# Asymptotically exact photonic analogues of chiral symmetric topological
tight-binding models

S. Palmer1, Y. Ignatov1, R.V. Craster2 & M. Makwana2 1 The Blackett
Laboratory, Imperial College London, London SW7 2AZ, United Kingdom  2
Department of Mathematics, Imperial College London, London, SW7 2AZ, United
Kingdom

###### Abstract

Topological photonic edge states, protected by chiral symmetry, are attractive
for guiding wave energy as they can allow for more robust guiding and greater
control of light than alternatives; however, for photonics, chiral symmetry is
often broken by long-range interactions. We look to overcome this difficulty
by exploiting the topology of networks, consisting of voids and narrow
connecting channels, formed by the spaces between closely spaced perfect
conductors. In the limit of low frequencies and narrow channels, these void-
channel systems have a direct mapping to analogous discrete mass-spring
systems in an asymptotically rigorous manner and therefore only have short-
range interactions. We demonstrate that the photonic analogues of topological
tight-binding models that are protected by chiral symmetries, such as the SSH
model and square-root semimetals, are reproduced for these void-channel
networks with appropriate boundary conditions. We anticipate, moving forward,
that this paper provides a basis from which to explore continuum photonic
topological systems, in an asymptotically exact manner, through the lens of a
simplified tight-binding model.

  *

##  1 Introduction

The field of topological materials has revealed exotic phenomena such as
robust, unidirectional edge states that occur at the interfaces between
materials that belong to two different topological phases. The Nobel Prize in
Physics 2016 was awarded to Thouless, Haldane, and Kosterlitz, [1, 2, 3] for
predicting such phases in electronic systems where such topological edge
states promise to revolutionise electronics and quantum computing [4, 5, 6, 7,
8]. Although topological phases were first discovered in electronic systems,
the underlying principles are applicable to wave systems in general, including
photonic and acoustic systems [9, 10]. There is now great interest in
reproducing topological phases in photonics using _photonic crystals_ :
periodic nanostructures with tunable photonic bands. In the short to medium
term, topological photonic materials may improve the performance of photonic
devices by reducing dissipation, especially when guiding light around sharp
corners, and in the longer term could offer a platform for fault-tolerant
quantum computers in photonics [10, 11, 12]. Applications that are unique to
topological photonic materials include the design of more efficient lasers,
where the topological edge mode acts as a cavity in which light propagates and
is amplified unidirectionally and coherently despite imperfections in the
crystal [13, 14, 15], and the cloaking of large photon sources from each other
using the polarisation of light [16].

Many topological materials are protected by chiral symmetry (also known as
sublattice symmetry) [17, 18, 19], both directly as in the SSH model [20] and
indirectly as in _square-root topological insulators_ [21] where the squared
Hamiltonian is block-diagonal and at least one of the blocks corresponds to a
known non-trivial system. Chiral symmetry acts on Bloch Hamiltonians as [5]

$\hat{\mathcal{S}}\hat{H}(\vec{k})\hat{\mathcal{S}}^{-1}=-\hat{H}(\vec{k}),$
(1)

where $\hat{\mathcal{S}}$ is a unitary operator that squares to $+1$. Chiral
symmetry is relatively common in tight-binding models of electronic systems,
however in photonics, chiral symmetry is often broken by long-range
interactions [22, 23]; despite this, it can be engineered in certain photonic
systems, examples include arrays of dielectric resonators [24] or grating
structures [25].

In this paper, we engineer chiral symmetric photonic systems where transverse-
electric polarised light propagates, in the voids and narrow connecting
channels, located between perfect conductors, as shown in figure 1(a). In
section 2, we review why at low frequencies, this photonic system behaves like
a _discrete_ classical network of inductors and capacitors (or equivalently as
a classical network of masses and springs, as shown in figure 1(b)), in the
limit where the inclusions are closely spaced [26, 27]. The void-channel
networks have vanishing long-range interactions, as the channels are made
increasingly narrow, thus certain configurations of the void-channel networks
have a chiral symmetric limit (up to a shift of constant frequency).

In tight-binding models, terminating the lattice freely does not change the
onsite potential and therefore preserves chiral symmetry. In mass-spring and
void-channel models, the free boundary condition generally breaks chiral
symmetry [28, 29]. We propose that the chiral symmetry can be restored at the
interfaces by “capping” the mass-spring/void-channel networks with heavy
masses/large voids, respectively. In section 3.1 we use the well known SSH
model [20] to demonstrate this principle, and verify that the void-channel SSH
geometry features topological edge states. Although in this article we do not
concentrate upon the asymptotic theory of mapping continuum systems to
discrete models we note that there is considerable advantage in being able to
accurately map between them: the entire machinery and theory for topological
tight-binding systems then carries across into continuum systems. In simpler
settings of connected acoustic tubes and straight channels [30, 31] and [32]
illustrate the power of being able to translate back-and-forth between
continuum and discrete networks; we use the asymptotic methodology of [26, 27]
showing that curved thin channels can be employed for closely spaced cylinders
(and other smooth objects) and noting that a three-dimensional network
extension [33] is also available.

The void-channel geometries are also a promising platform to realise square-
root topological systems [21]. For example, Maimaiti _et al_ [28] showed that
the nearest-neighbour tight-binding model of the honeycomb-kagome lattice is a
_square-root semimetal_ that inherits the topology from the honeycomb sector
of the squared Hamiltonian. The authors also proposed an analogous mass-spring
model using a gravitational potential energy term to adjust the onsite terms
of the equations of motion [28]; it is not apparent to us if an analogue of
this gravitational term exists for the void-channel geometries. In section
3.2, we produce a photonic analogue of the square-root semimetal by capping
our void-channel network with large voids, thereby ensuring that the chiral
symmetry of the squared Hamiltonian is not broken by the interfaces. We study
the interfaces in a ribbon and a triangular metaparticle of the honeycomb-
kagome lattice, and observe that topologically protected edge and corner
states can be excited.

##  2 Methodology

Figure 1: (a) Two voids of vacuum (white) and a narrow connecting channel
embedded in perfect conductor (grey) will behave like a pair of inductors and
a capacitor for transverse-electric polarised light at low frequencies and for
narrow channels [26, 27]. The voids act as inductors with inductance $L$
because the currents, $I$, circulate around the surface of the voids and
induce out-of-plane magnetic fields within the voids. We label the magnetic
field within the left and right voids as $H_{z}^{-}$ and $H_{z}^{+}$,
respectively. The channel acts as a capacitor because the difference of
magnetic field across the channel generates an electric field, $\vec{E}$,
across the gap. This analogy can be extended to larger networks of voids and
channels. (b) The void-channel or inductor-capacitor network is also analogous
to a mass-spring network, where masses $m$ are connected by spring constants
$k$, and the masses oscillate in and out of the page.

We apply the method of Vanel _et al_ [26] to map the TE-polarised Maxwell’s
equations, within networks of voids and channels formed between closely spaced
perfect conductors (as shown in figure 1(a)), to equivalent networks of
resonators in an asymptotically exact manner. The voids behave as inductors
and the channels behave as capacitors; the difference in the out-of-plane
magnetic field across a channel, $H_{z}^{+}-H_{z}^{-}$, results in an electric
field, $\vec{E}$, perpendicular to the channel [26, 27]. Equivalently, we may
consider a mass-spring network where the voids are mapped to masses and the
channels are mapped to springs, as shown in figure 1(b).

