Symmetry conserving configuration mixing method with cranked states

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Physics Letters B

www.elsevier.com/locate/physletb

Symmetry conserving configuration mixing method with cranked states

Marta Borrajo, Tomás R. Rodríguez

^∗

, J. Luis Egido

DepartamentodeFísicaTeórica,UniversidadAutónomadeMadrid,E-28049Madrid,Spain

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received1April2015

Receivedinrevisedform12May2015 Accepted13May2015

Availableonline15May2015 Editor:J.-P.Blaizot

Keywords:

Beyond-mean-fieldtheories GCM

Time-reversalsymmetrybreaking Densityfunctionals

Wepresentthefirstcalculationsofasymmetryconservingconfigurationmixingmethod(SCCM)using time-reversalsymmetrybreakingHartree–Fock–Bogoliubov(HFB)stateswiththeGognyD1Sinteraction.

Themethodincludesparticlenumberandtridimensionalangularmomentumsymmetryrestorationsas well as configuration mixingwithin thegenerator coordinatemethod(GCM) framework.The nucleus 32Mgischosentoshowtheperformanceandreliabilityofthecalculations.Additionally,0⁺₁,2⁺₁ and4⁺₁ statesarecomputedforthemagnesiumisotopicchain,whereanoticeablecompressionofthespectrum isobtainedbyincludingcrankedstates,leadingtoaverygoodagreementwiththeknownexperimental data.

©^{2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

Atrustworthy description of the spectra of the atomicnuclei is one of the main goals of low-energy nuclear structure the- ory. The interacting shell model [1,2] is likely the most widely usedandsuccessfultool tocomputeaccuratelylow-lying spectro- scopicproperties.However,shellmodelapplicationsarelimitedto regions not far away from shell closures where manageable va- lencespacescanbedefined. Ontheotherhand,microscopicself- consistentmeanfieldmethods(SCMF)[3]basedonnuclearenergy densityfunctionalssuchasSkyrme,Gognyand/orRelativisticMean Field (RMF) can be in principle used throughout the whole nu- clearchart.Inordertoapplythesemethodstothestudyofnuclear spectra,theyhavetobeextendedbyincludingbeyond-mean-field (BMF) correlations. In particular,symmetry conservingconfigura- tion mixing methods (SCCM) are the most naturalextensions of SCMFapproachesandhaveshownafairperformanceindescribing qualitativelynotonlynuclearspectra,butalsogroundstateprop- erties,electromagnetictransitionsanddecays.Unfortunately,quan- titativeaccuratepredictions havenotbeenreachedso far,mainly duetothelackoftime-reversalsymmetrybreakingintrinsicstates withintheexistingimplementationsoftheSCCMmethods.Inthis letterwepresentan extensionoftheSCCM framework,basedon thegeneratorcoordinatemethod(GCM)withparticlenumberand triaxialangularmomentumprojections,thatincludescrankingin-

*

Correspondingauthor.

E-mailaddresses:[email protected](M. Borrajo),[email protected] (T.R. Rodríguez),[email protected](J. Luis Egido).

trinsicstates.Inthenumericalapplicationsweusethefiniterange density dependent Gogny interaction (D1S parametrization [4]).

In theearliest implementations oftheSCCM methodonlyaxially symmetric intrinsicstateswere considered [5–9].A major break- throughtowardsabetterdescriptionofthenuclearspectrawithin the SCCM framework was the inclusion of the triaxial degree of freedom[10–12].Afurtherstepforwardwasthefirstimplementa- tionofan SCCMmethodbasedona Skyrmepseudo-potentialfor oddnucleibyB.Ballywhoobtainedverypromisingresultsinde- scribingthebenchmarknucleus²⁵Mg[13].

