Quantised Vortices: A Handbook of Topological Excitations

160 Pages·2019·26.476 MB·English

by  Tapio Simula

Quantised Vortices
A handbook of topological excitations

Quantised Vortices
A handbook of topological excitations
Tapio Simula
Swinburne University of Technology, Melbourne, Australia
Morgan & Claypool Publishers
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ISBN 978-1-64327-126-2(ebook)
ISBN 978-1-64327-123-1(print)
ISBN 978-1-64327-124-8(mobi)
DOI 10.1088/2053-2571/aafb9d
Version:20190701
IOPConcisePhysics
ISSN2053-2571(online)
ISSN2054-7307(print)
AMorgan&ClaypoolpublicationaspartofIOPConcisePhysics
PublishedbyMorgan&ClaypoolPublishers,1210FifthAvenue,Suite250,SanRafael,CA,
94901,USA
IOPPublishing,TempleCircus,TempleWay,BristolBS16HG,UK
Dedication
Ollille

Contents
Preface xiii
Acknowledgements xvi
Author biography xvii
Part I Vortices in Flatland
1 Vortices 1-1
1.1 Space–time symmetries 1-1
1.2 Quantum liquids 1-2
1.3 Vorticity in classical fluids 1-3
1.4 Vorticity in quantum liquids 1-4
2 Quasiparticle picture 2-1
2.1 Emergence of quasiparticles 2-1
2.2 Boson commutation relations 2-2
2.3 Fermion anticommutation relations 2-2
2.4 Majorana relations 2-3
2.5 Anyon quasiparticles 2-3
2.6 Non-abelian anyon quasiparticles 2-4
2.7 Bogoliubov–de Gennes equations 2-4
2.8 Time-reversal symmetry 2-5
2.9 Particle–hole symmetry 2-6
2.10 Chiral symmetry 2-7
2.11 Phonon spectrum 2-8
2.12 Landau critical velocity 2-9
2.13 Roton–maxon spectrum 2-9
2.14 Edge modes 2-10
2.15 Dipole, breathing, quadrupole and scissors modes 2-11
2.16 Kelvin mode vortex waves 2-12
2.17 Tkachenko mode vortex waves 2-14
2.18 Caroli–de Gennes–Matricon modes 2-15
2.19 Nambu–Goldstone zero mode 2-15
2.20 Majorana zero mode 2-16
2.21 Magnon spin waves 2-17
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QuantisedVortices
3 Cold atoms 3-1
3.1 Scalar Bose–Einstein condensates 3-1
3.2 Bose zero-temperature energy functional 3-3
3.3 Thomas–Fermi relations 3-4
3.4 Healing length 3-4
3.5 Thermodynamic relations 3-5
3.6 Quantum hydrodynamic equations 3-6
3.7 Two-component Bose–Einstein condensates 3-7
3.8 Spin-1 Bose–Einstein condensates 3-8
3.9 Spin-2 Bose–Einstein condensates 3-10
3.10 High-spin Bose–Einstein condensates 3-11
3.11 Representations of spinor Bose–Einstein condensates 3-12
3.12 Exotic interactions 3-13
3.13 Bardeen–Cooper–Schrieffer mean-field theory 3-14
3.14 Ultracold Fermi gases 3-15
3.15 Dirac–Bogoliubov–de Gennes systems 3-17
3.16 Gapless, massless, linear spectra 3-17
3.17 Gapped, massive, quadratic spectra 3-18
4 Topological invariants and quantities 4-1
4.1 Topology and ordered structures 4-1
4.2 A game of lines and loops 4-1
4.3 Maps and order parameters 4-2
4.4 Homotopy classification of defects 4-3
4.5 Burgers vector 4-4
4.6 Gauss–Bonnet theorem 4-4
4.7 Winding number 4-5
4.8 Berry phase, curvature, and connection 4-6
4.9 Chern number 4-7
4.10 Linking number, writhe and twist 4-8
4.11 Helicity 4-9
4.12 Enstrophy 4-10
4.13 Kauffman bracket polynomial 4-10
4.14 Jones polynomial 4-10
viii
QuantisedVortices
5 Topological excitations 5-1
5.1 Topological defects 5-1
5.2 Soliton 5-1
5.3 Bright soliton 5-2
5.4 Grey and dark soliton 5-2
5.5 Solitonic vortex 5-3
5.6 Plain vortex 5-4
5.7 Polynomial vortex 5-5
5.8 Coherence vortex 5-6
5.9 Fractional vortex 5-6
5.10 Baby skyrmion 5-7
5.11 Monopole 5-8
5.12 Fluxon, chargeon, and dyon 5-8
5.13 Alice vortex and Cheshire charge 5-9
6 Structure of a plain vortex 6-1
6.1 Vortex uncertainty principle 6-1
6.2 Kelvon 6-2
6.3 Circulation quantum 6-3
6.4 Vortex energy 6-3
6.5 Thermodynamic stability 6-4
6.6 Spectral, energetic stability 6-5
6.7 Dynamical Lyapunov stability 6-5
6.8 Inertial vortex mass 6-6
6.9 Gravitational vortex mass 6-7
6.10 Kelvon-based vortex mass 6-7
6.11 Hydrodynamic induced vortex mass component 6-8
6.12 Relativistic vortex mass component 6-8
6.13 Baym–Chandler vortex mass 6-9
6.14 Kopnin vortex mass 6-9
7 Vortex dynamics 7-1
7.1 Adiabatic vortex dynamics 7-1
7.2 Vortex force and velocity 7-2
7.3 Magnus effect and mutual induction 7-3
7.4 Vortex pair creation and annihilation 7-3
7.5 Onsager point vortex model 7-4
ix