Phase conventions

In the present study, we use four elementary building blocks to construct the EDF, namely, the scalar and vector nonlocal densities, $\rho(\vec {r},\vec {r}')$ and $\vec {s}(\vec {r},\vec {r}')$, along with the total derivative $\vec {\nabla}$ and relative momentum $\vec {k}$ (6). Spherical representations of the building blocks can be defined by using standard convention of spherical tensors [26] as

$$\rho_{00}(\vec {r},\vec {r}') = p_\rho \rho(\vec {r},\vec {r}') ,$$ (88)

$$s_{ 1,\mu=\left\{-1,0,1\right\}}(\vec {r},\vec {r}') = p_s\left\{\tfrac{ 1}{\sqrt{2}}\left(s_{ x}(\vec {r},\vec {r}') -i... ...}\left(s_{ x}(\vec {r},\vec {r}') +is_{ y}(\vec {r},\vec {r}')\right)\right\} ,$$ (89)

$$\nabla_{1,\mu=\left\{-1,0,1\right\}} = p_{\rule{0ex}{1.5ex}\nabla}\left\{\tfrac{ 1}{\sqrt{2}}\left(\nabl... ...right), \nabla_z, \tfrac{-1}{\sqrt{2}}\left(\nabla_x+i\nabla_y\right)\right\} ,$$ (90)

$$k_{1,\mu=\left\{-1,0,1\right\}} = p_{\rule{0ex}{1.5ex}k}\left\{\tfrac{ 1}{\sqrt{2}}\left(k_x-ik_y\right), k_z, \tfrac{-1}{\sqrt{2}}\left(k_x+ik_y\right)\right\} ,$$ (91)

where $p_\rho$, $p_s$, $p_{\rule{0ex}{1.5ex}\nabla}$, and $p_{\rule{0ex}{1.5ex}k}$, are arbitrary phase factors, $\vert p_\rho \vert=\vert p_s\vert=\vert p_{\rule{0ex}{1.5ex}\nabla}\vert=\vert p_{\rule{0ex}{1.5ex}k}\vert=1$. These phase factors define the phase convention of the building blocks, and can be used to achieve specific phase properties of densities and terms in the EDF, as discussed in this Appendix.

In order to motivate the best suitable choice of the phase convention, in Tables 21 and 22 we present relations between the spherical and Cartesian representations of densities and terms in the EDF, respectively. All NLO densities in the Cartesian representation, which are listed in Table 21, are real. It is then clear that the phase convention, which would render all NLO densities in the spherical representation real does not exist. However, for the phase factors $p_\rho$, $p_s$, $p_{\rule{0ex}{1.5ex}\nabla}$, and $p_{\rule{0ex}{1.5ex}k}$ equal to $\pm1$ or $\pm{i}$, in the spherical representation all NLO densities and terms in the EDF are either real or imaginary.

Table 21: Spherical and Cartesian representations of local densities (24) up to NLO. Only scalar densities and the $\mu=0$ components of vector densities are shown. Numbers in the first column refer to numbers of primary densities (23) shown in Tables 3 and 4. The last column shows factors preceding densities in the Cartesian representation evaluated for the phase conventions of Eq. (92). Time-even densities are marked by using the bold-face font.

