[figure]style=plain,subcapbesideposition=top

# Spin-Polarized Fractional Corner Charges and Their Photonic Realization

Ran Gladstein Gladstone  School of Applied and Engineering Physics, Cornell
University, Ithaca, NY 14853<EMAIL_ADDRESS>Minwoo Jung  Department of
Physics, Cornell University, Ithaca, NY 14853  Gennady Shvets  School of
Applied and Engineering Physics, Cornell University, Ithaca, NY 14853

###### Abstract

We demonstrate that a spin degree of freedom can introduce additional texture
to higher order topological insulators (HOTIs), manifesting in novel
topological invariants and phase transitions. Spin-polarized mid-gap corner
states of various multiplicities are predicted for different HOTI phases, and
novel bulk-boundary correspondence principles are defined based on bulk
invariants such as total and spin corner charge. Those are shown to be robust
to spin-flipping perturbations. Photonic realizations of spin-linked
topological phases are demonstrated in engineered systems using pseudo-spin.

††preprint: APS/123-QED

The interplay between spin and topology has long been of interest to
physicists Kane and Mele (2005); Bernevig and Zhang (2006); Jungwirth  _et
al._ (2012); Kimura  _et al._ (2007); Qi  _et al._ (2006). Shortly after the
prediction of time-reversal (TR) preserving Chern insulators Haldane (1988),
the concept was extended to spinful electrons, leading to the prediction
Bernevig  _et al._ (2006) and experimental realization of the quantum spin
Hall (QSH) effect Khang  _et al._ (2018); Jamali  _et al._ (2015). The
extension to spinful systems typically involves doubling of the Hilbert space
to include two similar copies of the original spinless system Bernevig  _et
al._ (2006); Asbóth  _et al._ (2016), and can have significant effect on the
topology of the system. For example, while the Chern insulator has a
$\mathbb{Z}$ topological invariant, a spin Chern insulator possesses a
$\mathbb{Z}_{2}$ topological invariant due to TR and inversion symmetries
Kitaev (2009). The addition of spin to the system has, therefore, changed its
topological classification.

More recently, these concepts have been extended to higher order topological
insulators (HOTIs) Benalcazar  _et al._ (2017); Schindler  _et al._ (2018);
Peterson  _et al._ (2018); Franca  _et al._ (2018); Schindler  _et al._
(2019); Kooi  _et al._ (2021); Freeney  _et al._ (2020); Kempkes  _et al._
(2019), and their photonic/phononic counterparts (PHOTIs) Xie  _et al._
(2018); Jung  _et al._ (2020); Xie  _et al._ (2019); Chen  _et al._ (2019,
2021); Noh  _et al._ (2018). A HOTI is a $d$-dimensional topological insulator
with a bulk topological invariant predicting topological states localized on
some of its $d-n$ dimensional terminations, where $n\in\mathbb{N}$ and $2\leq
n\leq d$. It is natural to ask if spinful HOTIs have any emerging topological
properties and classification that distinguish them from their spinless
counterparts. Most of research of spinful HOTIs has so far concentrated on the
systems with fermionic TR symmetry Kooi  _et al._ (2021); Schindler  _et al._
(2019) or time-dependent drive Franca  _et al._ (2018).

In this Letter we examine the effects of adding the spin degree of freedom
(DOF) to HOTIs when fermionic TR symmetry is broken yet bosonic TR symmetry is
maintained Maczewsky  _et al._ (2020a); Asbóth  _et al._ (2016). Our analysis
also applies to pseudo-spin DOFs, such as layer Qiao  _et al._ (2011); Park
_et al._ (2019); Pesin and MacDonald (2012); Lu  _et al._ (2018) and
polarization Ma  _et al._ (2015); Ma and Shvets (2017); Gladstone  _et al._
(2019). We use the latter to emulate spinful PHOTIs based on engineered
microwave structures. The specific 2D lattice models and their corresponding
periodic electromagnetic waveguides considered in this Letter possess $C_{6}$
symmetry, and are additionally endowed with: Kekulé texture Wu and Hu (2016);
Jung  _et al._ (2020), spin DOF, and a symmetry-preserving spin-flipping
perturbation coupling the spin-$\uparrow$ and spin-$\downarrow$ components.
The latter will be assumed in the form of Kane-Mele spin-orbit coupling (SOC)
Kane and Mele (2005). For spinless models, such topological crystalline
insulators (TCIs) have been shown to support a HOTI phase with a quantized
corner charge of $Q_{c}=1/2$ Benalcazar  _et al._ (2019).

Below we demonstrate that when an independent spin subspace is introduced, two
non-trivial phases can emerge: a phase possessing two quantized spin-polarized
corner charges $Q_{c}^{\uparrow,\downarrow}=1/2$ (spin-HOTI), and a phase
characterized by a single quantized corner charge $Q_{c}=1/2$ (HOTI). While in
the absence of SOC the spin-HOTI phase can be thought of as being
independently topological for each of the spin states Chen  _et al._ (2021),
such simple interpretation is no longer valid in the presence of a finite
spin-flipping SOC. Which one out of these two phases is realized depends on
the strength of the SOC that couples the two spin states, and on the
individual topological properties of each spin component in the absence of
such coupling. We further demonstrate that the distinct properties of the
spin-HOTI and HOTI phases manifest themselves in different multiplicities and
spin textures of their corresponding zero-energy corner states. When effective
fermionic TR symmetry is restored Wu and Hu (2016); Maczewsky  _et al._
(2020b), the corner states delocalize and merge with topologically-protected
edge states.

Our starting point is the tight binding (TB) model on a honeycomb lattice
schematically shown in Fig. 1(a):

$\begin{split}H=\sum_{\langle
ij\rangle}t_{\text{in}}^{\uparrow}c_{i\uparrow}^{\dagger}c_{j\uparrow}+\sum_{\langle
i^{\prime}j^{\prime}\rangle}t_{\text{out}}^{\uparrow}c_{i^{\prime}\uparrow}^{\dagger}c_{j^{\prime}\uparrow}+\sum_{\langle
ij\rangle}t_{\text{in}}^{\downarrow}c_{i\downarrow}^{\dagger}c_{j\downarrow}\\\
+\sum_{\langle
i^{\prime}j^{\prime}\rangle}t_{\text{out}}^{\downarrow}c_{i^{\prime}\downarrow}^{\dagger}c_{j^{\prime}\downarrow}+\sum_{\langle\langle
ij\rangle\rangle\alpha\beta}\frac{i}{3\sqrt{3}}\lambda_{\text{SOC}}\nu_{ij}s^{x}_{\alpha\beta}c_{i\alpha}^{\dagger}c_{j\beta},\end{split}$
(1)

where the first (third) term describes the nearest neighbor intra unit cell
hopping of the spin up (down) electrons, the second (fourth) term describes
the nearest neighbor inter unit cell hopping of the spin-$\uparrow$
($\downarrow$) electrons, and the fifth term describes the next-nearest
neighbor Kane-Mele SOC Kane and Mele (2005). Without loss of generality, we
assume that all hopping amplitudes are positive, and that the energy scale is
defined by setting
$\left(t_{\text{out}}^{\uparrow,\downarrow}+t_{\text{in}}^{\uparrow,\downarrow}\right)/2=1$.

The difference between inter- and intra-cell hopping amplitudes,
$\Delta_{\uparrow,\downarrow}\equiv
t_{\text{out}}^{\uparrow,\downarrow}-t_{\text{in}}^{\uparrow,\downarrow}$,
determines the topology of the two uncoupled spin sub-spaces in the absence of
SOC: each spin component’s subspace behaves as a HOTI for
$\Delta_{\uparrow,\downarrow}>0$, or a trivial insulator for
$\Delta_{\uparrow,\downarrow}<0$, as shown in Fig. 1(b). The topological phase
transitions at $\Delta_{\uparrow,\downarrow}=0$ are the consequences of the
band inversions between $p$\- and $d$-orbital mode profiles Jung  _et al._
(2020); Wu and Hu (2016): the energies of the $p$-orbital bands are below
(above) the bandgap for trivial (topological) insulators.

The unperturbed spin corner charges $\tilde{Q}_{c}^{\uparrow,\downarrow}$ are
separately defined for each spin subspace Benalcazar  _et al._ (2019) in a
manner similar to the spin Chern number Bernevig  _et al._ (2006). Corner
charges can be calculated based on the symmetry properties of the propagation
bands below the bandgap at the high-symmetry $\Gamma$ and $M$ points of the
Brillouin zone Benalcazar  _et al._ (2019):

$\tilde{Q}_{c}^{\uparrow,\downarrow}=\left(\\#M_{\uparrow,\downarrow}-\\#\Gamma_{\uparrow,\downarrow}\right)/4$
(2)

where $\\#M_{\uparrow}$ ($\\#\Gamma_{\uparrow}$) is the number of
$C_{2}$-invariant spin-$\uparrow$ modes below the band gap at the $M$
($\Gamma$) point (same for spin-$\downarrow$ modes). Under this definition of
$\tilde{Q}_{c}^{\uparrow,\downarrow}$, three distinct cases can be identified:
(i) $\tilde{Q}_{c}^{\uparrow}=\tilde{Q}_{c}^{\downarrow}=1/2$ (topological for
both spins when $\Delta_{\uparrow,\downarrow}>0$), (ii)
$\left[\tilde{Q}_{c}^{\uparrow}=0,\tilde{Q}_{c}^{\downarrow}=1/2\right]$ or
$\left[\tilde{Q}_{c}^{\uparrow}=1/2,\tilde{Q}_{c}^{\downarrow}=0\right]$
(topological for only one spin when $\Delta_{\uparrow}\Delta_{\downarrow}<0$),
and (iii) $\tilde{Q}_{c}^{\uparrow}=\tilde{Q}_{c}^{\downarrow}=0$ (trivial for
both spins when $\Delta_{\uparrow,\downarrow}<0$).

