# The Nature of Topological Protection in Spin and Valley Hall Insulators

Matthias Saba<EMAIL_ADDRESS>The Blackett Laboratory, Imperial College
London, London SW7 2AZ, UK  Adolphe Merkle Institute, University of Fribourg,
1700 Fribourg, Switzerland  Stephan Wong  The Blackett Laboratory, Imperial
College London, London SW7 2AZ, UK  School of Physics and Astronomy, Cardiff
University, Cardiff CF24 3AA, UK  Mathew Elman  The Blackett Laboratory,
Imperial College London, London SW7 2AZ, UK  Sang Soon Oh  School of Physics
and Astronomy, Cardiff University, Cardiff CF24 3AA, UK  Ortwin Hess
<EMAIL_ADDRESS>The Blackett Laboratory, Imperial College London, London
SW7 2AZ, UK

###### Abstract

Recent interest in optical analogues to the quantum spin Hall and quantum
valley Hall effects is driven by the promise to establish topologically
protected photonic edge modes at telecommunication and optical wavelengths on
a simple platform suitable for industrial applications. While first
theoretical and experimental efforts have been made, these approaches so far
both lack a rigorous understanding of the nature of topological protection and
the limits of backscattering immunity. We here use a generic group theoretical
methodology to fill this gap and obtain general design principles for purely
dielectric two-dimensional topological photonic systems. The method
comprehensively characterizes possible 2D hexagonal designs and reveals their
topological nature, potential and limits.

Since Haldane and Raghu Haldane and Raghu (2008); Raghu and Haldane (2008)
proposed the optical analogue of the integer quantum Hall effect in a 2D
photonic crystal (PhC) using gyromagnetic materials in 2008, the field of
topological photonics has been growing. The broad interest in topological
photonic systems mainly stems from the promise of unidirectional,
backscattering-free interface waves that are protected against material and
fabrication imperfections, and environmental fluctuations Lu  _et al._ (2014,
2016); Khanikaev and Shvets (2017). These properties bear the potential to
solve many of the state of the art problems connected with on-chip optical
computation and data processing.

The original proposal quickly sparked subsequent experimental studies using
gyromagnetic and gyroelectric effects Wang  _et al._ (2009). The need of
extremely large magnetic fields to observe the predicted behaviour at higher
frequencies, however, limits the scope of applications. A number of
workarounds to reach higher frequencies without the need of gigantic static
magnetic fields include emulating reciprocity breaking through coupled
resonators Fang  _et al._ (2012), and engineering of anti-symmetric scattering
matrices in parity-time-duality symmetric systems Silveirinha (2017);
Khanikaev  _et al._ (2013). These approaches, however, either require
structures that are much larger than the operation wavelength or exotic
constituent materials.

Designs based on purely dielectric platforms Wu and Hu (2015); Chen  _et al._
(2017); Ma and Shvets (2016), conceivably related to the Quantum Spin Hall
effect (QSHE) or the Quantum Valley Hall effect (QVHE), have recently become
an active field of investigation to overcome these limitations. While these
systems seem promising from a practical point of view, the nature and role of
topological protection is not well understood to date. Topological concepts
such as the spin and valley Chern numbers for the QSHE and QVHE have entered
the discussion. In electro-magnetic systems with reciprocity symmetry, the
former is, however, neither rigorous invariant nor is a bulk-edge
correspondence (in the Kane Mele sense Kane and Mele (2005)) established. The
valley Chern number as defined in the literature Chen  _et al._ (2017); Ma and
Shvets (2016)is not a topological invariant as shall become evident below,
while a bulk-boundary correspondence based on a lattice-folding Hamiltonian
has been recently established Qian  _et al._ (2018).

We here use a group theoretical pathway Saba  _et al._ (2017) to understand in
which sense all-dielectric QSHE and QVHE edge modes are unidirectional and
topologically protected. In particular, we show that QVHE edge modes in the
center of the band gap are weakly protected against moderate arbitrary
perturbations. In contrast to strongly $\mathbb{Z}$-protected integer quantum
Hall states and $\mathbb{Z}_{2}$-protected proper quantum spin Hall states,
their existence and robustness with respect to perturbations relies, however,
on the particular choice of crystal termination and inclination.
Backscattering immunity on the other hand requires orthogonality of counter-
propagating _chiral spin_ states which is here based on spatial symmetries of
the underlying bulk and shown to be approximately valid for small band gaps.
We illustrate our findings by means of the QSHE PhC introduced in Wu and Hu
(2015) and a new PhC design based on the kagome lattice Mekata (2003); Guo and
Franz (2009), and derive a list of general principles for the design of all-
dielectric QVHE insulators for applications at optical or near-infrared
frequencies.

Figure 1: Two hexagonal lattice designs with broken $C_{6v}$ symmetry made of
dielectric rods, indefinitely extended in the third direction. (a) QSHE
honeycomb structure with sites at the corners of the hexagons. The design is
perturbed by shrinking/expanding every third Wigner-Seitz cell labelled A Wu
and Hu (2015). (b) QVHE kagome structure with sites at the center of the
edges. The design is perturbed by moving the rods along the edges.

The structure of this manuscript is as follows: In section II, we explore the
possibility of topological band gaps in symmetry perturbed $C_{6v}$ lattices.
We group theoretically enumerate all possible spatial perturbations, excluding
only long range chirps. A band gap opens only in three cases, one of which
requires breaking of reciprocity symmetry, while the other two can be related
to the well known QSHE and QVHE like systems (canonical examples of each case
are shown in Fig. 1 and discussed in detail in Wong  _et al._ (2019)). The
resulting bandstructures are derived and discussed in section III. The
topological properties of the QSHE and QVHE scenarios are illuminated in
sections IV and V, respectively. It is found that QSHE systems, while
resembling a $\mathbb{Z}_{2}$ invariant edge state dispersion in first order
perturbation, do not lead to topological protection. QVHE systems on the other
hand lead to topologically protected edge modes that can be related to strong
Chern protection. Since the type of protection only exists in an extended
parameter space including geometrical perturbation, it depends on inclination
and termination, consistent with the findings presented in Qian  _et al._
(2018). The associated bulk-boundary correspondence is therefore not
comparable to strong protection through breaking of optical reciprocity
Haldane and Raghu (2008). The edge modes are, however, protected in the same
way against moderate random perturbations if corresponding design principles
are met.

##  I Perturbation theory on irreducible space group representations

Term | Space Group Element $\hat{g}$ | Hermiticity $\hat{O}_{\text{Herm}}$ | Time Reversal $\hat{O}_{\text{TR}}$ | Reciprocity $\hat{O}_{\text{Rec}}$
---|---|---|---|---
$f_{\lambda}$ | $\,\overline{\Lambda\vphantom{\bar{\Lambda}}}\,_{\bm{k}i}(g).f_{\lambda}.\Lambda_{\bm{k}i}^{T}(g)$ | $f_{\lambda}^{\dagger}$ | $\,\overline{\Lambda\vphantom{\bar{\Lambda}}}\,_{\bm{k}i}(I).\,\overline{f_{\lambda}\vphantom{\bar{f_{\lambda}}}}\,.\Lambda_{\bm{k}i}^{T}(I)$ | $\,\overline{\Lambda\vphantom{\bar{\Lambda}}}\,_{\bm{k}i}(I).f_{\lambda}^{T}.\Lambda_{\bm{k}i}^{T}(I)$
$\bm{f}_{a}$ | $\Lambda_{a}^{T}(g)\cdot\left(\,\overline{\Lambda\vphantom{\bar{\Lambda}}}\,_{\bm{k}i}(g).\bm{f}_{a}.\Lambda_{\bm{k}i}^{T}(g)\right)$ | $\bm{f}_{a}^{\dagger}$ | $\,\overline{\Lambda\vphantom{\bar{\Lambda}}}\,_{\bm{k}i}(I).\,\overline{\bm{f}_{a}\vphantom{\bar{\bm{f}_{a}}}}\,.\Lambda_{\bm{k}i}^{T}(I)$ | $\,\overline{\Lambda\vphantom{\bar{\Lambda}}}\,_{\bm{k}i}(I).\bm{f}_{a}^{T}.\Lambda_{\bm{k}i}^{T}(I)$
$\bm{f}_{k}$ | $R(p)\cdot\left(\,\overline{\Lambda\vphantom{\bar{\Lambda}}}\,_{\bm{k}i}(g).\bm{f}_{k}.\Lambda_{\bm{k}i}^{T}(g)\right)$ | $\bm{f}_{k}^{\dagger}$ | $-\,\overline{\Lambda\vphantom{\bar{\Lambda}}}\,_{\bm{k}i}(I).\,\overline{\bm{f}_{k}\vphantom{\bar{\bm{f}_{k}}}}\,.\Lambda_{\bm{k}i}^{T}(I)$ | $-\,\overline{\Lambda\vphantom{\bar{\Lambda}}}\,_{\bm{k}i}(I).\bm{f}_{k}^{T}.\Lambda_{\bm{k}i}^{T}(I)$
Table 1: Transformation of the different terms in Eq. 3 under different
symmetries of the system. Here, $f$ is understood as a second rank tensor in
the double index $(s\alpha)$, and $(.)$ is the matrix product with respect to
this double index; $\overline{f}$ denotes complex conjugation, $f^{T}$ the
matrix transpose and $f^{\dagger}\,{=}\,(\overline{f})^{T}$ the conjugate
transpose. $(\cdot)$ denotes the matrix product in the $\bm{a}$ and $\bm{k}$
vector space. $\Lambda_{a}(g)$ is the representation corresponding to the
geometrical perturbation, and $R(g)$ the Rodrigues matrix representation for
3D Euclidean vectors. The parity or inversion symmetry $I$ covers the action
of time reversal and reciprocity on the partners if $I$ is an element of the
space group; if not, the action of these two symmetries maps out of the
irreducible representation (irrep) in question and both do not yield any
additional restriction on the form of $f$. Note that $I\,{=}\,C_{2}$ if a 2D
planar group is considered instead of a 3D space group.

We here derive a procedure to obtain the eigenvalues and fields close to a
high symmetry point in the irreducible Brillouin zone (BZ), considering
geometrical perturbations that break the original symmetry. The general idea
is introduced in Saba  _et al._ (2017). We here extend the procedure to space
group symmetries (including translations) to deal with geometrical
perturbations that break translation symmetry. This further requires
simplification such that only group generators (rather than an infinite number
of elements) need to be considered.

