Fixed points of the SRG evolution and the on-shell limit of the nuclear force

3.6 MB·

by  E. Ruiz Arriola

Fixed points of the SRG evolution and the on-shell limit of the nuclear force
E.RuizArriolaa,S.Szpigelb,V.S.Timo´teo1c
aDepartamentodeF´ısicaAto´mica,MolecularyNuclearandInstitutoCarlosIdeFisicaTeo´ricayComputacional
UniversidaddeGranada,E-18071Granada,Spain
bCentrodeRa´dio-AstronomiaeAstrof´ısicaMackenzie,EscoladeEngenharia,UniversidadePresbiterianaMackenzie
01302-907,Sa˜oPaulo,SP,Brasil
cGrupodeO´pticaeModelagemNume´rica-GOMNI,FaculdadedeTecnologia-FT,UniversidadeEstadualdeCampinas-UNICAMP
13484-332,Limeira,SP,Brasil
6 Abstract
1
0 We study the infrared limit of the similarity renormalization group (SRG) using a simple toy model for the nuclear
2
force aiming to investigate the fixed points of the SRG evolution with both the Wilson and the Wegner generators.
r Weshowhowafullydiagonalinteractionatthesimilaritycutoffλ→0maybeobtainedfromtheeigenvaluesofthe
p
hamiltonian and quantify the diagonalness by means of operator norms. While the fixed points for both generators
A
areequivalentwhennobound-statesareallowedbytheinteraction, thedifferencesarisingfromthepresenceofthe

8 Deuteron bound-state can be disentangled very clearly by analyzing the evolved interactions in the infrared limit
2 λ → 0 on a finite momentum grid. Another issue we investigate is the location on the diagonal of the hamiltonian

in momentum-space where the SRG evolution places the Deuteron bound-state eigenvalue once it reaches the fixed
]
h point. This finite momentum grid setup provides an alternative derivation of the celebrated trace identities, as a by
t product. ThedifferenteffectsduetoeithertheWilsonortheWegnergeneratorsonthebindingenergiesofA=2,3,4
-
l systems are investigated and related to the ocurrence of a Tjon-line which emerges as the minimum of an avoided
c
u crossingbetweenEα = 4Et −3Ed andEα = 2Et. Allinfraredfeaturesoftheflowequationsareillustratedusingthe
n toymodelforthetwo-nucleonS-waves.
[

2
v 1. Introduction
0
6 The similarity renormalization group (SRG) approach was proposed independently by Glazek and Wilson and
3
Wegner[1–4]andwasinitiallyappliedinsolid-statephysicstosimplifymany-particlerandomlydisorderedsystems.
2
Since the solution of a many-particle problem requires diagonalization of the hamiltonian, a transformation which
0
. makesitmorediagonalwouldbeofgreatapplicability,speciallyiftheeigenvaluesarepreserved. Thisideawasthen
1 appliedbyWegnerinordertosuppresstheoff-diagonalmatrix-elementsofthehamiltonianbymeansofunitarytrans-
0
formations[3].Theunitarityofthetransformationsensuresisospectralityandthegeneratorofthetransformationscan
6
1 bechosentobediagonalifonewantstodrivetheoriginalhamiltoniantowardsaband-diagonalformandtosuppress
: itsoff-diagonalmatrix-elements. Asaby-productonegetsaframeworkwheretheseoff-diagonalcomponentsmaybe
v
betterhandledinperturbationtheory.
i
X
The application of the SRG approach to nuclear physics was proposed by Bogner, Furnstahl and Perry [5] with
r thesesimplificationsinmindandwillbeourconcernhere(forreviewsseee.g.[6–8]andreferencestherein).Thefirst
a
applicationsoftheSRGmethodconsideredhigh-precision[9,10]andchiraleffectivefieldtheory(ChEFT)[11,12]
nucleon-nucleon(NN)interactionsastheinputforthetwo-bodyflowequations. Despitedifferentdegreesoffreedom
andtypesofinteraction,thenuclearmany-bodyproblemhasthesamedifficultyasthemany-electronprobleminsolid-
state physics. More diagonal hamiltonians result in great simplification and faster convergence in nuclear structure
calculations. ThisfeaturegaverisetoawholeprogramofapplicationoftheSRGmethodsinnuclearphysics[5–8].
Theformalismwasthenextendedtothree-bodyforcesinseveralschemes[13–15].
1Correspondingauthor,tel.:[phone],e-mailaddress:[e-mail]
PreprintsubmittedtoAnnalsofPhysics April29,2016
Yetinthetwo-nucleonsystem,theinterplaybetweentheSRGandasubtractiverenormalizationapproach[16–19]
wasinvestigatedusingtheChEFT NN interactionatleading-order[20]. Also, theroleoflong-distancesymmetries
ineffectiveinteractionsobtainedviaSRGflowequationshavebeenrecentlyinvestigatedbyus,showingthatthereis
a particular SRG scale at which the SU(4) spin-isospin Wigner symmetry is realized almost exactly [21, 22]. This
isaremarkableresultwhichshowsthatdespitetheincreasingpopularityoftheSRGideas,techniquesandextensive
computational applications there is still much to be learned from dedicated analysis. The present paper provides
furtherinsightsalongtheselines.
