††thanks: Contributed equally††thanks: Contributed equally

# Sub-dimensional topologies, indicators and higher order phases

Gunnar F. Lange<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory, University
of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Adrien
Bouhon<EMAIL_ADDRESS>Nordic Institute for Theoretical Physics
(NORDITA), Stockholm, Sweden  Department of Physics and Astronomy, Uppsala
University, Box 516, SE-751 21 Uppsala, Sweden  Robert-Jan Slager
<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory, University of Cambridge,
J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom

###### Abstract

The study of topological band structures have sparked prominent research
interest the past decade, culminating in the recent formulation of rather
prolific classification schemes that encapsulate a large fraction of phases
and features. Within this context we recently reported on a class of
unexplored topological structures that thrive on the concept of sub-
dimensional topology. Although such phases have trivial indicators and band
representations when evaluated over the complete Brillouin zone, they have
stable or fragile topologies within sub-dimensional spaces, such as planes or
lines. This perspective does not just refine classification pursuits, but can
result in observable features in the full dimensional sense. In three spatial
dimensions (3D), for example, sub-dimensional topologies can be characterized
by non-trivial planes, having general topological invariants, that are
compensated by Weyl nodes away from these planes. As a result, such phases
have 3D stable characteristics such as Weyl nodes, Fermi arcs and edge states
that can be systematically predicted by sub-dimensional analysis. Within this
work we further elaborate on these concepts. We present refined representation
counting schemes and address distinctive bulk-boundary effects, that include
momentum depended (higher order) edge states that have a signature dependence
on the perpendicular momentum. As such, we hope that these insights might spur
on new activities to further deepen the understanding of these unexplored
phases.

##  I Introduction

Topological effects in band structures has received significant attention in
recent years Qi and Zhang (2011); Hasan and Kane (2010), leading to a myriad
of topological phases and effects Slager  _et al._ (2012); Fu (2011); Slager
(2019); Bzdušek  _et al._ (2016); Höller and Alexandradinata (2018); Shiozaki
_et al._ (2017); Fang  _et al._ (2012); Lee  _et al._ (2020); Mesaros  _et
al._ (2013); Kariyado and Slager (2020); Zhang and Liu (2015); Rhim  _et al._
(2018); Bouhon and Black-Schaffer (2017a). Of particular interest in this
regard however is recent progress on mapping out a considerable fraction of
topological materials. Namely, using constraints on band representations
between high symmetry point in the Brillouin zone (BZ) that can reproduce the
full K-theory in certain cases Kruthoff  _et al._ (2017), elaborate schemes
have emerged that upon comparing these condition in momentum space to real
space provide for direct indicators of topological non-triviality Po  _et al._
(2017); Bradlyn  _et al._ (2017); Watanabe  _et al._ (2018); Elcoro  _et al._
(2020); Bouhon  _et al._ (2020a). This is usually phrased in terms of so-
called symmetry indicators Po  _et al._ (2017) or elementary band
representations Bradlyn  _et al._ (2017). While the former is roughly obtained
by considering the constraints in momentum space as a vector space, which
delivers indicators upon dividing out Fourier transformed trivial atomic limit
configurations, the rational behind considering elementary band
representations (EBR) is that a split EBR must lead to a non-trivial behavior
Bradlyn  _et al._ (2017). That is, a topological configuration can by
definition not be represented in terms of Wannier functions (of the localized
kind) that also respect all symmetries.

This progress in itself has already sparked the discovery of new kinds of
topologies. It was for example found that mismatches between stable symmetry
indicators and split EBRs can be understood by the existence of fragile
topological phases Po  _et al._ (2018). Such fragile phases formally amount to
a difference of trivial phases and can consequently be trivialized by the
addition of trivial bands, rather than bands having opposite topological
invariants Bouhon  _et al._ (2019); Peri  _et al._ (2020); Song  _et al._
(2019). This concept of fragile topology in fact not only applies to symmetry
indicated phases (i.e. those that can be deduced from the irreducible
representation content at high symmetry points) but can be generalized by
taking into account multi-gap conditions Bouhon  _et al._ (2020b). Such phases
can physically be understood as arising by braiding non-Abelian frame charges
Wu  _et al._ (2019); Ahn  _et al._ (2019); Bouhon  _et al._ (2020c), leading
to novel types of invariants and physical effects Ünal  _et al._ (2020) .

In recent work Bouhon  _et al._ (2020a), we elucidated the idea of fragile
topology in a magnetic context Jo  _et al._ (2020); Qi  _et al._ (2008);
Wieder and Bernevig (2018); Liu  _et al._ (2020); Yang  _et al._ (2021); Mong
_et al._ (2010); Otrokov  _et al._ (2019); Zhang and Liu (2015); Elcoro  _et
al._ (2020) and outlined its connection to the concept of sub-dimensional
topology. The essential idea articulates around the fact that while EBRs can
be globally connected, thus appearing trivial in previous schemes, they can
still be splittable on sub-dimenional spaces such as planes in 3D Brillouin
zones. These sub-dimensional spaces can effectively be diagnosed in terms of
symmetry indicators and EBR content. This is not merely an esoteric
observation but results in real (in the 3D sense) consequences. The non-
trivial plane might for example host stable invariants such as Chern numbers
or spin Chern numbers that necessarily have to be compensated by Weyl nodes to
lead to a globally connected EBR. As result, there are phases that have stable
topological features (such as Chern planes and Weyl nodes) with distinct
physical observables (such as edge states and Fermi arcs) that depend on this
new concept, i.e the analysis of sub-dimensions that in turn reveal the
necessary existence of nodes in the 3D band structure. While we presented an
exhaustive table of all space groups in which the simplest form of this
mechanism occurs Bouhon  _et al._ (2020a), it was conjectured that this new
view can play a role in other settings as well.

In this work, we further develop the idea of fragile magnetic topology and
sub-dimensional topology by investigating the bulk-corner and bulk-hinge
correspondence, the twisted boundary conditions and the bulk-edge
correspondence for these phases. This elucidates the connection between
fragile magnetic topology, sub-dimensional topology and other well-studied
phases such as higher-order topological insulators (HOTI). In particular, we
explore the physical consequences of sub-dimensional topology by looking at
the hinge and edge state spectra of a sub-dimensional phase, and by examining
its bulk-edge correspondence. Interestingly, we find that the sub-dimensional
perspective can manifest itself by affecting the edge/hinge spectrum as
function of the momentum along the hinge or direction perpendicular to the
plane hosting the non-trivial sub-dimensional topology. We thus find as a main
result that this refined perspective can results in specific consequences at
the edge, momentum depended hinge spectra, thereby providing for new physical
signatures.

The paper is organised as follows. In section II we introduce the magnetic
space-groups (MSG) and representation content which we will be considering. In
section III, we present the spectra in various finite-dimensional geometries
and comment on their connection to corner/edge charges and to HOTI. In section
IV, we further corroborate our findings by connecting them to the recently
introduced idea of real-space invariants and twisted boundary conditions Song
_et al._ (2020); Peri  _et al._ (2020). In section V, we finally connect our
discussion to Wilson loops and a bulk-edge correspondence. We conclude in
section VI.

##  II Setup and topology for MSG75.5 and MSG77.18

###  II.1 Setup and topology for MSG75.5

The magnetic space-group (MSG) 75.5 is generated from the tetragonal (non-
magnetic) space group 75 (P4, generated by $C_{4}$ rotation) by including the
antiunitary symmetry $(E|\tau)^{\prime}$, with $E$ the identity,
$\tau=\boldsymbol{a}_{1}/2+\boldsymbol{a}_{2}/2$, $(\cdot)^{\prime}$ denoting
time-reversal and $\boldsymbol{a}_{i}$ the primitive vectors of a (primitive)
tetragonal Bravais lattice. Thus MSG75.5 is a Shubnikov type IV MSG which
hosts anti-ferromagnetic ordering Bradley and Cracknell (1972). We showed in
Bouhon  _et al._ (2020a) that starting from magnetic Wyckoff position (mWP)
$2b$, with sites $\boldsymbol{r}_{A}=\boldsymbol{a}_{1}/2$ and
$\boldsymbol{r}_{B}=\boldsymbol{a}_{2}/2$, the real-space symmetries
necessitate a minimum of four states to be present in the unit cell. Our
choice of unit cell is shown in Fig. 1a). We introduced a spinful model of
this MSG with two sites per unit cell, each hosting two orbitals, in Bouhon
_et al._ (2020a) (summarized in Appendix A.1). This model can be split into
two disconnected two-band subspaces over the entire BZ whilst respecting all
symmetries of MSG75.5, and thus realizes a split magnetic elementary band
representation (MEBR) Elcoro  _et al._ (2020). All symmetry indicators are
trivial in our model, and therefore one of these subspaces necessarily
realizes fragile topology, i.e. it can be trivialized by coupling to trivial
bands. The other two-band subspace realizes an atomically obstructed limit,
where the electrons localize at a mWP distinct from the mWP of the atomic
orbitals Bradlyn  _et al._ (2017). We note, however, that both subspaces
display non-trivial Wilson loop winding as discussed in section V. Thus whilst
the obstructed atomic insulator label is useful pictorially, a more careful
rigorous analysis based on the Wilson loop is needed in general, which we
carry out in section V. The split in our model can be written explicitly as:

$\mathrm{MEBR}^{2b}_{75.5}\rightarrow\underbrace{(\mathrm{MEBR}_{75.5}^{2b}\ominus\mathrm{MEBR}_{75.5}^{2a})}_{\mathrm{Lower\
subspace}}\oplus\underbrace{\mathrm{MEBR}_{75.5}^{2a}}_{\mathrm{Upper\
subspace}}.$

Where $\ominus$ denotes formal subtraction of MEBRs. In terms of the spinful
site-symmetry co-IRREPs (using notation from the Bilbao Crystallographic
Server Perez-Mato  _et al._ (2011); Aroyo  _et al._ (2006a, b)) this can be
written as:

$[\underbrace{(^{1}\overline{E}^{2}\overline{E})_{2b}\uparrow
G\ominus(^{1}\overline{E}_{1})_{2a}\uparrow G}_{\mathrm{Lower\
subspace}}]\oplus\underbrace{(^{1}\overline{E}_{1})_{2a}\uparrow
G}_{\mathrm{Upper\ subspace}}$ (1)

This decomposition can be determined directly from the momentum space IRREPs,
using the formalism of (magnetic) topological quantum chemsistry Bradlyn  _et
al._ (2017); Elcoro  _et al._ (2020). For MSG75.5, the unitary symmetries do
not dictate any $k_{z}$ dependence. It is therefore legitimate to consider the
planes containing the time-reversal invariant momentum points (TRIMPs) in the
BZ separately, which results in an effective 2D model. We choose to focus on
the plane $k_{z}=0$.

