A quartet BCS-like theory

We introduce a BCS-like theory for the quartet correlations induced by the isovector pairing interaction. It is based on a coherent state of BCS type and, unlike usual mean field approaches, it displays a vanishing pair anomalous density $\langle c^\dagger c^\dagger\rangle =0$. We find good agreement between our theory and the exact results. We discuss how the pairing and quarteting correlations share some similar qualitative features within the BCS approach. However, there is no sharp quarteting phase transition. We also present various ways in which our theory may be further developed.

Introduction. The advances in the experimental techniques of the last three decades have opened up new possibilities to investigate the nature of the nuclear interactions at the N = Z line. The significant overlap between neutron and proton orbitals in this region allows the the existence of proton-neutron (pn) pairing, which is suppressed away from N = Z.
Already in the first theoretical approach to pn pairing dating six decades ago, Belyaev, Zacharev and Soloviev [1] realized that "one must take into consideration the quadruple correlation of α particle-like nucleons in addition to pair correlations" [1]. Subsequent works proposed various ways to incorporate these quartet correlations into a functioning theory, e.g. those of Flowers and Vujičić [2] and of Brémond and Valatin [3]. More recently, Civitarese, Reboiro and Vogel [4] concluded that "an isospin-symmetric Hamiltonian, treated with the generalized Bogolyubov transformation, fails to describe the ground state properties correctly". This transformation naturally diagonalizes the Hamiltonian in the mean field approximation, i.e. upon the replacement of some operators associated with large matrix elements by their expectation value. In our case, the monopole pair operators P † are replaced by the pairing anomalous density P † → P † . The existence of only pair condensates in the case of pn pairing should thus be carefully considered.
Nevertheless, the standard mean field approach to pairing in N = Z nuclei has been for many years the generalized Hartree-Fock-Bogoliubov approximation [5], where all types of Cooper pairs are treated in a unified manner. In the last decade, the particle number and isospin conserving Quartet Condensation Model (QCM) was proposed for the study of isovector pairing and quarteting correlations in N = Z nuclei [6,7]. It was further developed in Refs. [8,9,10,11,12,13] to the case of isoscalar pairing and N > Z nuclei. General microscopic quartet models for a shell-model basis with an effective Hamiltonian were also recently proposed [14,15,16,17,18,19]. In these approaches, the basic building blocks are not the Cooper pairs anymore, but four-body structures composed of two neutrons and two protons coupled to the isospin T = 0 and to the angular momentum J = 0, denoted "α-like quartets". Interesting connections between the symmetry restricted pair condensate and the quartet descriptions have very recently been uncovered [20]. However, as to this day, "no symmetry-unrestricted mean-field calculations of pn pairing, based on realistic effective interaction and the isospin conserving formalism have been carried out" [20,21].
In this work, we take a step along this direction. We construct a solvable BCS-like theory of quartet correlations without assuming any mean field approximation, but only a coherent state ansatz of BCS type involving the isoscalar quartet operator. At variance with the quartet BCS theory of Schuck et al [22], solving our simpler model requires no approximation.
Formalism. The system of interest consists of a number N = Z of neutrons and protons moving outside a self-conjugate inert core, and interacting through a charge-independent pairing force. The isovector pairing Hamiltonian is suitable to describe both spherical and deformed nuclei, where i, j denote the single particle doubly-degenerate states and ǫ i refers to the single particle energies; a time conjugated state will be denoted byī. The N i,0 operator counts the total number of particles, N i,0 = N i,1 + N i,−1 = τ=ν,π c † i,τ c i,τ + c † i,τ c¯i ,τ , and the isovector triplet of pair operators is given by An almost exact solution for the Hamiltonian (1) may be obtained within the QCM. The formulation of this model is based on correlated four-particle structures known as "quartets". To obtain a quartet, one first defines a set of collective ππ, νν and πν Cooper pairs Γ † τ (x) ≡ N lev i=1 x i P † i,τ , which depend on a set of mixing amplitudes x i , i = 1, 2, ..., N lev . A collective quartet operator is then constructed by coupling two collective pairs to the total isospin T = 0, For N = Z nuclei, the ground state of the Hamiltonian (1) is taken to be a "condensate" of such α-like quartets, where n q is the number of quartets. The concept of a "condensate" denotes here the state obtained by acting with the same operator a number of times on the vacuum. It should not be confused with an ideal boson-type condensate. The model is solved by determining numerically the mixing amplitudes x i upon the minimization of the Hamiltonian expectation value subject to the unit norm constraint, i.e.