We start with a generic Hamiltonian
$H(\bm{a},\lambda):\mathcal{H}\rightarrow\mathcal{H}$ that is a linear
operator on a physical Hilbert space $\mathcal{H}$. An eigenvector
$\bm{v}\,{\in}\,\mathcal{H}$ satisfies $H(\bm{a},\lambda)\,\bm{v}\,{=}\,0$,
with the associated nonlinear eigenvalue $\lambda$ (the frequency for the
monochromatic Maxwell equations) and free (geometrical) system parameters
$a_{i}\,{\in}\,\mathbb{R}$. Let $H(\bm{a}{=}0,\lambda)$ be invariant under the
action of space group symmetries $g\in\mathcal{G}$, _i.e._
$[g,H(0,\lambda)]\,{=}\,0$ for all $g$ and $\lambda$, so that the eigenvectors
are a superposition of partners $\varphi_{\bm{k}i;s\alpha}$ of a single irrep
$\Lambda_{\bm{k}i}$ of the space group $\mathcal{G}$. For details on this
statement and the group theoretical terminology used in the following see
Bradley and Cracknell (2010); Dresselhaus (2013); Saba  _et al._ (2017) and
Appendix A. In summary, $\bm{k}$ labels a star with respect to the invariant
subgroup of translations $\mathcal{T}\,{\subset}\,\mathcal{G}$ and coincides
with a Bloch wave vector within the irreducible BZ, _i.e._ the asymmetric unit
cell of the associated reciprocal space; the index $i$ labels an irrep
$\Delta_{i}(\bm{k})$ of the little group of $\bm{k}$, with partners labeled by
$\alpha$. $\Lambda_{\bm{k}i}\,{=}\,\Delta_{i}(\bm{k})\uparrow\mathcal{G}$ is
thus an induced representation with respect to the left-coset expansion
$\mathcal{G}=\sum_{s}g_{s}\mathcal{T}\;(g_{s}\in\mathcal{G})$, with the
partner index $s$ iterating over the inequivalent representations of
$\mathcal{T}$ within the $\bm{k}$-star. These inequivalent representations are
one-dimensional and have the Bloch character
$\chi_{s}(T)\,{=}\,\exp\\{\imath(p_{s}\,\bm{k})\cdot\bm{T}\\}$
($T{\in}\mathcal{T}$). $p$ is the isogonal point group element corresponding
to $g$, _i.e._ in Seitz notation $g\,{=}\,\\{p,t\\}$, with $t$ a translation
(not necessarily in $\mathcal{T}$ for non-symmorphic space groups).

The goal is to use perturbation theory in order to expand around an evaluation
point $x=(\bm{a}\,{=}\,0,\lambda_{0},\bm{k}_{0})$. However, a perturbation
approach can only be applied if the states close to the evaluation point are
similar to the state at the evaluation point, or mathematically speaking that
the Bures distance $D(\phi,\psi)=\sqrt{1-|\langle\phi|\psi\rangle|^{2}}$ (for
properly normalized states $\langle\phi|\phi\rangle$ with the inner product
$\langle\cdot|\cdot\rangle:\mathcal{H}^{*}\times\mathcal{H}\rightarrow\mathbb{C}$
associated with $\mathcal{H}$) approaches zero:

$D\left(\psi(x),\,\psi(x+dx)\right)\overset{dx\rightarrow
0}{\rightarrow}0\text{ .}$

The orthogonality of the irrep’s partners, on the other hand, implies that the
distance of states (infinitesimally) apart in $\bm{k}$ is $1$, so that the
above requirement is clearly violated. Separating a Bloch phase off the
partners
$\phi_{\bm{k}i;s\alpha}:=u_{\bm{k}i;s\alpha}\,\exp\\{\imath(p_{s}\,\bm{k})\cdot\bm{r}\\}$
solves the dilemma if the new sesquilinear product is defined with $u$ instead
of $\phi$. This, however, generally renders the Hamiltonian
$\bm{k}$-dependent, with the mapped eigenproblem:

$\tilde{H}(\bm{a},\lambda,\bm{k})\,u_{\bm{k}i;s\alpha}\,{=}\,0\text{ .}$ (1)

To be precise, $\tilde{H}$ also depends on the partner index $s$. The
following arguments do, however, not depend on this index, so that we drop it
for the Hamiltonian in the vicinity of the evaluation point to avoid confusion
with $s$ belonging to the irrep $\bm{k}_{0}i$ at the evaluation point.

In zero order (degenerate) perturbation theory, the eigenvector at
$\bm{k}_{0}+\delta\bm{k}$ in the vicinity of the evaluation point can be
expressed as a linear combination
$\sum_{s,\alpha}c_{s\alpha}u_{\bm{k}_{0}i;s\alpha}$. Testing the eigenvalue
equation at $\bm{k}_{0}+\delta\bm{k}$ with the partners
$u_{\bm{k}_{0}i;s\alpha}$ thus yields a low-dimensional algebraic eigenvalue
problem with the coefficients $c_{s\alpha}$ forming the eigenvectors:

$\sum_{s^{\prime}\alpha^{\prime}}\langle
u_{s\alpha}\,|\,\tilde{H}(\delta\bm{k},\lambda_{0}+\delta\lambda,\bm{k}_{0}+\delta\bm{k})\,u_{s^{\prime}\alpha^{\prime}}\rangle
c_{s^{\prime}\alpha^{\prime}}=0\text{ ,}$ (2)

where we have dropped the now redundant irrep index $(\bm{k}_{0}i)$ and just
kept the corresponding partner index $(s\alpha)$. The solution of Eq. 2 yields
the bandstructure in the vicinity of the evaluation point, including the
eigenmodes via $c_{s\alpha}$ and thus the topological charge of the evaluation
point. In order to solve this equation without any explicit knowledge of the
$u_{s\alpha}$, we first Taylor expand the matrix element

$\displaystyle
f_{s\alpha}^{(s^{\prime}\alpha^{\prime})}(\bm{a},\lambda_{0}+\delta\lambda,\bm{k}_{0}+\delta\bm{k})$
$\displaystyle:=\langle
u_{s\alpha}\,|\,\tilde{H}(\delta\bm{k},\lambda_{0}+\delta\lambda,\bm{k}_{0}+\delta\bm{k})\,u_{s^{\prime}\alpha^{\prime}}\rangle$

in first order:

$\displaystyle f(\bm{a},\lambda_{0}+\delta\lambda,\bm{k}_{0}+\delta\bm{k})$
$\displaystyle=f_{0}+\bm{f}_{a}\cdot\bm{a}+\bm{f}_{k}\cdot\delta\bm{k}+f_{\lambda}\,\delta\lambda$
$\displaystyle=\left[\nabla_{a}f(\bm{a},\lambda_{0},\bm{k}_{0})\right]_{\bm{a}=0}\cdot\bm{a}$
$\displaystyle\,+\left[\nabla_{k}f(0,\lambda_{0},\bm{k})\right]_{\bm{k}=\bm{k}_{0}}\cdot\delta\bm{k}$
$\displaystyle\,+\left[\partial_{\lambda}f(0,\lambda,\bm{k}_{0})\right]_{\lambda=\lambda_{0}}\delta\lambda\text{
.}$ (3)

In the last line we have used the fact that $f_{0}\,{=}\,0$ by definition.

We now employ our knowledge on how the different terms in Eq. 3 transform
under symmetry operations to derive _selection rules_ on the matrix elements.
Consider a symmetry operation $\hat{O}$ that leaves the system Hamiltonian
invariant, _i.e._ $[\hat{O},H(\bm{a}\,{=}\,0,\lambda_{0})]\,{=}\,0$. This
implies that the matrix elements $f$ are also invariant under the action of
$\hat{O}$. $f$ thus must be in the kernel

$\left[\hat{O}-{\mathbbm{1}}\right]f=0\text{ .}$

Note that the symmetry operations act on $\delta\bm{k}$, but they leave
$\bm{k}_{0}$ invariant by virtue of its definition as a label of the space
group irreps (the action is instead implicitly included in the induced irreps
themselves). For convenience, we wrap the action on the vectorial nature of
the permutation parameters $\bm{a}$ and $\delta\bm{k}$ onto the matrix element
in the scalar product: For example, we use the identity
$\bm{f}_{k}\cdot(R^{T}(p)\delta\bm{k})\,{=}\,(R(p)\bm{f}_{k})\cdot\delta\bm{k}$,
so that the permutation parameter stays effectively invariant. This yields the
independent kernel equations that need to be satisfied for all system
symmetries $\hat{O}$:

$\displaystyle\left[\hat{O}-{\mathbbm{1}}\right]\bm{f}_{a}$ $\displaystyle=0$
$\displaystyle\left[\hat{O}-{\mathbbm{1}}\right]\bm{f}_{k}$ $\displaystyle=0$
$\displaystyle\left[\hat{O}-{\mathbbm{1}}\right]f_{\lambda}$
$\displaystyle=0\text{ .}$ (4)

Note that $f\,{\in}\,\text{GL}_{n}(\mathbb{C})$, so that
$\bm{f}_{k}\,{\in}\,\mathbb{R}^{d}\,{\otimes}\,\text{GL}_{n}(\mathbb{C})$,
where $d$ is the spatial dimension of the lattice (here $d\,{=}\,2$) and $n$
is the dimension of the space group irrep at the evaluation point. The system
symmetry $\hat{O}$ thus acts on both vector spaces in the tensor product
independently. The transformations under spatial and non-spatial symmetries
are listed in Tab. 1. The non-spatial symmetries all square to the identity
for spin-less systems. They are not independent and form a _trinity_ :
Application of any pair yields the remaining operation, for example
$\hat{O}_{\text{TR}}\hat{O}_{\text{Rec}}\,{=}\,\hat{O}_{\text{Herm}}$. For the
unitary spatial symmetries $\hat{g}$, we show in Elman (2017) that it suffices
for the permutation elements to satisfy