Figure 1: Spinful HOTI: the tight-binding (TB) model and its phase diagrams.
(a) Schematic of the TB model with definitions of intra-cell (red lines) and
inter-cell (blue lines) hopping amplitudes. Black circles: lattice sites,
dashed hexagons: unit cells, green dashed lines: spin-orbit coupling (SOC)
amplitudes $\lambda_{\text{SOC}}$ and their signs ($\pm$).(b,c) Phase diagram
at constant $\lambda_{\text{SOC}}=0$ (b) and $\lambda_{\text{SOC}}=0.25$ (c).
Solid curves: bulk bandgap closure at
$\lambda_{\text{SOC}}=\lambda_{\text{SOC}}^{\rm
th}(\Delta_{\uparrow},\Delta_{\downarrow})$, color bar: $Q_{c}^{\uparrow}$ for
trivial, spin-topological, and topological (not quantized) phases. Dashed
line: spin-degeneracy $\Delta_{\uparrow}=\Delta_{\downarrow}$ corresponding to
quantum spin-Hall (QSH) phase.

When the spin components are coupled through the SOC, the expressions for
$\tilde{Q}_{c}^{\uparrow,\downarrow}$ are no longer quantized, and must be
redefined because of the mixing between the spin-$\uparrow$ and
spin-$\downarrow$ contents of the propagation bands. Therefore, we modify the
definitions of the corner charges by weighing the contribution of each band by
its spin-$\uparrow/\downarrow$ content:

$Q_{c}^{\uparrow,\downarrow}=\frac{1}{4}\sum_{i}\left(\left|\langle\uparrow,\downarrow|\psi_{i}(M)\rangle\right|^{2}-\left|\langle\uparrow,\downarrow|\psi_{i}(\Gamma)\rangle\right|^{2}\rangle\right),$
(3)

where $|\psi_{i}(\mathbf{k})\rangle$ is the wave function of a
$C_{2}$-invariant band $i$ below the band gap at the $\mathbf{k}=M,\Gamma$
points.

It can be shown that $Q_{c}^{\uparrow,\downarrow}$ defined by Eq.(3) are
quantized at $Q_{c}^{\uparrow,\downarrow}=1/2$ for the spin-HOTI, and at
$Q_{c}^{\uparrow,\downarrow}=0$ for the trivial phase. Briefly, the
probability conservation for all the bands below the bandgap ensures that
$Q_{c}^{\uparrow,\downarrow}=\tilde{Q}_{c}^{\uparrow,\downarrow}$ for these
two cases. This is due to the lack of cross-band transitions generated by the
$C_{6}$ symmetry-preserving SOC: the $d$ ($p$) orbitals for both spins are on
the same side the bandgap for both spins, and the SOC only connects the
orbitals of one spin component to the same orbitals of the other component
(see the SOM for details).

On the contrary, $Q_{c}^{\uparrow,\downarrow}$ corner charges are not
separately quantized in the HOTI phase. The cause of this transition from
either trivial to HOTI (if $\Delta_{\uparrow\downarrow}<0$) phase, or from
spin-HOTI to HOTI (if $\Delta_{\uparrow\downarrow}>0$) is the band inversion
by the strong SOC coupling satisfying
$\lambda_{\text{SOC}}>\lambda_{\text{SOC}}^{\rm(th)}$. Here
$\lambda_{\text{SOC}}^{\rm(th)}\equiv\sqrt{\Delta_{\uparrow}\Delta_{\downarrow}}$
is the threshold value of the SOC strength corresponding to the closing of the
bulk bandgap, and followed by band inversion. However, the total spin-
independent corner charge $Q_{c}=Q_{c}^{\uparrow}+Q_{c}^{\downarrow}=1/2$
remains quantized in the HOTI phase, thus making it topologically nontrivial.
Phase diagrams in the $\left(\Delta_{\downarrow},\Delta_{\uparrow}\right)$
space showing one trivial and two topological (spin-HOTI and HOTI) phases are
shown in Fig. 1(b),(c) for constant $\lambda_{\text{SOC}}$. The spin-
degeneracy region $\Delta_{\uparrow}=\Delta_{\downarrow}$ represented by the
dashed line in Fig. 1(c) corresponds to the fermionic TR symmetric QSH phase
Wu and Hu (2016); Maczewsky  _et al._ (2020b) embedded in a bosonic TR
symmetric HOTI phase (see SOM).

To investigate the existence and multiplicity of zero-energy corner states
(ZCSs), we consider an interface between trivial and topological domains
containing a single $120^{\circ}$ corner, as shown in the upper-left inset in
Fig.2(a). In the case when the topological phase is a HOTI, its single
quantized topological charge $Q_{c}=1/2$ manifests itself as a single spin-
polarized ZCS. The spectrum for such corner-containing domain wall is
calculated using a finite-domain TB calculation, and the corresponding
spectrum comprised of a continuum of bulk/edge modes and a corner state is
shown in Fig.2(a) (see Table S1 for the TB parameters). The existence of the
ZCS is a direct consequence of the quantized bulk invariant $Q_{c}$ and of the
chiral symmetry present in a $C_{6}$ lattice with Kekule distortion, both of
which are preserved even with finite SOC (see SOM).

The resulting ZCS has a non-trivial spin texture because the SOC term mixes
the two spin components. It can be characterized by the spin imbalance
$S_{c}=Q_{c}^{\uparrow}-Q_{c}^{\downarrow}\neq 0$. Although the bulk quantity
$S_{c}$ is not quantized in the HOTI phase, our calculation of $\langle
S_{z}\rangle$ from the corner local density of states (LDOS) Peterson  _et
al._ (2020) (see SOM) shows that $S_{c}\approx\langle S_{z}\rangle$ for a wide
range of the SOC strengths $\lambda_{\rm SOC}$, as shown in Fig.2(b).
Therefore, the bulk quantity $S_{c}$ is a useful measure of the
experimentally-measurable corner deficit of the spin polarization $\langle
S_{z}\rangle$.

Figure 2: Properties of corner-shaped domain walls between a topological ((a)
HOTI and (c) spin-HOTI) and a trivial phase: a TB model. Upper-left inset in
(a): domain wall shape. Energy spectra: bulk (blue), edge (green), and corner
(red) modes for the HOTI/trivial (a) and spin-HOTI/trivial (c) interfaces.
Lower-right inset in (a): intensity of the single ZCS. (b) Comparison of the
bulk invariant $Q_{c}$ and $S_{c}$ (HOTI phase) and $\langle S_{z}\rangle$ for
the single ZCS in (a). (d) Intensities of the two ZCSs in (c). TB model
parameters: see Table S1.

The existence of the separate quantized spin charges
$Q_{c}^{\uparrow}=Q_{c}^{\downarrow}=1/2$ in the spin-HOTI phase guarantees a
degenerate pair of ZCSs (see Fig. 2(c,d)) with distinct spin textures, as long
as $\Delta_{\uparrow,\downarrow}>0$ and the SOC coupling strength satisfies
$\lambda_{\text{SOC}}<\lambda_{\text{SOC}}^{\rm(th)}$. When the latter
condition is violated, a phase transition into the HOTI phase takes place, the
quantization of the spin charges $Q_{c}^{\uparrow\downarrow}$ is lost, and the
two corner states merge into one ZCS. The localization lengths of the two
corner states are generally not equal (see Fig. 2(d)), and are determined by
the hopping amplitudes, $\lambda_{\text{SOC}}$, as well as the size of the
bandgap between the edge states. The latter can be controlled by ”roughening”
the corner-adjacent domain edges Jung  _et al._ (2020) (see SOM).

Next, we present a photonic platform that can be used for emulating a spinfull
HOTI described by the Hamiltonian given by Eq. (1). The structure shown in
Fig.3(a) is comprised of a photonic crystal (PhC) waveguide made of perfect
electric conductors (PEC) elements sandwiched between two PEC plates Ma  _et
al._ (2015); Ma and Shvets (2017); Gladstone  _et al._ (2019): see the SOM for
the geometric parameters of all PhC designs. The PhC possesses $C_{6}$
symmetry, thereby allowing a Kekulé distortion Hou  _et al._ (2007), which
opens a band gap at the $\Gamma$ point of the Brillouin Zone. In the case of
symmetric air-gaps ($g_{\rm top}=g_{\rm bot}$) between the PhC elements and
the plates, the mid-plane mirror symmetry $z\rightarrow-z$ ensures that the
electromagnetic modes can be classified as either TE-like or TM-like, and that
the two are decoupled from each other Joannopoulos  _et al._ (2008). The
TE(TM) modes are distinguished by having a non-vanishing $H_{z}$ ($E_{z}$)
field component at the $z=h/2$ midplane, as shown in Fig.3(d).

