Table Of ContentSeries in
Modern
Condensed
Matter
Physics
Vol. 1
EXOTIC PROPERTIES
OF SUPERFLUID He
G. E. Volovik
EXOTIC PROPERTIES
OF SUPERFLUID 3He
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Series In
Modern
Condensed
Metier
Physics
Vol. 1
EXOTIC PROPERTIES
OF SUPERFLUID 3He
G. E. Volovik
Landau Institute for Theoretical Physics, Moscow
World Scientific
U 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, NJ 07661
UK office: 73 Lynton Mead, Totteridge, London N20 8DH
EXOTIC PROPERTIES OF SUPERFLUID ’He
Copyright © 1992 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form
or by 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.
ISBN 981-02-0705-0
ISBN 981-02-0706-9 (pbk)
Printed in Singapore by JBW Printers & Binders Pte. Ltd.
Foreword
John Bardeen once compared condensed matter physics with a huge
deep-rooted tree. Some of its shoots may dry up, but there are always new
branches which will grow, prosper and blossom. In the last decades we
have witnessed so many exciting developments in condensed matter physics:
superfluidity in 3He, superconductivity in organic compounds and oxides,
integer and fractional quantum Hall effect, to name just a few examples.
At the same time, a number of new concepts like symmetry breaking, co
herence, scaling and fractional statistics as well as new powerful techniques
like renormalization group, topological methods and other field-theoretical
tools, have appeared. Such rapid development of modern condensed matter
physics calls for a wide activity in publications at different levels. In the
current literature there are plenty of conference proceedings and some com
prehensive review volumes accessible mainly to experts. On the other hand,
the time is not ripe yet for many of these new developments to be included
in standard textbooks.
The aim of this series is to provide an introduction to new areas of
research in condensed matter physics, mainly addressing a young readership:
graduate students and beginning research workers. The basic format is an
independent volume by an expert author of around 150 pages each. These
volumes could be lecture notes or short monographs of a lighter style. The
VI Foreword
emphasis will be on new physical concepts and interconnections with other
phenomena and fields, while the corresponding mathematical tools will be
used whenever necessary. In selecting topics we tried to pick those of broader
current scientific interest but avoiding too “hot” ones so that these volumes
can stay on the shelf for some time.
We would like to express our sincere thanks to the authors of individual
volumes who kindly agreed to take up this difficult project. We are sure
that every active researcher will deeply appreciate the authors’ willingness
to spend part of their precious research time for this educational purpose.
Hopefully, this will be a rewarding experience, as bright new minds will
come to the field, encouraged and well prepared by their predecessors. We
would highly appreciate comments and suggestions from readers and poten
tial authors.
Igor Dzyaloshinkii
Stig 0. Lundqvist
YuLu
Contents
Foreword v
1. Introduction 1
2. Broken Symmetry in Superfluid Phases of 3He 12
2.1. Symmetry G of the Normal State of Liquid 3He 12
2.2. Landau Theory of Superfluid Transition in 3He 13
2.3. Classes of Superfluids 15
2.4. Equilibrium Order Parameters for A- and B-phases
of 3He 16
2.5. Degeneracy of Equilibrium States 18
2.6. Manifold of Degenerate States in 3He-B 18
2.7. Magnetic Anisotropy in 3He-A 19
2.8. Liquid-crystal Anisotropy in 3He-A 20
2.9. Manifold of Degenerate States in 3He-B 22
2.10. Residual Symmetry H of 3He-B 23
2.11. Residual Symmetry H of 3He-A 24
2.12. Combined Spin-orbital Symmetry and Relative
Spin-orbitalA nisotropy in 3He-B 26
vii
viii Contents
3. Textures and Supercurrents in Superfluid Phases of 3He 27
3.1. Textures, Gradient Energy and Rigidity 27
3.2. Why Superfuids are Superfluid 30
3.3. Superfluidity and Response to a Transverse Gauge Field 32
3.4. Nonpotentia! Superflow in 3He-A 34
3.5. Perpetuum Motion of the A-phase 35
3.6. Textural Energy and Supercurrent in 3He-A 36
3.7. Spin Soliton in 3He-A 38
3.8. Order Parameter Textures 39
3.9. Coherence Length and London Limit 40
3.10. Disgyrations and Vortex. London Equations for the
y.
nl Orbital Texture 41
e o
us 3.11. Disgyrations and Vortex. Singularity in the Degeneracy
Parameters 44
3.12. Radial Disgyration. The Hard Core Structure 45
3.13. Pure Vortices. The Hard Core Structure 46
3.14. A-B Interface. Symmetry and Structure 48
3.15. Symmetry of the A-B Interface and Boundary Conditions 50
4. Bose Excitations in Superfluid Phases of 3He 53
4.1. Goldstone Bosons in 3He-A 53
4.2. Soft Modes Dynamics and Lie Algebra of Group G:
Spin Dynamics and SO^ Symmetry 54
4.3. Spin Waves. Goldstone and Quasi-Goldstone Modes 56
4.4. Nuclear Magnetic Resonance in 3He-A 57
4.5. NMR on Textures in 3He-A 58
4.6. Vacuum Symmetry H and Quantum Numbers of Bose and
Fermi Excitations in 3He-A 60
4.7. Bosonic Collective Modes in 3He and Irreducible
Representations of Group H 63
4.8. Bosonic Collective Modes in 3He-B 65
4.9. Dynamics of the Goldstone Fields in 3 He-B 66
4.10. Superfluid Hydrodynamics in 3He-B 68
4.11. Goldstone Bosons in 3He-B 69
Contents ix
5. Fermi Excitations in Superfluid Phases of 3He 71
5.1. Bogoliubov-Nambu Matrix for Fermions
in Pair-correlated Fermi Systems 71
5.2. Qnasiparticles in Conventional Superconductors 72
5.3. Representation for the Gap Function in 3He
and the Order Parameter 74
5.4. Quasiparticle Spectrum in 3He-B 75
5.5. Bogoliubov Hamiltonian for Quasiparticles in 3He-B
vs. Dirac Hamiltonian for Electrons 75
5.6. Lorentz Symmetry as Combined Symmetry,
View from 3He-B 77
5.7. Breaking of the Relative Lorentz Symmetry 78
5.8. Gap Nodes in the Quasiparticle Spectrum of 3He-A
Class of Intermediate Superfluids 79
5.9. Combined Gauge Symmetry and Gap Nodes 81
5.10. Stability of Gap Nodes in the A-phase.
Evolution of Fermi Points at A—>B Transition 82
5.11. Spectrum Near the Fermi Points and Relativistic
Massless Particles 84
5.12. Left-handed and Right-handed Fermions Near the
Fermi Points 86
5.13. Spin-orbit Waves and W Bosons 89
5.14. Mass of the W Bosons is Zero in the BCS Theory of
3He-A 90
5.15. Hidden Symmetry in the A-phase 91
5.16. Origin of the W Boson Mass in 3He-A 93
5.17. Gravitons in 3He-A 94
5.18. Cosomologicai Term in the Einstein Equations,
View from 3He-A 95
6. Orbital Dynamics and Anomalies in Quantum
Field Theory 98
6.1. Lie Algebra of Poisson Brackets for A-phase
Orbital Dynamics 98
