Statics and Dynamics of Nonlinear Systems: Proceedings of a Workshop at the Ettore Majorana Centre, Erice, Italy, 1–11 July, 1983

317 Pages·1983·6.15 MB·English

by  A. Bishop (auth.), Professor Dr. Giorgio Benedek, Professor Dr. Heinz Bilz, Dr. Roland Zeyher (eds.)

47 Springer Series in Solid-State Sciences
Edited by Peter Fulde
Springer Series in Solid-State Sciences
Editors: M. Cardona P. Fulde H.-J. Queisser
Volume 40 Semiconductor Physics - An Introduction By K Seeger
Volume 41 The LMTO Method Muffin-Tin Orbitals and Electronic Structure
By H.L. Skriver
Volume 42 Crystal Optics with Spatial Dispersion and the Theory of Excitations
By VM. Agranovich and VL. Ginzburg
Volume 43 Resonant Nonlinear Interactions.of Light with Matter
By V.S. Butylkin, A.E. Kaplan, Yu.G. Khronopulo, and E.I. Yakubovich
Volume 44 Elastic Media with Microstructure II Three-Dimensional Models
By I.A. Kunin
Volume 45 Electronic Properties of Doped Semiconductors
By B. I. Shklovskii and A L. Efros
Volume 46 Topological Disorder in Condensed Matter
Editors: F. Yonezawa and T. Ninomiya
Volume 47 Statics and Dynamics of Nonlinear Systems
Editors: G. Benedek, H. Bilz, and R Zeyher
Volume 48 Magnetic Phase Transitions
Editors: M. Ausloos and R 1. Elliott
Volume 49 Organic Molecular Aggregates, Electronic Excitation
and Interaction Processes
Editors: P. Reineker, H. Haken, and H. C. Wolf
Volume 50 Multiple Diffraction of X-Rays in Crystals
By Shih-Lin Chang
Volume 51 Phonon Scattering in Condensed Matter
Editor: W. Eisenmenger
Volumes 1 - 39 are listed on the back inside cover
Statics and Dynamics
of Nonlinear Systems
Proceedings of a Workshop
at the Ettore Majorana Centre,
Erice, Italy, 1-11 July, 1983
Editors: G. Benedek, H. Bilz, and R Zeyher
With 117 Figures
Springer-Verlag
Berlin Heidelberg New York Tokyo 1983
Professor Dr. Giorgio Benedek
Dipartimento di Fisica, Universita degli Studi di Milano, Via Celoria 16
1-20133 Milano, Italy
Professor Dr. Heinz Bilz
Max-Planck-Institut fUr Festkorperforschung, Heisenbergstrasse 1
D-7000 Stuttgart, Fed. Rep. of Germany
Dr. Roland Zeyher
Max-Planck-Institut fUr Festkorperforschung, Heisenbergstrasse 1
D-7000 Stuttgart, Fed. Rep. of Germany
Series Editors:
ProfessocDr. Manuel Cardona
Professor Dr. Peter Fulde
Professor Dr. Hans-Joachim Queisser
Max-Planck-Institut fUr Festkorperforschung, Heisenbergstrasse 1
D-7000 Stuttgart 80, Fed. Rep. of Germany
ISBN-13: 978-3-642-82137-0 e-ISBN-13: 978-3-642-82135-6
DOl: 10.1007/978-3-642-82135-6
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2153/3130-543210
Preface
The investigation of the properties of nonlinear systems is one of the fast deve
loping areas of physics. In condensed matter physics this 'terra incognita' is
approached from various starting points such as phase transitions and renormali
zation group theory, nonlinear models, statistical mechanics and others. The study
of the mutual interrelations of these disciplines is important in developing uni
fying methods and models towards a better understanding of nonlinear systems.
The present book collects the lectures and seminars delivered at the workshop
on "Statics and Dynamics of Nonlinear Systems" held at the Centre for SCientific
Culture "Ettore Majorana·" in Erice;· Italy, July 1 to 11, 1983, in the framework
of the International School of Materials Science and Technology. Experts and
young researchers came together to discuss nonlinear phenomena in condensed matter
physics.
The book is divided into five parts, each part containing a few general artic
les introducing the subject, followed by related specialized papers. The first
part deals with basic properties of nonlinear systems including an introduction
to the general theoretical methods. Contrfbutions to the nonlinear aspects of
phase transitions are collected in the second part. In the third part properties
of incommensurate systems are discussed. Here, competing interactions lead to
charge-density waves, soliton lattices and other complex structures. Another
point of special interest, illustrated in the fourth part, is the 'chaotic' be
havior of various systems such as Josephson junctions and discrete lattices. The
investigation of exactly solvable models is the subject of the fifth part, where
various classes of nonlinear excitations are described in some detail. First steps
towards two- and. three-dimensional systems are discussed. Throughout the Proceed
ings applications to specific materials such as ferroelectrics, transition-metal
compounds and polyacetylene are presented. The book should be useful to resear
chers and students who are interested in this exciting and rapidly changing field
of condensed matter physics.
