Analytical approach for the Quartet Condensation Model

V.V. Baran; D.S. Delion

arXiv:1901.04759·nucl-th·March 27, 2019

Analytical approach for the Quartet Condensation Model

V.V. Baran, D.S. Delion

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TL;DR

This paper derives explicit analytical formulas for the Quartet Condensation Model, simplifying calculations of isovector pairing correlations in nuclei by replacing complex recurrence relations with polynomial expressions.

Contribution

It provides the first complete analytical expressions for key quantities in the QCM, enabling straightforward numerical implementation.

Findings

01

Analytical formulas for the norm of quartet states

02

Explicit expressions for the mean value of the Hamiltonian

03

Simplified computational approach for isovector pairing calculations

Abstract

Within the Quartet Condensation Model (QCM), the isovector pairing correlations for $N = Z$ nuclei are described with a very high accuracy by a condensate of $α$ -like quartets. The usual approach involves cumbersome recurrence relations in order to compute numerically the relevant quantities of the model: the norm of the quartet states and the mean value of the isovector pairing Hamiltonian as functions of the pair mixing amplitudes. We present the final analytical expressions for the above mentioned quantities, for all cases up to four quartets in the valence shell. The analytical QCM expressions were obtained by a straightforward implementation of the SO(5) algebra in the symbolic computer algebra system Cadabra2. The norm of the quartet states and the mean value of the Hamiltonian are polynomial functions of the mixing amplitudes. The numerical implementation of the QCM model is…

Tables1

Table 1. Table 1: Single particle spectrum and mixing amplitudes for the nucleus S 32 superscript 𝑆 32 {}^{32}S , in both particle and hole formalisms.

s.p. state	$ϵ$	$x_{p}^{(q = 4)}$	$x_{h}^{(q = 2)}$	$x_{p}^{(q = 4)} \cdot x_{h}^{(q = 2)}$
$1 d_{5 / 2}$	-3.926	0.291	0.0457	0.0133
$2 s_{1 / 2}$	-3.208	0.260	0.0511	0.0133
$1 d_{3 / 2}$	2.112	0.0317	0.420	0.0133

Equations79

H = i = 1 \sum N_{lev} ϵ_{i} N_{i, 0} + τ = 0, \pm 1 \sum i, j = 1 \sum N_{lev} V_{ij} P_{i, τ}^{†} P_{j, τ},

H = i = 1 \sum N_{lev} ϵ_{i} N_{i, 0} + τ = 0, \pm 1 \sum i, j = 1 \sum N_{lev} V_{ij} P_{i, τ}^{†} P_{j, τ},

Γ_{τ}^{†} \equiv i = 1 \sum N_{lev} x_{i} P_{τ, i}^{†},

Γ_{τ}^{†} \equiv i = 1 \sum N_{lev} x_{i} P_{τ, i}^{†},

Q^{\dagger}\equiv\left[\Gamma^{\dagger}\Gamma^{\dagger}\right]^{T=0}_{S=0}\equiv 2\Gamma^{\dagger}_{1}\Gamma^{\dagger}_{-1}-\big{(}\Gamma^{\dagger}_{0}\big{)}^{2}

Q^{\dagger}\equiv\left[\Gamma^{\dagger}\Gamma^{\dagger}\right]^{T=0}_{S=0}\equiv 2\Gamma^{\dagger}_{1}\Gamma^{\dagger}_{-1}-\big{(}\Gamma^{\dagger}_{0}\big{)}^{2}

