Technology Innovation Institute, Masdar City 9639, Abu Dhabi, United Arab
Emirates Centre of Excellence ENSEMBLE3 sp. z o.o., Wolczynska 133, Warsaw,
01-919, Poland

# Long-range molecular energy transfer
mediated by strong coupling to
plasmonic topological edge states

Álvaro Buendía  Instituto de Estructura de la Materia, Consejo Superior de
Investigaciones Científicas, Serrano 121, 28006 Madrid, Spain
<EMAIL_ADDRESS>Jose A. Sánchez-Gil  Instituto de Estructura de la Materia,
Consejo Superior de Investigaciones Científicas, Serrano 121, 28006 Madrid,
Spain  Vincenzo Giannini  Instituto de Estructura de la Materia, Consejo
Superior de Investigaciones Científicas, Serrano 121, 28006 Madrid, Spain
William L. Barnes  Department of Physics and Astronomy, University of Exeter,
Stocker Road, Devon, EX4 4QL, UK  Marie S. Rider  Department of Physics and
Astronomy, University of Exeter, Stocker Road, Devon, EX4 4QL, UK
<EMAIL_ADDRESS>
###### Abstract

Strong coupling between light and molecular matter is currently attracting
interest both in chemistry and physics, in the fast-growing field of molecular
polaritonics. The large near-field enhancement of the electric field of
plasmonic surfaces and their high tunability make arrays of metallic
nanoparticles an interesting platform to achieve and control strong coupling.
Two dimensional plasmonic arrays with several nanoparticles per unit cell and
crystalline symmetries can host topological edge and corner states. Here we
explore the coupling of molecular materials to these edge states using a
coupled-dipole framework including long-range interactions. We study both the
weak and strong coupling regimes and demonstrate that coupling to topological
edge states can be employed to enhance highly-directional long-range energy
transfer between molecules.

##  1 Introduction

In recent years, the strong interaction of light and molecular materials has
attracted a growing wave of interest, due to possible applications in
photonics, photo-chemistry and molecular quantum optics 1, 2, 3, 4, 5, 6, 7,
8. Molecules and photonic modes are in the strong coupling regime when their
interaction strength exceeds their individual losses. This leads to a
hybridization of molecule and photon, producing polaritons, which are part-
matter, part-light. Early work focused on closed cavities as the source of
confined photonic modes, while more recent work has highlighted the merits of
open photonic systems such as planar surfaces, gratings and nanoparticle
arrays. In systems such as these, the molecular material is directly
accessible - crucial for direct measurement - whilst supporting photonic modes
that are generally dispersive or highly delocalised, which can be exploited
for enhancing energy transfer between molecules strongly coupled to the
photonic modes 9, 10, 11, 12, 13, 14.

In this work we focus on topological arrays of plasmonic nanoparticles 15, 16;
many studies have explored the strong coupling of molecules to plasmonic
modes, for example in single nanoparticles 17, 18, cavities 19, 20, gratings
21, surface lattice modes 22, chains of nanoparticles 23, 24, 25, and single
topological plasmonic nanoparticles 26. However, the combination of
topological states and collective strong coupling has yet to be explored to
its full potential, although the coupling of a single quantum emitter to a
topological photonic mode has been successfully demonstrated 27. Here we
present a theoretical study of molecules coupled to the topological edge
states of a metallic nanoparticle array, with the aim of enhancing and
protecting directional molecular energy transport.

Using a coupled-dipole framework 21, 28, we have developed the tools to study
strong coupling between molecules and arrays of nanoparticles. This model can
be applied to study any system which can be described as an heterogeneous
array of coupled dipoles.

In Section 2, we study the dispersion bands and optical properties of the 2D
Su-Schrieffer-Heeger (SSH) plasmonic array and its edge states. In Section
3.1, we calculate the molecular energy transfer enhancement in the weak
coupling regime. In Section 3.2 we study the strong coupling between molecules
and edge states and strong-coupling enhanced molecular energy transfer. We
demonstrate this by considering molecules as individual dipoles in the
coupled-dipole model, and verify this result by also using a continuum model
in which the molecular material is modelled as an effective medium.

