# Closed Shell States in Periodic Compounds

Milan Damnjanović<EMAIL_ADDRESS>NanoLab, Faculty of Physics, Uni
Belgrade, Studentski trg 12, 11000 Belgrade, Serbia  SANU, Kneza Mihaila 36,
11000 Belgrade, Serbia  Ivanka Milošević  NanoLab, Faculty of Physics, Uni
Belgrade, Studentski trg 12, 11000 Belgrade, Serbia

###### Abstract

Vanishing of the total angular momentum of the electrons occupying all
orbitals of a closed shell in an atom is a textbook fact. Understanding the
symmetry content of the atomic shell as irreducible representation of angular
momentum, enables straightforward transfer of the notion to (translational or
helically) periodic systems. More relevant generalizations naturally appear:
stratum shell is intermediate step to physically sound band representations,
including elementary and basic ones and connected components. We show that
nontrivial determinant representation indicates stable topology of band in
single colorless layer groups and obstructive limit in in single colorless
line groups.

##  I Introduction

Notion of the closed shell originates in atomic physics, where in the context
of angular momentum (orbital, spin, total) it describes a set of single
particle states of a multiplet
$\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{k,m}~{}\\!\\!\rangle$ ($m=k,\dots,-k$) of the
irreducible representation (IR) with weight $k$ of the angular momentum.
Angular momentum of the electrons completely occupying such a multiplet (a
closed shell) vanishes, and the total electronic angular momentum is
effectively contributed only by the populated states from the opened (not
fulfilled) multiplets. In fact, this means that the total electronic state,
the Slater determinant $\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{\Psi}~{}\\!\\!\rangle$
of a closed shell is invariant under rotations. Clearly, if there are several
fulfilled multiplets, they altogether have the same property, forming a closed
shell in a bit broadened sense.

Surprisingly, there is no analogous notion for the crystalline systems;
instead, as J. Birman [1] formulated ”crystal is normally considered to be in
the $\Gamma^{1+}$ electronic state (single Slater determinant with all
orbitals doubly occupied)”, it is assumed that the total electronic state is
invariant under the symmetry group. Here presented study, based on the common
properties of the periodic (space, layer and line) groups reveals the
complexity as well as the importance of this question.

Within single particle (tight-binding, mean-field) approximations (e.g.
Hartree-Fock, density functional) electron hamiltonian $H$ and a commuting
with it representation $D({\bm{G}})$ of the symmetry group ${\bm{G}}$ are
defined in the (single-particle) state space ${\cal S}$. The representation is
reducible, and decomposes into the irreducible components
$D^{(\mu)}({\bm{G}})$ with frequencies $f^{\mu}_{D}$:

$D({\bm{G}})=\sum_{\mu}f^{\mu}_{D}D^{(\mu)}({\bm{G}}).$ (1)

Each eigenergy $E_{{\mu t}}$ corresponds to the multiplet
$\\{\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{{\mu
tm}}~{}\\!\\!\rangle~{}|~{}m=1,\dots,|\mu|\\}$ such that
$H\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{{\mu tm}}~{}\\!\\!\rangle=E_{{\mu t}}$. Each
multiplet spans an irreducible subspace ${\cal S}^{(\mu)}_{t}$, and for all
$\mu$ and $t=1,\dots f^{\mu}$ these vectors form a stationary symmetry adapted
basis of ${\cal S}$.

A system of $N$ electrons occupies $N$ different single-particle states of
${\cal S}$ due to Pauli principle. This imposes a unique decomposition ${\cal
S}={\cal C}\oplus{\cal O}\oplus{\cal U}$ onto the maximal closed (fully
occupied; $|{\cal C}|\leq N$) ${\cal C}$, maximal unoccupied (empty) ${\cal
U}$ and the complemental open (partly occupied) ${\cal O}$ subspaces invariant
under $D({\bm{G}})$. This invariance implies that $D({\bm{G}})$ is reduced to
the corresponding subrepresentations in these subspaces:
$D({\bm{G}})=C({\bm{G}})+O(G)+U({\bm{G}})$. An $N$-electron state in a single-
particle model is a Slater determinant of the filled states. Such $N$-particle
Slater-determinants span the total state space, the antisymmetrized $N$-th
tensor power ${\cal S}^{N}_{-}$ of ${\cal S}$. The group acts in ${\cal
S}^{N}_{-}$ as the antisymmetrized power $[D^{N}]({\bm{G}})$ of $D({\bm{G}})$.
The effective single particle state ${\cal S}_{\mathrm{eff}}={\cal
C}\oplus{\cal O}$ yields the effective $N$-particle subspace ${\cal
S}^{N}_{\mathrm{eff}}=\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{1,\dots,|{\cal
C}|}~{}\\!\\!\rangle\otimes{\cal O}^{N-|{\cal C}|}_{-}<{\cal S}^{N}_{-}$
spanned by the states matching the decomposition. It is straightforward to
show that in this subspace the group is represented by

$D_{\mathrm{eff}}=\det{C}\ \,[O^{N-|{\cal C}|}].$ (2)

Note that the required invariance means that ${\cal C}$, ${\cal O}$ and ${\cal
U}$ contain only the whole multiplets, which prevents ${\cal C}$ to be simply
the span over the occupied states, with the complement ${\cal U}$ (thus with
no ${\cal O}$).

It is well established that the ground (many-electron) state to a great extent
determines properties of the condensed matter. Within the single particle
prescription it emerges by filing the lowest energy $N$ eigenstates; the
highest occupied one defines the Fermi level $E_{\mathrm{F}}$. Possible doping
changes the available number of electrons (and $E_{\mathrm{F}}$), but once it
is fixed ${\cal C}$ and ${\cal O}$ are well defined. Therefore, its
characterisation in terms of symmetry given by (2) has the role of the total
angular momentum in atomic or molecular physics.

Recall that a state $\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{\Psi}~{}\\!\\!\rangle$ of
$N$ electrons is from the antisymmetrized tensor power ${\cal S}^{N}_{-}$ of
the single electron state space ${\cal S}$ (infinite dimensional, but in
molecular physics sometimes modeled by a finite dimensional subspace). If
these electrons occupy states
$\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{\psi_{i}}~{}\\!\\!\rangle$ ($i=1,\dots,N$)
from ${\cal S}$, then the total state is the Slater determinant expressed in
terms of the permutations $\pi$ (with parity $\tilde{\pi}{}$) from the
symmetric group $S_{N}$, represented by the operators $\Delta(\pi)$ gathered
in the atisymmetrization operator $A$:

$\displaystyle\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{\Psi}~{}\\!\\!\rangle=\sqrt{N!}A\left(\operatorname*{\otimes}\limits^{N}_{i=1}\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{\psi_{i}}~{}\\!\\!\rangle\right),$
$\displaystyle A=\tfrac{1}{N!}\sum_{\pi\in
S_{N}}(-1)^{\tilde{\pi}}\Delta(\pi),\ \ \ \ \ $
$\displaystyle\Delta(\pi)\left(\operatorname*{\otimes}\limits^{N}_{i=1}\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{\psi_{i}}~{}\\!\\!\rangle\right)$
$\displaystyle\buildrel\rm def\over{=}$
$\displaystyle\left(\operatorname*{\otimes}\limits^{N}_{i=1}\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{\psi_{\pi^{-1}i}}~{}\\!\\!\rangle\right).$

##  II Determinant representations of crystal

Elements of a $\wp$-periodic (line, layer and space for $\wp=1,2,3$) group
${\bm{G}}$ are $g=(\,G\,|\,{\bm{g}}\,)$ (Koster’s notation): $G$ is element of
isogonal group ${\bm{P}}<O(3)$, while ${\bm{g}}$ is from the space
$\mathds{R}^{\wp}$ containing lattice spanned by translational (or helical for
$\wp=1$) invariant abelian subgroup ${\bm{T}}$, such that
${\bm{P}}={\bm{G}}/{\bm{T}}$.

