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Superfluid Analogies of Cosmological Phenomena

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Superfluid Analogies of Cosmological Phenomena Superfluid analogies of cosmological phenomena G.E. Volovik Low Temperature Laboratory, Helsinki University of Technology Box 2200, FIN-02015 HUT, Finland and L.D. Landau Institute for Theoretical Physics, 117334 Moscow, Russia In a modern viewpoint the relativistic quantum field theory is the emergent phenomenon arising in the low energy corner of the physical fermionic vacuum – the medium, whose nature remains un- known. The same phenomenon occurs in the condensed matter systems: In the extreme limit of low energy the condensed matter system of special universality class acquires all the symmetries, which we know today in high energy physics: Lorentz invariance, gauge invariance, general covariance, etc. The chiral fermions as well as gauge bosons and gravity field arise as fermionic and bosonic collective modes of the system. The inhomogeneous states of the condensed matter ground state – vacuum – induce nontrivial effective metrics of the space, where the free quasiparticles move along geodesics. This conceptual similarity between condensed matter and quantum vacuum allows us to simulate many phenomena in high energy physics and cosmology, including axial anomaly, bary- oproduction and magnetogenesis, event horizon and Hawking radiation, cosmological constant and rotating vacuum, etc., probing these phenomena in ultra-low-temperature superfluid helium, atomic Bose condensates and superconductors. Some of the experiments have been already conducted. Contents I Introduction. Physical vacuum as condensed matter. 4 II Landau-Khalatnikov two-fluid hydrodynamics as effective theory of gravity. 5 A Superfluidvacuumandquasiparticles.. .................. 5 B Dynamicsofsuperfluidvacuum. .............. 6 1 Absence of canonical Lagrangian formalism in effective theories. .................. 7 C Normalcomponent–“matter”. ............. 7 D Quasiparticlespectrum and effective metric. .................... 8 E Effective metric for bosonic collective modes in other systems. ..................... 8 F Effective quantum field and effective action . .................. 9 G Vacuum energy and cosmological constant. Nullification of vacuumenergy. 10 H Einstein action and higher derivative terms . .................... 12 III “Relativistic” energy-momentum tensor for “matter” moving in “gravitational” superfluid arXiv:gr-qc/0005091v3 3 Oct 2000 background in two fluid hydrodynamics 13 A Kinetic equation for quasiparticles (matter) . ...................... 13 B Momentum exchange between superfluid vacuum and quasiparticles.................. 13 C Covariancevsconservation.. ................ 14 D Energy-momentumtensorfor“matter”.. ................. 15 E Localthermodynamicequilibrium.. ................. 15 F Global thermodynamic equilibrium. Tolman temperature. Pressure of “matter” and “vacuum” pressure. .......................................... ......... 16 IV Universality classes of fermionic vacua. 17 1 Classes of fermionic quasiparticle spectrum . ................... 17 A Fermisurfaceastopologicalobject . ................. 17 1 LandauFermiliquid ................................. ......... 21 2 Non-LandauFermiliquids . .. .. .. .. .. .. .. .. .. .. .. ........... 22 B Fully gapped systems: “Dirac particles” in superconductors and in superfluid 3He-B......... 22 C SystemswithFermipoints ............................ ............. 23 1 ChiralparticlesandFermipoint . .............. 23 2 TopologicalinvariantforFermipoint . ................ 23 1 3 Topological invariant as the generalization of chirality......................... 24 4 Relativistic massless chiral fermions emerging near Fermipoint. .................. 25 D Gapped systems with nontrivial topology in 2+1 dimensions ...................... 27 V Fermi points: 3He-A vs Standard Model 27 A Superfluid 3He-A .............................................. 28 1 Fermiliquidlevel. ................................. ........... 28 2 BCSlevel. ......................................... ....... 28 3 “Relativistic”level. .............................. ............. 31 4 Hierarchyofenergyscales. ............. 32 5 Spinvsisospin. .................................... ......... 33 B Standard Model and its Momentum Space Topology . ................. 33 1 FermionsinStandardModel . ........... 33 2 Momentum-spacetopologicalinvariants . ................. 34 3 Generating function for topological invariants constrainedbysymmetry . 34 4 Discretesymmetryandmasslessfermions . ................ 35 5 Nullification of topological invariants below electroweak transition and massive fermions . 36 6 RelationtoAxialAnomaly . .. .. .. .. .. .. .. .. .. .. .. ........... 36 VI Effective relativistic quantum field theory emerging in a systemwithFermipoint. 