In what follows, the TE(TM) nature of the modes will be used to emulate the
$\uparrow$ ($\downarrow$) isospin components, while the mid-plane symmetry
breaking $(\Delta g\equiv|g_{\text{top}}-g_{\text{bot}}|\neq 0)$ will be used
to emulate the SOC Ma  _et al._ (2015); Ma and Shvets (2017). While it is not
possible to assign hopping amplitudes to the modes, we use the frequency
ordering of the $\bf{p}=\left(p_{x},p_{y}\right)$ and
$\bf{d}=\left(d_{xy},d_{x^{2}-y^{2}}\right)$ orbitals with respect to the
photonic bandgap at the $\Gamma$ point to classify the structures as trivial
or topological Wu and Hu (2015, 2016); Jung  _et al._ (2020).

Using COMSOL simulations, we designed the unit cells shown in Fig. 3(a) for
two types of PhCs satisfying the following criteria for the TE(TM) modes: (i)
frequency-degeneracy at the $\Gamma$-point for both PhCs, (ii) inverted
photonic band structures (topological for one mode, trivial for the other) for
a PHOTI-type PhC shown in Fig. 3(b), and (iii) similar photonic band
structures (topological or trivial for both modes) for the other type of a
PhC. A PHOTI corresponding to (ii) – trivial TE mode and a topological TM mode
– is shown in Fig. 3(b), while a spin-PHOTI corresponding to (iii) –
topological for both modes – is shown in Fig. 3(c). Both stuctures have
symmetric air-gaps, and the mid-plane orbital profiles of the TE ($H_{z}$) and
TM ($E_{z}$) modes corresponding to the four bands below the bandgap are shown
in Fig. 3(d), where the left (right) column corresponds to a PHOTI (spin-
PHOTI) structure.

Figure 3: Emulation of spinful PHOTIs using a photonic crystal waveguide with
Kekule distortion. (a) Top and side views of the PhC unit cell with inter-cell
distance $a$. (b,c) Band structures for PHOTI (b) and spin-PHOTI (c) designs.
TE (blue symbols) and TM (red symbols) are degenerate at the $\Gamma$ point.
(d) Field profiles of the $\bf{p}$ and $\bf{d}$ orbitals bands below the band
gap at the $\Gamma$ point for PHOTI (left column) and spin-PHOTI (right
column) for TE (dashed blue box) and TM (dashed red box) electromagnetic
modes. Parameters: symmetric air-gaps $g_{\rm top}=g_{\rm bot}$, see Table S2
for details.

To demonstrate that the designed PHOTI and spin-PHOTI structures emulate the
corresponding phases described by the TB model given by Eq.(1), we
investigated phase transitions produced by the SOC-emulating air-gap asymmetry
$\Delta g\neq 0$ by comparing the band structures calculated from the TB model
shown in Figs. 4(a),(b) to the first-principles photonic band structures shown
in Figs. 4(c),(d). No photonic bandgap closing occurs in the case of the PHOTI
(Fig. 4(d)) even as $\Delta g$ is progressively increased. On the other hand,
complete bandgap closing for $\Delta g=\Delta g^{\rm(th)}$ is observed for the
spin-PHOTI system, as indicated in Fig. 4(c). Further increase in the
effective SOC term ($\Delta g>\Delta g^{\rm(th)}$) reopens the bandgap Asbóth
_et al._ (2016) and induces a phase transition from the spin-PHOTI to PHOTI
phase (see SOM).

Figure 4: Comparison between HOTIs based on the TB model and their photonic
counterparts: bulk bands (a-d) and corner states (e-f). Left column: spin-HOTI
(a) and spin-PHOTI (c) phases. Right column: HOTI (b) and PHOTI (d) phases.
(a) Spin-HOTI bands without (with) the SOC term: solid (dashed:
$\lambda_{\text{SOC}}=\lambda_{\text{SOC}}^{\rm th}$) lines. (b) Same as (a),
but for HOTIs. (c) Photonic band structure for a spin-PHOTI phase with $\Delta
g=\Delta g^{\rm(th)}$. Inset: top view of the unit cell. (d) Same as (c), but
for a PHOTI phase. $p,d$ orbitals are marked with arrows. (e) Photonic corner
energy spectrum of a spin-PHOTI and (f) HOTI interfacing a trivial phase at
$120^{\circ}$ corner. Insets: $|\mathbf{E}|^{2}$ (top of (e) and (f)) and
$|\mathbf{H}|^{2}$ (bottom of (e)) profiles of the corner modes. Parameters:
Table S2.

Next, we demonstrate that interfacing a photonic structure possessing
quantized bulk invariants (either a PHOTI or spin-PHOTI) with a trivial PhC
produces corner states with correct multiplicities predicted by the TB model
of spinful HOTIs. For the PHOTI shown in Fig. 3(c), we indeed find a single
corner state marked by an arrow in Fig.4(f) and plotted at a $120^{\circ}$
corner in the inset. We used edge roughening Jung  _et al._ (2020) at the
interface between corner-adjacent domain walls to increase the band gap
between the edge modes, thereby further localizing the corner state.

Photonic analog of the degenerate ZCS pairs predicted by the TB model is
constructed by interfacing a trivial PhC with a spin-PHOTI that was chosen to
have the same parameters as the structure shown in Fig. 3(c), but a smaller
air-gap asymmetry $\Delta g<\Delta g^{\text{(th)}}$. Indeed, two nearly-
degenerate corner states marked with arrows in Fig.4(e) were found. Field
intensities $|\mathbf{E}|^{2}$ and $|\mathbf{H}|^{2}$ of the two corner states
plotted in the inset show distinct spatial profiles and polarizations, which
is equivalent to having spin texture.

Several differences between the predictions of the simplified TB model and the
results of the continuum electromagnetic calculation are notable. The edge
modes shown in Fig.4(e) and (f) as green dots appear only in the lower half of
the bulk photonic bandgap (compare with Fig. 2(a),(c)) due to slight mismatch
of the band gaps between the trivial and PHOTI phases. This mismatch is also
responsible for the spectral shift of the corner state away from the mid-gap
frequency and the lack of exact degeneracy between the corner states.

In summary, we have investigated higher-order topological insulating phases on
a hexagonal lattice with spin-dependent Kekulé textures. When spin-flipping
perturbations, such as spin-orbit coupling, are included in a tight-binding
model, two types of insulating topological phases are predicted: a spin-HOTI
possessing two independent fractionally-quantized bulk invariants
$Q_{c}^{\uparrow\downarrow}$, and a HOTI possessing just one such invariant
$Q_{c}=Q_{c}^{\uparrow}+Q_{c}^{\downarrow}$. A bulk-boundary correspondence
between such bulk invariants and the existence of corner states is
established, and their photonic analogues are proposed. Our results present an
opportunity for future development of novel photonic devices with active
switching of their topological corner states by controlling the mid-plane
mirror symmetry, thereby inducing a topological phase transition. Condensed
matter realization of the spin-HOTI phase with
$\Delta_{\uparrow}=\Delta_{\downarrow}$ is possible using a platform such as
CO molecules deposited on Cu Freeney  _et al._ (2020). An effective SOC can be
introduced by applying a magnetic field, which would introduce a phase
transition to a quantum Hall phase instead of a QSH phase.

This work was supported by the Office of Naval Research Award No.
N00014-21-1-2056, National Science Foundation Award No. NNCI-1542081, and the
Army Research Office Award W911NF2110180. M. J. also acknowledges the support
from the Kwanjeong Fellowship from the Kwanjeong Educational Foundation.

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Supplemental Materials: Title for main text

##  I High symmetry points mode profiles and calculation of the spin polarized
corner charge

In the main text we presented mode profiles only at $\Gamma$ because that is
where the topological band inversion occurs. However, it is important to
inspect all the relevant mode profiles of the TE-like and TM-like modes at
both the $\Gamma$ and $M$ points to verify that our PhC designs indeed have
the expected quantized corner charge. We will do that here for the PHOTI phase
of Fig. 3(b) and count the number of modes below the band gap with $+1$
eigenvalue to $C_{2}$ rotation. The same procedure can be repeated for all the
other designs.

We present the relevant field profiles at Fig. S1. We only present modes that
fit our definition of TE-like and TM-like from the main text. Notably, there
is a missing TE-like mode at $\Gamma$. The mode does not actually disappear,
but rather its profile no longer adheres to a TE-like mode profile due to its
low frequency. If the mode retained its profile it could only have an
eigenvalue of +1 to $C_{2}$, being the ground mode of that polarization.

Figure S1: Mode profiles for TM-like and TE-like modes of relevant bands below
the band gap, at $\Gamma$ and $M$ points. $H_{z}$ ($E_{z}$) field at $z=h/2$
plotted for TE-like (TM-like) modes.

For the TM-like modes we have 3 modes with +1 eigenvalues to $C_{2}$ at
$\Gamma$ and only 1 at $M$. Using the mode profiles to calculate $[M]$ defned
by Benalcazar et al. Benalcazar  _et al._ (2019) we find that
$[M]_{\uparrow}=-2$ and $Q_{c}^{\uparrow}=[M]/4=1/2\text{ (mod 1)}$.
Similarly, assuming the missing TE-like mode has a symmetric profile we have 0
modes with eigenvalues +1 to $C_{2}$ at $\Gamma$ and 1 at $M$. Therefore for
the TE-like modes $[M]_{\downarrow}=0$ and $Q_{c}^{\downarrow}=0\text{ (mod
1)}$. This is the expected result for this realization of the HOTI phase with
no SOC.