The parameters of the discrete inductor-capacitor/mass-spring networks are not
reliant upon lumped parameters and/or heuristic approximations; these
approaches are common in electrical engineering as lumped circuit models [34],
or as optimisation with databases [35], and these successfully to take complex
systems across to networks. The advantage of the alternative approach here is
that the effective parameters are simple and explicit. We proceed by utilising
matched asymptotic expansions, Vanel _et al_ [26] demonstrated that the
precise values of the masses and spring constants, corresponding to a
particular network of voids and channels, can be determined in a remarkably
simple manner at low frequencies and in the limit of narrow channels,
$h/a\rightarrow 0$. The masses are proportional to the area of the voids,

$m_{i}=A_{i}\cdot m_{0}/a^{2},$ (2)

whilst the spring constants are a function of the half-width of the channel,
$h$, in addition to the radius of curvature of the two sides of the channel,
$R_{1}$ and $R_{2}$,

$k=\frac{1}{\pi}\sqrt{\frac{h}{R_{1}}+\frac{h}{R_{2}}}.$ (3)

Equations (2) and (3) allow us to accurately model the continuous void-channel
network using the much simpler equations of motion of a discrete mass-spring
system, without the need for any fine tuning or parameter fitting. The reverse
mapping allows us to propose new photonic void-channel models where the
coupling of the field between different voids is highly controllable. This
afford us complete control over the symmetries of the Hamiltonian and, hence,
suggests that void-channel networks could be a powerful platform for realising
symmetry-protected photonic topological phases.

In this paper, the void-channel solutions were obtained using the open source
finite element solver FreeFem++ [36]; we used this to solve the Helmholtz
equation for our void-channel model,

$\frac{\partial^{2}H_{z}}{\partial t^{2}}-c^{2}\nabla^{2}H_{z}=0,$ (4)

where $c$ is the speed of light in the space between the perfect conductor and
where we notably applied Neumann boundary conditions along the surface of the
perfect conductor [26, 27]. This was achieved using a modified version of a
set of FreeFem++ scripts that were originally used to model phononic crystals
[37].

##  3 Results

###  3.1 The Su-Schrieffer-Heeger chain

Figure 2: Squared frequency spectrum (normalised by
$\omega_{0}=\sqrt{k_{0}/m_{0}}$) of a mass-spring chain with equal masses,
$m=0.1m_{0}$, and alternating spring constants $k_{1}=0.01k_{0}$ and
$k_{2}=0.02k_{0}$, as shown in the inset. The width of the unit cell is $a$.
The spectrum closely resembles the energy spectrum of an electronic SSH tight-
binding model [38, 20], except that the squared frequency has been shifted by
$(k_{1}+k_{2})/m=0.3\omega_{0}^{2}$.

To explore the symmetries of photonic void-channel networks, let us first
consider the equations of motion of an SSH-like mass-spring network consisting
of equal masses, $m$, and alternating spring constants, $k_{1}$ and $k_{2}$,
as shown in the inset of figure 2,

$\left[\begin{array}[]{cc}\frac{k_{1}}{m}+\frac{k_{2}}{m}&-\frac{k_{1}}{m}-\frac{k_{2}}{m}e^{-ika}\\\
-\frac{k_{1}}{m}-\frac{k_{2}}{m}e^{+ika}&\frac{k_{1}}{m}+\frac{k_{2}}{m}\end{array}\right]\left[\begin{array}[]{c}\big{|}u_{1}\big{>}\\\
\big{|}u_{2}\big{>}\end{array}\right]=\omega^{2}(k)\left[\begin{array}[]{c}\big{|}u_{1}\big{>}\\\
\big{|}u_{2}\big{>}\end{array}\right].$ (5)

We see in figure 2 that the squared frequency spectrum of this chain resembles
the energy spectrum of the electronic SSH tight-binding model. Unlike the
tight-binding model the diagonal terms in equation (5) are non-zero; however,
as they are equal, they merely correspond to a simple shift of frequency by
$\sqrt{k_{1}/m+k_{2}/m}$,

$\left[\begin{array}[]{cc}0&-\frac{k_{1}}{m}-\frac{k_{2}}{m}e^{-ika}\\\
-\frac{k_{1}}{m}-\frac{k_{2}}{m}e^{+ika}&0\end{array}\right]\left[\begin{array}[]{c}\big{|}u_{1}\big{>}\\\
\big{|}u_{2}\big{>}\end{array}\right]=\left(\omega^{2}(k)-\frac{k_{1}+k_{2}}{m}\right)\left[\begin{array}[]{c}\big{|}u_{1}\big{>}\\\
\big{|}u_{2}\big{>}\end{array}\right].$ (6)

This frequency shift allows for the chiral symmetry of the bulk equations to
be preserved.

Figure 3: An SSH-like mass-spring chain with free and “wall” boundary
conditions. The mass-spring chain consists of equal masses, $m$, connected by
springs of alternating spring constants $k_{1}$ and $k_{2}$. (a‑b) Schematic
and squared frequency spectrum of an SSH-like mass-spring chain with free
boundary conditions. The chain is 20 masses long and we take
$k_{1}=0.02k_{0}$, $k_{2}=0.01k_{0}$, and $m=0.1m_{0}$. Although the winding
number of the bulk Hamiltonian is non-trivial, no edge states are observed
because the free boundary condition breaks the chiral symmetry. (c‑d)
Schematic and squared frequency spectrum of the same chain but with the “wall”
boundary condition, where the edges of the SSH-like mass-spring chain are
attached to immovable walls with springs of spring constant $k_{2}$. The wall
boundary condition restores the chiral symmetry of the chain and symmetry-
protected edge states are observed in the bulk band gap.

The mass-spring model differs from the original SSH tight-binding model [38,
20] because the forces on the masses are proportional to the _differences_ of
the mass displacements, whereas in tight-binding models the hopping is
proportional to the wavefunction amplitudes themselves [39]. In particular,
while the original SSH model is chiral symmetric for a finite chain with free
boundary conditions, this is not the case for the mass-spring chain with free
boundary conditions. The chiral symmetry is broken, in the latter, by the non-
zero term along the diagonal of the matrix and therefore there are no
topological edge states in the frequency spectrum, (see figure 3(b)),

$\left[\begin{array}[]{ccccc}-\frac{k_{2}}{m}&-\frac{k_{1}}{m}\\\
-\frac{k_{1}}{m}&0&-\frac{k_{2}}{m}\\\ &-\frac{k_{2}}{m}&0&-\frac{k_{1}}{m}\\\
&&-\frac{k_{1}}{m}&0&\ddots\\\
&&&\ddots&\ddots\end{array}\right]\left[\begin{array}[]{c}u_{1}\\\ u_{2}\\\
u_{3}\\\ u_{4}\\\
\vdots\end{array}\right]=\left(\omega^{2}(k)-\frac{k_{1}+k_{2}}{m}\right)\left[\begin{array}[]{c}u_{1}\\\
u_{2}\\\ u_{3}\\\ u_{4}\\\ \vdots\end{array}\right].$ (7)

