# Projected spin texture as a bulk indicator of fragile topology

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>TCM Group, Cavendish Laboratory, University of
Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Nordic
Institute for Theoretical Physics (NORDITA), Stockholm, Sweden  Robert-Jan
Slager<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory, University of
Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom

###### Abstract

We study the relationship between projected momentum-space spin textures and
Wilson loop winding, proving a map between band topology of and spin topology
in certain restricted symmetry settings, relevant to fragile topology. Our
results suggest that the spin gap may act as a smoking gun bulk indicator for
fragile topology within specific scenarios.

Introduction.— The concept of spin has played a perpetually significant role
in physics. With the rise of topological materials this role was reinforced as
paradigmatic time reversal (TRS) protected topological insulators Qi and Zhang
(2011); Hasan and Kane (2010) thrive on having spinful TRS. Similarly, winding
numbers associated with real space spin textures, such as Skyrmions Fert  _et
al._ (2017), have garnered significant attention over the years, as have
Rashba/Dresselhaus spin-orbit coupling (SOC) terms in momentum space, which
display (pseudo-)spin momentum locking Dresselhaus  _et al._ (2007). These
terms usually arise in a two-band $\boldsymbol{k}\cdot\boldsymbol{p}$
description in presence of strong SOC, describing a twofold degeneracy at a
certain momentum. Generally, however, the topology of a gapped band structure
is not captured by a purely local analysis, but instead requires comparing
multiple momentum space points or paths (e.g. Wilson loops Bouhon  _et al._
(2019); Alexandradinata and Bernevig (2016); Alexandradinata  _et al._
(2014)), leading to the consideration of spin textures across Brillouin zones
(BZ).

Whilst spin is a fundamental parameter in band topology, the specific role of
spin textures are yet to be fully explored Schrunk  _et al._ (2022). Such
relations can however be motivated by specific examples. In a quantum spin
Hall (QSH) phase preserving TRS for example, the relevant topological
invariant quantifying the winding of the spin sector is the spin Chern number.
This spin Chern number was shown to be robust as long as the spin gap does not
close Prodan (2009) and in close correspondence with the Kane-Mele band-
topological invariant Kane and Mele (2005). In addition, spin plays an
important role in the irreducible representations (IRREPs) that determine the
gluing conditions Kruthoff  _et al._ (2017), which in turn fix the momentum
space configurations considered in recent symmetry-based schemes such as
topological quantum chemistry Bradlyn  _et al._ (2017) or symmetry indicators
Po  _et al._ (2017).

Recently, unified relations between band topology and spin textures have
received reinvigorated interest as reflected in work on describing spin-
textures around higher degeneracies Lin  _et al._ (2022a), as well as moving
away from local in momentum space (e.g. $\boldsymbol{k}\cdot\boldsymbol{p}$)
descriptions, to a global description taking into account space-group
symmetries Okuma (2018) and spin textures at various points in the BZ Gatti
_et al._ (2020); Mera Acosta  _et al._ (2021); Lin  _et al._ (2022b).
Similarly, we note that in Refs. Bouhon  _et al._ (2019, 2021) a relation
between certain fragile topology and spin-locking processes has been
previously reported. Given these appealing indications, we here discuss a
symmetry setting where, rather than coexisting, there is a direct mapping
between fragile topological phases and spin topology. In essence we are,
inspired by the above-mentioned symmetry based classifications schemes,
concerned with the question if there are connectivity rules amongst spin bands
and whether they impact band topology. Encouragingly, we are able to identify
spin textures, and in particular the spin gap, as a bulk signature of fragile
topological phases Po  _et al._ (2018); Bouhon  _et al._ (2020a). This is
useful, as fragile phases are generically difficult to diagnose Peri  _et al._
(2020). Concretely we find:

  * •

In the presence of $C_{2}\mathcal{T}$ symmetry, with
$[C_{2}\mathcal{T}]^{2}=1$, we can define a $\mathbb{Z}_{2}$ topological
invariant, distinct from the spin Chern number, that relates to a
superconducting symmetry class and characterizes the (vanishing of) the spin
gap over the BZ.

  * •

This invariant relates to the electronic band structure in two-band subspaces
corresponding to a fragile split elementary band representation (EBR) Po  _et
al._ (2018); Michel and Zak (2000); Bouhon  _et al._ (2019); Bradlyn  _et al._
(2017) within specific symmetry settings, giving a bulk topological
classification perspective of fragile electronic band topology. We find that
whenever the spin invariant is trivial we necessarily have a fragile
electronic phase.

  * •

We discuss possible realistic experimental schemes to measure the spin gap and
thus the $\mathbb{Z}_{2}$ invariant.

This paper proceeds by introducing these ideas from a general point of view
and subsequently applying and retrieving our findings in the context of two
models: a hexagonal model having TRS (non-magnetic) and a tetragonal model
with non-symmorphic TRS (magnetic).

(Pseudo-)spin connectivity and parallel transport.— We begin by defining spin
textures in 2D systems with SOC that lack inversion symmetry. In such systems,
spin is not a good quantum number across the BZ, and we cannot
straightforwardly associate a spin degree of freedom (DOF) to the bands. Even
in the presence of SOC, however, there are $n$-fold degeneracies in the band
structure to which we can associate an $\frac{n-1}{2}$ pseudo-spin DOF Lin
_et al._ (2022a). If such a degeneracy occurs at a high-symmetry point (HSP)
$\boldsymbol{k}_{0}$, whose little-group contains a rotation axis
perpendicular to the system’s basal plane, then this gives a natural spin-
up/spin-down quantization axis for the pseudo-spin. Namely, there exists a
canonical gauge choice at the degeneracy, consisting of the pseudo-spin
eigenstates that diagonalize the rotation symmetry representation. We will
mostly focus on twofold degeneracies which are spanned by a pseudo-spin
$\frac{1}{2}$ DOF. In this setting, the above gauge fixing reduces the gauge
freedom at the degeneracy from $\mathrm{U}(2)$ to
$\mathrm{U}(1)\times\mathrm{U}(1)$. The presence of (symmorphic or
nonsymmorphic) time reversal symmetry (TRS), see below, is then used to
further reduce the gauge freedom to $\mathrm{U}(1)$ by imposing that the
pseudo-spins be the image of each-other under TRS, i.e.
$\mathcal{T}(\chi_{\uparrow},\chi_{\downarrow})\propto(\chi_{\uparrow},\chi_{\downarrow})\sigma_{y}\mathcal{K}$,
where $\mathcal{K}$ is the complex conjugation.

To explore the relationship between the (pseudo-)spin states at the degenerate
points and topology, we need to consistently compare (pseudo-)spins at various
points in the BZ in a gauge-controlled fashion. To attain gauge-consistent
bands associated with (pseudo-)spin,
$(|\chi_{\uparrow},\boldsymbol{k}\rangle,|\chi_{\downarrow},\boldsymbol{k}\rangle)$,
we start from (pseudo-)spin eigenstates at a degenerate HSP
$\boldsymbol{k}_{0}$ written in the above canonical gauge. At
$\boldsymbol{k}_{0}$, we fix the remaining gauge phase factor of the
eigenstates, and parallel transport these states from the degeneracy at
$\boldsymbol{k}_{0}$ to some other $\boldsymbol{k}$ 111Explicitly making such
a construction across the entire BZ whilst preserving symmetries may be
challenging, though it is possible for some symmetry classes as discussed in
Ref. Soluyanov and Vanderbilt (2012). We will only be interested in the states
along specific lines in the BZ, however, where such a construction can easily
be performed.. We will see that this construction is related to how the
eigenvalues of the projected spin operator are used to disentangle degenerate
bands in the QSH Prodan (2009). We note that these parallel-transported states
are not generically eigenstates of the Hamiltonian away from the degeneracy.

