Symmetry restoration in the nuclear-DFT description of proton-neutron pairing

We show that the symmetry-restored paired mean-field states (quasiparticle vacuua) properly account for isoscalar vs. isovector nuclear pairing properties. Full particle-number, spin, and isospin symmetries are restored in a simple SO(8) proton-neutron pairing model, and prospects to implement similar approach in a realistic setting are delineated. Our results show that provided all symmetries are restored, the pictures based on pair-condensate and quartet-condensate wave functions represent two equivalent ways of looking at the physics of the nuclear proton-neutron pairing.

Do proton-neutron (pn) pairs form collective condensates in nuclei, similarly as the like-particle pairs do? Ever since the existence of like-particle nuclear pairing was suggested in 1958 by Bohr, Mottelson, and Pines [1], this simple question has been addressed in numerous studies [2]. As late as in 2004, the authors of Ref. [3] concluded that in spite of many attempts to extend the quasiparticle approach to incorporate the effect of pn correlations, no symmetry-unrestricted mean-field calculations of pn pairing, based on realistic effective interaction and the isospin conserving formalism have been carried out. This conclusion holds even today.
In this Letter, we show that sometimes contradicting conclusions about existence of the pn pair condensate may have resulted from using a mean-field formalism without full symmetry restoration. The physics of the pn pairing does require simultaneous breaking, and then restoration of three major symmetries: particlenumber, angular-momentum, and isospin. In the shellmodel framework, these symmetries are not broken and then do not have to be restored. A number of such studies already exist, see, e.g., Ref. [4]. However, the shellmodel rather interprets the pn pairing as an effect of a strong nucleon-nucleon isoscalar interaction, and is less concerned with analyzing the obtained wave functions in terms of collective condensates. In this sense, the question of existence of the putative pn condensate remains open.
Due to the attractive nature of the nuclear interaction, atomic nuclei are strongly correlated systems exhibiting superfluid properties. Theoretical description of nuclear superfluidity is directly related to the theory of electronic superconductivity, wherein Cooper pairs of electrons in time-reversed states condensate near the Fermi level. In the nuclear case, we may expect a possible formation of six types of pairs, corresponding to the four degrees of freedom of the nucleon: spin and isospin, up and down. More precisely, we may have scalar-isovector Cooper pairsP + ν , with three projections of the total isospin ν ≡ T z =0, ±1, and vector-isoscalar pairsD + µ , with three projections of the total spin µ ≡ S z =0, ±1. A condensation of spin-alignedD + µ pairs recently attracts an increased attention, see Refs. [5, 6] and references cited therein.
The most general pair condensate is represented by a quasiparticle vacuum, which can be written as the Thouless state [7,8], |Φ = N exp{Ẑ + }|0 , for the Thouless pairẐ + given bŷ (1) Here p ν and d ν are complex isovector and isoscalar amplitudes, respectively, |0 is the particle vacuum, and N is the normalization constant.
It is now obvious that in the Thouless state all symmetries: particle-number, spin, and isospin, are strongly mixed. Therefore, the standard paired-mean-field minimization of the average energy, which in nuclear physics is called Hartree-Fock-Bogolyubov (HFB) theory [8], may or may not give the best result. A great number of studies based on the HFB approach already exist, see, e.g., Refs. [9-13] and reviews in Refs. [2,3]. In this Letter, we argue that to analyze the problem of the pn pairing one has to employ a more sophisticated approach that is based on the minimization of energy after all symmetries are restored.
This corresponds to the so-called variation-afterprojection (VAP) [8] method, which employs the projected Thouless states, as variational trial states. The projection operators:P A on particle number A,P S MK on total spin S and its projection M , andP T N L on total isospin T and its projection N , involve, respectively, one-dimensional integration over the gauge angle, three-dimensional integration over the spin-rotation Euler angles, and three-dimensional integration over the isospin-rotation Euler angles [8,14]. In this Letter, we report on implementing, for the first time, a complete seven-dimensional integration, which allows us to fully restore all relevant symmetries, broken in an arbitrary symmetry-unrestricted Thouless state.
However, before embarking on a full-scale VAP calculations in a realistic nuclear DFT setting, one would like to know if such a complete, and demanding approach is capable of bringing better solutions when applied in a simple model. For that, in this Letter we perform a full VAP analysis of the well-known SO(8) model [2, [15][16][17][18][19], which assumes simple forms of the isovector and isoscalar pairs within a single-particle phase space of a few degenerate ℓ shells, that is, where a + are the creation operators of a particle with orbital angular momentum ℓ, spin 1 2 , and isospin 1 2 . The round brackets denote triple standard Clebsch-Gordan coupling to the total orbital angular momentum L, spin S, and isospin T , having, respectively, projections M , S z , and T z . The maximum number of particles allowed in this phase space equals to 4Ω for Ω = ℓ (2ℓ + 1). For deformed nuclei with spin-orbit coupling taken into account, the notion of spin should, in fact, be understood as that of the alispin [20], which pertains to a pair of deformed Kramers-degenerate single-particle states.
