Symmetries of local densities

Suppose that the Fock-space operator $\hat{\mathbf{U}}$ or $\hat{\cal{U}}$, belonging to D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$, respectively, represents a symmetry conserved by a mean-field many-particle state $\vert\Psi_+\rangle$ or $\vert\Psi_-\rangle$, in even or odd fermion spaces, i.e.,

   

$${\hat{\mathbf{U}}}\vert\Psi_+\rangle = u\vert\Psi_+\rangle,$$ (57)

$${\hat{\cal{U}}}\vert\Psi_-\rangle = u\vert\Psi_-\rangle.$$ (58)

As discussed in Sec. 2.5, eigenvalue u can be equal to $\pm1$ or $\pm{i}$, and moreover, in odd fermion systems $\hat{U}$cannot be equal to either $\hat{\cal{T}}$ or $\hat{\cal{P}}^T$, i.e, neither the time reversal nor the product of inversion and time reversal can be a conserved symmetry in odd systems. According to conventions introduced in Sec. 2.3, in odd systems the hat always denotes one of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators introduced in Sec. (2.1), and not one of their partners [Eqs. (33) and (34)]. Of course, if $\hat{\cal{U}}$ is a symmetry of $\vert\Psi_-\rangle$ then $\bar{\cal{U}}$ is a symmetry as well, so from the point of view of conserved symmetries, any extra study of partner operators is unnecessary.

 

Table 3: Symmetry properties of space-spin one-particle operators $\hat{O}$ belonging to different one-dimensional ircoreps of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ and D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ groups. The first column gives names of different ircoreps, the second column lists examples of operators $\hat{O}$, and the remaining columns give signs in the expression $\hat{U}^\dagger\hat{O}\hat{U}$=$\pm$$\hat{O}$, for operators $\hat{U}$given in the column headers. Note that these results do not depend on whether operators $\hat{U}$ act in even or odd spaces, and therefore the Fock-space notation is used for them.

$\mbox{Ircorep}$	$\mbox{Space-spin one-particle operators $\hat{O}$ }$	$\hat{T}$	$\hat{P}$	$\hat{R}_{x}$	$\hat{R}_{y}$
${\mbox{{invariant}s}}$	$x^2,y^2,z^2,\nabla_x^2,\nabla_y^2,\nabla_z^2; xi\nabla_y\hat{\sigma}_z, yi\nabla_z\hat{\sigma}_x, zi\nabla_x\hat{\sigma}_y$	+	+	+	+
${\mbox{{pseudo}{invariant}s}}$	$xyz, xi\nabla_yi\nabla_z, yi\nabla_zi\nabla_x, zi\nabla_xi\nabla_y; i\nabla_x\hat{\sigma}_x,i\nabla_y\hat{\sigma}_y,i\nabla_z\hat{\sigma}_z$	+	-	+	+
${\mbox{{anti}{invariant}s}}$	$xi\nabla_x,yi\nabla_y,zi\nabla_z; xy\hat{\sigma}_z, yz\hat{\sigma}_x, zx\hat{\sigma}_y$	-	+	+	+
${\mbox{{anti}{pseudo}{invariant}s}}$	$xyi\nabla_z,yzi\nabla_x, zxi\nabla_y; x\hat{\sigma}_x,y\hat{\sigma}_y,z\hat{\sigma}_z$	-	-	+	+
${\mbox{{{$x$ }-covariant}s}}$	$x;i\nabla_y\hat{\sigma}_z, i\nabla_z\hat{\sigma}_y$	+	-	+	-
${\mbox{{{$y$ }-covariant}s}}$	$y;i\nabla_x\hat{\sigma}_z, i\nabla_z\hat{\sigma}_x$	+	-	-	+
${\mbox{{{$z$ }-covariant}s}}$	$z;i\nabla_x\hat{\sigma}_y, i\nabla_y\hat{\sigma}_x$	+	-	-	-
${\mbox{{{$x$ }-{pseudo}covariant}s}}$	$yz;xi\nabla_y\hat{\sigma}_y, i\nabla_xy\hat{\sigma}_y, \hat{\sigma}_xyi\nabla_y$	+	+	+	-
${\mbox{{{$y$ }-{pseudo}covariant}s}}$	$xz;yi\nabla_z\hat{\sigma}_z, i\nabla_yz\hat{\sigma}_z, \hat{\sigma}_yzi\nabla_z$	+	+	-	+
${\mbox{{{$z$ }-{pseudo}covariant}s}}$	$xy;zi\nabla_x\hat{\sigma}_x, i\nabla_zx\hat{\sigma}_x, \hat{\sigma}_zxi\nabla_x$	+	+	-	-
${\mbox{{{$x$ }-{anti}covariant}s}}$	$i\nabla_x;y\hat{\sigma}_z,z\hat{\sigma}_y$	-	-	+	-
${\mbox{{{$y$ }-{anti}covariant}s}}$	$i\nabla_y;x\hat{\sigma}_z,z\hat{\sigma}_x$	-	-	-	+
${\mbox{{{$z$ }-{anti}covariant}s}}$	$i\nabla_z;x\hat{\sigma}_y,y\hat{\sigma}_x$	-	-	-	-
${\mbox{{{$x$ }-{anti}{pseudo}covariant}s}}$	$yi\nabla_z, zi\nabla_y;\hat{\sigma}_x$	-	+	+	-
${\mbox{{{$y$ }-{anti}{pseudo}covariant}s}}$	$xi\nabla_z, zi\nabla_x;\hat{\sigma}_y$	-	+	-	+
${\mbox{{{$z$ }-{anti}{pseudo}covariant}s}}$	$xi\nabla_y, yi\nabla_x;\hat{\sigma}_z$	-	+	-	-