$\displaystyle\sum_{g\in\mathcal{S}}\left[\hat{g}-{\mathbbm{1}}\right]\bm{f}_{a}$
$\displaystyle=0$
$\displaystyle\sum_{g\in\mathcal{S}}\left[\hat{g}-{\mathbbm{1}}\right]\bm{f}_{k}$
$\displaystyle=0$
$\displaystyle\sum_{g\in\mathcal{S}}\left[\hat{g}-{\mathbbm{1}}\right]f_{\lambda}$
$\displaystyle=0\text{ ,}$ (5)

where $\mathcal{S}$ denotes an arbitrary set of generators of the space group
$\mathcal{G}$. With this simplification, only a single kernel equation has to
be satisfied for spatial symmetries per perturbation matrix element (instead
of an infinite number of equations Eq. 4). Application of Eq. 5 for spatial
symmetries and subsequently Eq. 4 for non-spatial symmetries hence reduces the
degrees of freedom in Eq. 3 substantially and establishes its allowed form.
After this step, we can solve Eq. 2 that takes the compact form:

$f(\bm{a},\delta\lambda,\delta\bm{k})\,\bm{c}=0\text{ .}$ (6)

##  II 2D hexagonal lattices with $C_{6v}$ symmetry

The most systematic route towards topologically protected surface states in
classical systems with lattice symmetry is breaking of symmetry induced
(deterministic) degeneracies that carry a topological charge in (an extended)
reciprocal space. A topologically non-trivial band gap can thus be opened.
This idea has already been used in the original work by Haldane and Raghu
where simultaneous breaking of reciprocity, time reversal symmetry and spatial
symmetry has been suggested using a Faraday constituent material in a magnetic
field Raghu and Haldane (2008). It has been exploited ever since in various
systems Lu  _et al._ (2014).

In this work, we focus on systems with no broken reciprocity symmetry. There
are two possible starting points for 2D wallpaper groups: First, accidental
point-degeneracies with conical dispersion in square lattices with broken
$C_{4v}$ symmetry, arising through splitting of flat 2-fold degeneracies at
the $\Gamma$ and $X$ point in the original symmetry, are an intriguing
starting point Makwana  _et al._ (2019). They are, however, more cumbersome to
design, since the starting point is not a deterministic degeneracy (albeit
stemming from one). We here instead focus on deterministic point-degeneracies
with conical dispersion. These only exist at the $K$ ($K^{\prime}$) point of
hexagonal lattices, since all other deterministic degeneracies in 2D lattices
are located at high symmetry points that are mapped onto themselves,
immediately leading to flat bands if either reciprocity or time reversal
symmetry are present.

Representation | $C_{6}$ | $\sigma$ | $T$
---|---|---|---
$\Lambda_{\Gamma 1}$ | $1$ | $1$ | $1$
$\Lambda_{\Gamma 2}$ | $1$ | $-1$ | $1$
$\Lambda_{\Gamma 3}$ | $-1$ | $1$ | $1$
$\Lambda_{\Gamma 4}$ | $-1$ | $-1$ | $1$
$\Lambda_{\Gamma 5}$ | $R_{6}^{2}$ | $\sigma_{3}$ | ${\mathbbm{1}}_{2}$
$\Lambda_{\Gamma 6}$ | $R_{6}$ | $\sigma_{3}$ | ${\mathbbm{1}}_{2}$
$\Lambda_{K1}$ | $\sigma_{1}$ | ${\mathbbm{1}}_{2}$ | $\text{diag}\left(w^{2},w\right)$
$\Lambda_{K2}$ | $\sigma_{1}$ | $-{\mathbbm{1}}_{2}$ | $\text{diag}\left(w^{2},w\right)$
$\Lambda_{K3}$ | $\sigma_{1}\otimes R_{6}^{4}$ | ${\mathbbm{1}}_{2}\otimes\sigma_{3}$ | $\text{diag}\left(w^{2},w\right)\otimes{\mathbbm{1}}_{2}$
$\Lambda_{M1}$ | $P_{i+1}$ | $\text{diag}\left(\sigma_{1},1\right)$ | $\text{diag}\left(-{\mathbbm{1}}_{2},1\right)$
$\Lambda_{M2}$ | $P_{i+1}$ | $-\text{diag}\left(\sigma_{1},1\right)$ | $\text{diag}\left(-{\mathbbm{1}}_{2},1\right)$
$\Lambda_{M3}$ | $-P_{i+1}$ | $\text{diag}\left(\sigma_{1},1\right)$ | $\text{diag}\left(-{\mathbbm{1}}_{2},1\right)$
$\Lambda_{M4}$ | $-P_{i+1}$ | $-\text{diag}\left(\sigma_{1},1\right)$ | $\text{diag}\left(-{\mathbbm{1}}_{2},1\right)$
Table 2: The irreducible space group representations in $p6mm$ that correspond
to high symmetry points in the irreducible BZ. Listed are the representation
matrices for the three group generators $C_{6}$, $\sigma$, and $T$ defined in
the main text. For compactness of notation, $\sigma_{i}$ denotes the
respective Pauli matrix,
$R_{6}\,{:=}\,1/2\left({\mathbbm{1}}_{2}-\sqrt{3}\imath\sigma_{y}\right)$ the
planar Rodrigues $60$ degrees rotation matrix,
$w\,{:=}\,\exp\\{2\pi\imath/3\\}$ the Bloch phase $p_{K^{\prime}}(T)$,
$P_{i+1}$ the 3D cyclic permutation matrix, ${\mathbbm{1}}_{n}$ the
$n$-dimensional identity matrix, and $\text{diag}(\\{e_{i}\\})$ the block-
diagonal matrix with the tuple of diagonal entries as argument.

We start with degeneracies at the $K$ point (note that the $K^{\prime}$ point
is in the same star and hence the same irrep of the wallpaper group) in a
hexagonal wallpaper group $p6mm$ (17), which encompasses all other hexagonal
wallpaper groups of lower symmetry. We will enumerate all (geometrical)
perturbations $\bm{a}$ that break the $p6mm$ symmetry, and derive
corresponding matrix elements $f$ (Eq. 3) that yield the dispersion and the
bulk states in the vicinity of the $K$ point. To this end, we apply the
formalism derived in Sec. I to establish the allowed form of the perturbation
matrix $f(\bm{a},\delta\lambda,\delta\bm{k})$ in the vicinity of the
evaluation point $(\bm{a},\lambda,\bm{k})\,{=}\,(0,\lambda_{0},\bm{k}_{0})$.

Tab. 2 shows the irreps of $p6mm$ at the corners of the irreducible BZ. We
here adopt the labelling of the little group irrep underlying the induced
space group irreps from Bradley and Cracknell (2010), _i.e._ in $\Gamma 1$
correspond to $R_{1}$ of the little group $G^{3}_{12}$ in table 5.1 in Bradley
and Cracknell (2010) _etc_. At the $K$ point, the physical eigenfunctions that
transforms according to $\Lambda_{K3}$ correspond to a 2-fold degeneracy in
the bandstructure. As generators of the space group we choose the $C_{6}$
rotation, the mirror symmetry $\sigma$ that maps $y\mapsto-y$ and leaves $x$
invariant, and the translation $T$ along $x$, _i.e._ we choose the coordinate
system shown in Fig. 1. As coset representatives $r_{\alpha}$ in the expansion
$\mathcal{G}=\sum_{\alpha}r_{\alpha}\mathcal{G}_{K}$ (where $\mathcal{G}_{K}$
is the associated little group, _cf._ also App. A), we choose the identity $E$
and the $C_{2}$ rotation.

For the sake of completeness, we first apply Eq. 5 to $\Lambda_{\Gamma 5/6}$
which evidently describe 2-fold degeneracies at $\Gamma$. This procedure
results in an empty nullspace and hence a vanishing $f_{k}$ matrix element in
Eq. 3. In other words, the degeneracy does not split to first order in
$\bm{k}$. This finding is a consequence of spatial symmetry alone, but we note
that reciprocity does generally not allow linear splitting of a 2D degeneracy
along all directions at $\Gamma$ (or any other point in the BZ that maps to
itself under reciprocity). This can be seen from the fact that the first order
$\bm{k}$-dependence of the 2D matrix $f$ can always be expressed as
$f(\delta\bm{k})=\sum_{i=0}^{3}\sigma_{i}\,P_{i}(\delta\bm{k})$, with
$\sigma_{0}={\mathbbm{1}}$ and the respective Pauli matrix for $i>0$, and
$P_{i}$ a first order polynomial with vanishing constant term in the
components of $\delta\bm{k}$, or $P_{i}=\bm{c}\cdot\delta\bm{k}$. Only
$\sigma_{2}$ is thus commensurable with reciprocity (_cf._ Tab. 1). The
spectrum is hence given by
$\delta\lambda\propto\text{eigs}\\{\sigma_{2}P_{2}\\}=\pm P_{2}$, _i.e._ it
does not split at least along the line $P_{2}(\delta\bm{k})=0$ in $\bm{k}$
space.

Our main focus of interest is thus $\Lambda_{K3}$, for which application of
Eq. 5 yields $f_{k}\propto\gamma_{1}\delta k_{x}-\gamma_{2}\delta k_{y}$,
where we define a representation of the 4D Euclidean Clifford algebra as

$\displaystyle\gamma_{1}$ $\displaystyle:=\sigma_{3}\otimes\sigma_{3}$
$\displaystyle\gamma_{2}$ $\displaystyle:=\sigma_{3}\otimes\sigma_{1}$
$\displaystyle\gamma_{3}$ $\displaystyle:=\sigma_{1}\otimes{\mathbbm{1}}_{2}$
$\displaystyle\gamma_{4}$ $\displaystyle:=\sigma_{2}\otimes{\mathbbm{1}}_{2}$
$\displaystyle\gamma_{5}$ $\displaystyle:=\sigma_{3}\otimes\sigma_{2}\text{
.}$ (7)

These matrices evidently satisfy the anticommutator relation
$\\{\gamma_{i},\gamma_{j}\\}\,{=}\,2\delta_{ij}$. Note that $f_{k}$ is
invariant under non-spatial symmetries in Tab. 1, including time reversal
symmetry and reciprocity, if the proportionality constant is chosen to be
real. To see that we note that
$\Lambda_{K3}(C_{2})\,{=}\,\Lambda_{K3}^{3}(C_{6})\,{=}\,\gamma_{3}$ according
to Tab. 2. Therefore, application of Tab. 1 and the anticommutator relation
for the Clifford matrices yields
$\hat{O}_{\text{TR}}\gamma_{1/2}\,{=}\,\hat{O}_{\text{Rec}}\gamma_{1/2}\,{=}\,{-}\gamma_{3}.\gamma_{1/2}.\gamma_{3}\,{=}\,\gamma_{1/2}$.