The aim of this work is to explore the details underlying our recent results scanning all values of the SRG cut-
off and to extend them. In our previous letter [23] a connection between the infrared limit of the SRG evolution
withtheWegnergeneratorandLevinson’stheorem[24]wasestablished. Consequencesoftheinfraredinteractions
for few-nucleon systems and the nuclear many-body problem were discussed in Ref. [25] where a theoretical and
phenomenologicallysuccessfulpredictionfortheTjon-linehasbeenadvanced. Thisisawellknownexistinglinear
correlationbetweenthebindingenergiesofthetritonandtheα-particlewhichshouldbeexpectedonthebasisofscale
invariance [26] (see e.g. [27] for a review and references therein). Further consequences regarding unitary neutron
matterintheon-shelllimitwithacalculationoftheBertschparameterwereaddressedin[28]andanSRGdiscussion
oftheBCSpairinggaphasalsobeenundertakeninRef.[29].
InthisworkweinvestigatethefixedpointsoftheSRGevolutionindetailusingasimpletoymodelforthenuclear
force in the two-nucleon S-waves as a particular illustration which simplifies the computational effort considerably
and allows for detailed numerical studies in the infrared limit. However, in the infrared region we will suggest
that many features are fairly general and model independent. Here we focus on the Wilson and the Wegner SRG
generators,sincetheevolutionwithablock-diagonalgenerator[30]intheinfraredregionhasalreadybeenstudiedin
previousworks[31,32],wheretheexplicitrenormalizationofasimpleNN forceandtheimplicitrenormalizationof
apionlesseffectivefieldtheory(EFT)atnext-to-leadingorderwereshowntobeequivalentoverawiderangeofthe
renormalizationscale. TheSRGequationsaremostlysolvednumericallyonafinitemomentumgridwithsufficiently
many points as to approach the continuum and therefore the grid is viewed as an auxiliary means. Here, we will
analyze many effects which can only be clearly disentangled with the aid of this momentum grid, which by itself
hassomeimplicationsonitsownandprovidesbothaninfraredandultravioletcutoffsfeaturingtwobasicproperties
of finite nuclei, namely, the long wavelength character of weak binding systems as well as the finite size of atomic
nuclei. Kukulinandcollaboratorsinaseriesofrecentandremarkableworkshaveprofitedfromthesefinitegridsin
thefew-bodyproblemincludingbothboundandscatteringstatesbyanalyzingHamiltonianeigenvalues[33–36]. It
is conceivable that a judicious combination of SRG and this approach may provide useful insights into the nuclear
few-bodyproblem.
Thispaperisorganizedasfollows.InSection2wepresenttheSRGflowequationsinoperatorformandintroduce
someusefulfunctionalnotationbothinthecontinuumlimitandinthediscretizedformwhichallowsforadiscussion
offixedpointsandtheirstability. InSection3wereviewthetoymodelwhichprovidesaquitereasonabledescription
oftheNNsystemintheS-wavechannelsatlow-momentaandwillbeusedinpracticetocarryoutourinfraredanaly-
sis. InSection4wedealspecificallywiththeSRGflowequationsonafinitemomentumgrid. Thescatteringproblem
asaLippmann-Schwinger(LS)equationonthefinitemomentumgridisanalyzedinSection5. Thereweprovidea
motivation to define the phase-shift as an energy-shift which, unlike the conventional phase-shifts determined from
thesolutionoftheLSequation,isinvariantunderunitarytransformationsonthegrid. Weillustratetheusefulnessof
theanalysisbyre-derivingasetofgeneralizedtraceidentities,firstunveiledbyGraham,Jaffe,QuandtandWeigelin
Ref.[37],butstartingfromamomentumgrid. OurnumericalresultsontheseissuesarepresentedinSection6. There
wefocusonthecrucialissueoftheorderingofstatesalongtheSRGevolutiontrajectoryandtheremarkableconnec-
tiontoLevinson’stheoremthroughanenergy-shiftformula.Ourinterpretationcorrectstheerroneousimplementation
ofKukulinandcollaborators[33]. Someoftheconsequencesofthecorrectidentificationandorderingofstatesfor
thenuclearbindingenergieswhenapproachingtheinfraredlimitintheA=2,3,4systemsarediscussedinSection7.
TheretheSRGcutoffparameterstriggerstheavoidedcrossingpattern, familiarfrommolecularphysics, underlying
theTjon-linecorrelation. Finally,inSection8wepresentasummaryoftheresultsandourmainconclusions.
2
2. SRGflowequationsinoperatorform
Thesimilarityrenormalizationgroup(SRG)approach,developedbyGlazekandWilson[1,2]andindependently
by Wegner [3], has been intensively applied in the context of nuclear physics to handle multi-nucleon forces in
order to soften the short-distance core [5, 8] with a rather universal pattern for nuclear symmetries [21, 38] and
interactions [39]. The basic strategy underlying the application of the SRG methods to nuclear forces is to evolve
an initial (bare) interaction H, which has been fitted to NN scattering data, via a continuous unitary transformation
that runs a cutoff λ on energy differences. Such a transformation generates a family of unitarily equivalent smooth
interactionsH =U HU†withaband-diagonalstructureofaprescribedwidthroughlygivenbytheSRGcutoffλ.
λ λ λ
We employ the formulation for the SRG developed by Wegner [3], which is based on a non-perturbative flow
equationthatgovernstheunitaryevolutionofthehamiltonianwithaflowparametersthatrangesfrom0to∞,
H =U H U† , (1)
s s 0 s
where H0 ≡ Hs=0 istheinitialhamiltonianinthecenter-of-masssystem(CM)andUs istheunitarytransformation.