To explore the physical consequences of magnetic fragile topology in this
system, we analyze its edge/corner spectrum and also consider its evolution
under twisted boundary conditions (TBC). We therefore build a finite 2D
lattice version of our model which respects $C_{4}$ symmetry. We consider two
different ways of building this lattice, illustrated in Fig. 1 b)-c). As can
be seen in Fig. 1b), including an integer number of unit cells necessarily
violates $C_{4}$ symmetry for a cut along the crystallographic axis. We
therefore consider a half-integer number of unit cells in both direction for
such a cut. We also consider how these spectra depend on the real-space
termination, which leads us to also consider the cut in Fig. 1c). This cut is
also useful with regard to the TBC, which we discuss in section IV, as there
are no orbitals on the boundaries between regions related by $C_{4}$ symmetry.
We refer to this as the diagonal cut.

Figure 1: Illustration of the unit cell (panel a) and two possible lattice
cuts that respect $C_{4}$ symmetry (panels b and c) for MSG75.5. a) The unit
cell, with (m)WPs labelled, using both the convention of MSG75.5
($2a,2a^{\prime},2b,2b^{\prime}$) and of wallpaper group 10/p4
($1a,1b,2c,2c^{\prime}$). We place one spin up and one spin down orbital at
$2b$ and $2b^{\prime}$ respectively. b) An integer number of units cells
violates $C_{4}$ symmetry. This can be restored by considering a half-integer
number of unit cells, which corresponds to ignoring the sites circled in
green. c) To have $C_{4}$ symmetric sectors with no orbitals on the boundary,
we can perform the cut shown in the green square, with the boundary between
the sectors indicated.

Note that our finite model necessarily breaks the non-symmorphic antiunitary
symmetry $(E|\tau)^{\prime}$. This effectively reduces the symmetry to the
spinful ferro/ferrimagnetic phase 75.1 (Shubnikov type I), which in the 2D
case reduces to wallpaper group 10 (p4) with strong spin-orbit coupling (SOC)
and without time-reversal symmetry (TRS). This group has been well-studied in
e.g. Schindler  _et al._ (2019); Song  _et al._ (2020). We reproduce some of
their results to make the connection to the novel sub-dimensional topology in
MSG77.18 more transparent.

Wallpaper group 10 (p4) has three (non-magnetic) WPs: $1a$ and $1b$ with
point-group (PG) symmetry $C_{4}$ and $2c$ with PG symmetry $C_{2}$. This
labeling is also shown in Fig. 1a). This is similar to the model considered in
Song  _et al._ (2020), however, there the fragile phase originates from TRS.
In our case, the fragile phase results from magnetic symmetries which manifest
in the gluing of IRREPs in momentum space. In terms of the site-symmetry
IRREPs of MSG75.1, our model realizes the decomposition in equation (2),

$\underbrace{[(^{1}\overline{E})_{2c}(^{2}\overline{E})_{2c}\ominus(^{1}\overline{E}_{1})_{1a}(^{2}\overline{E}_{1})_{1b}]}_{\mathrm{Lower\
subspace}}\oplus\underbrace{(^{1}\overline{E}_{1})_{1a}(^{2}\overline{E}_{1})_{1b}}_{\mathrm{Upper\
subspace}}.$ (2)

As suggested by Eq. (1) and Eq. (2), in the following we refer to the
unoccupied band subspace as the obstructed phase because it is formally
compatible with an obstructed atomic limit in terms of its IRREPs content, and
we call the occupied band subspace the fragile phase because it can only be
written as a subtraction of two atomic limits.

###  II.2 Setup and topology for MSG77.18

As there are no symmetry constraints in the $k_{z}$ direction for MSG75.5, we
can freely consider topologies on the planes $k_{z}=0$ and $k_{z}=\pi$
independently. This is not the case in MSG77.18, as was discussed in Bouhon
_et al._ (2020a). This MSG is similar to MSG75.5, except that all $C_{4}$
rotations are replaced by screw rotations $C_{4_{2}}=(C_{4}|00\frac{1}{2})$,
and the non-symmorphic time-reversal is replaced by $(E|\tau_{d})^{\prime}$
with
$\tau_{d}=\boldsymbol{a}_{1}/2+\boldsymbol{a}_{2}/2+\boldsymbol{a}_{3}/2$. The
non-symmorphic screw symmetry imposes connectivity constraints along the
$k_{z}$ direction which prevent us from globally gapping the band structure in
the 3D BZ. However, we can still gap the band structure on the planes
$k_{z}=0$ and $k_{z}=\pi$. We constructed a model in Bouhon  _et al._ (2020a),
summarized in appendix A.2, which realizes a gapped band structures on these
planes. Our model is based on mWP $2a$ in MSG77.18, with sites
$\boldsymbol{r}_{A}=\boldsymbol{a}_{1}/2+z\boldsymbol{a}_{3}$,
$\boldsymbol{r}_{B}=\boldsymbol{a}_{2}/2+(z+1/2)\boldsymbol{a}_{3}$, where we
specialize to $z=0$. This model agrees with the model for MSG75.5 on the plane
$k_{z}=0$, and therefore everything we find for MSG75.5 holds on the plane
$k_{z}=0$ of MSG77.18 as well.

To satisfy the symmetry constraints, we must necessarily have two nodal points
along the $\overline{\Gamma\text{Z}}$ line and two other nodal points on the
$\overline{\text{MA}}$ line, resulting in a semi-metallic phase as studied by
e.g. Bouhon and Black-Schaffer (2017a, b); Mathai and Thiang (2017); Sun  _et
al._ (2018); Slager  _et al._ (2017); Yang and Nagaosa (2014); Slager  _et
al._ (2016); Zou  _et al._ (2019); Wieder  _et al._ (2020); Călugăru  _et al._
(2019); Lin and Hughes (2018). These Weyl nodes manifest through Fermi arcs in
the surface spectrum, and as gapless states in the hinge spectrum as we show
in Fig. 5. Furthermore, the pair of Weyl points on one vertical line are
related by $C_{2}T$ symmetry and must thus have equal chirality, say positive
chirality for the Weyl points on $\overline{\Gamma\text{Z}}$ Bouhon  _et al._
(2020a). We can thus define a vertical cylinder surrounding the $k_{z}$-axis
over which the Chern number must be $+2$. By the Nielsen-Ninomiya theorem this
must in turn be compensated by the Weyl points appearing on the
$\overline{\text{MA}}$ line contributing a Chern number of $-2$. We thus see
that upon employing the sub-dimensional analysis we can predicatively
enumerate in-plane non-triviality and, accordingly, necessarily present Weyl
nodes that are the consequence of the in-plane topology Bouhon  _et al._
(2020a). In the rest of this paper, we further explore the consequences of
these topological structures in MSG75.5 and MSG77.18.

##  III Corner charges

To elucidate the physical consequences of magnetic fragile phases in sub-
dimensional topologies, we begin by considering the corner charges present in
the system. Corner charges were studied in detail, for 2D non-magnetic
systems, in Benalcazar  _et al._ (2019); Schindler  _et al._ (2019). They can
be related to electric multipole moments, as described in Benalcazar  _et al._
(2017). For an obstructed atomic or fragile insulator it can happen that
charge neutrality and space-group symmetry are incompatible. Symmetry then
necessitates an imbalance of ionic and electronic degrees of freedom, which
gives rise to a filling anomaly $\chi$, defined as:

$\chi=(\\#\mathrm{Electronic\ sites}-\\#\mathrm{Ionic\ sites})\ \mathrm{mod}\
n$ (3)

Symmetry-allowed perturbations of the edges and corners can change this number
in general, but it is always well-defined modulo some number $n$ related to
the order of the symmetry. For the corner charges in MSG75.5 and MSG77.18,
$n=4$ due to the fourfold rotation symmetry.

If the edges are insulating, then the excess charge must localize at the
corners of the system. If $\chi$ is incompatible with the order of the
symmetry, then there will be fractional charges at the corners. This charge on
the corner is only well-defined in the absence of an edge state, as edge
states generically allow charge to flow away from the corner. Corner charges
are therefore closely linked with higher-order topological insulators (HOTIs),
as explored in Schindler  _et al._ (2018); Slager  _et al._ (2015); Wieder
_et al._ (2020); Călugăru  _et al._ (2019); Benalcazar  _et al._ (2019);
Khalaf  _et al._ (2018); Van Miert and Ortix (2018); Khalaf  _et al._ (2019).
We find that this allows for a counting procedure to determine the excess
charge in MSG75.5. We only consider half-filling, so that every ionic and
electronic site contributes a single charge.