|\Psi_{q}(x)\rangle=\big{(}Q^{\dagger}\big{)}^{q}|0\rangle~{},

|\Psi_{q}(x)\rangle=\big{(}Q^{\dagger}\big{)}^{q}|0\rangle~{},

δ ⟨ Ψ_{q} (x) ∣ H ∣ Ψ_{q} (x)⟩

δ ⟨ Ψ_{q} (x) ∣ H ∣ Ψ_{q} (x)⟩

⟨ Ψ_{q} (x) ∣ Ψ_{q} (x)⟩

⟨ 0∣ Γ_{1} ϵ_{i} N_{i, 1} Γ_{1}^{†} ∣0 ⟩

⟨ 0∣ Γ_{1} ϵ_{i} N_{i, 1} Γ_{1}^{†} ∣0 ⟩

= 2 ϵ_{i} x_{i} ⟨ 0∣ Γ_{1} P_{i, 1}^{†} ∣0 ⟩

= 2 ϵ_{i} x_{i} ⟨ 0∣ (P_{i, 1}^{†} Γ_{1} + x_{i} - x_{i} N_{i, 1}) ∣0 ⟩

= 2 ϵ_{i} x_{i}^{2} = 2 E_{2},

⟨ Ψ_{q} (x) ∣ Ψ_{q} (x)⟩

⟨ Ψ_{q} (x) ∣ Ψ_{q} (x)⟩

⟨ Ψ_{q} (x) ∣ H ∣ Ψ_{q} (x)⟩

Σ_{α}

Σ_{α}

V_{α β}

N_{1}

N_{1}

E_{1}

v_{1}

N_{2}

N_{2}

E_{2}

v_{2}

\displaystyle+28~{}\Sigma_{4}~{}\mathcal{V}_{1,3}+4~{}\mathcal{V}_{1,7}+4~{}\mathcal{V}_{3,5}+14~{}{\Sigma_{2}}^{2}~{}\mathcal{U}_{4}+4~{}\Sigma_{2}~{}\mathcal{U}_{6}-9~{}\Sigma_{4}~{}\mathcal{U}_{4}-~{}\mathcal{U}_{8}\big{)}~{}.

N_{3}

N_{3}

\displaystyle+26~{}{\Sigma_{2}}^{2}~{}\Sigma_{8}+22~{}{\Sigma_{2}}^{2}~{}{\Sigma_{4}}^{2}-120~{}\Sigma_{2}~{}\Sigma_{4}~{}\Sigma_{6}-24~{}{\Sigma_{2}}^{4}~{}\Sigma_{4}-8~{}{\Sigma_{2}}^{3}~{}\Sigma_{6}+44~{}{\Sigma_{4}}^{2}~{}{\Sigma_{2}}^{2}\big{)},

E_{3}

- 38 E_{8} Σ_{4} + 52 E_{8} Σ_{2}^{2} - 20 E_{10} Σ_{2} + 26 E_{2} Σ_{2} Σ_{8} + 22 E_{2} Σ_{2} Σ_{4}^{2} - 20 E_{2} Σ_{4} Σ_{6} - 8 E_{2} Σ_{6} Σ_{2}^{2}

\displaystyle-24~{}\mathcal{E}_{2}~{}\Sigma_{4}~{}{\Sigma_{2}}^{3}-40~{}\mathcal{E}_{4}~{}\Sigma_{2}~{}\Sigma_{6}+44~{}\mathcal{E}_{4}~{}\Sigma_{4}~{}{\Sigma_{2}}^{2}-60~{}\mathcal{E}_{6}~{}\Sigma_{2}~{}\Sigma_{4}\big{)}~{},

v_{3}

+ 176 Σ_{4} Σ_{2}^{2} V_{1, 3} + 208 Σ_{2}^{2} V_{1, 7} + 208 Σ_{2}^{2} V_{3, 5} + 44 Σ_{4}^{2} Σ_{2} V_{1, 1} + 52 Σ_{8} Σ_{2} V_{1, 1} - 160 Σ_{6} Σ_{2} V_{1, 3}

- 240 Σ_{4} Σ_{2} V_{1, 5} - 80 Σ_{2} V_{1, 9} - 120 Σ_{4} Σ_{2} V_{3, 3} - 80 Σ_{2} V_{3, 7} - 40 Σ_{2} V_{5, 5} - 40 Σ_{4} Σ_{6} V_{1, 1} - 8 Σ_{10} V_{1, 1}

+ 108 Σ_{4}^{2} V_{1, 3} - 76 Σ_{8} V_{1, 3} + 272 Σ_{6} V_{1, 5} - 152 Σ_{4} V_{1, 7} + 136 Σ_{6} V_{3, 3} - 152 Σ_{4} V_{3, 5} + 24 V_{1, 11} + 24 V_{3, 9}

+ 24 V_{5, 7} + 52 Σ_{2}^{4} U_{4} - 80 Σ_{2}^{3} U_{6} - 116 Σ_{4} Σ_{2}^{2} U_{4} - 116 Σ_{2}^{2} U_{8} + 24 Σ_{6} Σ_{2} U_{4} + 296 Σ_{4} Σ_{2} U_{6}

\displaystyle+48~{}\Sigma_{2}~{}\mathcal{U}_{10}-5~{}{\Sigma_{4}}^{2}~{}\mathcal{U}_{4}+45~{}\Sigma_{8}~{}\mathcal{U}_{4}-216~{}\Sigma_{6}~{}\mathcal{U}_{6}+102~{}\Sigma_{4}~{}\mathcal{U}_{8}-18~{}\mathcal{U}_{12}\big{)}~{}.

