3.1. Symmetry of Crystalline Systems Including a Surface
In crystalline systems, all the symmetry operations other than time reversal and particle-hole transformation are elements of a space group. Here we focus on the momentum line including the time-reversal-invariant momentum
along which the one-dimensional topological invariant is defined. Symmetry operations respecting the
point are (screw) rotation and (glide) reflection, which are classified into those preserving (type-I) and inverting (type-II) the surface of
. The type-I symmetry operations are two-fold (screw) rotation
along the
axis and mirror (glide) reflections
with respect to
plane and
with respect to
plane (
Figure 2a). The type-II symmetry operations, on the other hand, are two (screw) rotations [
and
] and one mirror (glide) reflection [
], as shown in
Figure 2b. Afterwards, we denote a type-I operation by
then we have
The spatial inversion
I is represented in terms of
as
for
, where
is a type-II symmetry operation, i.e.,
3.3. Topological Invariant
In the following, we derive necessary conditions for finite-valued topological invariants, which is defined by
indicates the direction normal to the surface, i.e.,
is the number of the Majorana zero modes on the surface perpendicular to
. Note that the type-II symmetries
may define an topological invariant but it does not correspond to the zero-energy surface states since the surface is not invariant against
. This is why only the type-I symmetries
are considered here. Glide reflection along the direction parallel to the surface, e.g.,
a-glide with respect to the
plane for the
surface, is one of the possible type-I
symmetries for the winding number. Screw rotation, however, is not used for the winding number because the surface is not invariant by the operation. Glide reflection that translates a system along the direction normal to the surface and screw rotation may define a bulk invariant although the bulk-edge correspondence does not hold, as type-II
symmetry. Henceforth, for the rotational symmetries, we suppress the suffix
as
, due to the uniqueness of the directions of the integrals, i.e.,
must be along the rotational axis.
Now we derive the constraint to
by the symmetries. One gets
These equations are derived by applying unitary transformations by
and by
. Here we introduce
as
Note that
includes the
n-fold rotation (if exist) in addition to the two-fold rotations. In consequence, the conditions of
and of
(Equation (
9)) are necessary for
. From the above condition,
is derived because of
.
Next, we prove that the two-fold symmorphic symmetry operations, rotations and reflections, satisfy the condition of Equation (
9) while the nonsymmorphic ones, glide reflections, do not on the Brillouin zone boundary. Symmetry operations are represented by the direct product of real-space part
and spin part
,
. For two-fold rotations and mirror reflections, the real-space part
is an orthogonal matrix with
and
. The spin part is given by Pauli matrices then
,
, and
. As a result, the chiral operator is given by
so that the condition of Equation (
9) holds. For glide reflections, on the contrary, the orbital part
on the Brillouin zone boundary is purely-imaginary matrix hence the condition Equation (
9) is not satisfied, i.e.,
with
being a glide reflection, where
is the translation vector of the glide reflection (for details, see
Appendix B).
For symmorphic space groups, the necessary condition for
is easily obtained as follows. The commutation relations of the representations for symmetry operations in a point group are uniquely determined to be
in spinful systems. With the help of the above relation, the condition Equation (
20) reduces to
Here,
is the character of
O hence the possible topological invariant is determined only from the representation theory of point group, irrespective of details of the system, as summarized in the tables in
Appendix C. An example for a nonsymmorphic space group is also shown in
Appendix C. The condition of
is extracted from the above equations. This means that the character of the spatial inversion
must be
for the existence of topological superconductivity. That is consistent with the absence of time-reversal-invariant Majorana fermion in even-parity superconductors [
32].
Finally, we show that two of
,
, and
always vanish. Here
is the two-fold (not screw) rotation,
is the mirror or glide reflection with respect to the
plane. The statement is immediately seen from Equation (
23) for symmorphic space groups:
and
are not simultaneously satisfied. This is also true at
for nonsymmorphic space groups since the commutation relations of symmetry operations are the same as those for the symmorphic space group. When
is the
-glide reflection, the commutation relation changes from the symmorphic one at the boundary
.
, however, vanishes from Equation (
21). In consequence, it is impossible that two of
,
, and
simultaneously take nontrivial values.