Proton-neutron pairing, a concise overview

A unique aspect of proton-rich nuclei with $N$$\approx$$Z$ is that neutrons and protons occupy the same single-particle orbitals. Consequently, due to the large spatial overlaps between neutron and proton single-particle wave functions, pn pairing is expected to be present in those systems.

So far, the strongest evidence for enhanced pn correlations around the $N$=$Z$ line comes from the measured binding energies [8,9,10,11,12,13,14,15,16] and the isospin structure of the low-lying states in odd-odd nuclei [17,18,19,20,21,22,23,24,,26,27,28,16]. The pn correlations are also expected to play some role in single-beta decay [29,30,31], double-beta decay [32,33,34,35,36,37,38], transfer reactions [39,40,41,42,43,44,45] (see, however, Ref. [16]), structure of low-lying collective states [46], alpha decay and alpha correlations [47,48,49,50,45,51,52], structure of high spin states [20,53,54,55,56,57,58,59,60,,62,63,64,65,66,67,,69,70,71,72,14,73,74,75,,77,78,79], and in properties of low-density nuclear matter [80,81,82,83,84,85,86,87,88,89,90].

Actually, the pn pairing is not ``the new kid on the block" but it has a long history and is ultimately connected to the charge invariance of the strong Hamiltonian. (For reference, in 1932 Heisenberg introduced isotopic spin [91] and in 1936 Wigner introduced the nuclear SU(4) supermultiplets [92].) An important step was the adaptation of Racah's concept of seniority by Racah and Talmi [93], and Flowers [94] in 1952. In the independent quasiparticle (BCS) picture [95], pairing condensate appears as a result of an attractive interaction between quasiparticles near the Fermi surface. The term ``nuclear superconductivity" was first used by Pines at the 1957 Rehovot Conference to point out that the new BCS theory might also apply to nuclei [96]. This was formally accomplished in the late fifties [97,98] and shortly afterwards the importance of pn pairing was emphasized [99,47,100] and a number of theoretical papers dealing with the generalization of the BCS theory to the pn pairing case appeared [101,102,103].

Independently, group-theoretical methods based on the quasi-spin formalism were developed. Many insights were gained by simple solvable models employing symmetry-dictated interactions [104,105,106,107,108,109,110,111]. Two families of models were used, one based on the $j$-$j$ coupling with the symmetry SO(5) (appropriate for the $T$=1 pairing) and the other based on the $L$-$S$ coupling with the symmetry SO(8) (appropriate for the $T$=0 and $T$=1 pairing). These models have been consecutively developed and applied to various physically interesting cases [42,44,112,36,45,113,114]. Among many techniques used to solve the problem of pn pairing with schematic interactions, worth mentioning are the exact methods [115,116,43] used to describe isovector states of a charge-independent pairing Hamiltonian.

Properties of pn pairing (at low and high spins, in cold and hot nuclei) have been studied within the large-scale shell model (diagonalization shell-model, variational shell model, and Monte Carlo shell model) [90,117,118,69,70,119,120,121,,64,123,20,21]. It was concluded that the isovector pairing in the dominating $J$=0 channel mainly acts between time-reversed states within the same shell. On the other hand, isoscalar pairing can also involve coupling (mainly $J$=1) between spin-orbit partners. Consequently, spin-orbit splitting plays a crucial role in understanding the $T$=0 pairing [20].

It is to be noted that it is by no means obvious how to extract ``pairing correlations" from the realistic shell-model calculations. The ``pairing Hamiltonian" is an integral part of the residual shell-model interaction. The shell-model Hamiltonian is usually written in the p-p representation, but it also can be transformed to the p-h representation by means of the Pandya transformation [124]. This means that the high-$J$ interaction between pairs can translate into the low-$J$ interaction in the p-h channel. It is only in the mean-field theory that the division into ``particle-hole" and ``particle-particle" channels appears naturally. One way of translating the shell-model results into mean-field language is by means of correlators, such as the number of $T$=0 and $T$=1 pairs in the shell-model wave function, [112,120,125,69].

The extension of the Interacting Boson Models (IBM) to the case of pn bosons had to wait until 1980, when IBM-3 (only $T$=1 pairs [126]) and IBM-4 [both ($T$=1, $S$=0) and ($T$=0, $S$=1) bosons [127]] were proposed. For recent applications of various algebraic models, see Refs. [128,129,,131,132,133,134,135,136,137,26,138].

An alternative strategy to the pn pairing problem is via the mean-field approach. Here, the major conceptional step was the proposition that quasiparticles are mixtures not only particles and holes but also protons and neutrons. The resulting HFB quasi-particle vacuum is a superposition of wave functions corresponding to even-even and odd-odd nuclei with different particle numbers. Unlike in the standard nn and pp pairing cases, the coefficients of the Bogoliubov transformation are, in general, complex. Generalized Bogoliubov transformation, generalized gap equations, and pn pairing fields are discussed in Refs. [101,139,140,141,142,143,144,,146,147,148,,150,151,152,,53,154,155,156,55,,57,157,62,65,158,67,68,159,160,161].

The problem of the spontaneous isospin breaking in the mean-field theory was realized soon after the development of the generalized quasiparticle approach [144,148,48]. The symmetry is broken by the independent (separate) treatment of $T$=1 proton and neutron pairing correlations and by the BCS quasiparticle mean field (the generalized product wave function is not an eigenstate of isospin). Several techniques have been developed to restore isospin. They include the Generator Coordinate Method, RPA, Kamlah expansion, iso-cranking, and exact projection [144,148,48,162,163,37,164,67,,165,166,15,16]. It is fair to say, however, 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.