On the other hand, the angular momentum projection with theenergydensityfunctionalsmentionedaboveisperformedafter theenergyvariation.Therefore,theconsiderationofonlyintrinsic wave functionswith zeroangular momentum content (< Jx>=

< Jy>=<^Jz>=⁰)inthe currentSCCMcalculationstendstofa- vor the groundstates with respectto other excited states anda stretchinginthespectraisusuallyfoundwithrespecttotheexper- imental values. The addition of time-reversalsymmetry breaking intrinsic states (< Jx>=⁰) obtained by the cranking procedure willthusincrease thevariationalspaceforexcitedstatesandwill provide a better description of the spectrum. Pioneering angular momentum projection of crankingstates has been reportedwith schematic pairing plus quadrupole interactions [14–16] andwith Skyrmeenergydensityfunctionals [17,18].However, neithercon- figuration–shape–mixingnor,inthecaseofSkymeinteractions, pairingcorrelations,weretakenintoaccount.Theaimofthisletter istopresentthefirstresultsoftheextensionoftheSCCMmethod http://dx.doi.org/10.1016/j.physletb.2015.05.030

0370-2693/©^{2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

now time-reversal-symmetry breaking intrinsic states introduced throughcrankingcalculations.

Thestartingpointistheconstructionofasetofintrinsicmany- body states having different deformations and intrinsic angular momentum. Such states, |β2,

γ

,Jc≡ |^{, have the structure of} Hartree–Fock–Bogoliubov(HFB)states[19]andarefoundbymini- mizingtheparticlenumberprojectedHFBenergy,¹i.e.:

E^_Jc

(β

2

, γ ) =  ˆ

^{H PNPZ}





^PNPZ

 − ω

J_c

 ˆ

^Jx

 − λ

q₂₀

 ˆ

^Q20

 − λ

q₂₂

 ˆ

^Q22



⁽¹⁾

where P^N(Z) is the neutron (proton) number projection oper- ator [19]. This is the so-called variation after particle number projection method (PN-VAP) [21]. The first term in the r.h.s. of Eq. (1) is the particle number projected energy and the last termscorrespond totheconstraints onthecrankingangularmo- mentum Jc and on the quadrupole deformation of the system (β2,

γ

). The Lagrange multipliers

ω

Jc, λq20 and λq22 ensure the conditions:

 ˆ

^Jx

 = 

J_c

(

J_c

+

¹

);  ˆ

^Q20

 =

^q20

;  ˆ

^Q22

 =

^q22 (2) where ˆJx is the x-component of the angular momentum op- erator and Qˆ2μ with

μ

= −²,−¹,..,2 is the

μ

component of the quadrupole operator. The deformation parameters mentioned abovearedefinedas:

q20

= β

₂cos

γ

C

;

^q22

= β

₂sin

γ

√

2C

;

^C

=



5

4

π

4

π

3r₀²A^5/3 (3) being A the massnumber andr0=¹.2 fm. In the presentwork we have imposed the parity conservation as a self-consistent symmetry of the intrinsic states: P|ˆ = |^{, being} Pˆ the par- ity operator. Therefore, neither negative parity states nor odd- multipoledeformation – such as the octupole– degreesof free- dom are explored here. Furthermore, these states are invariant under the so-called simplex-x, Sˆx|≡ ˆP^e−iπˆJx|= |^{, and the} T-simplex-y, Sˆy^T|≡ ˆP^e−iπˆJyˆT |= | symmetries, being ˆT ^the time-reversal operator. The last condition is chosen to have real coefficients in the HFB transformation, and the simplex-x sym- metry is very suitable to perform cranking calculations(Eq. (1)).

The set of operators { ˆP,Sˆx,Sˆ_y^T} ^{are the three generators of a} subgroup of the more general point group D_2h^T [22,23]. The lat- ter has an additional generator, e.g., the time-reversal operator.

