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William G. Unruh Ralf Schützhold (Eds.)
Quantum Analogues:
From Phase Transitions
to Black Holes
and Cosmology
ABC
Editors
WilliamG.Unruh RalfSchützhold
DepartmentofPhysics&Astronomy InstitutfürTheoretischePhysik
UniversityofBritishColumbia TechnischeUniversitätDresden
6224AgriculturalRoad 01062Dresden,Germany
Vancouver,B.C.V6T1Z1,Canada E-mail:[email protected]
E-mail:[email protected] dresden.de
W.G.UnruhandR.Schützhold,QuantumAnalogues:FromPhaseTransitionstoBlack
HolesandCosmology,Lect.NotesPhys.718(Springer,BerlinHeidelberg2007),DOI
10.1007/b11804185
LibraryofCongressControlNumber:2007921534
ISSN0075-8450
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ISBN-13 978-3-540-70858-2SpringerBerlinHeidelbergNewYork
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Acknowledgements
The present book contains a series of selected lectures1 from the interna-
tional workshop2 on “Quantum Simulations via Analogues” which took place
at the Max Planck Institute for the Physics of Complex Systems in Dresden
(Germany) from July 25th till 28th in 2005. Financial support from the pro-
gramme “Cosmology in the Laboratory” (COSLAB) of the European Science
Foundation(ESF)aswellasfromtheMaxPlanckInstituteforthePhysicsof
Complex Systems is gratefully acknowledged. Furthermore, we would like to
thankMandyLochar(fromtheMaxPlanckInstituteforthePhysicsofCom-
plex Systems) as well as Sarah Mostame, Friedemann Queisser, Dr. Gernot
Schaller, Markus Tiersch, Michael Uhlmann, and Dr. Yan Xu (all from the
Emmy-NoetherResearchGroupattheInstituteforTheoreticalPhysicsatthe
DresdenUniversityofTechnology,Germany)fortheirvaluablehelpregarding
the organisation of this workshop. The Emmy-Noether Research Group has
been supported by the the German Research Foundation (DFG). Finally, we
would like to thank the authors of the chapters for their efforts and – last
but not least – Friedemann Queisser and Markus Tiersch for combining the
various contributions into a book.
Vancouver, Dresden Ralf Schu¨tzhold
January 2007 William G. Unruh
1 The lecture given by J. Yngvason is not included here since it has already been
publishedelsewhereandcanbefoundinLect.NotesPhys.690,199–215(2006).
2 http://www.mpipks-dresden.mpg.de/~quasim05/
Contents
The Analogue Between Rimfall and Black Holes
W. G. Unruh.................................................... 1
Reference ....................................................... 4
Effective Horizons in the Laboratory
R. Schu¨tzhold ................................................... 5
1 Introduction ................................................. 5
1.1 Preliminaries............................................ 5
1.2 The Acoustic Analogy.................................... 6
1.3 Generalisations .......................................... 7
1.4 Geometric Concepts...................................... 7
2 Event Horizon ............................................... 8
2.1 Black Hole Thermodynamics .............................. 8
2.2 Hawking Effect and Trans-Planckian Problem ............... 9
2.3 De Laval Nozzle ......................................... 10
2.4 Impact of Dispersion Relation ............................. 11
3 Cosmic Horizons ............................................. 14
3.1 Particle Horizon ......................................... 16
3.2 Time-dependent Phase Transitions at Zero Temperature ...... 18
3.3 Similarities to Cosmic Inflation ............................ 21
3.4 Expanding Bose–Einstein Condensates ..................... 23
4 Summary and Outlook........................................ 24
References ...................................................... 28
Quantum Phase Transitions from Topology
in Momentum Space
G. E. Volovik.................................................... 31
1 Introduction ................................................. 31
2 Fermi Surface and Lifshitz Transition ........................... 34
2.1 Fermi Surface as a Vortex in p-space ....................... 34
VIII Contents
2.2 Lifshitz Transitions ...................................... 36
2.3 Metal-superconductor Transition........................... 37
3 Fermi Points................................................. 41
3.1 Fermi Point as Topological Object ......................... 41
3.2 Quantum Phase Transition in BCS–BEC Crossover Region.... 45
3.3 Quantum Phase Transitions in Standard Model.............. 47
4 Fermi Lines.................................................. 53
4.1 Nodes in High-T Superconductors ......................... 53
c
4.2 Z -Lines................................................ 54
2
4.3 Gap Induced by Interaction Between Layers................. 56
4.4 Reentrant Violation of “Special Relativity”
in Bilayer Graphene...................................... 57
4.5 Quantum Phase Transition in High-T Superconductor ....... 59
c
5 Topological Transitions in Fully Gapped Systems................. 59
5.1 Skyrmion in 2-Dimensional Momentum Space ............... 59
5.2 Quantization of Physical Parameters ....................... 61
5.3 Quantum Phase Transitions............................... 62
5.4 Quantum Phase Transition in 1D Quantum Ising Model ...... 66
6 Conclusion .................................................. 70
References ...................................................... 70
Superfluid 3He as a Model System for Cosmology –
Experimental Point of View
P. Skyba........................................................ 75
1 Introduction ................................................. 75
2 Basic Properties of the Superfluid 3He .......................... 76
3 States with Coherent Spin Precession in 3He-B
and their Cosmological Analogues .............................. 80
3.1 A Spin Wave Analogue of a Black Hole ..................... 82
3.2 A Persistent Precessing Domain as an Analogue of Q-Ball..... 83
4 Search for an Unruh Effect Analogue in 3He-B ................... 85
4.1 Vibrating Wire – An Accelerated Detector in Superfluid 3He-B 86
5 Discussion................................................... 91
References ...................................................... 92
Dynamical Aspects of Analogue Gravity:
Quantum Backreaction in Bose-Einstein Condensates
U. R. Fischer.................................................... 93
1 Analogue Gravity: An Overview................................ 93
1.1 The Concept of an Effective Space-time Metric .............. 93
1.2 The Metric in Bose-Einstein Condensates ................... 96
1.3 Pseudo-energy-momentum Tensor.......................... 98
2 Excitations in Bose-Einstein Condensates........................ 99