This distinction arises because the end masses are only connected to a
solitary spring; we can restore the chiral symmetry by anchoring the chain to
an immovable wall with a spring, of spring constant $k_{2}$, as shown in
figure 3(c). The equations of motion of this chain with the “wall” boundary
condition then become,

$\left[\begin{array}[]{ccccc}0&-\frac{k_{1}}{m}\\\
-\frac{k_{1}}{m}&0&-\frac{k_{2}}{m}\\\ &-\frac{k_{2}}{m}&0&-\frac{k_{1}}{m}\\\
&&-\frac{k_{1}}{m}&0&\ddots\\\
&&&\ddots&\ddots\end{array}\right]\left[\begin{array}[]{c}u_{1}\\\ u_{2}\\\
u_{3}\\\ u_{4}\\\
\vdots\end{array}\right]=\left(\omega^{2}(k)-\frac{k_{1}+k_{2}}{m}\right)\left[\begin{array}[]{c}u_{1}\\\
u_{2}\\\ u_{3}\\\ u_{4}\\\ \vdots\end{array}\right].$ (8)

We pictorially see, from figure 3(d), that the chiral symmetry is restored
and, hence, SSH-like edge states emerge at the mid-gap frequency,
$\sqrt{k_{1}/m+k_{2}/m}$.

Figure 4: (a) The “wall” boundary condition of the SSH-like mass-spring chain,
introduced in figure 3(b), can be approximated by replacing the wall with
heavy masses, $M$. (b) The average frequency of the pair of edge states for
the mass-spring chain capped with heavy masses (dots) compared with that for
the mass-spring chain with the wall boundary condition (solid line), for the
same values of $m$, $k_{1}$, and $k_{2}$ as before. Each chain contain 20
masses of mass $m$, and the capped chain has two capping masses of mass $M$ on
each end for a total of 22 masses. For capping masses that are an order of
magnitude heavier than the masses in the rest of the chain, $M\gtrsim 10m$,
the squared frequencies of the edge states of the two chains agree within
about 3% error.

The wall boundary condition can be well approximated by capping mass-spring
models with heavy masses, as shown in figure 4. This allows us to propose a
photonic analogue of the SSH model that consists of a one-dimensional network
of voids and channels as shown in figure 5(a).

Figure 5: (a) Schematic of an SSH-like void-channel chain that is analagous to
the capped mass-spring chain introduced in figure 4. The grey region is
perfect conductor, and the white is air. Only the beginning of the chain is
shown. The bulk of the chain consists of equally sized voids of half-width $H$
and length $L$ connected by channels of alternating half-widths $h_{1}$ and
$h_{2}$. The precise shapes of the voids and channels are defined in the main
text. Note that the bulk region runs from $x=-L/4$ to $(N+1/4)L$ such that the
curvature of the walls is well defined at the narrowest point of each channel.
The half-width at the end of the bulk region is $h_{\mathrm{end}}$; the chain
is then capped by larger voids that are roughly circular with diameter $L$.
(b) Squared frequency spectrum of the void-channel chain (red crosses,
$H=L/10$, $h_{1}=H/400$, $h_{2}=H/100$) and analagous mass-spring chain (black
dots, $k_{1}=0.02$, $k_{2}=0.01$, $m=0.1m_{0}$, $M=\pi/4m_{0}$). The
frequencies are normalised by $\omega_{0}=\sqrt{k_{0}/m_{0}}$ for the mass-
spring model and $\omega_{0}=2\pi c_{0}/L$ for the void-channel model. The
chains consist of 20 masses/voids (or 22 including the pair of larger
masses/voids at the end). (c-d) The magnetic field (red positive, blue
negative) of the two edge modes of the void-channel chain. We see that the
chain preserves chiral symmetry well: the magnetic field is relatively weak in
the large capping voids and each edge mode is well localised to just one
sublattice.

To emulate equal masses connected by alternating spring constants we require
equally sized voids connected by relatively thin channels of alternating
widths. A simple choice for the shape of the upper and lower walls of the
geometry is

$y(x)=\pm\left[H\sin^{2}(\pi x/L)+h_{1}\sin^{2}(\pi x/2L)+h_{2}\cos^{2}(\pi
x/2L)\right],$ (9)

where we take the half-width of the void as $H=L/10$, and the alternating
half-widths of the channels as $h_{1}=H/400$ and $h_{2}=H/100$. Note that the
upper and lower walls run from $x=-\tfrac{1}{4}L$ to $x=(N+\tfrac{1}{4})L$ to
ensure that the radius of curvature of each channel from $x=0$ to $x=NL$ are
well defined. The local radius of curvature of the walls of each channel is
$R=L^{2}/(2\pi^{2}H)$. As $h_{1},h_{2}\ll H$, the area of each void in the
bulk region is approximately $A_{\mathrm{bulk}}\approx
2\int_{0}^{L}H\sin^{2}(\pi x/L)\mathrm{d}x=HL$.

The walls are capped by roughly circular voids of diameter $L$. On the left
hand side,

$\displaystyle x_{\mathrm{L}}(\theta)$
$\displaystyle=\frac{L}{2}\cos(\theta)-\frac{3}{4}L,$ (10) $\displaystyle
y_{\mathrm{L}}(\theta)$
$\displaystyle=\frac{L}{2}\sin(\theta)+h_{\mathrm{end}}\cos(\theta/2),$ (11)

for $\theta=[0,2\pi]$ and on the right hand side,

$\displaystyle x_{\mathrm{R}}(\theta)$
$\displaystyle=\frac{L}{2}\cos(\theta)+\left(N+\frac{3}{4}\right)L,$ (12)
$\displaystyle y_{\mathrm{R}}(\theta)$
$\displaystyle=\frac{L}{2}\sin(\theta)-h_{\mathrm{end}}\sin(\theta/2),$ (13)

for $\theta=[-\pi,\pi]$, where the $h_{\mathrm{end}}=y(-L/4)=y(NL+L/4)$ term
is included to ensure that the walls of the geometry are continuous. As
$h_{\mathrm{end}}\lesssim L$, we can take the area of the large capping voids
as approximately $A_{\mathrm{cap}}=\pi L^{2}/4$. Note that this is a slight
underestimate of the true area of the caps because we do not account for the
region $-L/4\leq x\leq 0$ or for the extra height of the void, described by
Equations (10)-(13), compared to a circle of diameter L.