Theflowparametershasdimensionsof[energy]−2andintermsoftheSRGcutoffλwithdimensionofmomentumis
givenbytherelation s=λ−4. Asusual,wesplitthehamiltonianasH =T +V ,whereT ≡Tcm+Trel isthekinetic
s s
energy,whichweassumetobeindependentofs,andV istheevolvedpotential. Foratranslationalinvariantsystem,
s
i.e.V ≡Vrel,wecanseparatetheCMandconsideronlytheSRGevolutionofthehamiltonianfortherelativemotion,
s s
Hrel = Trel+Vrel,sincetheCMkineticenergyTcm doesnotcontribute[40]. Forsimplicity,inwhatfollowswewill
s s
dropthesuperscript“rel”. TheSRGflowequationinoperatorformcanthenbewrittenas
dH dV
s = s =[η ,H ], (2)
ds ds s s
with
dU
η = sU† =−η† , (3)
s ds s s
andistobesolvedwiththeboundaryconditionH | ≡ H =T +V . Theanti-hermitianoperatorη whichspecifies
ss→0 0 0 s
the unitary transformation U is usually taken as η = [G ,H ], where G is a hermitian operator which we will
s s s s s
call the SRG generator since it defines η and so the flow of the hamiltonian. The most popular choices for the
s
generatoraretherelativekineticenergyG =T (Wilsongenerator)[5],theevolvingdiagonalpartofthehamiltonian
s
G =diag(H )= HD(Wegnergenerator)[2]andthesocalledblock-diagonalgeneratorG = HBD = PH P+QH Q,
s s s s s s s
wheretheoperators Pand Q = 1−Pareorthogonalprojectors(P2 = P, Q2 = Q, QP = PQ = 0)forstatesbelow
andaboveagivenmomentumscaleΛ [30]2.
BD
SofartheSRGequationsarequitegeneralandcanbeusedtoevolveanyhamiltonian. Inwhatfollowswewill
restricttothecaseofNN interactionsinthecenter-of-mass(CM)system.
2.1. Partial-waveequations
After decomposition of the NN interaction in partial waves the structure of the SRG flow equations simplifies
considerably in the off-diagonal relative momentum-space basis (see e.g. [44] for an explicit derivation and details)
forwhichthefollowingnormalizationinthecompletenessrelationwillbeassumed(hereandinwhatfollowsweuse
unitssuchthat(cid:126)=c= M =1,whereMisthenucleonmass):
(cid:90)
2
p2dp|p(cid:105)(cid:104)p|=1. (4)
π
Inserting this into Eq. (2) with the Wilson generator, G = T, the flow equation for the SRG evolution of the NN
s
potentialisgivenby(wedropthepartial-wavequantumnumbersforsimplicity)
dV (p,p(cid:48)) 2(cid:90) ∞
s =−((cid:15) −(cid:15)(cid:48))2 V (p,p(cid:48))+ dqq2 ((cid:15) +(cid:15)(cid:48) −2(cid:15) )V (p,q)V (q,p(cid:48)), (5)
ds p p s π p p q s s
0
2ThisisaunitaryimplementationtoallenergiesofthepreviouslyproposedVlowk approach[41]. Novelgeneratorshavebeenproposedin
[42,43].
3
where(cid:15) =(cid:104)p|T|p(cid:105)= p2. TheflowequationfortheSRGevolutionwiththeWegnergenerator,G = HD,reads
p s s
dV (p,p(cid:48)) (cid:104) (cid:105) 2(cid:90) ∞ (cid:104) (cid:105)
s =−((cid:15) −(cid:15)(cid:48)) e (s)−e (s) V (p,p(cid:48))+ dqq2 e (s)+e (s)−2e (s) V (p,q)V (q,p(cid:48)), (6)
ds p p p p(cid:48) s π p p(cid:48) q s s
0
wheree (s) = (cid:104)p|HD|p(cid:105) = p2+V (p,p). Aswesee,theseareintegro-differentialequationswhichcannotgenerally
p s s
besolvedanalytically(seehowever[45,46])andrequireamassivecomputationalefforttobesolvednumerically. We
willtacklethisproblembelowbythetraditionaldiscretizationmethodofthecontinuum3.
2.2. Operatorspace
Most compact features of the SRG formalism can be best appreciated within an operator space setup. We start
withsomeremarksfromtheoperatortheoryforfinite-dimensionaloperatorsinordertointroducesomenotation(see
e.g. the standard textbook [47]). For operators acting on a Hilbert space endowed with a scalar product of states ψ
andϕsuchas(cid:104)ψ,ϕ(cid:105),wecanalsodefineafurtherscalarproductbetweenoperatorsAandB,namely
(cid:104) (cid:105)
(cid:104)A,B(cid:105)=Tr A†B . (7)
FromherewedefinetheFrobeniusnormasusual
(cid:104) (cid:105)
||A||2 =(cid:104)A,A(cid:105)=Tr A†A , (8)
andhencetheinduceddistancebetweenoperatorsas
d[A,B]=||A−B||. (9)
Thenormcanalsobedefinedas
||A||=sup ||Aψ||, (10)
||ψ||=1
(cid:112)
where the standard scalar product induced norm, ||ψ|| = + (cid:104)ψ,ψ(cid:105) has been introduced. Of course, this requires
finite norm operators ||A|| < ∞ which is unfortunately not the case for the usual unbounded operators in quantum
mechanics,suchasthekineticenergy. ThereforeanultravioletcutoffΛisgenerallyassumedtooperatehere. Using
thepartial-waverelativemomentum-spacenormalizationmetricgiveninEq.(4)wehave
2(cid:90) ∞
(cid:104)ψ,ψ(cid:105)= dp p2|ψ(p)|2 . (11)
π
0
Thus,thekineticenergywouldhaveinfinitenorm,||T||→Λ2/M →∞ifeverythingistakenliterally. However,note
thatthecontinuumSRGflowequationsdonotneedthekineticenergytobebound,butratherthepotentialenergy.So,
theimplicitassumptionisthatatveryhighenergiesthekineticenergydominatesoverthepotentialenergyandhence
SRGflowequationsarewelldefinedprovidedthevalueofthepotentialdoesnotboundlesslygrow.Thefirstproblems
weencounterwithallthesepropertiesare: i)thefactthatthemomentum-spacebasisspansacontinuumsetandii)
theoperatorsareunbounded. Thisdifficultyiscircumventedinpracticebyusingafinitegridinmomentum-spaceand
alsointroducingahigh-momentumcutoff p =Λwhichwillbeassumedbelow.
max
2.3. Isospectralflowandfixedpoints
InthissectionwebrieflyreviewtheconceptsofisospectralflowandfixedpointsincorporatedintheSRGevolution
andthevariationalinterpretationinoperatorspace.