If $\boldsymbol{k}^{*}$ is another HSP whose point-group contains the same
rotation axis as $\boldsymbol{k}_{0}$ then, if no gap closings with external
bands occur, the parallel-transported states will again be eigenstates of the
Hamiltonian at $\boldsymbol{k}^{*}$. We find numerically that the resulting
states are also rotation eigenstates at $\boldsymbol{k}^{*}$, although a
general proof of this property is still of active interest Bouhon  _et al._
(2019). We conclude that the (pseudo-)spin eigenvalues, i.e. spin-up or spin-
down, are good quantum numbers at both degeneracies. It is then meaningful to
compare the (pseudo-)spin eigenvalue of the parallel-transported states at the
HSPs $\boldsymbol{k}^{*}$ and $\boldsymbol{k}_{0}$ and consider whether it is
preserved, i.e. indicating a parallel spin configuration, or reversed, i.e.
indicating a spin-flip configuration. This method of connecting (pseudo-)spin
eigenvalues at distinct HSP across the Brillouin zone gives a characterization
of the parallel-transported bands which is reminiscent of the above-mentioned
symmetry-based classification methods Fu (2011); Slager  _et al._ (2012);
Kruthoff  _et al._ (2017); Bradlyn  _et al._ (2017); Po  _et al._ (2017);
Scheurer and Slager (2020); Elcoro  _et al._ (2021); Watanabe  _et al._ (2018)
for Bloch bands. We stress however that the characterization in terms of the
connectivity of pseudo-spin eigenvalues at degeneracies in the BZ is a priori
independent of these methods as we are concerned with how these (pseudo-)spin
eigenvalues connect for a fixed set of irreducible representations (IRREPs) at
the degeneracies, and this spin-connectivity can generically change even when
the IRREPs remain fixed.

Spin bands.— In what follows we assume that we are dealing with twofold-
degeneracies at which the spin-up and spin-down DOF are good quantum numbers.
It turns out, moreover, that the Bloch eigenstates of the systems we consider
realize the bare electron spin at these degeneracies. To compare the spins, we
could naively define the spin operator in the Bloch eigenframe
$\\{|u_{n},\boldsymbol{k}\rangle\\}_{n}$ (i.e. the Bloch energy eigenvectors)
as Okuma (2018)

$\boldsymbol{\hat{S}}_{s}(\boldsymbol{k})_{n}=\langle
u_{n},\boldsymbol{k}|\hat{\boldsymbol{S}}|u_{n},\boldsymbol{k}\rangle,$ (1)

where $\hat{\boldsymbol{S}}=\boldsymbol{n}\cdot\boldsymbol{\hat{\sigma}}$ is
the spin operator along $\boldsymbol{n}$ in the basis of the Hamiltonian (e.g.
tracing over orbital DOFs). This is the operator relevant for spin-ARPES
measurements. However, this operator is not invariant under mixing of the
Bloch eigenvectors of the occupied bands, as required by the parallel
transport gauge. We therefore consider a multi-band generalization of Eq. (1),
the projected spin operator,

$\boldsymbol{\hat{S}}_{P}(\boldsymbol{k})=P(\boldsymbol{k})\hat{\boldsymbol{S}}P(\boldsymbol{k}),$
(2)

considered in Refs. Prodan (2009); Lin  _et al._ (2022b). Here
$P(\boldsymbol{k})=\sum_{n=1}^{N_{\text{occ.}}}|u_{n},\boldsymbol{k}\rangle\langle
u_{n},\boldsymbol{k}|=\mathcal{U}(\boldsymbol{k})\mathcal{U}(\boldsymbol{k})^{\dagger}$
is the projector onto the occupied bands, where
$\mathcal{U}(\boldsymbol{k})=(|u_{1},\boldsymbol{k}\rangle,|u_{2},\boldsymbol{k}\rangle,\dots,|u_{N_{\mathrm{{occ}}}},\boldsymbol{k}\rangle)$
is the rectangular matrix of the occupied Bloch eigenvectors (we assume
$N_{\mathrm{occ}}=2$ in what follows). Diagonalizing this operator allows us
to define spin bands across the BZ. We note that the projected spin operator
is gauge-invariant with respect to mixing of the occupied bands eigenvalues.
At HSPs this operator has quantized eigenvalues $\pm\frac{1}{2}$, and two zero
eigenvalues due to the two unoccupied bands (i.e. the projector
$P(\boldsymbol{k})$ has rank 2, see the reduced spin operator below). When the
two occupied spin bands are gapped, there exists a continuous frame with the
same spin eigenvalue at $\Gamma$ and K. This is precisely the parallel
transported frame constructed above. When the spin gap closes instead, it
indicates an exchange of spin eigenvalues across the Brillouin zone.
Furthermore, when the spin spectrum is gapped, there is a one-to-one
correspondence between the winding of the Wilson loop of the occupied bands
and the spin Chern number Prodan (2009); Lin  _et al._ (2022b), which we
consider later. We conclude that the parallel transported frame and the
spectral decomposition of the projected spin operator are complementary ways
to relate the spin texture to electronic band topology.

Symmetry of the spin operator.— The existence of topological phases crucially
depend on symmetries. We are here primarily interested in 2D systems with
(symmorphic or non-symmorphic Bouhon  _et al._ (2021)) time-reversal symmetry
(TRS) $\mathcal{T}$ and out-of plane twofold rotation symmetry $C_{2z}$ with
their product satisfying $[C_{2z}\mathcal{T}]^{2}=+1$ (satisfied for both
examples considered below), which we note has direct consequences for defining
multi-gap invariants such as Euler class Bouhon  _et al._ (2019, 2020b);
Slager  _et al._ (2022); Ahn  _et al._ (2019); Bouhon  _et al._ (2020a); Ünal
_et al._ (2020); Jiang  _et al._ (2021). The out-of plane rotation gives a
natural spin quantization axis, and in what follows we focus mostly on the
$\hat{S}_{z}$ operator. The operators $\mathcal{T}$ and $C_{2z}$ acts on the
spin operator $\hat{\boldsymbol{S}}$ as

$\hat{\boldsymbol{S}}\xrightarrow[]{\mathcal{T}}-\hat{\boldsymbol{S}},$ (3)
$\\{\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}\\}\xrightarrow[]{C_{2z}}\\{-\hat{S}_{x},-\hat{S}_{y},\hat{S}_{z}\\}.$
(4)

Combining these transformations, and noting that $C_{2z}\mathcal{T}$ is local
in momentum, it follows that the single spin operator
$\hat{S}_{s,z}(\boldsymbol{k})_{n}=0$ away from degeneracy, suggesting that in
this symmetry setting, spin-ARPES will be unable to detect the spin texture.