##  2 Topological edge states in a finite array of plasmonic nanoparticles

The simplest model showing topological edge states is the Su-Schrieffer-Heeger
(SSH) model, initially introduced to explain the physics of polyacetalyene 29.
The SSH model is a tight binding 1D-model with two staggered nearest-
neighbours hoppings, $t_{1}$ and $t_{2}$, which represent the single and
double bonds in the polymer. When terminated at the strong bond, the system
hosts 0D edge states protected by the chiral symmetry of the lattice. The SSH
model is applicable to a wide range of analogue systems, from radio-frequency
metamaterials 30 to plasmonics. Furthermore, the plasmonic analogue of the SSH
model 31 offers much complexity and richness, due to non-Hermiticity, all-to-
all interactions 32 and tunable polarization 33.

The SSH2D model (see scheme in Fig. 1(a)) is the two-dimensional extension of
the SSH model. The particles are ordered in a square lattice, with alternating
distances $\beta d/2$ and $(2-\beta)d/2$ in the $x$ and $y$ directions, where
$d$ is the lattice parameter and $\beta$ is a dimensionless parameter that
ranges from 0 to 2. For $\beta>1$ the array is in the topologically non-
trivial phase and hosts both 1D (edge) and 0D (corner) states.

2D plasmonic arrays of nanoparticles with several particles per unit cell and
rotational symmetries have been shown to host topological edge and corner
states 34, 35, 36, 37, 21, which are robust to disorder and perturbations.
Topological corner states have recently been proposed for lasing applications
38, 37, while the quasi-1D character of the edge states makes them suitable
for energy transfer applications.

The edge states in the SSH2D model 39, 40, 41, 36, 42, 43, 44, 45, 46 are
protected by weak symmetries, namely the (generalized) sublattice symmetry and
rotational ($C_{4}$) and mirror symmetries - these do not offer robustness
against backscattering, however, the position in the unit cell and circular
polarization of the emitter can impose handedness in the transport 35.
Coupling achiral plasmonic modes to chiral molecules can also lead to chiral
behaviour, as Ref. 47, 48 showed for surface lattice resonances.

When metallic nanoparticles are small and roughly spherical they can be
described as single electric dipoles,
$\textbf{p}=\overleftrightarrow{\alpha}(\omega)\textbf{E}$, where
$\overleftrightarrow{\alpha}(\omega)$ is the electric polarizability of the
nanoparticle and E is the incident electric field. For a single nanosphere,
the polarizability tensor acts like a scalar. In the quasi-static limit, i.e.
when the particle is small compared to the wavelength of the light
($a\ll\lambda$), the polarizability $\alpha_{\mathrm{QS}}(\omega)$ is:

$\alpha_{\mathrm{QS}}(\omega)=4\pi
a^{3}\epsilon_{0}\frac{\epsilon(\omega)-\epsilon_{\mathrm{B}}}{\epsilon(\omega)+2\epsilon_{\mathrm{B}}},$
(1)

where $\epsilon_{B}$ is the background permittivity and $\epsilon(\omega)$ the
electric permittivity of the metal, which in the Drude model is:

$\epsilon(\omega)=\epsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega(\omega+i\gamma)}.$
(2)

where $\epsilon_{\infty}$ is the static dielectric constant, $\omega_{p}$ is
the plasma resonance and $\gamma$ is the optical loss of the metal.

The quasi-static polarizability neglects damping, so we make the radiative
correction:

$\alpha(\omega)=\frac{\alpha_{\mathrm{QS}}(\omega)}{1-i\frac{k^{3}}{6\pi}\alpha_{\mathrm{QS}}(\omega)}.$
(3)

Metallic nanospheres show an optical resonance at the frequency
$\omega_{\mathrm{sp}}=\omega_{p}/\sqrt{2\epsilon_{B}+\epsilon_{\infty}}$,
known as localized surface plasmon resonance (LSPR).

When instead of a single metallic nanoparticle we employ an ensemble of them,
the plasmonic modes of the particles hybridize, leading to collective
plasmonic modes. When the distances between the particles are at least $3a$,
the dipolar approximation holds and the array can be described as a system of
coupled dipoles.

