Quantum phase transitions from topology in momentu

arXiv:cond-mat/0601372v5[cond-mat.str-el]18Apr2006QuantumphasetransitionsfromtopologyinmomentumspaceG.E.Volovik1,21LowTemperatureLaboratory,HelsinkiUniversityofTechnology,P.O.Box2200,FIN-02015HUT,Espoo,Finland2LandauInstituteforTheoreticalPhysics,Kosygina2,119334Moscow,RussiaManyquantumcondensedmattersystemsarestronglycorrelatedandstronglyinteractingfermionicsystems,whichcannotbetreatedperturba-tively.However,physicswhichemergesinthelow-energycornerdoesnotdependonthecomplicateddetailsofthesystemandisrelativelysimple.Itisdeterminedbythenodesinthefermionicspectrum,whichareprotectedbytopologyinmomentumspace(insomecases,incombinationwiththevacuumsymmetry).Closetothenodesthebehaviorofthesystembecomesuniver-sal;andtheuniversalityclassesaredeterminedbythetoplogicalinvariantsinmomentumspace.Whenonechangestheparametersofthesystem,thetransitionsareexpectedtooccurbetweenthevacuawiththesamesymmetrybutwhichbelongtodifferentuniversalityclasses.DifferenttypesofquantumphasetransitionsgovernedbytopologyinmomentumspacearediscussedinthisChapter.TheyinvolveFermisurfaces,Fermipoints,Fermilines,andalsothetopologicaltransitionsbetweenthefullygappedstates.Theconsid-erationbasedonthemomentumspacetopologyoftheGreen’sfunctionisgeneralandisapplicabletothevacuaofrelativisticquantumfields.ThisisillustratedbythepossiblequantumphasetransitiongovernedbytopologyofnodesinthespectrumofelementaryparticlesofStandardModel.1Introduction.Therearetwoschemesfortheclassificationofstatesincondensedmatterphysicsandrelativisticquantumfields:classificationbysymmetry(GUTscheme)andbymomentumspacetopology(anti-GUTscheme).Forthefirstclassificationmethod,agivenstateofthesystemischaracter-izedbyasymmetrygroupHwhichisasubgroupofthesymmetrygroupGoftherelevantphysicallaws.ThethermodynamicphasetransitionbetweenequilibriumstatesisusuallymarkedbyachangeofthesymmetrygroupH.Thisclassificationreflectsthephenomenonofspontaneouslybrokensymme-try.Inrelativisticquantumfieldsthechainofsuccessivephasetransitions,inwhichthelargesymmetrygroupexistingathighenergyisreducedatlowenergy,isinthebasisoftheGrandUnificationmodels(GUT)[1,2].Incon-densedmatterthespontaneoussymmetrybreakingisatypicalphenomenon,andthethermodynamicstatesarealsoclassifiedintermsofthesubgroupH2G.E.VolovikoftherelevantgroupG(seee.g,theclassificationofsuperfluidandsupercon-ductingstatesinRefs.[3,4]).ThegroupsGandHarealsoresponsiblefortopologicaldefects,whicharedeterminedbythenontrivialelementsofthehomotopygroupsπn(G/H);cf.Ref.[5].Thesecondclassificationmethodreflectstheoppositetendency–theantiGrandUnification(anti-GUT)–wheninsteadofthesymmetrybreak-ingthesymmetrygraduallyemergesatlowenergy.Thismethoddealswiththegroundstatesofthesystematzerotemperature(T=0),i.e.,itistheclassificationofquantumvacua.Theuniversalityclassesofquantumvacuaaredeterminedbymomentum-spacetopology,whichisalsoresponsibleforthetypeoftheeffectivetheory,emergentphysicallawsandsymmetriesatlowenergy.ContrarytotheGUTscheme,wherethesymmetryofthevac-uumstateisprimarygivingrisetotopology,intheanti-GUTschemethetopologyinthemomentumspaceisprimarywhilethevacuumsymmetryistheemergentphenomenoninthelowenergycorner.Atthemoment,weliveintheultra-coldUniverse.AllthecharacteristictemperaturesinourUniverseareextremelysmallcomparedtothePlancken-ergyscaleEP.Thatiswhyallthemassivefermions,whosenaturalmassmustbeoforderEP,arefrozenoutduetoextremelysmallfactorexp(−EP/T).ThereisnomatterinourUniverseunlesstherearemasslessfermions,whosemasslessnessisprotectedwithextremelyhighaccuracy.Itisthetopologyinthemomentumspace,whichprovidessuchprotection.Forsystemslivingin3Dspace,therearefourbasicuniversalityclassesoffermionicvacuaprovidedbytopologyinmomentumspace[6,7]:(i)Vacuawithfully-gappedfermionicexcitations,suchassemiconductorsandconventionalsuperconductors.(ii)VacuawithfermionicexcitationscharacterizedbyFermipoints–pointsin3Dmomentumspaceatwhichtheenergyoffermionicquasiparticlevanishes.Examplesareprovidedbysuperfluid3He-Aandalsobythequan-tumvacuumofStandardModelabovetheelectroweaktransition,whereallelementaryparticlesareWeylfermionswithFermipointsinthespectrum.Thisuniversalityclassmanifeststhephenomenonofemergentrelativisticquantumfieldsatlowenergy:closetotheFermipointsthefermionicquasi-particlesbehaveasmasslessWeylfermions,whilethecollectivemodesofthevacuuminteractwiththesefermionsasgaugeandgravitationalfields.(iii)Vacuawithfermionicexcitationscharacterizedbylinesin3Dmo-mentumspaceorpointsin2Dmomentumspace.WecallthemFermilines,thoughingeneralitisbettertocharacterizezeroesbyco-dimension,whichisthedimensionofp-spaceminusthedimensionofthemanifoldofzeros.Linesin3Dmomentumspaceandpointsin2Dmomentumspacehaveco-dimension2:since3−1=2−0=2;comparethiswithzeroesofclass(ii)whichhaveco-dimension3−0=3.TheFermilinesaretopologicallystableonlyifsomespecialsymmetryisobeyed.ExampleisprovidedbythevacuumofthehighTcsuperconductorswheretheCooperpairingintoad-wavestateTopologyofquantumphasetransitions3qqcnochangeofsymmetryalongthepathT(temperature)Tne-Δ/TqclineoffirstordertransitionT(a)(b)Fig.1.Quantumphasetransitionb