3However,theusefulnessofdiscretizingthecontinuumequationsgoesbeyondthepracticalneedofnumericallyimplementingthescattering
problem;itprovidesatheoreticalbridgebetweentheenergy-shiftinthespectrumandthescatteringphase-shifts.Wewillseethatthisbecomesa
crucialaspectwhenthenumberofgridpointsisreducedtoaminimum.
4
The isospectrality of the SRG flow equation becomes evident from the trace invariance property of the evolved
hamiltonianH . ForanychoiceoftheSRGgeneratorG wehaveaunitarytransformationandhence
s s
TrHn =TrHn , (12)
s 0
for any integer n. This property follows directly from the commutator structure of the SRG flow equation plus the
regulatorassumption4. Indeed,usingthecyclicpropertiesofthetraceweget
(cid:32) (cid:33)
d dH (cid:16) (cid:17)
TrHn =nTr Hn−1 s =nTr Hn−1[η ,H ] =0. (13)
ds s s ds s s s
FixedpointsoftheSRGevolutioncorrespondtostationarysolutionsoftheflowequation,Eq.(1),
dH
s =[[G ,H ],H ]=0. (14)
ds s s s
Thisconditionimpliesthatthereisabasisinwhichboth[G ,H ]andH becomesimultaneouslydiagonalatafixed
s s s
point. ThequestioniswhatchoicesofthegeneratorG actuallydrivethehamiltonian H tothediagonalform. For
s s
generatorsG whichsatisfyd/ds(TrG2)=0andusingthecyclicpropertiesofthetraceandtheinvarianceofTrHn,
s s s
wegetthat
d (cid:104) (cid:105)
Tr(H −G )2 =2Tr[G ,H ]2 =−2Tr (i[G ,H ])†(i[G ,H ]) ≤0, (15)
ds s s s s s s s s
because A ≡ i [G ,H ] = A† is a self-adjoint operator and therefore A†A is a semi-definite positive operator. Since
s s
Tr (H −G )2 is positive but its derivative is negative, the limit s → ∞ exists and corresponds to the infrared fixed
s s
pointoftheSRGevolution(λ → 0), atwhichthehamiltonian H becomesdiagonal. Thus, theSRGflowequation
s
justprovidesacontinuousproceduretodiagonalizetheinitialhamiltonianH =T +V .
0 0
InthecaseoftheSRGevolutionwiththeWilsongenerator,G =T,theflowequationisgivenby
s
dH dV
s = s =[[T,H ],H ]=[[T,V ],T +V ]. (16)
ds ds s s s s
Wethengetfor||V ||≡TrV2that
s s
d (cid:104) (cid:105)
TrV2 =2Tr[T,V ]2 =−2Tr (i[T,V ])†(i[T,V ]) ≤0. (17)
ds s s s s
Asaconsequenceofthisandusingtheunitaryequivalence,H =T +V =U H U†,wegetthat
s s s 0 s
0<TrV2 ≤TrV2 . (18)
s 0
Thereforetheremustbeaminimumvalueobtainedatthelimits→∞whichalsoimpliesin[T,V ]=0duetothe
s→∞
stationaryconditiononthederivativefortheinfraredfixedpoint. Hence
(cid:12)
limTrV2 =minTrV2(cid:12)(cid:12) . (19)
s→∞ s Vs s(cid:12)Hs=T+Vs=UsH0Us†
Thus,intheinfraredlimit s → ∞(λ → 0)theSRGevolutionwiththeWilsongeneratoryieldsasymptoticallytothe
smallestpotential,intheFrobeniusnormsense,givingthesamespectrumastheinitialpotentialV andcommuting
0
withthekineticenergyT,i.e. beingdiagonalinmomentum-space. Thisisaratherinterestingresultasitprovidesa
workingdefinitiononthe“size”ofthepotential,andmoreoverdifferentpotentialscanactuallybecomparedusingthe
distancebetweentheoperatorsinducedbytheFrobeniusnorm. Furthermore,thiscanbeinterpretedasaquantitative
whi4cMhafothreamloatciaclalploy,tesnutciahlaVp(rr,orp(cid:48)e)r=tyVis(ri)llδ(dre−finre(cid:48)d)dinivtehregecsoanstitnhueummolmimeinttusimncceuetovffen.Afolrson,=the1toranceehoafsaTcro(Vmsm)u=ta(cid:82)t0o∞rTp2r[VAs,(Bp,]po)nl=y(cid:82)v0a∞nirs2hdersVw(rh,ern)
bothTr(AB)andTr(BA)arefinite,asthechoiceA=pandB=xclearlyillustrates,since[p,x]=−i(cid:126)andhenceTr[p,x]=−i(cid:126)TrI=∞.