The quartet state has, by construction, a well defined particle number and isospin. As such, it may be well suited to precisely describe various nuclear properties that are sensitive to the fluctuations of these quantum numbers. However, the presence of a condensate is not guaranteed in a particle number conserving theory (which always has a nontrivial solution). With this motivation, we look for a mean field BCS-like formulation of quartet correlations which may provide more insight into the quartet condensation phenomenon.
In analogy with the standard BCS case, we consider a coherent state involving the quartet operator Despite of the formal similarity with the standard BCS theory, this ansatz is considerably more complicated. To see this, consider a simple BCS state for like-particles, e.g. neutrons The BCS state can be factorized as the exponent contains a single sum over the single particle states. In the quartet case however, the exponent is expanded as . It is a highly entangled mixture of all single particle states which hinders a direct factorization. To develop a tractable BCS-like theory for quartets, we need a linearization procedure for the exponent.
We achieve this by first writing the coherent quartet as a quadratic form. Consider the rotated pair operators defined At this stage, the quartet-BCS state may be factorized as |QBCS = 3 a=1 exp[(γ † a ) 2 ]|0 . To proceed, for each of the three factors we use an identity relating it to a Gaussian integral having as the source term a coherent pair. We write the QBCS state as with z = (z 1 , z 2 , z 3 ) ∈ R 3 and γ = (γ 1 , γ 2 , γ 3 ). In this form, the exponent is a simple linear combination of single particle operators, z · γ † = N lev i=1 x i z · p † i , which may be factorized just as in the BCS case where we took into account that at most second order terms are generated in each factor; they involve the quartet operator that fills completely the level i, Let us note that while the QBCS state violates particle number conservation, it still has a good isospin T = 0. As such, the integral appearing in Eq. (6) may be interpreted as an isospin projection operation acting on a generalized Brémond-Valatin factorized state |ψ(z,ẑ) [3]). Here, at variance with the original Brémond-Valatin ansatz, the isovector pair may be oriented along any directionẑ in isospin space. Moreover, the Gaussian integration has the crucial role of performing configuration mixing on the Brémond-Valatin factorized ansatz (which by itself misses most of the relevant correlations). As such, the QBCS state is a superposition of all possible factorized states, each having the relative amplitude of finding a pair on the level i proportional to x i , and that of finding a quartet to x 2 i . Note that, while the factorized Brémond-Valatin may be treated by a quasi-particle transformation, the integration necessary to obtain the QBCS state completely destroys the quasi-particle picture.
As a first step, in the present work we limit ourselves to assessing the consequences of breaking only the particle number conservation; we leave the breaking of isospin conservation to future investigations.
We compute the ground state correlations within the QBCS model by minimizing the expectation value of E(x) = QBCS |H − λN 0 |QBCS / QBCS |QBCS subject to the particle number constraint QBCS |N 0 |QBCS / QBCS |QBCS = 4n q , where n q is the number of quartets, n q = (N + Z)/4.     Let us finally remark that the QBCS ground state of Eq. (3) contains only components where the particle number is a multiple of 4. Hence, there is no pair anomalous density in our theory, i.e. QBCS |P i,τ |QBCS = 0. There is, however, a fourbody anomalous density, which we analyze below.