Coupled electric dipole models are routinely employed to study periodic arrays
of plasmonic nanoparticles 49, 50. However, edge states are not present in the
infinite periodic array as they arise from the breaking of translational
symmetry, and so it is necessary here to work with finite arrays. For an array
of $N$ nanoparticles, we have a system of $N$ coupled-dipole equations:

$\overleftrightarrow{\alpha}^{-1}(\omega)\textbf{p}_{n}=\frac{k^{2}}{\epsilon_{0}}\sum_{m\neq
n}\overleftrightarrow{\textbf{G}}(\omega,\textbf{r}_{n},\textbf{r}_{m})\textbf{p}_{m},$
(4)

where $\overleftrightarrow{\textbf{G}}(\omega,\textbf{r}_{n},\textbf{r}_{m})$
is the Green dyadic’s function, which mediates the interaction between two
dipoles. The Green dyadic function is given by:

$\begin{split}{\overleftrightarrow{{\bf
G}}}(\omega,\textbf{r}_{n},\textbf{r}_{m})&=\frac{e^{ikR}}{4\pi
R}\bigg{[}\left(1+\frac{i}{kR}-\frac{1}{k^{2}R^{2}}\right)\mathcal{I}-\left(1+\frac{3i}{kR}-\frac{3}{k^{2}R^{2}}\right)\frac{\textbf{R}\otimes\textbf{R}}{R^{2}}\bigg{]},\end{split}$
(5)

where $\mathcal{I}$ is the identity matrix,
$\textbf{R}=\textbf{r}_{m}-\textbf{r}_{n}$ and $R=|\textbf{R}|$.

In the near-field ($kR\ll 1$), ${\overleftrightarrow{{\bf
G}}}(\omega,\textbf{r},\textbf{r}_{0})$ can be approximated by the quasi-
static Green dyadic:

$\begin{split}{\overleftrightarrow{{\bf
G}}}_{\mathrm{QS}}(\omega,\textbf{r}_{n},\textbf{r}_{m})&=\frac{1}{4\pi
k^{2}R^{3}}\bigg{[}-\mathcal{I}+3\frac{\textbf{R}\otimes\textbf{R}}{R^{2}}\bigg{]}.\end{split}$
(6)

However, we will consider the full Green dyadic and long-range interactions in
the array and will simply use this approximation to understand near-field
coupling.

The coupled-dipole system of equations (Eq. (4)) can be simply rewritten in a
matricial way as:

$\lambda_{i}(\omega_{i})=0,$ (7)

where $\lambda_{i}$ is the $i$-th eigenvalue of
$\overleftrightarrow{\alpha}_{\mathrm{eff}}^{-1}(\omega)$, which is the
inverse of the effective polarizability of the array, whose $3\times 3$ blocks
are given by:

$\displaystyle[\overleftrightarrow{\alpha}_{\mathrm{eff}}(\omega)]^{-1}_{3n-2;3n;3m-2;3m}=\begin{cases}\overleftrightarrow{\alpha}^{-1}(\omega)&\mbox{if
}n=m,\\\
\frac{k^{2}}{\epsilon_{0}}\overleftrightarrow{\textbf{G}}(\omega,\textbf{r}_{n},\textbf{r}_{m})&\mbox{if
}n\neq m.\end{cases}$ (8)

The dipolar moments of the normal modes are given by
$\textbf{P}_{i}=\begin{pmatrix}\textbf{p}_{1}\dots,\textbf{p}_{N}\end{pmatrix}^{T}$,
which is the $i$-th eigenvector of
$\overleftrightarrow{\alpha}^{-1}_{\mathrm{eff}}(\omega_{i})$.

When periodicity is broken, the normal modes no longer have a defined momentum
k, so the dispersion bands $\omega(\textbf{k})$ are not defined. The usual way
to obtain the dispersion bands in systems with broken periodicity is the
supercell method 28, where a larger ancillary unit cell is used to recover
periodicity. However, long-range interactions in periodic arrays imply
infinite lattice sums, and for large unit cells this approach would be
computationally very consuming and so we take a different route. Instead, we
use the spatial dependence of the eigenmodes to infer the dependence on k. By
classifying the eigenmodes $\textbf{P}_{i}$ by symmetries (see Appendix A for
more information) and localization (bulk, edge and corner states), we obtain
effective dispersion bands for a finite array:

$\omega_{\mathrm{sym}}(\textbf{k})=\frac{\sum_{i=\\{i_{\mathrm{sym}}\\}}\omega_{i}|\textbf{P}_{\textbf{k},i}|^{2}}{\sum_{i=\\{i_{\mathrm{sym}}\\}}|\textbf{P}_{\textbf{k},i}|^{2}},$
(9)

where
$\textbf{P}_{\textbf{k},i}=\textbf{P}_{i}\cdot\begin{pmatrix}e^{i\textbf{k}\textbf{r}_{1}}\cdots
e^{i\textbf{k}\textbf{r}_{N}}\end{pmatrix}^{T}$.