##  II Momentum space tight binding Hamiltonian

Fourier transforming the tight binding model of Eq. (1) from the main text,
one finds that the Hamiltonian can be written in four blocks. The two diagonal
blocks, one for each spin, are the standard Kekulé texture Hamiltonian Jung
_et al._ (2020) and the off diagonal blocks describe the SOC:

$H=\begin{pmatrix}H_{\uparrow}&H_{\text{SOC}}\\\
H^{\dagger}_{\text{SOC}}&H_{\downarrow}\end{pmatrix},$ (S1)

where

$H_{\uparrow,\downarrow}=\begin{pmatrix}0&-t_{\text{in}}^{\uparrow,\downarrow}&0&-t_{\text{out}}^{\uparrow,\downarrow}e^{-i\mathbf{k}\cdot\mathbf{a}_{1}}&0&-t_{\text{in}}^{\uparrow,\downarrow}\\\
-t_{\text{in}}^{\uparrow,\downarrow}&0&-t_{\text{in}}^{\uparrow,\downarrow}&0&-t_{\text{out}}^{\uparrow,\downarrow}e^{i\mathbf{k}\cdot\mathbf{a}_{2}}&0\\\
0&-t_{\text{in}}^{\uparrow,\downarrow}&0&-t_{\text{in}}^{\uparrow,\downarrow}&0&-t_{\text{out}}^{\uparrow,\downarrow}e^{-i\mathbf{k}\cdot\mathbf{a}_{3}}\\\
-t_{\text{out}}^{\uparrow,\downarrow}e^{i\mathbf{k}\cdot\mathbf{a}_{1}}&0&-t_{\text{in}}^{\uparrow,\downarrow}&0&-t_{\text{in}}^{\uparrow,\downarrow}&0\\\
0&-t_{\text{out}}^{\uparrow,\downarrow}e^{-i\mathbf{k}\cdot\mathbf{a}_{2}}&0&-t_{\text{in}}^{\uparrow,\downarrow}&0&-t_{\text{in}}^{\uparrow,\downarrow}\\\
-t_{\text{in}}^{\uparrow,\downarrow}&0&-t_{\text{out}}^{\uparrow,\downarrow}e^{i\mathbf{k}\cdot\mathbf{a}_{3}}&0&-t_{\text{in}}^{\uparrow,\downarrow}&0\end{pmatrix},$
(S2)

$\tiny
H_{\text{SOC}}=i\frac{\lambda_{\text{SOC}}}{3\sqrt{3}}\begin{pmatrix}0&0&-1-e^{-i\mathbf{k}\cdot\mathbf{a}_{1}}-e^{i\mathbf{k}\cdot\mathbf{a}_{3}}&0&1+e^{-i\mathbf{k}\cdot\mathbf{a}_{1}}+e^{i\mathbf{k}\cdot\mathbf{a}_{2}}&0\\\
0&0&0&-1-e^{-i\mathbf{k}\cdot\mathbf{a}_{1}}-e^{i\mathbf{k}\cdot\mathbf{a}_{2}}&0&1+e^{-i\mathbf{k}\cdot\mathbf{a}_{3}}+e^{i\mathbf{k}\cdot\mathbf{a}_{2}}\\\
1+e^{-i\mathbf{k}\cdot\mathbf{a}_{3}}+e^{i\mathbf{k}\cdot\mathbf{a}_{1}}&0&0&0&-1-e^{-i\mathbf{k}\cdot\mathbf{a}_{3}}-e^{i\mathbf{k}\cdot\mathbf{a}_{2}}&0\\\
0&1+e^{-i\mathbf{k}\cdot\mathbf{a}_{2}}+e^{i\mathbf{k}\cdot\mathbf{a}_{1}}&0&0&0&-1-e^{-i\mathbf{k}\cdot\mathbf{a}_{1}}-e^{i\mathbf{k}\cdot\mathbf{a}_{3}}\\\
-1-e^{-i\mathbf{k}\cdot\mathbf{a}_{2}}-e^{i\mathbf{k}\cdot\mathbf{a}_{1}}&0&1+e^{-i\mathbf{k}\cdot\mathbf{a}_{2}}+e^{i\mathbf{k}\cdot\mathbf{a}_{3}}&0&0&0\\\
0&-1-e^{-i\mathbf{k}\cdot\mathbf{a}_{2}}-e^{i\mathbf{k}\cdot\mathbf{a}_{3}}&0&1+e^{-i\mathbf{k}\cdot\mathbf{a}_{3}}+e^{i\mathbf{k}\cdot\mathbf{a}_{1}}&0\end{pmatrix},\normalsize$
(S3)

$\mathbf{a}_{1}=a(1,0)$, $\mathbf{a}_{2}=a(-1/2,\sqrt{3}/2)$ and
$\mathbf{a}_{3}=(-1/2,-\sqrt{3}/2)$. Defining a $C_{2}$ operator Jung  _et
al._ (2020) for a single spin as

$C_{2}=\begin{pmatrix}0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\
1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\ 0&0&1&0&0&0\\\ \end{pmatrix},$ (S4)

one can verify that the operator $C_{2}\otimes I_{2\times 2}$ is a symmetry
operator for Eq. (S1) at both $\Gamma$ and $M$, which means they remain
rotation invariant momenta points. The same procedure can also be applied to
the $C_{3}$ operator:

$C_{3}=\begin{pmatrix}0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
0&0&0&0&0&1\\\ 1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\ \end{pmatrix},$ (S5)

and $C_{3}\otimes I_{2\times 2}$, which shows that $\Gamma$ and $K$ remain
rotation invariant momenta. This is why the addition of SOC as a perturbation
does not change the eigenvalues of the system and why the addition of SOC does
not change the total corner charge $Q_{c}$.

When setting $t_{\text{in}}^{\uparrow}=t_{\text{in}}^{\downarrow}$ and
$t_{\text{out}}^{\uparrow}=t_{\text{out}}^{\downarrow}$, meaning the spin
subspaces are degenerate, transforming this Hamiltonian according to the
unitary transformation $H\rightarrow\exp{(i\pi s_{y}/4)}H\exp({-i\pi
s_{y}/4)}$, where $\hat{s}_{y}$ is the second Pauli matrix, results in a block
diagonal Hamiltonian, meaning we can still define a spin polarized corner
charge for each subspace using Eq. (2) of the main text without resorting to
Eq. (3).

##  III Derivation of phase transition condition
$\lambda^{2}_{\text{SOC}}=\Delta_{\uparrow}\Delta_{\downarrow}$ and coupling
of bands through SOC

Without SOC the Hamiltonian of Eq. (S1) has two independent spin subspaces.
Without loss of generality we assume the topological phase. Additionally, we
operate at the $\Gamma$ point because that is where the band gap closes. The
spin up subspace will have two $p$ orbital modes at $\Delta_{\uparrow}$ and
two $d$ orbital modes at $-\Delta_{\uparrow}$. Likewise, the spin down
subspace will have two $p$ orbital modes at $\Delta_{\downarrow}$ and two $d$
orbitals at $-\Delta_{\downarrow}$. The SOC perturbation couples spin up $p$
$(d)$ orbitals to spin down $p$ $(d)$ orbitals but does not couple orbitals of
different types, because it respects $C_{6}$ symmetry, and the coupling
strength is $\lambda_{\text{SOC}}$. This can be seen by transforming
$H_{\text{SOC}}$ to the orbital basis Jung  _et al._ (2020) defined according
to
$\left[s,p_{x}+ip_{y},p_{x}-ip_{y},d_{xy}+id_{x^{2}-y^{2}},d_{xy}-id_{x^{2}-y^{2}},f\right]$,
where:

$\begin{split}s&=[1,1,1,1,1,1]/\sqrt{6},\\\ p_{x}&=[1,0,0,-1,0,0]/\sqrt{2},\\\
p_{y}&=[0,1,1,0,-1,-1]/2,\\\ d_{xy}&=[2,-1,-1,2,-1,-1]/\sqrt{12},\\\
d_{x^{2}-y^{2}}&=[1,-1,-1,1,-1,-1]/\sqrt{6},\\\
f&=[1,-1,1,-1,1,-1]/\sqrt{6}.\end{split}$ (S6)

Performing this transformation one finds
$H_{\text{SOC}}=\lambda_{\text{SOC}}B^{\dagger}\text{diag}(0,-1,-1,1,1,0)B$,
where
$B=[s,p_{x}+ip_{y},p_{x}-ip_{y},d_{xy}+id_{x^{2}-y^{2}},d_{xy}-id_{x^{2}-y^{2}},f]$.
Therefore the eigenenergies after the SOC perturbation are the eigenvalues of
the matrices

$\begin{pmatrix}\Delta_{\uparrow}&\lambda_{\text{SOC}}\\\
\lambda_{\text{SOC}}&\Delta_{\downarrow},\end{pmatrix},\begin{pmatrix}-\Delta_{\uparrow}&\lambda_{\text{SOC}}\\\
\lambda_{\text{SOC}}&-\Delta_{\downarrow},\end{pmatrix}$ (S7)

for the qudrupole and dipole modes respectively. Solving for the eigenvalues
of these matrices we find that the quadrupole energies are

$E_{d\pm}=-\frac{\Delta_{\uparrow}+\Delta_{\downarrow}}{2}\pm\sqrt{\lambda_{\text{SOC}}^{2}+\left(\frac{\Delta_{\uparrow}-\Delta_{\downarrow}}{2}\right)^{2}}$
(S8)

and the dipole energies are

$E_{p\pm}=\frac{\Delta_{\uparrow}+\Delta_{\downarrow}}{2}\pm\sqrt{\lambda_{\text{SOC}}^{2}+\left(\frac{\Delta_{\uparrow}-\Delta_{\downarrow}}{2}\right)^{2}}.$
(S9)

Setting $E_{d-}=E_{p+}=0$ gives us the band gap closing condition of
$\lambda_{\text{SOC}}^{2}=\Delta_{\uparrow}\Delta_{\downarrow}$.