Mean-field state $\vert\Psi\rangle$ can be characterized by the single-particle density matrix $\rho$ (see Ref. [1] for the definition), for which the symmetry properties (57) imply

$${\hat{U}}^\dagger\rho\hat{U}= \rho,$$ (59)

independently of eigenvalue u. (Symmetry property (59) does not depend on whether the mean-field state belongs to the even or odd fermion space, and therefore, we use the Fock-space notation $\hat{U}$for the symmetry operators, see definitions in Sec. 2.1 and discussion in Sec. 2.3.) It then follows that the single-particle self-consistent Hamiltonian $h[\rho]$ is also symmetric with respect to operator $\hat{U}$ [1], namely,

$${\hat{U}}^\dagger h[\rho]\hat{U}= h[\rho].$$ (60)

Eq. (60) implies that if $\varphi$ is a normalized single-particle eigenfunction of $h[\rho]$, then $\hat{U}\varphi$ is also a normalized eigenfunction, both belonging to the same eigenvalue. As a consequence, it can be shown [19] that the symmetry is preserved during the standard self-consistent iteration, provided the entire multiplets of states belonging to the same eigenvalue of $\hat{U}$ are either fully occupied, or fully empty. In such a case Eqs. (59) and (60) are fulfilled repeatedly in the successive steps of iteration, and $\hat{U}$ is a self-consistent symmetry.

Since the one-body density is a fermion-number conserving one-body operator, it can be classified according to one-dimensional ircoreps of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$, and this can be done both in even and odd systems. This means that either the given operator $\hat{U}$ is a conserved symmetry, Eqs. (57) and (59), and the density matrix belongs to the given one-dimensional ircorep of the subgroup generated by $\hat{U}$, or $\hat{U}$ is a broken symmetry and the density matrix has two non-zero components in two different such one-dimensional ircoreps. It follows that in odd systems the density matrix has always non-zero components in two ircoreps corresponding to the time reversal.

This classification procedure is used below to enumerate properties of the density matrix when one or more D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators are conserved symmetries. Note also, that unlike for the many-body states $\vert\Psi\rangle$, one does not have a freedom to change the phase of the density matrix, because it is a hermitian operator independent of the phase of the mean-field state it corresponds to. Therefore, if the density matrix has non-zero components in two ircoreps corresponding to two different eigenvalues of an antilinear D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operator, it cannot be transformed to the form in which it would have been either even or odd with respect to this operator.

A definite symmetry of the density matrix, Eq. (59), implies certain symmetries for local densities and their derivatives. These symmetries are discussed and enumerated in the present section.

