Two kinds of nucleons

As one is dealing with $Z$ protons and $N$ neutrons, two gauge angles, $\phi_{n}$ and $\phi_{p}$, must enter the number projection operator:

$$P^{NZ}=\frac{1}{2\pi }\int d\phi_{n}\ e^{i\phi _{n}(\hat{N}-N)}\frac{1}{ 2\pi }\int d\phi_{p}\ e^{i\phi _{p}(\hat{Z}-Z)}.$$ (70)

Consequently, the total projected energy (53) becomes a double integral,

$$E^{N}=\int d\phi_{n}~d\phi_{p}~y_{n}(\phi_{n})~y_{p}(\phi_{p})~ E(\phi_{n},\phi_{p}),$$ (71)

where the transition energy density

$$E(\phi_{n},\phi_{p})=\int d{\bm r}~{\cal H}({\bm r},\phi_{n},\phi _{p})$$ (72)

depends on both gauge angles $\phi_{n}$, $\phi_{p}$.

To simplify notation, we use the isospin label $q$=$\tau_3$ ($q$=+1 for neutrons and -1 for protons) and $\bar{q}$=$-q$. In the following, we shall employ the convention $y(\phi_{q})$$\equiv$ $y_q(\phi_{q})$, $C(\phi_{q})$$\equiv$ $C_q(\phi_{q})$, and $Y(\phi_{q})$$\equiv$ $Y_q(\phi_{q})$. The isospin-dependent particle-hole and particle-particle fields (66), (67) can be written as:

$$h_{q}^{N} = \int d\phi_{q}~y(\phi_{q})~$$

$$\times \left[ Y(\phi_{q})\left( \int y(\phi_{\bar{q}})~E(\phi _{q},\phi_{\bar{q}})d\phi_{\bar{q}}~\right) \right.$$

$$+ \left. e^{-2i\phi_{q}}~C(\phi_{q})\left( \int y(\phi_{\bar{q }})~h_{q}(\phi_{q},\phi_{\bar{q}})d\phi_{\bar{q}}~\right) C(\phi _{q}) \right]$$

$$- \left[ \int d\phi_{q}~y(\phi_{q})\right. 2ie^{-i\phi _{q}}\sin(\phi _{q})\tilde{\rho}_q(\phi_{q})$$

$$\times \left. \left( \int y(\phi_{\bar{q}})\tilde{h}_{q}(\phi _{q},\phi_{\bar{q}})d\phi_{\bar{q}}\right) C(\phi _{q})+^{\;}h.c.\right] ,$$ (73)

$$\tilde{h}_{q}^{N} = \int d\phi_{q}~y(\phi_{q})~e^{-i\phi_{q}}$$

$$\times \left[ \left( \int y(\phi_{\bar{q}})\tilde{h}_{q}(\phi _{q},\phi_{\bar{q}})d\phi_{\bar{q}}\right) C(\phi _{q})+(...)^{T}\right].$$ (74)

In numerical applications, the two-dimensional integrals over the gauge angles are replaced by a sum over $L_n\times L_p$ points using the Gauss-Chebyshev quadrature method [34].