Uncertainty of the nuclear intrinsic states in the improved variation after projection calculations
UUncertainty of the nuclear intrinsic states in the improved variation after projectioncalculations
Zao-Chun Gao ∗ China Institute of Atomic Energy, Beijing 102413, People’s Republic of China (Dated: February 9, 2021)Projection is noninvertible. This means two different vectors may have the same projected com-ponents. In nuclear case, one may take the intrinsic state as a vector, and take the nuclear wavefunction as the projected component obtained by projecting the former onto good quantum num-bers. This immediately comes to the conclusion that, for a given nuclear state in the laboratoryframe of reference, the corresponding intrinsic state in the intrinsic frame of reference can not beuniquely determined. In this letter, I will show this interesting phenomenon explicitly based on theimproved variation after projection(VAP) method. First of all, it is found that, the form of the trialVAP wavefunction with spin J can be greatly simplified by adopting just one projected state ratherthan previously adopting all (2 J + 1) spin-projected states for each selected Slater determinant.This is crucial in the calculations of high-spin states with arbitrary intrinsic Slater determinants.Based on this simplified VAP, the present calculations show that orthogonal intrinsic states (dif-fered by K ) may have almost the same projected wavefunctions, indicating the uncertainty of thenuclear intrinsic states. This is quite different from the traditional concept of intrinsic state whichis expected to be unique. Projection is a powerful technique that has long beenused in various fields of physics, such as the particlephysics [1], the atomic physics [2] and the physics ofcondensed matter [3]. In nuclear physics, the projectionmethod has also been used by many authors to improvethe quality of the nuclear wavefunctions [4–13]. Thistechnique removes the spurious part of the trial wave-function and makes the projected wavefunction rathersuitable in describing various properties of nuclei andother quantum systems. The projected wavefunctionswith different forms have been varied by several authors[4, 7, 8, 11–13], so that they can be as close as possibleto the corresponding eigenfunctions obtained by the shellmodel(SM) [14]. Such variation of the projected wave-function is often called as variation after projection(VAP)[15]. The VAP methods are expected to be applicable inlarge model space where full shell model calculation cannot be performed presently or in the near future.Projection is noninvertible. In geometry, one can eas-ily find two different vectors whose projected componentsin a given direction are identical. In nuclear physics, amean-field state with symmetries broken can be consid-ered as a vector, and its projected states with good quan-tum numbers are the components of the vector. Similarly,one can imagine that there may exists two different in-trinsic states whose projected states are the same. Butunlike the case in geometry, it is not simple to find suchtwo different intrinsic states. However, the VAP withprojected wavefunction fully optimised may provide thepossibility that one can obtain different intrinsic stateswhose projected states are identical. Such phenomenonhas already been shown in our previous VAP calculations[11]. ∗ Electronic address: [email protected]