###  III.1 Corner charges in MSG75.5

As $\mathrm{MEBR}_{75.5}^{2b}$ corresponds to a trivial insulator, with the
electrons (center of band charges) localized at the ionic sites, its total
filling anomaly must be zero. It follows that the filling anomalies of the
occupied and unoccupied subspaces of the split EBR in MSG75.5 must sum to zero
(modulo 4). We can therefore determine the filling anomaly of the fragile
phase (i.e. the occupied subspace) at half-filling by studying the atomic
obstructed phase (i.e. the unoccupied subspace) and counting the ionic and
electronic sites in the system (this only works because of the vanishing bulk
polarization, see Schindler  _et al._ (2019)). For this purpose, we redraw
Fig. 1 with the sites of the magnetic WP$2a$ included. This corresponds to the
non-magnetic WPs $1a$ and $1b$ and is shown in Fig. 2.

Figure 2: Same cuts for the $k_{z}=0$ plane of MSG75.5/MSG77.18 as in Fig. 1b)
and c), but including the WPs where the electrons localize. The electrons
localize on the smaller (black/green) sites, but the ions sit on the larger
(red/blue) sites.

Counting the total number of electronic and ionic sites gives for the cut in
Fig. 2a) a filling anomaly of $\chi=41-40=1$. Note, however, that whether or
not we include the electronic sites on the boundary is a matter of convention,
as they only represent the localization centers of the electrons (they are not
real sites in our model). We can therefore discount charges on the edge, but
must do so in a $C_{4}$ symmetric fashion, as discussed above. We thus expect
the obstructed phase to have an excess charge of $1$ mod $4$ electrons, and
the fragile phase must then have a compensating excess charge of $3$ mod $4$
electrons. The same counting gives for the diagonal cut in 2b) an anomaly of
$\chi=25-16=9$, which gives the same $\chi$ of $1$ mod $4$ in the obstructed
phase.
To determine whether or not this filling anomaly gives rise to corner states,
we must determine whether there is any excess charge localized on the edge of
the system, which could result in a conducting edge. We show the counting of
charges on the edge in Fig. 3, together with the edge spectra for the relevant
cuts.

Figure 3: Counting of charges on the edge for MSG75.5. The arrows indicate
which directions are considered to be periodic. The greyed out orbitals stem
from adjacent unit cells and the black dotted lines indicate a possible choice
of cell for the diagonal case. This is shown for the straight cut in panels a)
and b) and the diagonal cut in panel c) and d), with associated edge band
structures. The edge bands were calculated using the PythTB package Coh and
Vanderbilt (2016), with 150 unit cells perpendicular to the edge.

To count the orbitals, we must take care not to overcount sites which are
periodic images of each other (indicated by greyed out orbitals). This gives a
filling anomaly of $\chi=7-7=0$ for Fig. 3a), and $14-12=2$ for Fig. 3b).
Note, however, that we can remove charges on the edge, as long as we remove
them symetrically from both edges, e.g. the edge charges are quantized mod 2.
We therefore do not expect any fractional charges on the edge for either cut,
and therefore expect quantized corner charges. We noted in Ref. Bouhon  _et
al._ (2020a), that we have a quantized Berry phase of $\pi$ in the fragile
subspace. This does not give rise to a topological edge state because the
ionic sites are shifted from the origin by the same amount as the electronic
sites, as discussed in section V. If this relation is violated, we expect an
edge state to arise. We show the effect of removing various orbitals on the
edge on the edge spectrum in Fig. 9 in Appendix C.1. This confirms that we can
get in-gap states by removing edge orbitals. When we remove pairs of orbitals
at one edge, there are no topological in-gap edge states.

We note that this counting can also be used to predict the split of states
into the occupied/unoccupied space in this case. If there are $N$ total states
in the system, we expect $N/2-1$ states below the gap, $4$ corner states in
the gap and $N/2-3$ states above the gap. Note, however, as shown in Fig. 3c)
and d), that there are residual, model-specific, edge branches in the system
which slightly extend into the gap but are not genuine topological in-gap
states. These likely come from a nearby symmetry which our model only breaks
weakly. We therefore expect some residual charge on the edges, but this charge
is not topologically protected. To compute the corner charges, we therefore
sum over the charge from the occupied bands contained in a region with finite
thickness including the edge. We show the total charge for both cuts in Fig.
4, together with the associated spectra and typical corner states.

Figure 4: Corner charges and spectrum for the straight cut a) and b) and
diagonal cut c) and d) respectively for MSG75.5. On the top, we show the
charge distribution, with Fermi level set at the top of the gap, above the
corner states. Red indicates excess charge (relative to the center), blue a
deficit of charge where we sum over all occupied states. We confirm
numerically that the excess electronic charge in a region away from the center
is $3$ for both cuts. In the lower panels we show the spectrum, including the
absolute value squared of a typical corner state. In both cuts, there are
$N/2-1$ occupied states, $4$ corner states and $N/2-3$ unoccupied states,
where $N$ is the total number of states. All calculations were done using the
PythTB package Coh and Vanderbilt (2016).

To confirm that these are indeed corner charges, we also plot the same system
with a single spin removed on the boundary in Fig. 11 in Appendix C.2. This
violates the integer quantization of charges (since we are removing one
orbital while we remain at half-filling) and naturally leads to the appearance
of an edge state with edge charges. In Fig. 11, we clearly see that the edge
states dominate the corner charges, illustrating that, in contrast, Fig. 4
displays corner charges. We also confirm numerically that the excess charge in
a region including the boundary is $3$ when fixing the Fermi energy above the
corner states but below the upper (conduction) subspace. The sub-dimensional
topology in MSG75.5 thus hosts corner charges. As the unitary symmetries do
not dictate the $k_{z}$ dependence for MSG75.5, we expect that these $C_{4}$
symmetry protected corner charges are present for all values of $k_{z}$,
leading to hinge states.

###  III.2 Corner charges in MSG77.18

The translational symmetry in the $z$-direction for MSG75.5 allows for the
existence of hinge states that can be traced to the corner states of the
fragile topology at $k_{z}=0$ and $k_{z}=\pi$. The screw symmetry in MSG77.18
breaks this translational symmetry, and connects the planes at $k_{z}=0$ and
$k_{z}=\pi$. Our model for MSG77.18 was introduced in Bouhon  _et al._
(2020a), and is discussed in Appendix A.2. It agrees with the model for
MSG75.5 on the plane $k_{z}=0$, and we therefore expect corner charges on this
plane. As we move along the $k_{z}$ direction, the screw symmetry necessitates
the existence of Weyl nodes along $\overline{\Gamma\mathrm{Z}}$ and
$\overline{\mathrm{MA}}$, which leads to Fermi arcs in the surface spectrum.
We denote the $k_{z}>0$ coordinates of these Weyl nodes as
$k_{\overline{\Gamma\mathrm{Z}}}$ and $k_{\overline{\mathrm{MA}}}$. In the
hinge spectrum, we then expect a gap closing for all
$k_{z}\in\big{[}\mathrm{min}\big{(}k_{\overline{\Gamma\mathrm{Z}}},k_{\overline{\mathrm{MA}}}\big{)},\mathrm{max}\big{(}k_{\overline{\Gamma\mathrm{Z}}},k_{\overline{\mathrm{MA}}}\big{)}\big{]}$.
We plot the hinge spectrum for the straight and diagonal cut in Fig. 5.

Figure 5: Hinge spectrum and edge spectrum in the $k_{z}=\pi$ plane for a slab
calculation of MSG77.18. Panels a) and b) correspond to straight cuts. Panels
c) and d) present data for the diagonal cut. All calculations were done using
the PythTB package Coh and Vanderbilt (2016)

We note that the symmetry $k_{z}\rightarrow-k_{z}$ is not maintained for
edge/corner states, as the non-symmorphic $C_{2}T$ symmetry (relating $k_{z}$
to $-k_{z}$) is broken at the hinge. We also note that we get clear in-gap
states for the hinge spectrum in the diagonal cut. We show the equivalent of
Fig. 4 for the $k_{z}=\pi$ plane of 77.18 in Fig. 6.

Figure 6: Corner charges for the straight cut a) and b) and diagonal cut c)
and d) respectively for the $k_{z}=\pi$ plane of MSG77.18. On the top, we show
the charge distribution, with Fermi level fixed to the same value as for
MSG75.5, shown in Fig. 4. Red indicates excess charge (relative to the
center), blue a deficit of charge. We confirm numerically that the excess
electronic charge in a region away from the center is $1$ for the straight
cut. In the lower panels we show the spectrum, including the absolute value
squared of a typical corner state. In the straight cut, there are $N/2-3$
occupied states, $4$ corner states, and $N/2-1$ empty states, with $N$ the
total number of states. All calculations were done using the PythTB package
Coh and Vanderbilt (2016).

We note that the diagonal cut actually hosts edge modes on this plane, and
that the corner charge differs from that on the $k_{z}=0$ plane. This
illustrates that the counting approach, which we adopted for MSG75.5 (or
equivalently the $k_{z}=0$ plane of MSG77.18), breaks down on the plane
$k_{z}=\pi$. Thus care must be taken when extending counting schemes in 2D
such as those discussed in Benalcazar  _et al._ (2019); Schindler  _et al._
(2019) to full 3D models. This difference between the planes can be understood
by considering more carefully where the electrons localize on the plane
$k_{z}=\pi$ of MSG77.18, as we describe in section V. The counting for the
$k_{z}=0$ plane of MSG77.18 only works because we have a direct map to an
explicit 2D model, where the obstructed limits are known.