Eq. 5 further trivially yields $f_{\lambda}\,{\propto}\,{\mathbbm{1}}$, while
$f_{a}$ evidently depends on the type of geometrical perturbation. We first
discuss the two cases of interest that lead to the QSHE and QVHE effect,
respectively, and then show that those are indeed the only non-trivial cases
among all geometrical perturbations without long-range chirp Hielscher  _et
al._ (2017), _i.e._ those that correspond to the high symmetry points in the
irreducible BZ listed in Tab. 2 with $\Lambda^{n}(T)\,{=}\,{\mathbbm{1}}$ for
some finite integer $n$.

Figure 2: The perturbation arrows of the six closest lattice sites from the
origin for the three different partners A,B,C belonging to the QSHE
perturbation. The honeycomb hexagon is shown as a guide to the eye. The whole
lattice is spanned by application of the reduced translation symmetry (_cf._
Fig. 1).

We start with the perturbation introduced in Wu and Hu (2015) and illustrated
in Fig. 1 (a). The representation $\Lambda_{a}$ corresponding to this
perturbation is most conveniently chosen to be three-dimensional, with
partners illustrated in Fig. 2. By inspection, the representation matrices for
the group generators are then obtained as

$\displaystyle\Lambda_{\text{QSHE}}(C_{6})$
$\displaystyle=\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}\text{,}$
$\displaystyle\Lambda_{\text{QSHE}}(\sigma)$
$\displaystyle={\mathbbm{1}}\text{, and}$
$\displaystyle\Lambda_{\text{QSHE}}(T)$ $\displaystyle=\begin{pmatrix}0&1&0\\\
0&0&1\\\ 1&0&0\end{pmatrix}\text{.}$

Application of Eq. 5 for the spatial symmetries leads to

$\displaystyle f_{a}$
$\displaystyle=d_{1}{\mathbbm{1}}_{4}\left(a_{A}+a_{B}+a_{C}\right)$
$\displaystyle+d_{2}\Big{[}\begin{pmatrix}0_{2x2}&{\mathbbm{1}}_{2}\\\
{\mathbbm{1}}_{2}&0_{2x2}\end{pmatrix}a_{A}$
$\displaystyle+\begin{pmatrix}0_{2x2}&w{\mathbbm{1}}_{2}\\\
w^{2}{\mathbbm{1}}_{2}&0_{2x2}\end{pmatrix}a_{B}$
$\displaystyle+\begin{pmatrix}0_{2x2}&w^{2}{\mathbbm{1}}_{2}\\\
w{\mathbbm{1}}_{2}&0_{2x2}\end{pmatrix}a_{C}\Big{]}\text{ ,}$

with $w\,{:=}\,\exp\\{2\pi\imath/3\\}$, ${\mathbbm{1}}_{n}$ the
$n{\times}n$-dimensional identity matrix, and $0_{n{\times}m}$ the
$n{\times}m$-dimensional zero matrix. This result is commensurable with the
non-spatial symmetries if $d_{1},d_{2}\,{\in}\,\mathbb{R}$ by application of
Tab. 1 and the fact that $\bm{a}\,{\in}\,\mathbb{R}^{3}$ by definition. We
disregard the trivial term that shifts all bands simultaneously up or down
without altering eigenvectors in the following and set $d_{1}=0$.
Additionally, $a_{B}$ and $a_{C}$ have only been introduced to obtain a full
set of partners under the symmetry operations of the unperturbed lattice,
while the deterministic perturbation in Wu and Hu (2015) only uses
$a_{A}\,{=:}\,a$. We will thus in the following further set
$a_{B}\,{=}\,a_{C}\,{=}\,0$, finally leading to the simplified form

$f_{\text{QSHE}}\left(a,\lambda,\delta\bm{k}\right)=\gamma_{1}\delta
k_{x}-\gamma_{2}\delta k_{y}+\gamma_{3}a-{\mathbbm{1}}\lambda\text{ ,}$ (8)

with the perturbation parameters rescaled such that the respective
coefficients are absorbed and Eq. 8 is in the most convenient form. A
different choice would not alter any of the findings below, and only lead to a
different global slope of the bands and a different sign of the topological
invariants which globally depends on the particular physical realisation and
the choice of origin and direction of the arrows in Fig. 2. We will
specifically discuss the dependence on the choice of origin below which
clearly distinguishes the topological protection found here from strong
protection in the integer quantum Hall sense.

Irrep | spatial only | including non-spatial | gap
---|---|---|---
$\Lambda_{\Gamma 1}$ | ${\mathbbm{1}}_{4}$ | ${\mathbbm{1}}_{4}$ | no
$\Lambda_{\Gamma 2}$ | ${\mathbbm{1}}_{2}\otimes\sigma_{2}$ | $-$ | (yes)
$\Lambda_{\Gamma 3}$ | $\sigma_{3}\otimes{\mathbbm{1}}_{2}$ | $-$ | no
$\Lambda_{\Gamma 4}$ | $\gamma_{5}$ | $\gamma_{5}$ | yes
$\Lambda_{\Gamma 5}$ | $({\mathbbm{1}}_{2}\otimes\sigma_{3},{\mathbbm{1}}_{2}\otimes\sigma_{1})$ | $({\mathbbm{1}}_{2}\otimes\sigma_{3},{\mathbbm{1}}_{2}\otimes\sigma_{1})$ | no
$\Lambda_{\Gamma 6}$ | $(\gamma_{1},\gamma_{2})$ | $-$ | no
$\Lambda_{K1}$ | $\gamma_{3}\mp\imath\gamma_{4}$ | $\gamma_{3}$ | yes
$\Lambda_{K2}$ | $(\sigma_{1}\mp\imath\sigma_{2})\otimes\sigma_{2}$ | $-$ | no
$\Lambda_{K3}$ | $\big{(}(\sigma_{1}-\imath\sigma_{2})\otimes\sigma_{3},$ | $\big{(}\sigma_{1}\otimes\sigma_{3},$ |
| $\phantom{\big{(}}(\sigma_{1}-\imath\sigma_{2})\otimes\sigma_{1},$ | $\phantom{\big{(}}\sigma_{1}\otimes\sigma_{1},$ | no
| $\phantom{\big{(}}(\sigma_{1}+\imath\sigma_{2})\otimes\sigma_{3},$ | $\phantom{\big{(}}\sigma_{2}\otimes\sigma_{3},$ |
| $\phantom{\big{(}}(\sigma_{1}+\imath\sigma_{2})\otimes\sigma_{1}\big{)}$ | $\phantom{\big{(}}\sigma_{2}\otimes\sigma_{1}\big{)}$ |
$\Lambda_{M1}$ | $-$ | $-$ | $-$
$\Lambda_{M2}$ | $-$ | $-$ | $-$
$\Lambda_{M3}$ | $-$ | $-$ | $-$
$\Lambda_{M4}$ | $-$ | $-$ | $-$
Table 3: List of the terms $f_{a}$ corresponding to geometrical perturbations
that transform according to irreducible planar group representations in Tab.
2. The first column lists the tuple of matrices corresponding to the partners
defined through the representation matrices in Tab. 2 for spatial symmetries
only. If non-spatial symmetries (time reversal and reciprocity) are included,
only a specific linear combination of partners might be admissible, thus
leading to a reduced tuple size (for example for $\Lambda_{K1}$, where only a
perturbation along $a_{1}\,{+}\,a_{2}$ leads to non-zero
$f_{a}\,{\propto}\,\gamma_{3}$). The coefficients of the tuples including non-
spatial symmetries further are assumed to be real numbers, and “$-$” indicates
that no perturbation matrix is found to satisfy all symmetries. The last
column lists the possibility of a topological bandgap under the particular
perturbation.

Substitution of Eq. 8 into Eq. 6 yields the $4$-dimensional linear eigenvalue
problem

$W_{\text{QSHE}}\,\bm{c}=\left(\gamma_{1}\delta k_{x}-\gamma_{2}\delta
k_{y}+\gamma_{3}a\right)\bm{c}=\lambda\,\bm{c}\text{ .}$ (9)

We will diagonalize the _perturbation Hamiltonian_ $W$ in the following
section. We here proceed with the QVHE case. Consider the perturbed kagome
lattice Mekata (2003) illustrated in Fig. 1 (b). The representation matrices
belonging to the perturbation are evidently one-dimensional and given by

$\displaystyle\Lambda_{\text{QVHE}}(C_{6})$ $\displaystyle=-1\text{ ,}$
$\displaystyle\Lambda_{\text{QVHE}}(\sigma)$ $\displaystyle=-1\text{ , and}$
$\displaystyle\Lambda_{\text{QVHE}}(T)$ $\displaystyle=1\text{ .}$

Following the same procedure as above, the corresponding perturbation
eigenproblem is shown to be

$W_{\text{QVHE}}\,\bm{c}=\left(\gamma_{1}\delta k_{x}-\gamma_{2}\delta
k_{y}+\gamma_{5}a\right)\bm{c}=\lambda\,\bm{c}\text{ .}$ (10)

Let us now revisit the nature of the two particular example systems that lead
to the QSHE and QVHE perturbation Hamiltonians Eq. 9 and Eq. 10 above. The
question we are asking here in particular is whether there are other possible
perturbations leading to a different Hamiltonian, respectively. Any arbitrary
perturbation can be understood as belonging to some vector space that is
spanned by a choice of basis vectors. These can be understood as partners of a
generally reducible representation of $\mathcal{G}$, and thus can be
decomposed into irreps. The completeness of irreps Dresselhaus (2013); Saba
_et al._ (2013) therefore guarantees that any perturbation transforms
according to a direct sum of irreps of $\mathcal{G}$, which reduces to the
irreps listed in Tab. 2 since we exclude long-range chirps that would lead a
system with no translational symmetry.