Let us note the symmetry action of $C_{2z}\mathcal{T}$ on the projector

$P(\boldsymbol{k})\rightarrow\hat{U}_{C_{2z}\mathcal{T}}P(\boldsymbol{k})^{*}\hat{U}_{C_{2z}\mathcal{T}}^{\dagger},$
(5)

where $\hat{U}_{C_{2z}\mathcal{T}}\mathcal{K}$ is the representation of
$C_{2z}\mathcal{T}$ in the orbital basis, and the action of
$C_{2z}\mathcal{T}$ symmetry on the Bloch eigenvectors

$\hat{U}_{C_{2z}\mathcal{T}}\mathcal{U}(\boldsymbol{k})^{*}=\mathcal{U}(\boldsymbol{k})\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k}).$
(6)

Working now with the projected spin-operator in the $z$-direction, Eq. (2), we
find the symmetry action (see Appendix A)

$\hat{S}_{P,z}(\boldsymbol{k})\xrightarrow[]{C_{2z}\mathcal{T}}-\hat{U}_{C_{2z}\mathcal{T}}(\boldsymbol{k})\hat{S}_{P,z}(\boldsymbol{k})^{*}\hat{U}_{C_{2z}\mathcal{T}}(\boldsymbol{k})^{\dagger}.$
(7)

Because we require that $[C_{2z}\mathcal{T}]^{2}=+1$, there exists a basis
choice that transforms $\hat{U}_{C_{2z}\mathcal{T}}$ to the identity matrix
(which can be found using an Autonne-Takagi decomposition, see Refs. Bouhon
_et al._ (2020b); Peng  _et al._ (2022) and Appendix A), such that
$C_{2z}\mathcal{T}$ symmetry implies

$\hat{S}_{P,z}(\boldsymbol{k})=-\hat{S}_{P,z}(\boldsymbol{k})^{*},$ (8)

i.e. $\hat{S}_{P,z}$ is purely imaginary and off-diagonal in the new basis.
Thus, in the presence of $C_{2z}\mathcal{T}$, the projected spin operator has
particle-hole symmetry which is local in $\boldsymbol{k}$ and squares to $+1$.
This places this operator in the superdconducting AZ+I (Altland-Zirnbauer +
inversion) class D, which is characterized by the zeroth homotopy group
$\pi_{0}=\mathbb{Z}_{2}$ Bzdusek and Sigrist (2017).

Topology of the spin operators.— To compute this nontrivial homotopy charge
explicitly, we work in the real basis where the projected spin operator is
purely off-diagonal. Then, the topological invariant is given by Bzdusek and
Sigrist (2017) $z_{2}^{s}=$sign(Pf[i$\hat{S}_{P,z}(\boldsymbol{k})])$, where
Pf is the Pfaffian. This invariant can only change when there is a gap closing
in the spin spectrum, which by $C_{2z}{\mathcal{T}}$ symmetry must happen when
the eigenvalues of the projected spin operator cross $0$, thus implying an
exchange of spin eigenvalues of the parallel transported frame from $\Gamma$
to K.

Note, however, as mentioned above, that the projected spin operator in systems
with $N$ bands generically has $N-N_{\mathrm{occ}}$ zero modes, associated
with the unoccupied bands Lin  _et al._ (2022b). These zero modes force
$z_{2}^{s}$ to be trivial. We can therefore instead work with the reduced spin
operator,

$\boldsymbol{S}^{r}_{P}(\boldsymbol{k})_{nm}=\langle
u_{n},\boldsymbol{k}|\hat{\boldsymbol{S}}|u_{m},\boldsymbol{k}\rangle=[\mathcal{U}(\boldsymbol{k})^{\dagger}\hat{\boldsymbol{S}}_{P}\mathcal{U}(\boldsymbol{k})]_{nm},$
(9)

As discussed in Lin  _et al._ (2022b), the spectrum of the reduced spin
operator is identical to that of the projected spin operator, except that it
excludes the zero modes associated with the unoccupied bands. The reduced spin
operator is gauge covariant (see Appendix A), so that the eigenvalues are
gauge invariant. We repeat the above symmetry analysis for the $z$-component
of the reduced spin operator, and find (see the derivation in Appendix A)

$\hat{S}^{r}_{P,z}(\boldsymbol{k})=-\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})\hat{S}^{r}_{P,z}(\boldsymbol{k})^{*}\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})^{\dagger},$
(10)

where we have used the representation of $C_{2z}\mathcal{T}$ in the occupied
Bloch eigenvectors, Eq. (6). As we discuss in Appendix A, upon a change to the
real basis $\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})$ goes to the
identity, so that Eq. (10) again makes the $z$-component of the projected spin
operator imaginary and off-diagonal. This allows us to compute the above
topological invariant from the Pfaffian of the reduced spin operator in the
real basis. By computing in the parallel-transport gauge defined above, we can
therefore define a (generically) non-trivial $\mathbb{Z}_{2}$ invariant
associated with the spin bands,

$z_{2,\mathrm{red}}^{s}=\mathrm{sign}(\mathrm{Pf}[i\hat{S}^{r}_{P,z}(\boldsymbol{k})]).$
(11)

Crucially, this invariant can only change when the spin gap closes, which
should happen along a spin nodal ring surrounding one of the HSPs. In the case
of a two-band parallel-transported frame discussed above, the nontrivial spin
invariant indicates that the spin eigenvalues cross zero an odd number of
times between the twofold degeneracies of the different HSPs. This invariant
therefore captures the spin connectivity between degeneracies discussed above.
We next show that this spin connectivity is in direct correspondence with the
fragile topology of the electronic bands.

Pseudo-spin and band topology.— A standard way to diagnose band topology is by
using the Wilson loop operator $W(\boldsymbol{k})$ Bouhon  _et al._ (2019);
Alexandradinata and Bernevig (2016); Alexandradinata  _et al._ (2014). We have
previously established an analytic relationship between the Wilson loop
winding and the pseudo-spin structure in two specific symmetry settings.
Namely, in Ref. Bouhon  _et al._ (2019) we studied the four-band EBR of the
honeycomb lattice populated with a $p_{z}$ orbital and two $1/2$-spins on each
sub-lattice site, assuming the hexagonal space group P6mm$1^{\prime}$, i.e.
with the point group $C_{6v}$ and TRS. We showed that when the EBR is split
into two occupied and two unoccupied bands, the two-band subspace that hosts
twofold degeneracies at $\Gamma$ and $K$ exhibits a rigid relationship between
the winding of Wilson loop $W$ and its spin structure. More precisely, we
found that all phases with SOC that are adiabatically connected to gapped
phases with inversion symmetry (i.e. with zero Rashba SOC) only admit spin
textures with aligned spins at $\Gamma$ and K and exhibit a Wilson loop
winding $W\in\pm 1+3\mathbb{Z}$. Moreover, we found phases with a spin-flip
configuration, from $\Gamma$ to K, that are not adiabatically connected to
inversion-symmetric phases and only admit a Wilson loop winding $W\in
3\mathbb{Z}$ (in practice, we only found models of such phases with $W=0$). We
recall that an odd Wilson loop winding in this system implies a nontrivial
$\mathbb{Z}_{2}$ QSH phase, while all other phases are fragile topological. In
addition, in Ref. Bouhon  _et al._ (2021), we found similar results in a
tetragonal, antiferromagnetic space group PC4, i.e. with point group $C_{4}$
and non-symmorphic TRS
$\mathcal{T}\\{E|[\boldsymbol{t}_{1}+\boldsymbol{t}_{2}]/2\\}$. Specifically,
we find the relationships shown in Tab. 1.

$\Gamma$ | K/M | Wilson loop winding $W$
---|---|---
$C_{6v}$+$\mathcal{T}$ |  $C_{4}$+$\mathcal{T}\\{E|[\boldsymbol{t}_{1}+\boldsymbol{t}_{2}]/2\\}$
$\\{\uparrow,\downarrow\\}$ | $\\{\uparrow,\downarrow\\}$ | $\pm 1+3\mathbb{Z}$ | $1+2\mathbb{Z}$
$\\{\uparrow,\downarrow\\}$ | $\\{\downarrow,\uparrow\\}$ | $3\mathbb{Z}$ | $2\mathbb{Z}$
Table 1: Relationship between the spin eigenvalues ($\uparrow$ for $+1/2$,
$\downarrow$ for $-1/2$) at $\Gamma$ and K/M of the parallel-transported
pseudo-spin eigenframe and Wilson loop winding for a hexagonal/tetragonal
system in the specific symmetry setting described in Refs. Bouhon  _et al._
(2019, 2021).