##  IV Real space invariants and twisted boundary conditions

Having explored the corner and edge spectrum, we will turn to relating the
bulk and boundary features from a Wilson flow perspective in the next Section.
However, before we move to this main topic, we shortly comment on the
connection with the recently introduced concepts of twisted boundary
conditions (TBC) Song  _et al._ (2020); Peri  _et al._ (2020) and real-space
invariants. In particular, we note that on a single plane (e.g. $k_{z}=0$) of
MSG75.5, everything we have discussed is exclusively protected by $C_{4}$
symmetry. As a result, we can directly relate to the results as discussed in
Ref. Song  _et al._ (2020), providing an alternative perspective on fragile
phases. This also serves to connect our work to other recent works, and can be
relevant in an experimental setting Peri  _et al._ (2020). This formalism is
most transparent when the phase being considered is gapless and when there are
no orbitals on the boundary between regions related by $C_{4}$ symmetry. We
therefore focus exclusively on the $k_{z}=0$ plane of MSG75.5/MSG77.18, using
the diagonal cut illustrated in Fig. 1c), as MSG77.18 hosts a gapless phase on
the $k_{z}=\pi$ plane for this cut.

This plane hosts wallpaper group p4 with SOC but without TRS. Using the
expressions for the real-space invariants (RSIs) found in Song  _et al._
(2020), we find that both the occupied and the unoccupied subspace have
nonzero RSIs, the expressions of which can be found in appendix D. This
implies a non-trivial flow under twisted boundary conditions (TBC), where the
coupling between the $C_{4}$ symmetric sectors is twisted by a factor
$\lambda=e^{i\theta}$. The spectrum under this twisting is shown in Fig. 7,
and we check that it agrees with the flow predicted from the RSIs. Note also
the presence of the corner states which, as they are not cut by the TBCs, do
not flow.

Figure 7: Twisted boundary conditions for a finite model of the $k_{z}=0$
plane of MSG75.5. The hoppings between adjacent regions (orange arrows) in the
clockwise/counterclockwise direction are multiplied by $\lambda=e^{\pm
i\theta}$. The hoppings between diagonal regions (purple arrows) are
multiplied by Re($\lambda^{2}$).

The RSIs can also be used to predict whether or not a set of bands are
fragile. We find, as anticipated from the above discussion, that the lower
band subspace is fragile, whereas the upper subspace is not. We refer for
further details to Appendix D.

##  V Bulk-edge correspondence

The Wilson loop spectrum provides a powerful tool for the bulk-edge
correspondence, which we here consider for the different edge terminations and
at the different $k_{z}$-planes. We recall that upon integrating the Wilson
loop along the momentum direction ($k_{\perp}$) perpendicular to the edge
direction ($x_{\parallel}$), the Wilson phases give the $x_{\perp}$-component
of the center of charge of the occupied band states, thus relating to a
specific WP, which can then be used to predict an atomic obstruction (i.e. the
displacement of band charges to a WP distinct from the WP of the atomic
orbitals) Zak (1989) and, subsequently, an edge-specific charge anomaly Su
_et al._ (1979). Importantly, this argument is conclusive only when the Wilson
loop spectrum is quantized as an effect of symmetries, since different WPs
only then relate to topologically distinct phases (thus avoiding the adiabatic
transfer of charges from one WP to an other). In the following we make use of
the $\\{0,\pi\\}$ Berry phases (given as the sum of all Wilson loop phases)
protected by $C_{2}T$ symmetry Bouhon  _et al._ (2019) and the
$\mathbb{Z}_{2}$ polarization protected by TRS and $C_{2}$ symmetry Lau  _et
al._ (2016) associated with Kramers degeneracies of the Wilson loop spectrum.

We start with a discussion of the effect of the symmetries of MSG75.5 and
MSG77.18 on the Wilson loops in relation to (i) the sub-dimensional bulk
topologies ($k_{z}=0,\pi$), and (ii) the two possible choices of geometries
(straight versus diagonal, see Fig. 8 a)). This allows us to determine which
topological invariant ($\boldsymbol{Z}_{2}$ Berry phase and
$\boldsymbol{Z}_{2}$ polarization) is associated with an edge geometry, as
well as its actual value indicated by symmetry. We then motivate the bulk-edge
correspondence for each case.

###  V.1 Topological insights from Wilson loop

We show in Fig. 8 the Wilson loop spectrum for MSG77.18 (and MSG75.5, see
below) corresponding to a straight geometry at the plane b) $k_{z}=0$ and d)
$k_{z}=\pi$, and corresponding to the $45^{\circ}$-rotated (diagonal) geometry
at the plane c) $k_{z}=0$ and e) $k_{z}=\pi$. Fig. 8 a) shows the directions
of integration in the momentum space with a full-line double arrow (blue) for
the straight geometry and a dashed double arrow (orange) for the diagonal
geometry. The Wilson loop spectrum for MSG75.5 is qualitatively the same as in
Fig. 8 b) for the straight geometry and Fig. 8 c) for the diagonal geometry,
both for the planes $k_{z}=0$ and $k_{z}=\pi$. Indeed, for MSG75.5 the sub-
dimensional topologies at $k_{z}=0$ and $k_{z}=\pi$ are the same.

Figure 8: a) Planes of the Brillouin zone at $k_{z}=0$ ($\Gamma$, X, M, X’)
and $k_{z}=\pi$ (Z, R, A, R’), along straight directions
$\\{\boldsymbol{b}_{1},\boldsymbol{b}_{2}\\}$, and diagonal directions
$\\{\boldsymbol{b}_{1}+\boldsymbol{b}_{2},-\boldsymbol{b}_{1}/2+\boldsymbol{b}_{2}/2\\}$.
b)-e) Wilson loop spectrum for MSG77.18 integrated (b,d) along the straight
direction $\boldsymbol{b}_{1}$ (while varying $k_{y}$), and (c,e) along the
diagonal direction $\boldsymbol{b}_{1}+\boldsymbol{b}_{2}$ (while varying
$k_{2}$), for the plane (b,c) $k_{z}=0$ and (d,e) $k_{z}=\pi$. The respective
paths of Wilson loop integration for the straight geometry is
$l_{k_{y}}=[(0,k_{y})+\boldsymbol{b}_{1}\leftarrow(0,k_{y})]$ (dashed orange),
and for the diagonal geometry is
$l_{k_{2}}=[(0,k_{2})+\boldsymbol{b}_{1}+\boldsymbol{b}_{2}\leftarrow(0,k_{2})]$
(full blue), shown as double arrows in a).

####  V.1.1 Sub-dimensional topologies: $k_{z}=0$ vs $k_{z}=\pi$

Let us now explain the differences between the Wilson loops in terms of the
sub-dimensional topologies, i.e. $k_{z}=0$ versus $k_{z}=\pi$, for MSG77.18.
The sub-dimensional topology of the $k_{z}=0$ plane for MSG77.18 is identical
to sub-dimensional topology of MSG75.5, with the high-symmetry points
$\\{\Gamma,\text{M}\\}$ as TRIMPs (time reversal invariant momentum points)
and with $[C_{2}T]^{2}=+1$ which protects the complete winding of two-band
Wilson loop, thus indicating a nontrivial Euler class (fragile) topology
Bouhon  _et al._ (2019, 2020a). The winding of Wilson loop does not depend on
the geometry (straight or diagonal) and we accordingly find complete Wilson
loop windings in Fig. 8 b) and c). At $k_{z}=\pi$, the sub-dimensional
topology for MSG77.18 is characterized by the TRIMPs
$\\{\text{R},\text{R'}\\}$ (which differ from the TRIMPs
$\\{\text{Z},\text{A}\\}$ for MSG75.5) and with $[C_{2}T]^{2}=-1$ which
implies the Kramers degeneracy of the energy bands over the whole momentum
plane, while discarding Euler class topology Bouhon  _et al._ (2020a).
Accordingly, we find that there is no complete winding of the Wilson loops in
Fig. 8 d) and e).

####  V.1.2 Symmetries and quantizations of Wilson loop: straight vs diagonal
geometry

We now turn to a detailed account of the quantizations and symmetries of
Wilson loop spectra protected by symmetries for the two geometries, i.e.
straight versus diagonal. Our aim is to determine which topological invariant
($\boldsymbol{Z}_{2}$ Berry phase and $\boldsymbol{Z}_{2}$ polarization) can
be associated with an edge geometry and when it is symmetry indicated (i.e.
with a definite value). These results combine the effects of $C_{2}$ symmetry,
and the non-symmorphic anti-unitary symmetries TRS and $C_{2}T$. The
derivation of the constraints on the Wilson loop due to the non-symmorphic
anti-unitary symmetries (TRS and $C_{2}T$) is given in Appendix B. Below we
write $\mathcal{W}[l_{k}]$ the Wilson loop with the spectrum (phases)
$\\{\varphi_{1}(k),\varphi_{2}(k)\\}$ evaluated over the non-contractible base
path $l_{k}$ that is parametrized by $k=k_{y}$ in the straight geometry, and
by $k=k_{2}$ in the diagonal geometry [Fig. 8 a)].

