Table Of ContentMAGNETIC SYSTEMS
WITH
COMPETING INTERACTIONS
(Frustrated Spin Systems)
edited by H T Diep
World Scientific
MAGNETIC SYSTEMS
WITH
COMPETING INTERACTIONS
(Frustrated Spin Systems)
This page is intentionally left blank
MAGNETIC SYSTEMS
WITH
COMPETING INTERACTIONS
(Frustrated Spin Systems)
edited by
HTDiep
Groupe de Physique Statistique
Universite de Cergy-Pontoise
Cergy-Pontoise, Cedex
France
World Scientific
Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
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UK office: 73 Lynton Mead, Totteridge, London N20 8DH
MAGNETIC SYSTEMS WITH COMPETING INTERACTIONS
(FRUSTRATED SPIN SYSTEMS)
Copyright © 1994 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form
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ISBN: 981-02-1715-3
Printed in Singapore by Uto-Print
PREFACE
Magnetic systems with competing interactions have been first investigated four
decades ago. Well-known examples include the Ising model on the antiferromag-
netic triangular lattice studied by G. H. Wannier in 1950 and the Heisenberg helical
structure discovered independently by A. Yoshimori and J. Villain in 1959. However,
extensive investigations on magnetic systems with competing interactions have really
started with the concept of frustration introduced almost at the same time by G.
Toulouse and J. Villain in 1977 in the context of spin glasses. Therefore, the title of
the book should have been Frustrated Spin Systems. The frustration was initially
defined for Ising spin systems. It was later generalized to vector spin systems. The
frustration is generated by the competition of different kinds of interaction and/or
by the lattice geometry. As a result, in the ground state a number of spins in a
frustrated Ising system behave as free spins. In the case of frustrated vector spin
systems, the ground-state configuration is usually non-collinear. These ground-state
properties (degeneracy and symmetry) give rise to spectacular and often unexpected
behaviors at finite temperatures.
Many properties of frustrated systems are still not well understood at present.
Recent studies shown in this book reveal that many established theories have never
before encountered so many difficulties as they do now in dealing with frustrated
systems. In some sense, frustrated systems are excellent candidates to test approxi
mations and to improve theories.
The contents of this book cover recent developments in the problem of frustration
effects in spin systems. The chapters of this book are written by researchers who have
actively contributed to the field. Many results are from recent works of the authors.
Some new, yet unpublished results and a number of unsettled questions are also
included.
The book is intended for post-graduade students as well as researchers in statis
tical physics, magnetism, materials science and various domains where real systems
can be described with the spin language. Explicit demonstrations of formulae and
full arguments leading to important results are given where it is possible to do so.
Pedagogical effort has been made to make each chapter to be self-contained, compre
hensible for researchers who are not really involved in the field. Basic methods are
given in detail.
V
vi Preface
The systems shown in this book include two- and three-dimensional systems of
Ising, XY or Heisenberg spins. For practical purposes, each chapter treats a different
aspect of the problem. The book is organised as follows.
The first three chapters treat frustrated vector spin systems. The two following
chapters study frustrated Ising spin systems. In these five chapters, only systems
periodically defined on a lattice are considered. Frustration effects are thus studied
without interaction disorder. The remaining two chapters are devoted to random
magnets and experimental studies of frustrated spin systems, respectively.
I summarize in the following the contents of each chapter.
Chapter I is devoted to critical properties of frustrated XY and Heisenberg spin
systems. A review on various models studied so far is presented. The nature of
the phase transition is discussed by mean-field theory and Monte Carlo simulations.
These methods are described in detail. The utility of mean-field theory is demon
strated, in particular the description of ordered states in terms of the spin density
s(r) written as a Fourier expansion, and analysis of Landau-type free energies F[s(r)].