5
measureoftheoff-shellnessoftheinteraction. Inthepartial-waverelativemomentum-spacebasistheorbitaldegen-
eracy induced by the (2/π)p2dp integration measure in the Frobenius norm has two complementary effects. While
itsuppressesthecontributiontothenormfromlow-energystatesitalsoenhancesthecontributionfromhigh-energy
components. Thus,minimizingthepotentialalongtheSRGevolutiontrajectorytransfersveryefficientlyhigh-energy
componentsintolow-energycomponents. Thisprovidesaworkingschemewhereanyshort-distance,orequivalently
high-momentum core, becomes softer. It is fair to say that this is the main reason why SRG methods have become
popular in realistic nuclear applications. For completeness let us mention that there is an alternative interpretation
of softness of the interaction not based on the Frobenius norm, and based on the insightful work of Weinberg [48]
and taken up by recent studies from several viewpoints [49, 50] where the repulsive character of the interaction at
short-distanceplaysakeyrole. Atpresenttheconnectionbetweenthesetwoalternatives,whilesuggestingdifferent
measures of the softness, is somewhat vague and we will not dwell into it here. In our case, we will deal with a
potentialtoymodelwheretherepulsivepieceisabsentfromthestart(seeSection3).
InthecaseoftheSRGevolutionwiththeWegnergenerator,G = HD,theflowequationisgivenby
s s
dH dV
s = s =[[HD,H ],H ]. (20)
ds ds s s s
Then,inthiscasewegetfor||H −HD||2 ≡Tr(H −HD)2that
s s s s
d (cid:20)(cid:16) (cid:17)†(cid:16) (cid:17)(cid:21)
Tr(H −HD)2 =2Tr[HD,H ]2 =−2Tr i[HD,H ] i[HD,H ] ≤0, (21)
ds s s s s s s s s
suchthat||H −HD||→0andso
s s
lim H = HD =min||H −HD||, (22)
s s s s
s→∞ Hs
whichjustshowsthattheWegnergeneratorminimizesthedistancetoitsdiagonalmatrix-elementskeepingtheeigen-
values ofthe originalHamiltonian. Ofcourse, if thehamiltonianbecomes diagonalthe eigenvectorscannot befree
momentumeigenstates.
Finally,fortheblock-diagonalgeneratorwedefinetwoorthogonalprojectionoperatorsP+Q=1whichsplitthe
statesbeloworaboveagivenmomentumscaleΛ . Inthiscasetheflowequationisgivenby
BD
dH
s =[[PH P+QH Q,H ],H ]. (23)
ds s s s s
Wethengetthattheevolutionmakestheasymptotichamiltonianblock-diagonalhenceminimizingtheoff-diagonal
matrix-elements,
lim H = PH P+QH Q=min||H −PH P−QH Q||. (24)
s s s s s s
s→∞ Hs
2.4. Discreteequations
OnlyinfewcasestheSRGoperatorequationscanbehandledinthecontinuum[45,46].Inthissectionweanalyze
thedetailsoftheimplementationoftheSRGwhenafinitedimensionalreductionofthemodelspaceisimposed. The
particularcaseofamomentumgriddiscretizationofthecontinuumwillbediscussedspecificallyinalatersection.
Forsimplicitywewillconsiderabasisofeigenstates|n(cid:105)ofthekineticenergyoperatorT onafinite N-dimensional
HilbertspaceH ,namelyT|n(cid:105)=(cid:15) |n(cid:105)(n=1,2,...,N),andassumethatthecorrespondingspectrumofeigenvalues
N n
(cid:15) isnon-degenerate(similartowhathappensinthepartial-waverelativemomentum-spacebasisforwhich(cid:15) = p2).
n n n
Thus,thematrix-elementsofthehamiltonianH =T +V inthisbasisreadH ≡(cid:104)n|H|m(cid:105)=δ (cid:15) +V .
nm nm n nm
ThediscreteSRGflowequationsforthematrix-elementsofthehamiltonianinthecaseoftheWilsongenerator
canbewrittenintheform
dH (s) (cid:88)
nm = ((cid:15) +(cid:15) −2(cid:15) )H (s)H (s)
ds n m k nk km
k
(cid:88)
= −((cid:15) −(cid:15) )[e (s)−e (s)]H (s)+ ((cid:15) +(cid:15) −2(cid:15) )H (s)H (s), (25)
n m n m nm n m k nk km
k(cid:44)n,m
6
whereH (s)=δ (cid:15) +V (s)ande (s)≡ H (s)=(cid:15) +V (s). InthecaseoftheWegnergeneratorthediscreteSRG
nm nm n nm n nn n n
flowequationsread
dH (s) (cid:88)
nm = [e (s)+e (s)−2e (s)]H (s)H (s)
ds n m k nk km
k
(cid:88)
= −[e (s)−e (s)]2H (s)+ [e (s)+e (s)−2e (s)]H (s)H (s). (26)
n m nm n m k nk km
k(cid:44)n,m
TheseequationsaretobesolvedwiththeboundaryconditionsH (s)| ≡ H (0)=δ (cid:15) +V (0).
nm s→0 nm nm n nm
ThefixedpointsoftheSRGevolutionwithagivengeneratorG correspondtothestationarysolutionsoftheSRG
s
flowequationsforthematrix-elementsofthehamiltonian,
dH (s)
nm =(cid:104)n|[[G ,H ],H ]|m(cid:105)=0, (27)
ds s s s
whichimplies,forboththeWilson(G = T)andtheWegner(G = HD)generators,thatintheinfraredlimit s → ∞
s s s
(λ→0)thehamiltonianH becomesdiagonal5. Thus,wehavethat
s
lim H (s)=δ E , (28)
nm nm n
s→∞
where{E }N denotesthespectrumofdiscreteeigenvaluesofthehamiltonianH obtainedintheinfraredlimits→∞,
n n=1 s
whicharegivenby
E ≡ lime (s)=(cid:15) +V (s→∞). (29)
n n n n
s→∞
2.4.1. OrderingofthespectruminducedbytheSRGevolution
Aswehaveshown,theSRGevolutionwithboththeWilsonandtheWegnergeneratorsonafiniteN-dimensional
discretespacehaveinfraredfixedpointsatwhichthehamiltonianbecomesdiagonal. Thus,wehavetwointerpolating
SRG trajectories between the initial bare hamiltonian, H , and the final one, H . From this point of view, there
0 s→∞
seems to be no conceptual difference between the SRG evolution with both generators, since the isospectrality of
the SRG flow equation guarantees the invariance of the spectrum of eigenvalues of the hamiltonian H . This naive
s
argumentoverlooksanimportantdetail: thefactthatthroughtheSRGevolutionthebasisofeigenstates|ψ (s)(cid:105)ofthe
α
hamiltonianH isactuallychanging,i.e.