In the past, much of the discussion related to properties of the pn pairing concentrated on the question of whether real or complex quasiparticle amplitudes have to be used. Therefore, to solve this problem in the context of the most general Thouless pairs (1), we briefly touch upon their symmetries. To begin, let us consider a system described by a scalar and isoscalar Hamiltonian, such as the one in the SO(8) model that we define shortly.
First, since an unabridged spin and isospin projections involve rotating the Thouless pair over the full spin and isospin SO(3) groups, we can freely chose its initial orientations. This means that the isovector and vector paircreation operators, Eqs. (3) and (4), can be arbitrarily aligned along one of the directions in space and isospace, respectively. Without any loss of generality, we can chose orientations along the z axes, that is, we can keep in Eq. (1) only spherical amplitudes p 0 and d 0 . Then, the Thouless states become eigenstates of spin and isospin projections with S z = T z = 0. Such a choice has an enormous advantage, namely, it allows for reducing the integrations over the spin and isospin Euler angles to one dimension only, which reduces seven dimensions of integration to three only. Nevertheless, we were able to test and benchmark all our results by also performing unrestricted integrations.
Second, we note that the particle-number projected state equals to |Φ A = N ′ (Ẑ + ) A/2 |0 . Therefore, an overall multiplicative factor of the Thouless pair, and its phase, can be absorbed in the normalization constant N ′ , and are thus irrelevant. This allows us to parametrize the most general Thouless pair (1) in terms of two angles 0 ≤ α < π and 0 ≤ ϕ < π only, that is, Angle α (β) thus controls the relative amplitude (phase) between the isovector and isoscalar pairs. Third, we have to take into account the fact that every scalar and isoscalar Hamiltonian is also invariant with respect to the spin and isospin signatures,Ŝ ≡ exp(iπŜ y ) = iσ y andT ≡ exp(iπT y ) = iτ y , respectively, which rotate spins and isospins by angle π about the corresponding y axes. It is easy to show that the isovector and isoscalar pairs, Eqs. (3) and (4), are S-even-T -odd and S-odd-Teven, respectively, and thus the Thouless pairs transform asŜẐ Finally, we have to fix the phase convention. Here we adopt the one of Condon-Shortley in the LS basis, by which all single-particle states transform under time-reversalT as, This convention carries over to the isovector and isoscalar pairs, Eqs. (3) and (4), which turn out to be time-even and time-odd, respectively. As a consequence, the Thouless pairs transform under time reversal aŝ Altogether, we see that the invariance of the Hamiltonian with respect to the spin or isospin signature renders the average energies periodic in ϕ with period of π 2 , whereas that with respect to the time reversal renders them symmetric with respect to the line at ϕ = π 4 . At this line, the Thouless pairs are time-even (up to an irrelevant phase factor).
Since our entire analysis of symmetries is performed for the Thouless states, we avoid any possible ambiguities related to definitions and phase conventions of quasiparticle states, density matrices, and pairing tensors, which can now consistently be determined from the Thouless pairs using generic expressions [8].
By considering Hamiltonian [19], strength is controlled by parameter g, whereas the relative importance of the isovector vs. isoscalar pairing is governed by the mixing parameter x. For x = +1(−1), the Hamiltonian has a pure isoscalar (isovector) character, whereas within the interval −1 < x < 1, we should expect a competition between the two possible types of pairing. Using the group-theory methods, Hamiltonian (10) can be diagonalized exactly [16,19]. In Fig. 1, we show average values of Hamiltonian (10) calculated for the unprojected (upper panels) and projected (2) on A = 24 and T = S = 0 (lower panels) Thouless states. Red dots and red band indicate the minima of energies, that is, in the upper and lower panels they indicate solutions of the HFB and VAP equations, respectively. We see that in all cases the minima of energies appear at ϕ = π 4 , that is, for time-even Thouless states.
For the unprojected states, for x < 0 the minima stay at α = π (purely isovector pairs) and then for x > 0 they flip over to α = 0 (purely isoscalar pairs). At x = 0, the HFB energy is entirely independent of α, that is, states with any isovector-isoscalar pair mixing are exactly degenerate. Our HFB results confirm observations of Ref. [18] that the unprojected mean-field states do not exhibit isovector-isoscalar pairing mixing.
However, as we see in the lower panels of Fig. 1, our VAP states do exhibit such a mixing perfectly well. Indeed, even a small departure from the pure isovector or isoscalar interaction moves the VAP solutions away from the unmixed states characterized by α = 0 or α = π. As expected, at x = 0 the VAP solution appears at α = 1 2 π, that is, the pairs are then maximally mixed.