In this Letter, I will present more interesting exam-ples to show such phenomenon, in which the obtainedintrinsic states with the same projected states can evenbe orthogonal to one another. The present calculationsare based on our newly improved VAP in which the formof trial wavefunction has been largely simplified. Thisimprovement is associated with the spin projection (orcalled as angular momentum projection) and is crucialin the calculations of high-spin states with arbitrary in-trinsic Slater determinants or HFB vacua, as will be in-troduced below.Let me start with the simplest variation of a Slaterdeterminant(SD), | Φ (cid:105) , i.e., the Hartree-Fock (HF) mean-field approximation. | Φ (cid:105) is assumed to be arbitrarilydeformed. One can vary | Φ (cid:105) so that the expectationalenergy, (cid:104) Φ | ˆ H | Φ (cid:105) , reaches a minimum. However, the min-imized HF energy and the corresponding | Φ (cid:105) are usuallyfar away from the exact SM ones [6].The HF mean-field approximation breaks the symme-tries of rotation and reflection. One may project | Φ (cid:105) toget a set of projected states, P JMK P π | Φ (cid:105) , where, P JMK and P π are the spin and parity projection operators,respectively. The state P JMK P π | Φ (cid:105) has good quantumnumbers of spin ( J ), the magnetic quantum number inthe laboratory frame ( M ) and parity ( π ). Notice that, K is not a good quantum number, as will be discussed later.Since the parity projection does not have any relation tothe present problem, | Φ (cid:105) is assumed to be an eigenstateof parity, and only the spin projection will be discussed.Generally, with given J and M , one can project | Φ (cid:105) onto 2 J +1 different projected states, P JMK | Φ (cid:105) , with K = − J , − J + 1, · · · , J . The nuclear wavefunction can beexpressed in terms of these projected states | Ψ JMα (cid:105) = J (cid:88) K = − J f JαK P JMK | Φ (cid:105) , (1) a r X i v : . [ nu c l - t h ] F e b where α is used to label the states with the same J and M . The coefficients f JαK and the corresponding en-ergy E Jα of Eq.(1) can be determined by solving the Hill-Wheeler(HW) equation of order 2 J + 1, J (cid:88) K (cid:48) = − J ( H JKK (cid:48) − E Jα N JKK (cid:48) ) f JαK (cid:48) = 0 , (2)where H JKK (cid:48) = (cid:104) Φ | ˆ HP JKK (cid:48) | Φ (cid:105) and N JKK (cid:48) = (cid:104) Φ | P JKK (cid:48) | Φ (cid:105) .For convenience, I assume E J ≤ E J ≤ · · · ≤ E J J +1 . Thecoefficients f JαK (cid:48) should satisfy the normalization condi-tion. J (cid:88) K,K (cid:48) = − J f JαK N JKK (cid:48) f JαK (cid:48) = 1 . (3)In Eq.(2), f JαK is independent of M . Since M doesnot carry any information of physics, it will no longerbe discussed and simply be regarded as a fixed quantumnumber.To solve Eq.(2), the first step is the diagonalization of N J and one has J (cid:88) K (cid:48) = − J N JKK (cid:48) R kK (cid:48) = n k R kK , (4)where n k ≥ R k with k = 1 , , ... J + 1 areeigenvalues and the corresponding eigenvectors, respec-tively. Here, I assume n ≥ n ≥ · · · ≥ n J +1 . Thenone can establish a new set of orthonormal basis states, | ψ Jk (cid:105) ( k = 1 , , · · · , J + 1), | ψ Jk (cid:105) = 1 √ n k J (cid:88) K = − J R kK P JMK | Φ (cid:105) , (5)and the HW equation (2) can be transformed into a nor-mal eigenvalue equation J +1 (cid:88) k (cid:48) =1 (cid:104) (cid:104) ψ Jk | ˆ H | ψ Jk (cid:48) (cid:105) − E Jα δ kk (cid:48) (cid:105) u Jαk (cid:48) = 0 . (6)Actually, the energies in Eq.(2) are obtained by solvingEq.(6) and the coefficients of the wavefunctions, f JαK , areobtained from u Jαk , i.e., f JαK = J +1 (cid:88) k =1 R kK u Jαk √ n k . (7)The problem is, the 2 J + 1 projected states are not or-thogonal to one another. Hence in practical calculations,it is possible that some n k values can be very tiny or evenzero. Consequently, the corresponding | ψ Jk (cid:105) basis statesmay not be precise enough to guarantee the stability ofthe calculated energies and wavefunctions. To avoid thistrouble, the | ψ Jk (cid:105) states with tiny n k values should be abandoned. This can be done by setting a