s
U |ψ (s=0)(cid:105)=|ψ (s)(cid:105), (30)
s α α
and thus the isospectrality does not necessarily fix the final ordering of the eigenvalues which is obtained in the
infraredlimit s→∞(λ→0). ThisisnotapeculiarfeatureoftheSRGevolution;itissharedbyanydiagonalization
procedure. IntheGausseliminationmethod[51], forinstance, onemakesanarbitrarychoiceonhowtoreducethe
originalmatrixtoadiagonalforminafinitenumberofsteps; there-orderingofeigenstateshastobeover-imposed
at the end by hand, arbitrarily choosing one of the possible permutations of the eigenvalues. Such a re-ordering
of eigenstates is equivalent to a unitary transformation. However, unlike the conventional diagonalization methods,
the SRG evolution can be interpreted as a continuum diagonalization through the flow parameter s, and thus an
infinite number of steps is involved. As a consequence, the SRG evolution induces a very specific ordering of the
eigenstates. This, of course, does not provide exact results since in practice the infrared limit s → ∞ is never
5Oneshouldnotethatthestationaryconditioninoperatorform,Eq.(14),inprinciplejustrequiresthatatafixedpointboth[Gs,Hs]andHs
becomediagonalinthesamebasis, notnecessarilytheoneinwhichthegeneratorGs isdiagonal. However, wecanshowthatthecondition
[[Gs,Hs],Hs]=0alsoimpliesthat[Gs,Hs]=0.TakingadiscretebasisinwhichHsisdiagonal,i.e.Hαβ(s)=δαβHα(s),wehave
(cid:88)(cid:104) (cid:105) (cid:104) (cid:105)
(cid:104)α|[Gs,Hs]|β(cid:105)= Gαγ(s)Hγβ(s)−Hαγ(s)Gγβ(s) =Gαβ(s) Hβ(s)−Hα(s) .
γ
Thus,forα(cid:44)βweget,intheabsenceofdegeneracies,that(cid:104)α|[Gs,Hs]|β(cid:105)=0onlyifGαβ(s)=0andsoboththehamiltonianHsandthegenerator
Gs arediagonalinthesamebasis. Ofcourse, thisbecomesatrivialresultforgeneratorsGs whichbydefinitionarediagonalinthebasisof
eigenstatesofthekineticenergyoperatorT,suchastheWilsonandtheWegnergenerators.
7
reached,buttheerrorsscaleexponentiallywiththeflowparametersandareoforderO[e−min(s(cid:15)n2)].Thecrucialaspect
is that since on a finite dimensional discrete space the SRG flow equation becomes a set of non-linear first-order
coupleddifferentialequationsforthematrix-elementsoftheevolvedhamiltonianH (s)withtheboundaryconditions
nm
H (s)| ≡ H (0),theuniquenessofthesolutionimpliesthatjustoneparticularorderingoftheeigenvaluestakes
nm s→0 nm
placeasymptoticallyintheinfraredlimits→∞,whichmaydependonthechoiceoftheSRGgeneratorG .
s
Letusconsidertheinitialbarehamiltonian H inthebasisofeigenstates|n(cid:105)ofthekineticenergyoperatorT on
0
a finite N-dimensional Hilbert space H . The spectrum of eigenvalues {(cid:15) }N of the operator T is assumed to be
N n n=1
non-degenerate and arranged in ascending order, i.e. (cid:15) < (cid:15) < ··· < (cid:15) . If we denote by {E0}N the spectrum of
1 2 N α α=1
N discreteeigenvaluesofH obtainedbyanyconventionalmatrixdiagonalizationmethodandarranged(byhand)in
0
ascendingordersimilarlytothespectrumofeigenvaluesofT,i.e. E0 < E0 < ··· < E0,thenwehavethatthefinal
1 2 N
orderingofthespectrumofeigenvalues{E }N oftheSRGevolvedhamiltonian H , obtainedasymptoticallyinthe
n n=1 s
infraredlimits→∞,isspecificallygivenby
{E }N ≡{E0 }N , (31)
n n=1 π(α) α=1
whereπ(α)isoneoftheN!possiblepermutationsofthespectrumofH .
0
It is important to note that in the continuum diagonalization of the hamiltonian through the SRG evolution, the
correspondencebetweenthekineticenergies(cid:15) andthediagonalmatrix-elementsofthepotentialV (s)ismaintained
n n
allthewayalongtheSRGtrajectory,asonecanclearlyseefromtheexpressionforthediagonalmatrix-elementsof
thehamiltonian,H (s) = (cid:15) +V (s). Thus,wehaveauniquewell-definedpairingofthekineticenergies(cid:15) withthe
nn n n n
eigenvalues E obtainedasymptoticallyintheinfraredlimit s → ∞,namely E = H (s → ∞) = (cid:15) +V (s → ∞),
n n nn n n
which is indeed what determines the specific ordering of the spectrum induced by the SRG evolution. One should
alsonotethatdependingonwhichparticularfinalorderingoftheeigenvaluestakesplaceintheinfraredlimit s→∞
theremaybecrossingamongstdiagonalmatrix-elementsofthehamiltonianH (s)alongtheSRGtrajectory. Onthe
nn
otherhand,intheconventionaldiagonalizationmethodsthereareN!differentwaysofpairingthekineticenergies(cid:15)
n
withtheeigenvaluesE0,correspondingtothepossibleorderingsofthespectrum. AswehavediscussedinRef.[23],
α
thisisacrucialissuetoestablishanisospectraldefinitionofthephase-shiftbasedonanenergy-shiftapproach,which
necessarilyinvolvesaprescriptiontoordertheeigenvaluesE0 andsettheirpairingwiththekineticenergies(cid:15) .