Such studies make clear the symmetries broken at phase transitions and can lead to
very good agreement with experimental results on magnetic phase diagrams. Most
of the results discussed in this chapter concern frustration caused by lattice geome
try (mostly triangular), or the effects of further-neighbor interactions. The authors
emphasize phase-transition phenomena in three-dimensional (3D) systems, induced
by varying temperature or an applied magnetic field. In particular, magnetic phase
diagrams which exhibit unusual multicritical-point structures are examined. Com
parison with experimental results is emphasized, for the most part from work on the
quasi-lD hexagonal insulators ABX (such as CsNiCl).
3 3
Chapter II deals with the renormalization of the effective theories relevant for
classical and quantum frustrated Heisenberg models. Both Landau-Ginzburg-Wilson
and Non-.Linear-Sigma models are studied by the e = 4 — D, e = D — 2 and large
N expansions. Since no agreement is found betweeen these perturbative approaches,
the authors discuss the possible scenarios for the physics of the classical systems in
D = 3. In particular, a detailed discussion is given on the controversial issue of the
nature of the phase transition in canted spin systems. In the quantum case, a study
is made of the phase transition that occurs in two dimensions at zero temperature
between a semi-classical Neel ordered phase and a quantum disordered phase as the
spin magnitude is varied. It is shown that one of the most effective signature at finite
size of the transition is the existence and the scaling of a tower of states that collapse
onto the ground state in the thermodynamical limit. At finite temperature it is
shown that the behavior of the correlation length as a function of the temperature
gives also a test of the nature of the order at zero temperature.
Chapter HI treats some quantum Heisenberg antiferromagnets in low dimension
at zero temperature. The so-called J\ — J1 model in one dimension and on the square
lattice are studied by finite-size scaling analysis. The model on a square lattice has
attracted most attention as a rather crude model of the effects of doping on coDDer
Preface vii
oxide planes. The square sites correspond to the copper sites in the two-dimensional
plane of copper and oxygen that is the common structural feature of the family of
superconducting copper oxides. A shorter review of work concerning more specifically
with triangular and Kagome lattices is also given. The interest of the last is that
there is a good experimental realization. Special attention is paid to the case of
spin 1/2 for which quantum fluctuations should be most noticeable. The aim of this
chapter is to discuss calculations of the last few years which attempt to give correct
answers to the question of what happens to a quantum Heisenberg antiferromagnet
when one introduces interactions or geometry such that in a classical picture not all
antiferromagnetic bonds may be satisfied.
Chapter IV shows the frustration effects in exactly solved two-dimensional Ising
models. The systems considered in this chapter are periodically defined (without
bond disorder). The frustration due to competing interactions will itself induce dis
order in the spin orientations. The results obtained can be applied to physical systems
that can be described by a spin language. After a detailed presentation of 16- and
32-vertex models, applications are made to some selected systems which possess most
of the spectacular features due to the frustration such as high ground-state degener
acy, reentrance, successive phase transitions and disorder solutions. In some simple
models, up to five transitions separated by two reentrant paramagnetic phases are
found. A conjecture is made on the origin of the paramagnetic reentrant phase. The
nature of ordering as well as the relation between the considered systems and the
random-field Ising model are discussed. The relevance of disorder solutions for the
reentrance phenomena is also pointed out.
Chapter V deals mainly with the Ising model on the antiferromagnetic triangular
and stacked triangular lattices. Ground-state properties and the nature of the phase
transition are studied by various methods, as functions of the spin magnitude S
and nearest- and next-nearest-neighbor interactions. It is shown in this chapter that
the symmetry of spin ordering is strongly dependent S. Furthermore, due to the
frustration, there exist "free" spins or "free" linear-chains, on which internal fields
are canceled out, in some frustrated Ising spin systems. These free spins and free
linear-chains play an important role as for spin orderings. Existence of these free
spins and free linear-chains is explicitly shown in this chapter and the role of them
is discussed. Another characteristic feature of frustrated Ising spin systems is that
various metastable states exist in these systems. Existence of metastable states is
closely related to the degeneracy of ground state and also to the excited states. These
metastable states may give rise to a first order phase transition as found in some
models introduced in this chapter. The effects of the far-neighbor interactions in the
Ising model on the antiferromagnetic triangular and stacked triangular lattices are
clarified.
