Non-Perturbative Renormalization Group Analysis in

arXiv:quant-ph/0208173v128Aug20021Non-PerturbativeRenormalizationGroupAnalysisinQuantumMechanicsKen-IchiAoki,∗AtsushiHorikoshi,∗∗MasakiTaniguchi∗∗∗andHaruhikoTerao†InstituteforTheoreticalPhysics,KanazawaUniversity,Kanazawa920-1192,JapanWeanalyzequantummechanicalsystemsusingthenon-perturbativerenormalizationgroup(NPRG).TheNPRGmethodenablesustocalculatequantumcorrectionssystemati-callyandisveryeffectiveforstudyingnon-perturbativedynamics.Westartwithanharmonicoscillatorsandproceedtoasymmetricdoublewellpotentials,supersymmetricquantumme-chanicsandmanyparticlesystems.§1.IntroductionThenon-perturbativerenormalizationgroup(NPRG)hasbeenformulatedthroughanalysesofcriticalphenomena1)andappliedtonon-perturbativestudiesofstatis-ticalmechanicsandquantumfieldtheories.Ithasbeenestablishedasapowerfultoolforanalysesofnon-perturbativedynamicsinsystemsofmany(infinite)degreesoffreedom,becauseitallowsfortheevaluationoffluctuationswithoutrecoursetoperturbationseries.Severaltypesofnon-perturbative(exact)renormalizationgroupequationshavebeenderivedbyintegrationwithscaledecompositionandhavebeenappliedtovarioussystems.2),3),4),5),6),7),8),9)Inthisarticle,weapplytheNPRGmethodtoquantummechanicalsystems,thatis,systemsoffinitelymanydegreesoffreedom2),6),10)toanalyzetheirnon-perturbativedynamics.Generally,therearetwotypesofnon-perturbativequantities.Onecorrespondstothesummationofallordersofaperturbativeseries,whichcouldberelatedtoBorelresummation.11)Theotherisanessentialsingularitywithrespecttoacouplingconstantλ0,whichhasastructurelikee−1λ0.12),13)Wearenotabletoexpandsuchasingularcontributionaroundλ0=0.Asingularityofthistypeappearsinthecaseofquantumtunneling.Forexample,inasymmetricdoublewellsystem,therearetwodegenerateenergylevelsateachminimum,whicharemixedthroughtunnelingtogenerateanenergygapΔE∼e−1λ0.Theexponentialfactorisknowntoresultfromthefreeenergyoftopologicalconfigurations,i.e.,instantons.Inthisarticle,wefirstsummarizehowtoanalyzequantummechanicalsystemsusingtheconceptofNPRGandchecktowhatextentNPRGcanbeusedtoevaluatenon-perturbativeeffectsquantitatively.TheNPRGequationweemployhereisalocalpotentialapproximatedWegner-Houghton(LPAW-H)equation,14),15)which∗E-mail:[e-mail]∗∗E-mail:[e-mail]∗∗∗E-mail:[e-mail]†E-mail:[e-mail],A.Horikoshi,M.TaniguchiandH.Teraoweusetoanalyzequantumanharmonicoscillatorsandasymmetricdoublewellsys-tems.Incontrasttothesymmetricdoublewellsystem,thestandardinstantonmethoddoesnotworkforanasymmetricpotential,andthemuchmoresophisti-catedmethodofthevalleyinstantonhasbeendevelopedfortheirtreatment.30),31)TheNPRGmethodisfoundtoworkforasymmetricpotentialsaswellasforsym-metricpotentials,becauseNPRGdoesnotrelyonparitysymmetry.Weproceedtoanalysesofmorecomplicatedsystems,supersymmetricquantummechanics(SUSYQM)andmanyparticlesystems.SUSYQMisatoymodelfordynamicalSUSYbreaking.33),34)Although,ingeneral,thereisnospontaneoussym-metrybreakinginsystemswithfinitelymanydegreesoffreedom,someextraordinarysymmetries,suchasSUSY,canbebrokeneveninquantummechanics.SUSYbreak-ingisahighlynon-perturbativephenomenonbecauseofthenon-renormalizationthe-orem,andwewillseethatNPRGshouldbeapplicablefornon-perturbativeSUSYbreaking.Inaddition,analysesofquantummanyparticlesystemshavebecomeveryim-portantwithrecentdevelopmentsinnano-technology.Solvingtheproblemofhowthequantumcoherenceofavariableofatargetsystemisaffectedbyothervariables(theenvironment)isquiteimportant.Forexample,itisnecessaryforrealizationofqubitforquantumcomputers.However,standardmethodsthatarewellsuitedfortreatingsystemsofonedegreeoffreedom,theSchr¨odingerequation,instanton,etc.,donotworkwellinsuchcomplicatedsystems.WebelievethatNPRGisversatileenoughtoanalyzesuchsystems.Asafirststep,weanalyzequantumtunnelingphenomenaintwoparticlequantumsystems.§2.Non-perturbativerenormalizationgroupInthissection,webrieflysummarizetheformulationofNPRGwithD-dimensionalrealscalarfieldtheory.2.1.ScaledecompositionIntheNPRGmethod,thetheoryisdefinedbytheWilsonianeffectiveactionSΛ[φ].ThisisaneffectivetheorywithanultravioletenergycutoffΛ:Z=ZDφe−SΛ[φ].(1)Wedecomposethepathintegrationvariableφ(p)intotwopartswithrespecttothemomentumscalepasφ(p)=(φ(p)0≤|p|Λ−ΔΛ:lowermodes,φs(p)Λ−ΔΛ≤|p|≤Λ:shellmodes,andtransformthepartitionfunctionZasfollows:Z=ZDφDφse−SΛ[φ+φs],=ZDφe−SΛ[φ]ZDφse−SΛ[φs]e−SintΛ[φ,φs],Non-PerturbativeRenormalizationGroupAnalysisinQuantumMechanics3=ZDφe−SΛ[φ]De−SintΛ[φ,φs]Eφs,=ZDφe−SΛ[φ]e−ΔSΛ[φ],=ZDφe−SΛ−ΔΛ[φ],(2)whereΔSΛ[φ]≡−logDe−SintΛ[φ,φs]Eφs≡ZDφse−SΛ[φs]e−SintΛ[φ,φs].(3)Weunderstandtheshellmodepathintegralh···iφsastherenormalizationtrans-formation.Ifweevaluateitbyperturbativeexpansionwithrespecttocouplingconstants,weobtaintheso-calledperturbativerenormalizationgroupequations.16)Ofcoursesuchequationsarevalidonlyintheweakcouplinglimit.Instead,wetakethelimitΔΛ→0todefineNPRGequation,whichisthefundamentalprocedure.14)2.2.DerivationoftheNPRGequationTakingthelimitΔΛ→0,wecanexpresstherenormalizationtransformationasadifferential