The alternating spring constants of the corresponding mass-spring network are

$\displaystyle k_{1}$ $\displaystyle=\frac{1}{\pi}\sqrt{2h_{1}/R}\cdot
k_{0}=0.01k_{0},$ (14) $\displaystyle k_{2}$
$\displaystyle=\frac{1}{\pi}\sqrt{2h_{2}/R}\cdot k_{0}=0.02k_{0},$ (15)

the masses in the bulk of the chain are

$\displaystyle m=A_{\mathrm{bulk}}\cdot m_{0}/L^{2}=0.1m_{0},$ (16)

and the larger capping masses are

$\displaystyle M=A_{\mathrm{cap}}\cdot m_{0}/L^{2}=\frac{\pi}{4}m_{0},$ (17)

such that $M/m\approx 8$, where $k_{1}$, $k_{2}$, $m$, and $M$ are as defined
earlier in figure 4. As the capping mass is roughly an order of magnitude
larger than the bulk masses, we expect that the chiral symmetry is largely
restored to the model. Indeed, figure 5(b) shows that the energy spectra of
both the void-channel network (red) and the mass-spring network (black) behave
as non-trivial SSH chains with a pair of topological edge states in the energy
gap.

Overall, there is good agreement between the void-channel and mass-spring
models; in particular we see similar band gaps and a pair of topological edge
states in each model. The models agree well at lower frequencies, because the
mapping between the models is valid for frequencies, below a cut-off frequency
that scales as $\omega_{\mathrm{cutoff}}^{2}\sim 1/h$ [27]. We shall explore
this error, in greater detail, when we study the honeycomb-kagome lattice.

The edge states are largely localised on separate sublattices and decay
quickly into the bulk, as shown for the void-channel model in figures 5(c‑d).
It is intuitive that in the mass-spring model the heavy capping masses will
oscillate with a smaller amplitude than the other masses, and we see that
correspondingly the fields in the capping voids of the void-channel model are
weak. The non-zero field within capping voids indicates that the chiral
symmetry is not perfectly restored. This could be improved by increasing the
size of the capping voids, however this does not seem necessary as the chiral
symmetry violation is weak enough that the squared frequencies of the edge
states remain well centered within the band gap.

###  3.2 Flat edge states in the honeycomb-kagome lattice

Having established that the chiral symmetry of the mass-spring/void-channel
networks can be restored by capping the interfaces with sufficiently heavy
masses/large voids, we now turn to a more complex case of a square-root
semimetal where the topology is protected by the chiral symmetry of the
honeycomb sector of the squared Hamiltonian [21, 40]. We shall see that
despite the differences between the mass-spring/void-channel models and the
original tight-binding model, capping the interfaces again allows the
protecting symmetry to be restored and for the topological edge states to be
observed.

####  3.2.1 Tight-binding model

Figure 6: (a) Schematic of the tight-binding Hamiltonian of the honeycomb-
kagome lattice, $\hat{H}^{\text{hk}}$, with nearest-neighbour hopping
parameter $t$, as introduced by [40]. The honeycomb and kagome sites are shown
in blue and red, respectively, and $\vec{a}_{1}$ and $\vec{a}_{2}$ are the
lattice parameters. (b) The energy spectrum of $\hat{H}^{\text{hk}}$ is
symmetric about $E=0$ because of the chiral symmetry. The flat band and Dirac
crossings at K are reminiscent of the nearest-neighbour tight-binding models
on the honeycomb and kagome lattices, $\hat{H}^{\mathrm{h}}$ and
$\hat{H}^{\mathrm{k}}$, respectively. This is because
$\left(\hat{H}^{\text{hk}}\right)^{2}$ is block-diagonal, with the blocks
proportional to the energy spectra of $\hat{H}^{\mathrm{h}}$ and
$\hat{H}^{\mathrm{k}}$, up to a constant shift of energy, as shown in (c‑d)
for the honeycomb and kagome sectors of
$\left(\hat{H}^{\text{hk}}\right)^{2}$, respectively. The energy spectrum of
$\hat{H}^{\text{hk}}$ therefore inherits features, including topology, from
$\hat{H}^{\mathrm{h}}$ and $\hat{H}^{\mathrm{k}}$. In particular,
$\hat{H}^{\mathrm{h}}$ is a topological semimetal, and $\hat{H}^{\text{hk}}$
is therefore known as a square-root topological semimetal [21, 40].

Before we introduce our mass-spring and void-channel models, let us review the
tight-binding model introduced by Mizoguchi _et al_ [40] and explain its
topological origins. Figure 6(a) shows the nearest-neighbour tight-binding
model of the honeycomb-kagome lattice, also known as the decorated honeycomb
lattice. The Hamiltonian has a block off-diagonal form [40],

$\underline{\underline{H}}_{\vec{k}}^{\text{hk}}\left[\begin{array}[]{c}u_{1}\\\
\vdots\\\
u_{5}\end{array}\right]=\left[\begin{array}[]{cc}\underline{\underline{0}}_{2\times
2}&t\underline{\underline{\Psi}}_{\vec{k}}^{\dagger}\\\
t\underline{\underline{\Psi}}_{\vec{k}}&\underline{\underline{0}}_{3\times
3}\end{array}\right]\left[\begin{array}[]{c}u_{1}\\\ \vdots\\\
u_{5}\end{array}\right],$ (18)

where $u_{1}$ and $u_{2}$ are amplitudes at the honeycomb sites, $u_{3}$,
$u_{4}$, and $u_{5}$ are amplitudes at the kagome sites, $t$ is the hopping
strength, $\underline{\underline{0}}_{n\times m}$ is an $n\times m$ matrix of
zeros, and

$\underline{\underline{\Psi}}_{\vec{k}}=\left[\begin{array}[]{cc}1&1\\\
e^{i\vec{k}\cdot\vec{a}_{1}}&1\\\
e^{i\vec{k}\cdot\vec{a}_{2}}&1\end{array}\right].$ (19)

As there are only hoppings between sites belonging to _different_ sublattices,
this tight-binding model is chiral symmetric. The unitary chiral symmetry
operator is

$\underline{\underline{\Gamma}}=\left[\begin{array}[]{cc}\underline{\underline{I}}_{2}&\underline{\underline{0}}_{2\times
3}\\\ \underline{\underline{0}}_{3\times
2}&-\underline{\underline{I}}_{3}\end{array}\right],$ (20)

where $\underline{\underline{I}}_{n}$ is an $n\times n$ identity matrix.

The energy bands of the tight-binding model, equation (18), are shown in
figure 6(b). The spectrum is symmetric about $E=0$ due to the chiral symmetry.
Curiously, the spectrum contains features of both the underlying honeycomb and
kagome lattices, such as the symmetry-protected Dirac cones at K [41] and the
flat band [42]. This is because equation (18) belongs to a class of
Hamiltonians known as square-root Hamiltonians, meaning that the square of
equation (18) is block diagonal,

$\left(\underline{\underline{H}}_{\vec{k}}^{\text{hk}}\right)^{2}=\left[\begin{array}[]{cc}t^{2}\underline{\underline{\Psi}}_{\vec{k}}^{\dagger}\underline{\underline{\Psi}}_{\vec{k}}&\underline{\underline{0}}_{2\times
3}\\\ \underline{\underline{0}}_{3\times
2}&t^{2}\underline{\underline{\Psi}}_{\vec{k}}\underline{\underline{\Psi}}_{\vec{k}}^{\dagger}\end{array}\right]=\left[\begin{array}[]{cc}\underline{\underline{H}}_{\vec{k}}^{\mathrm{h}}&\underline{\underline{0}}_{2\times
3}\\\ \underline{\underline{0}}_{3\times
2}&\underline{\underline{H}}_{\vec{k}}^{\mathrm{k}}\end{array}\right],$ (21)

where

$\underline{\underline{H}}_{\vec{k}}^{\mathrm{h}}=\left[\begin{array}[]{cc}3t^{2}&(1+e^{+i\vec{k}\cdot\vec{a}_{1}}+e^{+i\vec{k}\cdot\vec{a}_{2}})t^{2}\\\
(1+e^{-i\vec{k}\cdot\vec{a}_{1}}+e^{-i\vec{k}\cdot\vec{a}_{2}})t^{2}&3t^{2}\end{array}\right]$
(22)

is the tight-binding Hamiltonian of a honeycomb lattice with nearest-neighbour
hopping strength $t^{2}$ and on-site potential $3t^{2}$, and

