towards ab initio calculations with the dynamical vertex approximation anna galler资料.pdf

该【towards ab initio calculations with the dynamical vertex approximation anna galler资料 】是由【小舍儿】上传分享，文档一共【12】页，该文档可以免费在线阅读，需要了解更多关于【towards ab initio calculations with the dynamical vertex approximation anna galler资料 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..,041004(2018)SpecialTopicshttps:///.041004NewabinitioApproachestoExploringEmergentPhenomenainQuantumMatterTowardsabinitioCalculationswiththeDynamicalVertexApproximationAnnaGaller1,JosefKaufmann1,PatrikGunacker1,MatthiasPickem1,PatrikThunstr?m1,2,,andKarstenHeld11InstituteofSolidStatePhysics,TUWien,1040Wien,Austria2DepartmentofPhysicsandAstronomy,MaterialsTheory,UppsalaUniversity,75120Uppsala,Sweden(ReceivedMay15,2017;acceptedSeptember5,2017;publishedonlineJanuary26,2018)Whilekeye?ectsofthemany-bodyproblem—suchasKondoandMottphysics—canbeunderstoodintermsofon-sitecorrelations,non-local?uctuationsofcharge,spin,,wereviewrecentprogressindiagrammaticextensionstodynamicalmean-??rstrecapitulatethequantum?eldtheoreticalbackgroundbehindthetwo--particlequantitiesusingwormsamplingandvertexasymptotics,beforegivinganintroductiontotheabinitiodynamicalvertexapproximation(AbinitioDΓA).Finally,,,)TheItisparticularlychallengingtoperformabinitiocalcu-screeningoftheCoulombinteractionandthenon-localorrelationsareGWself-,thestandardapproachtomaterialspnictides,42–44)intermetallics,45)cuprates46)andothertran-calculations,densityfunctionaltheory(DFT)1–3)withex-–51)IncorporatingaGWcorrectionintochange–correlationfunctionalsbasedonthelocaldensityDFT+DMFTwas,.,essentialfordescribingtheredcolorapproximation()(GGA),4)-bodymethodsHowever,thenon-localcorrelationsofaGW+,othernon-Awell-establishedroutetothisendisHedin’s5)so-,theself-energyisgivenbytheexampleofsuchnon-localcorrelationsarespin?uctua-productoftheGreen’sfunctiontimesascreenedinteraction:tions,53)ic?(RPA),55),itisnaturaltoexpectittoAwaytoaddressnon-localcorrelationsandtokeep,atimproveontheexchangepartoftheexchange–correlationthesametime,,7)approximation(DCA)56)andcellularDMFT(CDMFT).57,58)Ontopofthisthereare—becauseofthescreening—ge-Here,aclusterofsites,oracoarse-grainedBrillouinzoneisnuinecorrelatione?ectssuchasquasiparticlerenormaliza-embeddedinanon-interactingDMFT-likebathinsteadofationsand?,9)However,,non-,GWisonlyappropriateforweaklycorrelatedestablishingthepresenceofpseudogaps,,orrelationsare,ontheotherhand,thesee,.,–62and,forareview,?eldtheory(DMFT).10–12)Whileabinitiocalculationswithmanyorbitalsontheotherhand,DMFTdoesnotcoverallcorrelations,itincludesthethenumericale?–,65)insulatortransitions,12)butalsoprovideforamorereliableInthisrespect,diagrammaticextensionsofDMFTsuchasism13–15)andkinksintheenergy-thedynamicalvertexapproximation(DΓA)66,67),17),andotherrelateddiagrammaticDMFT18,19)(forreviews,)hasbeenaextensions,68–73)?rstalocaltwo-,non-?rstDFT+DMFTcalculationsallowedareconstructedthroughtheBethe–Salpeter67,74)orthroughforabetterunderstandingoftheMott–,75,76)Short-andlong-rangedcorre-V2O3,22)ofδ-Pu,23)theCevolumecollapse,22)-,)Morerecently,amongothers,ironexponentscouldbecalculatedforthe?rsttimefortheicmoment,25–27)Hubbardmodel77,78)aswellasthequantumcriticalpointcomplexoxideheterostructures,28–32)nanoscopicstruc-)Intwodimensions,spin?uctuationstures,33,34)thermoelectricity,35–37)spin–orbitinteractions,38,39)icorderandgiverisetopseudogapandelectronicentanglement40),80,81)ic041004-1?2018ThePhysicalSocietyofJapan:...,041004(2018))andsuperconductiv-,83),ithasbeensug-gested85,86)touseasastartingvertexqthebarenon-localCoulombinteractionVqaswellasalllocalvertexcorrectionsloc,whichalsoincludesthelocalCoulomb(Hubbard)(G)(F),butalsonon-localcorrelationsconstitutethegeneralizedsusceptibility?0t0F,suchasspin?,thisAbinitioDΓAschemeincludesmorephysicsthanGW+DMFTbutrequireskytheadditionalcalculationofthelocaltwo-particlevertexandG;lmeThT?ckleTckme0T?i;e2TsolvingtheBethe–1;2;3TAbinitioDΓAhasbeenrecentlyimplementedandappliedto000000studynon-)hT?ceTcy0eTc0000eTcy0e0T?i;e3TInthispaperwereviewtheAbinitioDΓAmethodaswellkl1kqm2kqm3kl0000astherecentprogressforcalculatingthemulti-orbitallocalwhereTisthetime-orderingoperator,andbracketsh:,weintroducethemulti-orbitalHamiltonianthethermalexpectationvalueintheactioncorrespondingthatwewanttosolveandournotationregardingGreen’stotheHamiltonianinEq.