Althoughtheuseof self-consistentsymmetriesconstrains thein- clusion of correlations within the mean-field approach through the spontaneous symmetry-breaking mechanism, they are im- posed to reduce the computational burden. In the present case, we will also exploit theseself-consistent symmetries to test the performance ofthe methodsince they providenon-trivial checks that help to identify possible inconsistencies. For instance, the choice of the collective coordinates (β2,

γ

) divides the possi- ble quadrupole deformations in six sextants, depending on the range of the angle

γ

[19]. As a result, the values of

γ

equal to 0^◦(60^◦), 120^◦(180^◦) and 240^◦(300^◦) correspond to prolate (oblate) axial deformation and they are related by the differ- ent orientations of the principal axes of inertia with respect to the z-axis [19]. If J_c =^{0, the intrinsic wave functions do not}

1 Forthe sakeofsimplicity,we willwritedownthroughoutthe textanyen- ergy kernelasanexpectation value ofa hamiltonian operator.However, Gogny interactionscontainadensity-dependenttermwhichpreventssuchanotationrig- orously[20].Nevertheless,thistermishandledseparatelyinasimilarfashionasin Refs.[6,11],andthefollowingnotationcanbestillused.

on the orientation of the axes, being all of the sextants com- pletely equivalent. However, if J_c=^{0, the energy will depend} on the orientation of the principal axes of inertia with respect to theintrinsicrotation axis,in ourcase, thex-axis(see Eq.(1)).

Therefore, the intrinsic states are only invariant under the sub- group of the D_2h^T point group mentioned above andthesextants are now symmetric only with respect to the

γ

= (^120◦,300^◦) direction.

We check this symmetry by performing PN-VAP calculations in the(β2,0^◦≤

γ

_≤360^◦) plane forthreevaluesof thecranking angular momentum J_c=0 (time-reversal symmetry conserving), 2 and4,selectingthenucleus³²Mgasanexample.InFig. 1(a)–(c) we show such potential energy surfaces (PES) – first term in the r.h.s. of Eq. (1). Here, the intrinsic states were expanded in ninemajorsphericalharmonicoscillatorshellsandthenumberof points included in the mesh of each PES is 502.We notice first theequivalencebetweenall ofthesextantsinthecasewherethe time-reversalsymmetryispreserved( Jc=^0,Fig. 1(a)).Suchare- dundancy is reduced to the half of the plane separated by the

γ

= (^120◦,300^◦)axisfor J_c=^{2 and4(Figs.1(b)–(c))asexpected.}

Wefindtheabsoluteminimumofthe Jc=^{0 PESinthespherical} point asitis presumed fromtheneutronmagic number N=^20.

Theenergygrowsmorerapidlyalongtheoblatedirectionsthanin theprolateones.Additionally,asecondminimum–around1 MeV higher–isobtainedataxialprolateconfigurationswithβ2=⁰.45.

For largervalues than β2≈⁰.7 theenergyincreases quicklyalso along theprolatelines.For Jc=²,4 thePESresemble the Jc=⁰ one except for the values along

γ

₌120^◦, where the energy is not as favoredas in

γ

=^{0◦ and}

γ

=^{240◦.The minima ofthese} surfaces appearatsuch prolate configurations withβ2=⁰.45,as in the Jc =^{0 case but shifted to higher energy values, around}

≈^{2 MeV and}≈^{3 MeV for J}c=2 and 4 respectively.

Theintrinsicmany-bodystates,|^{,breakalsotherotationalin-} variance ofthe hamiltonian andthese quantumnumbers can be restored by projecting onto good number ofparticles (N,Z ) and angularmomentum( J,M):

|

J M

;

N Z

; σ ; β

2

, γ ,

Jc

 = 

K

g_K^Jσ_(β

2,γ,Jc)

|

J M K

;

N Z

; β

2

, γ ,

Jc



(4) whereK= −^J,−^J+¹,. . . ,J isthecomponentoftheangularmo- mentum inthebody-fixed z-axisandthestatesgiveninther.h.s.

aredefinedas:

|

^{J M K}

;

^{N Z}

; β

2

, γ ,

J_c

 =

^P_{M K}^{J PNPZ}

|β

2

, γ ,

J_c



⁽⁵⁾

being P_{M K}^J the angularmomentum projection operator [19].Ad- ditionally, the coefficients g^Jσ

K(β2,γ,Jc) and the projected energies E^Jσ

β2,γ,Jc are found variationally by solving the so-called Hill–

Wheeler–Griffin(HWG)equationsinthe K -subspace[19]:



K^



H_{K K}^{J;N Z}(β2,γ,Jc)