For example, we can immediately identify
$\Lambda_{\text{QVHE}}=\Lambda_{\Gamma 4}$. For $\Lambda_{\text{QSHE}}$, we
introduce the similarity transform $S^{-1}.\Lambda_{\text{QSHE}}(g).S$, with

$S:=\begin{pmatrix}1&1&1\\\ 1&w&w^{2}\\\ 1&w^{2}&w\end{pmatrix}\text{ .}$

This simultaneously brings the generator matrices (and thus all representation
matrices) into an irreducible block form. Comparison of the blocks with Tab. 2
immediately reveals the direct sum partition
$\Lambda_{\text{QSHE}}=\Lambda_{\Gamma 1}\oplus\Lambda_{K1}$. We can now
interpret our previous result for the QSHE $f_{a}$ in the following way: The
trivial identity matrix part originates from the trivial planar group
representation $\Lambda_{\Gamma 1}$ (the same that lead to
$f_{\lambda}\,{\propto}\,{\mathbbm{1}}$), while the non-trivial part stems
from $\Lambda_{K1}$.

Using the above conclusion that any geometrical perturbation of interest is a
direct sum of the planar group representations listed in Tab. 2, we can now
build the most general perturbation matrix by considering these irreps
separately. Note that the reverse is not necessarily true, _i.e._ not all
direct sums of the irreducible representations in Tab. 2 can be interpreted as
a geometrical perturbation, as the latter requires
$\Lambda_{a}(g)\in\text{GL}_{n}(\mathbb{R})$. For example, the $\Lambda_{Ki}$
on their own cannot form a geometrical perturbation as the generator matrix
$\Lambda_{Ki}(T)\,{\notin}\,\text{GL}_{n}(\mathbb{R})$ (and there is no
similarity transform that makes all generator matrices real).

##  III Diagonalisation of perturbation Hamiltonians

Consider the general perturbation Hamiltonian

$W=\gamma_{1}\delta k_{x}-\gamma_{2}\delta k_{y}+f_{a}a$ (11)

as derived in section II. Generally, the eigenvalues $\lambda_{\alpha}$
corresponding to $W$ are solutions of the characteristic equation
$P(\lambda,\delta k_{x},\delta k_{y},a)=0$, where $P$ is a quartic polynomial.
Since the main goal is to obtain frequency isolated topological edge states, a
necessary requirement on $f_{a}$ is that it produces a gapped dispersion
relation, for which $\lambda_{\alpha}(\delta k_{x},\delta k_{y},a)\,{\neq}\,0$
for all $\alpha$ and $\delta\bm{k}$ if $a\,{\neq}\,0$.

Figure 3: Geometrical perturbation bandstructures in the vicinity of the $K$
space group irrep. (a)-(f) perturbations that are trivial under translations,
(g)-(i) perturbations that act similar to $K$ point irreps as involved in the
QSHE perturbation Hamiltonian in Eq. 9 (_cf._ also Tab. 3). The bandstructures
shown are for the cases without time-reversal symmetry and reciprocity
enforced, and arbitrary perturbation. A band gap for finite perturbation only
opens up in case (b), (d), and (g), and only for the latter two cases in
systems with time-reversal symmetry.

Generally, any $4{\times}4$ matrix $M$ over the complex numbers can be
represented by $M\,{=}\,\sum_{n,m=0}^{3}\gamma_{(nm)}\,c_{(nm)}$, with
$\gamma_{(nm)}\,{:=}\,\sigma_{n}\otimes\sigma_{m}$,
$\sigma_{0}\,{:=}\,{\mathbbm{1}}_{2}$, and $c_{(nm)}\in\mathbb{C}$. This can
be easily seen by considering $M_{(\alpha\beta)}$ and $c_{(nm)}$ as
$16$-dimsensional vectors, respectively. The $16\times 16$ matrix
$\gamma_{(\alpha\beta),(nm)}$ above has evidently full rank (the four $\sigma$
matrices are linearly independent) so that the $c_{(nm)}$ can be explicitly
constructed by inversion of $\gamma$. However, the derived perturbation
Hamiltonian matrices summarized in Tab. 3 only contain a single of those
$\gamma$ matrices per perturbation partner. This substantially reduces the
complexity of the eigenproblem. In the following, we use that two $\gamma$
matrices either commute or anti-commute, and the fact that if
$\lambda\in\text{eig}(W)$ we also have $\lambda^{2}\in\text{eig}(W^{2})$. Let
us now study the Hamiltonian $W\,{=}\,\gamma x+\gamma^{\prime}y$, and discuss
the two cases I: $[\gamma,\gamma^{\prime}]\,{=}\,0$ and II:
$\\{\gamma,\gamma^{\prime}\\}\,{=}\,0$. Case I:

$\displaystyle W^{2}$
$\displaystyle=\left(x^{2}+y^{2}\right){\mathbbm{1}}+2xy\gamma\gamma^{\prime}$
$\displaystyle\Rightarrow\lambda^{2}$ $\displaystyle=x^{2}+y^{2}+2\chi
xy\text{ ,}$

where $\chi\in\text{eig}(\gamma\gamma^{\prime})$ is a fourth root of $1$. If
$W$ is hermitian, we immediately see that $\chi=\pm 1$, so that

$\lambda=\pm\sqrt{x^{2}+y^{2}\pm 2xy}=\pm|x\pm y|\text{ .}$

Since $W$ is also chirally symmetric with respect to some
$\gamma^{\prime\prime}$ that anti-commutes with $\gamma$ and $\gamma^{\prime}$
(we can always find $5$ matrices satisfying a Clifford algebra), _i.e._
$\gamma^{\prime\prime}\gamma\gamma^{\prime\prime}=-\gamma$ and equivalent for
$\gamma^{\prime}$, we further know that the eigenvalues come in pairs
$\text{eig}(W)\,{=}\,\\{\pm\lambda_{1},\pm\lambda_{2}\\}$. In other words, the
eigenvalues above split symmetrically into two positive and two negative
branches that meet along the lines $x\,{=}\,\pm y$ in the 2D parameter space,
respectively, with eigenvalue $\lambda_{1/2}=0$. We thus immediately see that
a combination of commuting $\gamma$ matrices does not allow for the desired
complete bandgap in the vicinity of the evaluation point. Case II yields the
much simpler result

$\displaystyle W^{2}$ $\displaystyle=\left(x^{2}+y^{2}\right){\mathbbm{1}}$
$\displaystyle\Rightarrow\lambda^{2}$ $\displaystyle=x^{2}+y^{2}\text{ ,}$

which corresponds to a doubly-degenerate and conical dispersion in the
vicinity of the evaluation point. To avoid confusion, we remind the reader
that we are working on the basis of space group representations with index
$\bm{k}$, which is closely related to, but not identical to the Bloch wave
vector in the first BZ. Thus, if a doubly-degenerate hyperconic dispersion is
described here without breaking translation symmetry (as in $\Lambda_{\Gamma
n}$), this translates to two single hypercones at $K$ and $K^{\prime}$ in the
standard Bloch function picture, respectively.

The same analysis can be performed for a three or four term Hamiltonian, as
appearing in Eq. 11, leading to typical bandstructures for finite fixed
perturbations as shown in Fig. 3. Noticeably, only in three cases (only two
with time-reversal symmetry) a bandgap opens for all $\bm{k}$ close to the
evaluation point: Both of the time-reversal/reciprocity symmetric scenarios
correspond to the previously discussed QVHE ($\Gamma 4$) and QSHE ($K1$)
cases, while the third case ($\Gamma 2$) is the well-known reciprocity-
breaking scenario Haldane and Raghu (2008). We have thus rigorously shown that
there are only the two well-known systems that have a chance to lead to
frequency isolated topological edge modes in hexagonal lattices (as long as
long-range compressions of the whole structure that destroy periodicity are
excluded). While this might not come as a surprise given the amount of
published work on the topic, our analysis clearly shows that looking for
alternative routes based on geometrically perturbed hexagonal symmetries is an
endeavour doomed to failure.

##  IV Ostensible $\mathbb{Z}_{2}$ topological charge in QSHE systems

In this section, we study the QSHE Hamiltonian in-depth. In this case, the
original translation symmetry of the unperturbed $p6mm$ lattice is broken. The
interpretation of the partners is thus that these correspond to the trivial
representation with respect to the translation group of the perturbed
structure, _i.e._ $\delta\bm{k}$ is with respect to the $\Gamma$ point for all
partners in the standard bandstructure picture of the perturbed symmetries BZ.
The bulk dispersion for finite perturbation strength $a$ is reminiscent of
that in Fig. 3 (g). In order to understand the occurrence of surface bands and
the associated topology, we need to examine the eigenvectors $\bm{c}$ of
$W_{\text{QSHE}}$ that solve $Eq.~{}\ref{eq:QSHE_eigen}$.

In the following, we make use of the fact that $W_{\text{QSHE}}$ evidently
commutes with $S\,{=}\,\sigma_{1}\otimes\sigma_{2}$ for any $\left(\delta
k_{x},\delta k_{y},a\right)$, and that the eigenstates can now be obtained
from the eigenfunctions of $S$. In Fermionic systems $S$ has the meaning of
time reversal invariance, corresponding to an anti-unitary operator with
$S^{2}\,{=}\,{-}1$ and associated Kramers pairs. We here have a simple linear
operator that plays a similar role that is, however, understood entirely
within the framework of linear maps. The spin channels are spanned by the two
separate eigenvalues of geometric multiplicity of two of $S$, respectively. In
the following, we need to bear in mind, however, that $S$ only commutes with
the first order perturbation Hamiltonian, and the associated Spin Chern number
is thus no genuine topological invariant of the system.

