Building blocks

Let $\rho\left(\vec {r}\sigma,\vec {r}'\sigma'\right)$ denote the one-body density matrix in space-spin coordinates. In what follows, in order to simplify the notation, we omit the isospin degree of freedom, because in the particle-hole channel all densities appear in the isoscalar and isovector forms [24], and generalization to proton-neutron systems does not present any problem. Within this assumption, the EDF we consider has the form:

$${\cal E} = \int {\rm d}^3\vec {r} {\cal H}_E(\vec {r}),$$ (1)

where the energy density ${\cal H}_E(\vec {r})$ can be represented as a sum of the kinetic and potential energies,

$${\cal H}_E(\vec {r}) = \frac{\hbar^2}{2m}\tau_0 + {\cal H}(\vec {r}) .$$ (2)

In the present study, we focus on the potential energy density ${\cal H}(\vec {r})$ only.

First, using the Pauli matrices $\sigma_a$, where index $a=\left\{x,y,z\right\}$ enumerates the Cartesian components of a vector, the density matrix is separated into the standard scalar and vector parts [25],

$$\rho\left(\vec {r}\sigma,\vec {r}'\sigma'\right) \!=\!\tfrac{1}{2... ...vert\sigma_a\right\vert\sigma'\right\rangle s_a\left(\vec {r},\vec {r}'\right),$$ (3)

where

$$\rho\left(\vec {r},\vec {r}'\right) = \sum_{\sigma}\rho\left(\vec {r}\sigma,\vec {r}'\sigma\right) ,$$ (4)

$$\vec {s}\left(\vec {r},\vec {r}'\right) = \sum_{\sigma\sigma'}\rho\left(\vec {r}\sigma,\vec {r}'\sigma'\right)\left\langle \sigma'\left\vert\vec {\sigma}\right\vert\sigma\right\rangle.$$ (5)

These two nonlocal densities will be used as building blocks of the functional together with the derivative operator $\vec {\nabla}$ and the relative momentum operator $\vec {k}$,

$$\vec {k}=\frac{1}{2i}\left(\vec {\nabla}-\vec {\nabla}'\right).$$ (6)

To most easily satisfy the constraints imposed by the rotational invariance, in our method, the building blocks are represented as spherical tensor operators [26], i.e., $\rho_{\lambda\mu}\left(\vec {r},\vec {r}'\right)$ for $\lambda=0$ and $s_{\lambda\mu}\left(\vec {r},\vec {r}'\right)$, $\nabla_{\lambda\mu}$, and $k_{\lambda\mu}$ for $\lambda=1$. In this notation, $\lambda$ is the rank of the tensor, and $\mu=-\lambda,\ldots,+\lambda$ is its tensor component. In the present study we use the following definitions of the building blocks in the spherical representation:

$$\rho_{00}(\vec {r},\vec {r}') = \rho(\vec {r},\vec {r}') ,$$ (7)

$$s_{ 1,\mu=\left\{-1,0,1\right\}}(\vec {r},\vec {r}') = -i \left\{\tfrac{ 1}{\sqrt{2}}\left(s_{ x}(\vec {r},\vec {r}') -i... ...}\left(s_{ x}(\vec {r},\vec {r}') +is_{ y}(\vec {r},\vec {r}')\right)\right\} ,$$ (8)

$$\nabla_{1,\mu=\left\{-1,0,1\right\}} = -i \left\{\tfrac{ 1}{\sqrt{2}}\left(\nabla_x-i\nabla_y\right), \nabla_z, \tfrac{-1}{\sqrt{2}}\left(\nabla_x+i\nabla_y\right)\right\} ,$$ (9)

$$k_{1,\mu=\left\{-1,0,1\right\}} = -i \left\{\tfrac{ 1}{\sqrt{2}}\left(k_x-ik_y\right), k_z, \tfrac{-1}{\sqrt{2}}\left(k_x+ik_y\right)\right\} .$$ (10)

In what follows, we most often omit indices and arguments of these spherical tensors and we simply write $\rho$, $s$, $\nabla$, and $k$ to lighten the notation.

In principle, arbitrary phase factors could be used in front of the spherical tensors. In Appendix B, we discuss possible choices of such phase conventions, and determine the particular ones selected in Eqs. (7)-(10). These phase conventions, which are not the standard ones, are used throughout the paper and define the phase properties of all other objects that we construct by using the building blocks above.