−

E_β^Jσ

2,γ,JcN_{K K}^{J;N Z}(β2,γ,Jc)



g_K^Jσ_(β

2,γ,Jc)

=

0 (6)

where

σ

=¹,2,. . . labels the possiblesolutions of such general- izedeigenvalueproblemsand:

N_{K K}^{J;N Z}(β2,γ,Jc)

= 

^{J M K}

;

^{N Z}

; β

2

, γ ,

J_c

|

^{J M K}

;

^{N Z}

; β

2

, γ ,

J_c



⁽⁷⁾ H_{K K}^{J;N Z}(β2,γ,Jc)

= 

^{J M K}

;

^{N Z}

; β

2

, γ ,

J_c

| ˆ

^H

|

^{J M K}

;

^{N Z}

; β

2

, γ ,

J_c



(8) are the norm and hamiltonian overlaps. From the above defini- tions, we can obtain some usefulpropertiesof the state P^NP^Z|

Fig. 1. (Coloronline.)(a)–(c)PN-VAPand(d)–(f)particle numberandangularmomentumprojectedpotentialenergysurfacesfordifferentvaluesofthecrankingangular momentum Jcforthenucleus³²Mg.GognyD1Sinteractionisusedhere.Thecontourplotsareseparatedinenergyby1.0 MeV.EachPESisnormalizedtotheenergyof theircorrespondingminima,i.e.,(a)−249.902 MeV,(b)−247.910 MeV,(c)−246.789 MeV,(d)−252.924 MeV,(e)−252.021 MeV and(f)−250.463 MeV.Theblackbullets in(a)–(c)arethestatesincludedintheGCMcalculationwhiletheyellowsquaresin(c)arethestatesanalyzedindetailinFigs. 2–3.

such as [19,16]: a) the probability distribution, W_K^J(β2,

γ

,Jc), of finding an eigenstate of the angular momentum |^{J K}^{; b)} the total probability distribution, W^J(β2,

γ

,Jc), of finding a value of the angular momentum J ; and c) the projected energy E_K^J(β2,

γ

, Jc).

W_K^J

(β

2

, γ ,

Jc

) =

^N

J;^{N Z} K K(β₂,γ,Jc)

β

2

, γ ,

Jc

|

^PNPZ

|β

2

, γ ,

Jc



⁽⁹⁾ W^J

(β

2

, γ ,

Jc

) = 

K

W_K^J

(β

2

, γ ,

Jc

)

(10)

E_K^J

(β

2

, γ ,

Jc

) =

^H

J;N Z K K(β2,γ,Jc)

N_{K K}^{J;N Z}_(β2,γ,Jc) ⁽¹¹⁾

Thedecomposition W_K^J andtheenergy E_K^J arequantitiesthat dependontheorientationoftheprincipalaxesofinertiawithre- spect to the (x,y,z)-axes. Nevertheless,the following properties are deduced fromthe self-consistent symmetries imposed tothe intrinsicstates:W_K^J=^W₋^JK,E_K^J =^E₋^JK,and,if J isodd,theK=⁰ component is forbidden.The dependence on K is removed once the K -mixing is performed since the norm W^J and the energy E^Jσ

β2,γ,Jc are scalarquantities[19,10,11].Inaddition, if J_c=^{0 the}

Fig. 2. (Coloronline.)ProbabilitydistributionsofprojectionsK – W_K^J(β2,γ,Jc)– for even(leftpanel)and odd(rightpanel)values ofthe angular momentum J fortheintrinsicstates:(a)–(b)(β2,γ,Jc)= (0.5,10^◦,4)and(c)–(d)(β2,γ,Jc)= (0.5,230^◦,4).Distributionofprobabilitiesof(e)evenvaluesand(f)oddvaluesof theangularmomentum J –W^J(β2,γ,Jc)–forthesameintrinsicHFB-typewave functions.