The eigensolutions of $S$ are (where $n$ accounts for geometric multiplicity
and we denote the two induced orthogonal channels by $\uparrow$, $\downarrow$
respectively):

$[\lambda_{S};\bm{v}_{n}]=\begin{cases}\left[-1;\,\frac{1}{\sqrt{2}}\begin{pmatrix}\imath\\\
0\\\ 0\\\ 1\end{pmatrix},\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ -\imath\\\ 1\\\
0\end{pmatrix}\right],&\uparrow\\\
\left[\phantom{-}1;\,\frac{1}{\sqrt{2}}\begin{pmatrix}-\imath\\\ 0\\\ 0\\\
1\end{pmatrix},\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ \imath\\\ 1\\\
0\end{pmatrix}\right],&\downarrow\end{cases}\text{ .}$

The two different eigenvalues thus indeed induce two orthogonal 2D subspaces.
Due to $[S,W_{\text{QSHE}}]\,{=}\,0$, there hence exists a linear combination
of the two states in each channel that solves the original eigenproblem Eq. 9,
_i.e._ the associated kernel of the overdetermined problem

$\displaystyle\text{ker}\left[(W_{\text{QSHE}}-\lambda_{\pm}{\mathbbm{1}}_{4}).(\bm{v}_{1},\bm{v}_{2})\right]=\begin{cases}\begin{pmatrix}\lambda_{\pm}+\delta
k_{x}\\\ \delta k_{y}+\imath a\end{pmatrix},&\uparrow\\\
\begin{pmatrix}\lambda_{\pm}+\delta k_{x}\\\ \delta k_{y}-\imath
a\end{pmatrix},&\downarrow\end{cases}$

is not empty. In the above, $\lambda_{\pm}\,{=}\,\pm\sqrt{\delta
k_{x}^{2}+\delta k_{y}^{2}+a^{2}}$ are the established eigenvalues of the
original eigenproblem. Comparing with Eq. 6, the calculation of the Berry
connection or curvature of the physical Hilbert space collapses to the 4D
space spanned by the irreducible partners Saba  _et al._ (2017), and here to
the 2-dimensional vector space induced by $S$ since the spin basis vectors are
$\bm{k}$-independent and ortho-normalized. The 2D vectors above are, on the
other hand, the (non-normalized) eigenvectors of a Weyl Hamiltonian of
opposite chirality and Chern integer $C_{\uparrow\pm}\,{=}\,\mp 1$ and
$C_{\downarrow\pm}=\pm 1$.

A $\mathbb{Z}_{2}$ Kane Mele bulk-boundary correspondence would thus be
guaranteed by the fact that the Berry curvature vector is symmetric with
respect to the mirror operation $a\,{\mapsto}\,{-}a$, so that the spin Chern
number is $C\,{=}\,C_{\uparrow\pm}-C_{\downarrow\pm}\,{=}\,{\mp}1$ when
integrated over the finite perturbation plane $a=\text{const.}>0$.111We should
of course integrate over the finite hexagonal BZ, but the argument does not
change by this slight complication since the total Berry curvature of all Weyl
points through the $a=a_{0}$ plane in an extended zone scheme is $\pm N/2$ for
$N$ BZs, so that the integrated Berry curvature over one BZ evaluates to $\pm
1/2$ after taking the limit $N\rightarrow\infty$ and using translation
symmetry. As stated above, however, the whole argument is based on the vector-
space spanned by the irreps of the evaluation point and thus valid in zero
order perturbation theory only. The problem can be easily seen when
considering the bandstructure of a mixed QSHE-QVHE Hamiltonian
$W\,{=}\,\gamma_{1}k_{x}-\gamma_{2}k_{y}+\gamma_{3}a+\gamma_{5}b$ within the
2D parameter space $(a,b)\in\mathbb{R}^{2}$. It is well known that the (spin)
Chern number’s phase boundaries coincide with a closing of the associated band
gap Raghu and Haldane (2008). If the spin Chern number is a proper topological
invariant of the bulk, we can thus assign a phase space function
$C(a,b){:}\,\mathbb{R}^{2}\,{\rightarrow}\,\\{-1,1\\}$. The bandgap, however,
only closes at $(a,b)\,{=}\,0$ because all $4$ terms in $W$ are part of the
Clifford algebra Eq. 7. Hence, $C(a,b)$ is constant in
$\mathbb{R}^{2}\setminus 0$. But we have computed $C\,{=}\,{\pm}1$ for
$a\,{\lessgtr}\,0$ on the $b\,{=}\,0$ axis above. The paradox is trivially
resolved by realizing that the spin Chern number $C$ above is only an integer
for $b\,{=}\,0$ in the first place (since $[S,\gamma_{5}]\neq 0$). We have
thus shown that the spin Chern number is generally no longer a topological
invariant if the system is perturbed in any other way additionally to the QSHE
direction (even when staying within first order perturbation theory).

We nevertheless expect surface states following a QSHE dispersion in the
vicinity of the $\Gamma$ point if the perturbation strength $a$ is small,
_i.e._ for small band gaps, with counter-propagating modes that are
approximately orthogonal and hence non-interacting. To elaborate on this
statement, we consider the surface of two crystals with perturbation $\pm a$
in the positive/negative half-space in $x$. Note that we did not assume the
parameters to be real numbers during the derivation above. They hold for
imaginary parameters, and in particular for half-space solutions obtained by
Floquet theory, _i.e._ a transfer matrix approach. Let us thus assume a
complex wave number in the positive [negative] half-space such that the
associated fields are normalizable, with $\Im\\{\delta
k_{x}\\}\geq\left[\leq\right]0$. A minimum requirement for the existence of a
surface state is that the half-space solutions match at the interface $x{=}0$.
In the upper spin channel they are proportional to (in the basis of upper spin
vectors, and with $\delta k_{x}$ the wave number in the positive half-space)

$\begin{pmatrix}\lambda+\delta k_{x}\\\ \delta k_{y}+\imath
a\end{pmatrix}\quad\text{and}\qquad\begin{pmatrix}\lambda-\delta k_{x}\\\
\delta k_{y}-\imath a\end{pmatrix}$

in the positive and negative half-space, respectively. Coupling thus yields
$\lambda\,{=}\,\delta k_{y}$ and $\delta k_{x}\,{=}\,\imath a$. For the lower
spin channel we have

$\begin{pmatrix}\lambda+\delta k_{x}\\\ \delta k_{y}-\imath
a\end{pmatrix}\quad\text{and}\qquad\begin{pmatrix}\lambda-\delta k_{x}\\\
\delta k_{y}+\imath a\end{pmatrix}$

and obtain $\lambda\,{=}\,-\delta k_{y}$ instead. The two associated
normalized eigenvectors are

$\frac{1}{\sqrt{2}}\begin{cases}\begin{pmatrix}1\\\
1\end{pmatrix},&\uparrow\\\ \begin{pmatrix}1\\\
-1\end{pmatrix},&\downarrow\end{cases}\text{ .}$

Two orthogonal surface states with linear surface band dispersion of opposite
slope, and a decay length that is inversely proportional to the perturbation
strength, are thus predicted. The same result applies to any orthogonal pair
of $(k_{\parallel},k_{\perp})$ in the reciprocal plane, easily seen by
substituting $\delta k_{x}\,{=}\,k_{\perp}\cos\varphi-
k_{\parallel}\sin\varphi$ and $\delta
k_{y}\,{=}\,k_{\perp}\sin\varphi+k_{\parallel}\cos\varphi$ for some
$\varphi\in[0,2\pi)$.

##  V Genuine $\mathbb{Z}$ charge in QVHE systems

The QVHE Hamiltonian

$W_{\text{QVHE}}=\gamma_{1}\delta k_{x}-\gamma_{2}\delta k_{y}+\gamma_{5}a$

is algebraically similar to the QSHE Hamiltonian. It thus leads to the same
hyperconic dispersion, and similar spin separation. The spin separation,
however, is mathematically trivial here, as $W_{\text{QVHE}}=\sigma_{3}\otimes
W_{2}$ naturally separates through a Kronecker product so that the outer
vector space can be associated with the two spin channels. In other words, the
generating spin operator might be chosen as
$S=\sigma_{3}\otimes{\mathbbm{1}}$. The eigenpairs are:

$[\lambda_{S};\bm{v}_{n}]=\begin{cases}\left[-1;\,\begin{pmatrix}0\\\ 0\\\
1\\\ 0\end{pmatrix},\begin{pmatrix}0\\\ 0\\\ 0\\\
1\end{pmatrix}\right],&\downarrow\\\ \left[\phantom{-}1;\,\begin{pmatrix}1\\\
0\\\ 0\\\ 0\end{pmatrix},\begin{pmatrix}0\\\ 1\\\ 0\\\
0\end{pmatrix}\right],&\uparrow\end{cases}\text{ .}$

The derivation of surface state dispersion close to the $K(K^{\prime})$ point
is formally equivalent to the last section. The crucial difference lies in the
interpretation of this finding: Translation symmetry is not broken in the QVHE
scenario, so that, going back to the traditional bandstructure picture, the
two spin channels correspond to the $K$ and $K^{\prime}$ points in the BZ,
respectively.

Importantly, we can now look at the question of topological protection from a
different perspective. Before proceeding with this idea, let us formalise the
concept of topological Chern (or $\mathbb{Z}$) protected edge states. We here
consider hermitian Hamiltonians only.
Bulk invariant: Consider a physical eigenproblem with $d$-dimensional lattice
symmetry, with an associated family of (normalized) eigenvectors $v(\bm{k})$,
where $\bm{k}$ is an element of the reciprocal d-torus ($\bm{k}$-space)
induced by the lattice. The family of eigenvectors thus forms a smooth vector
bundle over $\bm{k}$-space. Any closed 2D manifold $\mathcal{M}$ within
$\bm{k}$-space (with tangent space parametrization $(k_{1},k_{2})$) thus
satisfies the generalized Gauss-Bonnet theorem Chern (1945)

$\int_{M}\Omega=2\pi\,\mathcal{C}$

for the associated Berry curvature

$\quad\Omega=-2\,dk_{1}\wedge
dk_{2}\;\Im\langle\partial_{k_{1}}v,\partial_{k_{2}}v\rangle\text{ ,}$

Figure 4: Setup for QVHE strong protection. Two Semi-infinite half-space
domains are perturbed with opposite perturbation strength $a_{g}$, separated
by a (red) domain wall pointing in $y$ direction.

with the Chern (aka Euler) number $\mathcal{C}\,{\in}\,\mathbb{Z}$ that is a
topological integer.
Boundary invariant: Boundary or surface bands may emerge at an interface
between two half-spaces made of two bulk systems with a mutual band gap, that
is an interval of eigenvalues within which no bulk solution can be found in
either domain. The parallel wave vector $\bm{k}_{\parallel}$ is an element of
the surface BZ that depends on lattice inclination and topologically
constitutes a $d{-}1$-torus. The projected bulk bandstructure
$\Lambda(\bm{k}_{\parallel})\,{=}\,\cup_{i}\Lambda_{i}(\bm{k}_{\parallel})\,{=}\,\cup_{i}\left\\{\lambda_{i}(\bm{k}_{\parallel}+k_{\perp}\hat{\bm{n}}):\,k_{\perp}\in\mathbb{R}\right\\}$
($\hat{\bm{n}}$ is the unit vector normal to the domain wall, $i$ iterates
over the sorted eigenvalues) defines the intervals where bulk solutions exist.
Conversely, the region $\mathbb{R}\setminus\Lambda(\bm{k}_{\parallel})$
describes the union of all surface band gaps. A gap between $\Lambda_{i}$ and
$\Lambda_{i+1}$ is called partial if there is a $\bm{k}_{\parallel}$ so that
$\Lambda_{i}(\bm{k}_{\parallel})\cap\Lambda_{i+1}(\bm{k}_{\parallel})\neq\emptyset$,
and otherwise total. A topological invariant can be assigned to any closed
path $\gamma$ within the surface BZ along which there is a total bandgap: The
signed number of crossings of the surface band dispersion with an arbitrary
continuous function $f(k_{\gamma})$ within the band gap
($f(k_{\gamma})\notin\Lambda(k_{\gamma})$) is a topological integer
$\mathcal{N}$.