Chapter VI is devoted to the problem of reentrant spin glass. The previous chap
ters of this book have largely focused on the novel and interesting competing effects
induced in magnetic systems via the presence of frustration. Frustration is also ubiq-
viii Preface
uitous in random systems. For example, a change in concentration of some species in a
magnetic ferromagnet alloy can introduce randomly located antiferromagnetic bonds
and, consequently, generate "random frustration". In the limit of large random frus
tration a magnetic system cannot accomodate a percolating ordered magnetic cluster
at zero temperature labelled by a magnetic Bragg peak at a well defined wavevector
q. However, in that case, a randomly frustrated magnet often displays a cooperative
freezing transition into a spin glass state. In the case of weak disorder and small
random frustration, there is competition between conventional magnetic order and
the spin glass ordering. In that regime, one observes experimentally the reentrant
spin glass (RSG) behavior. In the RSG regime, a system develops partial collinear
ferromagnetic (or antiferromagnetic) order at some temperature T. For a range of
c
temperature T < T < T, the system behaves largely like a conventional unsatu-
g c
rated magnet. However, below T, the system shows magnetic hysteresis and displays
g
properties analogous to those found in conventional spin glasses, hence the name
"reentrant spin glasses". The physics at the origin of this two-transition behavior has
remained for twenty years a subject of intense controversy. Chapter VI reviews the
progress made in the past five years.
Chapter VII describes experimental results on geometrically-frustrated magnetic
systems. In particular, results on stacked triangular lattice antiferromagnets and he-
limagnets are shown and compared to theoretical predictions presented in Chapters I
and II. Materials containing antiferromagnetically-coupled magnetic moments which
reside on geometrical units can inhibit a frustrated state at low temperatures. The
materials of interest may undergo phase transitions with novel properties to an un
usual ordered state, or they may not undergo a conventional phase transition at all.
This chapter reviews recent experimental progress in understanding the phases and
phase transitions displayed by several such magnetic materials. Much of it will focus
on neutron-scattering studies, as such studies have played a central role in charac
terizing the properties of these antiferromagnets and their phase transitions. The
review presented here is not intended to be exhaustive in nature, but rather to give
an overview of several materials with which the author is familiar, and to discuss
some of the experimental challenges which must be dealt with in such studies.
As a number of issues treated in this book are still debated, I alert the reader
that the authors of each chapter have taken the liberty to express their viewpoint on
each unsettled issue. This concerns specially the nature of the phase transition in
canted spin systems discussed in Chapters I, II and VII. Since the domains treated
in this book are currently investigated, it is clear that in a few years from now,
progress will be made and some of the unsettled questions discussed in this book will
be understood.
H. T. Diep
University of Cergy-Pontoise, France.
Spring 1994-
CONTENTS
Preface v
Chapter I. Critical Properties of Frustrated Vector Spin
Systems
M. L. Plumer, A. CailU, A. Mailhot and H. T. Diep
1. Introduction 1
2. Various models and predictions 4
2.1. Triangular lattices and helical spin structures 5
2.2. Fee, hep and rhombohedral antiferromagnets: order by disorder 9
2.3. Kagome and pyrochlore antiferromagnets: more disorder 11
3. Mean-field Landau models and Monte Carlo simulations 13
3.1. Phenomenological Landau-type free energy 14
3.2. Mean-field derivation of a Landau free energy 17
3.3. Monte Carlo simulations 19
4. Simple hexagonal antiferromagnet: magnetic phase
diagrams and multicritical points 23
4.1. Planar model with antiferromagnetic interlayer coupling 25
4.2. Planar model with ferromagnetic interlayer coupling 31
4.3. Heisenberg model with axial anisotropy 35
4.4. Next-nearest-neighbor interactions 39
5. Other problems of interest 41
6. Concluding remarks 43
7. Acknowledgements 45
8. References 45
V