Fig. 3. (Coloronline.) Particlenumberand angularmomentum projectedenergy overlaps E_K^J(β2,γ,Jc) as a function of K for the intrinsic wavefunctions (a) (β2,γ,Jc)= (⁰.5,10^◦,4)(filledsymbols)and(b)(β2,γ,Jc)= (⁰.5,230^◦,4)(empty symbols).Thelastcolumncorrespondstothelowestenergiesforagiven J afterK− mixingE_β^Jσ

2,γ ,J_c(Eq.(6))forthesameintrinsicstatesasin(a)and(b).

sameprobability distribution W^J andthesame angularmomen- tumprojectedenergyarefoundinthe sixsextantsofthe(β2,

γ

) plane.However,ifthecrankingtermisnon-zero,the(β2,

γ

)plane

line.

Wenowexploitthesesymmetriestoperformconsistencytests oftheresultsandchecktheimplementationofthemethod.There- fore,weselectfirsttwointrinsicstates,|β2=⁰.5,

γ

=^10◦,Jc=⁴ and|β2=⁰.5,

γ

=^230◦,Jc=⁴ ^{that aresymmetric withrespect} to the(

γ

=^120◦,300^◦)line(see yellowsquaresinFig. 1(c)).We representthedecompositionofthosestatesincomponentsofthe angularmomentum J andintrinsicz-projectionK inFig. 2(a)–(d), normalizedtothetotal probabilityinagiven J .Here weobserve that the decomposition in K is differentdepending on thevalue of

γ

. For

γ

=^{10◦ the} probability decreases in general rapidly with increasing K for a fixed value of J . Furthermore, the rel- ative weight of the components with large K tends to increase with larger angular momentum J , while the K =0 component foreven J and K=¹,2 components forodd J slightly decrease.

Theseresults areconsistent withhavingtheintrinsiclong inertia axisnearlyalongthez-axis.Ontheotherhand,theprobabilityfor a given J isdistributedin alarger numberof K componentsfor

γ

=^{230◦ andthesecomponentsaremuchflatterthaninthepre-} vious case when the angular momentum J is increased. In this case, the intrinsiclong inertia axis isalmost orientedperpendic- ular to the z-axis. Nevertheless,the decompositionin J of both states, summing all of the K components, are identical. We ob- serve two separate distributionsforeven J (Fig. 2(e))andodd J (Fig. 2(f)),beingtheabsolutescalelargerfortheformer.Theeven (odd) distribution probability increases from J=^{0 (1) until the} maximumat J=^{4 (5)isreached.Then, WJ decreases,obtaining} practicallyzeroprobabilityforeven(odd)angularmomentalarger than16(13).InFig. 3werepresenttheE_K^J(β2,

γ

,Jc)energiesde- finedabove.Weseeasinthepreviouscasenoticeabledifferences depending on the

γ

values.For

γ

=^{10◦ theenergies rise rather} quicklyforlargevaluesof K while for

γ

=^{230◦ theenergies are} flatter.Thesedifferencesare completelyremoved when K -mixing is performed through solving the HWG equations (Eq. (6)) as it is shown in Fig. 3(c). There, three bands can be distinguished, namely, agroundstate rotationalband withJ=^{2 builton top} of J=⁰⁺₁^,andtwoJ=^{1 bands,being J}=²⁺₁ ^{and J}=¹⁺₁ ^the correspondingband-heads.

We cantest even furthertheperformance of theangularmo- mentum projection by projecting the whole (β2,

γ

) plane as it is plotted in Fig. 1(d)–(f). For the Jc=^{0 case, the} equivalence between the six sextantsis preserved when angular momentum projection is performed. However, the angular momentum pro- jected PES attained by restoring the rotational symmetry of the J_c=^{0 statesaresymmetriconlyaroundtheaxis}(

γ

=^120◦,300^◦) (Fig. 1(e)–(f)).InFig. 1(d)–(f)onlythePESfor(J= ^Jc,

σ

₌1)are shown although the same equivalence is obtained forother val- ues of (J,

σ

). Apart from the symmetries discussed above, the angular momentum projection modifies significantly the surfaces obtainedatthePN-VAPapproach.Ingeneral,theminimafoundin the PES atthe PN-VAPlevel are nowwider anda slightly larger deformation is obtainedwhenever the angularmomentum is re- stored. As a matter offact, these beyondmean-field correlations move thegroundstatefromthesphericalpointtoprolateconfig- urations with β2≈⁰.5 (Fig. 1(d)), that was formerlya secondary minimuminthePN-VAPcalculation(Fig. 1(a)).Thiseffectwasal- ready obtainedwithaxial calculations[6]andisa self-consistent way to obtain the deformed ground state for the nucleus ³²Mg, i.e.,asbelongingtothe‘islandofinversion’withanerosionofthe N=^{20 magicnumber.}