The topological Chern bulk-boundary correspondence is the statement that the
boundary invariant $\mathcal{N}$ is equal to the difference in gap Chern
numbers of the two bulk domains. The gap Chern number is the sum over the
Chern numbers of all bands below the bandgap with respect to the unique closed
2D manifold222Such a manifold is closed due to the fact that
$k_{\perp}\in\mathbb{R}/G_{\perp}\mathbb{Z}$ with $G_{\perp}$ the smallest
reciprocal lattice vector along $\hat{\bm{n}}$. A conceptual problem arises
only if $\hat{\bm{n}}$ is at an irrational angle with respect to the primitive
lattice vectors so that $G_{\perp}$ does not exist. We exclude these lattice
inclinations here. $\mathcal{M}$ that projects onto $\gamma$ via parallel
projection onto the surface BZ. This version of the Chern bulk boundary
correspondence is a generalization of Hatsugai’s original statement Hatsugai
(1993), and is applied frequently in topological photonics Lu  _et al._
(2014). Rigorous proofs of this and other bulk-boundary correspondences beyond
tight-binding models require non-elementary mathematical concepts (see for
example Prodan (2016)).

Importantly, within the context of this work, the established Chern bulk
boundary correspondence introduced here does not rely on the fact that the
parameter space is the reciprocal space of a d-dimensional lattice. It must
only contain it such that the projection direction $\hat{\bm{n}}$ is along the
reciprocal space direction perpendicular to the surface BZ. Consider thus the
extended 3D parameter space $(k_{x},k_{y},a_{g})$ that contains the 2D BZ of
the hexagonal lattice. We define $a_{g}$ as the global geometrical
perturbation strength, with $a_{\pm}\,{=}\,\pm a_{g}$ in the two half spaces
(labeled $\pm$) right and left of the domain wall. Note that the symmetry in
the definition of $a_{\pm}$ is chosen for convenience only. In fact, the
following argument is in principle valid for any continuous functional
dependence on either side.

Figure 5: Topologically protected surface states in the extended parameter
space corresponding to Fig. 4. (a) The path $\gamma$ encompassing the
evaluation point $(K,a_{g}\,{=}\,0)$ in the extended edge BZ parameter space.
(b) Along $\gamma$, the signed number of crossings between the topologically
protected surface bands (red lines) and an arbitrary continuous curve within
the band gap (dashed line) is $\mathcal{N}\,{=}\,2$.

We have already established in the previous section that the perturbation
Hamiltonian is a Weyl Hamiltonian in each spin channel and carries a
topological charge of $\pm 1$. Here, the spin channels are separated in
$\bm{k}$-space. In other words, any closed surface that contains the Weyl
point at say $K$ only will lead to a Chern invariant of
$\mathcal{C}\,{=}\,\text{sgn}(a)$ for the band immediately below. This non-
trivial Chern invariant is equivalent to the gap Chern number as all other
bands further below will either be trivial or pair up in form of a similar
Weyl point. Let us consider the domain wall along $k_{y}$ without loss of
generality, as illustrated in Fig. 4. Consider the closed path $\gamma$ in the
extended $(k_{y},a_{g})$ parameter space shown in Fig. 5 (a). Generally, the
corresponding (closed) manifold $\mathcal{M}$ (that projects onto $\gamma$ via
parallel projection along $k_{x}$) sustains a jump in gap Chern number of
$\Delta\mathcal{C}_{\text{gap}}\,{=}\,2$, and thus predicts that two strongly
protected surface bands cross the bandgap with positive slope along $\gamma$.
This statement remains correct for any other $\gamma^{\prime}$ encompassing
only $K$, _i.e._ as long as the inclination is not along the $K$ direction
itself and $\delta k_{y}^{0}$ is small enough so that $K^{\prime}$ is outside
of $\mathcal{M}$.

Figure 6: Topologically protected surface states along the path $\gamma$
encompassing a finite area in $(k_{\parallel},a_{g})$ parameter space, _cf._
Fig. 5 (a).

In the close vicinity of the evaluation point, we can test the topological
prediction analytically, using the same method as in section IV. Due to the
hyperconic dispersion relation, the projected bulk band structure is bounded
by the bulk bands for $\delta k_{x}\,{=}\,0$, _i.e._ by
$\lambda_{B}\,{=}\,\pm\sqrt{a^{2}+\delta k_{y}^{2}}$. Matching the fields on
both sides of the domain wall close to $K$ on the other hand yields the
surface band dispersion $\lambda_{s}=-\text{sgn}(a_{g})\delta k_{y}$. This
result is shown in Fig. 5 (b): The predicted boundary invariant
$\mathcal{N}\,{=}\,2$ corresponding to
$\Delta\mathcal{C}_{\text{gap}}\,{=}\,2$ is clearly observed (the red surface
dispersion positively crosses the arbitrary dashed gap function twice). For
$\gamma$ encompassing a finite area, a simple analytical prediction based on
our group theoretical findings is not possible, but the topological protection
of the edge states of course still holds, as demonstrated in Fig. 6. The edge
bandstructure is here extracted from a supercell calculation for a 2D photonic
crystal based on a perturbed kagome lattice (see Wong  _et al._ (2019) for
details).

We conclude in noting that no genuine topological protection can be associated
with the optical QSHE case, although surface bands that resemble those of
proper $\mathbb{Z}_{2}$ protection can be found for small perturbations in
first order perturbation. For the QVHE case, we showed that genuine
topological $\mathbb{Z}$ protected edge states exist within the extended
parameter space, containing the reciprocal lattice and geometrical
perturbations. Within the scope of this manuscript, this protection is only
defined within the parameter space of a certain type of a deterministic
geometrical perturbation. The above line of thought is, however, applicable to
an extended parameter space of any dimension. We note in this context that
another pathway to understand topological protection in the QVHE case based on
lattice folding at the domain wall has recently been reported Qian  _et al._
(2018). The requirement imposed in order to be able to map to a topological
reference system there is that the two sub-lattices on either side of the
domain wall are related by a mirror symmetry. Conceivably, breaking of this
mirror symmetry by random perturbations would not allow to define a similar
reference system. In contrast, the method presented here is also valid for
inclinations where such a mirror symmetry is not possible, and it provides the
opportunity to extend the study to a deterministic QVHE system with
superimposed arbitrary small random perturbations along other directions in a
larger extended parameter space.

## Acknowledgements

This work was supported by the EPSRC through program grant EP/L027151/1 and by
the European Regional Development Fund through the Welsh Government
(80762-CU145 (East)).

## References

  * Haldane and Raghu (2008) F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
  * Raghu and Haldane (2008) S. Raghu and F. D. M. Haldane, Phys. Rev. A 78, 033834 (2008).
  * Lu  _et al._ (2014) L. Lu, J. D. Joannopoulos, and M. Soljačić, Nature Photonics 8, 821 (2014).
  * Lu  _et al._ (2016) L. Lu, J. D. Joannopoulos, and M. Soljačić, Nat Phys 12, 626 (2016).
  * Khanikaev and Shvets (2017) A. B. Khanikaev and G. Shvets, Nature Photonics 11, 763 (2017).
  * Wang  _et al._ (2009) Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, Nature 461, 772 (2009).
  * Fang  _et al._ (2012) K. Fang, Z. Yu, and S. Fan, Nature photonics 6, 782 (2012).
  * Silveirinha (2017) M. G. Silveirinha, Phys. Rev. B 95, 035153 (2017).
  * Khanikaev  _et al._ (2013) A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nature materials 12, 233 (2013).
  * Wu and Hu (2015) L.-H. Wu and X. Hu, Phys. Rev. Lett. 114, 223901 (2015).
  * Chen  _et al._ (2017) X.-D. Chen, F.-L. Zhao, M. Chen, and J.-W. Dong, Physical Review B 96, 020202 (2017).
  * Ma and Shvets (2016) T. Ma and G. Shvets, New Journal of Physics 18, 025012 (2016).
  * Kane and Mele (2005) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).
  * Qian  _et al._ (2018) K. Qian, D. J. Apigo, C. Prodan, Y. Barlas, and E. Prodan, Phys. Rev. B 98, 155138 (2018).
  * Saba  _et al._ (2017) M. Saba, J. M. Hamm, J. J. Baumberg, and O. Hess, Phys. Rev. Lett. 119, 227401 (2017).
  * Mekata (2003) M. Mekata, Physics Today 56, 12 (2003).
  * Guo and Franz (2009) H.-M. Guo and M. Franz, Phys. Rev. B 80, 113102 (2009).
  * Wong  _et al._ (2019) S. Wong, M. Saba, O. Hess, and S. S. Oh, arXiv preprint arXiv:1907.04310 (2019).
  * Bradley and Cracknell (2010) C. Bradley and A. Cracknell,  _The Mathematical Theory of Symmetry in Solids: Representation Theory for Point Groups and Space Groups (Oxford Classic Texts in the Physical Sciences)_ (Oxford University Press, 2010).
  * Dresselhaus (2013) M. S. Dresselhaus,  _Group Theory_ (Springer, 2013).
  * Elman (2017) M. Elman,  _A Rubrik for Fully Connected Topological Photonic Crystals in 3D_ , Ph.D. thesis (2017).
  * Makwana  _et al._ (2019) M. Makwana, R. Craster, and S. Guenneau, Opt. Express 27, 16088 (2019).
  * Hielscher  _et al._ (2017) J. Hielscher, C. Pouya, P. Vukusic, and G. E. Schröder-Turk, Opt. Express 25, 5001 (2017).
  * Saba  _et al._ (2013) M. Saba, M. D. Turner, K. Mecke, M. Gu, and G. E. Schröder-Turk, Phys. Rev. B 88, 245116 (2013).
  * Chern (1945) S.-s. Chern, Annals of Mathematics , 674 (1945).
  * Hatsugai (1993) Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993).
  * Prodan (2016) E. Prodan,  _Bulk and boundary invariants for complex topological insulators : from K-Theory to physics_ (Springer, Switzerland, 2016).
  * Rössler (2009) U. Rössler,  _Solid State Theory: An Introduction_ (Springer, 2009).
  * Zhen  _et al._ (2015) B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, Nature 525, 354 (2015).
  * Huang  _et al._ (2011) X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, Nature Materials 10, 582 (2011).