No.	$\rho_{mI,nLvJ,Q\mu}$					Cartesian representation [25,30,24]	Phase
1	$\bm{\rho_{00,0000,00}}$	=	$[ \rho ]_{00}$	=	$\phantom{-}\phantom{i}p_\rho$	$\rho$	$+1$
	$\bm{\rho_{11,0000,10}}$	=	$[\nabla \rho ]_{10}$	=	$\phantom{-}\phantom{i}p_{\rule{0ex}{1.5ex}\nabla}p_\rho$	$\nabla_z\rho$	$-i$
	$\bm{\rho_{20,0000,00}}$	=	$[[\nabla\nabla]_0\rho ]_{00}$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}\nabla}^2 p_\rho$	$\tfrac{1}{\sqrt{3}}\Delta\rho$	$+1$
2	${}{\rho_{00,1101,10}}$	=	$[k \rho ]_{10}$	=	$\phantom{-}\phantom{i}p_{\rule{0ex}{1.5ex}k}p_\rho$	$j_z$	$-i$
	${}{\rho_{11,1101,00}}$	=	$[\nabla[k \rho]_1]_{00}$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}\nabla}p_{\rule{0ex}{1.5ex}k}p_\rho$	$\tfrac{1}{\sqrt{3}}\vec {\nabla}\cdot\vec {j}$	$+1$
	${}{\rho_{11,1101,10}}$	=	$[\nabla[k \rho]_1]_{10}$	=	$\phantom{-}i p_{\rule{0ex}{1.5ex}\nabla}p_{\rule{0ex}{1.5ex}k}p_\rho$	$\tfrac{1}{\sqrt{2}}\left(\vec {\nabla}\times\vec {j}\right)_z$	$-i$
3	$\bm{\rho_{00,2000,00}}$	=	$[[kk]_{0} \rho ]_{00}$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}k}^2 p_\rho$	$\tfrac{1}{\sqrt{3}}\left(\tau-\tfrac{1}{4}\Delta\rho\right)$	$+1$
17	${}{\rho_{00,0011,10}}$	=	$[ s ]_{10}$	=	$\phantom{-}\phantom{i}p_s$	$s_z$	$-i$
	${}{\rho_{11,0011,00}}$	=	$[\nabla s ]_{00}$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}\nabla}p_s$	$\tfrac{1}{\sqrt{3}}\vec {\nabla}\cdot\vec {s}$	$+1$
	${}{\rho_{11,0011,10}}$	=	$[\nabla s ]_{10}$	=	$\phantom{-}i p_{\rule{0ex}{1.5ex}\nabla}p_s$	$\tfrac{1}{\sqrt{2}}\left(\vec {\nabla}\times\vec {s}\right)_z$	$-i$
	${}{\rho_{20,0011,10}}$	=	$[[\nabla\nabla]_0 s ]_{10}$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}\nabla}^2p_s$	$\tfrac{1}{\sqrt{3}}\Delta s_z$	$-i$
	${}{\rho_{22,0011,10}}$	=	$[[\nabla\nabla]_2 s ]_{10}$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}\nabla}^2p_s$	$\tfrac{1}{\sqrt{15}}\left(3\nabla_z\vec {\nabla}\cdot\vec {s}-\Delta s_z\right)$	$-i$
18	$\bm{\rho_{00,1110,00}}$	=	$[k s ]_{00}$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}k}p_s$	$\tfrac{1}{\sqrt{3}}J^{(0)}$	$+1$
	$\bm{\rho_{11,1110,10}}$	=	$[\nabla[k s]_0]_{10}$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}\nabla}p_{\rule{0ex}{1.5ex}k}p_s$	$\tfrac{1}{\sqrt{3}}\nabla_z J^{(0)}$	$-i$
19	$\bm{\rho_{00,1111,10}}$	=	$[k s ]_{10}$	=	$\phantom{-}i p_{\rule{0ex}{1.5ex}k}p_s$	$\tfrac{1}{\sqrt{2}}J_z$	$-i$
	$\bm{\rho_{11,1111,00}}$	=	$[\nabla[k s]_1]_{00}$	=	$- i p_{\rule{0ex}{1.5ex}\nabla}p_{\rule{0ex}{1.5ex}k}p_s$	$\tfrac{1}{\sqrt{6}}\vec {\nabla}\cdot\vec {J}$	$+1$
	$\bm{\rho_{11,1111,10}}$	=	$[\nabla[k s]_1]_{10}$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}\nabla}p_{\rule{0ex}{1.5ex}k}p_s$	$\tfrac{1}{2}\left(\vec {\nabla}\times\vec {J}\right)_z$	$-i$
21	${}{\rho_{00,2011,10}}$	=	$[[kk]_{0} s ]_{10}$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}k}^2p_s$	$\tfrac{1}{\sqrt{3}}\left(T_z-\tfrac{1}{4}\Delta s_z\right)$	$-i$
22	${}{\rho_{00,2211,10}}$	=	$[[kk]_{2} s ]_{10}$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}k}^2p_s$	$\tfrac{1}{\sqrt{15}}\left(3F_z-\tfrac{3}{4}\nabla_z\vec {\nabla}\cdot\vec {s}-T_z+\tfrac{1}{4}\Delta s_z\right)$	$-i$

