12 March 1998

Ž .Physics Letters B 422 1998 1–5

A particle-number-conserving solution to the generalizedpairing problem

Feng Pan 1, J.P. Draayer, W.E. OrmandDepartment of Physics and Astronomy, 202 Nicholson Hall, Louisiana State UniÕersity, Baton Rouge, LA 70803-4001, USA

Received 9 October 1997Editor: W. Haxton

Abstract

An exact, number-conserving solution to the generalized, orbit-dependent pairing problem is derived by introducing aninfinite-dimensional algebra. A method for obtaining eigenvalues and eigenvectors of the corresponding Hamiltonian is alsogiven. The relevance of the orbit-dependent pairing solution is demonstrated by comparing predictions of the model withshell-model calculations. q 1998 Elsevier Science B.V.

PACS: 21.60.-n; 21.60.Cs; 02.20.Tw; 03.65.FdKeywords: Generalized pairing; Infinite dimensional algebra; Exact solution

Pairing has long been considered an importantinteraction in physics. The concept can be tracedback to the seniority scheme introduced by Racah in

w xatomic physics 1 . Its physical significance was firstw xrealized in the study of superconductivity 2 . Fol-

lowing the suggestions of Bohr, Mottelson, and Pinesw x3 , the first detailed application of pairing in nuclei

w xwas made by Belyaev 4 . The concept has sincebeen applied to other phenomena: high T supercon-c

w xductivity 5,6 , applications using the Hubbard modelw x w x7 , and pairing phenomena in liquids 8 and metal

w xclusters 9 .BCS methods have yielded major successes in

studies of superconductivity. When applied to nuclei,however, one must remember that the number of

1 On leave from the Department of Physics, Liaoning NormalUniversity, Dalian 116029, P.R. China.

spectroscopically active nucleons is typically tooŽ .small n;10 to support underlying assumptions of

the theory, namely d nrn is not negligible, and as aconsequence particle-number nonconservation ef-fects can enter and give rise to spurious states,nonorthogonal solutions, etc. Although this challengecan be partially remedied by making use of particle-

w xnumber projection techniques 10 , the elegance andsimplicity of the BCS method are then compromised.Secondly, an essential feature of pairing correlationsare even-odd differences, which are driven mainly byPauli blocking. It is difficult to treat these differencesin the BCS formalism because different quasi-par-ticle bases must be introduced for different blockedlevels. After an investigation into the accuracy of theBCS approximation, Kerman and Lawson suggestedthat an exact diagonalization of the pairing Hamilto-

w xnian is necessary 10 . Based on these observations, aparticle-number-conserving method for treating the

0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0370-2693 98 00034-3

( )F. Pan et al.rPhysics Letters B 422 1998 1–52

pairing problem in well-deformed nuclei was putw xforward 11 . The method uses a configuration-en-

ergy truncation scheme and takes the strength of thepairing interaction to be the same for all orbitals.Unfortunately, because of the deformation, each or-bital can only accommodate a single pair of particlesand this limits the applicability of the theory. Com-plementary mean-field techniques with approximateLipkin-Nogami number projection and non-trivial

w xpairing interactions 12,13 have also been found tobe a useful means for handling pairing correlations

w xin such systems 14–16 . However, exact shell-modelsolutions, when feasible, even if only approximate,are the best way to probe the true nature of many-body correlation effects.

The purpose of this contribution is to give anexact, particle-number-conserving solution to thegeneralized, orbit-dependent pairing problem. Themethod can be applied to any number of pairs in anymodel space, even to cases where exact diagonaliza-tion is computationally prohibitive.

The generalized pairing Hamiltonian for sphericalnuclei can be written as

ˆ † X q y X< <Hs e a a y G c S j S j , 1Ž . Ž . Ž .Ý Ýj jm jm j jXjm jj

"Ž .where the e are single-particle energies and S jj0Ž .and S j are the pairing operators for a single-j

shell defined by

jymq † †S j s y a a ,Ž . Ž .Ý jm jymm)0

jymyS j s y a a ,Ž . Ž .Ý jym jmm)0

10 † †S j s a a qa a y1Ž . Ž .Ý jm jm jym jym2 m)0

1ˆs N yV , 2Ž .ž /j j2

where V ' jq1r2 is the maximum number ofjˆpairs in the j-th shell, N is the j-th shell particlej

number operator, and the c X measure the orbit-orbitj j

pairing strength. Note that Hamiltonians with degen-Žerate single-particle energies e se for all j-j

. Ž X ) X .orbitals and separable pairing strengths c sc cj j j j

are special cases of the general theory.Ž .In general, for N pairs, Eq. 1 can be diagonal-

ized with bases states that are products of the single-jshell pairing wave functions:

< : qk 1 qk 2 qk 3 qk p < :N s B S S S PPP S 0 , 3Ž .Ý k k PPP k j j j j1 2 p 1 2 3 pki

where the summation is restricted so that

p

k sN , 4Ž .Ý iis1

and the B are expansion coefficients to bek k PPP k1 2 p

< :determined. Here, 0 is the pairing vacuum state thatsatisfying the condition

y< :S 0 s0 for all j. 5Ž .j

The dimensionality of the Hamiltonian matrix in thisbasis increases rapidly with increasing N and thenumber of shells p. Due to the Pauli Principle, it isless than or equal to the dimension of the irreducible

˙Ž . w x Ž .representation irrep N0 of the unitary group U p ,

pqNy1 !Ž .dim F . 6Ž .