cutoff param-eter (cid:15) > | ψ Jk (cid:105) states with n k > (cid:15) aretaken to form the nuclear wavefunction.Such basis cutoff works if | Φ (cid:105) remains unchanged.However, if one tries to vary | Φ (cid:105) in attempt to get thebest nuclear wavefunction | Ψ JMα (cid:105) , in other words, toperform the VAP calculation, new trouble arises due tothe changes of n k values during VAP iteration. When afixed cutoff parameter (cid:15) is used, the number of selected | ψ Jk (cid:105) states likely changes as VAP iteration goes on. Thiswill destroy the smoothness of the energy as a functionof the variational parameters. On the other hand, if onefixes the number of selected | ψ Jk (cid:105) states, then some ofthem might have too small n k values at certain VAP it-eration. Certainly, both treatments of the | ψ Jk (cid:105) selectiondo not guarantee the stability of the VAP iteration.One can imagine that the safest VAP calculation mightbe the one that only | ψ J (cid:105) with the largest n is taken.But | ψ J (cid:105) still includes several projected states P JMK | Φ (cid:105) .Sometimes, some of them might have very tiny norms, (cid:104) Φ | P JKK | Φ (cid:105) , or two projected states might be almost thesame. For instance, P JMK | Φ (cid:105) = ( − I − K P JM, − K | Φ (cid:105) when | Φ (cid:105) is time-even [17]. These problems may still affect theprecision of the calculated nuclear wavefunctions. Fi-nally, it looks the only way of ensuring the VAP stabilityis that only one projected state P JMK | Φ (cid:105) ( K is arbitrary)is taken. Then, the nuclear wavefunction can be simpli-fied as | Ψ JM ( K ) (cid:105) = P JMK | Φ (cid:105) (cid:113) (cid:104) Φ | P JKK | Φ (cid:105) . (8) K is put into the bracket since it is not a good quantumnumber. Notice that | Φ (cid:105) is non-axial.Now, there have two forms of nuclear wavefunctions,Eq.(1) and Eq.(8), to be optimized. One may expectEq.(1) is better than Eq.(8) if they have the same | Φ (cid:105) .Actually, almost all VAP authors prefer to take the for-mer Eq.(1) [7, 8, 12, 13]. However, if one do the VAP cal-culations with these two different forms of nuclear wave-functions, independently, the results are quite interset-ing.As a first example of such calculations, the USDB in-teraction [16] defined in the sd shell is used. The lowest J π = 2 + and 7 + energies in Mg are calculated. Theenergy of Eq.(8) is written as E J ( K ) = (cid:104) Ψ JM ( K ) | ˆ H | Ψ JM ( K ) (cid:105) . (9)For each K , E J ( K ) will be minimized independently andone can get 2 J + 1 converged E J ( K ) energies at spin J .The VAP method of Ref.[12] is applied to minimize E J in Eq.(2) and E J ( K ) in Eq(9), respectively. The resultsare shown in Fig.1. It is seen that the E J energies arevery close to the exact ones calculated by the full SMwhich has been discussed in Ref. [12]. Surprisingly, onecan also see that all the E J ( K ) energies are almost thesame as E J energies for both J π = 2 + and 7 + without - 8 - 6 - 4 - 2 0 2 4 6 8- 8 5- 8 0- 7 5- 7 0 J p = 7 + J p = 2 + Energies (MeV) K S h e l l M o d e l V A P e n e r g y E J V A P e n e r g y E J ( K ) FIG. 1: (Color online) Calculated VAP energies, E J (redlines) and E J ( K )(filled dots), for the yrast 2 + and 7 + statesin Mg, with wavefunctions Eq.(1) and Eq.(8), respectively.The corresponding exact shell model energies are shown asblack lines. The USDB interaction is adopted. exception. Correspondingly, the 2 J + 1 converged wave-functions | Ψ JM ( K ) (cid:105) for E J ( K ) are obtained. If K isa good quantum number, these | Ψ JM ( K ) (cid:105) states (withcommon fixed J and M ) should be orthogonal to eachother. This is clearly not possible because it is knownthe norm matrix, N J , in Eq.