The, workshop nas been held under the auspices of the European Physical Society,
with the ~ponsorship and the financial support of the European Research Office
(ERO) and the European Office of Aerospace Research and Development (EOARD); IBM
Italia; the Italian Ministries of Education and of Scientific and Technological
Research; the Technological Committee of the Italian National Research Council
(CNR); the National Group for the Structure of Matter (GNSM) of CNR, and the Sici
lian Regional Government. All these Institutions are gratefully acknowledged. We
would also like to express our thanks to Prof. A. Zichichi, the Director of the
Majorana Centre, and to Dr. Pinola Savalli, Dr. A. Gabriele and all the Staff
members of the Centre for their help in organizing the workshop. We also thank
Prof. M. Balkanski, Director of the International School of Materials Science and
Technology for his advice during the organization of the workshop.
Erice, Stuttgart, July 1983 C. Benedek' H. Bi/z· R. Zeyher
v
Contents
Part 7 Basic Aspects of Nonlinear Systems
1.1 Introduction. By A. Bishop ••.•.••.•••••••••••.••.•••.••..••.•••.. 3
1.2 Spectral Transform and Solitons. By F. Calogero ....••.•...••..... 7
1.3 Linear and Nonlinear Aspects in Lattice Dynamics
By E. Magyari and H. Thomas (With 3 Figures) •..•.•..•••.••.•••••. lB
1.4 The Thermoelastic-Plastic Transition in Metals: Thermal Emis-
sion as a Probe to Identify the Yield Point. By C.E. Bottani
and G. Caglioti (With 2 Figures) •••.•....•......••.••............ 28
1.5 Solvable Many-Body Problems. By F. Calogero ••••...•..••.........• 35
1.6 Stability of Hydrated M~-DNAs: A Challenge in the Theory of Non
linear Systems. By G.F. Nardelli, M. Bracale, C. Signorini, and
G. Zucchelli (With 12 Figures) .•.....•••.••..•.•..••....••...••.. 37
Part 2 Phase Transitions
2.1 Anharmonic Properties Near Structural Phase Transitions: An Update
By K.A. MUller (With 6 Figures) .•...•.•••••.•••.•.•..•.••.•.•...• 68
2.2 Critical Phenomena from Wilson's Exact Renormalization-Group
Equation. By E.K. Riedel and K.E. Newman (With 1 Figure) ••••••... 80
2.3 Quantum Flu~tuations of a 3-Dimensional ~4 Model-Quantum
Ferroelectrics. By D. Schmeltzer •.....••.•••••.••••..••.•.•..•••. 85
2.4 Stochastic Quantization and Critical Dynamics. By P. Ruggiero and
M. Zannett i (Wi th 2 Fi gures) • • . • . • • • . . • • . • . • . . . . • • • . . . • . • • • . . . . • . 90
Part 3 Incommensurate Phases and Charge-Density Waves
3.1 The Physical Mechanisms Leading to Incommensurate Phases
By V. Heine (With 8 Figures) ..•••..•••....•••..•..••••••...•••... 98
3.2 Excitations and Chaotic States in Incommensurate Multi-Soliton
Lattices: Experiments. By R. Blinc (With 10 Figures) •••.•...•.... 113
3.3 The Transition by Breaking of Analyticity in Incommensurate
Structures and the Devil's Staircase; Application to Metal-Insula-
tor Trans itions in Peierl s Cha ins. By S. Aubry (With 5 Figures) .,. 126
VII
3.4 Non-Linearity Induced by Charge-Density Wave Motion
By P.- Monceau (With 6 Figures) •••.••..••..•••.••..••....•••...... 144
3.5 Low-Frequency Dynamics of Soliton Lattices
By R. Zeyher (With 1 Figure) ••••..•••••••.•••••...•••.••...••..•. 153
Part 'I Chaos in Condensed Matter
4.1 Universality and Fractal Dimension of Mode-Locking Structure in
Systems with Competing Periodicities. By P. Bak (With 8 Figures) .• 160
4.2 Chaos and Solitons in Josephson Junctions
By Y. Imry (With 6 Figures) •..••..•.••••....•••.••.•........•..•. 170
4.3 Transition to Deterministic Chaos in·a Hydrodynamic System
By M. Giglio, S. Musazzi, and U. Perini (With 6 Figures) •...•.•.. 189
4.4 Chaos and Solitons in Dissipative Nonlinear Systems
By A. Bishop (With 1 Figure) .•.•••••••••••...•.•...•..•••.•...••. 197
4.5 Chaos~Induced Diffusion. By S. Thomae (With 3 Figures) ••...•.••.. 204
Part 5 Solitons and Other Exact Solutions of Nonlinear Equations
5.1 Classical Statistical Mechanics of Lattice Dynamic Model Systems:
Transfer Integral and Molecular-Dynamics Studies
By T. Schneider (With 19 Figures) .•...••....•..•.•...•.•.•••...•. 212