2.1 Particle-number-conserving Mean-field Expansion............ 99
Contents IX
2.2 Gross-Pitaevskiˇı and Bogoliubov-de Gennes Equations........100
3 Quantum Backreaction........................................101
3.1 Calculation of Backreaction Force From Microscopic Physics ..101
3.2 Comparison with Effective-action Technique.................102
4 Failure of Effective-action Technique ............................104
5 Cutoff Dependence of Effective Action ..........................106
6 Static Example for the Backreaction Force.......................107
7 Conclusion ..................................................109
References ......................................................111
Analogue Space-time Based
on 2-Component Bose–Einstein Condensates
S. Weinfurtner, S. Liberati, M. Visser...............................115
1 Introduction and Motivation...................................115
2 Theory of the 2-Component BEC...............................117
2.1 Gross–Pitaevskii Equation ................................117
2.2 Dynamics...............................................118
2.3 Healing Length ..........................................121
3 Emergent Space-time at Low Energies ..........................123
3.1 Pseudo-Finsler Geometry ................................124
3.2 Bi-metric Geometry .....................................127
3.3 Mono-metric Geometry ..................................129
3.4 Merging Space-time Geometry with Mass Eigenmodes .......130
3.5 Special Case: Ξ =constant ...............................131
4 Application to Quantum Gravity Phenomenology.................133
4.1 Specializing the Wave Equation............................136
4.2 Hydrodynamic Approximation.............................137
4.3 Beyond the Hydrodynamical Approximation ................141
4.4 The Relevance for Quantum Gravity Phenomenology.........145
5 Outlook, Summary and Discussion .............................152
Appendix A Finsler and co-Finsler Geometries.......................154
A.1 Basics..................................................154
A.2 Connection with the Quasi-particle PDE Analysis............155
A.3 Lorentzian Signature Finsler Geometries....................158
A.4 Summary ...............................................160
Appendix B Some Matrix Identities ................................160
B.1 Determinants ...........................................160
B.2 Hamilton–Cayley Theorems ...............................161
References ......................................................161
Links. Relating Different Physical Systems Through
the Common QFT Algebraic Structure
G. Vitiello ......................................................165
1 Introduction .................................................165
X Contents
2 Doubling the Degrees of Freedom...............................168
2.1 The Two-slit Experiment .................................169
3 Unitarily Inequivalent Representations in QFT...................171
3.1 Quantum Dissipation.....................................171
3.2 The Thermal Connection and the Arrow of Time ............174
4 Two-mode Squeezed Coherent States ...........................176
5 Quantum Brownian Motion....................................177
6 Dissipative Noncommutative Plane .............................179
7 Thermal Field Theory ........................................183
8 The q-deformed Hopf Algebra and QFT.........................188
9 Entropy as a Measure of the Entanglement ......................192
10 Trajectories in the H Space....................................193
11 Deterministic Dissipative Systems and Quantization ..............196
12 Conclusions..................................................201
References ......................................................202
The Classical and Quantum Roots
of Pauli’s Spin-statistics Relation
B. Kuckert ......................................................207
1 Introduction .................................................207
2 Setting......................................................209
2.1 The Klein-Gordon Equation...............................209
2.2 The Hermitian Scalar Field ...............................210
2.3 The General Setup.......................................212
3 The Unruh Effect and the Bisognano-Wichmann Theorem.........214
3.1 States of Quantum Systems ...............................215
3.2 N-point Functions .......................................216
3.3 Rindler Wedges and the Unruh Effect ......................216
3.4 The Bisognano-Wichmann Theorem........................218
4 Modular P CT-symmetry, the Spin-statistics Connection,
1
and Modular PCT-symmetry ..................................220
4.1 Covering Groups of SO(3) and L .........................221
1
4.2 The Spin-statistics Connection ............................223
4.3 PCT-symmetry..........................................225
5 Conclusion ..................................................226
References ......................................................227
Black Hole Lasers Revisited
U. Leonhardt, T. G. Philbin.......................................229
1 Introduction .................................................229
2 Dispersion...................................................232
2.1 Bogoliubov Dispersion....................................233
3 Numerical Results ............................................238
4 Black Hole Amplifier..........................................240
Contents XI
References ......................................................245
Cosmic Strings
M. Sakellariadou.................................................247
1 Introduction .................................................247
2 Topological Defects...........................................251
2.1 Topological Defects in GUTs ..............................251
2.2 Spontaneous Symmetry Breaking ..........................252
2.3 Thermal Phase Transitions and Defect Formation............255
2.4 Cosmic String Dynamics..................................257
2.5 Cosmic String Evolution..................................261
2.6 String Thermodynamics ..................................264
2.7 Genericity of Cosmic Strings Formation within SUSY GUTs ..266
3 Cosmic Microwave Background Temperature Anisotropies .........268
3.1 Mixed Models ...........................................270
3.2 Supersymmetric Hybrid Inflation ..........................271
4 Cosmic Superstrings ..........................................280
5 Conclusions..................................................285
References ......................................................285
Index..........................................................289