α n
Insection6wewillillustratethroughnumericalcalculationsthatwhenbound-statesareallowedbytheinteraction
thephase-shiftsevaluatedusingtheenergy-shiftapproachdonotcomplytoLevinson’stheorem[24]atlow-energies
ifanaivepairingissetjustbyorderingthespectrumofeigenvaluesE0 inascendingorderasthekineticenergies(cid:15) ;
α n
wefurthershowthatthespecificorderingofthespectruminducedbytheSRGevolutionwiththeWegnergenerator
in the infrared limit s → ∞ remarkably provides a prescription to evaluate the phase-shifts using the energy-shift
approachwhichallowstoobtainresultsthatfulfillLevinson’stheoreminthepresenceofbound-states. Insection7
wedicusstheinequivalentbehaviourofboththeWilsonandtheWegnergeneratorsinsimplevariationalcalculations
beyondsomecriticalSRGcutoffapproachingtheinfraredlimit.
2.4.2. Stabilityanalysisoftheinfraredfixedpoints
As we have pointed out, the uniqueness of the solution of the SRG flow equations on a finite N-dimensional
discrete space implies that a very specific final ordering of the eigenvalues of the hamiltonian is obtained in the
infraredlimit s → ∞(λ → 0). Ofcourse,theuniquenessofthesolutionfurtherimpliesthattheinfraredfixedpoint
towhichtheSRGevolvedhamiltonianissteadilydrivenmustbeasymptoticallystable. Inthissectionwewillcarry
outaperturbativestabilityanalysisoftheinfraredfixedpointsforboththeWilsonandtheWegnergenerators,which
isbasedonalinearizationoftheSRGflowequationssimilartothatdescribedinRef.[52],asanattempttodetermine
a priory the final ordering of the spectrum induced by the SRG evolution in the infrared limit. As we will see, the
perturbative analysis is well-succeeded in predicting the ordering of the spectrum only in the case of the Wilson
generator, although the results for both generators are consistent with the analytical proof of diagonalization of the
SRGevolvedhamiltonianpresentedinRefs[7,40].
Let us consider a perturbation of the matrix-elements of the SRG evolved hamiltonian H (s) near an infrared
nm
fixedpointH (s→∞)=δ E ,namely
nm nm n
H (s)=δ E +∆H (s), (32)
nm nm n nm
8
withthematrix-elementsoftheperturbation∆H (s)requiredtosatisfythecondition∆H (s→∞)=0.
nm nm
ByinsertingtheperturbedhamiltonianintotheSRGflowequationandtakingonlythetermstofirst-orderinthe
perturbation we can obtain a set of linearized flow equations for the matrix-elements ∆H (s). In the case of the
nm
WilsongeneratorG =T,wegetfromEq.(25)
s
d∆H (s)
nm =−((cid:15) −(cid:15) )(E −E )∆H (s). (33)
ds n m n m nm
Thesolutionsoftheseequationsforthediagonalmatrix-elements(n=m)arejustconstantswhichactuallyvanishdue
tothecondition∆H (s → ∞) = 0,namely∆H (s) ≡C = 0,whileforthenon-diagonalmatrix-elements(n (cid:44) m)
nn nn nn
thesolutionsaregivenby
∆Hnm(s)=Cnme−s((cid:15)n−(cid:15)m)(En−Em) , (34)
where the integration constantsC will depend on the initial conditions set for the matrix-elements H (s) of the
nm nm
perturbedhamiltonian. Thus,wehave
Hnm(s)= Enδnm+Cnme−s((cid:15)n−(cid:15)m)(En−Em)+... . (35)
Clearly, in the absence of degeneracies the off-diagonal matrix-elements will be ensured to monotonically decrease
withsprovided((cid:15) −(cid:15) )(E −E )>0. ThisimpliesthatfromallN!possiblefinalorderingsofthespectrumonlythe
n m n m
oneinwhichtheeigenvaluesE arearrangedaccordingtothekineticenergies(cid:15) ,i.e. inascendingorder,corresponds
n n
toanasymptoticallystableinfraredfixedpoint. Thus,intheWilsongeneratorcasetheperturbativestabilityanalysis
to first-order in the perturbation allows to predict beforehand the specific final ordering of the spectrum induced by
theSRGevolutionintheinfraredlimit.
InthecaseoftheWegnergeneratorG = HD,wegetfromEq.(26)
s s
d∆H (s)
nm =−(E −E )2∆H (s). (36)
ds n m nm
whichyieldsthesolution
∆Hnm(s)=Cnme−s(En−Em)2 , (37)
suchthat
Hnm(s)= Enδnm+Cnme−s(En−Em)2 +... . (38)
As one can see, in this case the off-diagonal matrix-elements will monotonically decrease with s (in the absence
of degeneracies) regardless the ordering of the eigenvalues and so in principle all N! possible final orderings of the
spectrum correspond to asymptotically stable infrared fixed points. Thus, in the Wegner generator case the specific
final ordering of the spectrum induced by the SRG evolution in the infrared limit cannot be determined a priory
throughtheperturbativestabilityanalysistofirst-orderintheperturbationandwehavetorelyonnumericalanalysis.
