Proxy-SU(3) symmetry in the shell model basis

Abstract

The proxy-SU(3) symmetry has been proposed for spin-orbit like nuclear shells using the asymptotic deformed oscillator basis for the single particle orbitals, in which the restoration of the symmetry of the harmonic oscillator shells is achieved by a change of the number of quanta in the z-direction by one unit for the intruder parity orbitals. The same definition suffices within the cartesian basis of the Elliott SU(3) model. Through a mapping of the cartesian Elliott basis onto the spherical shell model basis, we translate the proxy-SU(3) approximation into spherical coordinates, proving, that in the spherical shell model basis the proxy-SU(3) approximation corresponds to the replacement of the intruder parity orbitals by their de Shalit–Goldhaber partners. Furthermore it is shown, that the proxy-SU(3) approximation in the cartesian Elliott basis is equivalent to a unitary transformation in the z-coordinate, leaving the x–y plane intact, a result which in the asymptotic deformed oscillator coordinates implies, that the z-projections of angular momenta and spin remain unchanged. The present work offers a microscopic justification of the proxy-SU(3) approximation and in addition paves the way, for taking advantage of the proxy-SU(3) symmetry in shell model calculations.

Appendices

Appendix A: Prerequisites for the unitarity of the operators

Within the pseudo-SU(3) approach it is well known, that there is a unitary transformation connecting the normal parity orbitals in a nuclear shell to their pseudo-SU(3) counterparts. A pedagogical description of this unitary transformation is given in [41], applied to the case of the one-dimensional harmonic oscillator (1D-HO) (Sect. 4 of [41]). Subsequently, the unitary transformation for the three-dimensional harmonic oscillator (3D-HO) from spherical coordinates to the pseudo-SU(3) basis is given in Sect. 5 of [41], while the unitary transformation, appropriate for the spherical Nilsson Hamiltonian (with deformation $\epsilon =0$), which includes a spin-orbit interaction term and an $\mathbf {l}^2$ term, where $\mathbf {l}$ is orbital angular momentum, is given in Sect. 6 of [41]. A concise version of this work is given in [42]. A generalization of this unitary transformation to the case of the deformed Nilsson Hamiltonian has been given in [43].

Let us consider an operator $G(\mathbf {r}, \mathbf {p}, \mathbf {S})$, where $\mathbf {r}$, $\mathbf {p}$, $\mathbf {S}$ are the position, momentum, and spin respectively. The operator G will not be in general unitary, but it can be made unitary through the use of the Hermitian operator [41]

$$\begin{aligned} \mathcal {O}(\mathbf {r}, \mathbf {p}, \mathbf {S}) = G^\dagger (\mathbf {r}, \mathbf {p}, \mathbf {S}) G(\mathbf {r}, \mathbf {p}, \mathbf {S}), \end{aligned}$$

(57)

by taking

$$\begin{aligned} U = G(\mathbf {r}, \mathbf {p}, \mathbf {S}) [\mathcal {O} (\mathbf {r}, \mathbf {p}, \mathbf {S})]^{-1/2}. \end{aligned}$$

(58)

Since $\mathcal {O}$ is a Hermitian operator, it is possible, to find a basis, in which this operator is diagonal. In this basis the operator $[\mathcal {O} (\mathbf {r}, \mathbf {p}, \mathbf {S})]^{-1/2}$ can be obtained by taking the square roots of the inverse terms in the diagonal, provided that the eigenvalues of $\mathcal {O}$ are positive. Square roots of positive operators have been considered repeatedly in the mathematical literature [60,61,62,63].

In the case of the original 1D-HO Hamiltonian one has [45]:

$$\begin{aligned} H_0 = a^\dagger a, \quad a^\dagger = {1\over \sqrt{2}}(x-ip), \quad a= {1\over \sqrt{2}}(x+ip), \end{aligned}$$

(59)

where x and p are the coordinate and the corresponding momentum, while $a^\dagger$ and a denote the usual creation and annihilation operators. The unitary operator is obtained from the usual annihilation operator a and has the form [41]

$$\begin{aligned} U = a (a^\dagger a)^{-1/2}. \end{aligned}$$

(60)

It is clear, that the above mentioned prerequisites are fulfilled, since the operator $a^\dagger a$ in the basis considered is diagonal and has positive eigenvalues, so that the square roots of their inverses can be calculated. The Hamiltonian resulting from the unitary transformation is [41]

$$\begin{aligned} H_0' = a^\dagger a +1. \end{aligned}$$

(61)

This implies, that the levels characterized by $\mathcal {N}$ quanta in the old Hamiltonian carry $\mathcal {N}-1$ quanta in the new one, sharing the same energy, while the ground state of the old Hamiltonian has no counterpart. In other words, in the new Hamiltonian $\mathcal {N}$ is replaced by $\mathcal {N}-1$, but the loss of energy is counterbalanced by the $+1$ term appearing in Eq. (61). This $+1$ term plays a major role in the proxy-SU(3) approximation, as discussed in Sect. 7.

In the above the spectra of the old and the new Hamiltonian are the same, as they should be for a unitary transformation. It should also be noticed, that we have the complete set of states of the harmonic oscillator both before and after the unitary transformation, i.e., the ground state is included in both cases. Therefore the U(1) symmetry characterizing the 1D-HO before the unitary transformation is conserved after the unitary transformation.

Appendix B: Unitary transformations in the pseudo-SU(3) scheme

For the sake of comparison, the original 3D-HO Hamiltonian for a particle with spin 1/2 is:

$$\begin{aligned} H_0 = \mathbf {a^\dagger } \cdot \mathbf {a}, \quad \mathbf{{a^\dagger }}= {1\over \sqrt{2}}(\mathbf {r}-i \mathbf {p}), \quad \mathbf{a}= {1\over \sqrt{2}}(\mathbf{r}+i\mathbf{p}), \end{aligned}$$

(62)

where $\mathbf{r}$ and $\mathbf{p}$ are the coordinates and the corresponding momenta, while $\mathbf {a^\dagger }$ and $\mathbf{a}$ denote the usual creation and annihilation operators. The unitary operator is obtained from $(\mathbf{a} \cdot \mathbf{S})$, where $\mathbf{S}$ is the spin operator, and reads [41]

$$\begin{aligned} U =2(\mathbf {a} \cdot \mathbf {S}) (\mathbf {a^\dagger } \cdot \mathbf {a} -2 \mathbf {L} \cdot \mathbf {S}) ^{-1/2}. \end{aligned}$$

(63)

The Hamiltonian resulting from the unitary transformation is [41]:

$$\begin{aligned} H_0' = \mathbf{a^\dagger }\cdot \mathbf{a} +1. \end{aligned}$$

(64)

This again implies, that the levels characterized by $\mathcal {N}$ quanta in the old Hamiltonian carry $\mathcal {N}-1$ quanta in the new one, sharing the same energy, while the ground state of the old Hamiltonian has no counterpart. As in the case of the 1D-HO, the loss of one quantum of excitation energy by the fact that $\mathcal {N}$ of the original Hamiltonian is mapped onto $\mathcal {N}-1$ in the final Hamiltonian is counterbalanced by the $+1$ term in Eq. (64). This $+1$ term plays a major role in the proxy-SU(3) approximation, as discussed in Sect. 7. The unitary transformation considered above is quite general, since it involves the spin and the orbital angular momentum.