In the straight geometry, the Wilson loop phases at $k_{y}=0$ and $k_{y}=\pi$
are quantized to $(\varphi_{1},\varphi_{2})=(0,\pi)$ by the $C_{2}$ symmetry,
i.e. it follows from the $C_{2}$ eigenvalues of the band eigenstates at the
high-symmetry points $\Gamma$ and M Alexandradinata  _et al._ (2014); Bouhon
and Black-Schaffer (2017c); Bouhon  _et al._ (2019), i.e. the Berry phase must
be
$\gamma_{B}[l_{k_{y}=0,\pi}]=\varphi_{1}+\varphi_{2}\>\mathrm{mod}\>2\pi=\pi\>\mathrm{mod}\>2\pi$.
This is true on both planes $k_{z}=0,\pi$. Then on the $k_{z}=0$ plane, by the
$\boldsymbol{Z}_{2}$ quantization of the Berry phase protected by $C_{2}T$
($[C_{2}T]^{2}=+1$), we conclude that the Berry phase must be $\pi$ for all
$k_{y}$. On the $k_{z}=\pi$ plane, we have $[C_{2}T]^{2}=-1$ and there is the
question of the quantization of the Berry phase. We have shown in Appendix B
that the non-symmorphic $C_{2}T$ imposes
$\varphi_{2}(k_{y})=-\varphi_{1}(k_{y})+\pi\>\mathrm{mod}\>\ 2\pi$ on the
spectrum of the Wilson loop. We thus get
$\gamma_{B}(k_{y})=\varphi_{1}(k_{y})+\varphi_{2}(k_{y})=\pi\>\mathrm{mod}\>2\pi$
for all $k_{y}$. We note that the non-symmorphic TRS furthermore requires
$\varphi_{2}(-k_{y})=\varphi_{1}(k_{y})+\pi\>\mathrm{mod}\>\ 2\pi$ (see
Appendix B) which explains the global structure of the Wilson loops in 8 b)
and d). Also, since there is no path $l_{k_{y}}$ for which TRS squares to $-1$
for all $k_{x}$, no $\boldsymbol{Z}_{2}$ polarization can be defined in the
straight geometry.

In the diagonal geometry, the Wilson loop is quantized by $C_{2}$ at
$k_{2}=\pm\pi/\sqrt{2}$ to $(\varphi_{1},\varphi_{2})=(0,0)$, for $k_{z}=0$
and $k_{z}=\pi$ (this follows from the fact that the $C_{2}$ eigenvalues of
the occupied-band eigenstates at X and at R are all equal to $\mathrm{i}$ or
$-\mathrm{i}$), such that the Berry phase is zero. (At $k_{2}=0$, and
$k_{z}=0$ or $k_{z}=\pi$, on the contrary, there is no quantization of the
Wilson loop from $C_{2}$.) At $k_{z}=0$, the $\boldsymbol{Z}_{2}$ quantization
of the Berry phase (protected by $C_{2}T$ with $[C_{2}T]^{2}=-1$) implies it
must be zero for all $k_{2}$, in agreement with Fig. 8 c). At $k_{z}=\pi$, the
non-symmorphic $C_{2}T$ symmetry requires
$\varphi_{2}(k_{2})=-\varphi_{1}(k_{2})$ (see Appendix B), hence again the
Berry phase must be zero for all $k_{2}$, in agreement with Fig. 8 e). We now
address the effect of non-symmorphic TRS which requires
$\varphi_{2}(k_{2})=\varphi_{1}(-k_{2})$ for $k_{z}=0$ and $k_{z}=\pi$ (see
Appendix B), which explains the global structure of the Wilson loops in Fig. 8
c) and e). Furthermore, from the square of TRS,
$T^{2}=-\mathrm{e}^{-\mathrm{i}(k_{x}+k_{y}+k_{z})}$, we predict one diagonal
per $k_{z}$-plane along which $T^{2}=-1$. We find $T^{2}=-1$ for
$k_{x}=k_{y}\>\mathrm{mod}\>2\pi$ at $k_{z}=0$, i.e. on the diagonal
$\overline{\Gamma\text{M}}$ (connecting the TRIMPs $\Gamma$ and M), and
$T^{2}=-1$ for $k_{x}=k_{y}+\pi\>\mathrm{mod}\>2\pi$ at $k_{z}=\pi$, i.e. on
the diagonal $\overline{\text{RR'}}$ (connecting the TRIMPs R and R’). This
results in the presence of Kramers degeneracies of the Wilson loop spectrum at
$k_{2}=0$ for $k_{z}=0$, namely $(\varphi_{1},\varphi_{2})=(\pi,\pi)$, and at
$k_{2}=\pm\pi/\sqrt{2}$ for $k_{z}=\pi$, namely
$(\varphi_{1},\varphi_{2})=(0,0)$. Moreover, the combination of $C_{2}$ and
TRS leads to the definition of a $\nu[l]\in\boldsymbol{Z}_{2}$ polarization on
these diagonals which is directly indicated by the phases of the Kramers
degeneracies, i.e.

$\displaystyle\nu^{(k_{z}=0)}[l_{k_{2}=0}]$ $\displaystyle=1,$ (4)
$\displaystyle\mathrm{and}~{}\nu^{(k_{z}=\pi)}[l_{k_{2}=\pi/\sqrt{2}}]$
$\displaystyle=0.$

We show below that these have important consequences for the bulk-edge
correspondence.

###  V.2 Sub-dimensional charge anomalies and edge states

Let us first summarize the findings of the previous section. The Berry phase
is $\pi$ in the straight geometry on both planes $k_{z}=0$ and $k_{z}=\pi$.
The Berry phase is $0$ in the diagonal geometry on both planes $k_{z}=0$ and
$k_{z}=\pi$. There is no $\boldsymbol{Z}_{2}$ polarization in the straight
geometry. In the diagonal geometry, there a nontrivial $\boldsymbol{Z}_{2}$
polarization on $l^{(k_{z}=0)}_{k_{2}=0}=\overline{\Gamma\text{M}}$, and a
trivial one on $l^{(k_{z}=\pi)}_{k_{2}=0}=\overline{\text{RR'}}$.

####  V.2.1 Straight geometry

We consider here the bulk-edge correspondence at $k_{z}=0$ for which the
phases for MSG75.5 and MSG77.18 are the same. We use MSG75.5 to give a
detailed account of the bulk-edge correspondence in that momentum plane.

Let us start with a discussion of the straight edge geometry. Let us argue
explicitly for MSG75.5 that the $\pi$-Berry phase does not indicate a charge
anomaly. Indeed, the Wyckoff position $2b$ of the atomic orbitals and the
Wyckoff positions $2a$ of the obstructed band charges both have, component-
wise, one site centered in the unit cell and one site shifted to the unit cell
boundary. I.e. defining the (unordered) sets of component-wise positions
relative to the unit cell center

$\displaystyle\\{\boldsymbol{r}^{(2b)}_{A,i},\boldsymbol{r}^{(2b)}_{B,i}\\}$
$\displaystyle=\\{0,a/2\\}\>\mathrm{mod}\>a,$ (5)
$\displaystyle\mathrm{and}~{}\\{\boldsymbol{r}^{(2a)}_{C,i},\boldsymbol{r}^{(2a)}_{D,i}\\}$
$\displaystyle=\\{0,a/2\\}\>\mathrm{mod}\>a,$

for $i=x,y$, there is no difference. Therefore, even in the case of an
obstruction (from WP $2b$ to WP $2a$) there is no charge anomaly, i.e. no
mismatch between the number of charges localized at the distinct WPs, in a
ribbon system with straight edge cuts and assuming that a slide of the ribbon
contains an integer number of unit cells (see Appendix C.1 with the results
for a fractional number of unit cells). This can also be readily checked
through direct counting. We thus conclude that there is no topological edge
branch along the straight edges. We can readily transpose this conclusion is
directly for MSG77.18 at $k_{z}=0$, as well as at $k_{z}=\pi$ for MSG75.5.

Since the Berry phase is also $\pi$ at $k_{z}=\pi$ for MSG77.18, we find
similarly to $k_{z}=0$ that there is no edge state in the straight geometry
(assuming an integer number of unit cells per slice of the ribbon). We
conclude that in the straight geometry there no effect of the sub-dimensional
topology (i.e. comparing $k_{0}=\pi$ and $k_{z}=\pi$) on the bulk-edge
correspondence for MSG77.18.

####  V.2.2 Diagonal geometry

To fix ideas, we assume that $x_{1}=x_{\perp}$ is the direction perpendicular
to the edge, and $x_{2}=x_{\parallel}$ the direction parallel to it. Then, we
define the diagonal unit cell through
$\boldsymbol{a}_{\perp}=\boldsymbol{a}_{1}/2+\boldsymbol{a}_{2}/2=a/\sqrt{2}\hat{e}_{1}$
and
$\boldsymbol{a}_{\parallel}=-\boldsymbol{a}_{1}+\boldsymbol{a}_{2}=\sqrt{2}a\hat{e}_{2}$,
such that invariance under a translation by $\boldsymbol{a}_{\parallel}$ is
satisfied. Note that this unit cell is different from the edge unit cell in
figure 5. Writing the atomic positions in the diagonal axes, i.e.
$\boldsymbol{r}=(x_{1},x_{2})$, the perpendicular component ($x_{1}$) of the
Wyckoff positions in the diagonal geometry are

$\displaystyle(\boldsymbol{r}^{(2b)}_{A,1},\boldsymbol{r}^{(2b)}_{B,1})$
$\displaystyle=\dfrac{a}{2\sqrt{2}}(1,1)\>\mathrm{mod}\>\dfrac{a}{\sqrt{2}},$
(6)
$\displaystyle\mathrm{and}~{}(\boldsymbol{r}^{(2a)}_{C,1},\boldsymbol{r}^{(2a)}_{D,1})$
$\displaystyle=(0,0)\>\mathrm{mod}\>\dfrac{a}{\sqrt{2}}.$

The zero Berry phase obtained for the diagonal geometry thus indicates that
there must be an even number of charges at WP $2b$ and at WP $2a$. Assuming an
obstruction of the charges from WP $2b$ to WP $2a$, the diagonal edge cut
leads to a total charge anomaly of $\pm 2e$ for the ribbon if we assume that a
slide of the ribbon contains an integer number of diagonal unit cells (see
Appendix C.1 with the results for a fractional number of unit cells). This
means that we have a charge anomaly of $\pm e$ per diagonal edge.