5.2 The Spectral Transform: Methods for the Fourier Analysis of Non
linear Wave Data. By A.R. Osborne (With 11 Figures) ••••..••....•• 242
5.3 Kink- and Polaron-Solitons in Models of Polyacetylene and Other
Peierls-Distorted Materials. By A. Bishop (With 3 Figures) 263
5.4 Counting Solitons and Phonons in the Toda Lattice
By N. Theodorakopoulos •.....••.•...•••.••....•..••.......•.....•. 271
5.5 Structure of Interfaces. By H. MUller-Krumbhaar (With 1 Figure) 278
5.6 Commensurate Structures in Solids. By H. BUttner, G. Behnke,
H. Frosch, and H. Bilz (With 2 Figures) ••......•.....•.•....•.... 281
5.7 Periodon Solutions in Two- and Three-Dimensional Lattices
By U. Schroder, W. Kress, and H. Bilz (With 1 Figure) ••.••••••. ·.294
5.8 Intrinsic Non-Linearity (in Phonon Numbers) of the Response Func
tions of Elastic Systems: A Proposal of Quantum Acoustics
By N. Terzi •...•....••••••••..••••..••••.•••.•..•....••.•.••..... 300
Index of Contributors ••••••••••••••••••••••••••••••••••••.•••.••••••••• 311
VIII
Part 1
Basic Aspects of Nonlinear Systems
The first group of lectures dealt with some general aspects and basic proper
ties of the statics and dynamics of nonlinear systems. A. Bishop presented
a broad introductory survey of the different phenomena where solitons or,
more generally, solitary waves play an important role and reviewed the dif
ferent theoretical methods and models used, at present, for the analysis of
data. F. Calogero gave an introduction to the spectral transform, its ap
plication to certain classes of nonlinear evolution equations and the proper
ties of the exact solutions obtained by this method. Some general basic
properties and methodical aspects of nonlinear lattices were investigated
by H. Thomas and E. Magyari discussing also applications to Jahn-Teller
systems. In the following contribution C. Bottani and G. Caglioti in
vestigated the transition from the thermoelastic to the plastic region of
deformation which can be described as the critical point of a dynamical
instability. In his second contribution F. Calogero looked into the non
linear aspects of many-body problems and focussed on the mathematical e
lements of exactly solvable models. Finally G.F. Nardelli et al. discussed
applications of the theory of nonlinear models to biological systems.
1.1 Introduction
A. Bishop
Theoretical Division and Center for Nonlinear Studies
Los Alamos National Laboratory, MS-B262, Los Alamos, NM 87545, USA
Despite impressive progress in recent years, it remains almost impossible
to define the evolving "subject" of nonlinear physics. Certainly the para
digm [1] of "solitons" plays a central role in the condensed matter contexts
which are emphasized in these Proceedings. However nonlinear science is es
sentially interdisciplinary, both in terms of common underlying science and
the importance of the interplay of experimental, analytical and numerical
investigations [2-4]. This wider setting includes important advances in nu
merical techniques (e.g., adaptive grid methods), combustion, nonlinear dif
fusion (e.g., interface instabilities and dynamics throughout the natural
sciences), etc.
Nonlinear science is mostly a focus on old (and complex!) strongly nonlinear
phenomena but with new techniques, systematics and extensions-including use
of aomputers and synergistic approaches (as exemplified by the Fermi-Pasta
Ulam experiments or the modern history of soliton equations); concepts (so
litons and integrability, topology, frustration, chaos and the universality
of maps, fractals, etc.); generiaity of nonlinear equations (i.e., very dif
ferent phYSical systems can be described by the same basic equations if they
share underlying physics); physical scales of conceptually or technically
similar phenomena (e.g., astrophysical fronts and vortices; atmospheric
blocking states; internal solitons and gulf-stream eddies in oceans; clumps
and cavitons in turbulent plasmas; self-focusing in lasers; self-induced
transparency in optical devices; vortex configurations in two-dimensional
magnets; polarons and excitons in polymers; etc.).