(2) is usually not diago-nal. On the contrary, it is expected they are the samestate since all E J ( K ) energies converge to the same level,which wavefunction should be unique, thus one shouldhave |(cid:104) Ψ JM ( K ) | Ψ JM ( K (cid:48) ) (cid:105)| ≈ K and K (cid:48) . All possible overlaps, |(cid:104) Ψ JM ( K ) | Ψ JM ( K (cid:48) ) (cid:105)| , for both J π = 2 + and 7 + stateshave been calculated. Most of them are above 0.99 andthe worst overlap is still over 0.98. This clearly tellsus that one may arbitrarily select a K number and useEq.(8) to do the VAP calculation without losing goodapproximation.Before performing further calculations, it is necessaryto understand the reason why E J ( K ) energies are al-most the same as E J . At a given spin J , one can assumethat the nucleus is rotating. Once one has the optimizedSD | Φ (cid:105) for Eq.(1), E J is then obtained from the Eq.(2).This also determines the orientation of rotational axisrelative to | Φ (cid:105) for E J . Since E J is the lowest, the mo-ment of inertia along with this rotational axis should bethe largest. On the other hand, one can first fix the ori-entation of the rotational axis relative to | Φ (cid:105) and thendo the energy minimization, which is actually done bytaking Eq.(8). It is expected that the moment of inertiaalong this fixed rotational axis can also reach the samemaximum as that for E J , and E J ( K ) converges to E J ,simultaneously. However, since spin is quantized, theremay have slight difference between E J and E J ( K ). Thus it is shown that the VAP calculations with wave-functions in Eq.(1) and Eq.(8) are almost equivalent,which seems to be a universal phenomenon. If this istrue, Eq.(8) can be used to simplify the VAP calculationand make the VAP iteration more stable. This is crucialin extending the present VAP calculations with arbitrarySD to high-spin states.Usually, | Φ (cid:105) is considered to be the intrinsic state andthe nuclear wavefunction should be taken as the Eq.(1).However, if one takes Eq.(8) as the nuclear wavefunction,then it looks more reasonable that the K -projected state, P K | Φ (cid:105) , is regarded as the intrinsic state since the restpart, (1 − P K ) | Φ (cid:105) , has never been used to construct thenuclear wavefunction. Here P K = π (cid:82) π e i ( K − ˆ J z ) φ dφ isthe K -projection operator. Those P K | Φ (cid:105) states with dif-ferent K are strictly orthogonal to one another. However,their projected states can be almost identical accordingto Eq.(10).So far, there is only one SD in Eq.(8). To improvethe VAP approximation, Eq.(8) should be generalized byincluding more SDs, and the new form of VAP wavefunc-tion can be written as, | Ψ JMα ( K ) (cid:105) = n (cid:88) i =1 f Jαi P JMK | Φ i (cid:105) , (11)where, n is the number of included | Φ i (cid:105) SDs. f Jαi isdetermined by the following Hill-Wheeler equation, n (cid:88) j =1 ( H Jii (cid:48) − E Jα N Jii (cid:48) ) f Jαi (cid:48) = 0 , (12)where H Jii (cid:48) = (cid:104) Φ i | ˆ HP JKK | Φ i (cid:48) (cid:105) and N Jii (cid:48) = (cid:104) Φ i | P JKK | Φ i (cid:48) (cid:105) .One may expect Eq.(11) can also be used to describethe non-yrast states. To check this validity, a newly de-veloped VAP algorithm [13] is applied, in which the low-lying state wavefunctions with the same spin and paritycan be varied on the same footing. This can be safelyrealized by minimizing the sum of the corresponding low-lying energies, (cid:80) mi =1 E Jα . Here m is the number of calcu-lated states.It should be reminded there are two more potentialproblems with the projected basis, which may seriouslyaffect the stability of the VAP calculation. The one is thenorm, N Jii , could become very tiny as the VAP iterationproceeds. The other is the possibility of