##  IV Derivation of photonic effective Hamiltonian and chiral symmetry

In this section we show the correspondence between the photonic PEC
bianisotropic waveguide and the tight binding model of Eqs. (S1-S3) and
discuss its limitations. We first deal with the TE-like and TM-like subspaces
independently. We designed our waveguides such that they all posses eigenmodes
with field profiles that have $s$, $p_{x}$, $p_{y}$,$d_{xy}$,$d_{x^{2}-y^{2}}$
and $f$ orbital profiles and are separated well enough in energy from any
additional bands that may interact with the relevant bands. Imposing $C_{6}$
symmetry on each subspace, i.e. $C_{6}HC_{6}=H$ we find that $H$ is a
circulant matrix at the rotation invariant momenta of $\Gamma$ and $M$. This
holds exactly for the TB model of Eq. (S2) as well as the photonic waveguide.

Chiral symmetry of the form $SHS=-H$ with $S=\text{diag}(1,-1,1,-1,1,-1)$
applies to $H_{\uparrow,\downarrow}$ but for the photonic system it is not
exact if one considers possible hopping terms that are not exclusively nearest
neighbor. We therefore have to assume that if we take the photonic eigenmodes
and transform them to the sublattice basis, where each mode is localized at an
effective lattice site in the unit cell Jung  _et al._ (2020), the overlap
integrals between next-nearest neighbors modes of the same polarization are
negligible compared to nearest neighbor overlap integrals. Phenomenologically,
this seems to apply because the frequency of the corner modes is very close to
the mid-gap frequency in all our designs. We note that one can design photonic
systems where chiral symmetry breaks (see supplementary information of Jung
_et al._ (2020)), hence chiral symmetry for the photonic system is by design,
not an intrinsic symmetry like in the lattice models.

The meaning of chiral symmetry in the lattice model is that the lattice can be
divided into two sublattices, one containing the odd index sites and the other
the even index sites. In the orbital basis the meaning of the chiral symmetry
operator is seen by its action on the basis vectors $\left|s\right>$,
$\left|p_{\pm}\right>$, $\left|d_{\pm}\right>$, $\left|f\right>$:

$\begin{split}S\left|s\right>=\left|f\right>,\\\
S\left|f\right>=\left|s\right>,\\\
S\left|p_{\pm}\right>=\left|d_{\pm}\right>,\\\
S\left|d_{\pm}\right>=\left|p_{\pm}\right>,\end{split}$ (S10)

where $\left|p_{\pm}\right>=\left|p_{x}\right>\pm i\left|p_{y}\right>$ and
$\left|d_{\pm}\right>=\left|d_{xy}\right>\pm i\left|d_{x^{2}-y^{2}}\right>$.

Assuming chiral symmetry applies to the photonic waveguide, one finds that the
even indices of the circulant matrix $H_{\uparrow,\downarrow}$ at $\Gamma$ and
$M$ must all be 0. The form of the effective Hamiltonian is therefore:

$H_{\uparrow,\downarrow}=\begin{pmatrix}0&c_{1}&0&c_{3}&0&c_{5}\\\
c_{5}&0&c_{1}&0&c_{3}&0\\\ 0&c_{5}&0&c_{1}&0&c_{3}\\\
c_{3}&0&c_{5}&0&c_{1}&0\\\ 0&c_{3}&0&c_{5}&0&c_{1}\\\
c_{1}&0&c_{3}&0&c_{5}&0\\\ \end{pmatrix}.$ (S11)

Demanding bosonic time-reversal symmetry $KHK=H$ where $K$ is the complex
conjugation operator means all elements of the Hamiltonian are real at the
time-reversal invariant momenta $\Gamma$ and $M$. Demanding that the system is
lossless constrains the Hamiltonian to be Hermitian, i.e. $H^{\dagger}=H$, and
since it is also real $H^{T}=H$, hence $c_{1}=c_{5}$, giving the final form of

$H_{\uparrow,\downarrow}=\begin{pmatrix}0&c_{1}&0&c_{3}&0&c_{1}\\\
c_{1}&0&c_{1}&0&c_{3}&0\\\ 0&c_{1}&0&c_{1}&0&c_{3}\\\
c_{3}&0&c_{1}&0&c_{1}&0\\\ 0&c_{3}&0&c_{1}&0&c_{1}\\\
c_{1}&0&c_{3}&0&c_{1}&0\\\ \end{pmatrix}.$ (S12)

which is identical to the values obtained by Eq. (S2) at $\Gamma$ and $M$ if
one sets $c_{1}=-t_{\text{in}}$ and $c_{3}=-t_{\text{out}}$.

Next, we turn to consider the effects of breaking mid-plane mirror symmetry
which couples the TE-like modes and the TM-like modes. From Slater cavity
perturbation theory Ma  _et al._ (2015), we find the effect of the
perturbation on the $H_{\uparrow,\downarrow}$ diagonal blocks is simply a
frequency shift of the TE-like and TM-like subspaces that can be engineered to
be the same magnitude for both polarizations by optimizing the energy overlap
of the TE-like modes and TM-like modes after the introduction of the mirror
symmetry breaking perturbation. The perturbation also introduces off-diagonal
blocks, whose form we analyze below.

Imposing $C_{6}$ symmetry of the form $\tilde{C_{6}}=C_{6}\otimes I_{2\times
2}$, i.e. $\tilde{C6}H\tilde{C_{6}}=H$, we find that the off-diagonal blocks
$H_{\text{SOC}}$ are circulant matrices, just like the diagonal blocks. Since
the systems is still lossless, the combined Hamiltonian is still Hermitian, so
the off-diagonal blocks are Hermitian conjugates of each other, and the system
still has bosonic time-reversal symmetry, so its elements are all real. We can
therefore construct one off-diagonal block and the other is determined
according to the first.

Assuming the frequency shift of both polarizations is 0, then due to time-
reversal property of a bianistropic waveguide with real
$\underline{\underline{\varepsilon}}$,$\underline{\underline{\mu}}$ and
$\underline{\underline{\chi}}$ Xiong  _et al._ (2017) we know that a chirality
operator $\tilde{S}$ must exist such that $\tilde{S}H\tilde{S}=-H$.
Furthermore, we know this operator must flip the sign of the TM-like modes
because it flips the sign of the transverse magnetic field Xiong  _et al._
(2017). $\tilde{S}$ must therefore take the form
$\tilde{S}=S\otimes\sigma_{z}$, where $S$ is some operator acting on the TE-
like and TM-like subspaces separately. Using the assumption of chiral symmetry
for $H_{\uparrow,\downarrow}$ before the introduction of mirror-symmetry
breaking we find that $S$ can only be identical to
$S=\text{diag}(1,-1,1,-1,1,-1)$ introduced before.

The geometric meaning of this chiral symmetry in the sublattice basis is that
the lattice can still be divided into two sublattices even with the
introduction of spin and hopping terms that involve spin. The first new
sublattice is the spin up / TE-like modes at sites 1,3,5 and spin down / TM-
like modes at sites 1,3,5 and the second sublattice is spin down / TM-like at
sites 2,4,6 and TE-like at sites 2,4,6.

Imposing $\tilde{S}H\tilde{S}=-H$ constraints the off-diagonal blocks to obey
$SH_{\text{SOC}}S=H_{\text{SOC}}$, where the negative sign is gone due to the
$\sigma_{z}$ component of $\tilde{S}$ for the off-diagonal blocks. One then
finds that the circulant matrix $H_{\text{SOC}}$ odd index elements vanish.
After a change of basis $S_{x}\rightarrow-S_{y}$ all elements become
imaginary. Therefore at $\Gamma$ and $M$ it has the form:

$H_{\text{SOC}}=i\begin{pmatrix}c_{0}&0&c_{2}&0&c_{4}&0\\\
0&c_{0}&0&c_{2}&0&c_{4}\\\ c_{4}&0&c_{0}&0&c_{2}&0\\\
0&c_{4}&0&c_{0}&0&c_{2}\\\ c_{2}&0&c_{4}&0&c_{0}&0\\\
0&c_{2}&0&c_{4}&0&c_{0}\\\ \end{pmatrix}.$ (S13)

Kane-Mele SOC of Eq.(S3) is a particular example of such allowed coupling
matrix at $\Gamma$ and $M$, which is why the photonic model and the TB model
are so similar in their behavior.