$\underline{\underline{H}}_{\vec{k}}^{\mathrm{k}}=\left[\begin{array}[]{ccc}2t^{2}&(1+e^{-i\vec{k}\cdot\vec{a}_{1}})t^{2}&(1+e^{-i\vec{k}\cdot\vec{a}_{2}})t^{2}\\\
(1+e^{+i\vec{k}\cdot\vec{a}_{1}})t^{2}&2t^{2}&(1+e^{-i\vec{k}(\vec{a}_{2}-\vec{a}_{1})})t^{2}\\\
(1+e^{+i\vec{k}\cdot\vec{a}_{2}})t^{2}&(1+e^{+i\vec{k}(\vec{a}_{2}-\vec{a}_{1})})t^{2}&2t^{2}\end{array}\right]$
(23)

is the tight-binding Hamiltonian of a kagome lattice with nearest-neighbour
hopping strength $t^{2}$ and on-site potential $2t^{2}$ [40]. The squared
energy spectrum of the honeycomb and kagome sectors, of equation (21), are
shown in figures 6(c‑d), respectively.

Arkinstall _et al_ [21] introduced a class of topological materials whose non-
trivial topology is inherited from the squared Hamiltonian. They named these
materials square-root topological insulators. The nearest-neighbour tight-
binding model of the honeycomb lattice is a topological semi-metal. The
honeycomb-kagome lattice is therefore a square-root topological semi-metal,
with the non-trivial topology inherited from the honeycomb sector of the
squared Hamiltonian [40, 43]. In the following sections, we introduce mass-
spring and void-channel analogues of the square-root topological semimetal on
the honeycomb lattice and study the symmetry protected edge states.

####  3.2.2 Mass-spring and void-channel models

Figure 7: Mass-spring and void-channel models of a honeycomb-kagome lattice
with nearest-neighbour coupling. (a) In the mass-spring model, the masses at
honeycomb sites ($m_{h}$, blue) and kagome sites ($m_{k}$, red) are connected
by springs of equal spring constants $k$. (b) In the void-channel model, voids
and channels are formed between flower-shaped perfectly conducting particles
arranged on a triangular lattice of lattice parameter $a$. The flower shapes
consist of six cylinders of radius $r$ arranged in a ring of radius $R$. The
voids at honeycomb the honeycomb sites have area $A_{h}$ (blue voids) and the
voids at kagome sites have area $A_{h}$ (red voids). The channels have equal
half-widths $h$. (c) Bulk frequency bands of the mass-spring model (black
points, $m_{h}=0.01104m_{0}$, $m_{k}=0.00736m_{0}$, $k=0.01978k_{0}$) and the
void-channel model (red points, $A_{h}=0.01104a^{2}$, $A_{k}=0.00736a^{2}$,
$2h=0.001a$). The frequencies are normalised by
$\omega_{0}=\sqrt{k_{0}/m_{0}}$ for the mass-spring model and $\omega_{0}=2\pi
c_{0}/a$ for the void-channel model. There is good agreement between the two
models, particularly at the lower frequencies. We chose the masses and areas
such that $m_{h}/m_{k}=A_{h}/A_{h}=3/2$ in order that the models resemble the
tight-binding model of figure 6(c) but with a shift of frequency, as discussed
in the main text.

Figure 7(a) shows a mass-spring model of the honeycomb-kagome lattice where
the honeycomb masses, $m_{h}$, and kagome masses, $m_{k}$, are connected by
springs of equal spring constant $k$,

$\left[\begin{array}[]{cc}\frac{3k}{m_{h}}\underline{\underline{I}}_{2\times
2}&-\frac{k}{m_{h}}\underline{\underline{\Psi}}_{\vec{k}}^{\dagger}\\\
-\frac{k}{m_{k}}\underline{\underline{\Psi}}_{\vec{k}}&\frac{2k}{m_{k}}\underline{\underline{I}}_{3\times
3}\end{array}\right]\left[\begin{array}[]{c}u_{1}\\\ u_{2}\\\ u_{3}\\\
u_{4}\\\
u_{5}\end{array}\right]=\omega^{2}(\vec{k})\left[\begin{array}[]{c}u_{1}\\\
u_{2}\\\ u_{3}\\\ u_{4}\\\ u_{5}\end{array}\right].$ (24)

First, note that the unsquared equations have eigenvalue $\omega^{2}$, and the
squared equations would have eigenvalue $\omega^{4}$. Next, we note that in
the mass-spring model the block-diagonal terms are non-zero and the two off-
diagonal block matrices are scaled by different factors, namely $m_{h}$ and
$m_{k}$. In a recent study of tight-binding and mass-spring honeycomb-kagome
lattices, Mizoguchi _et al_ [40] reproduced the tight-binding model by letting
$m_{h}=m_{k}$ and setting the block-diagonal of the matrix to zero by adding a
gravitional potential term in which the masses roll around in dents on a
floor. Our interest is in mapping the mass-spring models to void-channel
networks but no analogue of these dents for the void-channel network was
apparent to us. Regardless, we shall demonstrate the squared tight-binding and
mass-spring models are analagous. First, we decompose the unsquared mass-
spring matrix equation as

$\left[\begin{array}[]{cc}\frac{3k}{m_{h}}\underline{\underline{I}}_{2\times
2}&-\frac{k}{m_{h}}\underline{\underline{\Psi}}_{\vec{k}}^{\dagger}\\\
-\frac{k}{m_{k}}\underline{\underline{\Psi}}_{\vec{k}}&\frac{2k}{m_{k}}\underline{\underline{I}}_{3\times
3}\end{array}\right]=\frac{\alpha k}{m_{0}}\underline{\underline{I}}_{5\times
5}+\left[\begin{array}[]{cc}+\frac{\beta
k}{m_{0}}\underline{\underline{I}}_{2\times
2}&-\frac{k}{m_{h}}\underline{\underline{\Psi}}_{\vec{k}}^{\dagger}\\\
-\frac{k}{m_{k}}\underline{\underline{\Psi}}_{\vec{k}}&-\frac{\beta
k}{m_{0}}\underline{\underline{I}}_{3\times 3}\end{array}\right],$ (25)

where

$\displaystyle\alpha$ $\displaystyle=\frac{3/m_{h}+2/m_{k}}{2}m_{0},$ (26)
$\displaystyle\beta$ $\displaystyle=\frac{3/m_{h}-2/m_{k}}{2}m_{0}.$ (27)