Let us now discuss the VAP solutions, that is, properties of states |AST that are projected on good particle number A, spin S, and isospin T with energies minimized over α at ϕ = π 4 . Figure 2 summarizes our VAP results obtained for different isospins (left panels) and particle numbers (right panels) [21]. As one can see in the top panels of the figure, the VAP energies (symbols) are indistinguishable from the exact values (lines). Only by plotting their relative differences in the logarithmic scale (upper middle panels), one can appreciate the fact that at x = 0 the VAP energies are precise up to 1.5%, and with growing |x| their precision rapidly improves by many orders of magnitude. In addition, the VAP results obtained for A = 4 and 6, Fig. 2(d), are for all values of x exact, that is, precise up to the numerical accuracy, see discussion below.
The lower middle panels show norms of the VAP Thouless isoscalar pairs |d 0 | 2 = cos 2 ( 1 2 α min ), cf. Eqs. (1) and (5). Again we see that for arbitrary strengths of the isoscalar vs. isovector interactions, the VAP isoscalar and isovector pairs do coexist. As illustrated in Fig. 2(e), at a given interaction strength x > 0, the role of the isoscalar pairs gradually decreases with isospin T , however, even for high values of T , their contributions are still significant.
A possible experimental evidence of coexistence between isoscalar and isovector pairing can be the observation and analysis of the deuteron transfer [22][23][24], which On the one hand, the relative deuteron transfer amplitudes increase with the strength of the isoscalar interaction, but this increase is fairly gradual, especially at higher isospins. On the other hand, absolute values of these amplitudes gradually decrease with the isospin. So, as expected, the observation of the strong deuteron transfer is most likely in N = Z nuclei, however, for N = Z, the effect does not abruptly disappear. Evidently, the SO(8) model is too simplistic to draw quantitative conclusions, and an analysis performed in a realistic shellstructure setting is very much required.
Indeed, it is now obvious that the effects of pn-pair condensation should be analyzed in a more sophisticated setting than that envisaged up to now. Within a meanfield approach, it appears that only by performing the VAP calculations one can fully account for a subtle balance between the isovector and isoscalar pairing correlations.
Methods to obtain full VAP results for realistic density functionals have already been formulated [25], and implemented for the simplest case of the particle-number restoration [26]. When combined with the full restoration of rotational and isospin symmetries, which were implemented without pairing in Ref. [27], and with the seven-dimensional symmetry restoration implemented in this Letter, a complete approach is possible, and is now being constructed.
For the Coulomb isospin mixing included together with pairing, a reduction of the three-dimensional isospin restoration to one dimension is not possible, and, moreover, the former will anyhow be required if the isocranking technology [10,11,28,29] is used to control the isospin degree of freedom. However, for axial nuclei, a one-dimensional symmetry integration suffices, so altogether we are then faced with five-dimensional integrals -fully manageable a task. Moreover, before attacking the full VAP approach, the results of this Letter indicate that a restricted minimization of the projected energies with respect to relative amplitudes of the isovector and isoscalar pairs could be a viable simplifying option.
The fact that the projected pair condensates properly describe isovector and isoscalar pairing correlations can be best seen by analyzing the simplest case of four particles. Then, the particle-number projected condensate is given by the square of the Thouless pair (1), that is, by |Φ 4 = N ′ (Ẑ + ) 2 |0 . However, the square of the Thouless pair is equal to a linear combination of five quartets: (P + P + ) (00) , (D + D + ) (00) , (P + D + ) (11) , (P + P + ) (02) , and (D + D + ) (20) , where superscripts (ST ) denote values of the total spin S and isospin T .
Restoration of the spin and isospin symmetries corresponds in this case to keeping only the first two scalar-isoscalar quartets and removing the other three. Thus the symmetry-projected |AST = |400 state becomes an exact linear combination of the two basic quartets [30]. Similarly, the symmetry-projected |610 state corresponds to an exact linear combination of these same two basic quartets supplemented by one vector-isoscalar pair D + (4). As a result, the A = 4 and 6 VAP solutions shown in Fig. 2(d) are identical to the exact ones. For larger particle numbers or isospins, the success of the VAP approach in describing the pair condensation relies on the fact that it properly accounts for the main components of the wave functions being given by the two basic scalar-isoscalar quartets.
In conclusion, within a simple SO(8) pairing model, we showed that the symmetry-projected pair condensates very accurately describe properties of the exact so-lutions, including the coexistence of the isovector and isoscalar pairing. Lack of the symmetry restoration thus explains a limited success in describing such a coexistence in standard mean-field approaches. Symmetry restoration is also key to reconciling the pair-condensation and quartet-condensation pictures of paired systems. Our study suggests that further work on properties of the proton-neutron nuclear pairing should be, and can be carried out within the variation-after-projection approach to mean-field pairings methods.
This work was partially supported by the STFC Grants No. ST/M006433/1 and No. ST/P003885/1.