##  V Additional phase diagrams

We chose to present $Q_{c}^{\uparrow}$ as the color for Fig. 1(c), but there
is some additional value in presenting additional phase diagrams with the
color encoding the additional bulk quantities we have defined in the main
text. Fig. S2(a) shows the phase diagram with $Q_{c}^{\uparrow}$ as the color.
In Fig. S2(b) we use $Q_{c}^{\downarrow}$ as the color content. We can see
both are quantized at 1/2 for the topological phase and 0 for the trivial
phase, but lose quantization for the HOTI phases as SOC is added. In Fig.
S2(c) we plot $S_{c}$ as the color content and notice that it retains
$S_{c}=0$ along the black dashed QSH line. This is expected since the spin
subspaces are identical for that parameter choice and therefore
$Q_{c}^{\uparrow}=Q_{c}^{\downarrow}$. Comparing to Fig. S2(a) one can see
that $S_{c}$ is a function of only one of $Q_{c}^{\uparrow}$ or
$Q_{c}^{\downarrow}$, i.e. they are not independent. This is because for the
HOTI phase $Q_{c}^{\downarrow}=1/2-Q_{c}^{\uparrow}$ due to the quantization
of $Q_{c}$. In Fig. S2(d) we plot $Q_{c}$ as the color content, which is
always quantized for all phases. Additionally, we plot the 3D phase diagram in
Fig. S3, whose cut planes are plotted in Fig. S2.

Figure S2: Phase diagrams with constant $\Delta_{\uparrow}$ with additional
color content for bulk quantities not shown in the main text. (a)Color content
is $Q_{c}^{\uparrow}$. Black dashed line marks the QSH phase. (b) Color is
$Q_{c}^{\downarrow}$. (c) Color is $S_{c}$. (d) Color is $Q_{c}$. Note that
the color bars for different sub-figures have different ranges. Figure S3: 3D
phase diagram. Phase boundaries are plotted using meshed surface. HOTI phase
captures all the volume that is not contained in the trivial and spin-HOTI
surfaces. QSH planes are in transparent red.

##  VI Quantization of $Q_{c}^{\uparrow}$ and $Q_{c}^{\downarrow}$ in spin-
HOTI phase with a finite $\lambda_{\text{SOC}}$

In this section, we provide a brief proof on how the spin-projected corner
charge indices $Q_{c}^{s_{z}=\uparrow}$ and $Q_{c}^{s_{z}=\downarrow}$ are
well-quantized in the spin-HOTI phase $Q_{c}^{\uparrow\downarrow}=\frac{1}{2}$
and in the trivial phase $Q_{c}^{\uparrow\downarrow}=0$ even in the presence
of an off-diagonal spin-orbit coupling ($H_{\text{SOC}}\otimes s_{x}$). In the
section III, we established that
$H_{\text{SOC}}=\left[\lambda_{\text{SOC}}B^{\dagger}\text{diag}(0,-1,+1,-1,+1,0)B\right]$,
where $B=[s,p_{-},p_{+},d_{-},d_{+},f]$ ($p_{\pm}=p_{x}\pm p_{y}$,
$d_{\pm}=d_{xy}\pm d_{x^{2}-y^{2}}$). In other words, $H_{\text{SOC}}\otimes
s_{x}$ can induce coupling only between $\left|{p_{\pm},\uparrow}\right>$ and
$\left|{p_{\pm},\downarrow}\right>$ or between
$\left|{d_{\pm},\uparrow}\right>$ and $\left|{d_{\pm},\downarrow}\right>$.

In the spin-topological phase ($\Delta_{\uparrow\downarrow}>0$), before we
introduce a finite $\lambda_{\text{SOC}}$, we have
$\left|{s,\uparrow}\right>$, $\left|{d_{\pm},\uparrow}\right>$,
$\left|{s,\downarrow}\right>$, $\left|{d_{\pm},\downarrow}\right>$ orbitals
below the bandgap and $\left|{f,\uparrow}\right>$,
$\left|{p_{\pm},\uparrow}\right>$, $\left|{f,\downarrow}\right>$,
$\left|{p_{\pm},\downarrow}\right>$ orbitals above the bandgap at
$\bf{\Gamma}$-point, thereby featuring $\\#\Gamma_{\uparrow,\downarrow}=3$
(for both spin subspaces, there are three eigenstates with +1 eigenvalue to
$C_{2}$ rotation below the band gap). We can see that a finite
$H_{\text{SOC}}$ term cannot induce any coupling between the states that are
placed across the bandgap. Then, given a finite $\lambda_{\text{SOC}}$, the
eigenstates at $\bf{\Gamma}$-point below the bandgap will be
$\left|{s,\uparrow}\right>$, $\left|{s,\downarrow}\right>$,
$A\left|{d_{+},\uparrow}\right>+B\left|{d_{+},\downarrow}\right>$,
$B\left|{d_{+},\uparrow}\right>-A\left|{d_{+},\downarrow}\right>$,
$C\left|{d_{-},\uparrow}\right>+D\left|{d_{-},\downarrow}\right>$, and
$D\left|{d_{-},\uparrow}\right>-C\left|{d_{-},\downarrow}\right>$. The linear
coefficients $A$ and $B$ are given by solving the eigenbasis for
$[\Delta_{\uparrow},\lambda_{\text{SOC}};\lambda_{\text{SOC}},\Delta_{\downarrow}]$,
and $C$ and $D$ for
$[\Delta_{\uparrow},-\lambda_{\text{SOC}};-\lambda_{\text{SOC}},\Delta_{\downarrow}]$.
To be specific,
$\frac{A}{B}=-\frac{C}{D}=\frac{\Delta_{\uparrow}-\Delta_{\downarrow}+\sqrt{(\Delta_{\uparrow}-\Delta_{\downarrow})^{2}+4\lambda_{\text{SOC}}^{2}}}{2\lambda_{\text{SOC}}}$
with $A^{2}+B^{2}=1$ and $C^{2}+D^{2}=1$ (they can be set to be real numbers).
Then, we get $\\#\Gamma_{\uparrow\downarrow}=1+A^{2}+B^{2}+C^{2}+D^{2}=3$.
Therefore, upon a finite $\lambda_{\text{SOC}}$,
$\\#\Gamma_{\uparrow\downarrow}$ is preserved to be at the same value obtained
with vanishing $\lambda_{\text{SOC}}$. This same analysis applies for the
trivial phase as well that $\\#\Gamma_{\uparrow\downarrow}=1$ is preserved
with or without $\lambda_{\text{SOC}}$.

Basically, since the Kane-mele type spin-orbit coupling induces coupling only
between the same orbital types, in the trivial or spin-HOTI phase, the
exchange of wavefunction amplitudes is happening only within the bands that
are below the bandgap or only within the bands that are above the bandgap.
Thus, the sum of spin-projected wavefunction amplitudes over all eigenstates
(with $C_{2}$ rotation eigenvalue of +1) below the bandgap remains the same
unless $\lambda_{\text{SOC}}$ is too strong to cause the band inversion.

##  VII Fractional corner-localized charge and corner-localized spin as
boundary observables

We emphasize that $Q_{c}$, $Q_{c}^{\uparrow}$, $Q_{c}^{\downarrow}$, and
$S_{c}$ are (when properly quantized) bulk topological invariants even though
these quantities are often treated interchangeably as boundary observables due
to well-established bulk boundary correspondence (BBC) Benalcazar  _et al._
(2017, 2019); Noh  _et al._ (2018). To be specific, as a result of
$Q_{c}=\frac{1}{2}$ defined in the bulk, a terminated $120^{\circ}$-angled
corner would possess a corner-localized $\frac{1}{2}$ deficit of local density
of states (LDOS) accumulated for all states below the mid-bandgap energy
(here, zero energy), calculated according to:

$\rho_{\uparrow}\left(\mathbf{R}\right)=\sum_{E<0}\sum_{i=1}^{6}\left|\psi_{n}\left(\mathbf{R},i,\uparrow\right)\right|^{2},$
(S14)

where the summation is done on all negative energy states excluding the zero
energy mode that is at small negative energy due to finite size effects,
$\psi_{n}$ is the eigenstate for the open boundary system, $\mathbf{R}$ is the
position of the unit cell and $i$ is the sublattice index. Thus
$\psi_{n}\left(\mathbf{R},i,\uparrow/\downarrow\right)$ is the wavefunction
value at position $\mathbf{R}$ sublattice $i$ with spin up or down. Fig. S4(a)
shows the accumulated LDOS $\rho$ pattern around an open corner in the HOTI
phase ($Q_{c}=\frac{1}{2}$); we obtain the corner charge of half by
integrating the deviation of accumulated LDOS from the value deep inside the
bulk (i.e. the number of bands below the bandgap; $\rho_{\rm bulk}=6$ in this
case). Since this is a topological property, this corner charge of
$\frac{1}{2}$ doesn’t change for the case of embedding the topological domain
inside surrounding trivial domain, see Fig. S4(b). Embedding into the trivial
domain is not necessary for the tight-binding model, but it is necessary for a
photonic emulation system, as a true open termination with vacuum is difficult
to realize in our photonic platform. The embedding naturally decreases the
size of the band gap between the edge states, thereby weakening the
localization of the corner state along the edges.

