Symmetry-invariant energy density

Based on the derivation after separation of symmetries, which we introduced above, it is clear that we can proceed by separating densities into irreducible representations of all required symmetries and then building the energy density by taking scalar products separately in each of the representations. Such a construction gives a symmetry-invariant energy density,

$${\cal H}^S(\vec {r}) = {\cal H}(\vec {r}),$$ (80)

where ${\cal H}^S(\vec {r})$ denotes the energy density calculated for a many-body state transformed by the symmetry operator $S$. This guarantees the invariance of the EDF and total energy (1) with respect to all considered symmetries. Such a strategy would also allow for using arbitrary, unrelated to one another coupling constants in each of the irreducible representations.

However, in practical applications, such a strategy was never up to now fully implemented--neither at N$^3$LO, for which the present study is the first attempt in the literature, nor at NLO, which corresponds to the standard Skyrme functionals (see Sec. 3.1 below). Only the time reversal and isospin symmetries were up to now treated in this way, and below we are going to follow the same path.

For the time reversal, all local densities discussed in Sec. 2.3 are either time-even or time-odd. Indeed, this simply follows from the facts [25] that

$$\begin{array}{rllll} \rho ^{T}(\vec {r},\vec {r}') &= &\rho ^... ...c {r},\vec {r}') &= -&\vec {s}(\vec {r}',\vec {r}), \end{array}$$ (81)

which give the time-even and time-odd parts in Eqs. (65) and (66) as

$$\begin{array}{rllll} \rho_+(\vec {r},\vec {r}') &= & \rho_+^*... ...r},\vec {r}') &= & \vec {s}_-(\vec {r}',\vec {r}) , \end{array}$$ (82)

i.e., $\rho_+(\vec {r},\vec {r}')$ and $\vec {s}_-(\vec {r},\vec {r}')$ are real symmetric functions and $\rho_-(\vec {r},\vec {r}')$ and $\vec {s}_+(\vec {r},\vec {r}')$ are imaginary antisymmetric functions. Moreover, the relative momentum operator $\vec {k}$ (6), which defines derivative operators $K_{nL}$, is imaginary and antisymmetric with respect to exchanging variables $\vec {r}$ and $\vec {r}'$. Altogether, it is easy to see that $T$-parities of primary densities $\rho_{nLvJ}(\vec {r})$ (23) are equal to $(-1)^{n+v}$, see Eq. (25) and columns denoted by $T$ in Tables 3 and 4. Similarly, $T$-parities of secondary densities $\rho_{mI,nLvJ,Q}(\vec {r})$ (24) are also equal to $(-1)^{n+v}$. Construction of the $T$-invariant energy density (80) can now be realized by multiplying densities that have identical $T$-parities.