N ! py1 !Ž .Ž .The equal sign in Eq. 6 holds when all the single-j

shell pairing wave functions in the summation of Eq.Ž . Ž .3 are Pauli allowed. From Eq. 6 , it is clear thatthe problem quickly becomes intractable becausethere are no analytical expressions or recursion rela-tions for determining the B coefficients.k k PPP k1 2 p

w xAs in the quasi-spin case 17 , we consider aslightly simpler Hamiltonian

ˆ q q y< <Hse a a y G S S , 7Ž .Ý jm jm 0 0jm

that is, one with degenerate single-particle energiesand a separable pairing interaction,

Sq s c)Sq j , Sy s c Sy j 8Ž . Ž . Ž .Ý Ý0 j 0 jj j

with the coefficients c satisfying the conditioni

< < 2c s1. 9Ž .Ý ii

This defines generalized pairing as proposed by Talmiw x18 . The separability assumption, though strong, isphysically motivated as it links the pair-pair interac-tion strength to the individual pair formation proba-

( )F. Pan et al.rPhysics Letters B 422 1998 1–5 3

Ž . X ) Xbility. In the notation of Eq. 1 , c sc c withj j j j< < 2c giving the percentage of single-j shell pairing inj

the Hamiltonian. In what follows, the c are taken toj

be real. Because the total number of particles is agood quantum number in a number-conserving the-

Ž .ory, the single-particle term in Eq. 7 is a constantŽ .and can be dropped without loss. Hence, Eq. 7

reduces to

ˆ q y< <Hsy G S S . 10Ž .0 0

To diagonalize this Hamiltonian, consider an algebragenerated by

S0 s c2 mS0 j ,Ž .Ým jj

S" s c2 mq1S" j . 11Ž . Ž .Ým jj

It is easy to show that these generators satisfy thefollowing commutation relations:

q y 0S ,S s2S ,m n mqnq1

0 " "S ,S s"S . 12Ž .m n mqn

� mTherefore, the S , m s 0,q,y; m s 0," 1,"m42, PPP form an infinite-dimensional algebra, one

that differs only slightly from a general Lie-algebraof the affine type without central extension.

The unique lowest-weight state of this algebra issimply the product of the single-j shell pairing vacuawith arbitrary seniority quantum numbers. Therefore,it suffices to consider the total seniority zero case.The lowest-weight state satisfies

y< :S 0 s0; ms0, "1, "2, PPP , 13Ž .m

and

12 m0 < : < < < : < :S 0 sy c V 0 sL 0 . 14Ž .Ým j j m2 j

Furthermore, it can be proven that the eigenvectorsˆof H for any N and non-zero energy eigenvalue can

be written as

< : q q q q < :N sNNS S S PPP S 0 , 15Ž .0 x x x1 2 Ny1

where NN is a normalization constant and

cjq qS s S j . 16Ž . Ž .Ýx 2i 1yc xj ij

� 4To obtain the variables x ; is1,2, PPP , Ny1 ,iŽ .Eq. 15 can be expanded in terms of x aroundi

x s0,i

< : n1 n2 nNy 1 q q q q < :N sNN x x PPP x S S S PPP S 0 ,Ý 1 2 Ny1 0 n n n1 2 Ny1ni

17Ž .

where the Sq are the Fourier-Laurent coefficients inni

the expansion of Sq, namelyx i

1q n qiS s dx x S . 18Ž .En i i xi i2p i 0

Ž .Using Eq. 17 and the commutation relations of Eq.Ž .12 , it is easily shown that the x , with is1,2, PPP ,i

Ny1, satisfy the relations

p1 1 1 12y V c a s q ,Ý Ýj j 22 y y yy1ya y c i i ki jjs1 k/i

is1, 2, PPP , Ny1, 19Ž .

with

Ny1 1s1, 20Ž .Ý

yiis1

where

2< <y sx ra , asy , h'Er y G . 21Ž . Ž .i i hq2 L1

We note that although these relations were derivedfor x f0, they are nonetheless valid in the entirei

Ž .complex plane except at the singularities in Eqs. 19Ž . Žand 20 . Therefore, the coefficients x i si

.1, 2, PPP , Ny1 and eigenvalues of the pairing en-ergy E/0 are simultaneously determined by the