Figure S4: Accumulated LDOS $\rho$ pattern around an open or embedded corner.
(a) HOTI phase
($t_{\text{in}}^{\uparrow}=-0.8$,$t_{\text{out}}^{\uparrow}=-1.2$,$t_{\text{in}}^{\downarrow}=-1.2$,$t_{\text{out}}^{\downarrow}=-0.8$)
with a finite SOC $\lambda_{\text{SOC}}=0.1$. (b) HOTI domain (green hexagons)
embedded in trivial domain (red hexagons); the trivial domain is set without
SOC. (c) Accumulated net spin $\rho_{\uparrow}-\rho_{\downarrow}$ pattern in
the HOTI phase considered in (a); the corner spin is not well-quantized as
discussed in the main text.

In Fig. S4(c), we also demonstrate the accumulated net spin texture
$\rho_{\uparrow}-\rho_{\downarrow}$ around an open corner. The LDOS-based
calculation of spin corner shown in Fig. 2(b) of the main text was obtained by
$\langle S_{z}\rangle=\sum(\rho_{\uparrow}-\rho_{\downarrow})$. As discussed
in the main text, this spin corner is not well quantized in the HOTI phase in
the presence of SOC. However, at a reasonable strength of SOC
$\lambda_{\text{SOC}}=0.1$, $\langle S_{z}\rangle$ retains a value very close
to the bulk quantity $S_{c}$ and both are close to 0.5 featuring a moderately
(not perfectly) robust preservation of corner-localized spin.

##  VIII Bulk boundary correspondence

In the main text, we briefly argued that $Q_{c}$, $Q_{c}^{\uparrow}$ and
$Q_{c}^{\downarrow}$ can be used to predict the number of corner states at the
interface of two different domains. Here, we elaborate on the details of these
crucial bulk boundary correspondences (BBC) in our spin-dependent kekule
lattice model in the presence of spin-orbit coupling. For the spinless kekule
lattice, it has been well-established that the topological origin of a corner
state can be explained as a direct result of $Q_{c}=\frac{1}{2}$ and the
chiral symmetry $S=\text{diag}(1,-1,1,-1,1,-1)$ Noh  _et al._ (2018); Jung
_et al._ (2020). $Q_{c}=\frac{1}{2}$ gives rise to half corner-localized
deficit of LDOS accumulated for the bands below zero energy, and the chiral
symmetry ensures that the same deficit of half from the bands above zero
energy. The two halves or, equally, whole of corner-localized LDOS deficit
should be compensated by a state localized around a corner, and this corner
state must be pinned at zero energy again due to the chiral symmetry. This
line of arguments effectively proves BBC in Kekulé lattices.

In our spinful system, the simple extension of the previous chiral symmetry
$S\otimes I_{2\times 2}$ does not satisfy the condition of a chiral symmetry
$H(S\otimes I_{2\times 2})=-(S\otimes I_{2\times 2})H$ with a finite
$\lambda_{\text{SOC}}$; here, $H$ is from Eq. (S1). In fact, a spin-dependent
extension $S_{\text{SOC}}=S\otimes\sigma_{z}$ satisfies the given condition
$HS_{\text{SOC}}=-S_{\text{SOC}}H$. Thus, our spinful system in the presence
of spin-orbit coupling possesses the chiral symmetry given as
$S_{\text{SOC}}$. Note that this new chiral symmetry operator, when applied to
an eigenvector, still preserves the spin contents of the eigenvector;
$\left<{\psi}\right|s_{z}\left|{\psi}\right>=\left<{\psi^{\prime}}\right|s_{z}\left|{\psi^{\prime}}\right>$
for $\left|{\psi^{\prime}}\right>=S_{\text{SOC}}\left|{\psi}\right>$. This
spin-preservation property is very crucial, as it allows us to apply the BBC
of kekule lattices to each spin subspace.

Now that we have properly defined the spin-preserving chiral symmetry for our
spinful Hamiltonian, let’s establish BBCs for the following cases:

###  VIII.1 Topological phase (for generic $\lambda_{\text{SOC}}$) with an
open corner or embedded corner interfacing trivial phase

With $\lambda_{\text{SOC}}=0$, the system is a simply decoupled stack of two
topological kekule lattices of each spin species. Therefore, we have
$Q_{c}^{\uparrow}=\frac{1}{2}$ and $Q_{c}^{\downarrow}=\frac{1}{2}$. Suppose
that we now turn on the spin-orbit-coupling $\lambda_{\text{SOC}}>0$. Recall
that we have established in the previous section that our spin-orbit coupling
only induces couplings between the same orbital (dipolar or quadrupolor)
states. Thus, there cannot be any coupling between the modes below zero energy
and the modes above zero energy in topological phase, because all
quadrupole(dipole)-like states of both spin species are below(above) zero
energy. As a result, $Q_{c}^{\uparrow}=\frac{1}{2}$ and
$Q_{c}^{\downarrow}=\frac{1}{2}$ are conserved even under a finite
$\lambda_{\text{SOC}}>0$, since the exchange of spin expectation values is
being done within the bands below zero energy.

Therefore, with and without spin-orbit-coupling, we always have well-quantized
$Q_{c}^{\uparrow}=\frac{1}{2}$ and $Q_{c}^{\downarrow}=\frac{1}{2}$, and each
of these invariants, along with the spin-preserving chiral symmetry
$S_{\text{SOC}}$, predicts a whole deficit of corner-localized accumulated
LDOS, which gives rise to total two corner states. There are two options for
the energy of these corner states: They can either both be at zero energy,
each being a chiral partner of itself, or they can be at $\pm E$ as chiral
partners of each other. For the perturbations we consider the corner states
are always chiral partners of themselves and are therefore at zero energy.

###  VIII.2 HOTI phase (for generic $\lambda_{\text{SOC}}$) with an open
corner or embedded corner interfacing trivial phase

$Q_{c}=\frac{1}{2}$ and the chiral symmetry $S_{\text{SOC}}$ ensures a corner
state at zero energy.

###  VIII.3 HOTI phase (for generic $\lambda_{\text{SOC}}$) with an embedded
corner interfacing topological phase

$Q_{c}=\frac{1}{2}$ and the chiral symmetry $S_{\text{SOC}}$ ensures a corner
state at zero energy ($Q_{c}=0$ in topological phase).

###  VIII.4 HOTI with $Q_{c}^{\uparrow}=1/2$, $Q_{c}^{\downarrow}=0$
interfaced with HOTI with $Q_{c}^{\uparrow}=0$, $Q_{c}^{\downarrow}=1/2$ (for
vanishing $\lambda_{\text{SOC}}=0$)

With vanishing spin-orbit-coupling $\lambda_{\text{SOC}}=0$,
$Q_{c}^{\uparrow}$ and $Q_{c}^{\downarrow}$ are well-defined even in the HOTI
phase; $Q_{c}^{\uparrow}=\frac{1}{2}$ and $Q_{c}^{\downarrow}=0$ for HOTI I,
and $Q_{c}^{\uparrow}=0$ and $Q_{c}^{\downarrow}=\frac{1}{2}$ for HOTI II.
Thus, for each spin subspace, two domains are topologically distinct with
$Q_{c}$ difference of half, and therefore we obtain a corner state at zero
energy.

We note that the BBC for the case of HOTI I interfacing with HOTI II is not
established for a finite spin-orbit coupling. We empirically find that the
spin corner $S_{c}$, even though not perfectly quantized, is nearly conserved
to be $+\frac{1}{2}$ for HOTI I and $-\frac{1}{2}$ for HOTI II, thus providing
an effective distinction between these two phases for small values of SOC.

##  IX Topological indices in QSH phase
($\Delta_{\uparrow}=\Delta_{\downarrow}$;
$|\lambda_{\text{SOC}}|>|\Delta_{\uparrow}|$)

When two spin subspaces are degenerate
($t^{\uparrow}_{\text{in/out}}=t^{\downarrow}_{\text{in/out}}\equiv
t_{\text{in/out}}$; thus, $\Delta_{\uparrow}=\Delta_{\downarrow}\equiv\Delta$
and $H_{\uparrow}=H_{\downarrow}\equiv H_{0}$), by changing the spin basis
$s_{x}\rightarrow s_{z}$, the system hamiltonian can be made to be block-
diagonal, where one block (spin up) is $H_{0}+H_{\text{SOC}}$ and the other
block (spin down) is $H_{0}-H_{\text{SOC}}$. Basically, in this basis, we have
a more conventional form of Kane-Mele spin-orbit coupling where it induces an
effective magnetic field to each of two decoupled spin subspaces with opposite
signs. In the chern-insulator perspective, we get
$C_{s_{x}=\uparrow}=C_{s_{x}=\downarrow}=0$ when
$|\lambda_{\text{SOC}}|<|\Delta_{0}|$ and $C_{s_{x}=\uparrow}=+1$ and
$C_{s_{x}=\downarrow}=-1$ when $|\lambda_{\text{SOC}}|>|\Delta_{0}|$ (here,
the chern number is the composite chern number of the bands below the
bandgap). Thus, the QSH phase $|\lambda_{\text{SOC}}|>|\Delta_{0}|$ carries
the spin-chern number of $C_{s_{x}=\uparrow}-C_{s_{x}=\downarrow}=2$.