Brillouin’s zone $\mho$ is torus $T^{\wp}$ (elementary cell of the inverse
lattice $\overline{{\bm{K}}}{}$), with points ${\bm{k}}$ (wave vectors)
parameterizing IRs of ${\bm{T}}$. Representatives of the orbits (stars
${\bm{k}}^{*}$) of the group in Brillouin’s zone form the irreducible domain
(ID) $\mho_{\bm{G}}$. It is partitioned into strata $\mho^{K}_{\bm{G}}$
gathering equivalent representatives (with conjugated, thus isomorphic,
stabilizers ${\bm{F}}^{\bm{k}}\cong{\bm{F}}^{K}$). Each stratum is a manifold,
with boundary (if not empty) consisting of strata with larger stabilizers. In
particular, generic stratum occupies interior of ID,
$\mho^{\mathrm{in}}_{{\bm{G}}}$, while the other, _special_ strata (planes,
lines, points) are on its boundary. Irreducible representations of ${\bm{G}}$
(Appendix B.1) associated to ${\bm{k}}$ are induced from the _allowed_ IRs of
the stabilizer,
$D^{(K{\bm{k}}\kappa)}({\bm{G}})=d^{(K{\bm{k}}\kappa)}({\bm{F}}^{K})\uparrow{\bm{G}}$;
accordingly, $\mu=(K,{\bm{k}},\kappa)$, where $\kappa$ counts
${\bm{k}}$-independent (projective) representations
$d^{(K\kappa)}({\bm{F}}^{K}/{\bm{T}})$ of the stratum little co-group; the
dimension of $D^{(\mu)}$ is $|\mu|=|{\bm{k}}^{*}||\kappa|$.

Each determinant representation is one dimensional; besides irreducibility,
this imposes two conditions to the associated ${\bm{k}}$ point: (a) its
stabilizer is the whole group (as $|{\bm{k}}^{*}|=1$), and (b) it is
simultaneously the allowed representation for which the factor system 111DO
SADA ZAOBIDJENE PROJRKTIVNE; ako mora pomenuti ranije; ovde navesti referencu,
zasto kod 1D nema netriv fac syst  (depending on ${\bm{k}}$ only) is (gauge)
equivalent to the trivial (not projective) one. With $g=(\,G\,|\,{\bm{g}}\,)$
and $g^{\prime}=(\,G^{\prime}\,|\,{\bm{g}}^{\prime}\,)$ these conditions are:

$\displaystyle
G\bm{\gamma}=\bm{\gamma}+{\bm{K}}^{\bm{\gamma}}_{G},\quad\forall
G\in{\bm{P}}\quad({\bm{K}}^{\bm{\gamma}}_{G}\in\overline{{\bm{K}}}{}),$ (4a)
$\displaystyle\mathrm{e}^{\mathrm{i}\bm{\gamma}\cdot(G{\bm{g}}^{\prime}-{\bm{g}}^{\prime})}=\mathrm{e}^{\mathrm{i}{\bm{K}}^{\bm{\gamma}}_{G}\cdot{\bm{g}}^{\prime}}=1,\quad\forall
g,g^{\prime}\in{\bm{G}}.$ (4b)

Such, to be called $\bm{\gamma}$ points, form set $\mho^{0}_{\bm{G}}$. Hence,
general form of determinant representation is:

$\overline{D}{}(g)\buildrel\rm
def\over{=}\det{D(g)}=\mathrm{e}^{\mathrm{i}\bm{\gamma}\cdot{\bm{g}}}d^{(\phi)}(G),\quad\bm{\gamma}\in\mho^{0}_{\bm{G}},$
(5)

where $d^{(\phi)}$ is one-dimensional representation of the isogonal group.
Clearly, in all groups one $\gamma$ point is $\Gamma$-point (${\bm{k}}=0$).
For abelian group ${\bm{G}}$ (pure translational, some layer and the first
family line groups) all IRs are one-dimensional, and each ${\bm{k}}$ is
$\bm{\gamma}$-point: $\mho^{0}_{\bm{G}}=\mho_{\bm{G}}=\mho$. Otherwise,
$\mho^{0}_{\bm{G}}$ is a proper lower dimensional subset of $\mho_{\bm{G}}$.
For symmorphic groups, (4b) is automatically fulfilled for arbitrary fixed
point (as then ${\bm{g}}$ is a lattice vector). Also, each fixed point within
ID interior is a $\bm{\gamma}$ point, as then ${\bm{K}}^{\bm{\gamma}}_{G}=0$.
Thus, only for the fixed points on the boundary of ID of nonsymmorphic groups
(4b) is an effective restriction, which involves the whole group ${\bm{G}}$,
and not only ${\bm{P}}$. The form (5) will be used for the forthcoming cases
(details on derivation are in Appendices).

##  III Crystal shells

Archetypic closed shell is that of electrons filling a multiplet of the
angular momentum ${\bm{k}}$ (one of ${\bm{l}}$, ${\bm{s}}$ or ${\bm{j}}$) in
${\cal S}$. For a fixed value $k$ of ${\bm{k}}$, states
$\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{\psi_{i}}~{}\\!\\!\rangle$ are standard basis
$\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{k,m}~{}\\!\\!\rangle$ ($m=-k,\dots,k$). For
these $N=2k+1$ electrons, $K=0$ is the value of total angular momentum[3]
${\bm{K}}=\sum_{i}{\bm{k}}_{i}$; common state,
$\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{\Psi}~{}\\!\\!\rangle=\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{K=0,M=0}~{}\\!\\!\rangle$,
is invariant under rotations. Indeed, arbitrary rotation $U$ acts on the
states $\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{k,m}~{}\\!\\!\rangle$ by matrix
$D^{(k)}(U)$ of the IR. But in ${\cal S}^{N}_{-}$ it acts as
$D(U)A=A\otimes^{k}_{m=-k}D^{(k)}(U)$, and
$D(U)\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{\Psi}~{}\\!\\!\rangle=\det{D^{(k)}(U)}\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{\Psi}~{}\\!\\!\rangle=\\!\mid~{}\\!\\!\\!\\!\\!\\!~{}{\Psi}~{}\\!\\!\rangle$,
since determinants of IRs of SU(2) are 1. Consequently, the total angular
momentum is equal to the angular momentum of the electrons outside the closed
shells.

Strict analogy with angular momentum establishes crystal shell as a multiplet
of an IR of ${\bm{G}}$. The permutational-block-form of its matrices (Appendix
B.2) yields sum ${\bm{k}}^{o}=\sum^{|{\bm{k}}^{*}|}_{p=1}{\bm{k}}_{p}$ of the
star vectors (phases of the blocks) in their determinants’ form (5)):

$\overline{D}{}^{(K{\bm{k}}\kappa)}(\,G\,|\,{\bm{g}}\,)=\mathrm{e}^{\mathrm{i}\bm{\gamma}\cdot{\bm{g}}}d^{(\phi_{\kappa})}(G),\quad\bm{\gamma}=|\kappa|{\bm{k}}^{o}.$
(6)

This representation, depending on $\kappa$ and $K$ (but not on ${\bm{k}}$) is
not necessarily totally symmetric.

