Bridging the quartet and pair pictures of isovector proton-neutron pairing
V.V. Baran, D. R. Nichita, D. Negrea, D. S. Delion, N. Sandulescu, P. Schuck
aa r X i v : . [ nu c l - t h ] S e p Bridging the quartet and pair pictures of isovector proton-neutron pairing
V.V. Baran , , ∗ D. R. Nichita , , D. Negrea , D. S. Delion , N. Sandulescu , and P. Schuck Faculty of Physics, University of Bucharest, 405 Atomi¸stilor,POB MG-11, Bucharest-M˘agurele, RO-077125, Romania ”Horia Hulubei” National Institute of Physics and Nuclear Engineering,30 Reactorului, RO-077125, Bucharest-M˘agurele, Romania Universit´e Paris-Saclay, CNRS, IJCLab, IN2P3-CNRS, 91405 Orsay, FranceUniversit´e Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France
The formal implications of a quartet coherent state ansatz for proton-neutron pairing are analyzed.Its nonlinear annihilation operators, which generalize the BCS linear quasiparticle operators, arecomputed in the quartetting case. Their structure is found to generate nontrivial relationshipsbetween the many body correlation functions. The intrinsic structure of the quartet coherent state isdetailed, as it hints to the precise correspondence between the quartetting picture and the symmetryrestored pair condensate picture for the proton-neutron pairing correlations.
Introduction.
After more than sixty years sincepairing effects were first considered in nuclear physics[1], the microscopic pairing models are still facing thechallenge of consistently describing the subtle inter-play between the isovector (T=1) and the isoscalar(T=0) proton-neutron pairing in nuclear systems [2].One of the first studies of the isovector pairing Hamil-tonian was performed by Beliaev et al in the frame-work of the generalised BCS approach in which theprotons and the neutrons are mixed through the Bo-goliubov transformation [3]. Since then, the BCS ap-proach was employed in the majority of studies andit was further extended to include also the isoscalarproton-neutron pairing interaction [4]. However, ashas been noticed already by Beliaev et al, the BCStreatment is not complete because one must take intoconsideration the quadruple correlation of α particle-like nucleons in addition to pair correlations. The firstinvestigation of these correlations has been done bySoloviev [5], who related them to a 4-body interac-tion term. Later on, the 4-body “quartet” correlationshave been discussed in relation to the standard two-body isovector pairing interaction by Br´emond andValatin [6] and by Flowers and Vujicic [7]. They pro-posed a BCS-like function in which the pairs are re-placed by quartets, but the calculations with this trialstate turned out to be too complicated and it wasnever applied to realistic cases. The first proof thatthe quartets are essential degrees of freedom for theisovector pairing Hamiltonian was given by Dobes andPittel [8] for the particular case of degenerate shells.They have shown that in this case the exact solution ofthe isovector pairing Hamiltonian for even-even N=Zsystems can be expressed as a quartet condensate,with the quartet defined as two isovector pairs cou-pled to total isospin T = 0. Later on quartet conden-sation models (QCM) have been proposed for non-degenerate levels and applied for realistic isovectorpairing Hamiltonians [9, 10]. Recently it was shown ∗ Corresponding author: vvbaran@fizica.unibuc.ro that the exact solution for the non-degenerate levelscan be also expressed in terms of quartets [11] andthat this solution turns to a quartet condensate in thestrong coupling coupling [12]. All these studies havedemonstrated that the α -like quartets are indispens-able for a proper description of isovector pairing.At this point it is worth stressing that the quartetcondensation in the pairing context mentioned aboveshould not be confused with the other ‘quartet con-densate’ concept, based on a similar wave