In the corner charge index perspective, we get
$Q_{c}^{\uparrow\downarrow}=\frac{1}{4}$,
$Q_{c}=Q_{c}^{\uparrow}+Q_{c}^{\downarrow}=\frac{1}{2}$, and
$S_{c}=Q_{c}^{\uparrow}-Q_{c}^{\downarrow}=0$. The spin-projected corner
charges $Q_{c}^{\uparrow\downarrow}$ are well-quantized as $\frac{1}{4}$ in
any spin-projection basis for the following reasons. First, let’s consider
$s_{x}$ basis where the system is favorably block-diagonalizable. Also, let’s
consider the $s_{x}=\uparrow$ subspace only. Across the phase boundary
$|\lambda_{\text{SOC}}|=|\Delta|$ between the spin-HOTI phase and QSH phase,
$\\#\Gamma_{s_{x}=\uparrow}$ changes from $3$ to $2$, due to the band-
inversion between $d_{+}$ and $p_{-}$ orbitals, see Fig. S5. In the meantime,
$\\#M_{s_{x}=\uparrow}$ stays as $1$. Thus, we observe the transition from
$Q_{c}^{\uparrow}=\frac{\\#\Gamma_{s_{x}=\uparrow}-\\#M_{s_{x}=\uparrow}}{4}=\frac{2}{4}$
to $\frac{1}{4}$. For the $s_{x}=\downarrow$ subspace, the same transition
occurs due to the band-inversion between $d_{-}$ and $p_{+}$ orbitals. The
same principle applies for the phase transition between the trivial phase and
QSH phase, resulting the change of $Q_{c}^{\uparrow\downarrow}=0$ to
$\frac{1}{4}$.

Figure S5: Band structures of a single $s_{x}=\uparrow$ spin subspace when the
system is block-diagonalizable under a spin-degeneracy condition
$t^{\uparrow}_{\text{in}}=t^{\downarrow}_{\text{in}}\equiv 0.8$,
$t^{\uparrow}_{\text{out}}=t^{\downarrow}_{\text{out}}\equiv 1.2$; the
strength of magnetic flux $\lambda_{\text{SOC}}$ is varied across the phase
transition $\lambda_{\text{SOC}}=\Delta$ ($\Delta=0.4$ in this case).

##  X Parameters for tight binding models and photonic crystal designs

Since our designs includes many parameters as shown in Fig. 3(a) of the main
text, and not wishing to trouble the general reader, we include the design
parameters of all phases plus SOC in the following tables, including what
figure and subfigure they appear at, as can be seen in Tables 1 and 2. For the
TB models the transition amplitudes between domain walls were set to be
$\tilde{t}_{\text{in}}^{\uparrow,\downarrow}=(t_{\text{in, outer
domain}}^{\uparrow,\downarrow}+t_{\text{in, inner
domain}}^{\uparrow,\downarrow})/2+\alpha(t_{\text{in, outer
domain}}^{\uparrow,\downarrow}-t_{\text{in inner
domain}}^{\uparrow,\downarrow})/2$, where $\alpha$ is a real scalar
representing the edge roughening. For $\alpha=1$ there is no edge roughening
and for $\alpha>1$ the amplitudes between domain walls take some value between
the amplitudes of the inner and outer domains. We’ve used $\alpha=2$ in all
our TB calculations. For the photonic first principle simulations the edge
roughening used was setting the cylinders separating the domains to either
$1.4\langle r_{1}\rangle$ or $0.7\langle r_{2}\rangle$ where $\langle
r_{1}\rangle,\langle r_{2}\rangle$ are the respective average radii of $r_{1}$
and $r_{2}$ of the inner and outer domains. The air gaps between said
cylinders and the top and bottom plates also took the average values of the
air gaps of both domains.

Phase | Figures | $t_{\text{out}}^{\uparrow}$ | $t_{\text{in}}^{\uparrow}$ | $t_{\text{out}}^{\downarrow}$ | $t_{\text{in}}^{\downarrow}$ | $\lambda_{\text{SOC}}$
---|---|---|---|---|---|---
HOTI | Fig.2(a) | 0.7 | 1.3 | 1.2 | 0.8 | 0.8
Spin-HOTI | Fig.2(b) | 1.3 | 0.7 | 1.2 | 0.8 | 0.4
Trivial | Fig.2(a),(b) | 0.7 | 1.3 | 0.8 | 1.2 | 0
Spin-HOTI | Fig.4(a) | 1.1 | 0.8 | 1.4 | 0.8 | 0 (solid lines) 0.42 (dashed lines)
HOTI | Fig.4(b) | 0.8 | 1.1 | 1.4 | 0.8 | 0 (solid lines) 0.42 (dashed lines)
Table 1: Parameters for tight binding models. Phase | Figures | $h$ | $h_{\text{mid}}$ | $r_{1}$ | $r_{1}^{\text{mid}}$ | $r_{2}$ | $r_{2}^{\text{mid}}$ | $g_{\text{top}}$ | $g_{\text{bot}}$
---|---|---|---|---|---|---|---|---|---
HOTI | Fig.3(b) | a | 0 | 0.24a | 0.24a | 0.136a | 0.136a | 0.045a | 0.045a
Spin-HOTI | Fig.3(c), Fig.S8(a) | a | 0.2a | 0.22a | 0.08a | 0.11a | 0.21a | 0.02a | 0.02a
Spin-HOTI | Fig.4(e) | a | 0.2a | 0.22a | 0.08a | 0.11a | 0.21a | 0.02a | 0.04a
Spin-HOTI and HOTI transition | Fig.4(c), Fig.S8(b) | a | 0.2a | 0.22a | 0.08a | 0.11a | 0.21a | 0.02a | 0.075a
Spin-HOTI | Fig.S8(c) | a | 0.2a | 0.22a | 0.08a | 0.11a | 0.21a | 0.02a | 0.12a
HOTI | Fig.4(d) | a | 0 | 0.24a | 0.24a | 0.136a | 0.136a | 0.045a | 0.12a
Trivial | Fig.4(f) | a | 0.28a | 0.13a | 0.25a | 0.17a | 0.1a | 0.045a | 0.045a
Trivial | Fig.4(e) | a | 0.06a | 0.08a | 0.236a | 0.17a | 0.1a | 0.045a | 0.045a
QSH | Fig.S6(b) | a | 0.2a | 0.22a | 0.04a | 0.132a | 0.22a | 0.015a | 0.1a
Trivial | Fig.S6(b) | a | 0.06a | 0.11a | 0.263a | 0.175a | 0.1a | 0.045a | 0.045a
Table 2: Parameters for photonic crystal designs.

##  XI QSH phase and edge states

As mentioned in the main text, when the TB model is tuned such that
$\Delta_{\uparrow}=\Delta_{\downarrow}$ and
$\lambda_{\text{SOC}}>\lambda_{\text{th}}$ a QSH phase is obtained and the
corner states become delocalized. This was confirmed in a TB and photonic
supercell simulation as shown in Fig. S6.

Figure S6: QSH phase. (a) TB model dispersion for
$t_{\text{out}}^{\uparrow,\downarrow}=1.2$,
$t_{\text{in}}^{\uparrow,\downarrow}=0.8$ and $\lambda_{\text{SOC}}=1$ for the
QSH domain and $t_{\text{out}}^{\uparrow,\downarrow}=0.8$,
$t_{\text{in}}^{\uparrow,\downarrow}=1.2$ and $\lambda_{\text{SOC}}=0$ for the
trivial domain, with $\alpha=2$ for cross domain hopping amplitudes. (b)
Photonic supercell simulation dispersion results. Bulk modes in blue and edge
modes in red and black. Inset: QSH phase in yellow interfacing a trivial phase
in purple. See Table II for parameters.

Additionally we investigated the effect of the edge roughening, i.e. changing
the transition amplitudes along the domain wall between the QSH and trivial
domains on the edge states. Edge roughening can gap the QSH edge states if the
band gap is small enough, as can be seen in Fig. S7(a),(b). However, deep
inside the QSH phase where the bulk band gap is large, edge roughening has no
effect on the edge states and does not gap them, as can be seen in Fig.
S7(c),(d).

Figure S7: Effect of edge roughening on QSH edge states. (a) QSH with small
bulk band gap and no roughening. (b) same as (a) but with edge roughening. (c)
QSH phase with large bulk band gap and no edge roughening. (d) same as (c) but
with edge roughening.

##  XII Photonic transition from spin-PHOTI to PHOTI

Here we show that increasing $\Delta g$ can cause a phase transition from the
spin-HOTI phase to the HOTI phase. We do this by taking the PhC design of Fig.
3(c) and Fig. 4(e) and further increasing $\Delta g$ to push the PhC to the
HOTI phase. As can be seen in Fig. S8(a), the PhC is in a spin-HOTI phase
since all bands below the band gap are $d$ orbitals. An introduction of
effective SOC by $\Delta g\neq 0$ closes the band gap in Fig. S8(b), and a
further increase of $\Delta g$ reopens the band gap, only now two of the bands
below the band gap are $p$ orbitals. We note that the $p$ and $d$ orbital band
gaps are not centered at the same frequency. This can be compensated by
changing other geometric parameters of the design, but for the sake of the
example we did not wish to change any parameters besides $g_{\text{bot}}$.

Figure S8: Photonic transition from spin-HOTI phase to HOTI phase. (a) Same as
Fig. 3(c) from the main text showing HOTI phase with mid-plane mirror
symmetry. All bands below the band gap are $d$ orbitals. (b). Same as Fig.
4(e) from the main text showing tuning of mid-plane mirror symmetry breaking
to close the band gap. (c) Further increase of the mid-plane mirror symmetry
breaking causes the band gap to reopen with two $d$ and two $p$ orbital bands,
indicating HOTI phase.