Some generalizations of crystal shell are natural for band theory; namely,
then the state space carries a _band representation_ [4, 5, 6, 7] $D$, with
${\bm{k}}$-independent, constant along the associated strata, frequencies of
the irreducible components $f^{(K{\bm{k}}\kappa)}_{D}=f^{(K\kappa)}_{D}$:

$D({\bm{G}})=\sum\nolimits_{K\kappa}f^{(K\kappa)}_{D}\sum\nolimits_{{\bm{k}}\in\mho^{K}_{\bm{G}}}D^{(K{\bm{k}}\kappa)}({\bm{G}});$
(7)

actually, this property structures the spectrum of Hamiltonian in the form of
bands consisted of _band-patches_ (parts over strata, having constant
degeneracy) $E^{t}_{K\kappa}({\bm{k}})$ (with $t=1,\dots,f^{(K\kappa)}_{D}$).
Each band-patch defines _stratum shell_ as the subspace assigned by _stratum
representation_
$D^{[K\kappa]}=\sum_{{\bm{k}}\in\mho^{K}_{\bm{G}}}D^{(K{\bm{k}}\kappa)}$.
Except for the point strata, or when $D^{(K{\bm{k}}\kappa)}$ is unit
representation, the corresponding determinant representation is mathematically
ill defined continual product of determinants (6) over the stratum. Still,
coincident results of the functional determinants method222Functional
determinant method for an infinite dimensional operator $A$ uses
$\det{A}=\mathrm{e}^{-\zeta^{\prime}_{A}(0)}$, with the Riemann function
$\zeta_{A}(x)=\mathrm{Tr}\,{A^{-x}}$. In the considered case
$A=\sum_{\bm{k}}\mathrm{e}^{\mathrm{i}(|\kappa|{\bm{k}}^{o}\cdot{\bm{g}})}d^{(\beta)}(G)$
is diagonal infinite-dimensional direct sum (i.e. integral) of the
determinants (6) over the domain $\mho^{K}_{\bm{G}}$. This gives (8), because
$\zeta_{A}(x)=\int_{\mho^{K}_{\bm{G}}}\mathrm{e}^{-\mathrm{i}(|\kappa|{\bm{k}}^{o}\cdot{\bm{g}})x}d^{(\beta)^{-x}}(G)\mathrm{d}{\bm{k}}$.
with the cellular decomposition starting with intuitive form

$\overline{D}{}^{[K\kappa]}(g)=\mathrm{e}^{\mathrm{i}\bm{\gamma}^{K}_{|\kappa|}.{\bm{g}}}d^{(\Phi^{K}_{\kappa})}(G),\quad\bm{\gamma}_{K}=|\kappa|\int_{\mho^{K}_{\bm{G}}}{\bm{k}}^{o}\mathrm{d}{\bm{k}},$

may justify both heuristics. Namely, $\phi_{\kappa}$ from (6) is fixed along
the stratum of integer measure $|\mho^{K}_{\bm{G}}|$ (”number of points”), and
$d^{(\Phi^{K}_{\kappa})}(G)=d^{(\phi_{\kappa})^{|\mho^{K}_{\bm{G}}|}}(G)$ is
continual power for a stratum of nonzero dimension. Further,
$d^{(\phi_{\kappa})}(G)$ is a root of 1 (represents an element of a finite
group ${\bm{P}}$); hence, $d^{(\phi_{\kappa})}(G)=\mathrm{e}^{2\mathrm{i}\pi
z/Z}$ for some (depending on $G$ and $\kappa$, but not on ${\bm{k}}$) integers
$Z$ (a divisor of the order of $G$) and $z\in[0,Z)$, giving
$d^{(\Phi^{K}_{\kappa})}(G)=\mathrm{e}^{2\mathrm{i}\pi
z|\mho^{K}_{\bm{G}}|/Z}$. If $A$ is (topologically open) interior of
$\mho^{K}_{\bm{G}}$, it can be partitioned into $Z$ mutually equal open
subdomains $A_{i}$ ($i=1,\dots,Z$) and boundaries between them. The
contributions of the subdomains cancel and
$d^{(\Phi^{K}_{\kappa})}(G)=\mathrm{e}^{2\mathrm{i}\pi z|B^{K}_{1}|/Z}$, where
union of all boundaries
$B^{K}_{1}=\mho^{K}_{\bm{G}}\setminus(\bigcup^{Z}_{i}A_{i})$ is of the lower
(then stratum) dimension $b^{K}_{1}$. This can be calculated if $B^{K}_{1}$ is
empty or finite set ($b^{K}_{1}=0$). Otherwise, the procedure is iterated
(partition $B^{K}_{1}$ into $Z$ open subdomains gives again lower dimensional
boundary $B^{K}_{2}$, etc.) until $n$ such that $b^{K}_{n}=0$ (examples in
Fig. 1)

$\overline{D}{}^{[K\kappa]}(g)=\mathrm{e}^{\mathrm{i}\bm{\gamma}_{K}\cdot{\bm{g}}}\mathrm{e}^{\tfrac{2\mathrm{i}\pi
z|B^{K}_{n}|}{Z}},\quad\bm{\gamma}_{K}=|\kappa|\int_{\mho^{K}_{\bm{G}}}{\bm{k}}^{o}_{K}\mathrm{d}{\bm{k}};$
(8)

recall that $z$ and $Z$ are $G$\- and $\kappa$-dependent333Actually, only
$|B^{K}_{n}|$ modulo $Z$ is relevant, and finer partitions do not change the
result (e.g. simultaneous partition of all $A_{i}$ into the same number of
open subsets enlarges $B^{K}_{1}$), meaning that common partition for all $G$
and $\kappa$ can be found, as a characteristic of stratum. For example, $Z$
can be taken as the least common multiple of orders of all elements of $P$..

A band structure comprises of the glued band patches; corresponding
$D^{[K\kappa]}$ are summed in the _band representation_ $D({\bm{G}})$. Its
filled carrier space is final generalized notion, _band shell_. Thus, (7) with
(8) gives:

$\overline{D}{}(g)=\mathrm{e}^{\mathrm{i}\bm{\gamma}_{D}\cdot{\bm{g}}}\prod_{K\kappa}d^{(\beta^{K}_{\kappa})^{f^{K\kappa}_{D}}}(G),\
\bm{\gamma}_{D}=\sum_{K\kappa}f^{K\kappa}_{D}\bm{\gamma}_{K}.$ (9)

Note that this notion can be applied for connected structures of bands
(otherwise open shell appearS); each insulating band structure is a
combination of band shells. Indeed, there is a set of _basic band
representations_ (BBRs) $B_{i}$ sufficient to decompose (not always in a
unique way) arbitrary band representation as $D=\sum_{b}c_{b}B_{b}$.
Simultaneously, the determinant representation of $D$ is factorized into
determinant representations of BBRs (DBBRs):
$\overline{D}{}=\otimes_{b}\overline{B}{}^{c_{b}}_{b}$, which shows how DBBRs
determine all band-shell representations.

##  IV Discussion

Symmetry enables to develop notion of the crystal closed shell hierarchically,
from the spaces of IRs (like the atomic shell of angular momentum), over the
strata shell to the connected components shells and basic band representations
shells. Each of them is a determinant representation, an IR associated to one
of the so called $\bm{\gamma}$-points in BZ, with trivial factor system and
fixed by the whole group (obviously including $\Gamma$-point); (6) shows that
the ray through the center of the star ${\bm{k}}^{*}$ contains $\gamma$-points
of determinant representations of IRS associated to ${\bm{k}}$. This result
enable easy derivation of the higher band structure shells.

Determinant representations and closed shells appear in the context of Pauli
antisymmetrization for fermions. This invokes half-integer representations (of
double groups), although the determinant representation of even- or infinite-
dimensional shell is an integer IR. However, shells of single groups and
determinant representations are useful in the analysis of influence of spin-
orbit coupling, since the total representation is tensor product of the
orbital shell with (two-dimensional) natural spin representation[10]) of unit
determinant. Consequently (see (14)), the orbital determinants (e.g. of BBRs)
are simply squared in total space. Besides these products, half-integer
representations allow other band representations, which are related to spin-
orbit interaction. We have calculated shells (of all types) determinants of
line and layer single and double groups.