In Fig. 1(b) we plot the dispersion bands for the out-of-plane
($\textbf{p}_{z}$) polarization. This corresponds to considering only $zz$
terms in previous tensorial equations. We choose out-of-plane polarizations
for the sake of simplicity, as $xx$ and $yy$ modes are hybridized, which
duplicates the number of bands to study. Additionally, the out-of-plane
polarization respects all the spatial symmetries of the SSH2D lattice, which
facilitates the existence of corner and edge states.

Figure 1: SSH2D array of metallic nanoparticles: (a) Scheme for a finite
$6\times 6$ SSH2D array. (b) Bulk ($B_{1}-B_{4}$, blue dotted lines), edge
($E_{1}-E_{4}$, green dotted lines) and corner ($C_{1}-C_{4}$, red dotted
lines) dispersion bands of an array of $30\times 30$ metallic nanoparticles,
with $a=10$ nm, $d=150~{}$nm, intra-cell distance $\beta d/2=0.8d$ and
$\hbar\omega_{\mathrm{sp}}=2.50$ eV and losses $\hbar\gamma=1$ meV, overlaid
on top of optical extinction $\sigma_{\mathrm{ext}}$. (c) Main edge state
($E_{1},E_{3}$) mode contributions at $\Gamma$, where all the dimers over a
single edge oscillate in-phase.

Out-of-plane bands can be spectrally isolated from in-plane bands using
elongated particles (e.g. spheroids) with the long axis oriented parallel to
$\textbf{u}_{z}$ 51. Here, for simplicity, we ignore the in-plane
contribution. We choose a SSH2D array of $15\times 15$ unit cells (i.e.
$30\times 30$ nanoparticles), with parameters $a=10$ nm, $d=150$ nm,
$\beta=1.6$ and $\hbar\omega_{\mathrm{sp}}=2.50eV$ with losses $\hbar\gamma=1$
meV. We consider reduced losses compared to silver or gold, as large losses
can make edge, bulk and corner states overlap spectrally. This approximation
allows for a clear analysis of the phenomena described in this work, but
should be noted when comparing to an experimental system using common metals.
We choose the largest $\beta$ within the dipolar approximation limits
$(2-\beta)d/2\geq 3a$ to maximize gaps and avoid this overlapping. Red, green
and blue lines represent, respectively, corner, edge and bulk bands. Light
lines are shown as white dashed lines.

In Fig. 1(b) the dispersion bands are overlaid on top of the optical
extinction, which is given by 35:

$\displaystyle\sigma_{\mathrm{ext}}(\omega,\textbf{k})=4\pi
k\Im\left(\frac{\hat{\textbf{E}}_{\mathrm{inc}}^{*}\overleftrightarrow{\alpha}_{\mathrm{eff}}(\omega)\hat{\textbf{E}}_{\mathrm{inc}}}{N|\textbf{E}_{0}|^{2}}\right),$
(10)

where $\hat{\textbf{E}}_{\mathrm{inc}}$ is the incoming electric field (a
plane wave) at the positions of the nanoparticles, i.e.
$\left[\hat{\textbf{E}}_{\mathrm{inc}}\right]_{3n-2,3n}=\textbf{E}_{0}e^{i(\textbf{k}\textbf{r}_{n}-\omega
t)}$. We consider both propagating and evanescent solutions with real or
imaginary $k_{z}=\sqrt{k^{2}-k_{x}^{2}-k_{y}^{2}}$ to probe the whole
reciprocal space. We see the upper bands are brighter and spectrally broader
than the lower frequency bands. This is due to the symmetries of the bands
(see Appendix A for more information), as dipoles within a single unit cell in
$B_{4}$ and $E_{3,4}$ states are in-phase, their emitted fields interfere
constructively in the far-field.

The plasmonic bands, specially the lower ones, are nearly flat. This is due to
long-range interactions and to the clustering of nanoparticles as
$\beta\rightarrow 2$, which become quadrumers in the bulk, dimers along the
edge, and monomers in the corners.