large overlapsamong the projected basis states. Both problems maydamage the precision of the calculated energies and thecorresponding wavefunctions, and may cause the collapseof the VAP iteration. Fortunately, such troubles do notappear in the calculations of Fig.1. However, to ensurethe stability, in the following calculations, two constraintterms are attached to the energy sum, and the final min-imized quantity is, Q = m (cid:88) α =1 E Jα + χ n (cid:88) i =1 N Jii + χ n (cid:88) i,j =1 i (cid:54) = j N Jij N Jji N Jii N Jjj , (13) +5 +4 +3 Energies(MeV) +2 N e
N a
M g
N a
M g
A l
N a
M g
A l
S i
A l
S i P S h e l l m o d e l V A P : K = 0 V A P : K = 8 +1 FIG. 2: (Color online) Calculated lowest five J π = 8 + ener-iges in some sd shell nuclei with the Shell model and theVAP, respectively. The five panels from low to high showthe energies from the yrast 8 +1 ones to the forth excited 8 +5 ones, respectively. VAP: K = 0 and VAP: K = 8 refer tothe results of the VAP calculations with | Ψ JMα ( K = 0) (cid:105) and | Ψ JMα ( K = 8) (cid:105) , respectively. For each nucleus, all calculatedenergies are shifted by the same quantity so that the yrast SMenery is zero (see the lowest panel). The USDB interaction isadopted. where the second term tends to push the norms, N Jii , tolarge values, and the third term tends to guide the pro-jected basis states to be orthogonal to one another. Thevalues of the last two terms should be as small as possibleprovided that the VAP iteration is stable. So the param-eters χ ≥ χ ≥ χ = 10 − MeV and χ = 1MeV. This makes the included projected states al-most orthogonal to one another after the VAP calculationconverges. It turns out that both quantities of the lasttwo terms in Eq.(13) are within 100keV in the presentcalculations.In the second example, the USDB interaction is stilladopted. The wavefunction of Eq.(11) with n = 10 and J = 8 was taken into the VAP calculation. The lowestfive J π = 8 + energies in some sd shell nuclei are cal- culated. Notice that the calculated nuclei include botheven-even and odd-odd ones. In principle, the K num-ber in | Ψ JMα ( K ) (cid:105) can be arbitrarily chosen within therange | K | ≤ J . Here, K = 0 and K = 8 are chosen andthe corresponding VAP energies, denoted by VAP: K = 0(red dots) and VAP: K = 8 (blue circles), respectively,are shown in Fig.2. It is seen that all the calculatedenergies of VAP: K = 8 perfectly coincide with those ofVAP: K = 0. This implies that the results of VAP: K = 0and VAP: K = 8 are the same. One can imagine VAP re-sults with other K should also be the same as VAP: K = 0(or VAP: K = 8). Comparing with the full shell model,the energies of both VAP: K = 0 and VAP: K = 8 arevery close to the same exact ones. Thus it is expectedthat both | Ψ JMα ( K = 0) (cid:105) and | Ψ JMα ( K = 8) (cid:105) are closeto the same shell model wavefunction for each calculatedstate. This means | Ψ JMα ( K = 0) (cid:105) and | Ψ JMα ( K = 8) (cid:105) are almost the same. Indeed, for all calculated states inFig.2, it is found that |(cid:104) Ψ JMα ( K = 0) | Ψ JMα ( K = 8) (cid:105)| ≥ . . (14)Actually, most of such overlaps are above 0 . K with | K | ≤ J , if the num-ber of selected | Φ i (cid:105) SDs, n , is large enough, then thespace spanned by the ˆ P JMK | Φ i (cid:105) states may fully coverthe whole J -scheme shell model configuration space. To-gether with the present calculations, it seems that thereis no need to consider the complicate K -mixing in allVAP calculations.As a more practical application, the high-spin states in Cr are calculated, which has been studied by the shellmodel[18], the projected shell model[19], and the pro-jected configuration interaction(PCI) method[20]. Herethe KB3 interaction [21] is taken, as has been