functionas the QCM one but dealing with in medium boundstates of four fermions as, e.g., alpha particles, andtheir Bose-Einstein condensation [13, 14] in finite nu-clei and infinite nuclear matter. While no alpha par-ticle condensate survives at saturation density, onemay develop a theory for quartet condensation whichin many aspects is similar to the BCS approach forthe condensation of pairs [15].Very recently we introduced such a BCS-like ap-proach involving a quartet coherent state in the con-text of proton-neutron pairing [16]. The aim of thiswork is to further explore the implications of a quar-tet coherent state ansatz for the proton-neutron pair-ing problem. This leads us to establish the generalrelation between the quartet models and the BCS-based models, which was previously investigated onlyfor particular cases [8–10, 17, 18].We consider the general isovector pairing Hamilto-nian H = N lev X i =1 ǫ i N i, + X τ =0 , ± N lev X i,j =1 V ij P † i,τ P j,τ , (1)where i, j denote the single particle four-fold degener-ate states and ǫ i refers to the single particle energies;a time conjugated state will be denoted by ¯ i . The firstpart is the standard single-particle term while the sec-ond part is the isovector pairing interaction expressedby the neutron-neutron ( τ = 1), proton-proton ( τ = −
1) and proton-neutron ( τ = 0) pairs operators de-fined by P † i,τ = [ c † i c † ¯ i ] T =1 τ . In the discussion below, wewill frequently refer to the set of collective ππ , νν and πν Cooper pairs Γ † τ ( x ) ≡ P N lev i =1 x i P † i,τ , which dependon a set of mixing amplitudes x i , i = 1 , , ..., N lev . Wedenote by q † i = ν † i, ↑ ν † i, ↓ π † i, ↑ π † i, ↓ the isoscalar quartetoperator that fills completely the level i .The BCS-like quartet coherent state ansatz intro-duced in Ref. [16] is written in terms of the QCMcollective quartet operator Q † ( x ) ≡ √ (cid:2) Γ † Γ † (cid:3) T =0 ≡ † ( x )Γ †− ( x ) − [Γ † ( x )] as | QBCS i = exp[ Q † ] | i = X n n ! [ Q † ] n | i . (2)Below, we shall explore in more detail the particularconsequences of its coherent state character.The paper concentrates on formal aspects of pair-ing and quartetting. We will first show that QBCS ofEq. (2) can be annihilated by a non-linear transforma-tion of fermion operators (mixing singles and triplesand/or doubles with quadruples). This is in analogyto the well-known, simpler case where a quasi-particleoperator annihilates the BCS state. We will discusshow the former can open very interesting possibili-ties of calculus with the quartet coherent states. Aninteresting aspect will be that QBCS can be writtenas a Hubbard-Stratonovich transformation of a singleparticle field. This will help to show that the num-ber projected QBCS is analytically equivalent to thenumber projected BCS for T = 0 states. For T >
QBCS annihilation operators.
One of the ma-jor advantages of the BCS approach is the possibilityof describing the paired system in a picture of weaklyinteracting “quasiparticles”, whose associated opera-tors obey an annihilation condition with respect to thecorrelated BCS vacuum. Despite its nonlinear charac-ter, the above quartet-BCS state still admits a gener-alized class of annihilation operators, due to its coher-ent state nature [15]. However, at variance with thelinear quasiparticles of the BCS case, the annihilationoperators in the quartetting case do not obey simplelinear equations of motion. For a specific particle op-erator c and for a specific pair operator P ∼ c c , thegeneral annihilation operators may be computed as α = c + [ Q † , c ] ,β = P + [ Q † , P ] (2) + 12 [ Q † , P ] (3 , , (3)where we used the decomposition [ Q † , P ] ≡ [ Q † , P ] (2) + [ Q † , P ] (3 , which is of the form c † c † + c † c † c † c . Explicitly, a proton-like annihilation opera-tor of the QBCS state has the form α i, ↑ = π i, ↑ − x i π † i, ↓ Γ † ( x ) + √ x i ν † i, ↓ Γ † ( x ) , (4)involving the annihilation of a particle and the cre-ation of a particle dressed by a collective pair. Anal-ogous relations hold for the other spin-isospin combi-nations. The specific form of these nonlinear annihilationoperators has interesting consequences; one in par-ticular is the existence of a nontrivial connection be-tween the two-body normal densities and the four-body anomalous densities. To see this, evaluate theaverage h QBCS | π † i, ↑ α i, ↑ + π † i, ↓ α i, ↓ | QBCS i = 0 by us-ing Eq. (4). This leads to the occupations expressedin terms of the quartetting tensor as n i = h π † i, ↑ π i, ↑ + π † i, ↓ π i, ↓ i = 2 x i X j x j h [ P † i P † j ] T =0 i , (5)where the averaging is performed on the QBCS state.It follows that the total number of quartets may beexpressed as the average of the collective quartet op-erator as n q = h QBCS | Q † ( x ) | QBCS i . (6)This is a generalization of the simple BCS case withthe ground state | BCS i ∼ exp[Γ † ( x )] | i and the an-nihilation operators α i, ↑ = c i, ↑ − x i c † i, ↓ . Here the oc-cupations may be computed from 0 = h c † i, ↑ α i, ↑ i = n i / − x i h c † i, ↑ c † i, ↓ i . It follows that the number of pairsis given by the average of the collective pair operator n p = X i n i / h BCS | Γ † ( x ) | BCS i . (7)For the BCS case, we may also introduce the occu-pation and unoccupation amplitudes and recover thefamiliar form n p = P i x i h c † i, ↑ c † i, ↓ i = P i ( v i /u i ) u i v i = P i v i .Returning to the second class of pair-like QBCSannihilation operators, the expressions resulting fromEq. (3), for each isovector pair P k,τ , are β k, ± = P k, ± − x k P † k, ∓ − x k Γ †∓ + x k Γ †∓ N k, ± − x k Γ † T k, ∓ ,β k, = P k, + x k P † k, + 2 x k Γ † + x k Γ † T k, − − x k Γ †− T k, − x k Γ † N k, , (8)involving pair creation and annihilation terms, to-gether with a nonlinear pair dressed by the particlenumber and isospin operators, T i, = − ( π † i, ↑ ν i, ↑ + π † i, ↓ ν i, ↓ ) / √ T i, − = − T † i, . Remarkably, thereis another nonlinear combination that commutes ex-actly with the quartet operator. Explicitly, with η k ≡ Γ † T k, − − Γ †− T k, + 12 Γ † N k, − x k P † k, X j N j, , (9)we have [ Q † , η ] = 0 and thus η k | QBCS i = 0. Be-cause the isospin operators, T ± = P k T k, ± , obey[ Q † , T ± ] = 0, they also annihilate the isospin con-serving QBCS state, T ± | QBCS i = 0.The annihilation of the QBCS state by the opera-tors η k and T ± leads to the fact that the operators inEq. (8) are not actually uniquely defined. We couldadd to any of the β ’s an arbitrary combination of η and P T and still obtain a valid pair-like annihilation oper-ator. This freedom could allow for new treatments tobe consistently developed for the pairing Hamiltonian,in analogy with Refs. [15, 19], as will be explored infuture works.
Structure of the QBCS state.
Computationswith the nonlinear QBCS ansatz are made tractablein Ref. [16] by a linearization procedure for the ex-ponent. The quartet operator is first expressed as thesquare of a rotated collective pair γ , Q = ~γ † · ~γ † , de-fined by γ † τ = P N lev j =1 x j p † j,τ , where p † j, = i ( P † j, − P † j, − ) / √ , p † j, = ( P † j, + P † j, − ) / √ ,p † j, = − iP † j, . (10)Note that this choice is not unique. A Hubbard-Stratonovich transformation is then used to representthe quartet coherent state as a combination of generalisovector pair BCS states,exp( Q † ) = exp( ~γ † · ~γ † ) = Z d z exp (cid:0) − ~z / ~z · ~γ † (cid:1) = Z d z e − ~z / N lev Y i =1 (1 + x i ~z · ~p † i + x i ~z q † i / , (11)where we omitted the overall normalization factor. Inthis way, we obtain a superposition of standard BCSstates, each factorized as a product over the singleparticle levels.To better understand this specific pattern of partialsymmetry breaking, it is instructive to pass to spher-ical coordinates in Eq. (11) and write the quartetcoherent