Notice that the determinant representation is not a priori unit, as it is
”normally considered” [1]. Actually, situation is quite interesting, and
appears to be related to topology. To remind, there are two phenomena
originating in topology.

Symmetry indicators[11] and topological quantum chemistry[12] are based on the
band representations (clearly, these generate all semiconducting shells). To
remind, _elementary band representations_ , $E=\\{E_{1},\dots,E_{|E|}\\}$, are
tabulated for space[12], layer[13] and line[14] groups. Within Grothendieck’s
extension to representation ring [11, 15], they form abelian group ${\bm{A}}$
of _atomic insulators_ (with localized electrons), which is (normal) subgroup
in the _band structures group_ ${\bm{B}}$, of the general band representations
(satisfying only compatibility and degeneracy conditions). The factor group of
_symmetry indicators_
$\bm{SI}={\bm{B}}/{\bm{A}}=\mathds{Z}_{1}\otimes\cdots\otimes\mathds{Z}_{C}$
is finite, with $C$ cyclic generators, and set of a _stable_ band
representation $S=\\{S_{1},\dots,S_{C}\\}$, each corresponding to a $\bm{SI}$
generator, complementing EBRs to the set of _basic_ band representations
(BBRs). The elements of ${\bm{B}}$ are combinations
$D=\sum_{e}f^{D}_{e}E_{e}+\sum_{s}f^{D}_{s}S_{s}$, and band representations
are those with non-negative IRs’ frequencies $f^{K\kappa}_{D}$. Within
$K$-theory, _trivial_ bands are non-negative combinations of EBRs
($f^{D}_{e}\geq 0,f^{D}_{s}=0$), _fragile topological bands_ are expandable in
EBRs ($f^{D}_{s}=0$) but with some negative EBRs’ frequencies, while _stable
topological_ bands necessarily contain stable components. For a given
representation there may exist (topologically) different (e.g. in
connectivity) band structures, which is related to Hamiltonian (dynamical
details).

Another topological aspect is _obstructed atomic limit_ (OAL). In fact, in the
usual models state space is inductive, with some orbitals associated to
different atoms (orbits) sitting in Wyckoff’s positions $x$. Corresponding
orbitals span the spaces invariant under representation $d^{x}({\bm{F}}_{x})$
of the stabilizer ${\bm{F}}_{x}$. Summing the induced representations
$d^{x}({\bm{F}}_{x})\uparrow{\bm{G}}$ over the orbit representatives, a BR $D$
is obtained, and it can be expanded over EBRs. It is expected that the
electrons are localized (Wannier centers) in the atoms. However, it may appear
that this expansion is not unique, in which case the localization may be
different (this is related to the change of Hamiltonian, i.e. to the change of
phase). So, various expansions in EBRs of a BR is an indication of a possible
OAL.

Namely, in single (colorless and grey) line groups, nonunit determinants
appear exactly in the groups enabling OALs. In the layer groups trivial and
fragile BRs are with unit determinants. In single groups nonunit shell
corresponds to stable connected topological band; closed shell with several
stable closed shells may be unit. However, in double layer colorless groups,
this is true only when $\bm{SI}({\bm{G}})=\bm{SI}(\tilde{{\bm{G}}}{})$
($\tilde{{\bm{G}}}{}$ denotes double group); in these cases (groups 2, 3, 6,
7, 49, 50, 51, 52, 65, 66, 73, 74 and 75) natural spin representation is
reduced into two 1D components, and tensor product of stable BR of ${\bm{G}}$
or $\tilde{{\bm{G}}}{}$ with any of these components gives a stable BR of
$\widetilde{{\bm{G}}}{}$ and ${\bm{G}}$, respectively. When only
$\bm{SI}(\tilde{{\bm{G}}}{})$ is nontrivial, closed shells are unit,
independent of the BR topology.

It should be noted that the non-unit closed shell representation may be
important when correspond to the ground states of DMI-systems. Then it is
obviously indication of the vibronic instability.

###### Acknowledgements.

This work is funded by Ministry of Science, Technological Development and
Innovation, Projects 451-03-47/2023-01 and SANU-F93.

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##  Appendix A Induced representations

###  A.1 Form of the induced representation

Let ${\bm{R}}=\\{{\bm{r}}_{p}~{}|~{}p=1,2,\dots\\}$ be an orbit of the
symmetry group ${\bm{G}}$, i.e. all the sites are generated by the the action
of the group transformations on a representative one, ${\bm{r}}_{1}$. There is
a stabilizing subgroup ${\bm{F}}={\bm{G}}_{{\bm{r}}_{0}}$ of ${\bm{G}}$, such
that its elements $f$ fix ${\bm{r}}_{0}$, i.e. $f{\bm{r}}_{1}={\bm{r}}_{1}$.
Accordingly, there is a coset decomposition of the group with a selection
(_transverzal_) ${\bm{Z}}=\\{z_{p}~{}|~{}p=1,2,\dots\\}$ of
$Z=|{\bm{G}}|/|{\bm{F}}|=|{\bm{R}}|$ coset representatives $z_{p}$, such that
$z_{p}{\bm{r}}_{1}={\bm{r}}_{p}$, i.e. each $z_{p}$ corresponds to the site
${\bm{r}}_{p}$. Each group element belongs to some coset of ${\bm{F}}$,
meaning that it is uniquely factorized as $g=z(g)f(g)$. Then the site mapping
$g{\bm{r}}_{p}={\bm{r}}_{gp}$ corresponds to the equality

$gz_{p}=z_{\pi(g)p}f(g,p),$ (10)

and defines uniquely the integer $\pi(g)p$ (coset ordinal) and the stabilizer
element $f(g,p)$. Associativity of group multiplication gives (for each $p$):

$\pi(gh)=\pi(g)\pi(h),\ f(gh,p)=f(g,hp)f(h,p).$ (11)

The first property shows that $\pi(g)$ is a group homomorphism into the
permutational group $S_{Z}$. It describes the permutational action of
${\bm{G}}$ over the sites. Matrices $E(g)=\sum_{p}E^{\pi(g)p}_{p}$
($E^{p}_{q}$ is matrix with elements
$(E^{p}_{q})_{ij}=\delta_{ip}\delta_{jq}$), are permutational, with elements
$E_{ij}(g)=\sum_{p}\delta_{i,\pi(g)p}\delta_{j,p}$ (column $j$ is zero, except
that 1 is in the row $\pi(g)i$), and represent this action, assuming the site
${\bm{r}}_{p}$ is associated to the column $E^{p}$ (all zeros except 1 in the
$p$-th row). These matrices, forming a permutational (ground) representation
$E({\bm{G}})$ of the group, relate group elements to permutations of the
sites, identifying (as far as this action is considered) $g$ with a
permutation $\pi(g)$ from permutational group $S_{Z}$. As a permutation each
element $g$ obtains its parity $\tilde{g}{}=\tilde{\pi}{}(g)=\det E(g)$. In
the following $\pi(g)p$ is shortened to $gp$ (e.g.
$E^{gp}_{p}=E^{\pi(g)p}_{p}$).

Properties (11) provide that for arbitrary representation $d({\bm{F}})$ of the
stabilizer, the matrices

$D(g)=\sum^{Z}_{p=1}E^{gp}_{p}\otimes d(f(g,p))$ (12)

are representation of ${\bm{G}}$, called _induced_ and denoted as
$D({\bm{G}})=d({\bm{F}})\uparrow{\bm{G}}$.