N_{4}

N_{4}

- 312 Σ_{4}^{3} Σ_{2}^{2} + 1504 Σ_{6}^{2} Σ_{2}^{2} + 360 Σ_{4} Σ_{8} Σ_{2}^{2} + 528 Σ_{12} Σ_{2}^{2} - 336 Σ_{4}^{2} Σ_{6} Σ_{2} - 2352 Σ_{6} Σ_{8} Σ_{2}

+ 2016 Σ_{4} Σ_{10} Σ_{2} - 288 Σ_{14} Σ_{2} + 321 Σ_{4}^{4} + 944 Σ_{4} Σ_{6}^{2} + 1395 Σ_{8}^{2} - 1206 Σ_{4}^{2} Σ_{8} - 1056 Σ_{6} Σ_{10}

- 312 Σ_{4} Σ_{12} + 90 Σ_{16}) .

E_{4}

E_{4}

+ 40 E_{2} Σ_{6} Σ_{2}^{4} + 276 E_{2} Σ_{4}^{2} Σ_{2}^{3} - 600 E_{10} Σ_{2}^{3} - 456 E_{6} Σ_{4} Σ_{2}^{3} - 304 E_{4} Σ_{6} Σ_{2}^{3} + 204 E_{2} Σ_{8} Σ_{2}^{3}

- 234 E_{4} Σ_{4}^{2} Σ_{2}^{2} + 396 E_{12} Σ_{2}^{2} + 180 E_{8} Σ_{4} Σ_{2}^{2} + 1128 E_{6} Σ_{6} Σ_{2}^{2} - 456 E_{2} Σ_{4} Σ_{6} Σ_{2}^{2} + 90 E_{4} Σ_{8} Σ_{2}^{2}

- 360 E_{2} Σ_{10} Σ_{2}^{2} - 78 E_{2} Σ_{4}^{3} Σ_{2} - 126 E_{6} Σ_{4}^{2} Σ_{2} + 376 E_{2} Σ_{6}^{2} Σ_{2} - 252 E_{14} Σ_{2} + 1260 E_{10} Σ_{4} Σ_{2}

- 1176 E_{8} Σ_{6} Σ_{2} - 168 E_{4} Σ_{4} Σ_{6} Σ_{2} - 882 E_{6} Σ_{8} Σ_{2} + 90 E_{2} Σ_{4} Σ_{8} Σ_{2} + 504 E_{4} Σ_{10} Σ_{2}

+ 132 E_{2} Σ_{12} Σ_{2} + 321 E_{4} Σ_{4}^{3} - 603 E_{8} Σ_{4}^{2} + 236 E_{4} Σ_{6}^{2} - 234 E_{12} Σ_{4} - 42 E_{2} Σ_{4}^{2} Σ_{6} - 660 E_{10} Σ_{6}

+ 708 E_{6} Σ_{4} Σ_{6} + 1395 E_{8} Σ_{8} - 603 E_{4} Σ_{4} Σ_{8} - 294 E_{2} Σ_{6} Σ_{8} - 396 E_{6} Σ_{10} + 252 E_{2} Σ_{4} Σ_{10}

- 78 E_{4} Σ_{12} - 36 E_{2} Σ_{14} + 90 E_{16}) .

v_{4}

v_{4}

- 876 Σ_{4} U_{4} Σ_{2}^{4} + 180 U_{8} Σ_{2}^{4} + 80 Σ_{6} V_{1, 1} Σ_{2}^{4} + 1104 Σ_{4} V_{1, 3} Σ_{2}^{4} + 816 V_{1, 7} Σ_{2}^{4} + 816 V_{3, 5} Σ_{2}^{4} + 464 Σ_{6} U_{4} Σ_{2}^{3}

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Full text

Analytical approach for the Quartet Condensation Model

V.V. Baran1,2 and D.S. Delion1,3,4

1”Horia Hulubei” National Institute of Physics and Nuclear Engineering,

30 Reactorului, RO-077125, Bucharest-Măgurele, România

2 Department of Physics, University of Bucharest, 405 Atomiştilor, POB MG-11, Bucharest-Măgurele, RO-077125, România

3Academy of Romanian Scientists, 54 Splaiul Independenţei, RO-050094, Bucharest, România

4Bioterra University, 81 Gârlei, RO-013724, Bucharest, România

Abstract

Background

Within the Quartet Condensation Model (QCM), the isovector pairing correlations for $N=Z$ nuclei are described with a very high accuracy by a ”condensate” of $\alpha$ -like quartets. The usual approach involves cumbersome recurrence relations in order to compute numerically the relevant quantities of the model: the norm of the quartet states and the mean value of the isovector pairing Hamiltonian as functions of the pair mixing amplitudes.

