orthogonality constraints and entropy in the SO(5)-Theory of
High Tc 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 Veff based on his Hamiltonian
Ha 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.
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.
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.
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.
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(n2+4kn4), where
n2 is the number of edges, where two plaquettes meet at a right
angle and n4 is the number of edges, where 4 plaquettes meet.
This model can be represented as a Z2-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 x38 (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 x44 for the
magnetization, susceptibility and specific heat in 3 dimensions yield critical
exponents, which are in agreement with previous results.
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.
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.
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.
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.
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.
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 Λν,μ=
Λμ,ν, μ≠&nu =1, ...d and one Ising spin Γ
on every link (ξ,ξ+eμ). 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.
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.
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.
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.
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.
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.
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(ε2). 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.
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.
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.
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.
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.
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).
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(ε2).
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.
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.
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.
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.
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/n3.
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.
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(m1+m2)/G(m1)*G(m2) 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.
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(m1+m2)/G(m1)*G(m2) 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(m1)*G(m2). 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 CPn-1 model, and the HPn-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 B1 of
Hikami's four-loop β-function (Bernreuther and 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.
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.
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
φa=2ν+O(ε3) in d=2+ε dimensions.
If both symmetries are broken simultaneously, then the (leading) crossover
exponent is in general φs=2ν+3+O(ε3).
This holds in particular for local spin-scattering potentials.
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.
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.
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.
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.
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∝tθ, 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∝tθ, θ larger tan 2, is obtained
as observed in turbulence.
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.
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 Ec 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.
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.
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 Ec 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.
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 Ec 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 [].
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
Φ4 interaction problem.
The inverse procedure and the equivalence of disordered harmonic systems
with interacting systems of superfields is used to give a mapping of
interacting systems 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.
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.
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=Σr,r' vr,r' |r> <r'|
may have localized and extended eigenstates depending on the energy of the
particle. The energy Ec 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 vr,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].
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.
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.
Relations between Nonlinear σ-Models of Various
Symmetries
Nucl. Phys. B180 [FS2] (1981) 77
In a formal expansion in powers of T, m1 and m2
it is shown that the correlation functions of the non-linear σ-model
with unitary symplectic symmetry
Sp(m1+m2)/Sp(m1)*Sp(m2) at
temperature T equal (apart from an overall factor) the correlations of the
model with orthogonal symmetry
O(-2m1-2m2)/O(-2m1)*O(-2m2)
at temperature -½T. Similarly the non-linear σ-model with unitary
symmetry U(m1+m2)/U(m1)*U(m2)
yields correlations which are invariant under a simultaneous change of sign in
T, m1, and m2.
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/E2 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.
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
S2(x1,x'1,x2,x'2,E1,E2)
of a Gaussian-disordered electron system in the limit of deeply localized
states and small energy differences ω=E1-E2.
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 E1, E2. 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 |E1+E2|→∞,
ω/|E1+E2|→=const«1. We also apply it
to the hydrodynamic limit ω/|E1+E2|→0,
|E1+E2| 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.
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.
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.
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.
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-Ec)-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.
Renormalization Group and the Anderson Model of Disordered
Systems
in International Summer School "Recent Advances in Statistical
Mechanics", Brasov (1979) 63
(without abstract)
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
dσ=2σ/(σ-1) and the exponent η are
reported in order ε and ε², respectively.
The Critical State, General Aspects
in C. Domb and M.S. Green (eds.) Phase Transitons and Critical Phenomena,
Academic Press, 6 (1976) 7
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
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
A. The nonlinear term
B. The nontrivial fixed point in order ε
C. Exponent η
D. Isotropic n-component model
A. Scaling fields
B. Invariance properties
C. Universality
D. Coexistence curve
E. Logarithmic anomalies
F. The limit case yE=0: Phase transitions of infinite
order
A. Order parameter correlations
B. Recursion equation for correlation functions
C. Correlation functions for finite wave-lengths near Tc
D. Scaling fields for inhomogeneous perturbations, universality of
scaling
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 Oi. 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 <Oi>. 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.
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.
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.
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.
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.
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.
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.
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.
On the Magnetic Phase Diagram of (Mn,Fe)WO4
Wolframite
Solid State Comm. 12 (1973) 785
The magnetic phase diagram of mixed crystals of type
(Mn,Fe)WO4 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.
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.
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.
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.
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 He3-He4 mixtures)
is discussed by using renormalization group techniques. The tricritical
exponents and scaling fields are calculated for three dimensions.
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.
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.
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.
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.
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/γ0=β/β0=ν/ν0
=1-½α, δ/δ0=η/η0=1,
with the subscripts zero denoting the index values for the ordinary
two-dimensional Ising model.
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 Mdn 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.
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.
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.
Crossover Effects in Dynamical Critical Phenomena in MnF2
and FeF2
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 MnF2 and FeF2.
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≥Tc.
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.
Anomaly of the Ferromagnetic Susceptibility χq near
Tc
Phys. Lett. 29A (1969) 77
Observing that the ferromagnetic phase transition in an inhomogeneous
field hq occurs along a λ-line, a correction to the
wavenumber-dependent susceptibility χq, proportional to
the entropy of the system, is derived.
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.
On the Dynamics of the Heisenberg Antiferromagnet at
TN
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.
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 Tc numerical results are given.
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 gD(r) is calculated for large distances r in a
low-temperature approximation (taking into account short-range order):
g1(r)=exp(-C1Tr),
g2(r)=r-C2T,
limr→∞ g3(r)=exp(-C3T).
In two dimensions this model exhibits infinite susceptiblity χ=1/2T
Σr g2(r) at low temperatures.
Comparison is made of g1 with the exact result and of g3
with a spinwave-treatment showing agreement of ln g(r) within order T.
Spontaneous magnetization for D=1 and 2 is ruled out exactly.
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.