Crucially, we find in this work that the two-band subspaces with zero Wilson
loop winding, i.e. necessarily hosting a spin-flip configuration between
$\Gamma$ and K/M, precisely exhibit a change in the $z_{2,\mathrm{red}}^{s}$
invariant, Eq. (11), so that we obtain in the hexagonal case:

$\displaystyle z_{2,\mathrm{red}}^{s}=0\ \mathrm{mod\ }2$ $\displaystyle\iff
W\in\pm 1+3\mathbb{Z},$ (12) $\displaystyle z_{2,\mathrm{red}}^{s}=1\
\mathrm{mod\ }2$ $\displaystyle\iff W=0+3\mathbb{Z}.$

A similar result can be verified for the tetragonal case. We therefore see
that the $\mathbb{Z}_{2}$ spin invariant acts as a complement to the
electronic topology: the two-band subspace of a split EBR with trivial Wilson-
loop winding in this specific symmetry setting is only allowed when there is a
non-trivial $\mathbb{Z}_{2}$ spin invariant, implying a stable spin-spectrum
nodal ring. Conversely, a trivial $\mathbb{Z}_{2}$ spin invariant implies non-
zero Wilson loop winding. This constitutes a main finding of this paper and we
illustrate these results for two phases with different Wilson-loop windings in
the hexagonal case in Fig. 1. Similar results for the antiferromagnetic
tetragonal counterpart are detailed in Appendix C.

Figure 1: Relevant figures for the P6mm$1$’ symmetric case. In a)-d) we show
the fragile case (where the Wilson loop of the occupied and unoccupied bands
winds), whereas e)-h) shows the trivial case (where the Wilson loop of the
occupied bands does not wind). In a) and e) we show the band structures, with
IRREPs of P6mm$1$’ indicated. In b) and f) we show the eigenvalues of the
reduced spin operator along high-symmetry paths of the BZ. Note that spin is
not a good quantum number at M, but is a good quantum number at $\Gamma$ and
K. In c) and g), we show the Pfaffian and Wilson loop of the occupied bands
over the BZ, with
$\Gamma^{\prime}=\Gamma+\boldsymbol{b}_{1}+\boldsymbol{b}_{2}$. Comparing the
plots, we see that a non-trivial Wilson loop corresponds to a trivial Pfaffian
invariant and vice-versa. Finally in d) and h) we show the spin expectation
value over the BZ of the lowest spin band [as shown along high-symmetry lines
in b) and f)]. This shows that the case with trivial Wilson loop has spin
nodal lines. For more details on the models, see Appendix B and Ref. Bouhon
_et al._ (2019).

Experimental detection of spin operator.— As discussed above, spin-ARPES is
generically not capable of resolving the projected spin texture. However, if
the spin invariant $z_{2,\mathrm{red}}^{s}$ is non-trivial, then the spin gap
necessarily vanishes somewhere along the relevant high-symmetry line.
Conversely, this implies that whenever the reduced spin operator is gapped
across the entire BZ, we necessarily have a non-trivial winding of the Wilson
loop within the split EBR context and symmetry settings we are considering. In
general, the spin gap can close at points away from the high-symmetry line, or
it can close twice along the line, without changing the spin invariant and
therefore the Wilson loop quantization. Thus, when the spin is gapless, we may
still be in a phase where the Wilson loop winds (note, however, that Ref. Lin
_et al._ (2022b) showed that the spin gap is perturbatively stable). However,
when the spin spectrum is gapped, we are necessarily in a phase where the
Wilson loop winds.

We can therefore detect topology by measuring the gap in the projected spin
spectrum. As pointed out in Lin  _et al._ (2022b), the spin gap provides a
bound on the excitation spectrum of a system when quenching with a
$\hat{z}$-directed Zeeman field. Similarly, we also expect this gap to show up
in the spin structure factor of neutron scattering experiments. For the Zeeman
quenched case, the density of excited states is (as shown in Ref. Lin  _et
al._ (2022b)):

$\Gamma_{ex}=\sum_{\boldsymbol{k}\in\mathrm{BZ}}\mathrm{Tr}\big{[}P(\boldsymbol{k})\hat{S}_{z}[1-P(\boldsymbol{k})]\hat{S}_{z}P(\boldsymbol{k})\big{]}.$
(13)

Importantly, we find that $\Gamma_{ex}$ is an order of magnitude larger in the
case where the spin spectrum is gapless.

Relation to spin Chern number.— When the gap in the projected spin operator is
open across the BZ, one can compute the Wilson loop of the spin bands. This
corresponds to the spin-Chern number Prodan (2009). We define the Chern number
of the upper/lower spin bands as $C^{\pm}$, and define the spin-Chern number
as $C_{s}=C^{+}-C^{-}$. As discussed in Appendix D and shown in Fig. 3, we
find in the non-magnetic hexagonal case that $C_{s}=4$, whereas in the
magnetic tetragonal case, $C_{s}=2$. In non-magnetic systems, the spin Chern
number is related to the Kane-Mele $\mathbb{Z}_{2}$ invariant as
$\nu_{\mathrm{KM}}=(1/2)(C_{s}\ \mathrm{mod}\ 4)$. Thus, the hexagonal case
does not have a strong invariant associated with the electronic band structure
for even winding of the Wilson loop, in agreement with ”fragile” topology (in
fact, Ref. Bouhon  _et al._ (2019) showed that the fragile topology with
$W\in\pm 4+12\mathbb{Z}$ can be trivialized by coupling to trivial bands, see
also Fig. 4). In the magnetic case, this corresponds to having a QSH phase
without TRS, which has also been investigated in Ref. Yang  _et al._ (2011);
Xiao  _et al._ (2022), where it was found that the QSH edge states can be
generically gapped when TRS is broken. We showed in Ref. Lange  _et al._
(2021) that the surface spectrum in the magnetic case agrees with that
expected of a fragile topological insulator, and the IRREP content is also
consistent with fragile topology, justifying the label of fragile topological
phase.

Conclusions.— We have discussed a general scheme to investigate refined band
topologies beyond symmetry indicators, and how it relates to (pseudo-)spin
DOFs. We have further shown that, for certain symmetry settings, the existence
of a gap in a certain spin operator directly indicates the presence of fragile
topology. This provides a bulk indicator for fragile topology, and we discuss
concrete experimental schemes to measure this spin gap.

Outlook.— An interesting remaining question concerns the more general
relationship between the spin-invariant and the band topology, beyond two-band
degeneracies. Note, in particular, that the $\mathbb{Z}_{2}$ invariant
$z^{s}_{2,\mathrm{red}}$ is defined for any even number of bands in the
occupied space, in any system with $C_{2z}\mathcal{T}$ symmetry. However, the
pseudo-spin connectivities, and their relationship to Wilson loop were
specifically derived in a two-band subspace. The connection with higher
degeneracies, or additional subspaces, comprise an interesting future pursuit.
Furthermore, it is interesting to note that we uncovered a similar
relationship between Wilson loop windings and pseudo-spin orderings in two
very different space groups. This points to a more generic structure. As
Wilson loops are difficult to measure, such a relationships to an observable
would be of particular interest. We also note that our results can apply to
any physical operator that can be associated with the degeneracies in the BZ.
Finally, it would also be interesting to more systematically consider the role
that SOC, and the vanishing of spin-flip terms at certain HSPs, plays as well
as the role of the spin Chern number in these systems and generalizations to
different quantities such as sub-lattice winding numbers.