Purpose

We present the final analytical expressions for the above mentioned quantities, for all cases up to four quartets in the valence shell.

Method

The analytical QCM expressions were obtained by a straightforward implementation of the SO(5) algebra in the symbolic computer algebra system Cadabra2, as described below.

Results

The norm of the quartet states and the mean value of the Hamiltonian are polynomial functions of the mixing amplitudes. The numerical implementation of the QCM model is thus made trivial as a matter of copying and pasting the presented formulas.

Conclusions

We introduce in this work a method of computer aided analytical calculus for a many body setting. In particular, we provide precise and easy to use tools for the description of isovector pairing correlations.

Quartet Condensation, Isovector Pairing

pacs:

21.60.−n, 21.60.Gx

The $\alpha$ -particle is the nucleus with the largest binding energy in nature. For this reason this structure survives as an $\alpha$ -cluster, as can be seen from the binding energy analysis of nuclei. Therefore the $\alpha$ -cluster model of the nucleus was proposed in the early years of the nuclear structure theory Haf38 . The $\alpha$ -like structure is hindered by the Pauli principle and various approaches were proposed to account for it Flo63 ; Bri66 ; Ari71 ; Wil77 ; Gam83 ; Del02 . The $\alpha$ -like structures were experimentally evidenced in light nuclei Ike68 and therefore they were extensively analysed in the low-lying energy region Fre97 ; Fre07 ; Hor04 ; Fun09 ; Toh17 , as well as in dipole resonance area He14 ; Chi16 . In medium and heavy nuclei $\alpha$ -clustering can experimentally be correlated with the $\alpha$ -decay phenomenon Del10 . It was understood that an $\alpha$ -clustering component is necessary in addition to the single-particle basis in order to describe the absolute value of the $\alpha$ -decay width Var92 ; Del04 . This can be explained by the fact that $\alpha$ -particles can appear only at relative low nuclear densities Rop98 , a situation which is realised on the nuclear surface of $\alpha$ -decaying nuclei Del13 .

Recently the Quartet Condensation Model (QCM) was proposed for the study of isovector pairing correlations in $N=Z$ nuclei San12a ; danielphd and further developed in San12 ; Neg14 ; San14 ; San15 ; Neg17 ; Sam17 ; Neg18 to the case of isoscalar pairing and $N>Z$ nuclei. Here, the building blocks are not the Cooper pairs, but four-body clusters composed of two neutrons and two protons coupled to the isospin $T=0$ and to the angular momentum $J=0$ . The standard QCM procedure uses involved recurrence relations in order to compute the norm of the quartet states and the mean value of the isovector pairing Hamiltonian as functions of the pair mixing amplitudes. Our purpose it to give closed analytical expressions for the above mentioned quantities, for all cases up to four quartets in the valence shell.

We consider the isovector pairing Hamiltonian applicable to both spherical and deformed nuclei

[TABLE]

where $i,j$ denote the single particle doubly-degenerate states and $\epsilon_{i}$ refers to the single particle energies; a time conjugated state will be denoted by $\bar{i}$ . The $N_{i,0}$ operator counts the total number of particles, $N_{i,0}=\sum_{\tau=\pi,\nu}\left(c^{\dagger}_{i,\tau}c_{i,\tau}+c^{\dagger}_{\bar{i},\tau}c_{\bar{i},\tau}\right)$ , whereas the isovector triplet of pair operators is given by $P^{\dagger}_{i,\tau}=\left[c^{\dagger}_{i,\tau}c^{\dagger}_{\bar{i},\tau}\right]^{T=1}_{S=0}$ . Explicitely, $P^{\dagger}_{i,1}=c^{\dagger}_{i,\nu}c^{\dagger}_{\bar{i},\nu},P^{\dagger}_{i,-1}=c^{\dagger}_{i,\pi}c^{\dagger}_{\bar{i},\pi},P^{\dagger}_{i,0}=\dfrac{1}{\sqrt{2}}\left(c^{\dagger}_{i,\nu}c^{\dagger}_{\bar{i},\pi}+c^{\dagger}_{i,\pi}c^{\dagger}_{\bar{i},\nu}\right)$ . The $V_{ij}$ coefficients represent the matrix elements of the pairing interaction in the $\pi\pi,\nu\nu$ and $\pi\nu$ channels.