Taking the $\alpha k/m_{0}$ term to the right hand side of the equations of
motion, we obtain

$\left[\begin{array}[]{cc}+\frac{\beta
k}{m_{0}}\underline{\underline{I}}_{2\times
2}&-\frac{k}{m_{h}}\underline{\underline{\Psi}}_{\vec{k}}^{\dagger}\\\
-\frac{k}{m_{k}}\underline{\underline{\Psi}}_{\vec{k}}&-\frac{\beta
k}{m_{0}}\underline{\underline{I}}_{3\times
3}\end{array}\right]\left[\begin{array}[]{c}u_{1}\\\ \vdots\\\
u_{5}\end{array}\right]=\left(\omega^{2}(\vec{k})-\frac{\alpha
k}{m_{0}}\right)\left[\begin{array}[]{c}u_{1}\\\ \vdots\\\
u_{5}\end{array}\right].$ (28)

Note that the equations of motion are only chiral symmetric about
$\omega^{2}=\alpha k/m_{0}$ if we choose $2m_{h}=3m_{k}$ such that $\beta=0$.

The matrix of equation (28) squares to

$\displaystyle\left[\begin{array}[]{cc}+\frac{\beta
k}{m_{0}}\underline{\underline{I}}_{2\times
2}&-\frac{k}{m_{h}}\underline{\underline{\Psi}}_{\vec{k}}^{\dagger}\\\
-\frac{k}{m_{k}}\underline{\underline{\Psi}}_{\vec{k}}&-\frac{\beta
k}{m_{0}}\underline{\underline{I}}_{3\times 3}\end{array}\right]^{2}$
$\displaystyle=\left(\frac{\beta
k}{m_{0}}\right)^{2}\underline{\underline{I}}+\left[\begin{array}[]{cc}\frac{k^{2}}{m_{k}m_{h}}\underline{\underline{\Psi}}_{\vec{k}}^{\dagger}\underline{\underline{\Psi}}_{\vec{k}}&\underline{\underline{0}}_{2\times
3}\\\ \underline{\underline{0}}_{3\times
2}&\frac{k^{2}}{m_{k}m_{h}}\underline{\underline{\Psi}}_{\vec{k}}\underline{\underline{\Psi}}_{\vec{k}}^{\dagger}\end{array}\right],$
(33)

such that the ensuing squared equations of motion are

$\left[\begin{array}[]{cc}\frac{k^{2}}{m_{k}m_{h}}\underline{\underline{\Psi}}_{\vec{k}}^{\dagger}\underline{\underline{\Psi}}_{\vec{k}}&\underline{\underline{0}}_{2\times
3}\\\ \underline{\underline{0}}_{3\times
2}&\frac{k^{2}}{m_{k}m_{h}}\underline{\underline{\Psi}}_{\vec{k}}\underline{\underline{\Psi}}_{\vec{k}}^{\dagger}\end{array}\right]\left[\begin{array}[]{c}u_{1}\\\
\vdots\\\ u_{5}\end{array}\right]=\left[\left(\omega^{2}(\vec{k})-\frac{\alpha
k}{m_{0}}\right)^{2}-\left(\frac{\beta
k}{m_{0}}\right)^{2}\right]\left[\begin{array}[]{c}u_{1}\\\ \vdots\\\
u_{5}\end{array}\right],$ (34)

which is analagous to the squared tight-binding equation (21) with
$t^{2}\leftrightarrow k^{2}/(m_{k}m_{h})$ and
$E^{2}\leftrightarrow\left(\omega^{2}(\vec{k})-\frac{\alpha
k}{m_{0}}\right)^{2}-\left(\frac{\beta k}{m_{0}}\right)^{2}$. Note that for
the squared equations of motion, $\beta\neq 0$ simply corresponds to another
shift in frequency and does not break any symmetries of the squared equations.

Next, we propose a photonic analogue of this mass-spring network. Our network
of voids and channels is formed between “flower” shaped particles of perfect
conductors arranged on a triangular lattice with lattice parameter $a$; each
flower consists of six cylinders of radius $r$ that are distributed along a
ring of radius $R$ (see figure 7(b)). The voids at the honeycomb and kagome
sites (shown in blue and red, respectively) are connected by narrow channels
of half-width $h$, where

$R\cos\frac{\pi}{6}=\frac{a}{2}-r-h.$ (35)

We fixed the surface-to-surface gap as 2h=a/1000, this is similar to the ratio
we used for the SSH model in the previous section. While this is quite a small
ratio of $h/a$, we see in A that less extreme ratios of $h/a$ would also be
viable. For a given value of $r$, we can then determine
$R=(a/2-h-r)/\cos(\pi/6)$ and numerically calculate the areas of the honeycomb
and kagome voids, $A_{h}$ and $A_{k}$. We settled on $r=0.259a$, for which
$R=0.27771a$, $A_{h}=0.01104a^{2}$, and $A_{k}=0.00736a^{2}$, such that
$A_{h}/A_{k}=3/2$. As shown in figure 7(c), there is good agreement between
the system of voids and channels (red) and the discrete system of masses and
springs (black) where

$\displaystyle m_{h}$ $\displaystyle=A_{h}\cdot m_{0}/a^{2}=0.01104m_{0},$
(36) $\displaystyle m_{k}$ $\displaystyle=A_{k}\cdot
m_{0}/a^{2}=0.00736m_{0},$ (37) $\displaystyle k$
$\displaystyle=\frac{1}{\pi}\sqrt{\frac{2h}{r}}=0.01978k_{0}.$ (38)

We also verify in A that the agreement between the mass-spring and void-
channel networks improves as the channels are made more narrow; this is in
line with expectations from the asymptotic model [26]. We have chosen these
particular parameters such that $A_{h}/A_{k}=m_{h}/m_{k}=3/2$ and therefore
$\beta=0$ in order that the mass-spring and void-channel models more closely
resemble the tight-binding model shown in figure 6. When we study the edge
states in the next section, we shall see that the topological edge states
persist even if $m_{h}/m_{k}\neq 3$ and $\beta\neq 0$.

Now that we have introduced our photonic geometry, let us compare and contrast
our work with some recent studies of the honeycomb-kagome lattice in photonics
and acoustics. Maimaiti _et al_ [28] studied the response of a triangular
array of metallic cylinders to microwave radiation. They noted that the voids
between the cylinders lie on a honeycomb-kagome lattice and they used Monte
Carlo methods to fit their model to a honeycomb-kagome tight-binding model.
Crucially, however, the cylinders were not closely spaced and the authors did
not consider any topological aspects of the array; it is likely that the
quality of the edge states in this system would be reduced by longer-ranged
coupling between voids and their next-nearest-neighbours. On the other hand,
Yan _et al_ [44] studied a honeycomb-kagome array of acoustic resonators
connected by narrow channels and considered the symmetry protected topology.
However, in their work the width of the channels were alternated to produce a
square-root topological insulator where the topology was inherited from the
breathing kagome sector of the squared Hamiltonian, whereas we study the
lattice with equal channel widths, which is akin to the mass-spring/tight-
binding models of Mizoguchi _et al_ [40], where the non-trivial topology is
inherited from the honeycomb sector of the squared Hamiltonian.