The last step in the present SCCM many-body method is the configurationmixing:

Fig. 4. (Coloronline.)(a)Excitationenergiesoftheyraststatescalculatedforthe nucleus³²Mg with the GCM method with 17axial states and Jc=0 (S1),49 axial+^{triaxialstatesand J}c=^{0 (S}2),81axial+^{triaxialstatesand J}c=⁰,2 and113 axial+^{triaxialstatesand J}c=⁰,2,4.(b) Excitationenergiesforthethreelowest statesforeach J -value,calculatedwiththeGCMmethodwith113axial+^triaxial statesand Jc=⁰,2,4 (fullsymbols),largescaleshellmodelcalculations(opencir- cles)andexperimentaldata(asterisks).

|

J M

;

N Z

; σ  = 

β2,γ,K,J_c

f_β^Jσ

2,γ,K,Jc

|

J M K

;

N Z

; β

2

, γ ,

Jc



(12)

Again, the coefficients f^Jσ

β2,γ,K,Jc in Eq. (12), and the final spec- trum,E^Jσ ,areobtainedbysolvingthegeneralHWGequations:



{α^}



H_{^J_α^{;N Z}_};{α}

−

^EJσN_{^J_α^;_};{^{N Z}_α}



f_{^J_α^σ_}

=

^{0 (13)}

where{

α

_{}≡ {β}2,

γ

,K,Jc}^{nowencodesall the}constraintsand K in a single index. H and N are the energy and norm overlaps respectively:

N_{^J_α^;_};{^{N Z}_α}

= 

^{J M K}

;

^{N Z}

; β

2

, γ ,

J_c

|

^{J M K}

;

^{N Z}

; β

2^

, γ

^

,

J^_c



⁽¹⁴⁾ H_{^J_α^{;N Z}_};{α}

= 

^{J M K}

;

^{N Z}

; β

2

, γ ,

J_c

| ˆ

^H

|

^{J M K}

;

^{N Z}

; β

₂^

, γ

^

,

J_c^



⁽¹⁵⁾

Toshedlight ontheimpact ofincludingtime-reversalsymmetry breakingstatesinthespectrum,thenucleus³²Mghasbeencom- putedwiththeGCMmethodusingfoursetsofintrinsicwavefunc- tions.All ofthemare computedwithnine major oscillatorshells (No.s.=^{9) inthe workingbasis.2 The simplest one (S}1) ismade ofthe17axialandtime-reversalsymmetricstates.Suchstatesare markedinFig. 1(a)withdots alongthe(

γ

₌0^◦,180^◦)axis.Then, the S2 set is defined by adding 32 more time-reversal conserv- ingstates(J_c=⁰)inthe(β2,

γ

)plane(theremaining dotsinthe samefigure). Finally, two more batchesof states, S₃ and S₄, are establishedby adding 32time-reversal symmetry breakingstates with Jc=^{2, and 32 more with J}c=^{4 (see the dots in Fig. 1).}

Therefore, S1⊂^S2⊂^S3⊂^S4,beingthetotalnumberofstatesin thelargestsetequalto113.

Theground state bandscalculated withthe GCMmethod im- plemented with the differentsets described above are shownin Fig. 4(a). Firstly, the ground state energies obtained for the dif-

2 Wehavecheckedtheconvergenceoftheenergyspectrabyperformingaxial calculationsincludinguptothirteenmajoroscillatorshells.Wehaveobtainedprac- ticallythesameresultsastheNo.s.=^{13 onesalreadywithN}o.s.=^{7,thusshowing} agoodconvergenceoftheresultswithrespecttothesizeoftheworkingbasis.

ferentcalculationsareprettycloseexceptforthepure axialcase, namely:

E

(

0⁺₁

) = −

²⁵³

.