The spin structure of the density matrix is given by

 

$$\rho(\mbox{{\boldmath {$r$ }}}\sigma,\mbox{{\boldmath {$r$ }}}'\sigma') {\textstyle{\frac{1}{2}}}\rho(\mbox{{\boldmath {$r$ }}},\mbox{{\boldmath {$r$ }}}')\delta_{\sigma\sigma'}$$ =

$${\textstyle{\frac{1}{2}}}\sum_{k=x,y,z}s_k(\mbox{{\boldmath {$r$ }}},\mbox{{\boldmath {$r$ }}}') <\sigma\vert{\hat{\sigma}}_k\vert\sigma'>,$$ + (61)

where $\mbox{{\boldmath {$r$ }}}$=(x,y,z) and $\mbox{{\boldmath {$r$ }}}'$= (x',y',z')represent three-dimensional position vectors. When the rotational symmetry is preserved one often refers to $\rho(\mbox{{\boldmath {$r$ }}},\mbox{{\boldmath {$r$ }}}')$ and $s_k(\mbox{{\boldmath {$r$ }}},\mbox{{\boldmath {$r$ }}}')$ as the scalar and vector densities, respectively. In our case, the rotational symmetry is broken, and we will avoid using these terms. Instead, we classify the densities according to the ircoreps of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group. As discussed above, for the one-body operators only the one-dimensional ircoreps are relevant for the classification. There are 16 characteristic transformation properties of the bases for one-dimensional ircoreps. In Table 3 we list all these ircoreps, illustrated by examples of space-spin operators of interest, e.g., powers of coordinates, x, y, z and gradients, $\nabla_x$, $\nabla_y$, $\nabla_z$.

The Table also lists explicitly the transformation properties of operators belonging to every type of symmetry. For example, the sign "-" which appears in row denoted by "y-covariants" and column denoted by $\hat{R}_{x}$ means that $\hat{R}_{x}^\dagger y\hat{R}_{x}$=-y. It can be easily checked that the Pauli matrices, $\hat{\sigma}_x$, $\hat{\sigma}_y$, $\hat{\sigma}_z$ transform under the signatures as the x, y, z coordinates, respectively, do not change under the inversion, and change their signs under the time reversal. Therefore, these can be classified as k-antipseudocovariants for k= x,y,z, respectively. Spin-dependent operators belonging to other ircoreps can also be constructed from the Pauli matrices. Therefore, examples of spin-dependent operators are also listed in the Table. In Table 3 we have introduced the same names for operators as for the bases of one-dimensional ircoreps (see Sec. 2.2).

Similarly as in Ref.[20], we consider the following local densities:

-

particle and spin densities

   

$$\rho(\mbox{{\boldmath {$r$ }}}) \rho(\mbox{{\boldmath {$r$ }}},\mbox{{\boldmath {$r$ }}}),$$ = (62)

$$s_k(\mbox{{\boldmath {$r$ }}}) s_k(\mbox{{\boldmath {$r$ }}},\mbox{{\boldmath {$r$ }}}),$$ = (63)

-

kinetic and spin-kinetic densities

   

$$\tau_{kl}(\mbox{{\boldmath {$r$ }}}) \left[\nabla_k{{\nabla}'}_l \rho(\mbox{{\boldmath {$r$ }}},\mbox{... ...dmath {$r$ }}}')\right]_{\mbox{{\boldmath {$r$ }}}=\mbox{{\boldmath {$r$ }}}'},$$ = (64)

$${T}_{klm}(\mbox{{\boldmath {$r$ }}}) \left[{\nabla}_k{{\nabla}'}_l s_m(\mbox{{\boldmath {$r$ }}},\mbox... ...dmath {$r$ }}}')\right]_{\mbox{{\boldmath {$r$ }}}=\mbox{{\boldmath {$r$ }}}'},$$ = (65)

-

current and spin-current densities

   

$${j}_k(\mbox{{\boldmath {$r$ }}}) \frac{1}{2i} \left[{\nabla}_k-{{\nabla}'}_k) \rho(\mbox{{\boldmat... ...dmath {$r$ }}}')\right]_{\mbox{{\boldmath {$r$ }}}=\mbox{{\boldmath {$r$ }}}'},$$ = (66)

$$J_{kl}(\mbox{{\boldmath {$r$ }}}) \frac{1}{2i} \left[(\nabla_k-\nabla'_k) s_{l}(\mbox{{\boldmath {$... ...dmath {$r$ }}}')\right]_{\mbox{{\boldmath {$r$ }}}=\mbox{{\boldmath {$r$ }}}'},$$ = (67)

where each index k, l, or m may refer to either of x, y, or z. It follows from the hermiticity of the density matrix $\rho$ that all the above local densities are real functions of $\mbox{{\boldmath {$r$ }}}$. Usually only the traces of kinetic densities,

 

$$\tau(\mbox{{\boldmath {$r$ }}}) \sum_k \tau_{kk}(\mbox{{\boldmath {$r$ }}}),$$ = (68)

$${T}_{m}(\mbox{{\boldmath {$r$ }}}) \sum_k {T}_{kkm}(\mbox{{\boldmath {$r$ }}}),$$ = (69)

are used in applications.

