Density matrices in the isospin space

We begin with the discussion of the building blocks of the HFB theory: one-body density matrices. In the HFB theory, expectation values of all observables and, in particular, of the nuclear Hamiltonian can be expressed as functionals of the density matrix $\hat{\rho}$ and the pairing tensor $\hat{\kappa}$ defined as [167]

$$\hat{\rho}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't') = \langle \Psi \vert a_{\mbox{{\boldmath {$r$}}}'s't'}^{+}a_{\mbox{{\boldmath {$r$}}}st}\vert\Psi \rangle ,$$ (1)

$$\hat{\kappa}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't') = \langle \Psi \vert a_{\mbox{{\boldmath {$r$}}}'s't'}a_{\mbox{{\boldmath {$r$}}}st}\vert\Psi \rangle ,$$ (2)

where $a_{\mbox{{\boldmath {$r$}}}st}^{+}$ and $a_{\mbox{{\boldmath {$r$}}}st}$ create and annihilate, respectively, nucleons at point $\mbox{{\boldmath {$r$}}}$, spin $s$= $\pm{\textstyle{\frac{1}{2}}}$, and isospin $t$= $\pm{\textstyle{\frac{1}{2}}}$, while $\vert\Psi \rangle$ is the HFB independent-quasiparticle state. Instead of using the antisymmetric pairing tensor it is more convenient to introduce the p-p density matrices that can be defined in two forms, $\hat{\tilde{\rho}}$ or $\hat{\breve{\rho}}$, denoted by ``tilde'' and ``breve'', respectively:

$$\hat{\tilde{\rho}}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't') = -2s'\langle \Psi \vert a_{\mbox{{\boldmath {$r$}}}'-s't'}a_{\mbox{{\boldmath {$r$}}}st}\vert\Psi \rangle ,$$ (3)

$$\hat{\breve{\rho}}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't') = 4s't'\langle \Psi \vert a_{\mbox{{\boldmath {$r$}}}'-s'-t'}a_{\mbox{{\boldmath {$r$}}}st}\vert\Psi \rangle .$$ (4)

In Ref. [5], p-p density matrix $\hat{\tilde{\rho}}$ was used to treat the n-n and p-p pairing correlations without the proton-neutron mixing. It was then shown that for conserved time-reversal symmetry $\hat{\tilde{\rho}}$ is hermitian, and leads to p-p local densities that have the structure which is analogous to that of the p-h local densities. However, in the case of the proton-neutron mixing studied here, we decided to use the p-p density matrix $\hat{\breve{\rho}}$, because it allows a more transparent treatment of the isoscalar and isovector pairing channels. Detailed discussion of this point will be presented in Sec. 3.3 below.

With each of density matrices of Eqs. (1) and (4) three other matrices are associated:

-

the hermitian conjugate matrices defined as:

$$\hat{\rho}^{+}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't') = \hat{\rho}^{\ast} (\mbox{{\boldmath {$r$}}}'s't',\mbox{{\boldmath {$r$}}}st),$$ (5)

$$\hat{\breve{\rho}}^{+}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't') = \hat{\breve{\rho}}^{\ast} (\mbox{{\boldmath {$r$}}}'s't',\mbox{{\boldmath {$r$}}}st)$$ (6)

-

the time-reversed matrices defined as:

$$\hat{\rho}^{T}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't') = 4ss'\hat{\rho}^{\ast}(\mbox{{\boldmath {$r$}}}\,\mbox{$-s$}t,\mbox{{\boldmath {$r$}}}'\,\mbox{$-s$}'t'),$$ (7)

$$\hat{\breve{\rho}}^{T}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't') = 4ss'\hat{\breve{\rho}}^{\ast}(\mbox{{\boldmath {$r$}}}\,\mbox{$-s$}t,\mbox{{\boldmath {$r$}}}'\,\mbox{$-s$}'t'),$$ (8)