####  3.2.3 Edge states in a ribbon

Figure 8: (a) Schematic of a ribbon of the honeycomb-kagome void-channel
network introduced in figure 7, but terminated by slabs of perfect conductor
at the top and bottom. The large magenta voids at the boundary have area
$A_{\mathrm{cap}}$ and reduce the chiral symmetry breaking at the interfaces.
(b) Frequency bands of a ribbon that is $N_{\mathrm{cells}}=10$ unit cells
long. The frequencies are normalised by $\omega_{0}=\sqrt{k_{0}/m_{0}}$ for
the mass-spring model and $\omega_{0}=2\pi c_{0}/a$ for the void-channel
model. (c-e). Visualisations of the labelled eigenmodes in panel b. For each
shown here, this is also an energy degenerate inversion symmetric partner at
the other edge. (c) The lowest pair of bands are excitations in the large
voids, whereas (d-e) are topological edge states protected by the chiral
symmetry of the squared Hamiltonian.

In order to produce topological edge states, we must introduce interfaces in a
manner that preserves (i) the block-diagonal nature of the squared equations
and (ii) the chiral symmetry of the honeycomb sector of the squared equations
[40]. When we take the square of the tight-binding model with free boundary
conditions, the sites at the edge of the model gain a different onsite
potential when compared to the sites of the same sublattice, all be it, in the
bulk. In order to retain the chiral symmetry of the honeycomb lattice, we
impose that the kagome sites, in the tight-binding model, are located at the
edge of the model. The edges of the mass-spring model therefore consist of
kagome sites capped by heavy masses to emulate the free boundary condition.

Figure 8(b) shows a comparison between the void-channel model (red) and the
mass-spring model (black, with the same values of $k$, $m_{h}$, and $m_{k}$ as
before, and capped with masses of $M=A\cdot m_{0}/a^{2}=0.12247m_{0}$ at the
honeycomb sites along the interface). As before, there is good agreement
between the mass-spring model and void-channel models however the accuracy
decreases as the frequency increases. We observe several new edge states, that
were not present in the bulk eigenmodes; these are marked, in the dispersion
curves, by the red circles c, d and e in figure 8(b), and visualised in
figures 8(c-e), respectively.

We see from figure 8(c) that the lowest pair of bands correspond to
excitations within the large voids. As we increase the size of the capping
voids, these bands would flatten to zero frequency. On the other hand, figures
8(d-e) show the pair of topological edge states arising from the non-trivial
topology of the honeycomb sector of the squared equations [40].

If the squared system was exactly chiral symmetric then the topological edge
states should be flat [40]; instead there is a slight tilt indicating a weak
symmetry breaking. Interestingly, the field within the large voids is weaker
for the higher energy eigenmode in figure 8(e), suggesting that the chiral
symmetry of the squared system is better preserved at the higher frequencies.
This is because the frequency and character of the unwanted excitation in the
capping voids (see figure 8(c)) is more similar in character to the edge state
with lower frequency (figure 8(d), where honeycomb and kagome sites are in
phase) than the edge state with higher frequency (figure 8(e), where honeycomb
and kagome sites are out of phase). The unwanted mode therefore hybridises
more strongly with the lower frequency edge state.

Although we have chosen $m_{h}/m_{k}=3$, such that the mass-spring and void-
channel models more closely resemble the tight-binding model of figure 6(e),
we verify in C that the edge states persist even if $m_{h}/m_{k}\neq 3$ such
that $\beta\neq 0$. We also verify in B that the edge states are not protected
without the presence of the large capping masses/voids which restore the
chiral symmetry of the squared equations of motion.

####  3.2.4 Edge and corner states in a triangular metaparticle

Figure 9: (a) Frequency eigenspectrum of a triangular metaparticle built from
the mass-spring honeycomb-kagome network introduced in figure 7. The edges are
capped with extremely heavy masses ($M=1000m_{0}$, black points) or realistic
masses ($M=0.12247m_{0}$, as in figure 8). The frequencies are normalised by
$\omega_{0}=\sqrt{k_{0}/m_{0}}$. The upper left inset shows a schematic of the
triangular metaparticle with the heavy masses shown in magenta. The schematic
shows a particle with $N_{\mathrm{cells}}=7$ unit cells along each edge, but
$N_{\mathrm{cells}}=19$ was used in the calculations. The lower right inset
shows a zoom of the lower frequency set of edge and corner states. (b)‑(e)
show steady state fields, upon driving the system with a harmonic force at the
honeycomb site at the center of the highlighted blue region, at the
frequencies indicated in the lower right inset of panel a.

We now study corner and edge states in a large but finite “triangular
metaparticle” of the honeycomb-kagome lattice, as shown in the upper-left
inset of figure 9(a). Having established the validity of the mass-spring model
for the bulk and at the edges, we model the system using only the discrete
mass-spring equations as these are more accessible, far faster to solve and
still retain the crucial physics we are interested in. As with the ribbon, we
cap the ends with heavy masses at honeycomb sites to reduce the breaking of
the chiral symmetry in the honeycomb sector of the squared equations.

The main panel of figure 9(a) shows the energy spectrum of a triangular
metaparticle with $N_{\mathrm{cells}}=19$ unit cells along each edge for a
realistic capping mass ($M=0.12247m_{0}$, red) and for a very large capping
mass where the chiral symmetry of the honeycomb sector of the squared
equations is near-perfectly restored ($M=1000m_{0}$, black). We identify the
large flat region of eigenmodes at $\omega\approx 2.4\omega_{0}$ as the bulk
flat band inherited from the kagome lattice, and the smaller flat regions of
eigenmodes at $\omega\approx 1.25\omega_{0}$ and $\omega\approx 3.1\omega_{0}$
as the topological edge states inherited from the honeycomb lattice. The
lower-right inset of figure 9(a) shows the energy eigenmodes of the lower
frequency edge state in more detail. Although the edge state is extremely
flat, for the unrealistically large value of $M$, there is an advantage to
using a more realistic value of $M$ for which the protecting symmetry is
weakly broken.

Figure 10: Visualisation of the triply degenerate eigenmodes of the triangular
metaparticle for the mode indices (a) 248, (b) 249, (c) 250 of figure 9(a).
The fields reveal that these eigenmodes are corner states. The modes are
degenerate because of the $\mathrm{C}_{3}$ symmetry of the triangle; the
eigensolver has therefore returned arbitrary linear superpositions of the
three corner eigenmodes.

Figures 9(b-e) show the steady-state solutions of the triangular mass-spring
metaparticles capped by realistic masses and driven by time-harmonic forces,
centered at the honeycomb sites highlighted in light blue, for the four
frequencies labelled in the lower-right inset of figure 9(a). Note that we
forced the system at frequencies just below the resonances because the energy
of the closed mass-spring system solution diverges if we drive exactly at a
resonant frequency. In figure 9(b) the energy propagates freely through the
particle. This is because the two eigenmodes that are closest to the driving
frequency are actually bulk eigenmodes corresponding to the Dirac cones at
$\mathrm{K}$ and $-\mathrm{K}$ of figure 7(c). In figure 9(c-d) we see that
the energy propagates around the edge of the particle but not into the bulk.
As the energy increases, the modes become more localised to the edges. Figure
9(e) shows that the field is localised in all directions, when driving at the
edge of the triangle, at the frequency residing slightly below the group of
triply degenerate eigenstates. This is because these eigenstates are corner
eigenstates, as shown in figure 10. Crucially, the weak breaking of the chiral
symmetry of the squared equations has lifted the degeneracies between the bulk
states and the edge/corner states, allowing these to be excited at different
frequencies.