(1).Looselyspeaking,theone--particlepropagatordescribestheamplitudefortheprocessinous-timequantumMonteCarlocalculation(CT-QMC)ofthewhichoneparticleisaddedattime0toanensembleofNlocaltwo-particlevertex,usingvertexasymptotics87)andparticles,thesystemisthenlettoevolvewithNt1particles,),werecapitulatetheAbini-,89)whichstartswiththelocalvertexplusThetwoparticlepropagatordescribesasimilarprocess,thebarenon-localinteraction,constructsthroughtheBethe–exceptthattwoparticlesareaddedandremovedatfourSalpeterequationthenon-localfullvertexandthroughdi?–Dysonequationthenon-localself--andtwo-,wepresenttheresultsobtainedforSrVO3andHamiltonianconservesspin(.,intheabsenceofspin–orbit()thedi?erenceinself-energywhenusingcoupling),thereareonlysixnon-bina-thevertexasymptoticsinCT--particlepropagator:1;2;3TGlmm0l0e1;2;3T;000qkk0qkk0Ourstartingpointisthefollowingmulti-orbitalGlmm0l0e1;2;3TGlmm0l0e1;2;3T:e5THamiltonianofinteractingelectronsinasolid:000,posedasfollows:H?kllmm00l0kmklmXXqkk0kk0qkk0qyyGlmm0l0?q0G;lmG0;m0l0t00;;lmm0l0teUlm0ml0tVlm0ml0Tck0qm0ckl0ckqm0ck0l0:0kk0qll0mm0X0000tqkkFqkk00qkk00000:e6T0;;lmhnnhh0n0;;nhmle1Tnn0hh0Herecy(c)creates(annihilates)anelectronwithHere,βistheinversetemperature,andwehaveFourierklkl?rstterminEq.(1),ordingtoconsistingoftheone-particledispersions,isusuallyobtainedZqkk0i1ie!T2tie0!T3fromGW-orDFT-?d1d2d3e00low-energysubsetoftheresultingband-structureisidenti-qkk0?ed,thehoppingsareexpressedinasuitablebasistoyieldGlmm0l0e1;2;3Te7Ttheaboveamplitudes-handfour-vectornotationk?ek;inTandThematrixelementshavebeenseparatedintolocalq?eq;i!nTforfermionic[in?e2nt1T=]andbosonic(Hubbard=Hund-like)UPcontributionsandpurelynon-local(i!n?2n=)variables,,.,Vq?)Interactionstrengthsofthebosonicfrequencyωcorrespondstotheparticle–holeqcorrespondingtothechosenorbitalsubsetcanbeobtainedchannelandiscoinedparticle–)orAdditionally,onecande?neaparticle–hole-)putesenergycostsforparticle–particlefrequencynotationas,.,,whilethelattersimulates94,-0;;lmm0l0G;ll0G;m0mkk0;e8Tpropertiesforthemany-bodyHamiltoniande?nedinEq.(1).Thisinformationcanbeextractedfromtheone-andtwo-andtookinEq.(6)ountthat0isdiagonalinparticleGreensfunctions,whicharerespectivelygivenbyk,-2?2018ThePhysicalSocietyofJapan:...,041004(2018)?rsttwotermsinEq.(6),esstotheself-“unconnected”contributionsofthetransversalandlongi-energyΣ:tudinalparticle–holechannel(laterreferredtoas“ph”andXq?00k0?eUmlm0l0tV00Tnk00“ph”),respectively:thetwoparticlespropagateindepend-;mm0mlml;llll0k,the“connected”termdescribestheactualXeU00tVq00Tnkq0scatteringprocessesandconsistsofthefullvertexFwithfourmllmmllm;ll“legs”,whichcanbeexpressedbytwobaretwo-particlell0qpropagators,herewrittenintheph-eUmlhntVT00F0000G0:twotermstogether,?tF(),formthemlhn0;nllnd;nlhmhh000ll0nn0hh0generalizedsusceptibilityfromwhich,.,measurableqk0charge-orspin-,96)Here,the?rsttwotermsinvolvingthedensitymatrixnk0are;llicandspin–orbitcouplingthe(static)HartreeandFockcontributions,,thereareonlytwoindependentspincrucialobservationisthatEq.(14)allowsforanalternativecon?gurationsforthetwo--energy:Insteadconvenienttointroducethe(d)ensityand(m)icofselectingdiagramsfortheself-energyΣ(ortheone-combinations,.,forF:),thediagrammaticscanbeliftedtotheFFtF;e9Ttwo-particlelevel,.,approximationsaremadeontheleveld"""#F""F"#:e10Tcontextofmany-,67,69,72,Thisway,thespinindicesarereplacedbythelabel74,97,,wearegoingtodescribeAbinitio-r2fd;mg;forspin-independentquantities,suchas0,theDΓA,86)thedynamicalvertexapproximationforrealisticσ--,wediscusstainedinthefullvertexFrcanfurtherbedividedintohoweverrecentadvancesforcalculatingthelocalFandΓbyqkk0qkk0qkk0CT-;lmm0l0?r;lmm0l0tr;lmm0l0e11Twhereistheph-irreduciblevertex,-alculationoftheLocalVertexrdiagramscontainedinΓcannotbeseparatedintotwopartsThetwo-