056

, −

²⁵³

.

477

, −

²⁵³

.

486

, −

²⁵³

.

498 MeV forS1,S2,S3andS4respectively.Thatshowsthatthegroundstate energyisconvergedwithrespecttoaddingtime-reversalsymme- try breaking components. However, the excited states are more affectedbytheinclusionoftriaxialandcrankingstates.Hence,we first observe a moderate compression of the spectrum from the axial (K =⁰)to triaxialcalculationswith Jc=^{0.The decreasein} energy is larger withincreasing the angular momentum, mainly duetothepossibilityofhavingmore K -mixingintheGCMstates.

However,thevariationalspacefortheexcitedstatesaremuchbet- terexplorediftime-reversalsymmetrybreakingisallowed.There- fore,asignificantcompressionofthespectrumisobtainedforthe S3andS4setsandthedifferences,onceagain,arebiggerforlarger angularmomentum. In addition,we can infer that theexcitation energiesforthe2⁺₁ and4⁺₁ statesarealreadyconvergedwiththe S₃ calculation sincethey donot varysignificantly fromincluding Jc=^{4 states tothe J}c=⁰,2 ones.Thisisnot thecaseforlarger valuesoftheangularmomentum, whereprobablyintrinsicstates with Jc=⁶,8,. . . should be also included in the GCM. For the sakeofcompleteness,thefullspectrumcomputedwiththe S4 set isrepresentedinFig. 4(b).Here,thefirsttwobandsdisplayarota- tionalcharacter, witha parabolictrendintheexcitation energies, 0⁺ band-heads andJ=^{2 spacing. Athird band starting at2+}₃ withJ=^{1 isalsoobtained,showingaslightodd-even J stagger-} ing.Inaddition,largescaleshellmodel(LSSM)results[24,25]and experimentaldata[26–30]arealsorepresentedinFig. 4(b).Thanks to thecompression ofthe spectrum produced by theaddition of crankingstates,aremarkableagreementbetweentheexperimen- talandtheoreticalvaluesforthe 2⁺₁ and4⁺₁ energiesisobtained.

Inaddition,thepresentSCCMcalculationspredictverysimilarex- citation energies to the LSSM values for the g.s. band. However, the low excitation energyof the0⁺₂ state [30] isnot reproduced here. LSSM calculations have shown that this state is very sen- sitive to a subtle mixing of spherical 0p–0h and superdeformed 4p–4hconfigurations[25].Inthepresentframework,theinclusion ofpairing fluctuations[9]and/or explicitquasiparticleexcitations could help to solve thisproblem since the excited 0⁺ statesare mainlyaffectedbysuchadegreeoffreedom,loweringtheexcita- tionenergiesofthosestates.

Finally,weexploresystematically theeffectoftheinclusion of time-reversalsymmetrybreakingstatesinthemagnesiumisotopic chain^24–34Mg.TheresultsareobtainedwithNs.o.=^{7 –themini-} mumthatguaranteesagoodconvergencewithrespecttothesize of the basis in this isotopic chain – and the sets of wave func- tions definedabove S₁,S₂,S₃, i.e., axial andtriaxial shapeswith Jc=^{0 and 2are included. InFig. 5 we plotthe excitation ener-} giesforthe2⁺₁ and4⁺₁ calculatedwiththesedifferentapproaches compared to the experimental values.We see that the axial cal- culations describe the trends of the experimental data but the energiesarelargelyoverestimated.Includingthetriaxialdegreeof freedomwithoutbreakingthetime-reversalsymmetryreducesthe excitationenergiesbutthepredictedvaluesarestilltoohighwith respect to the experiments. Finally, adding Jc=^{2 states to the} GCMset ofwave functionscompresses furtherthespectrum and an outstandingagreementwiththeexperimental valuesisfound.

