Magnetic Systems With Competing Interactions PDF
Preview Magnetic Systems With Competing Interactions
MAGNETIC 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. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NT 07661 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 orby any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, USA. 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
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