-

the charge-reversed matrices defined as:

$$\hat{\rho}^{C}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't') = 4tt'\hat{\rho}(\mbox{{\boldmath {$r$}}}s\,\mbox{$-t$},\mbox{{\boldmath {$r$}}}'s'\,\mbox{$-t$}'),$$ (9)

$$\hat{\breve{\rho}}^{C}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't') = 4tt'\hat{\breve{\rho}}(\mbox{{\boldmath {$r$}}}s\,\mbox{$-t$},\mbox{{\boldmath {$r$}}}'s'\,\mbox{$-t$}'),$$ (10)

where the asterisk stands for the complex conjugation.

Here and below we present full sets of expressions even in those cases when they could, in principle, be replaced by verbal descriptions. We do so in order to avoid possible confusion at the expense of a slight increase in the length of the paper. We think that such an approach is highly beneficial to the reader, because in many cases small but significant differences appear in expressions that otherwise could have seemed analogous to one another.

The charge-reversal operation $C$ defined in Eq. (9) exchanges the neutron and proton charges, or equivalently, flips their isospin projections. Note that the time reversal is antilinear while the charge reversal is a linear operation, and that they commute with one another. Symmetries of the density matrices can be conveniently expressed in terms of just the hermitian conjugation, and time and charge reversals. Namely, it follows from definitions (1) and (4) that

$$\hat{\rho}^{+} = \hat{\rho},$$ (11)

$$\hat{\breve{\rho}}^{+} = -\hat{\breve{\rho}}^{TC},$$ (12)

where the superscript $TC$ denotes superposition of the time (7) and charge (9) reversals.

For $\vert\Psi \rangle$ being an independent-quasiparticle state the density matrices fulfill the following kinematical conditions

$$\hat{\rho}\bullet \hat{\breve{\rho}} = \hat{\breve{\rho}}\bullet \hat{\rho}^{TC} ,$$ (13)

$$\hat{\rho} = \hat{\rho}\bullet \hat{\rho} +\hat{\breve{\rho}}\bullet \hat{\breve{\rho}}^{+},$$ (14)

where $\bullet$ stands for integration over spatial coordinates and summation over spin and isospin indices, denoted by ${\textstyle\int\hspace{-0.9em}\sum}{\rm d}x$, e.g.:

$$(\hat{\rho} \bullet \hat{\breve{\rho}})(\mbox{{\boldmath {$r... ...(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}_2s_2t_2),$$ (15)

where we also abbreviated the space-spin-isospin variables by $x$$\equiv$ $\{\mbox{{\boldmath {$r$}}}st\}$. Equations (13) secure the projectivity of the generalized density matrix:

$$\hat{\breve{\mathcal R}}=\hat{\mathcal W}\hat{\mathcal R}\ha... ...\breve{\rho}}^{+} & \hat{1}-\hat{\rho}^{TC}\end{array}\right),$$ (16)

where $\hat{1} := \delta(x-x') := \delta (\mbox{{\boldmath {$r$}}}-\mbox{{\boldmath {$r$}}}')\delta_{ss'}\delta_{tt'}$ and the unitary matrix $\hat{\mathcal W}$,

$$\hat{\mathcal W} =\left(\begin{array}{cc}\hat{1}&0\\ 0 & -\... ...mbox{{\boldmath {$\sigma$}}}}_y\hat{\tau}_2\end{array}\right),$$ (17)

transforms the standard generalized density matrix $\hat{\mathcal R}$ (cf. Ref. [167]) to the ``breve'' representation.

When the pairing correlations of only like nucleons are taken into account, none but the diagonal (off-diagonal) matrix elements of density matrix $\hat{\rho}$ ( $\hat{\breve{\rho}}$) in isospin indices are considered. However, in a general case of pairing correlations between both, like and unlike nucleons, the remaining matrix elements become relevant as well. Therefore, in the following subsections we specify the spin-isospin structure of the p-h and p-p density matrices explicitly.