Turning to condensed matter contexts specifically, the array of applications
of the basic soliton paradigm is already huge. Rather than attempting even
a partial listing we will allow this to emerge from the excellent contri
butions to these Proceedings.
The "definition" of "soliton" remains regrettably nonstandard. We should
certainly distinguish between the exact solitons of fully integrable systems
with their remarkable collision and separability properties (CALOGERO); and
similar one-dimensional systems which lack perfect integrability (e.g., the
~-four equation). In either case "solitons" come in three varieties only
kinks (e.g., sine-Gordon), pulses (e.g., Toda) , or envelopes (e.g., nonlinear
Schrodinger). It has become practice to extend the one-dimensional soliton
notion to any static or dynamic, finite energy, long-lived, spatially lo
calized structure ("inhomogeneous" or "instrinsic defect" state). It might
well be preferable to be more specific and carefully distinguish vortices,
disgyrations, disclinations, etc. On the other hand the more catholic usage
does serve to emphasize the over-riding soliton paradigm. In addition to
soliton equations, nonlinear diffusion equations continue to be important
condensed matter growth areas - e. g., i nterfacia 1 instabil iti es, pattern se
lection, crystal growth in metallurgical contexts [4] (MOLLER-KRUMBHAAR).
3
Our somewhat arbitrary collection of the most active areas of strongly non
linear studies in condensed matter and statistic physics reads:
(1) Soliton equations are a unifying key to the great majority of one- and
two-dimensional soluble models in many-body and statistical physics and
field theory, and to equivalences between them. In particular (one-di
mensional) quantum soliton systems are directly related to models for
which exact Bethe Ansatze can be constructed [5]. Important examples
include spin 1/2 Ising-Heisenberg chains, model field theories, and the
Kondo model. Of course mappings between representations (fermions, spin
1/2, Coulomb gas, etc.) are not necessarily an advantage but in some
cases they result in simplifications and point the way toward exact
solutions.
(2) Exactly soluble models (quantum or classical) are important because they
can pinpoint the influence of basic ingredients (e.g., symmetries) in
complicated phenomena and formalisms (BISHOP, CALOGERO). However, in
rea 1i sti c condensed matter contexts perturbations are i nescapab 1e - im
purities, grain boundaries, forcing terms, damping, lattice discreteness,
dimensionality, external probes, etc. There are now many perturbation
schemes investigating these and other influences. In many cases the
notion'of "particle-like" collective coordinates associated with soli
tons remains a valid and physical interpretation. This emphasizes the
importance of soliton repre,sentations as an appropriate starting ap
proximation-in contrast to a linear basis. However, perturbations must
always be carefully considered if an over-simplistic view of solitons
and their consequences is to be avoided, tempting as these can be.
(3) Perhaps the most important 1e sson to students of soliton phys i cs (be
yond the importance of soliton bases themselves) is the necessity for
considering fluctuations with respect to the bare solitons [6]. The
fluctuations may be thermal, quantum, critical, etc., or formal, as in
the case of stability analysis. The analysis is precisely the same in
each case and is so essential because there is, of course, no superpo
sition. In some cases, as for exact (fully integrable) soliton systems,
the lack of superposition (i.e., "interaction") between solutions is
purely a phase or space shift with no mode conversion (CALOGERO). Even
here, however, the asymptotic phase shifts are responsible for density
of-states changes. Again, any bound states in the fluctuation spectra
about inhomogeneous structures need special consideration, reflecting
instability, localized vibrations, and (in the case of "zero-frequency"
modes) underlying symmetries. Density-of-states and zero-frequency modes
are crucially important in the diverse applications of fluctuation ana
lysis (BISHOP, THOMAS): stability; classical or quantum statistical
mechanics; quantization (including Bethe Ansatz techniques); nucleation
theory and metastable state decay; transport theory; renorma1ization
at critical points; etc.
(4) Topological (and nontopo10gica1) intrinsic defects and their classifi
cation (e.g., by homotopy theory) are also very important ingredients
in our more unified understanding of nonlinearity [7]. Important appli
cations in condensed matter have included he1ium-3 (A and B phases),
liquid crystals, Heisenberg magnets, etc. Topological classifications
do not give energies but at least indicate where to look. Clearly, the
variety of defect states is much greater in higher spatial dimensions
or with order-parameter spaces. It is worthwhile emphasizing the most
generic importance of identifying instrinsic defect states -namely to
understand transport and relaxation (a generalization of the familiar
notion of slippage via dislocations).
4