###  A.2 Determinant of induced representation

For any representation $D$ of a group ${\bm{G}}$, the determinants
$\det{D({\bm{G}})}$ preserve homomorphism, and form one-dimensional (thus
irreducible) representation of ${\bm{G}}$. For direct sum and tensor product
of representations the determinant representations are

$\displaystyle\det({\oplus_{i}D_{i}})$ $\displaystyle=$
$\displaystyle\prod\nolimits_{i}\det{D_{i}}$ (13)
$\displaystyle\det({\otimes_{i}D_{i}})$ $\displaystyle=$
$\displaystyle\prod\nolimits_{i}\det{{}^{\tfrac{\prod\nolimits_{i}|D_{i}|}{|D_{i}|}}}{D_{i}}.$
(14)

Due to single nonzero block in each block-column and block-row, determinant of
the matrix (12) factorizes,
$\det{D(g)}=\det^{|d|}{E(g)}\prod_{p}\det{d(f(g,p))}$, yielding:

$\det{D(g)}=\tilde{g}{}^{|d|}\prod^{Z}_{p=1}\det{d(f(g,p))}.$ (15)

Independence of the choices of the first element in a cycle and of the orbit
representatives follows from the invariance of the determinant under the
similarity transform.

###  A.3 Determinant of the ground representation

As a permutation, element $g$ (i.e. $\pi(g)$) is partitioned into $C^{g}$
cycles, $g=g_{1}\cdots g_{C^{g}}$. Precisely, denote any $p$ as $s^{1}_{1}$,
$gp$ as $s^{1}_{2}$, $g^{2}p=gs^{1}_{2}=s^{1}_{3}$, and so on, unless
$g^{C^{g}_{1}}s^{1}_{1}=s^{1}_{1}$ for minimal integer $C^{g}_{1}$; then from
complement of the subset $s^{1}=\\{s^{1}_{1},\dots,s^{1}_{C^{g}_{1}}\\}$
another $p$ is chosen as $s^{2}_{1}$, and the procedure is repeated, unless
the whole set $\\{1,\dots,Z\\}$ is exhausted in the last $C^{g}$-th cycle. In
this way the permutational action of $g$ is factorized into the independent
permutations, cycles of $g$ of the length $C^{g}_{c}$, each of them permuting
only the subset $s^{c}$; clearly, $\sum^{C^{g}}_{c=1}C^{g}_{c}=Z$ and
$g^{C^{g}_{1}\cdots C^{g}_{C^{g}}}=\mathds{1}$. If the sites are indexed such
that the members of these subsets are one after another, then the matrix
$E(g)$ is block-diagonal, with the cyclic permutational blocks of the
dimensions $C^{g}_{c}$ on the diagonal, and

$\det{E(g)}=\prod_{c}\tilde{g}{}_{c}=(-1)^{Z+C^{g}}.$ (16a) From
$g^{C^{g}_{c}}s^{c}_{1}=s^{c}_{1}$ (10) implies
$f(g^{C^{g}_{c}},s^{c}_{1})=z^{-1}_{s^{c}_{1}}gz_{s^{c}_{1}}$, and (11) gives
$f(g^{C^{g}_{c}},s^{c}_{1})=f(g^{C^{g}_{c}-1},s^{c}_{2})f(g,s^{c}_{1})$;
iterations end up by:
$z^{-1}_{s^{c}_{1}}g^{C^{g}_{c}}z_{s^{c}_{1}}=f(g^{C^{g}_{c}},s^{c}_{1})=f(g,s^{c}_{C^{g}_{c}})\cdots
f(g,s^{c}_{1}).$ (16b)

##  Appendix B Periodic groups

IRs of $\wp$-periodic group ${\bm{G}}$ are induced[16] from the allowed IRs of
the stabilizers of the ${\bm{k}}$-points in the irreducible domain
$\mho_{\bm{G}}$. A brief reminder on their form is focused to analysis of
their determinants with help of the above results.

###  B.1 Irreducible representations

The translational invariant abelian subgroup ${\bm{T}}$ of ${\bm{G}}$
(analogously, helical subgroup of line groups), as the direct product of $\wp$
(infinite) cyclic groups generated by translations for ${\bm{a}}_{i}$
($i=1,\dots,\wp$) has IRs

$\Delta^{({\bm{k}})}({\bm{a}})=\mathrm{e}^{\mathrm{i}{\bm{k}}\cdot{\bm{a}}},\quad{\bm{k}}\in\mho=T^{\wp}.$
(17)

Brillouin’s zone, the set of different representations, is $\wp$ torus, since
change of ${\bm{k}}$ for a vector of inverse lattice $\overline{{\bm{A}}}{}$
(with basis $\overline{{\bm{a}}}{}_{i}$, such that
$\overline{{\bm{a}}}{}_{i}\cdot{\bm{a}}_{i}=2\pi\delta_{ij}$), gives the same
representation. The action $g{\bm{k}}=G{\bm{k}}$ of $g=(\,G\,|\,{\bm{g}}\,)$
(effectively $G$ reduces to its restriction into the space spanned by lattice)
from each ${\bm{k}}$ generates an orbit, the star ${\bm{k}}^{*}$. The orbits
with conjugated stabilizers are gathered into strata; orbit representatives of
the strata form representative strata $\mho^{K}_{\bm{G}}$ filling ID.

Whole ${\bm{T}}$ is invariant subgroup of the ${\bm{k}}$-point stabilizer
${\bm{F}}^{\bm{k}}$, and the factor group
${\bm{P}}^{\bm{k}}={\bm{F}}^{\bm{k}}/{\bm{T}}$ is a subgroup of the isogonal
group ${\bm{P}}={\bm{G}}/{\bm{T}}$. Thus, the elements of ${\bm{F}}^{\bm{k}}$
are
$f=(\,F\,|\,{\bm{f}}\,)=(\,\mathds{1}\,|\,{\bm{a}}_{F}\,)(\,F\,|\,\bm{\phi}\,)$,
where $(\,F\,|\,\bm{\phi}\,)$ is arbitrary chosen ${\bm{T}}$-coset
representative with element $F$ of ${\bm{P}}^{\bm{k}}$. Also, any transversal
$\\{z_{p}=(\,Z_{p}\,|\,\bm{\zeta}_{p}\,)\\}$ of ${\bm{F}}^{\bm{k}}$ in
${\bm{G}}$ is determines the transverzal $Z_{p}$ of ${\bm{P}}^{\bm{k}}$ in
${\bm{P}}$: ${\bm{k}}^{*}$ vectors are ${\bm{k}}_{p}=Z_{p}{\bm{k}}$;
conventionally, ${\bm{k}}={\bm{k}}_{1}$ is from $\mho_{\bm{G}}$, and
$(\,Z_{1}\,|\,\bm{\zeta}_{1}\,)=(\,\mathds{1}\,|\,0\,)$.

Irreducible representations [16] of ${\bm{G}}$ associated to ${\bm{k}}$ are
the representations (12) induced from the stabilizer’s ${\bm{F}}^{\bm{k}}$
_allowed_ representations (irreducible, restricted to ${\bm{T}}$ decompose to
a multiple of $\Delta^{({\bm{k}})}({\bm{T}})$). The latter are

$d^{(K{\bm{k}}\kappa)}(\,F\,|\,{\bm{f}}\,)=\mathrm{e}^{\mathrm{i}{\bm{k}}\cdot{\bm{f}}}d^{(K\kappa)}(F);$
(18)

here $d^{(K\kappa)}$ is projective IR of ${\bm{P}}^{\bm{k}}$ for the factor-
system
$\varphi_{\bm{k}}(F,F^{\prime})=\mathrm{e}^{\mathrm{i}{\bm{k}}\cdot(F\bm{\phi}^{\prime}-\bm{\phi}^{\prime})}$,
which compensates fractional translations in the products
$(\,F\,|\,\bm{\phi}\,)(\,F^{\prime}\,|\,\bm{\phi}^{\prime}\,)$):