used inRef.s [18, 20]. This time, the wavefunctions | Ψ JMα ( K =0) (cid:105) and | Ψ JMα ( K = J ) (cid:105) are used to minimize the yrastenergies( α = 1) in Cr, respectively. Both forms ofwavefunctions include 5 projected SDs ( n = 5). The cal-culated results have been shown in Fig.3. From Fig.3(a),the VAP energies with both K = 0 and K = J are veryclose to the SM ones. The B(E2) values calculated withthe wavefunctions corresponding to Fig.3(a) are shownin Fig.3(b). All the B(E2) values are bunched up tightly.The B(E2) values with wavefunctions | Ψ JM ( K = 0) (cid:105) and | Ψ JM ( K = J ) (cid:105) are almost coincide with each other.This again implies | Ψ JM ( K = 0) (cid:105) and | Ψ JM ( K = J ) (cid:105) are almost the same. The overlap, |(cid:104) Ψ JM ( K =0) | Ψ JM ( K = J ) (cid:105)| , as a function of spin J has beenshown in Fig.3(c). Indeed, the overlaps for all spins areabove 98%. This confirms our previous conclusion inRef.[20] that one can not identify the intrinsic structureof the yrast states in Cr.Therefore, it looks the uncertainty of the nuclear in-trinsic state is a universal phenomenon in all VAP cal-culations. If one prefers to use HFB vacua rather thantaking the SDs in VAP, it is likely such uncertainty stillappears. This conclusion from VAP method is quite dif-ferent from that in the traditional nuclear collective mod- - 3 5- 3 0- 2 5- 2 0- 1 5
S h e l l M o d e l (cid:1)(cid:4)(cid:2)(cid:3)(cid:5)
K = 0 (cid:1)(cid:4)(cid:2)(cid:3)(cid:5)
K = J ( a )
01 0 02 0 03 0 0 ( b )
Energies(MeV)
B(E2) (e2fm4) ( c )
Overlap
S p i n J ( (cid:1) ) C r
FIG. 3: (Color online) Calculated results of the yrast statesin Cr with the VAP and the shell model. (a) The calcu-lated energies. (b) The B(E2) values from the wavefunc-tions corresponding to (a). (c) The values of the overlap, |(cid:104) Ψ JM ( K = 0) | Ψ JM ( K = J ) (cid:105)| . The KB3 interaction isadopted. els, in which the intrinsic state corresponding to a nuclearstate is expected to be unique.This work is supported by the National Natu-ral Science Foundation of China under Grant Nos.11975314,11575290, and by the Continuous Basic Scien-tific Research Project Nos. WDJC-2019-13, BJ20002501. [1] E. G. Lubeck, M. C. Birse, E. M. Henley, and L. Wilets,Phys. Rev. D , 234(1986).[2] M. Bylicki, Phys. Rev. A , 2079(1992).[3] Q. B. Yang and W. D. Wei, Phys. Rev. Lett. ,1020(1987).[4] Y.Kanada-En’yo, Phys. Rev. Lett. , 5291(1998).[5] T. Otsuka, M. Honma, T. Mizusaki, N. Shimizu, and Y.Utsuno, Prog. Part. Nucl. Phys. , 319 (2001).[6] Z.-C. Gao, and M. Horoi, Phys.Rev.C , 014311(2009).[7] K. W. Schmid, Prog. Part. Nucl. Phys. , 565 (2004).[8] N. Shimizu, Y. Tsunoda, Y. Utsuno, and T. Otsuka,Phys. Rev. C , 014312(2021).[9] K. Hara, Y. Sun, Int. J. Mod. Phys. E , 637 (1995).[10] J. M. Yao, J. Meng, P. Ring, and D. Vretenar, Phys. Rev.C , 044311(2010).[11] Z.-C. Gao, M. Horoi, Y. S. Chen, Phys. Rev. C ,064310(2015).[12] T. Ya, Y. He, Z.-C. Gao, J.-Q. Wang, and Y. S. Chen,Phys. Rev. C , 064307(2017). [13] J.-Q. Wang, Z.-C. Gao, Y.-J. Ma, and Y. S. Chen, Phys.Rev. C , 021301(R)(2018).[14] E. Caurier, G. Mart´ınez-Pinedo, F. Nowacki, A. Poves,A. P. Zuker, Rev. Mod. Phys. , 427(2005).[15] P. Ring and P. Schuck, The Nuclear Many-Body Problem(Springer Verlag, New York/Heidelberg/Berlin, 1980).[16] B. A. Brown and W. A. Richter, Phys. Rev. C , 034315(2006).[17] Z.-C. Gao, Y. S. Chen and J. Meng, Chin. Phys. Lett. , 650(2002)[18] E. Caurier, J. L. Egido, G. Mart´ınez-Pinedo, A. Poves, J.Retamosa, L. M. Robledo, and A. P. Zuker, Phys. Rev.Lett. , 2466 (1995).[19] K. Hara, Y. Sun, and T. Mizusaki, Phys. Rev. Lett. ,1922(1999).[20] Z.-C. Gao, M. Horoi, Y. S. Chen, Y. J. Chen and Tuya,Phys. Rev. C , 057303 (2011).[21] A. Poves and A. P. Zuker, Phys. Rep.71