36 A Collective modes of fermionic vacuum – electromagnetic andgravitationalfields. 36 B Physical laws in vicinity of Fermi point: Lorentz invariance, gauge invariance, general covariance, conformalinvariance. .. .. .. .. .. .. .. .. .. .. .. .. .. ............. 37 C Effectiveelectrodynamics. ............... 37 1 Effectiveactionfor “electromagnetic”field . .................. 38 2 Running coupling constant: zero charge effect. .................. 38 D Effective SU(N)gaugefields fromdegeneracyofFermipoint. ........ 39 1 Massof W -bosons, flat directions and supersymmetry.. ........... 40 VII Chiral anomaly in condensed matter systems and Standard Model. 40 A Adler-Bell-Jackiwequation. ................ 40 B Anomalous nonconservation of baryonic charge. .................... 42 C Analog of baryogenesis in 3He-A: Momentum exchange between superfluid vacuum and quasiparticle matter............................................. ........ 43 D Axial anomaly and force on 3He-Avortices. ............................... 44 E Experimental verification of Adler-Bell-Jackiw equation in rotating 3He-A. .............. 46 VIII Macroscopic parity violating effects. 48 A Helicityinparityviolatingsystems. ................... 48 B Chern-Simonsenergyterm.. .............. 48 1 Chern-SimonsterminStandardModel. .............. 48 2 Chern-Simons energy in 3He-A. .................................... 50 3 Kinetic energy of counterflow in 3He-Aanditsanalogforchiralfermions. 51 4 Massofhyperphoton. ................................ .......... 51 C Helical instability and “magnetogenesis” by chiral fermions. ...................... 52 1 Helical instability condition. ................ 52 2 Experimental “magnetogenesis” in rotating 3He-A. ......................... 53 D Mixed axial-gravitational Chern-Simons term. ..................... 53 1 Parityviolatingcurrent . ............. 53 2 Orbitalangularmomentumandfreeenergy. ............... 53 3 Effective Chern-Simons action for µa = 0 and T =0......................... 55 4 Finite density of states and Chern-Simons term6 in the presenceofcounterflow. 56 5 Unification of conventional and mixed CS terms. ................. 57 6 Possibleexperimentsincondensedmatter. ................. 57 2 IX Fermion zero modes and spectral flow in the vortex core. 58 A Fermionzeromodesonvortices . .............. 58 1 Anomalousbranchofchiralfermions . .............. 58 2 Integer vs half-odd-integer angular momentum of fermion zeromodes ............... 61 3 Hamiltonianforfermionsinthecore. ............... 61 4 Quasiclassical low-energy states on anomalous branch . ..................... 63 5 Quantum low-energy states and W -parity. .............................. 63 6 Majorana fermion with E =0onhalf-quantumvortex. 64 7 Fermionsonasymmetricvortices. .............. 64 B Spectral flow in singular vortices: Callan-Harvey mechanismofanomalycancellation . 65 1 Restrictedspectralflowinthevortexcore. ................. 67 2 Measurement of Callan-Harvey effect in 3He-B............................ 68 X Interface between two different vacua and vacuum pressure in superfluid 3He. 68 A Interface between vacua of different universality classes andAndreevreflection. 68 1 Fermionsintwoneighboringvacua. .............. 68 2 Andreevreflectionattheinterface. ............... 69 B Force acting on moving mirror from thermal relativistic fermions. ................... 71 1 Relativisticcase.................................. ............ 71 2 ForceactingonmovingABinterface. .............. 72 C Vacuum pressure and vacuum energy in 3He................................ 72 1 Interplay between vacuum pressure and pressure of matter. ..................... 72 2 Vacuum energy in 3He. ......................................... 73 3 Cosmological term in 3He. ....................................... 74 XI Vierbein defects. 75 A Vierbeindomainwall................................ ............. 75 1 Vierbein wall in 3He-Afilm. ...................................... 76 2 FermionsacrossVierbeinWall.. .............. 77 3 Communication across the wall via superluminal nonlinear dsipersion.. 79 B Conicalspaces..................................... ............ 80 1 Antigravitatingstring. .. .. .. .. .. .. .. .. .. .. .. ............. 80 2 EstimationofNewtonconstant. ............. 81 C Vortexvsspinningcosmicstring. ................ 82 XII Gravitational Aharonov-Bohm effect and Iordanskii force. 84 A GravitationalAharonov-Bohmeffect. ................. 84 B Iordanskiiforceonspinningstring. .................. 86 XIII Horizons, ergoregions, degenerate metric, vacuum instability and all that. 86 A Event horizons in vierbein wall and Hawking radiation. ...................... 86 B Landaucriticalvelocityandergoregion . ................... 88 C Painlev´e-Gullstrand metric in effective gravity in superfluids. Vacuum resistance to formation of horizon............................................ ......... 89 D
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