$d^{(K\kappa)}(F)d^{(K\kappa)}(F^{\prime})=\mathrm{e}^{\mathrm{i}{\bm{k}}\cdot(F\bm{\phi}^{\prime}-\bm{\phi}^{\prime})}d^{(K\kappa)}(FF^{\prime}).$
(19)

As the strata of ID contain ${\bm{k}}$-points with the conjugated, thus
isomorphic stabilizers, the isomorphic isogonal groups ${\bm{P}}^{\bm{k}}$
(projective) have the same IRs. Finally, the allowed representation of
stabilizer associates to a group element
$g=(\,G\,|\,{\bm{g}}\,)=z_{gp}f_{p}z^{-1}_{p}$ (here
$f_{p}=(\,F_{p}\,|\,{\bm{f}}_{p}\,)$) the matrix (12):

$D^{(K{\bm{k}}\kappa)}(g)=\sum_{p=1}^{|{\bm{k}}^{*}|}E^{gp}_{p}\otimes\mathrm{e}^{\mathrm{i}{\bm{k}}\cdot{\bm{f}}_{p}}d^{(K\kappa)}(F_{p}).$
(20)

Notably, all the blocks vanish except $D^{(K{\bm{k}}\kappa)}_{gp,p}(g)$. Since
$F^{-1}{\bm{k}}-{\bm{k}}$ is an inverse lattice vector, for arbitrary
${\bm{a}}_{F^{\prime}}$ holds
$\mathrm{e}^{\mathrm{i}{\bm{k}}\cdot(F\bm{\phi}^{\prime}-\bm{\phi}^{\prime})}=\mathrm{e}^{\mathrm{i}{\bm{k}}\cdot(F{\bm{f}}^{\prime}-{\bm{f}}^{\prime})}$.

The IRs of double periodic group are of the same form, since ${\bm{T}}$ is
their invariant subgroup, as well, and Brillouin’s zone, strata and ID are the
same. Only the stabilizers are doubled, $\widetilde{{\bm{F}}}{}^{\bm{k}}$,
with half-integer (extra) allowed representations; however, orbits and
transverzals are the same, giving the same permutational structure of the
block-matrices of the IRs [17, 10].

###  B.2 Determinant of an irreducible representation of periodic groups:
explicit derivation

For arbitrary group element $g=(\,G\,|\,{\bm{g}}\,)$ its powers are
$g^{n}=(\,G^{n}\,|\,{\sum^{n}_{i=1}G^{n-i}{\bm{g}}}\,)=(\,G^{n}\,|\,{\sum^{n-1}_{i=0}G^{i}{\bm{g}}}\,)$.
The product of several elements, denoted as
$\operatorname*{\overleftarrow{\prod}}^{n}_{i=1}g_{i}\buildrel\rm
def\over{=}g_{n}\cdots g_{1}$, is
$\operatorname*{\overleftarrow{\prod}}^{n}_{i=1}(\,G_{i}\,|\,{\bm{g}}_{i}\,)=(\,\operatorname*{\overleftarrow{\prod}}^{n}_{i=1}G_{i}\,|\,\sum^{n}_{i=1}\operatorname*{\overleftarrow{\prod}}^{n}_{t=i+1}G_{t}{\bm{g}}_{i}\,)$.
For $z_{p}=(\,Z_{p}\,|\,\bm{\zeta}_{p}\,)$ and
$z_{cj}=(\,Z_{cj}\,|\,\bm{\zeta}_{cj}\,)$ it follows
$z^{-1}_{p}g^{i}z_{p}=(\,Z^{-1}_{p}G^{i}Z_{p}\,|\,Z^{-1}_{p}(G^{i}\bm{\zeta}_{p}-\bm{\zeta}_{p}+\sum^{i-1}_{j=0}G^{j}{\bm{g}})\,)$
and
$z^{-1}_{c1}g^{C^{c}_{g}}z_{c1}=(\,F^{c}\,|\,{\bm{f}}^{c}\,)=(\,Z^{-1}_{c1}G^{C^{c}_{g}}Z_{c1}\,|\,Z^{-1}_{c1}(G^{C^{c}_{g}}\bm{\zeta}_{c1}-\bm{\zeta}_{c1}+\sum^{{C^{c}_{g}}-1}_{j=0}G^{j}{\bm{g}})\,)$.
Substututing (20) in (15), with ${\bm{k}}^{o}\buildrel\rm
def\over{=}\sum^{|{\bm{k}}^{*}|}_{p=1}{\bm{k}}_{p}$ being the sum of the star
wave vectors (and sum over cycle ${\bm{k}}^{o}_{c}\buildrel\rm
def\over{=}\sum^{C^{g}_{c}}_{j=1}{\bm{k}}^{c}_{j}$), one gets:

$\displaystyle\det{D^{(K{\bm{k}}\kappa)}(g)}$ $\displaystyle=$
$\displaystyle(-1)^{|\kappa|(|{\bm{k}}^{*}|+C^{g})}\mathrm{e}^{\mathrm{i}|\kappa|{\bm{k}}\cdot\sum^{|{\bm{k}}^{*}|}_{p=1}{\bm{f}}_{p}}\prod^{|{\bm{k}}^{*}|}_{p=1}\det{d^{(K\kappa)}(F_{p})}$
$\displaystyle=$
$\displaystyle(-1)^{|\kappa|(|{\bm{k}}^{*}|+C^{g})}\prod^{C^{g}}_{c=1}\det{d^{({\bm{k}},\kappa)}(\,Z^{-1}_{c1}G^{C^{g}_{c}}Z_{c1}\,|\,Z^{-1}_{c1}(G^{C^{g}_{c}}\bm{\zeta}_{c1}-\bm{\zeta}_{c1}+\sum^{{C^{g}_{c}}-1}_{j=0}G^{j}{\bm{g}})\,)}$
$\displaystyle=$
$\displaystyle(-1)^{|\kappa|(|{\bm{k}}^{*}|+C^{g})}\prod^{C^{g}}_{c=1}\det{\mathrm{e}^{\mathrm{i}{\bm{k}}\cdot
Z^{-1}_{c1}(G^{C^{g}_{c}}\bm{\zeta}_{c1}-\bm{\zeta}_{c1}+\sum^{C^{g}_{c}-1}_{j=0}G^{j}{\bm{g}})}d^{(K\kappa)}(Z^{-1}_{c1}G^{C^{g}_{c}}Z_{c1})}$
$\displaystyle=$
$\displaystyle(-1)^{|\kappa|(|{\bm{k}}^{*}|+C^{g})}\mathrm{e}^{\mathrm{i}|\kappa|\sum^{C^{g}}_{c=1}Z_{c1}{\bm{k}}\cdot(G^{C^{g}_{c}}\bm{\zeta}_{c1}-\bm{\zeta}_{c1}+\sum^{C^{g}_{c}-1}_{j=0}G^{j}{\bm{g}})}\prod^{C^{g}}_{c=1}\det{d^{(K\kappa)}(Z^{-1}_{c1}G^{C^{g}_{c}}Z_{c1})}$
$\displaystyle=$
$\displaystyle(-1)^{|\kappa|(|{\bm{k}}^{*}|+C^{g})}\mathrm{e}^{\mathrm{i}|\kappa|\sum^{C^{g}}_{c=1}({\bm{K}}^{g}_{c1}\cdot\bm{\zeta}_{c1}+{\bm{k}}_{c1}\cdot\sum^{C^{g}_{c}-1}_{j=0}G^{j}{\bm{g}})}\prod^{C^{g}}_{c=1}\det{d^{(K\kappa)}(Z^{-1}_{c1}G^{C^{g}_{c}}Z_{c1})}.$

