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Lecture Notes in Physics EditorialBoard R.Beig,Wien,Austria W.Beiglböck,Heidelberg,Germany W.Domcke,Garching,Germany B.-G.Englert,Singapore U.Frisch,Nice,France P.Hänggi,Augsburg,Germany G.Hasinger,Garching,Germany K.Hepp,Zürich,Switzerland W.Hillebrandt,Garching,Germany D.Imboden,Zürich,Switzerland R.L.Jaffe,Cambridge,MA,USA R.Lipowsky,Golm,Germany H.v.Löhneysen,Karlsruhe,Germany I.Ojima,Kyoto,Japan D.Sornette,Zürich,Switzerland S.Theisen,Golm,Germany W.Weise,Garching,Germany J.Wess,München,Germany J.Zittartz,Köln,Germany TheLectureNotesinPhysics TheseriesLectureNotesinPhysics(LNP),foundedin1969,reportsnewdevelopments in physics research and teaching – quickly and informally, but with a high quality and theexplicitaimtosummarizeandcommunicatecurrentknowledgeinanaccessibleway. Bookspublishedinthisseriesareconceivedasbridgingmaterialbetweenadvancedgrad- uatetextbooksandtheforefrontofresearchtoservethefollowingpurposes: •tobeacompactandmodernup-to-datesourceofreferenceonawell-definedtopic; •toserveasanaccessibleintroductiontothefieldtopostgraduatestudentsandnonspe- cialistresearchersfromrelatedareas; • to be a source of advanced teaching material for specialized seminars, courses and schools. Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Pro- ceedingswillnotbeconsideredforLNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive is available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks,subscriptionagencies,librarynetworks,andconsortia. ProposalsshouldbesenttoamemberoftheEditorialBoard,ordirectlytothemanaging editoratSpringer: Dr.ChristianCaron SpringerHeidelberg PhysicsEditorialDepartmentI Tiergartenstrasse17 69121Heidelberg/Germany [email protected] 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 ISBN-10 3-540-70858-8SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-70858-2SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2007 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorsandtechbooksusingaSpringerLATEXmacropackage Coverdesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper SPIN:11804185 54/techbooks 543210 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