Ž . Ž .system of equations Eqs. 19 and 20 .Similarly, the eigenvectors for Es0 can be ex-

pressed as

< : q q q < :N , 0 sNNS S PPP S 0 . 22Ž .x x x1 2 N

Using the same technique as in the E/0 case, it can

( )F. Pan et al.rPhysics Letters B 422 1998 1–54

be shown that in this case the x with is1,2, PPP , N,i

are determined by the following set of equations

c2j

V s0 for Ns1, 23Ž .Ý j 21yx c1 jj

1 c2 1jV s ,Ý Ýj 22 x yxx c y1 i ki jj k/i

is1, 2, PPP , N , for NG2. 24Ž .At this point, we reiterate that our solution is for a

separable, orbit-dependent pairing interaction as de-Ž .fined by Eq. 7 , and not for the generalized pairing

Ž .Hamiltonian of Eq. 1 . If the separability assump-tion is valid, the solutions should be good approxi-mations to the more general theory. The power of themethod lies in the fact that it gives the exact, parti-cle-number-conserving solution of the pairing prob-lem for any number of pairs in any model space,even for cases where full matrix diagonalizationscannot be carried out. Next we show, by example,that the separability assumption is consistent with‘‘realistic’’ shell-model results and hence that theory

Fig. 1. Comparison between the spectra obtained from the shellŽ . Ž .model SM and the separable, generalized pairing SGP method

as a function of the number of pairs, N. The dashed lines indicatethe correspondance between the levels in the two models.

Table 1Percent overlap between the eigenstates obtained with the separa-ble, generalized pairing method and the shell model as a functionof the number of pairs, N.

eigenstate _ N 1 2 3

1 96.2 99.4 97.82 94.7 97.6 97.03 99.4 97.2 94.34 98.8 95.85 99.2 96.86 97.0

can be used with confidence to study many-particlecorrelation phenomena in pair-rich systems.

As a test of the method, comparisons with shell-model calculations for like particles interactingthrough a ‘‘realistic’’ generalized pairing interactionwere carried out. The shell-model calculations wereperformed within the configuration space defined by

Žthe nuclear ds-shell 0d , 0d , and 1s or-5r2 3r2 1r2.bitals . The generalized pairing Hamiltonian of Eq.

Ž .1 was defined using the Js0 two-body matrixelements of the universal ds-shell Hamiltonian of

w xWildenthal 19 , where the mass scaling factorŽ .0.318rA was not included in the analysis. Since thenumber of Js0 states for any two-particle system is

Ž .small three for the ds-shell , it is reasonable toŽ .define the generalized pair in Eq. 8 as the wave

function of the shell-model ground state. For ourds-shell example this means the coefficients are:c s 0.570243, c s 0.626089, and c sd d s5r2 3r2 1r2

0.531823. The strength of the interaction was thenadjusted to reproduce the two-particle, ground-state

< <energy, that is, G s3.12778. With the Hamiltonianso defined, solutions were obtained for Ns1, 2, and3 pairs of like particles. If a diagonalization of thetwo-particle system is not feasible, an alternativeprocedure that work almost as well is to determinethe coefficients by the relative weight of the diagonalmatrix elements. With this choice, the coefficientswould be c s0.766482, c s0.512492, andd d5r2 3r2

< <c s0.714841 with G s2.28278. Shown in Fig.s1r2

1is a comparison between the eigenenergies obtainedŽ .from the shell model SM and the separable, gener-

alized pairing method. For reference, the quasi-spinŽ .QSA approximation gives -6.389, -10.684, and-12.788, respectively, for the Ns1, 2, and 3 pair

( )F. Pan et al.rPhysics Letters B 422 1998 1–5 5

cases. In this case, the parameters c are all thejŽ .same. The fully paired seniority zero solution is

< < Ž .well-known: eigenenergy Esy G N VyNq11r2 q NŽ .V y N ! Ž . < :with eigenstate S 0 , where N is

N !V !q qŽ .the number of pairs, VsÝ V and S sÝ S j .j j j

A further measure of how well the generalized pair-ing solutions correspond to the shell-model results isillustrated in Table 1 where the percent overlapbetween the shell-model and generalized pairingwave functions are given. In all cases, the wavefunctions obtained with the method presented here

Žhave better than a 94% overlap 96% for the ground.state with the shell-model states, which indicates

that the assumption of separability is sound.An exact, particle-number-conserving solution to

the generalized, orbit-dependent pairing problem hasbeen derived. A key feature of the derivation is theuse of an infinite-dimensional algebra, which is newin nuclear physics. Because the theory is number-conserving, it goes well-beyond what has been doneeven for the case of equal pairing strengths whichincludes to quasi-spin as a special limit. The rele-vance of the theory was shown by comparing eigen-values and eigenvectors with ‘‘realistic’’ shell-modelresults.

This work was supported by the National ScienceFoundation through Grant No. 9603006 and Cooper-ative Agreement No. EPS-9550481 which includesmatching from the Louisiana Board of Regents Sup-port Fund. Additional support for WEO from DOE

contract DE–FG02-96ER40985 is also acknowl-edged.

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