Table Of ContentSTRONG INTERACTIONS IN LOW DIMENSIONS
Physics and Chemistry of Materials
with Low-Dimensional Structures
VOLUME 25
Editor-in-Chief
F. LÉVY, Institut de Physique Appliquée, EPFL,
Département de Physique, PHB-Ecublens, CH-1015 Lausanne, Switzerland
Honorary Editor
E. MOOSER, EPFL, Lausanne, Switzerland
International Advisory Board
J. V. ACRIVOS, San José State University, San José, Calif., U.S.A.
R. GIRLANDA, Università di Messina, Messina, Italy
H. KAMIMURA, Dept. of Physics, University of Tokyo, Japan
W. Y. LIANG, Cavendish Laboratory, Cambridge, U.K.
P. MONCEAU, CNRS, Grenoble, France
G. A. WIEGERS, University of Groningen, The Netherlands
The titles published in this series are listed at the end of this volume.
STRONG INTERACTIONS IN
LOW DIMENSIONS
Edited by
D. Baeriswyl
Department of Physics,
University of Fribourg,
Fribourg, Switzerland
and
L. Degiorgi
Solid State Physics Laboratory,
ETH Zürich, Switzerland
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON/ LONDON
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 1-4020-1798-7 (HB)
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TABLE OF CONTENTS
Chapter 1 – Introduction
Strong interactions in low dimensions: introductory remarks 1
D. Baeriswyl and L. Degiorgi
[email protected], [email protected]
Chapter 2
Dynamic correlations in quantum magnets 21
C. Broholm and G. Aeppli
[email protected], [email protected]
Chapter 3
Angle resolved photoemission in the high temperature
superconductors 63
J.C. Campuzano
[email protected]
Chapter 4
Luttinger liquids: the basic concepts 93
K. Scho¨nhammer
[email protected]
Chapter 5
Photoemission in quasi-one-dimensional materials 137
M. Grioni
marco.grioni@epfl.ch
Chapter 6
Electrodynamic response in “one-dimensional” chains 165
L. Degiorgi
[email protected]
v
vi Strong interactions in low dimensions
Chapter 7
Optical conductivity and correlated electron physics 195
A.J. Millis
[email protected]
Chapter 8
Optical signatures of electron correlations in the cuprates 237
D. van der Marel
[email protected]
Chapter 9
Charge inhomogeneities in strongly correlated systems 277
A.H. Castro Neto and C. Morais Smith
[email protected], [email protected]
Chapter 10
Transport in quantum wires 321
A. Yacoby
[email protected]
Chapter 11
Transport in one dimensional quantum systems 347
X. Zotos and P. Prelovˇsek
xenophon.zotos@epfl.ch
Chapter 12
Energy transport in one-dimensional spin systems 383
A.V. Sologubenko and H.R. Ott
[email protected], [email protected]
Chapter 13
Duality in low dimensional quantum field theories 419
M.P.A. Fisher
[email protected]
Subject Index 439
Materials Index 441
Chapter 1
STRONG INTERACTIONS IN LOW
DIMENSIONS: INTRODUCTORY
REMARKS
D. Baeriswyl
D´epartement de Physique, Universit´e de Fribourg, P´erolles, CH-1700 Fribourg,
Switzerland.
[email protected]
L. Degiorgi
Laboratorium fu¨r Festko¨rperphysik, ETH Zu¨rich, CH-8093 Zu¨rich, Switzerland.
[email protected]
The physical properties of low–dimensional systems have fascinated
researchersforagreatpartof thelastcentury, andhaverecently become
one of the primary centers of interest in condensed matter research. At
the beginning, this field appeared much more like a playground for cre-
ative theorists than a serious domain of solid–state physics. In fact, the
exact treatment of the one–dimensional Lenz–Ising model and Bethe’s
ingeniousdiagonalization oftheantiferromagneticHeisenbergchainwere
considered at most as firststeps towards a theory of electronic and mag-
neticpropertiesofreal,three–dimensionalcrystals. Similarly,Tomonaga
presented his study of sound waves in a one–dimensional system of in-
teracting fermions as mathematically interesting but physically not very
useful. Acollection ofimportantearlydevelopments hasbeenassembled
andcommenteduponbyLiebandMattis[1], whoemphasizedthatwhile
exact solutions of one–dimensional models provide useful tests for ap-
proximate methods, “in almost every case the one–dimensional physics
is devoid of much structure, and describes a colorless universe much
less interesting than our own”. This point of view was based on the
observation that “in one dimension bosons do not condense, electrons
do not superconduct, ferromagnets do not magnetize, and liquids do
not freeze”. Fortunately, it has since been demonstrated that the one–
1
D. Baeriswyl and L. Degiorgi (eds.), Strong Interactions in Low Dimensions, 1–19.
© 2004 by Kluwer Academic Publishers, Printed in the Netherlands.