state asexp( Q † ) = Z ∞ d z z e − z / Z S dˆ n exp( z ˆ n · ~γ † ) . (12)Naturally, the isospin projection is already imple-mented by the angular integration. To see this, con-sider the coherent state of the isovector pair ~γ inte-grated over all directions in isospace, j † ≡ Z S dˆ n exp(ˆ n · ~γ † )= ∞ X k =0 ( ~γ † · ~γ † ) k (2 k + 1)! = ∞ X k =0 ( Q † ) k (2 k + 1)! = j ( i p Q † ) , (13)which is formally the expansion of a spherical Besselfunction of imaginary argument (hence the name).The basic information about the quartet correlationsis thus already contained in this simpler ansatz; by projecting onto good particle number, we always re-cover the QCM state, P n q exp( Q † ) | i = P n q j † | i = ( Q † ) n q | i . (14)We interpret now the role of the radial integral inEq. (12) as just changing the mixing between thecomponents having different particle numbers.The analytic expressions of the norm function andof the Hamiltonian average on the j † state may beobtained simply by dropping the radial integrals fromthe QBCS expressions (see Ref. [16], SupplementalMaterial). Remarkably, identical expressions were re-ported in Refs. [20, 21], in the context of the sym-metry restored BCS approach. The definition itself ofthe j † state hints at a precise relationship with theprojected BCS state, which we detail below. BCS Symmetry restoration for T = 0 . Thegeneralised BCS equations for isovector pairing ineven-even N = Z systems present two degenerate so-lutions with gap parameters ∆ ν = ∆ π = ∆ , ∆ πν = 0,and ∆ ν = ∆ π = 0 , ∆ πν = ∆ (for a proof, see [17]).The corresponding BCS states are given by | BCS I i = exp[Γ † ( x )] exp[Γ †− ( x )] | i , | BCS II i = exp[Γ † ( x )] | i . (15)Techniques for projecting these solutions onto goodparticle number and isospin have been developed in[20–26], with their connection to the quartet modelsonly being mentioned for particular cases in Refs. [8,17, 18].Here, we establish the correspondence in the generalcase by analytically performing the projection opera-tion on the BCS state, and recovering a version of the j † ansatz of Eq. (13). For simplicity, we consider theaxially symmetric state | BCS II i with T z = 0 and weemploy the isospin projection operator [27] P T ; T z =0 = Z S dˆ n D T ∗ (ˆ n ) R (ˆ n ) , (16)written in terms of a Wigner D -matrix and of the ro-tation operator in isospin space R (ˆ n ), which may befactorized as R (ˆ n ) = Q N lev i =1 R i (ˆ n ). Given the isoscalarcharacter of the fully occupied single particle level q † i | i , the only nontrivial term involves the rotationof the one-pair state. The isospin rotation operator R i (ˆ n ) = exp( − i ϕ ˆ T z ) exp( − i θ ˆ T y ) acting on a T z = 0pair state is effectively R i (ˆ n ) P † i, R i (ˆ n ) − = i ˆ n · ~p † i , (17)involving the same rotated pairs ~p † i of Eq. (10) usedto bring the collective quartet operator to a diagonalform. The isospin rotated BCS state becomes R (ˆ n ) | BCS II i = N lev Y k =1 (1 + i x k ˆ n · ~p † k − x k q † k / | i = exp( i ˆ n · ~γ † ) . (18)This implies that the isospin projected BCS may bewritten as P T ; T z =0 | BCS II i = Z S dˆ n D T ∗ (ˆ n ) exp( i ˆ n · ~γ † ) . (19)In particular, the T = 0 component is simply P T =0 | BCS II i = Z S dˆ n exp( i ˆ n · ~γ † )= ∞ X k =0 ( − ~γ † · ~γ † ) k (2 k + 1)! = ∞ X k =0 ( − Q † ) k (2 k + 1)! = j ( p Q † ) , (20)which is nothing else than Eq. (13) evaluated withimaginary mixing amplitudes or, equivalently, origi-nating from the ansatz exp( − Q † ).This proves the general equivalence of the projectedBCS and QCM approaches, for the isovector pairingcorrelations in the T = 0 ground state of N = Z even-even nuclei, i.e. P N =4 n q T =0 | BCS i = ( Q † ) n q | i = | QCM i . (21)Before detailing with the N > Z case below, weremark the possibility of establishing nontrivial con-nections between the correlation functions also forthe particle number projected