Lattice translations $g=(\,\mathds{1}\,|\,{\bm{a}}\,)$ fix each ${\bm{k}}$,
being thus identity permutation $\pi(g)=\mathds{1}$. Thus, they have
$C^{g}=|{\bm{k}}^{*}|$ cycles ($p=c$) of the lengths $c^{g}_{p}=1$; further,
$gz_{p}=(\,Z_{p}\,|\,\bm{\zeta}_{p}\,)(\,\mathds{1}\,|\,Z^{-1}_{p}{\bm{a}}\,)$
$z_{gp}=z_{p}$,
$f_{p}=z^{-1}_{gp}gz_{p}=(\,\mathds{1}\,|\,Z^{-1}_{p}{\bm{a}}\,)$, i.e.
$F_{p}=\mathds{1}$ and ${\bm{f}}_{p}=Z^{-1}_{p}{\bm{a}}$, giving:

$\det{D^{(K{\bm{k}}\kappa)}(\,\mathds{1}\,|\,{\bm{a}}\,)}=(-1)^{2|\kappa||{\bm{k}}^{*}|}\mathrm{e}^{\mathrm{i}|\kappa|\sum^{|{\bm{k}}^{*}|}_{p=1}Z_{p}{\bm{k}}\cdot{\bm{a}}}\prod^{|{\bm{k}}^{*}|}_{p=1}\det{\mathds{1}}=\mathrm{e}^{\mathrm{i}|\kappa|{\bm{k}}^{o}\cdot{\bm{a}}}$
(21)

###  B.3 $\bm{\gamma}$-points

This reveals that the determinants form a representation associated to the
$\bm{\gamma}$-point $|\kappa|{\bm{k}}^{o}$, which allows one-dimensional
representations. For the pure translational and other Abelian groups all IRs
are one-dimensional, and each ${\bm{k}}$ is a $\gamma$-point:
$\mho^{0}_{\bm{G}}=\mho_{\bm{G}}=\mho$. In other groups $\mho^{0}_{\bm{G}}$ is
discrete subset of $\mho_{\bm{G}}$.

According to (20), the one-dimensionality implies that each $\bm{\gamma}$ is a
fixed point for the whole group (otherwise the induction gives larger
dimension), ${\bm{F}}^{\bm{\gamma}}={\bm{G}}$ (and
${\bm{P}}^{\bm{\gamma}}={\bm{P}}$), as well as that the allowed representation
of ${\bm{P}}$ is an ordinary representation. Therefore $\mho^{0}_{\bm{G}}$
belongs to the maximally symmetric (regarding stabilizers) strata of ID, which
are of the minimal dimension, and at the boundary of all other strata. Since
$\bm{\gamma}$ is fixed (up to the inverse lattice) by the whole isogonal
group, it obeys (4a). Obviously, $\Gamma$-point, $\bm{\gamma}=0$, is a
solution. For $\bm{\gamma}$ in the interior of $\mho_{\bm{G}}$, only
${\bm{K}}^{\bm{\gamma}}_{P}=0$ is possible; thus, the fixed points of
${\bm{P}}$ in $\mho_{\bm{G}}$, besides those fixed by the Euclidean action (in
the lattice subspace), may be only on the boundary of $\mho$.

Still, there are fixed points with no one-dimensional representations. In
fact, as irreducible, such a representation is of the form (18), with
${\bm{P}}^{\bm{\gamma}}={\bm{P}}$, and the factor system satisfying (19). On
the other side, one-dimensional representations are projectively equivalent to
the unit representation. Taking in (18) $d^{(K\kappa)}(P)=1$ (as a projective
unit representation), and multiplying each of them by the corresponding matrix
$u(P)=\Delta^{(\beta)}(P)$ of one-dimensional (ordinary) representation of
${\bm{P}}$, the new multipliers
$\varphi^{\prime}_{\bm{\gamma}}(P,P^{\prime})=u(P)u(P^{\prime})\varphi_{\bm{\gamma}}(P,P^{\prime})/u(PP^{\prime})$
must be trivial. Thus, fixing a representative $p=(\,P\,|\,{\bm{p}}\,)$ for
each element $P$ of ${\bm{P}}$, and applying (19), in the view of the
assumption that $\Delta^{(\beta)}({\bm{P}})$ is a representation, the
sufficient condition for a fixed point to be a $\bm{\gamma}$-point becomes
(4b). Note the independence of $\Delta^{(\beta)}({\bm{P}})$, i.e. all one-
dimensional representations are simultaneously allowed. For symmorphic groups
this is automatically fulfilled (as then ${\bm{p}}$ is a lattice vector), as
well as within interior of $\mho$ (then ${\bm{K}}^{\bm{\gamma}}_{P}=0$).

To resume, $\bm{\gamma}$-points are points fixed by the whole group, with one-
dimensional associated IRs; they satisfy (4). The fixed points within interior
of ID are automatically $\bm{\gamma}$-points (actually, in this case the
corresponding $d^{(K\kappa)}({\bm{P}}^{\bm{k}})$ in (18) are ordinary
representations). As for symmorphic groups, all fixed points are
$\gamma$-points. Thus, only for the fixed points on the boundary of ID for
nonsymmorphic groups (4b) is an effective restriction, depending not only on
${\bm{P}}$, but on the whole group ${\bm{G}}$ (only they can differ for the
groups with the same ${\bm{P}}$). Finally, when (4) are fulfilled, the
determinant representation is

$\det{D^{(K{\bm{k}}\kappa)}(g)}=\mathrm{e}^{\mathrm{i}\bm{\gamma}\cdot{\bm{g}}}d^{(\beta)}(G),\quad\bm{\gamma}=|\kappa|{\bm{k}}^{o},$
(22)

where $d^{(\beta)}({\bm{P}})$ is one of the one-dimensional (ordinary)
representation of the isogonal group.

####  B.3.1 Layer groups

Figure 1: For each irreducible domain (as given in Ref. 18, 19, the cellular division into $d$ equivalent parts is presented. Point strata are red, and line ones are green (with green included edges and remaining points). Interior is generic stratum (grey, with included edges emphazized by black lines) is divided in with remaining lines (the equivalent ones are in the same color), which are then divided into open intervals and remaining points. Table 1: For each irreducible domain ID (not bolded if for colorless groups only), (standard [19]) of layer groups the isogonal group ${\bm{P}}$, $n$ orders of the elements $n$, division order (LCM of the orders of the double group generators) $d$, and number $c$ of contributing points for strata $S$ (underlined if $\bm{\gamma}$-point) are listed. ID | ${\bm{P}}$ | $n$ | $d$ |  $cS$ and $c\underline{\bm{\gamma}}{}$
---|---|---|---|---
1 | ${\bm{C}}_{1}$ | 1 | 4 | $0\underline{G}{}$
$\mathbf{2}$ | ${\bm{C}}_{2}$ | 2 | 4 | $1\underline{\Gamma}{},1\underline{X}{},1\underline{Y}{},1\underline{S}{},0G$
3 | ${\bm{D}}^{2}_{1}$ | 2 | 4 | $0\underline{\Phi}{},0\underline{\Lambda}{},0G$
4 | ${\bm{D}}^{1}_{1}$ | 2 | 4 | $0\underline{\Delta}{},0\underline{\Omega}{},0G$
5 | ${\bm{D}}^{0}_{1}$ | 2 | 4 | $0\underline{\Sigma}{},0G$
6 | ${\bm{C}}_{3}$ | 3 | 12 | $1\underline{\Gamma}{},1\underline{K}{},1\underline{L}{},5G$
$\mathbf{7}$ | ${\bm{C}}_{4}$ | 4 | 8 | $1\underline{\Gamma}{},1X,1\underline{S}{},7G$
$\mathbf{8}$ | ${\bm{D}}^{1}_{2}$ | 2 | 4 | $1\underline{\Gamma}{},1\underline{X}{},1\underline{Y}{},1\underline{S}{},3\Delta,3\Phi,3\Omega,3\Lambda,1G$
$\mathbf{9}$ | ${\bm{D}}^{0}_{2}$ | 2 | 4 | $1\underline{\Gamma}{},1Y,1\underline{S}{},3\Sigma,3\Xi,0G$
$\mathbf{10}$ | ${\bm{C}}_{6}$ | 6 | 12 | $1\underline{\Gamma}{},1M,1K,11G$
11 | ${\bm{D}}^{1}_{3}$ | 2,3 | 12 | $1\underline{\Gamma}{},1K,0\Delta,11\Phi,0G$
12 | ${\bm{D}}^{0}_{3}$ | 2,3 | 12 | $1\underline{\Gamma}{},1\underline{K}{},1\underline{L}{},11\Lambda,11\Sigma,11\Xi,1G$
$\mathbf{13}$ | ${\bm{D}}_{4}$ | 4 | 8 | $1\underline{\Gamma}{},1X,1\underline{S}{},7\Delta,7\Lambda,7\Sigma,1G$
$\mathbf{14}$ | ${\bm{D}}_{6}$ | 6 | 12 | $1\underline{\Gamma}{},1M,1K,11\Delta,11\Lambda,11\Sigma,1G$