Numerical results. We test our formalism against the well established QCM in the nuclei above 100 Sn and above 16 O. We consider the same model spaces and interactions as in Refs. [6,8], namely the spherical spectrum ǫ 2d 5/2 = 0.0MeV, ǫ 1g 7/2 = 0.2MeV, ǫ 2d 3/2 = 1.5MeV, ǫ 3s 1/2 = 2.8MeV together with the effective Bonn A potential of Ref. [23] for the sdg shell and the spectrum ǫ 1d 5/2 = −3.926MeV, ǫ 2s 1/2 = −3.208MeV, ǫ 1d 3/2 = 2.112MeV together with the USDB interaction of Ref. [24] for the sd shell. We solve the QCM model using the analytical method of Refs. [25,26]. We show in Fig. (1) the results for the correlations energies defined as E c = E 0 − E, where E is the energy of the ground state and E 0 is the energy calculated without taking into account the isovector pairing interaction. In both cases, we observe good agreement between the QBCS and the particle number conserving QCM, with relative errors of at most 6% for the 16 O core and 10% for the 100 Sn core (note that, at variance with the particle number projected BCS, the QCM offers practically the exact solution, within 1% error). A very good agreement is obtained also for the average level occupancies, shown in Fig.  (2) for the nuclei at shell half-filling. In Fig. (3), the relative particle number fluctuation ∆N/N = N 2 − N 2 /N is shown to exhibit the standard decrease when going to heavier systems.
As mentioned above, there is no anomalous pair density in the QBCS theory, P = 0. As an indicator of the four body correlations, we choose to study the quartet anomalous density  interaction strength. It is well known that the standard BCS has a nontrivial solution only for interaction strengths greater than a critical value dependent on the model space.
In the QBCS case we observe no evidence of a sharp transition from a quarteting to a normal phase as we decrease the interaction strength. As seen from Fig. 5, when scaling the interaction matrix elements by a factor κ ∈ [0, 1] the correlation energy varies smoothly, with no jumps of its derivative. Instead, there is a relatively narrow interval towards small values of the scaling factor κ where the derivative experiences a more pronounced variation. This behaviour of the QBCS (which does not conserve the particle number) is exhibited also by the number projected BCS theory. We conclude that the restoration of the isospin symmetry is enough to smoothen the transition from the quarteting to the normal phase.
Conclusions. We proposed a BCS-like theory for quartet correlations based on a quartet coherent state analogous to the famous |BCS state. Our ansatz is unique as it does not contain any pairing anomalous density P (as opposed to usual mean field treatments for pn pairing), but instead one of quartet type Q . We evidenced that the standard pairing and quarteting correlations share similar qualitative features, but there is no sharp quarteting transition. Additionally, we uncovered new connections between the quartet models and some of the early BCSlike attempts to pn pairing.
The QBCS theory is flexible and there are numerous ways in which it may be easily expanded. On the one hand, extra neutron pairs may be included for the study of N > Z nuclei by considering an ansatz of the form containing both neutron pairs and quartets as building blocks.
On the other hand, combined isovector and isoscalar interactions may be treated with the ansatz where Q iv is the quartet considered in this work (built from two isovector pairs) while Q is is the quartet operator built from two isoscalar pairs. The excited states of the QBCS model may be computed by minimizing the energy function subject to the additional constraint of zero overlap with the ground state. Although a quasiparticle picture would be preferred, recent results also indicate that it may be inappropriate for quartet correlations [27]. Nevertheless, an approximate quasiparticle description may be obtained within a boson formalism along the lines of Ref. [28]. Note that, in deriving the expressions for the averages of the various operators on the QBCS states starting from Eq. (6), one may as well treat the pair operators p i as bosons and still obtain the exact fermionic results.
Lastly, let us remark that the coherent quartet ansatz of Eq. (2) is still rather restrictive. More generally, one should consider the case of non-separable mixing coefficients, i.e. Q † = N lev i, j=1 X i j (2P † i,1 P † j,−1 − P † i,0 P † j,0 ) with X i j x i x j .This more complex case may also be treated by taking advantage of the properties of Gaussian integration. In this case we obtain |QBCS = exp(Q † )|0 = exp( i, j p i X i j p j )|0 where a scalar product is understood in all expressions involving two vectors. The norm and the averages of the various operators on the QBCS state may be automatically computed as Wick contractions of various X i j 's by using the Wick theorem for multidimensional Gaussian integrals [29].