CollaboratorsJorgeDukelskyInstitutodeEstructuradelaMateria,MadridStuartPittelBartolResearchInstitute,UniversityofDelaware,USAMarioStoitsovInstituteforNuclearResearch&NuclearEnergy,SofiaContainsIntroductionWilson’sRenormalizationGroupMethodDensityMatrixRenormalizationGroupMethodp-hDMRGbasicsApplicationtonuclearshellmodelproblemsOutlookWilson’sRenormalizationGroup(1974)Thegoal:tosolvetheKondoproblem(describestheantiferromagneticinteractionoftheconductionelectronswithasinglelocalizedimpurity)aftermappingitontoa1Dlatticeinenergyspace.Theassumption:low-energystatesmostimportantforlaw-energybehavioroflargequantumsystemsWilson’sRenormalizationGroup(1974)Theidea:numericallyintegrateouttheirrelevantdegreesoffreedomThealgorithm:isolatefinitesubspaceofthefullconfigurationspacediagonalizenumericallykeepmlowestenergyeigenstatesaddasiteiterateSamplingtheconfigurationspacemsssInfiniteprocedure“theonionpicture”superblockenvironment

thesizeofthesuperblockstaysthesamewhiletheenvironmentshrinksFromWRGtoDMRGTheWRGwasthefirstnumericalimplementationoftheRGtoanon-perturbativeproblemliketheKondomodel,forwhichithadenormoussuccess.WRGcannotbeappliedtootherlatticeproblems.For1DHubbardmodelsitbeginstodeviatesignificantlyfromtheexactresults.Theproblemresidesinthefactthatthetruncationstrategyisbasedsolelyonenergyarguments.ThesolutiontothisproblemwasproposedbyWhitewhointroductedtheDMRG:PRL69(1992)2863andPRB48(1993)10345.From

1DlatticestofiniteFermisystemsS.WhiteintroducedtheDMRGtotreat1Dlatticemodelswithhighaccuracy.PRL69(1992)2863andPRB48(1993)10345.S.WhiteandD.HussestudiedS=1HeisenbergchaingivingtheGSenergywith12significantfigures.PRB48(1993)3844.T.Xiangproposedthek-DMRGforelectronsin2Dlattices.PRB53(2019)R10445.S.WhiteandR.L.Martinusedthek-DMRGforquantumchemicalcalculation.J.Chem.Phys.110(2019)4127.SincethenapplicationsinQuantumChemistry,smallmetallicgrain,nuclei,quantumHallsystems,etc…reviewarticle:U.Schollw?ck,Rev.Mod.Phys.77(2019)259Theparticle-holeDMRGIntroducedbyJ.DukelskyandG.SierratostudysystemsofutrasmallsuperconductinggrainsPRL83(2019)172andPRB61(2000)12302Motivation:

BCSbreaksparticlenumber.PBCSimprovesthesuperconductingstate.Fluctuationdominatedphase?Levelordering:InFermisystems,theFermileveldefinesholeandparticlespstates.MostofthecorrelationstakeplaceclosetotheFermolevelp-hDMRGbasicsFLet'sconsiderforsimplicityaxially-symmetricNilsson-likelevels,whichadmitfourstates(s=4):Whenweaddthenextlevel:numberofparticlestatesgoesfrommtos×mnumberofholestatesgoesfrommtos×mnumberofstatesinvolvingparticlescoupledtoholesalsogoesup.F…F…FBasicideaofDMRGmethod:Ftruncatefromthes×mstatesforparticlestotheoptimum

mofthem,andlikewisefromthes×mstatesforholestotheoptimum

mofthem.Finiteprocedure

mediumenvironmentsuperblock

m

x

s

xmstartingpoint:infiniteproceduresizeofsuperblockandmediumstaythesamewhileenvironmentblockshrinksmediumblockstoredfrompreviousiteration“zipping”backandforth→iterativeconvergencewarmup1stsweep2ndsweepSamplingcriterion:FAQQ:Howtoconstructoptimalapproximationtothegroundstatewavefunctionwhenweonlyretaincertainnumberofparticleandholestates?A:Choosethestatesthatmaximizetheoverlapbetweenthetruncatedstateandtheexactgroundstate.Q:Howtodothis?A:DiagonalizetheHamiltonian…DefinethereduceddensitymatricesforparticlesandholesDiagonalizethesematrices:

representtheprobabilityoffindingaparticular-stateinthefullgroundstatewavefunctionofthesystem;…Optimaltruncationcorrespondstoretainingafixednumberofeigenvectorsthathavelargestprobabilityofbeingingroundstate,i.e.,havelargesteigenvalues;Parameteroftheprocedure:numberofstatesretainedaftereachinteraction;Bottomline:

DMRGisamethodforsystematicallybuildingincorrelationsfromallsingle-particlelevelsinproblem.Aslongasconvergenceissufficientlyrapidasafunctionofnumberofstateskept,itshouldgiveanaccuratedescriptionofthegroundstateofthesystem,withoutuseverhavingtodiagonalizeenormousHamiltonianmatrices;MustcalculatematrixelementsofallrelevantoperatorsateachstepoftheprocedureThehighestmemoryconsumingoperatorswithinablockareTheycanbecontractedwiththeinteractionandbereducedtoO(1)andO(L)Subtleties:Subtleties:Thismakesitpossibletosetupaniterativeprocedurewherebyeachlevelcanbeaddedstraightforwardly.Mustofcourserotatesetofstoredmatrixelementstooptimal(truncated)basisateachiteration.Procedureasdescribedguaranteesoptimizationofgroundstate.Togetoptimaldescriptionofmanystates,wemayneedtoconstructdensitymatricesthatsimultaneouslyincludeinfoonseveralstatesofthesystem.LegezaandSolyomusedquantuminformationconceptslikeblockentropyandentanglementtoconcludethattheDMRGisextremelysensitivetothelevelorderingandtheinitializationprocedure.ph-DMRG:modelcalculationsHamiltonian40particlesinj=99/2shellsizeofthesuperblockndim~1026parameters:ph-DMRG:realisticnuclearstructurecalculationsHamiltonianph-DMRG:realisticnuclearstructurecalculationsconfigurationspaceph-DMRG:24Mginm-scheme

sd-shell4valentprotons4valentneutronsUSDinteractionSphHFph-DMRG:Infinitevs.finiteprocedureph-DMRG:48Crinthej-schemeph-DMRG:48Crinthem-schemeTheOakRidgeDMRGprogramThomasPapenbrockfromORNLdevelopedanalternativeprogramfordoingnuclearstructurecalculationswiththeDMRG:DMRGwithsweepinginthem-schemeAxialHFbasis.ThelevelsfromtheFermienergy.Inthewarmup,protonsarethemediumforneutronsandviceversa.Inthesw