Let us now make use of the $\nu\in\boldsymbol{Z}_{2}$ polarization, which we
have seen is well defined in the diagonal geometry. At $k_{z}=0$, we have
found $\nu_{k_{2}=0}=1$ which corresponds to the Kramers degenerate Wilson
loop phases $(\varphi_{1},\varphi_{2})=(\pi,\pi)$. This implies that the band
charges are not obstructed, i.e. they can be located at WP $2b$. There is thus
no charge anomaly and following there is no edge state. This conclusion holds
for MSG75.5 at $k_{z}=0$ and $k_{z}=\pi$, and for MSG77.18 at $k_{z}=0$ only.

Considering now MSG77.18 at $k_{z}=\pi$, we have found
$\nu_{k_{2}=\pi/\sqrt{2}}=0$ which corresponds to the Kramers degenerate
Wilson loop phases $(\varphi_{1},\varphi_{2})=(0,0)$. This implies an
obstruction of the band charges, i.e. shifted from the atomic mWP $2a$ of
MSG77.18. We have argued above that with an obstruction there is a charge
anomaly of $\pm e$ per diagonal edge (assuming an integer number of diagonal
unit cells in one slice of the ribbon). We also know that the
$\boldsymbol{Z}_{2}$ polarization predicts the presence of an odd number
integer-valued electronic charge at a single edge Lau  _et al._ (2016), which
by virtue of the Kramers degeneracy means a half-integer-valued charge per
spin. The topological (helical) edge states take the form of an odd number of
edge Kramers pairs per edge. This is fully consistent with the numerical
results shown in Fig. 5 where we find one Kramers pair of edge states per
edge.

We finally conclude that there is a non-trivial effect of the sub-dimensional
topology on the bulk-edge correspondence that is observable in the diagonal
geometry.

##  VI Conclusion

In this work we have further explored the concept of sub-dimensional topology.
While these topologies have a connected EBRs and thus appear trivial, they
feature split EBRs on sub-dimensional spaces, such as planes, in the Brillouin
zone. These in-plane topologies are subsequently compensated by Weyl nodes and
have full dimensional non-trivial features. We in particular find that these
concepts can be related to more refined counting and symmetry indicator
arguments. Most notably, however, is the connection of these insights to
consequences on the edge. We find that the sub-dimensional topology results in
distinctive bulk-boundary signatures. These include hinge spectra and edge
states that have a distinct dependence on the perpendicular momentum,
underpinning the physical significance of this new topological concept. We
therefore hope that our results will result in the further exploration of
these features.

###### Acknowledgements.

R.-J. S. acknowledges funding from the Marie Skłodowska-Curie programme under
EC Grant No. 842901 and the Winton programme as well as Trinity College at the
University of Cambridge. G.F.L acknowledges funding from the Aker Scholarship.

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##  Appendix A Summary of the models

Here we reproduce the models used for MSG75.5 and MSG77.18. These models were
originally introduced in Bouhon  _et al._ (2020a), and are written in the
Bloch basis:

$|\varphi_{\alpha,\sigma},\boldsymbol{k}\rangle=\sum_{\boldsymbol{R}\in\boldsymbol{T}}\mathrm{e}^{\mathrm{i}\boldsymbol{k}\cdot(\boldsymbol{R}+\boldsymbol{r}_{\alpha})}|w_{\alpha,\sigma},\boldsymbol{R}+\boldsymbol{r}_{\alpha}\rangle,$
(7)

Where $\alpha\in\\{A,B\\}$ labels the sites in the unit cell, and
$\sigma\in\\{\uparrow,\downarrow\\}$ labels spin components. We choose to
order our basis as
$\boldsymbol{\varphi}=(\varphi_{A,\uparrow},\varphi_{A,\downarrow},\varphi_{B,\uparrow},\varphi_{B,\downarrow})$.
A more thorough analysis of these models in momentum space, including band
structures, can be found in Bouhon  _et al._ (2020a).

###  A.1 The model for MSG75.5

Our model for MSG75.5 is defined as:

$\displaystyle
H(\boldsymbol{k})=t_{1}f_{1}(\boldsymbol{k})\sigma_{z}\otimes\sigma_{z}$ (8)
$\displaystyle+t_{2}f_{2}(\boldsymbol{k})\sigma_{y}\otimes\mathbb{1}+t_{3}f_{3}(\boldsymbol{k})\sigma_{x}\otimes\mathbb{1}$
$\displaystyle+\lambda_{1}g_{1}(\boldsymbol{k})\mathbb{1}\otimes\sigma_{+}+\lambda_{1}^{*}g_{1}^{*}(\boldsymbol{k})\mathbb{1}\otimes\sigma_{-}$
$\displaystyle+\lambda_{2}g_{2}(\boldsymbol{k})\sigma_{x}\otimes\sigma_{+}+\lambda_{2}^{*}g_{2}^{*}(\boldsymbol{k})\sigma_{x}\otimes\sigma_{-},$

with $\sigma_{\pm}=(\sigma_{x}\pm\mathrm{i}\sigma_{y})/2$ and lattice form
factors

$\begin{array}[]{ll}f_{1}=\cos\boldsymbol{a}_{1}\boldsymbol{k}-\cos\boldsymbol{a}_{2}\boldsymbol{k},&g_{1}=\sin\boldsymbol{a}_{1}\boldsymbol{k}-\mathrm{i}\sin\boldsymbol{a}_{2}\boldsymbol{k},\\\
f_{2}=\cos\boldsymbol{\delta}_{1}\boldsymbol{k}-\cos\boldsymbol{\delta}_{2}\boldsymbol{k},&g_{2}=\sin\boldsymbol{\delta}_{1}\boldsymbol{k}-\mathrm{i}\sin\boldsymbol{\delta}_{2}\boldsymbol{k},\\\
f_{3}=\cos\boldsymbol{\delta}_{1}\boldsymbol{k}+\cos\boldsymbol{\delta}_{2}\boldsymbol{k},&\end{array}$
(9)

These are defined in terms of the bond vectors
$\boldsymbol{\delta}_{\left(\begin{subarray}{c}1\\\
2\end{subarray}\right)}=(\boldsymbol{a}_{1}\left(\begin{subarray}{c}-\\\
+\end{subarray}\right)\boldsymbol{a}_{2})/2$. We have assumed that
$\\{t_{1},t_{2},t_{3}\\}$ are real, while $\\{\lambda_{1},\lambda_{2}\\}$ can
be complex. We fix $t_{1},t_{2},t_{3}=1$, and
$\lambda_{1},\lambda_{2}=(1/2)\mathrm{e}^{\mathrm{i}\pi/5}$.

###  A.2 The model for MSG77.18

Our model for MSG77.18 is defined by adding an extra term $H^{\prime}$ to the
above model for MSG75.5:

$H^{\prime}(\boldsymbol{k})=H[f_{1},f_{2}^{\prime},f_{3}^{\prime},g_{1},g_{2}^{\prime}](\boldsymbol{k})+\\\
\rho_{1}h_{1}(\boldsymbol{k})\sigma_{x}\otimes\sigma_{z}+\rho_{2}h_{2}(\boldsymbol{k})\sigma_{y}\otimes\sigma_{z},$
(10)

Where $H(\boldsymbol{k})$ is given in Equation 8, and the lattice form factors
have been extended to 3D momentum space,

$\displaystyle f^{\prime}_{2}(\boldsymbol{k})$
$\displaystyle=\left(\cos\boldsymbol{\delta}^{\prime}_{1}\boldsymbol{k}-\cos\boldsymbol{\delta}^{\prime}_{2}\boldsymbol{k}+\cos\boldsymbol{\delta}^{\prime}_{3}\boldsymbol{k}-\cos\boldsymbol{\delta}^{\prime}_{4}\boldsymbol{k}\right)/2,$
(11) $\displaystyle f^{\prime}_{3}(\boldsymbol{k})$
$\displaystyle=\left(\cos\boldsymbol{\delta}^{\prime}_{1}\boldsymbol{k}+\cos\boldsymbol{\delta}^{\prime}_{2}\boldsymbol{k}+\cos\boldsymbol{\delta}^{\prime}_{3}\boldsymbol{k}+\cos\boldsymbol{\delta}^{\prime}_{4}\boldsymbol{k}\right)/2,$
$\displaystyle g^{\prime}_{2}(\boldsymbol{k})$
$\displaystyle=\left(\sin\boldsymbol{\delta}^{\prime}_{1}\boldsymbol{k}-\mathrm{i}\sin\boldsymbol{\delta}^{\prime}_{2}\boldsymbol{k}-\sin\boldsymbol{\delta}^{\prime}_{3}\boldsymbol{k}+\mathrm{i}\sin\boldsymbol{\delta}^{\prime}_{4}\boldsymbol{k}\right)/2,$
$\displaystyle h_{1}(\boldsymbol{k})$
$\displaystyle=\left(\sin\boldsymbol{\delta}^{\prime}_{1}\boldsymbol{k}+\sin\boldsymbol{\delta}^{\prime}_{2}\boldsymbol{k}+\sin\boldsymbol{\delta}^{\prime}_{3}\boldsymbol{k}+\sin\boldsymbol{\delta}^{\prime}_{4}\boldsymbol{k}\right)/2,$
$\displaystyle h_{2}(\boldsymbol{k})$
$\displaystyle=\left(\sin\boldsymbol{\delta}^{\prime}_{1}\boldsymbol{k}-\sin\boldsymbol{\delta}^{\prime}_{2}\boldsymbol{k}+\sin\boldsymbol{\delta}^{\prime}_{3}\boldsymbol{k}-\sin\boldsymbol{\delta}^{\prime}_{4}\boldsymbol{k}\right)/2,$

with
$\boldsymbol{\delta}^{\prime}_{1,2}=\boldsymbol{\delta}_{1,2}+\boldsymbol{a}_{3}/2$,
and
$\boldsymbol{\delta}^{\prime}_{3,4}=-\boldsymbol{\delta}_{1,2}+\boldsymbol{a}_{3}/2$,
and with new real parameters $\rho_{1},\rho_{2}\in\mathbb{R}$. We fix
$\rho_{1}=-1$ and $\rho_{2}=-2/5$.

