Skyrmions and Hall Transport

394 Pages·2023·23.41 MB·English

by  Bom Soo Kim

Skyrmions and
Hall Transport

Skyrmions and
Hall Transport
Bom Soo Kim
Published by
Jenny Stanford Publishing Pte. Ltd.
101 Thomson Road
#06‐01, United Square
Singapore 307591
Email: [email protected]
Web: www.jennystanford.com
British Library Cataloguing‑in‑Publication Data
A catalogue record for this book is available from the British Library.
Skyrmions and Hall Transport
Copyright © 2023 Jenny Stanford Publishing 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
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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
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required from the publisher.
ISBN 978‐981‐4968‐34‐8 (Hardcover)
ISBN 978‐1‐003‐37253‐0 (eBook)
To Myung‑Duk, Young‑Kyung, and Dong‑Yon

Contents
Preface xi
Acknowledgements xv
Introduction xvii
1 Symmetries of Magnetic Skyrmions 1
1.1 Symmetry 1
1.2 Action formalisms for localized spins 6
1.2.1 Landau‐Lifshitz equation 6
1.2.2 Landau‐Lifshitz revisited 11
1.2.3 Spin Lagrangian 15
1.3 Stability of magnetic skyrmions in chiral magnet 19
1.3.1 Helicoidal state in Nematic liquid crystals 19
1.3.2 Stability of spin spiral state 23
1.3.3 Stability of skyrmion crystals 29
1.3.4 Topological nature and skyrmion charge 37
1.4 Symmetries of skyrmion action 40
1.4.1 Stress energy tensor 41
1.4.2 Translation symmetry 44
1.4.3 Rotation symmetry 47
1.4.4 (Angular) Momentum of the skyrmion 51
1.5 Mutually compatible observables 53
2 Hydrodynamics 57
2.1 Introduction: Relativistic hydrodynamics 57
2.1.1 Ideal hydrodynamics 58
2.1.2 First derivative order 60
2.2 Hydrodynamics: New developments 62
2.2.1 Hydrodynamics with broken parity 62
viii Contents
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2.2.2 Hydrodynamics with broken boost 64
2.3 Hall viscosity: Introduction 66
2.3.1 Another view of stress tensor 66
2.3.2 Hall viscosity: A geometric picture 71
2.4 Charged hydrodynamics 74
2.4.1 With broken parity 78
2.4.2 With broken boost 81
2.5 Green’s function and Kubo formula 83
2.5.1 Linear response theory 83
2.5.2 Background field methods 88
2.5.3 Thermodynamic response functions 92
2.6 Irreversible thermodynamics 94
2.6.1 Onsager’s reciprocal relation 96
2.6.2 Seebeck, Peltier, and Thompson effects 100
2.6.3 Thermo‐electromagnetic effects 104
3 Hall Viscosity 109
3.1 Charged particles in electromagnetic fields 109
3.1.1 Quantum Hall fluid 113
3.1.2 Wave function 115
3.2 Landau Hamiltonian on torus 120
3.2.1 Flat torus 120
3.2.2 Deformed torus 124
3.3 Berry phase and Hall viscosity 128
3.3.1 Adiabatic process and Berry phase 128
3.3.2 Kubo formula for Hall viscosity 130
3.3.3 Hall viscosity of quantum Hall states 132
3.4 Quantum Hall systems and Hall conductivity 135
3.4.1 Conductivity for a single particle 137
3.4.2 Kubo formula for Hall conductivity 141
3.4.3 Topology and Hall conductivity 144
3.4.4 Hall conductivity with momentum dependence 148
3.5 Hall conductivity and Hall viscosity 148
3.5.1 Non‐homogeneous electric field 148
3.5.2 Hall viscosity in terms of Hall conductivity 151
4 Spin Dynamics 155
4.1 Landau‐Lifshitz‐Gilbert equation 156
4.1.1 Ferromagnetism 156
Contents ix
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4.1.2 Basic understanding of spin torque 159
4.1.3 Domain Wall illustration of LLG equation 162
4.2 Spin torques 164
4.2.1 Spin transfer torque 165
4.2.2 Domain wall illustration of STT 168
4.2.3 Spin‐orbit torque 173
4.2.4 Domain Wall illustration of SOT 177
4.2.5 Spin Hall torque 180
4.2.6 Domain Wall illustration of SHT 185
4.2.7 Emergent electromagnetic field 187
4.3 Generalized LLG equation 187
4.4 Thiele equation 189
4.4.1 Thiele’s original derivation 189
4.4.2 Generalizations with various spin torques 192
4.4.3 Generalization with transverse velocity 196
4.5 Spin effects on transport 197
4.5.1 Spin Seebeck effect 200
4.5.2 Spin Peltier effect 203
5 Ward Identity 207
5.1 Symmetries and Ward identity 208
5.1.1 Symmetry and equation of motion 208
5.1.2 Geometric understanding of Ward identities 210
5.2 An example 214
5.2.1 Ward identity for global currents 214
5.2.2 Ward identity in momentum space 217
5.3 Quantum field theory Ward identities 220
5.3.1 Ward identities for stress energy tensor 221
5.3.2 Ward identities based on symmetries 223
5.3.3 Ward identity with conserved current 234
5.3.4 Galilean invariant Ward identities 243
5.3.5 Ward identities with dissipative terms 248
5.4 Hall viscosity from Ward identities 250
5.4.1 Spectral representation 253
5.4.2 Fermions in magnetic field 256
5.5 Ward identity with topological charge 262
5.5.1 Topological charge as a central extension 262
5.5.2 Topological Ward identity 267