##  4 Conclusions

In this paper we have shown that networks of voids and narrow connecting
channels between perfect conductors are a promising platform for mimicking
chiral or square-root topological tight-binding models within photonics. This
was done by mapping the tight-binding models to mass-spring models, and then
mapping these mass-spring models to their asymptotically exact continuum
analogue [26, 27], comprised of void-channel networks.

We found that although introducing interfaces to the mass-spring/void-channel
networks could break the symmetries that protected the topological edge
states, these symmetries could be restored by capping the interfaces with
heavy masses/large voids. We were able to create a photonic analogue of the 1D
SSH model [20] with a chain of equally sized voids connected by channels of
alternating widths, and a photonic analogue of a square-root topological
semimetal [21, 40] with voids positioned on a honeycomb-kagome lattice and
narrow channels connecting the nearest-neighbour voids.

More broadly, we hope that the asymptotic network approximations espoused here
will provide a direct mapping to other complex photonic crystal phenomena,
including and beyond topological physics. Discrete models are able to
encompass highly non-trivial phenomenology and hence our approach provides a
systematic and simplified route to engineer exotic responses in continuum
photonic structures in an asymptotically exact manner.

##  5 Acknowledgements

S.J.P. acknowledges his studentship from the Centre for Doctoral Training on
Theory and Simulation of Materials at Imperial College London funded by EPSRC
Grant Number EP/L015579/1. The support of the UK EPSRC through grants
EP/L024926/1 is acknowledged by RVC and MM as is that of the ERC H2020 FETOpen
project BOHEME under grant agreement No. 863179.

##  Appendix A Validity of the mass-spring networks as an analogue of the
void-channel networks

###  A.1 Convergence with decreasing gap size

Figure 11: Normalised frequency of the flat band of honeycomb-kagome networks
of masses and springs (black line) and the equivalent networks of voids and
channels (red points). The agreement between the two models increases as the
gap size is decreased.

Figure 11 shows the frequency of the flat band as a function of gap size in
the honeycomb-kagome network of masses and springs (black line) and voids and
channels (red points) that were originally introduced in figure 7. The
agreement between the two models increases as the gap size decreases, as
expected from the asymptotic analysis of Vanel _et al_ [26, 27].

###  A.2 Operating frequencies and length scales

Let us consider the feasibility of manufacturing the honeycomb-kagome network
of voids and channels, and the frequencies and length scales at which it could
operate. We must find a balance between the size of the gaps, the size of the
particles, the frequencies at which the edge states occur, and the frequencies
at which the metals may be treated as perfect conductors.

For example, let us consider the parameters required to obtain edge states at
$\omega=$1\text{\,}\mathrm{THz}$$. The lower frequency edge states have a
normalised frequency of $\omega/\omega_{0}\approx 1.3$, where $\omega_{0}=2\pi
c_{0}/a$ and $c_{0}$ is the speed of light in vacuum, corresponding to a
lattice parameter of $a\approx 1.3\cdot 2\pi
c_{0}/\omega=$2.4\text{\,}\mathrm{mm}$$. This would correspond to channel
half-widths of $h=a/2000=$1.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ using
the ratio from earlier, although we have seen in the previous section that
this could be relaxed somewhat without the mapping between the void-channel
and mass-spring networks breaking down. Both $a$ and $h$ are orders of
magnitude greater than the skin depth of gold which is on the order of
$50\text{\,}\mathrm{nm}$ for $\omega=$1\text{\,}\mathrm{THz}$$, and it would
therefore be reasonable to treat the gold particles as perfectly conducting.

##  Appendix B No edge states in mass-spring/void-channel networks with free
boundary conditions

Figure 12: (a) Schematic of the same void-channel chain as in figure 5 but
without the large capping voids. (b) There is good agreement between the
squared frequency spectrum of the void-channel chain (red crosses) and the
corresponding mass-spring chain with free boundary conditions (black points).
The frequencies are normalised by $\omega_{0}=\sqrt{k_{0}/m_{0}}$ for the
mass-spring model and $\omega_{0}=2\pi c_{0}/L$ for the void-channel model.
Without the large capping voids/heavy capping masses chiral symmetry is broken
at the edges of these chains and there are no topological edge states in the
band gap.

We show in figure 12(a) the same SSH-like chain, as in figure 5, but without
the capping voids. Figure 12(b) shows the squared frequency spectrum of the
void-channel model (red crosses) and the corresponding mass-spring model with
free boundary conditions (black points). As expected, there are no edge states
because the chiral symmetry is strongly broken at the ends of the chains.
Similarly, we verify in figure 13 that the topological edge states are not
present in the ribbon of the honeycomb-kagome lattice without heavy capping
masses/large capping voids.

Figure 13: Edge states of the ribbon of honeycomb-kagome mass-spring model
introduced in figure 8 with $2m_{h}=3m_{k}$ for (a) the ideal ‘wall’ boundary
condition on the kagome sites (infinitely heavy capping masses at the
honeycomb sites) and (b) the free boundary condition on the kagome sites (no
capping masses). The edge states in (b) are not pinned to a particular energy
because the honeycomb sector of the squared equations of motion is not chiral
symmetric.

##  Appendix C Chiral symmetry of the squared honeycomb-kagome lattice for
$m_{h}\neq m_{k}$

Figure 14: The edge states of the square-root semimetal are protected by the
chiral symmetry of the squared equations, and may survive even as the chiral
symmetry of the unsquared equations is broken. We plot the frequencies
(normalised by $\omega=\sqrt{k_{0}/m_{0}}$) of the mass-spring model of the
mass-spring ribbon from figure 8 for (a) $2m_{h}<3m_{k}$, (b) $2m_{h}=3m_{k}$,
and (c) $2m_{h}>3m_{k}$, where $m_{h}$ and $m_{k}$ are the masses at the
honeycomb and kagome sites, respectively. The edge states are robust in all
three systems even though the unsquared equations are not chiral symmetric for
$2m_{h}\neq 3m_{h}$. This is because the topological edge states are protected
by the chiral symmetry of the honeycomb sector of the squared equations [40],
which can be preserved even as the chiral symmetry of the unsquared equations
is lost.

In figure 14 we plot the frequency bands of the mass-spring honeycomb-kagome
network with perfect wall boundary conditions, but relax the constraint
$2m_{h}=3m_{k}$ such that $\beta\neq 0$ where $\beta$ is defined in equation
(27)). This breaks the chiral symmetry of the unsquared equations, yet the
edge states remain flat and robust against this perturbation, because the
honeycomb sector of the _squared_ equations remains chiral symmetric.

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