Table 22: Spherical and Cartesian representations of terms in the EDF (30) up to NLO. The last column shows factors preceding terms in the Cartesian representation evaluated using the phase conventions of Eq. (92). Integration by parts was used to transform $\vec {s}\cdot\vec {\nabla}\vec {\nabla}\cdot\vec {s}$ into $-(\vec {\nabla}\cdot\vec {s})^2$, which is the term used previously in Refs. [30,24]. Coupling constants corresponding to terms that depend on time-even densities are marked by using the bold-face font. Bullets ($\bullet$) mark coupling constants corresponding to terms that do not vanish for conserved spherical, space-inversion, and time-reversal symmetries, see Sec. 4.

No.		$~C^{n'L'v'J'}_{mI,nLvJ}$		${[}\rho_{n'L'v'J'}{[}D_{mI} \rho_{nLvJ}{]}_{J'}{]}_0$		$\phantom{-}$	$$ Cartesian representation	Phase
1	$\bullet$	$~\bm{C^{0000}_{00,0000}}$		${[}\rho \rho {]}_0$	=	$\phantom{-}\phantom{i}p_\rho ^2$	$\rho^2$	+1
2	$$	$~{}{C^{0011}_{00,0011}}$		${[}s s {]}_0$	=	$- \phantom{i}p_s^2$	$\tfrac{1}{\sqrt{3}} \vec {s}^2$	+1
3	$\bullet$	$~\bm{C^{0000}_{20,0000}}$		${[}\rho {[}{[}\nabla\nabla{]}_0 \rho {]}_0{]}_0$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}\nabla}^2p_\rho ^2$	$\tfrac{1}{\sqrt{3}} \rho\Delta\rho$	+1
4	$\bullet$	$~\bm{C^{0000}_{00,2000}}$		${[}\rho {[}{[}kk{]}_0\rho{]}_0 {]}_0$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}k}^2p_\rho ^2$	$\tfrac{1}{\sqrt{3}} \left(\rho\tau-\tfrac{1}{4}\rho\Delta\rho\right)$	+1
5	$$	$~\bm{C^{1110}_{00,1110}}$		${[}{[}ks {]}_0 {[}k s{]}_0 {]}_0$	=	$\phantom{-}\phantom{i}p_{\rule{0ex}{1.5ex}k}^2p_s^2$	$\tfrac{1}{ 3} \left(J^{(0)}\right)^2$	+1
6	$\bullet$	$~\bm{C^{1111}_{00,1111}}$		${[}{[}ks {]}_1 {[}k s{]}_1 {]}_0$	=	$\phantom{-}\phantom{i}p_{\rule{0ex}{1.5ex}k}^2p_s^2$	$\tfrac{1}{\sqrt{12}}\vec {J}^2$	+1
7	$$	$~\bm{C^{1112}_{00,1112}}$		${[}{[}ks {]}_2 {[}k s{]}_2 {]}_0$	=	$\phantom{-}\phantom{i}p_{\rule{0ex}{1.5ex}k}^2p_s^2$	$\tfrac{1}{\sqrt{5}} \sum_{ab}J_{ab}^{(2)}J_{ab}^{(2)}$	+1
8	$\bullet$	$~\bm{C^{0000}_{11,1111}}$		${[}\rho {[}\nabla {[}k s{]}_1{]}_0{]}_0$	=	$- i p_{\rule{0ex}{1.5ex}\nabla}p_{\rule{0ex}{1.5ex}k}p_sp_\rho$	$\tfrac{1}{\sqrt{6}} \rho\vec {\nabla}\cdot\vec {J}$	+1
9	$$	$~{}{C^{1101}_{00,1101}}$		${[}{[}k\rho{]}_1 {[}k \rho{]}_1 {]}_0$	=	$- \phantom{i}p_{\rule{0ex}{1.5ex}k}^2p_\rho ^2$	$\tfrac{1}{\sqrt{3}} \vec {j}^2$	+1
10	$$	$~{}{C^{0011}_{20,0011}}$		${[}s {[}{[}\nabla\nabla{]}_0 s {]}_1{]}_0$	=	$\phantom{-}\phantom{i}p_{\rule{0ex}{1.5ex}\nabla}^2p_s^2$	$\tfrac{1}{ 3} \vec {s}\Delta\vec {s}$	+1
11	$$	$~{}{C^{0011}_{22,0011}}$		${[}s {[}{[}\nabla\nabla{]}_2 s {]}_1{]}_0$	=	$\phantom{-}\phantom{i}p_{\rule{0ex}{1.5ex}\nabla}^2p_s^2$	$\tfrac{-1}{\sqrt{5}} \left(\left(\vec {\nabla}\cdot\vec {s}\right)^2+\tfrac{1}{3}\vec {s}\Delta\vec {s}\right)$	+1
12	$$	$~{}{C^{0011}_{00,2011}}$		${[}s {[}{[}kk{]}_0 s{]}_1 {]}_0$	=	$\phantom{-}\phantom{i}p_{\rule{0ex}{1.5ex}k}^2p_s^2$	$\tfrac{1}{ 3} \left(\vec {s}\cdot\vec {T}-\tfrac{1}{4}\vec {s}\Delta\vec {s}\right)$	+1
13	$$	$~{}{C^{0011}_{00,2211}}$		${[}s {[}{[}kk{]}_2 s{]}_1 {]}_0$	=	$\phantom{-}\phantom{i}p_{\rule{0ex}{1.5ex}k}^2p_s^2$	$\tfrac{1}{\sqrt{5}} \Big(\vec {s}\cdot\vec {F}+\tfrac{1}{4}\left(\vec {\nabla}\cdot\vec {s}\right)^2$
							$-\tfrac{1}{3}\vec {s}\cdot\vec {T}+\tfrac{1}{12}\vec {s}\Delta\vec {s}\Big)$	+1
14	$$	$~{}{C^{1101}_{11,0011}}$		${[}{[}k\rho{]}_1{[}\nabla s {]}_1{]}_0$	=	$- i p_{\rule{0ex}{1.5ex}\nabla}p_{\rule{0ex}{1.5ex}k}p_sp_\rho$	$\tfrac{1}{\sqrt{6}} \vec {j}\cdot\vec {\nabla}\times\vec {s}$	+1