When operator $\hat{U}$ represents a conserved symmetry of the density matrix, Eq. (59), the transformation rules for gradients and spin operators, given in Table 3, imply definite transformation rules for the local densities. These are listed in Table 4, for all the one-dimensional ircoreps of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ as indicated in the first column. In the second column we show the local densities in forms defined by Eqs. (62)-(66), while the third column gives, when possible, the local densities in the traditional vector-tensor notation, e.g.,

 

$$\mbox{{\boldmath {$s$ }}}$$ = (s_x,s_y,s_z), (70)

$$\mbox{{\boldmath {$T$ }}}$$ = (T_x,T_y,T_z), (71)

 

$$\sum_k J_{kk},$$ J = (72)

$$(\stackrel{\leftrightarrow}{J})_{kl} \frac{1}{2}(J_{kl}+J_{lk})-\frac{1}{3}J\delta_{kl},$$ = (73)

$$(\mbox{{\boldmath {$J$ }}})_k \sum_{lm}\varepsilon_{klm}J_{lm}.$$ = (74)

Derivatives of densities up to the second order are also included in the Table.

From Table 4 one can read off the symmetry properties of various densities. Suppose d(x,y,z) is a generic name of one of the densities listed in the second or third column, and $\hat{U}$ is a generic name of one of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators listed in the first row. We use the convention that index i may take any value among x, y or z, while indices $k\neq l\neq m$ are arbitrary permutations of x, y, and z. If $\hat{U}$ represents a conserved symmetry, one has the following symmetry rule for the density d(x,y,z):

$$d(\epsilon_xx,\epsilon_yy,\epsilon_zz)=\epsilon d(x,y,z),$$ (75)

where $\epsilon$ is the sign listed in Table 4 in the row denoted by d and column denoted by $\hat{U}$. Signs $(\epsilon_x,\epsilon_y,\epsilon_z)$ are given in the last row of Table 3, and pertain to two D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators (viz. $\hat{U}$ and $\hat{U}^T$) in two adjacent columns. These latter signs give changes of coordinates (x,y,z) under the action of $\hat{U}$. As the time reversal does not affect spatial coordinates, these signs are the same for any pair of operators $\hat{U}$ and $\hat{U}^T$. One generic Table of signs determines, therefore, symmetry properties of any local density for any of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ symmetries being preserved.

For example, symmetry properties of density J_xy can be found by using indices l=x and m=y (which requires k=z) in the row pertaining to k-covariants. For the conserved $\hat{R}_{z}$= $\hat{R}_{k}$ symmetry we then find in the corresponding column $\epsilon$=+ and $\epsilon_x$= $\epsilon_l$=-, $\epsilon_y$= $\epsilon_m$=-, and $\epsilon_z$= $\epsilon_k$=+, which gives J_xy(-x,-y,z)= J_xy(x,y,z).

 

Table 4: Symmetry properties of various local densities belonging to different one-dimensional ircoreps of the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ groups. For instructions on using the Table see the text explaining Eq. (75).

$\mbox{Ircorep}$

Local densities

$\hat{P}$

$\hat{P}^T$

$\hat{R}_{k}$

$\hat{R}^T_{k}$

$\hat{S}_{k}$

$\hat{S}^T_{k}$

$\hat{R}_{l}$

$\hat{R}^T_{l}$

$\hat{S}_{l}$

$\hat{S}^T_{l}$

$\hat{R}_{m}$

$\hat{R}^T_{m}$

$\hat{S}_{m}$

$\hat{S}^T_{m}$

${\mbox{{invariant}s}}$

$\rho,\tau_{ii},\nabla^2_i\rho,\nabla_kJ_{lm}$

$\rho,\tau,\Delta\rho,\mbox{{\boldmath {$\nabla$ }}}\cdot\mbox{{\boldmath {$J$ }}}$