##  Appendix B Symmetries of the Wilson loop due to non-symmorphic TRS and
$C_{2}T$ symmetry

We here derive the effect of non-symmorphic TRS and $C_{2}T$ symmetry on the
Wilson loop for the different geometries, i.e. straight versus diagonal shown
in Fig. 8 a). We do it explicitly for MSG77.18 but the final results apply to
MSG75.5 as well. Let us write
$\mathcal{W}[l_{\boldsymbol{k}_{0}}]=\langle\boldsymbol{u},\boldsymbol{k}_{0}+\boldsymbol{K}|\prod\limits_{\boldsymbol{k}}^{\boldsymbol{k}_{0}+\boldsymbol{K}\leftarrow\boldsymbol{k}_{0}}\mathcal{P}_{\boldsymbol{k}}|\boldsymbol{u},\boldsymbol{k}_{0}\rangle$
the Wilson loop over the occupied Bloch eigenvectors
$\\{|u_{n},\boldsymbol{k}_{0}\rangle\\}_{n=1,2}$ integrated over the base loop
$l_{\boldsymbol{k}_{0}}=[\boldsymbol{k}_{0}+\boldsymbol{K}\leftarrow\boldsymbol{k}_{0}]$
that crosses the Brillouin zone with $\boldsymbol{K}$ defined as the smallest
reciprocal lattice vector in that direction. We define the anti-unitary
representation of the non-symmorphic TRS for MSG77.18, $(E|\tau_{d})^{\prime}$
where $\tau_{d}=(\boldsymbol{a}_{1}+\boldsymbol{a}_{2}+\boldsymbol{a}_{3})/2$,
in the basis of the occupied Bloch eigenstates through
$\mathcal{T}(\boldsymbol{k})=\langle\boldsymbol{\psi},-\boldsymbol{k}|^{(E|\tau_{d})^{\prime}}|\boldsymbol{\psi},\boldsymbol{k}\rangle=\mathrm{e}^{\mathrm{i}\boldsymbol{k}\cdot\tau_{d}}\langle\boldsymbol{u},-\boldsymbol{k}|(\sigma_{x}\otimes-\mathrm{i}\sigma_{y})\mathcal{K}|\boldsymbol{u},\boldsymbol{k}\rangle=\mathrm{e}^{\mathrm{i}\boldsymbol{k}\cdot\tau_{d}}\langle\boldsymbol{u},-\boldsymbol{k}|(\sigma_{x}\otimes-\mathrm{i}\sigma_{y})|\boldsymbol{u}^{*},\boldsymbol{k}\rangle\mathcal{K}=\mathcal{U}\mathcal{K}$
where $\mathcal{K}$ is the complex conjugation and $\mathcal{U}$ is unitary
Bouhon  _et al._ (2020a). It is then convenient to write the unitary
representation with the phase factor of the non-symmorphicity removed, i.e.
$U(\boldsymbol{k})=\mathrm{e}^{-\mathrm{i}\boldsymbol{k}\cdot\tau_{d}}\mathcal{U}(\boldsymbol{k})$
(this will correspond below to taking the “periodic gauge” Wang  _et al._
(2016); Bouhon and Black-Schaffer (2017c, b) in the Wilson loop over a non-
contractible path of the torus Brillouin zone). Similarly to the symmorphic
case Alexandradinata  _et al._ (2014), the constraint imposed by
$(E|\tau_{d})^{\prime}$ on the Wilson loop is found to be
$U^{*}(\boldsymbol{k}_{0}+\boldsymbol{K})^{-1}\mathcal{W}^{T}[l_{-\boldsymbol{k}_{0}-\boldsymbol{K}}]U^{*}(\boldsymbol{k}_{0})=\mathcal{W}[l_{\boldsymbol{k}_{0}}]$
with
$l_{-\boldsymbol{k}_{0}-\boldsymbol{K}}=[-\boldsymbol{k}_{0}\leftarrow-\boldsymbol{k}_{0}-\boldsymbol{K}]$,
which we rewrite as

$\mathrm{e}^{-\mathrm{i}\boldsymbol{K}\cdot\tau_{d}}\mathcal{U}^{*}(\boldsymbol{k}_{0}+\boldsymbol{K})^{-1}\mathcal{W}^{T}[l_{-\boldsymbol{k}_{0}-\boldsymbol{K}}]\mathcal{U}^{*}(\boldsymbol{k}_{0})=\mathcal{W}[l_{\boldsymbol{k}_{0}}].$
(12)

We thus conclude that in the straight geometry, i.e. taking
$\boldsymbol{k}_{0}=(0,k_{y})$ and $\boldsymbol{K}=\boldsymbol{b}_{1}$, the
Wilson loop spectrum satisfies the following symmetry

$\left\\{\varphi_{n}[l_{(0,-k_{y})}]\right\\}_{n=1,2}=\left\\{\varphi_{n}[l_{(0,k_{y})}]+\pi\>\mathrm{mod}\>2\pi\right\\}_{n=1,2},$
(13)

while in the diagonal geometry, i.e. taking $\boldsymbol{k}_{0}=(0,k_{2})$ and
$\boldsymbol{K}=\boldsymbol{b}_{1}+\boldsymbol{b}_{2}$, the symmetry reads

$\left\\{\varphi_{n}[l_{(0,-k_{2})}]\right\\}_{n=1,2}=\left\\{\varphi_{n}[l_{(0,k_{2})}]\right\\}_{n=1,2}.$
(14)

We now consider the effect of the non-symmorphic $C_{2}T$ symmetry of
MSG77.18, $(C_{2}|\tau_{d})^{\prime}$, on the Wilson loop spectrum. Its anti-
unitary representation in the occupied Bloch eigenstates reads
$\mathcal{A}(\boldsymbol{k})=\langle\boldsymbol{\psi},-C_{2}\boldsymbol{k}|^{(C_{2}|\tau_{d})^{\prime}}|\boldsymbol{\psi},\boldsymbol{k}\rangle=\mathrm{e}^{\mathrm{i}C_{2}\boldsymbol{k}\cdot\tau_{d}}\langle\boldsymbol{u},-C_{2}\boldsymbol{k}|(\sigma_{x}\otimes\mathrm{i}\sigma_{x})|\boldsymbol{u}^{*},\boldsymbol{k}\rangle\mathcal{K}=\mathcal{R}\mathcal{K}$
where $\mathcal{R}$ is unitary. Defining
$R(\boldsymbol{k})=\mathrm{e}^{-\mathrm{i}C_{2}\boldsymbol{k}\cdot\tau_{d}}\mathcal{R}(\boldsymbol{k})$,
we then find, similarly to the case of TRS above,
$R^{*}(\boldsymbol{k}_{0}+\boldsymbol{K})^{-1}\mathcal{W}^{*}[l_{\boldsymbol{k}_{0}}]R^{*}(\boldsymbol{k}_{0})=\mathcal{W}[l_{\boldsymbol{k}_{0}}]$,
which we rewrite as

$\mathrm{e}^{-\mathrm{i}C_{2}\boldsymbol{K}\cdot\tau_{d}}\mathcal{R}^{*}(\boldsymbol{k}_{0}+\boldsymbol{K})^{-1}\mathcal{W}^{*}[l_{\boldsymbol{k}_{0}}]\mathcal{R}^{*}(\boldsymbol{k}_{0})=\mathcal{W}[l_{\boldsymbol{k}_{0}}].$
(15)

We therefore conclude that in the straight geometry
($\boldsymbol{k}_{0}=(0,k_{y})$, $\boldsymbol{K}=\boldsymbol{b}_{1}$) the
Wilson loop spectrum satisfies the following symmetry

$\left\\{\varphi_{n}[l_{(0,k_{y})}]\right\\}_{n=1,2}=\left\\{-\varphi_{n}[l_{(0,k_{y})}]+\pi\>\mathrm{mod}\>2\pi\right\\}_{n=1,2},$
(16)

while in the diagonal geometry ($\boldsymbol{k}_{0}=(0,k_{2})$,
$\boldsymbol{K}=\boldsymbol{b}_{1}+\boldsymbol{b}_{2}$) the symmetry reads

$\left\\{\varphi_{n}[l_{(0,k_{2})}]\right\\}_{n=1,2}=\left\\{-\varphi_{n}[l_{(0,k_{2})}]\right\\}_{n=1,2}.$
(17)

We emphasize that these Wilson loop symmetries hold similarly at $k_{z}=0$ and
at $k_{z}=\pi$ since $\boldsymbol{K}$ has no $k_{z}$ component. This is also
the reason why these results also apply to MSG75.5.