Among many options of choosing the phase convention, in the present study we set

$$p_\rho =+1, ~~p_s=-i, ~~p_{\rule{0ex}{1.5ex}\nabla}=-i, ~~ ~p_{\rule{0ex}{1.5ex}k}=-i.$$ (92)

This choice is unique in the fact that all scalar densities and all terms in the EDF are then characterized by phase factors $+1$ connecting the spherical and Cartesian representations, see the last columns in Tables 21 and 22. This allows for the closest possible relationships between both representations, which may facilitate the use of the spherical representation as it is introduced in the present study. In particular, relations between coupling constants up to NLO (Table 22) and standard coupling constants in the Cartesian representation [24] then read, for terms depending on time-even densities:

$$\bm{C^{0000}_{00,0000}} = C^{\rho},$$ (93)

$$\bm{C^{0000}_{20,0000}} = \sqrt{3} \left(C^{\Delta\rho} + \tfrac{1}{4}C^{\tau}\right) ,$$ (94)

$$\bm{C^{0000}_{00,2000}} = \sqrt{3} C^{\tau} ,$$ (95)

$$\bm{C^{1110}_{00,1110}} = 3 C^{J0} ,$$ (96)

$$\bm{C^{1111}_{00,1111}} = \sqrt{12} C^{J1} ,$$ (97)

$$\bm{C^{1112}_{00,1112}} = \sqrt{5} C^{J2} ,$$ (98)