2 Strong interactions in low dimensions
dimensionalworld has its own richness, and thatthere arereal materials
which to a good approximation may be considered as consisting of un-
coupled chains, at least for temperatures or frequencies which are not
too low. Some of the specific features predicted for interacting one–
dimensional electron systems, such as charge– and spin–density waves,
haveindeedbeenobserved inmanyquasi–one–dimensional materials [2],
and under particular circumstances, such as Fermi–surfacenesting,can
also be found in higher dimensions.
In this book we attempt to convey the colorful facets of condensed
matter systems with reduced dimensionality. We are of course aware
of the fact that many important aspects must be left aside in such a
collection of specific subjects; some of the most regretful omissions will
be mentioned later. The following introductory remarks are intended as
an aid to identifying some of the essential concepts which will reappear
at several places in the subsequent chapters. At the same time this
introductionmayhelptoconnectthedifferenttopicstreatedinthebook,
some of which might at first sight appear rather disparate.
1. Ordering in low dimensions
Inourthree–dimensionalworldweareaccustomedtothespontaneous
appearance of order at sufficiently low temperatures. Liquids condense
to form periodic solids, magnetic moments are aligned in ferromagnetic
orantiferromagneticconfigurations,andFermiliquidsturnintosuperflu-
ids. Atzerotemperaturethestateoflowestenergydeterminesthestable
configuration of a system, but at finite temperatures it is the minimum
of the free energy which determines whether the order parameter – the
magnetization in a ferromagnet, the condensate fraction in a superfluid
or the intensity of Bragg peaks in a periodic solid – remains finite or
is completely suppressed due to thermal fluctuations. The destabilizing
effects of temperature are particularly strong in one dimension. While
in two or higher dimensions the Ising model exhibits long–range order
below a finite critical temperature, this is no longer true in one dimen-
sion, where thermal fluctuations destroy the spin correlations beyond
a finite correlation length. These fluctuations are even more effective
in the case of a continuous order parameter. Thus, according to the
Mermin–Wagner theorem, the classical two–dimensional XY model has
no true long–range order at finite temperatures [3]. Nevertheless this
model – which can be used for representing the phase fluctuations of
a complex order parameter or the spin configurations in an easy–plane
Heisenberg ferromagnet – shows a transition from a disordered high–
temperature phase with exponentially decaying correlations to a phase
Strong interactions in low dimensions: introductory remarks 3
withquasi–long–range order below theKosterlitz–Thouless temperature
T . The appropriate quantity describing this transition is the phase
KT
stiffness or superfluid density, which is finite below T and shows a
KT
universal discontinuity at T [4].
KT
At zero temperature classical spin models on a bipartite lattice have
long–range order, but this is not necessarily true for the corresponding
quantummodels. Thusforboththespin–1 quantumXYandHeisenberg
2
modelsthespincorrelationsdecayalgebraically, andeventheIsingchain
becomes disordered under a sufficiently strong transverse field. The
latter case is very illuminating [5, 6] as the external field allows one to
drivethesystemthroughaquantumcritical point,asecond–orderphase
transition at T = 0 where quantum fluctuations are relevant [7]. For
dimensions higher than 1 the quantum XY model has long–range order
for all values of the spin, at least on a hypercubic lattice [8]. This result
has also been proven for the quantum Heisenberg antiferromagnet for 3
or higher dimensions (and any spin) [9]. For dimension D=2 a rigorous
proof still seems not to be available for S = 1, but both numerical
2
simulations and analytical calculations indicate that long–range order
does exist, albeit with a reduced moment [10].
Low–dimensional systems not only experience strong quantum and
thermal fluctuations, but also admit ordering tendencies which are diffi-
cult to realize in three–dimensional materials. Prominent examples are
spin– and charge–density waves in quasi–one–dimensional organic com-
pounds and spontaneous circulating currents (leading to “orbital anti-
ferromagnetism”) in two dimensions. The competition among several
possibleorder parameters leads to rich phase diagrams and an enhanced
sensitivity to disorder or applied external fields. Some of these order
parameters are very difficult to observe directly. As an example, while
phases with spontaneous spin or charge currents around the plaquettes
of a square lattice occur naturally in models of interacting fermions in
twodimensions[11], theirunambiguousdetection appearstobeverydif-
ficult. Thus it is at present not clear whether the so–called “pseudogap
phase” in the layered cuprates is related to such a hidden order para-
meter [12].
2. Dimensional crossover
A real material is not truly one–dimensional (1D), butat most quasi–
one–dimensional, i.e. a collection of weakly coupled chains. (Notable
exceptions include quantum wires and nanotubes.) Thus one of the
important questions will be the extent to which the coupling between
chains is relevant. Anillustrative exampleis theIsingmodelon asquare