Brillouin zone of the _layer_ groups is 2-dimensional (the lattice space is
perpendicular to $z$-axis), and the isogonal groups are the axial point groups
with the crystallographic orders of principle axis: $n=1,2,3,4,6$. As the
horizontal mirror plane acts as identity, effective action of these point
groups is reduced to ${\bm{C}}_{n}$, ${\bm{C}}_{2n}$, ${\bm{C}}_{n}$,
${\bm{D}}_{n}$, ${\bm{D}}_{n}$, ${\bm{D}}_{2n}$ and ${\bm{D}}_{n}$,
respectively. For $q>1$ the group ${\bm{C}}_{q}$ in the interior of $\mho$
fixes only $\Gamma$-point; thus this is the only interior fixed and
$\bm{\gamma}$-point for the layer groups with nontrivial principle axis
rotations. When $q=1$, ${\bm{C}}_{1}$ fixes the whole $\mho$; if it is the
only effective element of ${\bm{P}}$, then $\mho^{\bm{\gamma}}=\mho_{\bm{G}}$,
and otherwise the fixed points are within axis (of the group ${\bm{D}}_{1}$)
or vertical mirror plane (for ${\bm{C}}_{1\textrm{v}}$)). Winding numbers of
the axis of ${\bm{D}}_{n}$ give altogether 14 types of irreducible domains
[19]. Here we present data for each of them. For each ID the least common
multiple $d$ of the generators of the associated (double) groups is
calculated, and the ID partition into $d$ equivalent parts is found (Fig. 1).
Then the number of the remaining points for each stratum of ID (Table 1) is
found. All the integrals appearing in the calculation of strata determinants
of IDs 1, 3, 4 and 5 (when $\gamma$ points are not isolated) are equal to 1.
These data suffice to calculate all the strata determinant.

####  B.3.2 Line groups

Table 2:  Thirteen families of the line groups. For each family (F) in the column Type H or L is for helical or linear factorization, $S$ or $N$ for (non)symmorphic groups and $+$ and $-$ for the colorless groups with/without $z$-reversing elements (grey group families are $-$), which corresponds to ID ($(-\pi,\pi]$ or $[0,\pi]$. Then follow columns $n$ (orders of the isogonal group generators)and $d$ (division order of the generic stratum LCM of the orders of the double group generators)/ Finally, number $c$ of contributing points for strata $S$ (underlined if $\bm{\gamma}$-point) are listed for colorless (C) and grey (G) groups. $x=4n-1$. F | Type | $n$ | $d$ | C: $cS,c\underline{\gamma}{}$ | G: $cS,c\underline{\gamma}{}$
---|---|---|---|---|---
1 | H S $+$ | $?,n$ | $?$ | $?\underline{G}{}$ | $1\underline{\Gamma}{},xG,1\underline{\Pi}{}$
2 | L S $-$ | $2n$ | $4n$ | $1\underline{\Gamma}{},xG,1\underline{\Pi}{}$ | $1\underline{\Gamma}{},xG,1\underline{\Pi}{}$
3 | L S $-$ | $n,2$ | $4n$ | $1\underline{\Gamma}{},xG,1\underline{\Pi}{}$ | $1\underline{\Gamma}{},xG,1\underline{\Pi}{}$
4 | L N $-$ | $2n,n,2$ | $4n$ | $1\underline{\Gamma}{},xG,1\Pi$ | $1\underline{\Gamma}{},xG,1\Pi$
5 | H S $-$ | $?,n,2$ | $?$ | $1\underline{\Gamma}{},?G,1\underline{\Pi}{}$ | $1\underline{\Gamma}{},xG,1\underline{\Pi}{}$
6 | L S $+$ | $n,2$ | $4n$ | $x\underline{G}{}$ | $1\underline{\Gamma}{},xG,1\underline{\Pi}{}$
7 | L N $+$ | $n,2$ | $4n$ | $x\underline{G}{}$ | $1\underline{\Gamma}{},xG,1\Pi$
8 | L N $+$ | $2n,n,2$ | $4n$ | $x\underline{G}{}$ | $1\underline{\Gamma}{},xG,1\Pi$
9 | L S $-$ | $2n,n,2$ | $4n$ | $1\underline{\Gamma}{},xG,1\underline{\Pi}{}$ | $1\underline{\Gamma}{},xG,1\underline{\Pi}{}$
10 | L N $-$ | $2n,n,2$ | $4n$ | $1\underline{\Gamma}{},xG,1\Pi$ | $1\underline{\Gamma}{},xG,1\Pi$
11 | L S $-$ | $n,2$ | $4n$ | $1\underline{\Gamma}{},xG,1\underline{\Pi}{}$ | $1\underline{\Gamma}{},xG,1\underline{\Pi}{}$
12 | L N $-$ | $n,2$ | $4n$ | $1\underline{\Gamma}{},xG,1\underline{\Pi}{}$ | $1\underline{\Gamma}{},xG,1\Pi$
13 | L N $-$ | $2n,n,2$ | $4n$ | $1\underline{\Gamma}{},xG,1\Pi$ | $1\underline{\Gamma}{},xG,1\Pi$

There are 13 infinite families of the line groups[20]. We assume periodicity
(translational or helical) along $z$-axis. Therefore, principle and helical
rotations (around $z$-axis) and vertical mirror planes act in Brillouin’s zone
as identity, and the isogonal groups ${\bm{C}}_{n}$ and
${\bm{C}}_{n\mathrm{v}}$ act as ${\bm{C}}_{1}$ (identity only) giving
$\mho_{\bm{G}}=\mho$, while other ones as ${\bm{D}}_{1}$ (identity and a
$z$-reversal operation), with $\mho_{\bm{G}}=[0,\pi/a]$ stratified into
interval $G=(0,\pi)$ and points $\Gamma$ and $\Pi$ (for $k=0,\pi$,
respectively). In the first case (families 1,6,7,8) $\mho^{\bm{\gamma}}=\mho$,
since the whole BZ is fixed, while (4b) is satisfied automatically. In other
families due to the $z$-reversing symmetries, the fixed points are $\Gamma$
and $\Pi$; (4b) is satisfied for $\Gamma$, and for $\Pi$ in the symmorphic
families 2, 3, 5 (helical axis used instead of the translational group, with
helical QNs), 9, 11, and nonsymmorphic 10 and 12. In nonsymmorphic families 4
and 13, the only $\bm{\gamma}$-point is $\Gamma$. When time reversal is added
(grey groups), all the irreducible domain are $[0,\pi]$; $\Gamma$ is
$\gamma$-point in all groups, while $\Pi$ only in symmorphic; in double grey
groups all the determinant representations are unit.