3. AsimpletoymodelfortheNNinteraction
We intend to explore the infrared limit of the SRG evolution (λ → 0). Quite generally, the equations to be
discussedinvolveheavynumericalcalculationswithadiscretizedcontinuumspectruminthecaseofnuclearphysics.
The problem is that most of the high-precision potentials, which fit NN scattering data up to the pion-production
√
threshold ( m M ∼ 350 MeV), have a very long tail in momentum space which requires many points and large
π N
momentum cutoffs not to miss important contributions. As a consequence the flow equation gets extremely stiff as
the SRG cutoff λ approaches zero, such that the computational effort becomes unduly expensive. Actually, there is
currentlyagapinSRGcalculationsbelowλ∼1fm−1forhigh-precision[9,10]andChEFT[11,12]NN potentials.
Therefore,wewillillustratemostofourpointsbyusingasimpletoymodelfortheNN interactionwhichreduces
the computational time and allow us to push the SRG evolution towards the infrared limit in a way which is not
9
practical with realistic interactions. The simplicity of the toy model does not affect the main features of the NN
interactionintheS-wavechannelsandgivesustheopportunitytoinvestigatetheinfraredfixedpointoftheSRGflow
equationswithanygenerator. HereweconcentrateontheWilsonandtheWegnergenerators.
Our framework is defined by a toy model for the NN force in the 1S and the 3S channels which consists of a
0 1
separablegaussianpotential,givenby
(cid:104) (cid:16) (cid:17) (cid:105)
V(p,p(cid:48))=Cg (p)g (p(cid:48))=C exp − p2+p(cid:48)2 /L2 . (39)
L L
The parameters C and L are determined from the solution of the LS equation for the on-shell transition matrix T
byfittingtheexperimentalvaluesoftheparametersoftheEffectiveRangeExpansion(ERE)tosecondorderinthe
on-shell momentum, i.e. the scattering length a and the effective range r . Namely, we solve the partial-wave LS
0 e
equationfortheT-matrixwiththetoymodelpotential,
2 (cid:90) ∞ V(p,q)
T(p,p(cid:48);E)=V(p,p(cid:48))+ dqq2 T(q,p(cid:48);E), (40)
π E−q2+i(cid:15)
0
whereEisthescatteringenergy,andmatchtheresultingon-shellT-matrixtotheEREexpansion,
(cid:34) (cid:35)
1 1
T−1(k,k;k2)=− − + r k2+O(k4)−ik =−[kcotδ(k)−ik] , (41)
a 2 e
0
√
wherek = E istheon-shellmomentumintheCMframeandδ(k)standsforthephase-shifts. Inordertoavoidthe
numericalintegrationonacontourinthecomplexplane,weswitchtotheLSequationforthepartial-wavereactance
matrixK withstanding-waveboundaryconditions,
2 (cid:90) ∞ V(p,q)
K(p,p(cid:48);k2)=V(p,p(cid:48))+ P dqq2 K(q,p(cid:48);k2), (42)
π k2−q2
0
wherePdenotestheCauchyprincipalvalue. TherelationbetweentheK-matrixandtheT-matrixon-shellisgivenby
K−1(k,k;k2)=T−1(k,k;k2) − ik=−kcotδ(k). (43)
FollowingthemethodintroducedbySteeleandFurnstahl[53],wefitthedifferencebetweentheinverseon-shell K-
matricescorrespondingtothetoymodelpotentialandtheEREexpansiontoaninterpolatingpolynomialofdegreek2
foraspreadofverysmallon-shellmomenta(k≤0.1fm−1),namely
∆K−1 = K−1(k,k;k2)−K−1 (k,k;k2)= A +A k2 . (44)
ERE 0 2
andthenminimizethecoefficientsA andA withrespecttothevariationsoftheparametersCandL.
0 2
Inthecaseoftheseparablegaussianpotentialtoymodel,givenbyEq.(39),itisstraightforwardtodeterminethe
phase-shiftsδ(k)fromthesolutionoftheLSequationfortheT-matrixusingtheansatz
T(p,p(cid:48);k2)=g (p)t(k)g (p(cid:48)), (45)
L L
where t(k) is called the reduced on-shell T-matrix. This leads to the simple relation (valid for separable potenials
only)
1 (cid:34) 2 (cid:90) ∞ 1 (cid:35)
kcotδ(k)=− 1− P dqq2 V(q,q) . (46)
V(k,k) π k2−q2
0
InTable1wedisplaythevaluesoftheparametersforthetoymodelpotentialinthe1S andthe3S channelsused
0 1
inournumericalcalculations,whichareadjustedtoreproducethecorrespondingexperimentalvaluesofa andr by
0 e
themethoddescribedabove.
Thephase-shiftsforthetoymodelpotentialinthe1S andthe3S channelsevaluatedfromEq.(46)areshownin
0 1
Fig.1,togetherwiththeresultsobtainedfromthe1993Nijmegenpartial-waveanalysis(PWA)[54]orthemorerecent
2013upgrades[55–58]. Asonecansee,despitethesimplicityofthepotentialandthefactthatthe3S channelisnot
1
treatedasacoupledchannel,ourtoymodelfortheNNinteractionprovidesareasonablequalitativedescriptionofthe
S-wavephase-shifts. Moreover,theon-shellT-matrixforthe3S channeltoymodelpotentialhasapolelocatedatan
1
imaginarymomentumk=iγ=i0.2314fm−1,correspondingtoasatisfactoryDeuteronbinding-energyB (cid:39)2MeV.
d
10