QCM state, basedon the above annihilation operators. We write Eq.(4) in schematic form α = c + c † c † c † , and projectthe annihilation condition α exp[ Q † ] | i = 0 onto afixed particle number, which singles out two terms.A proper particle-like annihilation operator for the QCM state may then be expressed in terms of theinverse amplitude coherent quartet, which satisfies Q (1 /x ) Q † ( x ) | QCM i = λ | QCM i , with λ a numeri-cal factor (for details see Appendix A of [28]). Weobtain e.g., for the proton-like annihilation operator, (cid:20) π i, ↑ + n q λ [ Q † , π i, ↑ ] Q (cid:18) x (cid:19)(cid:21) | QCM i = 0 (22)where the commutator can be read off Eq. (4). Inanalogy with Eq. (5) for the quartet coherent state,we may obtain a relation between the particle andthe quartet densities on the QCM state of the form h QCM | c † c | QCM i = h QCM | c † c † c † c † cccc | QCM i .This is perfectly analogous to the simple single-species BCS case, where the quasiparticle action onthe BCS state ( c i, ↑ − x i c † i, ↓ ) exp[Γ † ( x )] | i = 0 may beprojected to obtain the nonlinear annihilation relation (cid:20) c i, ↑ − x i N lev − n + 1 c † i, ↓ Γ (cid:18) x (cid:19)(cid:21) [Γ † ( x )] n | i = 0(23) We may then find the connection between the parti-cle and the pair densities on the projected BCS state | P BCS i = [Γ † ( x )] n | i as h c † i, ↑ c i, ↑ i = x i N lev − n + 1 N lev X j =1 x j h P † i P j i (24)Similar relationships may be established also forhigher order correlation functions, which could enablenew ways of solving the pairing problem, e.g. withinthe recent many body bootstrap approach [29, 30]. QCM vs projected BCS for
N > Z . In theQCM quartetting approach, the states for
N > Z systems are constructed by appending to the N = Z ansatz additional coherent pairs [9]. A state with n p excess neutron pairs and n q quartets, having T = T z = n p is defined as the particular combination | QCM ( T = T z = n p ) i = [Γ † ( y )] n p [ Q † ( x )] n q | i . (25)Here, one allows the extra collective pairs Γ † ( y ) tohave a different structure than the pairs Γ † ( x ) formingthe quartets. The same idea may be applied to theBCS ansatz: below, we consider the pair condensatesof Eq. (15) to have different mixing amplitudes. Notethat we also have to append a νν pair condensate tothe πν condensate in this N > Z case. In this section,we define | BCS II i = exp[Γ † ( y )] exp[Γ † ( x )] | i . Weconsider as illustrative examples an N = 4 , Z = 2system and an N = 6 , Z = 2 system. The particlenumber and isospin projected combinations are P N =6 T = T z =1 | BCS I i = (Γ † ,x Q † y − † ,y [Γ † x Γ † y ] T =0 ) | i , (26a) P N =6 T = T z =1 | BCS II i = (2Γ † ,y Q † x − Γ † ,x [Γ † x Γ † y ] T =0 ) | i , (26b) P N =8 T = T z =2 | BCS I i =(5 [Γ † ,y ] [Γ † y Γ † x ] T =0 − † ,y Γ † ,x Q † y ) | i , (26c) P N =8 T = T z =2 | BCS II i = (11 [Γ † ,y ] Q † x + 4 [Γ † ,x ] Q † y −
12 Γ † ,x Γ † ,y [Γ † x Γ † y ] T =0 ) | i , (26d)with the notation Γ † x = Γ † ( x ), Q † y = Q † ( y ) etc. Nat-urally, there are multiple options of coupling variouspairs to a given total isospin, and the QCM ansatz ofEq. (25) is just a particular choice. Interestingly, theQCM choice does not appear in all previous expres-sions.With the states (26), we performed variation-after-projection calculations for a picket-fence model ofeight doubly degenerate levels, of single particle en-ergies ǫ k = k −
1, and with a state independent inter-action of strength G . The analytical expressions for ( a ) =
6, T = T z = Γ x Q x Γ x [ Γ x Γ y ] P NT | BCS 〉 P NT | BCS 〉 G ( - E c / E c , Q C M )( % ) ( b ) =