###### Acknowledgements.

G.F.L is funded by the Aker Scholarship. We thank the Marie Skłodowska-Curie
programme under EC Grant No. 84290, the Winton Programme for the Physics of
Sustainability, a New Investigator Award, EPSRC grant EP/W00187X/1 and Trinity
College at the University of Cambridge

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##  Appendix A Symmetry of spin operators under $C_{2z}\mathcal{T}$

###  A.1 Defining the spin operators

We define the Bloch orbital basis in which the Bloch Hamiltonian is defined

$|\varphi_{\alpha},\boldsymbol{k}\rangle=\dfrac{1}{\sqrt{N}}\sum\limits_{\boldsymbol{k}}e^{i\boldsymbol{k}\cdot(\boldsymbol{R}_{m}+\boldsymbol{r}_{\alpha})}|w_{\alpha},\boldsymbol{R}_{m}+\boldsymbol{r}_{\alpha}\rangle\,,$
(14)

with the atomic Wannier function
$\langle\boldsymbol{r}|w_{\alpha},\boldsymbol{R}_{m}+\boldsymbol{r}_{\alpha}\rangle=w_{\alpha}(\boldsymbol{r}-\boldsymbol{R}_{m}-\boldsymbol{r}_{\alpha})$
for the $\alpha$-th atomic orbital. The Bloch eigenstates are then defined
through

$|\psi_{n},\boldsymbol{k}\rangle=|\boldsymbol{\varphi},\boldsymbol{k}\rangle[\mathcal{U}_{t}(\boldsymbol{k})]_{n}\,,$
(15)

with $\mathcal{U}_{t}(\boldsymbol{k})$ the matrix of column Bloch eigenvectors
$[\mathcal{U}_{t}(\boldsymbol{k})]_{n}=|u_{n},\boldsymbol{k}\rangle$, i.e.

$\mathcal{U}_{t}(\boldsymbol{k})=(|u_{1},\boldsymbol{k}\rangle,|u_{2},\boldsymbol{k}\rangle,\dots,|u_{N},\boldsymbol{k}\rangle).$
(16)

Where $N$ is the total number of bands, which we assume to be even. From Eq.
(2), the projected spin operator in the occupied Bloch band basis is given by:

$\hat{\boldsymbol{S}}_{P}(\boldsymbol{k})=P(\boldsymbol{k})\hat{\boldsymbol{S}}P(\boldsymbol{k})=\sum_{n,m\in\mathrm{occ}}\langle
u_{n},\boldsymbol{k}|\hat{\boldsymbol{S}}|u_{m}\boldsymbol{k}\rangle|u_{n},\boldsymbol{k}\rangle\langle
u_{m},\boldsymbol{k}|$ (17)

Where $\hat{\boldsymbol{S}}$ is the spin operator when tracing over all
orbital DOFs and $|u_{n},\boldsymbol{k}\rangle$ is the cell-periodic part of
occupied band $n$. We can write a multiband gauge-transformation of the
occupied bands as:

$|\tilde{u}_{n},\boldsymbol{k}\rangle=\mathcal{U}(\boldsymbol{k})G(\boldsymbol{k})=\sum_{m\in\mathrm{occ}}G_{mn}(\boldsymbol{k})|u_{m},\boldsymbol{k}\rangle$
(18)

with $G(\boldsymbol{k})\in\mathsf{U}(N_{\mathrm{occ}})$, and
$\mathcal{U}(\boldsymbol{k})$ is the rectangular matrix of occupied Bloch
eigenvectors

$\mathcal{U}(\boldsymbol{k})=(|u_{1},\boldsymbol{k}\rangle,|u_{2},\boldsymbol{k}\rangle,\dots,|u_{N_{\mathrm{occ}}},\boldsymbol{k}\rangle),$
(19)

which satisfies

$\displaystyle\mathcal{U}\mathcal{U}^{\dagger}$ $\displaystyle=P$ (20)
$\displaystyle\mathcal{U}^{\dagger}\mathcal{U}$
$\displaystyle=\mathbb{1}_{N_{\mathrm{occ}}}\times\mathbb{1}_{N_{\mathrm{occ}}}.$

We find that the projection operator is gauge-invariant, so that the projected
spin operator $\hat{\boldsymbol{S}}_{P}$ is also gauge-invariant. We are also
interested in the reduced spin operator:

$\hat{\boldsymbol{S}}_{P}^{r}(\boldsymbol{k})_{nm}=\langle
u_{n},\boldsymbol{k}|\hat{\boldsymbol{S}}|u_{m},\boldsymbol{k}\rangle=(\mathcal{U}^{\dagger}\hat{\boldsymbol{S}}_{P}\mathcal{U})_{nm}$
(21)

This operator is gauge-covariant, as under a gauge-transformation
$G(\boldsymbol{k})\in\mathsf{U}(N_{\mathrm{occ}})$:

$\hat{\boldsymbol{S}}_{P}^{r}(\boldsymbol{k})\rightarrow\tilde{\hat{\boldsymbol{S}}}_{P}^{r}(\boldsymbol{k})=G(\boldsymbol{k})^{\dagger}\boldsymbol{\hat{S}}_{P}^{r}(\boldsymbol{k})G(\boldsymbol{k})$
(22)

So that the spin eigenvalues are gauge-invariant. We now wish to understand
how these operators behave under $C_{2z}\mathcal{T}$ symmetry.

###  A.2 Action of $C_{2z}\mathcal{T}$

Letting $\hat{U}_{\mathcal{T}},\hat{U}_{C_{2z}}$ be the unitary actions of
time-reversal and $C_{2z}$ respectively, we define
$\hat{U}_{C_{2z}\mathcal{T}}=\hat{U}_{C_{2z}}\hat{U}_{\mathcal{T}}$. We
require that $(C_{2z}\mathcal{T})^{2}=+1$, which gives that
$\hat{U}_{C_{2z}\mathcal{T}}U_{C_{2z}\mathcal{T}}^{\dagger}=\mathbb{1}$ and
$\hat{U}_{C_{2z}\mathcal{T}}\hat{U}_{C_{2z}\mathcal{T}}^{*}=\mathbb{1}$, so
that $\hat{U}_{C_{2z}\mathcal{T}}=[\hat{U}_{C_{2z}\mathcal{T}}]^{T}$. We now
derive the symmetry properties of the projected and reduced spin operator. We
first give the action of the $C_{2z}\mathcal{T}$ symmetry on the occupied
Bloch eigenstate basis,

$\displaystyle{}^{C_{2z}\mathcal{T}}|\psi_{n},\boldsymbol{k}\rangle$
$\displaystyle=^{C_{2z}\mathcal{T}}|\boldsymbol{\varphi},\boldsymbol{k}\rangle[\mathcal{U}(\boldsymbol{k})]_{n}\,,$
(23)
$\displaystyle=|\psi_{m},\boldsymbol{k}\rangle[\mathcal{U}(\boldsymbol{k})^{\dagger}]_{m}\hat{U}_{C_{2z}\mathcal{T}}[\mathcal{U}(\boldsymbol{k})^{*}]_{n}\mathcal{K}\,,$
$\displaystyle=|\psi_{m},\boldsymbol{k}\rangle[\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})]_{mn}\mathcal{K}\,,$

where $\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})$ is a
$N_{\mathrm{occ}}\times N_{\mathrm{occ}}$ unitary matrix that represents the
$C_{2z}\mathcal{T}$ symmetry on the occupied-band eigen-subspace under
consideration (we assume that we can isolate the occupied subspace from the
other bands). For later reference, it is also useful to rewrite it as an
action on the occupied Bloch eigenframe

$\hat{U}_{C_{2z}\mathcal{T}}\mathcal{U}(\boldsymbol{k})^{*}=\mathcal{U}(\boldsymbol{k})\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})\,.$
(24)

The action of $C_{2z}\mathcal{T}$ on the two-band projector is then

$P(\boldsymbol{k})\rightarrow\hat{U}_{C_{2z}\mathcal{T}}P(\boldsymbol{k})^{*}\hat{U}_{C_{2z}\mathcal{T}}^{\dagger},$
(25)

Now note that:

$\begin{split}\hat{U}_{C_{2z}\mathcal{T}}^{\dagger}\hat{S}_{x}\hat{U}_{C_{2z}\mathcal{T}}=\hat{S}_{x}\\\
\hat{U}_{C_{2z}\mathcal{T}}^{\dagger}\hat{S}_{y}\hat{U}_{C_{2z}\mathcal{T}}=\hat{S}_{y}\\\
\hat{U}_{C_{2z}\mathcal{T}}^{\dagger}\hat{S}_{z}\hat{U}_{C_{2z}\mathcal{T}}=-\hat{S}_{z}\end{split}$
(26)

E.g., in the orbital basis $C_{2z}\mathcal{T}$ flips the $z$-component of spin
whilst leaving the others unchanged. Thus, the action on the projected spin-z
operator is:

$\displaystyle\hat{S}_{P,z}(\boldsymbol{k})$
$\displaystyle\rightarrow\hat{U}_{C_{2z}\mathcal{T}}P(\boldsymbol{k})^{*}(\hat{U}_{C_{2z}\mathcal{T}}^{\dagger}\hat{S}_{z}\hat{U}_{C_{2z}\mathcal{T}})P(\boldsymbol{k})^{*}\hat{U}_{C_{2z}\mathcal{T}}^{\dagger}$
(27)
$\displaystyle=-\hat{U}_{C_{2z}\mathcal{T}}P(\boldsymbol{k})^{*}\hat{S}_{z}P(\boldsymbol{k})^{*}\hat{U}_{C_{2z}\mathcal{T}}^{\dagger}$
$\displaystyle=-\hat{U}_{C_{2z}\mathcal{T}}\hat{S}_{P,z}^{*}\hat{U}_{C_{2z}\mathcal{T}}^{\dagger}\,.$

The $C_{2z}\mathcal{T}$ symmetry then requires

$\hat{S}_{P,z}(\boldsymbol{k})=-\hat{U}_{C_{2z}\mathcal{T}}\hat{S}_{P,z}^{*}\hat{U}_{C_{2z}\mathcal{T}}^{\dagger}\,.$
(28)

This implies that the z-component of the projected spin operator has a local
in $\boldsymbol{k}$, antiunitary, antisymmetry, squaring to $+1$. Following
Bzdusek and Sigrist (2017) this puts this operator in AZ+$\mathcal{I}$
symmetry class D, characterized by $\pi_{0}=\mathbb{Z}_{2}$ and spectral
symmetry around $s_{z}=0$.

To find an explicit expression for this $\mathbb{Z}_{2}$ topological invariant
associated with the spin spectrum, we note that, due the properties of
$\hat{U}_{C_{2z}\mathcal{T}}$ discussed previously, there exists a basis
transformation (given by an Autonne-Takagi decomposition Bouhon  _et al._
(2020b)), such that $\hat{U}_{C_{2z}\mathcal{T}}=\mathbb{1}$. In this basis,
$\hat{S}_{P,z}(\boldsymbol{k})$ is a purely imaginary, skew-antisymmetric
matrix at all $\boldsymbol{k}$, from which it follows that
Pf($i\hat{S}_{P,z}$) is well-defined. Then, in a smooth gauge, the Pfaffian
evolves continuously, and only switches sign through a spin gap closing. It
follows that $\mathrm{sign}[\mathrm{Pf}(i\hat{S}_{P,z})]$, evaluated at two
different points, gives the required invariant. Note, however, that
Pf($A$)${}^{2}=$Det($A$) and Det($\hat{S}_{P,z})=0$ because $\hat{S}_{P,z}$
always has $N-N_{\mathrm{occ}}$ zero eigenvalues, associated with the
eigenstates of $Q=1-P$ (physically: the conduction bands). Thus, the
$\mathbb{Z}_{2}$ invariant is always ill-defined for $\hat{S}_{P,z}$.

To circumvent this issue, we instead look at the reduced spin operator, Eq.
(9), where the zero eigenvalues are projected out. Rewriting the projected
spin operator as

$\hat{S}_{P,z}(\boldsymbol{k})=\mathcal{U}(\boldsymbol{k})\hat{S}^{r}_{P,z}(\boldsymbol{k})\mathcal{U}(\boldsymbol{k})^{\dagger}\,,$
(29)

we find from Eq. (28) and Eq. (24)

$\displaystyle\hat{S}_{P,z}(\boldsymbol{k})$
$\displaystyle\rightarrow-\hat{U}_{C_{2z}\mathcal{T}}P(\boldsymbol{k})^{*}\hat{S}_{z}P(\boldsymbol{k})^{*}\hat{U}_{C_{2z}\mathcal{T}}^{\dagger}$
(30)
$\displaystyle=\hat{U}_{C_{2z}\mathcal{T}}\mathcal{U}(\boldsymbol{k})^{*}[-\mathcal{U}(\boldsymbol{k})^{\dagger}\hat{S}_{z}\mathcal{U}(\boldsymbol{k})]^{*}P(\boldsymbol{k})^{*}$
$\displaystyle=-\mathcal{U}(\boldsymbol{k})\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})\hat{S}^{r}_{P,z}(\boldsymbol{k})^{*}\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})^{\dagger}\mathcal{U}(\boldsymbol{k})^{\dagger}.$

Since $C_{2z}\mathcal{T}$ symmetry requires that the last line must equal
$\hat{S}_{P,z}(\boldsymbol{k})$ in the form of Eq. (29), we finally get

$\hat{S}^{r}_{P,z}(\boldsymbol{k})=-\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})\hat{S}^{r}_{P,z}(\boldsymbol{k})^{*}\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})^{\dagger}.$
(31)

Which is very similar to the particle-hole symmetry in Eq. (28), except that
the symmetry operator now depends on $\boldsymbol{k}$, as a consequence of
projecting out the zero eigenvalues. Note that:

$\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})[\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})]^{*}=\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})[\breve{W}_{C_{2z}\mathcal{T}}(\boldsymbol{k})]^{\dagger}=\mathbb{1}\\\
$ (32)

Because $\hat{U}_{C_{2z}\mathcal{T}}=[\hat{U}_{C_{2z}\mathcal{T}}]^{T}$. It is
therefore possible to perform an Autonne-Takagi basis change at each
$\boldsymbol{k}$ to make $\hat{S}_{P,z}^{r}(\boldsymbol{k})$ antisymmetric and
off-diagonal. This decomposition will however amount to a local rotation of
the basis, rather than a global rotation. This may induce additional
topological structure. We now show that this does not happen when working in
the basis that makes the Hamiltonian real.

To show this, we now perform the Autonne-Takagi transformation Bouhon  _et
al._ (2020b). Because
$\hat{U}_{C_{2z}\mathcal{T}}=[\hat{U}_{C_{2z}\mathcal{T}}]^{T}$, we can write
$\hat{U}_{C_{2z}\mathcal{T}}=V_{C_{2z}\mathcal{T}}D[V_{C_{2z}\mathcal{T}}]^{T}$
for a unitary $V_{C_{2z}\mathcal{T}}$ and a diagonal $D$. Defining
$V_{\mathbb{R}}=\sqrt{D^{*}}V_{C_{2z}\mathcal{T}}^{\dagger}$, the Hamiltonian
is made real by the basis change:

$H_{\mathbb{R}}(\boldsymbol{k})=V_{\mathbb{R}}H(\boldsymbol{k})V_{\mathbb{R}}^{\dagger}$
(33)

The eigenstates of the Hamiltonian in this basis can be chosen to be real, as
we numerically verified, and are related to the eigenstates in the original
basis by:
$|u_{n}^{\mathbb{R}},\boldsymbol{k}\rangle=e^{i\varphi}V_{\mathbb{R}}|u_{n},\boldsymbol{k}\rangle$,
for some arbitrary gauge phase $\varphi$. Under $C_{2z}\mathcal{T}$:

$\begin{split}H(\boldsymbol{k})=V_{\mathbb{R}}^{\dagger}H_{\mathbb{R}}(\boldsymbol{k})V_{\mathbb{R}}=\hat{U}_{C_{2z}\mathcal{T}}H^{*}(\boldsymbol{k})\hat{U}_{C_{2z}\mathcal{T}}^{\dagger}\\\
=\hat{U}_{C_{2z}\mathcal{T}}V_{\mathbb{R}}^{T}H_{\mathbb{R}}(\boldsymbol{k})V_{\mathbb{R}}^{*}\hat{U}_{C_{2z}\mathcal{T}}^{\dagger}\end{split}$
(34)

And thus:

$H_{\mathbb{R}}(\boldsymbol{k})=V_{\mathbb{R}}\hat{U}_{C_{2z}\mathcal{T}}V_{\mathbb{R}}^{T}H_{\mathbb{R}}(\boldsymbol{k})V_{\mathbb{R}}^{*}\hat{U}_{C_{2z}\mathcal{T}}^{\dagger}V_{\mathbb{R}}^{\dagger}$
(35)

Defining
$\hat{U}_{C_{2z}\mathcal{T},\mathbb{R}}=V_{\mathbb{R}}\hat{U}_{C_{2z}\mathcal{T}}V_{\mathbb{R}}^{T}$,
we note that
$\hat{U}_{C_{2z}\mathcal{T},\mathbb{R}}=\sqrt{D^{*}}D\sqrt{D^{*}}=\mathbb{1}$,
so that $C_{2z\mathcal{T}}$ acts diagonally in the real basis. We can
similarly define the projected spin operator in occupied bands of the real
basis:

$\hat{\boldsymbol{S}}_{P,\mathbb{R}}=\sum_{n,m\in\mathrm{occ}}\langle
u^{\mathbb{R}}_{n},\boldsymbol{k}|\hat{\boldsymbol{S}}_{\mathbb{R}}|u^{\mathbb{R}}_{m},\boldsymbol{k}\rangle|u_{n}^{\mathbb{R}},\boldsymbol{k}\rangle\langle
u^{\mathbb{R}}_{m},\boldsymbol{k}|=V_{\mathbb{R}}\hat{\boldsymbol{S}}_{P}V_{\mathbb{R}}^{\dagger}$
(36)

With:

$\hat{\boldsymbol{S}}_{\mathbb{R}}=V_{\mathbb{R}}\hat{\boldsymbol{S}}V_{\mathbb{R}}^{\dagger}$
(37)

Being the spin operator in the real basis. Specializing to the z-direction,
this spin operator enjoys particle-hole symmetry:

$\hat{S}_{P,z,\mathbb{R}}=-\hat{S}^{*}_{P,z,\mathbb{R}}$ (38)

By virtue of $\hat{U}_{C_{2z}\mathcal{T},\mathbb{R}}$ acting diagonally. We
can similarly define the reduced spin operator in this basis:

$\hat{S}_{P,z,\mathbb{R}}^{r}=\mathcal{U}_{\mathbb{R}}^{\dagger}\hat{S}_{P,z,\mathbb{R}}\mathcal{U}_{\mathbb{R}}$
(39)

Where
$\mathcal{U}_{\mathbb{R}}(\boldsymbol{k})=(|u^{\mathbb{R}}_{1},\boldsymbol{k}\rangle,|u^{\mathbb{R}}_{2},\boldsymbol{k}\rangle,\dots|u^{\mathbb{R}}_{N_{\mathrm{occ}}},\boldsymbol{k}\rangle)$.
This operator will also satisfy
$\hat{S}_{P,z,\mathbb{R}}^{r}=-\hat{S}_{P,z,\mathbb{R}}^{r*}$, due to the
diagonal action of $\hat{U}_{C_{2z}\mathcal{T},\mathbb{R}}$ and the reality of
the eigenstates because:

$\breve{W}_{C_{2z}\mathcal{T},\mathbb{R}}=\mathcal{U}_{\mathbb{R}}^{\dagger}\hat{U}_{C_{2z}\mathcal{T},\mathbb{R}}\mathcal{U}_{\mathbb{R}}^{*}=\mathcal{U}_{\mathbb{R}}^{T}\mathcal{U}_{\mathbb{R}}=\mathbb{1}$
(40)

Thus, $\hat{S}_{P,z,\mathbb{R}}^{r}$ is an antisymmetric matrix related to the
reduced spin operator in the original band basis by:

$[\hat{S}^{r}_{P,z,\mathbb{R}}]_{nm}=e^{-i(\varphi_{n}-\varphi_{m})}[\hat{S}_{P,z}^{r}]_{nm}$
(41)

Where $\varphi_{n}$ are phases such that:

$|u_{n}^{\mathbb{R}},\boldsymbol{k}\rangle=e^{i\varphi_{n}(\boldsymbol{k})}V_{\mathbb{R}}|u_{n},\boldsymbol{k}\rangle$
(42)

This is again a local (not global) unitary equivalence, but in a smooth gauge
we expect them to not add additional topological structure. Thus, by choosing
a smooth gauge $|\tilde{u}_{1}^{\mathbb{R}}\rangle$, we can get a well-defined
Pfaffian invariant.

We can fix the gauge at $\Gamma$ by recognizing that $H$ and $\hat{S}_{z}$
commute at $\Gamma$ in our two-band subspace, so that we can choose
simultaneous eigenstates of $H$ and $\hat{S}_{z}$, which defines the rotation-
symmetric state. Similarly, we can choose simultaneous eigenstates of
$H_{\mathbb{R}}$ and $\hat{S}_{z,\mathbb{R}}$, which will be real as both
matrices are real. Starting from this basis, we can then perform a local band
mixing by taking the SVD of the overlap matrices, to get a gauge
transformation $U$. This will maintain the chiral symmmetry, as long as $U$ is
purely real, which it will be if we start from real eigenstates. This follows
from gauge covariance:

$U^{\dagger}S_{P,z,\mathbb{R}}^{r}U=-U^{\dagger}S^{r*}_{P,z,\mathbb{R}}U=-(U^{\dagger}S_{P,z,\mathbb{R}}^{r}U)^{*}$
(43)

Note that this does not quite correspond to choosing a smooth gauge, as we
have not ironed out the discontinuity at the end of the loop, arising from the
non-trivial Berry phase Soluyanov and Vanderbilt (2012). Such an ironing out
procedure would break the reality condition. Luckily, this is not necessary
for the Wilson loop quantization argument, which only relies on the Wilson
loop being diagonal.

In this basis, finally, the Pfaffian will be a smooth function of
$\boldsymbol{k}$ which changes sing across a nodal line. When
$N_{\mathrm{occ}}=2$ the Pfaffian is exactly equal to the spin eigenvalue of
one band, tracked smoothly through any degeneracies. However, the Pfaffian
also explains the topological protection of the nodal line more generally.

We note finally, that we find numerically that there exists an alternative
choice of gauge, without going through the real basis, where
$\hat{S}_{P,z}^{r}$ has a global particle-hole symmetry, at least in the
tetragonal magnetic case. At $\Gamma$, we can choose states such that the
rotation operator is diagonal and so that $\varphi_{1}=\varphi_{2}$. In this
case, $C_{2z}\mathcal{T}$ is represented by $-\sigma_{x}$ at $\Gamma$.
Parallel-transporting these states, we find that $C_{2z}\mathcal{T}$ is given
by $-\sigma_{x}$ along the entire path $\Gamma$ to M.

##  Appendix B Further information on the models

###  B.1 Hexagonal system

The models for the sixfold non-magnetic case were first considered in Bouhon
_et al._ (2019). They were constructed using a generalization of the
Dresselhaus method to construct tight-binding models for a specific
crystallographic space group Dresselhaus  _et al._ (2007). Specifically, they
are constructed by starting with layer-group 77 (corresponding to the $z=0$
plane of P6mm$1$’), and placing a $p_{z}$-orbital with both spin-components at
Wyckoff position 2b, and then expanding the Hamiltonian, including all
symmetry-allowed terms. This gives four bands, with twofold Kramers
degeneracies at the TRIMs $\Gamma$ and M. We focus on gapped phases, where
there are two occupied and two unoccupied bands.