In the following we limit ourselves to a short description of the model’s features for self-consistency. Within the QCM, one first defines a set of collective $\pi\pi$ , $\nu\nu$ and $\pi\nu$ Cooper pairs

[TABLE]

where the mixing amplitudes $x_{i}$ are the same in all cases due to isospin invariance. A collective quartet operator is then constructed by coupling two collective pairs to the total isospin $T=0$

[TABLE]

Finally, the ground state of the Hamiltonian (1) is described as a ”condensate” (although an actual $\alpha$ condensate appears only at low densities) of such $\alpha$ -like quartets

[TABLE]

where $q$ is the number of quartets. By construction, this state has a well defined particle number and isospin. Its structure is defined by the mixing amplitudes $x_{i}$ , which are determined numerically by the minimization of the Hamiltonian expectation value, subject to the unit norm constraint, i.e.

[TABLE]

In order to compute these quantities, the method proposed in San12 ; danielphd makes use of the recurrence relations obeyed by the matrix elements of the pairing interaction in the auxiliary basis $|n_{1}n_{2}n_{3}\rangle=\Gamma_{1}^{\dagger n_{1}}\Gamma_{-1}^{\dagger n_{2}}\Gamma_{0}^{\dagger n_{3}}|0\rangle$ of states having a well defined number of $\pi\pi,\nu\nu$ and $\pi\nu$ pairs. The advantage of this method lies in its generality: the same numerical code is able to compute the relevant quantities in all cases of interest. On the downside, within this framework a large number of numerical evaluation steps are required in order to obtain the values of the norm and Hamiltonian average. Taking into account the fact that the minimization procedure itself requires multiple evaluations of the functions, the code running times may be considerable, especially in the case of coupled mean-field $+$ quarteting self-consistent approaches (a possible interesting generalization of the relativistic mean field $+$ projected-BCS of Ref.Las18 ). Furthermore, the recurrence relations themselves are rather involved and thus challenging to derive and to implement numerically.

We address this issues by choosing to evaluate analytically rather than numerically the expressions of the norm and Hamiltonian average. On the one hand, a direct numerical implementation of the final formulas considerably shortens the code running times. On the other hand, the problem of obtaining the numerical implementation itself is made trivial, being a matter of copying and pasting the formulas (with some minor syntax modification to make them compatible with the chosen programming language).

The basic idea of our approach is that a single run of the symbolic evaluation code for an expression renders unnecesary the alternative of an arbitrary number of possible numerical evaluations.

To this purpose we employ the Cadabra2 symbolic computer algebra system cdb1 ; cdb2 ; cdb3 , capable of analytically handling operations with non-commuting objects. We have implemented the SO(5) algebra (e.g. presented in danielphd ) as a set of substitution rules which are repeatedly used in order to evaluate the averages of the relevant operators on the quartet states. The substitution operations are performed until convergence is achieved for the considered expression. As an illustrative schematic example of the procedure, consider an average of the single particle energy term

[TABLE]

where (some of) the substitution rules employed are derived directly from the SO(5) algebra: $N_{i,1}\Gamma^{\dagger}_{1}\rightarrow 2x_{i}P^{\dagger}_{i,1}+\Gamma^{\dagger}_{1}N_{i,1}$ and $\Gamma_{1}P^{\dagger}_{i,1}\rightarrow P^{\dagger}_{i,1}\Gamma_{1}+x_{i}-x_{i}N_{i,1}$ . Also, we use the standard vacuum annihilation conditions $N_{i,1}|0\rangle\rightarrow 0$ and $\Gamma_{1}|0\rangle\rightarrow 0$ and the notation $\epsilon_{i}x_{i}^{2}\rightarrow\mathcal{E}_{2}$ . At each step, it is also necessary to invoke the routines performing the distribution of terms and the sorting of each expression. The largest running time of our brute force implementation of the SO(5) algebra are of the order of a few tens of CPU hours for the most complicated case analyzed, that of four quartets (see Eqs. (12)-(14) below). However, one needs to keep in mind that the code needs to be executed only once. Also, it is not difficult to concieve further optimizations in order to reduce the execution times and as such to easily approach the cases of five or more quartets.