##  Appendix A A glossary for representation theory

We here define and explain the applied concepts of representation theory for
finite groups, mainly following Bradley and Cracknell (2010). It should be
noted that some of the findings from here on are restricted to unitary group
elements (expressed by linear operators $\hat{G}$ acting on $\mathcal{H}$),
_i.e._ they satisfy the inner product relation
$\langle\hat{G}v|\hat{G}w\rangle=\langle v|w\rangle$ for any any two vectors
$|v\rangle,|w\rangle\in\mathcal{H}$. Non-linear symmetries such as
hermiticity, reciprocity and time inversion are treated separately. The latter
$\hat{\tau}$ is for anti-unitary, with
$\langle\hat{\tau}v|\hat{\tau}w\rangle=\,\overline{\langle
v|w\rangle\vphantom{\bar{\langle v|w\rangle}}}\,$
($\,\overline{\,\cdot\,\vphantom{\bar{\,\cdot\,}}}\,$ denotes complex
conjugation), while the other two are fully non-linear and do not have a
linear/antilinear operator representation.

To obtain a formally finite group, we replace the translation group
$\mathcal{T}$ by the quotient group
$\mathcal{T}\mapsto\mathcal{T}/[T]_{\approx}$, $T\approx T^{\prime}$ if
$\hat{T}^{\prime}=\hat{T}\prod_{i}\hat{\bm{a}}_{i}^{n_{i}N_{i}}$, $i=1,2,3$,
with $\hat{\bm{a}}_{i}$ the $i$-th primitive translation,
$n_{i}\in\mathbb{Z}$, and $1\ll N_{i}\in\mathbb{N}$. This is a standard
simplification technique (_cf._ Rössler (2009)) and equivalent to imposing
periodic boundary conditions on a large but finite crystal, or supercell.

Representation (rep): Consider a homomorphism
$\Phi_{d}:\mathcal{G}\rightarrow\mathcal{D}_{d}$, with $G\mapsto D_{d}(G)$,
that maps any group element $G$ onto a linear $d$-dimensional map
$\mathcal{D}_{d}(G):\Omega_{d}\rightarrow\Omega_{d}$. We call $\Phi_{d}$, or
equivalently the set of $D_{d}(G)$, a representation of $\mathcal{G}$, with an
associated $d$-dimensional vector space $\Omega_{d}$.

Character: A representation can be uniquely characterised by the traces of the
associated maps $\mathcal{D}(G)$. These traces play a central role in a number
of powerful theorems, so that an established synonym for the trace of
$\mathcal{D}_{d}(G)$ is the character of the representation $\mathcal{D}$ with
respect to a group element $G$, with symbol $\chi(G)$.

Partner: It is convenient to choose a particular set of basis vectors that
span the vector space $\Omega_{d}$. These basis vectors are also known as
partners of the representation. All $D_{d}(G)$ are then given by unitary
square matrices of dimension $d$.

Irreducible representation (irrep): Consider a vector subspace $V$ of
$\Omega_{d}$. The representation $\mathcal{D}(G)$ is called irreducible, if
any such $V$ satisfies the following:

$\forall G\in\mathcal{G}:\;D_{d}(G)V\subset V\Rightarrow V={0}\lor
V=\Omega_{d}\text{ .}$

In other words, if we choose a set of partners, there is no similarity
transform that brings all matrices $D_{d}(G)$ into block diagonal form.
Irreducible representations can therefore be considered as a generalisation of
the concept of simultaneous diagonalization to non-Abelian groups.

The partners of all irreducible representations span the whole associated
Hilbert space. This statement is equivalent to the fact that any
representation of $\mathcal{G}$ can be written as a direct sum of irreducible
representations. Finding the irreps and the partners within the physical
Hilbert space is useful in the particular case, because the eigenfunctions of
the Hamiltonian $H$ can be identified with a superposition of partners of a
single irrep only. The exception to this rule are accidental degeneracies,
including exceptional points, which may occur when the eigenvalues of $H$ are
_accidentally_ (by coincidence for a particular set of parameters of the
physical system) glued together. These degeneracies are unpredictable from
first principles and rare, we will not consider them in the following.
Nevertheless, both cases can be engineered to occur at points or lines in the
BZ if the physical properties are desired Makwana  _et al._ (2019); Zhen  _et
al._ (2015); Huang  _et al._ (2011).

If we thus are able to determine the irreps of a whole space group, we can
gain additional insight into the nature of the physical modes, such as
prediction of degeneracies and selection rules for field integrals without
actual calculation of the eigensystem. Further, we can potentially decrease
the dimension of the particular problem for direct computation. For a
straight-forward recipe to determine the irreps of $\mathcal{G}$ that are of
interest for this work, three additional definitions are essential.

Star: Consider a group $\mathcal{G}$ with an invariant (not necessarily
Abelian) subgroup $\mathcal{T}$, _i.e._ $G\mathcal{T}G^{-1}=\mathcal{T}$ for
any $G\in\mathcal{G}$. Denote the irreps of $\mathcal{T}$ by $\Delta_{i}(T)$.
The star of $\mathcal{G}$ with respect to the representation $\Delta_{i}(T)$
of $\mathcal{T}$ is a maximal set of inequivalent representations
$\Delta_{i}^{(\alpha)}(T):=\Delta_{i}(G_{\alpha}^{-1}TG_{\alpha})$.

Little group: With the above definitions, the little group of $\mathcal{G}$
with respect to a particular representation $\Delta_{i}(T)$ is
$\mathcal{G}_{i}:=\left\\{G\in\mathcal{G}:\Delta_{i}(GTG^{-1})\equiv\Delta_{i}(T)\right\\}$.
Note that the little group is evidently a subgroup of $\mathcal{G}$ (albeit
not invariant in general).

Induced representation: Consider the expansion of a group $\mathcal{G}$ into
left cosets with respect to a subgroup $\mathcal{S}$, that is
$\mathcal{G}=\sum_{\alpha}r_{\alpha}\mathcal{S}$. Denote the irreps of
$\mathcal{S}$ by $\Gamma_{j}(S)$, where $S\in\mathcal{S}$ and $j$ is an index
(not the dimension of the irrep as before). The induced representation of
$\Gamma_{j}$ in $\mathcal{G}$ is written as $\Gamma_{j}\uparrow\mathcal{G}$.
It is most conveniently defined via a set of partners $|n\rangle$, so that the
irreps take the matrix form given by
$s_{a}|n\rangle=\Gamma_{j}^{(nm)}(s_{a})|m\rangle$. We introduce a new set of
partners $|\mu n\rangle:=r_{\mu}|n\rangle$, so that the induced representation
is given by:

$\Gamma_{j}\uparrow\mathcal{G}^{(nm)}_{\mu\nu}(G):=\begin{cases}\Gamma_{j}^{(nm)}(G_{\mu\nu})&\\!\text{if
}G_{\mu\nu}\\!:=\\!r_{\mu}^{-1}Gr_{\nu}\in\mathcal{S}\\\
0&\\!\text{else}\end{cases}$

Note that there is a unique pair of $(\mu,n)$ for each space group element
$G=r_{\mu}s_{n}$, and a (generally different) unique pair $(\nu,m)$, so that
the same element is written as $G=s_{m}r_{\nu}^{-1}$. Therefore, for a given
$\mu$ (or a given $\nu$) there is only one pair $(\mu,\nu)$, for which the
induced representation becomes non-zero, and its outer structure is hence that
of a permutation matrix. Importantly, we do not mean here that the induced
representation can be written as a tensor product $A_{\mu\nu}\otimes B_{nm}$.
Given three cosets and an element $G$ that is not in $\mathcal{S}$, it could
for example look like:

$\Gamma_{j}\uparrow\mathcal{G}^{(nm)}_{\mu\nu}(G)=\begin{pmatrix}0&\Gamma_{j}(G_{12})&0\\\
\Gamma_{j}(G_{21})&0&0\\\ 0&0&\Gamma_{j}(G_{33})\end{pmatrix}$

Small representation: The small representations of a group $\mathcal{G}$ with
respect to a subgroup representation $\mathcal{S}$ are those irreducible
representations $\Gamma_{j}^{(i)}$, that contain only a single irrep
$\Delta_{i}$ in $\mathcal{S}$, when restricted to $S\in\mathcal{S}$, _i.e._ we
may write $\Gamma_{j}(S)=\Delta_{i}(S)\oplus\Delta_{i}(S)\oplus\dots$, where
$\oplus$ denotes a direct sum.

With these definitions, one can show Bradley and Cracknell (2010) that the
irreps of a group $\mathcal{G}$, with an invariant subgroup $\mathcal{T}$ can
be obtained by the following steps:

  1. 1.

distribute the reps of $\mathcal{T}$ into stars and select one rep
$\Delta_{i}$ in each star

  2. 2.

find the small representations for each associated little group
$\Gamma_{j}^{(i)}$, that contain only $\Delta_{i}$ in $\mathcal{T}$

  3. 3.

the irreps of $\mathcal{G}$ are the induced representations
$\Gamma_{j}^{(i)}\uparrow\mathcal{G}$