+

+

+

+

+

+

+

+

+

+

+

+

+

+

${\mbox{{pseudo}{invariant}s}}$

Jii

J

-

-

+

+

-

-

+

+

-

-

+

+

-

-

${\mbox{{anti}{invariant}s}}$

$T_{klm}, \nabla_ij_i, \nabla_k\nabla_ls_m$

$\mbox{{\boldmath {$\nabla$ }}}\cdot\mbox{{\boldmath {$j$ }}}$

+

-

+

-

+

-

+

-

+

-

+

-

+

-

${\mbox{{anti}{pseudo}{invariant}s}}$

$\nabla_is_i$

$\mbox{{\boldmath {$\nabla$ }}}\cdot\mbox{{\boldmath {$s$ }}}$

-

+

+

-

-

+

+

-

-

+

+

-

-

+

${\mbox{{{$k$ }-covariant}s}}$

$J_{lm},\nabla_k\rho$

$(\mbox{{\boldmath {$J$ }}})_k,(\stackrel{\leftrightarrow}{J})_{lm}, (\mbox{{\boldmath {$\nabla$ }}}\rho)_k$

-

-

+

+

-

-

-

-

+

+

-

-

+

+

${\mbox{{{$k$ }-{pseudo}covariant}s}}$

$\tau_{lm},\nabla_l\nabla_m\rho,\nabla_kJ_{ii},$

$(\mbox{{\boldmath {$\nabla$ }}}J)_k,$

+

+

+

+

+

+

-

-

-

-

-

-

-

-

$\nabla_iJ_{ki},\nabla_iJ_{ik}$

$(\mbox{{\boldmath {$\nabla$ }}}\cdot \stackrel{\leftrightarrow}{J})_k, (\mbox{{\boldmath {$\nabla$ }}}\times \mbox{{\boldmath {$J$ }}})_k$

${\mbox{{{$k$ }-{anti}covariant}s}}$

$j_k,\nabla_ls_m$

$(\mbox{{\boldmath {$j$ }}})_k, (\mbox{{\boldmath {$\nabla$ }}}$$\times$ $\mbox{{\boldmath {$s$ }}})_k$

-

+

+

-

-

+

-

+

+

-

-

+

+

-

${\mbox{{{$k$ }-{anti}{pseudo}covariant}s}}$

$s_k,T_{iik},T_{kii},\nabla_lj_m,$

$(\mbox{{\boldmath {$s$ }}})_k,(\mbox{{\boldmath {$T$ }}})_{k}, (\mbox{{\boldmath {$\nabla$ }}}$$\times$ $\mbox{{\boldmath {$j$ }}})_k%$

+

-

+

-

+

-

-

+

-

+

-

+

-

+

$\nabla_k\nabla_is_i,\nabla_i^2s_k$

$(\mbox{{\boldmath {$\nabla$ }}}(\mbox{{\boldmath {$\nabla$ }}}$$\cdot$ $\mbox{{\boldmath {$s$ }}}))_k, (\Delta \mbox{{\boldmath {$s$ }}})_k$

$(\epsilon_k,\epsilon_l,\epsilon_m)=$

(--)

(+-)

(-++)

(-+-)

(+-+)

(-+)

(++-)

It is worth noting that symmetry properties (75) which correspond to various D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators, are related to one another only by the corresponding group multiplication rules. Therefore, a specific choice of the conserved generators, either for the complete D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ groups or for any of their subgroups [14], leads to a specific set of symmetry properties of local densities.

Symmetry properties (75) can be used for the purpose of a continuation of densities from one semi-space into the second semi-space, i.e., one can use only space points for, e.g., x$\geq$0. For two symmetry properties (75), coming from two different symmetry operators (but not from the pair $\hat{U}$and $\hat{U}^T$), one can restrict the space to a quarter-space, where two coordinates have definite signs, e.g., x$\geq$0 and y$\geq$0. Finally, three conserved symmetries allow for a restriction to one eighth of the full space with all the coordinates having definite signs, e.g., x$\geq$0, y$\geq$0, and y$\geq$0. The time-reversal symmetry does not lead to restrictions on the space properties of densities, but, when conserved, gives the vanishing of all the antiinvariant, antipseudoinvariant, anticovariant and antipseudocovariant densities, viz., s_k, j_k, T_klm for arbitrary k,l,mas well as their derivatives (see Table 4). The possibilities of simultaneously conserving one, two, three, or four symmetry operators from the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ or D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ groups will be discussed in Ref.[14].

Since density matrix $\rho$ and single-particle Hamiltonian $h[\rho]$ are always simultaneously invariant under any conserved symmetry $\hat{U}$, Eqs. (59) and (60), the discussion above can be repeated for self-consistent local fields appearing in a local mean-field Hamiltonian. Explicit formulas for symmetry properties of local fields are identical to those listed in Table 4, and will not be repeated here. In applications, these symmetries appear automatically when the self-consistent mean fields are calculated in terms of densities, cf. Ref [20].