6, T = T z = Γ x Q x Γ x [ Γ x Γ y ] P NT | BCS 〉 P NT | BCS 〉 - - G - | 〈 Q C M | ψ 〉 | FIG. 1: Error in the correlation energy E c = E ( G ) − E ( G = 0) (a) and overlaps (b) for the states (26) relative to the QCMstate (25), versus the interaction strength G (in units of the level spacing). The results are for two protons and four neu-trons on eight equidistant levels. To indicate the results we use the notations: “Γ x Q x ” for the state Γ † ( x ) Q † ( x ) | i of Eq.(25) with equal pair and quartet amplitudes x = y ; “Γ x [Γ x Γ y ] ” for the state Γ † ( x )[Γ † ( x )Γ † ( y )] T =0 | i ; “ P NT | BCS i , ”for the projected BCS states of Eqs. (26a,26b). the average of the isovector pairing Hamiltonian onthe states (26) were derived with the Cadabra2 com-puter algebra system [31] using the method presentedin Refs. [32, 33].In all cases, we obtained a very good agreement be-tween the projected BCS and the QCM results. Forthe chosen model, the overlaps do not decrease lowerthan 0 . N = 4 , Z = 2 system; the agree-ment between projected BCS and QCM improves forheavier systems (we note that the QCM ansatz givesa higher correlation energy in all cases).Note that even in the case of equal pair and quartetmixing amplitudes ( x = y ) the results are still good:the obtained overlaps with the QCM state (having x = y ) are always greater than 0 .
98, and the errors inthe correlation energies are always smaller than 8%.In this case, all analytical expressions for the projectedBCS states reduce to the QCM ansatz of Eq. (25) with x = y .In constructing the QCM ansatz for N > Z sys-tems, Ref. [9] mentions the necessity of a differentstructure for the excess collective neutron pairs withrespect to the collective pairs forming the quartet, asto reproduce the Hartree-Fock limit. However, thepresent results indicate that while the x = y choiceintroduces significant errors, it preserves the correctbehaviour in the weak pairing regime. Indeed, theHartree-Fock vacuum may be obtained as a limit ofthe x = y QCM ansatz by suitably scaling the mixing amplitudes. For the N = 4 , Z = 2 and N = 6 , Z = 2systems with the scalings w ε = (1 /ε, ε , , , . . . ) and z ε = (1 /ε, ε, ε, , , . . . ), we obtainΓ † ( w ǫ ) Q † ( w ε ) ∼ q † P † , + O ( ε ) , [Γ † ( z ε )] Q † ( z ε ) ∼ q † P † , P † , + O ( ε ) . (27)which reduce to the exact Hartree-Fock state in the ε → Summary and Conclusions.
We presented anattempt at bridging the descriptions of the proton-neutron isovector pairing correlations in the symmetrypreserving quartet picture and in the mean-field pair-condensate picture.For both the coherent and the projected state, thenonlinear annihilation operators are shown to gen-erate nontrivial connections between the many-bodycorrelation functions. A possible application of theserelations would be to consider the novel quantummany-body bootstrap approach [29, 30] and to imple-ment the condensate property of the ansatz in termsof these constraints for the correlation functions. Thiswould enable a numerically unified description, basedon a quartet coherent state, of both nuclear matterand finite nuclei. The same framework could be gen-eralized to quartetting in condensed matter systemse.g., to the study of bi-exciton condensation in semi-conductors or trapped fermionic atoms in optical lat-tices.Then, inspired by the structure of the quartet co-herent state, we have shown that the QCM ansatz forthe ground state of even-even N = Z systems can beobtained by projecting out the particle number andthe isospin from a proton-neutron BCS state. For the N > Z systems the P NT BCS and QCM states arenot analytically equivalent. However, their overlapsare very close to one. The numerical P NT BCS cal-culations indicate that the particular way of couplingvarious pairs to the total isospin of the
N > Z systemdoes not influence much the final results as long as thetrial states obeys the correct symmetry constraints.An interesting question is whether these facts hold in the case of an isovector-isoscalar pairing Hamiltonian.This issue we intend to address in a future study.
Acknowledgments
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