When we only consider nearest-neighbor hoppings, we find a phase whose IRREP
content is compatible with a fragile phase Po  _et al._ (2018); Bouhon  _et
al._ (2019). This is confirmed numerically in Bouhon  _et al._ (2019) and Fig.
4 by showing that the Wilson loop of both the occupied and the unoccupied
subspace winds, but that they can be unwound when embedded in a higher-
dimensional space. By contrast, when going up to 10th order neighbors in the
hoppings, Ref.Bouhon  _et al._ (2019) finds a phase that is still gapped over
the BZ but now the IRREPs at $\Gamma$ have inverted. This phase onyl displays
winding in the upper subspace. We note that in either case, a simple IRREP
analysis would suggest that one of the band subspaces corresponds to an
obstructed atomic limit, whereas the other one corresponds to a fragile limit,
though whether the occupied or unocupied subspce is fragile/atomic obstructed
is interchanged between the two phases. These phases are depicted in Fig. 1.

In the phase with few hoppings, Ref. Bouhon  _et al._ (2019) finds that the
Wilson loop winds by $2$ over the BZ in both subspaces. By contrast, in the
phase with long-range hoppings, the Wilson loop does not wind in the lower
subspace, but winds by $4$ in the upper subspace. This difference is not
captured in an IRREP analysis, but can be detected from the spin texture. In
particular we find that the spin texture is trivial for both subspaces in the
phase with Wilson loop windings of $2$, whereas it is nodal for both subspaces
in the phase with Wilson loop windings of $0$ and $4$. Mote that the Wilson
loop winding of $4$ for the upper subspace seems to contradict the results in
Tab. 1. However, those results assume that there is a twofold degeneracy at
$\Gamma$ and K, which is only satisfied for the lower subspaces in Fig. 1.
Thus, the upper subspace has a nodal spin texture, but this is not captured by
our link to connectivities between degeneracies.

###  B.2 Tetragonal system

The tetragonal system with winding Wilson loop was first introduced in Ref.
Bouhon  _et al._ (2021). Starting from magnetic space group PC4, (No. 75.5 in
the BNS convention), we place an s-orbital with both spin-components at
Wyckoff position 2b, again giving $4$ bands. There are twofold Kramers
degeneracies at $\Gamma$ and M, but not at X due to the magnetic structure of
the space group. Thus, the lowest possible connectivity is $2$. We again
construct a model using a generalized Dresselhaus procedure, though here we
also use MagneticTB Zhang  _et al._ (2022) for easier construction of higher-
order hopping models.

We again focus on the case where there are two occupied and two unoccupied
bands that are gapped across the BZ (for a discussion of other phases, see
Bouhon  _et al._ (2021); Lange  _et al._ (2021)). For nearest-neighbor
hoppings, we find a phase with Wilson loop winding $1$ both subspaces. Going
up to the tenth order hoppings, we find a phase where the IRREPs at X have
inverted and the occupied subspace has Wilson loop winding $0$, whereas the
unoccupied subspace has Wilson loop winding $2$. Similar to the hexagonal
case, a pure IRREP analysis would suggest that in either case there is one
fragile subspace and one atomic obstructed subspace, though they again
interchange.

In the case where the Wilson loop winding is $1$ in either subspace, the spin
texture is trivial across the BZ. However, when the windings are $0$ in the
occupied and $2$ in the unoccupied, we find a nodal-line spin structure in
both subspaces, as shown in Fig. 2 (though see the comment about the Wilson
loop winding of $0$ in Appendix C).

##  Appendix C Results for the fourfold magnetic case

We show the results for the fourfold magnetic case in Fig. 2, in complete
analog to Fig. 1. Note that it is difficult to tell the winding of the lower
subspace [Fig. 2g)], as it is not clear whether the Wilson bands form an
avoided crossing. We can check this by choosing a different Wilson loop
geometry (note that whether or not the Wilson loop winds should be independent
of geometry). We therefore look at the winding as we deform our base path in a
$C_{4}$ symmetric manner, starting from the path $\Gamma$M $\Gamma^{\prime}$
with $\Gamma^{\prime}=\Gamma+\boldsymbol{b}_{1}$ and smoothly deforming it to
the path $\Gamma$X$\Gamma^{\prime}$. This covers a quarter of the BZ, and
should therefore correspond to 1/4 of the total winding (we considered the
same construction in Bouhon  _et al._ (2021)). In doing so, we find the
expected result: The Wilson loop does not wind in Fig. 2g), so that the Wilson
bands form an avoided crossing.

Figure 2: Relevant figures for the P${}_{\mathrm{C}}4$ symmetric case (similar
to Fig. 1). In a)-d) we show the fragile case (where the Wilson loop of both
subspaces winds), whereas e)-h) shows the trivial case (where only the
occupied subspace has a winding Wilson loop). In a) and e) we show the band
structure, with IRREPs of P${}_{\mathrm{C}}4$ indicated. In b) and f) we show
the eigenvalues of the reduced spin operator along high-symmetry paths of the
BZ. Note that spin is not a good quantum number at X, but is a good quantum
number at $\Gamma$ and M. In c) and g), we show the Pfaffian and Wilson loop
over the BZ of the occupied bands, where
$\Gamma^{\prime}=\Gamma+\boldsymbol{b}_{1}$. The Wilson loop winds in c) but
not in g) (see comment in Appendix C). Finally in d) and h) we show the spin
expectation value over the BZ of the lowest spin band [as shown along high-
symmetry lines in b) and f)]. This illustrates that when the Wilson loop
winding is even (including $0$) the system displays spin nodal lines. For more
details on the models, see Appendix. B and Ref. Bouhon  _et al._ (2021).

##  Appendix D Spin Wilson loops

Whenever the spin spectrum is gapped (as happens in our models when the
pseudo-spin connectivities are trivial and the electronic band topology is
non-trivial), we can compute spin-resolved Wilson loops. Following Lin  _et
al._ (2022b), we compute the Wilson loops projected into the upper and lower
spin bands respectively. We present the results in Fig. 3 for the lower
electronic subspace only, though we find the same results for the upper
electronic subspace. For completeness, we also show the spin Wilson loops for
a case where we directly unwind the Wilson loop in Fig. 4.

Figure 3: Wilson loop winding for occupied band-subspace projected onto spin-
space, with colors indicating the component of spin responsible for the
winding. a) Shows the hexagonal case, whereas b) shows the tetragonal case. In
a), we have a spin Chern number of 4, whereas in b) we have a spin-Chern
number of 2 Figure 4: Band topology of hexagonal system when coupled to two
trivial DOFs (for details of this coupling, see Ref. Bouhon  _et al._ (2019)).
This is for the case where, originally, both the occupied and the unoccupied
space have a winding of $4$. We consider the topology of the four top bands.
We check that the energy gap to the lower two bands, and the spin gap of the
four-dimensional band subspace, is open across the BZ. In a)-c) we show a case
where the trivial bands are not coupled to the winding bands. We show the band
structure in a), with bands with trivial winding in the middle. In b), we show
the Wilson loop spectrum for the top four bands, which is decoupled in this
limit. In c), we show the spin Wilson loop, for the two top/bottom spin bands
in red/blue respectively, with a total spin-Chern number of 8. In d)-f), we
show the same results when coupling the bands with trivial and non-trivial
winding. We see that the Wilson loop gap, as expected in fragile topology.
However, the spin-Wilson loop does not gap, suggesting that the spin-Chern
number is preserved. Note, however, that it maps to a trivial Kane-Mele
invariant in either case Kane and Mele (2005).