We conclude by noting that these results are fully consistent with the
computed Wilson loops in Fig. 8.

##  Appendix C Additional figures

In this Appendix we present additional figures to further detail the findings
presented in the main text. In particular, we present the edge spectrum which
results from removing various orbitals on the edges in appendix C.1, and we
present the charge distribution when removing a single spin on the entire edge
in appendix C.2.

###  C.1 Alternative edge termination

We noted in Section III of the main text that removing a single orbital at the
edge of the system can induce edge states. In Fig. 9 we show a selection of
ways this can be done for MSG75.5 (equivalently the $k_{z}=0$ plane of
MSG77.18). In Fig. 10, we show the corresponding plot for the $k_{z}=\pi$
plane of MSG77.18.

Figure 9: Edge spectra for a selection of removed edge orbitals in the
straight cut (panel a - panel d) and diagonal cut (panel e - panel h) for
MSG75.5/the $k_{z}=0$ plane of MSG77.18. A full circle indicates the presence
of both spin components, a single arrow indicates that the other spin has been
removed and the lack of a site indicates that both spin components on this
site have been removed. a) Removing a down spin from the top edge. b) Removing
an up spin from the bottom edge. c) Removing a down spin from both edges. d)
Removing a down spin from the top edge and an up spin from the bottom edge. e)
Removing a down spin from the A site on the top edge. f) Removing both spin
components from the A site on the top edge. g) Removing a down spin from the A
site on the top edge and an up spin from the A site at the bottom edge. h)
Removing all spins from the A and B site at the top edge. All in-gap states
are singly degenerate. The states extending slightly into the gap in h) are
doubly degenerate. Figure 10: Edge spectra for a selection of removed edge
orbitals in the straight cut (panel a - panel d) and diagonal cut (panel e -
panel h) for the $k_{z}=\pi$ plane of MSG77.18. A full circle indicates the
presence of both spin components, a single arrow indicates that the other spin
has been removed and the lack of a site indicates that both spin components on
this site have been removed. a) Removing a down spin from the top edge. b)
Removing an up spin from the bottom edge. c) Removing a down spin from both
edges. d) Removing a down spin from the top edge and an up spin from the
bottom edge. e) Removing a down spin from the A site on the top edge. f)
Removing both spin components from the A site on the top edge. g) Removing a
down spin from the A site on the top edge and an up spin from the A site at
the bottom edge. h) Removing all spins from the A and B site at the top edge.
The in-gap bands in d), g) and h) are doubly degenerate.

###  C.2 Removing a single spin

We show the effect of removing a single spin on the boundary in Fig. 11.

Figure 11: Corner charges for the straight cut (panels a and b) and diagonal
cut (panels c and d) respectively for MSG75.5/the $k_{z}=0$ plane of MSG77.18.
On the top, we show the charge distribution, with Fermi level fixed to the
same value as for MSG75.5, shown in Fig. 4. Red indicates excess charge
(relative to the center), blue a deficit of charge, where we sum over all
occupied bands. On the bottom we show the spectrum, together with a state just
below the Fermi level. All calculations were done using the PythTB package Coh
and Vanderbilt (2016).

##  Appendix D Real-space invariants and twisted boundary conditions for
MSG75.5

In this appendix, we provide further detail on the RSIs and TBCs discussed in
section IV. Our notation follows that of Song  _et al._ (2020) closely, and we
use their tables throughout.

###  D.1 Real space invariants

Real-space invariants are defined at every WP as quantities which do not
change as we move orbitals in a symmetry-preserving fashion from a high-
symmetry WP to a lower-symmetry WP or vice versa. In the finite model of the
$k_{z}=0$ plane of MSG75.5, where the symmetries are broken down to wallpaper
group p4, the relevant (non-magnetic) WPs are $1a$, $1b$ and $2c$, as shown in
Fig. 1a). Removing an orbital from WP $1a$ or $1b$ (with site-symmetry
$C_{4}$) requires a minimum of four orbitals to come together, as this is the
only way to consistently subduce to the trivial position. Thus, the imbalance
between the four site-symmetry orbitals at WP $1a$ or $1b$ is protected, but
not the total number of orbitals. This allows for the definition of three
independent invariants (the real-space invariants) for each of these WP. For
(non-magnetic) WP $2c$, with site-symmetry group $C_{2}$, pairs of orbitals
must come together, so there is a single RSI. A full enumeration of the RSIs
for wallpaper groups can be found in Song  _et al._ (2020). For our model, we
find the RSIs for the lower/upper subspace as:

$\displaystyle\delta_{1a}$ $\displaystyle=$ $\displaystyle\pm 1$
$\displaystyle\delta_{2a}$ $\displaystyle=$ $\displaystyle\pm 1$
$\displaystyle\delta_{3a}$ $\displaystyle=$ $\displaystyle\pm 1$
$\displaystyle\delta_{1b}$ $\displaystyle=$ $\displaystyle\mp 1$
$\displaystyle\delta_{2b}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\delta_{3b}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\delta_{1c}$ $\displaystyle=$ $\displaystyle 0$

where $\delta_{iw}$ denotes RSI $i$ for the WP with label $w$. These RSIs can
be used to directly confirm the fragility of our occupied manifold. In
particular, for our symmetry setting (spinful $C_{4}$ without TRS), the
criterion for fragility is that it be impossible to find numbers
$N_{a},N_{b},N_{c}$ summing to $N_{\mathrm{bands}}=2$ whilst satisfying the
constraints tabulated in Song  _et al._ (2020) and reproduced here for
convenience:

$\displaystyle N_{a/b}$ $\displaystyle=$
$\displaystyle\delta_{1a/1b}+\delta_{2a/2b}+\delta_{3a/3b}\ \mathrm{mod}\ 4$
$\displaystyle N_{a/b}$ $\displaystyle\geq$
$\displaystyle-3\delta_{1a/1b}+\delta_{2a/2b}+\delta_{3a/3b}$ $\displaystyle
N_{a/b}$ $\displaystyle\geq$
$\displaystyle\delta_{1a/1b}-3\delta_{2a/2b}+\delta_{3a/3b}$ $\displaystyle
N_{a/b}$ $\displaystyle\geq$
$\displaystyle\delta_{1a/1b}+\delta_{2a/2b}-3\delta_{3a/3b}$ $\displaystyle
N_{a/b}$ $\displaystyle\geq$
$\displaystyle\delta_{1a/1b}+\delta_{2a/2b}+\delta_{3a/3b}$ $\displaystyle
N_{c}$ $\displaystyle=$ $\displaystyle\delta_{1c}\ \mathrm{mod}\ 2$
$\displaystyle N_{c}$ $\displaystyle\geq$ $\displaystyle-\delta_{1c}$
$\displaystyle N_{c}$ $\displaystyle\geq$ $\displaystyle\delta_{1c}$

In the occupied subspace, $N_{a}$ and $N_{b}$ have to equal $3\ \mathrm{mod\
4}$, whereas $N_{c}$ has to equal $0\ \mathrm{mod\ 2}$. However, this is not
possible to satisfy if $N_{a}+N_{b}+N_{c}=2$ Thus, the occupied subspace is
indeed fragile. The RSIs in the unoccupied subspace, on the other hand,
require $N_{a}$ and $N_{b}$ to equal $1\ \mathrm{mod\ 4}$, whilst $N_{c}$
still equals $0$ mod 2. This can be satisfied with $N_{a}=N_{b}=1$ and
$N_{c}=0$. As can be easily checked, this also satisfies all other conditions
implying that the unoccupied subspace is not fragile as expected.

###  D.2 Twisted boundary conditions

The twisted boundary conditions (TBC) are designed as perturbations to the
Hamiltonian which leave the RSIs invariant but exchange the $C_{4}$
eigenvalues of bands Song  _et al._ (2020). As the RSIs are defined in terms
of the $C_{4}$ eigenvalues, if the relative balance of states with differing
$C_{4}$ eigenvalues changes between the occupied and the unoccupied subspace,
then there is necessarily a gap closing under this perturbation to ensure that
the RSIs are invariant.

More concretely, as our system has $C_{4}$ symmetry, we can divide it into
four regions which are related by symmetry, shown in Fig. 7a). As we choose
the cut illustrated in Fig. 1c), there are no sites on the boundary between
regions so each site is uniquely assigned to a region. Multiplying the hopping
between the regions by some well-chosen parameter $\lambda$ (as described Song
_et al._ (2020)) amounts to a gauge change in the $C_{4}$ symmetry operator,
permuting the site-symmetry representation. These are the TBC for this system.
Using the RSIs, which are invariant under such transformations, we can predict
which orbitals are exchanged. If chosen correctly, we can force an exchange
between the occupied and the unoccupied spaces. The result of implementing the
TBC by tuning $\lambda:1\rightarrow i$ is shown in Fig. 7. Note that the
corner states (localized in the gap) are not cut by the boundary conditions
and therefore do not flow. Checking the IRREPs of the states that exchange
under TBC confirms that this pattern corresponds to our expected fragile
phases, as explained in Song  _et al._ (2020). This is thus another physical
signature of the fragile magnetic topology in MSG75.5, which in this case is
equivalent to fragility in the p4 wallpaper group. A similar pattern holds on
the $k_{z}=0$ plane of MSG77.18.