$$\bm{C^{0000}_{11,1111}} = \sqrt{6} C^{\nabla J} ,$$ (99)

and for terms depending on time-odd densities:

$${}{C^{0011}_{00,0011}} = \sqrt{3} C^{s} ,$$ (100)

$${}{C^{1101}_{00,1101}} = \sqrt{3} C^{j} ,$$ (101)

$${}{C^{0011}_{20,0011}} = 3 \left(C^{\Delta s} + \tfrac{1}{4} C^{T}\right) +\tfrac{1}{4}C^{F} - C^{\nabla s} ,$$ (102)

$${}{C^{0011}_{22,0011}} = \sqrt{5}\left(\tfrac{1}{4}C^{F} - C^{\nabla s}\right) ,$$ (103)

$${}{C^{0011}_{00,2011}} = 3C^{T} + C^{F} ,$$ (104)

$${}{C^{0011}_{00,2211}} = \sqrt{5} C^{F} ,$$ (105)

$$C^{1101}_{00,0011} = \sqrt{6} C^{\nabla j}.$$ (106)

At the same time, all vector densities in Table 21 and vector operators in Eqs. (89)-(91) are consistently characterized by phase factors $-i$ connecting the spherical and Cartesian representations.

Phase conventions (92) also lead to very simple phase properties, which our spherical tensors have with respect to complex conjugation. Indeed, spherical tensors (88)-(91) obey standard transformation rules under complex conjugation [26],

$$A^*_{\lambda\mu} = P_A(-1)^{\lambda-\mu}A_{\lambda,-\mu},$$ (107)

where $P_A=\pm1$. For nonlocal densities (88)-(89), Eq. (107) holds separately for their time-even and time-odd parts, split as in Eqs. (63) and (64). Using Eqs. (82) we then have

$$\begin{array}{rcl} P_{\rho_+}=+p_\rho ^2&,& P_{\rho_-}=-p_\rh... ...x}{1.5ex}\nabla}^2&,& P_k=p_{\rule{0ex}{1.5ex}k}^2, \end{array}$$ (108)

which for the phase convention of Eq. (92) reads

$$\begin{array}{rcl} P_{\rho_+}=+1&,& P_{\rho_-}=-1, \\ P_{s_+}=-1&,& P_{s_-}=+1, \\ P_\nabla=+1&,& P_k=-1. \end{array}$$ (109)

Standard rule (107) propagates through the angular momentum coupling, i.e., if signs $P_A$ and $P_{A'}$ characterize tensors $A_\lambda$ and $A'_{\lambda'}$, respectively, then the coupled tensor,

$$A''_{\lambda''\mu''} = [A_\lambda A'_{\lambda'}]_{\lambda''\mu''}... ...'} C^{\lambda''\mu''}_{\lambda\mu\lambda'\mu'} A_{\lambda\mu}A'_{\lambda'\mu'},$$ (110)

is characterized by the product of signs $P_{A''}= P_A P_{A'}$. Therefore, coupled higher-order densities (24) are characterized by signs,

$$P_{\rho_{mI,nLvJ,Q}} = P_\nabla^m P_k^n P_{vT},$$ (111)

where $v=0$ or 1 denotes the scalar or vector density, $\rho$ or $\vec {s}$, respectively, and $T=+1$ or $T=-1$ denotes the time-even or time-odd density. However, symmetry conditions (82) require that powers of the $k$ derivative determine the time-reversal symmetry of each local density, so that $T=(-1)^{n+v}$. From Eqs. (108) we then obtain

$$P_{\rho_{mI,nLvJ,Q}} = (-1)^{m+n+v}p_{\rule{0ex}{1.5ex}\nabla}^{2m} p_{\rule{0ex}{1.5ex}k}^{2n} p^2_v.$$ (112)

which for the phase convention of Eq. (92) reads

$$P_{\rho_{mI,nLvJ,Q}} = +1$$ (113)

for all densities. Therefore, the phase convention of Eq. (92) ensures that scalar densities and all terms in the EDF are always real.