We present below the results for the cases corresponding to a number of $q=1,2,3$ and 4 quartets in the valence space. The numerical results obtained using the expressions given below were confirmed to be identical to those obtained using the standard recurrence relations approach danielprivate .

The norms of the quartet states and the Hamiltonian averages as functions of the mixing amplitudes may be expressed as

[TABLE]

As expected, the above mentioned quantities are polynomial functions of the mixing amplitudes of degree $4q$ . It is convenient to express them in terms of the sums

[TABLE]

where $x_{i}^{\alpha}$ is the amplitude $x_{i}$ to the power $\alpha$ and ${N_{\text{lev}}}$ is the number of levels in the valence space. The generalization to the case of degenerate (spherical) levels is made trivial by the fact that the mixing amplitudes and interaction matrix elements are equal within each degenerate subspace.

For $q=1$ , we obtain

[TABLE]

For $q=2$ the results are

[TABLE]

The expressions for $q=3$ of the norm function, single particle energy and interaction terms read

[TABLE]

The final formulas below correspond to a number of $q=4$ quartets, for a total of 16 particles in the valence shell

[TABLE]

The above formulas may be employed to compute directly the ground state correlations in $N=Z$ nuclei with up to 16 particles in the valence shell. However, by exploiting the particle-hole symmetry, the same expressions may be applied to cases involving a larger number of particles: a system with a number of quartets $q$ may be mapped to an equivalent system with $N_{lev}-q$ hole-quartets. Let us note shortly that recently the particle-hole formalism has been used to elaborate an improved approximate treatment of pairing correlations Duk16 . Here, the starting point is the reformulation of the PBCS condensate in the particle-hole basis. A detailed study of the generalization of the particle-hole approach to quartet correlations was very recently developed phb .

In this paper we limit ourselves to confirm that the particle-hole symmetry is manifest in the framework of the QCM, using the above analytical expressions. We consider as testing ground the nucleus ${}^{32}S$ , which contains 4 quartets, or equivalently 2 hole-quartets, in the valence $sd$ shell. We use the same spherical single particle spectrum as in Ref. San12 , as displayed in Table I, and assume a constant isovector pairing strength $V_{ij}=-24/A$ MeV, with $A=32$ Che78 . The transition from particles to holes degrees of freedom may be perfomed in the standard way, resulting in the isovector pairing Hamiltonian in the hole representation

[TABLE]

where the number of holes operator is $\tilde{N}_{i,0}=4-N_{i,0}$ and the pair operators for holes are defined as $\tilde{P}^{\dagger}_{i,\tau}={P}_{i,\tau}$ . Similarly to the pairing case, it turns out that the original quartet ”condensate” may be related to a hole-quartet ”condensate” of inverse amplitudes (see the Appendix A of phb for details):

[TABLE]

We have confirmed this inverse proportionality by first computing the ground state of ${}^{32}S$ in a $q=4$ description using the expressions (12)-(14), and then a $q=2$ hole-quartet description with the formulas of Eq. (10), with the corresponding modifications of Eq. (15). In both cases we have obtained a correlation energy $E_{\text{corr}}=10.36$ MeV. The numerical results regarding the mixing amplitudes are presented In Table I with three significant digits, indicating explicitely the inverse proportionality of particle and hole amplitudes.

Let us finally note that, on the numerical side, as opposed to the standard recurrence relations method where the running times are of the order of a few minutes San12 , our timings are more than two orders of magnitude smaller (having used the same minimization routine of the NAG library).

In conclusion, we introduce in this paper a method of computer aided analytical calculus for many body problems where it is not only possible, but also advantageous, to perform some algorithmic computations symbolically instead of numerically. This approach presents a twofold benefit: on the numerical side, the computational time may be significantly reduced, and on the implementation side the effort is made negligible. Moreover, it may be applied to a wide class of many body models. In this work, we analyzed the particular example of the Quartet Condensation Model which precisely describes the isovector pairing correlations in $N=Z$ nuclei. The corresponding analytical formulas can easily be implemented in any programming language.

The extensions of the QCM model to $N>Z$ nuclei and also to the case of isoscalar pairing are currently under consideration from an analitical perspective and will be presented in future works.

Acknowledgements.

We thank the anonymous referee for the valuable suggestion of analyzing the particle-hole symmetry within our approach. This work was supported by the grants of the Romanian Ministry of Research and Innovation, CNCS - UEFISCDI, PN- III-P4-ID-PCE-2016-0092, PN-III-P4-ID-PCE-2016-0792, within PNCDI III, and PN-19060101/2019.

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