The only nucleuswhere the theoretical values tend to be lower than theexperimental onesis thenucleus ²⁴Mg. Since thisisan N=^{Z nucleus,somealphaclusteringand/or}proton–neutronpair- ing correlations could be missing within the present framework whichassumesastructureoftheintrinsicstatesgivenbyadirect productofprotonsandneutronswavefunctions.However,mixing

Fig. 5. (Coloronline.)2⁺₁ and4⁺₁ excitationenergiesfortheMgisotopicchaincal- culatedwiththe GCMmethodincludingaxialstates(redsquares),axial+^triaxial with Jc=0 states(bluediamonds)andaxial+triaxialwith Jc=0,2 states(magenta opendots).Experimentalvalues(blackdots)aretakenfromRef.[31]andreferences therein.

protons and neutrons to take into account such proton–neutron pairingcorrelationsisbeyondthescopeofthepresentstudy.

In any case, we have to underline that these results consti- tute the first explicit evidence of the compression of the spec- trumwhentime-reversalsymmetrybreakingistakenintoaccount in GCM calculations with particle number and angular momen- tumprojection.Globalcalculationsperformedwiththesemethods assuming axial symmetry have displayed a systematic overesti- mation of the 2⁺₁ excitation energies around a factor ∼¹.2–1.4 withrespecttotheexperimentalvalues,bothforSkyrme[32]and Gogny[33] functionals.Thepresentresultsshow thatsuch adis- agreement could be corrected by including triaxial and J_c=⁰ statesin the GCM framework. Infact, the incorporation of J_c in the GCMansatz (Eq. (12)) is a generalization ofthe double pro- jectionmethodofPeierlsandThouless[34,35].Thedoubleprojec- tion methodis knownto providethe exact translational massin thecaseoftranslationsby takingascoordinatesthe positionand thelinearvelocitiesinageneratorcoordinatemethod.Weexpect, therefore,thatthemomentofinertiaofourtheorywillbesimilar tothe oneprovided by theangularmomentum projectionbefore variationapproach,insteadoftheYoccozmomentofinertiagiven bytheangularmomentumprojectionaftervariationmethodused inearlierapproaches.Thisexpectationisconfirmedbyourresults that provide moments of inertia very close to the experimental ones.

Insummary,wehavepresentedthefirstGCMcalculationswith particle number and angular momentum projection of HFB-like statesconsideringdifferentquadrupoledeformations(axialandtri- axial)andintrinsiccrankingangularmomentum.Theperformance ofthemethodhasbeencheckedby takingadvantageoftheself- consistentsymmetriesimposed totheintrinsicmany-bodystates.

Since such wave functions were chosen to be eigenstates of a D^T_2h sub-groupgeneratedbythe parity,simplex-xandT-simplex- ysymmetryoperators ({ ˆP,Sˆx,Sˆ_y^T}^{),thepotential energysurfaces} (particle number and particle number plus angular momentum projected) must be symmetric in the (β2,

γ

) plane withrespect tothe (

γ

₌120^◦,300^◦) axis.Wehavecheckedsuch a non-trivial

takingthenucleus Mgasanexample.Theeffectofincludingin- crementally intrinsicstates withmore symmetries brokenin the GCMframeworkhasbeenalsoanalyzedin³²Mgandinthemag- nesiumisotopicchain^24–34Mg.Theresultshaveshownthatadding ( Jc=⁰⁾ time-reversalsymmetrybreakingstatessqueezes notably thespectraduetoabetterdescriptionoftheexcitedstatesfroma variationalpointofview.Sucha compressionputs thetheoretical valuesontopoftheexperimentalonesforthelowest2⁺ and4⁺ statesinthechain.Thenextstepwillbethecalculationofelectro- magneticpropertieswithinthepresentSCCMapproachandsome workisinprogressalongtheselines.

Acknowledgements

We acknowledge the support from GSI-Darmstadt and CSC- Loewe-Frankfurt computing facilities. T.R.R. thanks A. Poves and F. Nowacki for fruitful discussions and for providing us with the shell model results of ³²Mg. This work was supported by the Ministerio de Economía y Competitividad under contracts FPA2011-29854-C04-04,BES-2012-059405andProgramaRamóny Cajal2012number11420.

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