The edge states are doubly degenerate, and disperse similarly to the bulk
states of 1D SSH chains 42. For the upper gap edge states, the particles in
dimers oscillate in phase. These are symmetric or bright edge states. However
for the lower gap edge states, the particles in each dimer oscillate out of
phase. These correspond to antisymmetric or dark edge states. In Fig. 1(c) we
represent the main contribution at $\Gamma$ for a dark ($E_{1}$) and a bright
($E_{3}$) edge state, i.e. the modes where the dimers along a single edge
oscillate in phase.

The corner state bands are flat and quadruply-degenerate, and are fixed at
$\omega_{\mathrm{sp}}$ due to the generalized sublattice symmetry 43.

The notation of dark and bright bands refers to the extent to which a
particular mode will couple to a plane wave; dark modes can in fact be excited
and probed by near-field local coupling 52, despite their name. In our system
the dark bands are particularly of interest due to their flatness, as flat
bands can pose a large local density of states (LDOS). To derive the LDOS of
the metasurface 53, 52, first we introduce an effective Green function
$\overleftrightarrow{\textbf{G}}_{\mathrm{eff}}(\omega,\textbf{r},\textbf{r}_{0})$
that represents the interaction between two dipoles at position r and
$\textbf{r}_{0}$ via excitation of the array of plasmonic nanoparticles:

$\displaystyle\overleftrightarrow{\textbf{G}}_{\mathrm{eff}}(\omega,\textbf{r},\textbf{r}_{0})=\frac{k^{2}}{\epsilon_{0}}\sum_{i,j=1}^{N}\overleftrightarrow{\textbf{G}}(\omega,\textbf{r},\textbf{r}_{m})\left[\overleftrightarrow{\alpha}_{\mathrm{eff}}(\omega)\right]_{nm}\overleftrightarrow{\textbf{G}}(\omega,\textbf{r}_{n},\textbf{r}_{0}).$
(11)

The local density of states at position $\textbf{r}_{0}$, in terms of the
effective Green function, is then 53:

$\frac{\rho(\textbf{r}_{0},\omega)}{\rho_{0}}=1+\frac{2\pi}{k}\Im\bigl{\\{}\operatorname{Tr}\left[\overleftrightarrow{\textbf{G}}_{\mathrm{eff}}(\omega,\textbf{r}_{0},\textbf{r}_{0})\right]\bigr{\\}},$
(12)

where $\rho_{0}=\omega^{2}/\pi c^{3}$ is the density of electromagnetic modes
in blackbody radiation. In order to calculate the LDOS, we choose an out-of-
plane polarization for the probe too, as the near-field dipole-dipole
interaction (EQ. 6) between the probe and the $n$th nanoparticle will be
maximum when their dipoles are parallel and oriented in the direction of the
vector between their positions $\textbf{R}=\textbf{r}_{n}-\textbf{r}_{0}$. The
electric field hotspots, and therefore the maxima of LDOS, are expected to be
over nanoparticles, where $\textbf{R}=R\textbf{u}_{z}$.

This is another justification to focus on the out-of-plane modes of the
plasmonic array, as the hotspots of in-plane modes will be in-plane too
($z=0$), which make them more challenging to probe and couple to.

While the frequency of the probe will generally be fixed, the plasmonic array
resonance frequencies can be fine-tuned by changing the size, shape and
material of the nanoparticles and the lattice parameters. However, since we
are not trying to simulate the response of any specific real molecule, we fix
the plasmonic array parameters and vary the frequency of the probe
$\omega_{mol}$.

Figure 2: Local density of states and energy transfer enhancement via weak
coupling to dark and bright edge states: (a) Local density of states for a
$30\times 30$ array of metallic nanoparticles, with $a=10$ nm, $d=150$ nm,
intra-cell distance $\beta d/2=0.8d$, $\hbar\omega_{\mathrm{sp}}=2.50$ eV and
losses $\hbar\gamma=1$ meV. LDOS varies with $\omega$ for out-of-plane
polarization, and is given for three instances of $\mathbf{r}_{\mathrm{D}}$:
over the edge (green line), bulk (blue line), and corner (red line) at
$z=0.2d$. (b) Local density of states of the same array at the frequency of
the bright edge state $\omega=1.018\omega_{\mathrm{sp}}$. The local maxima of
the LDOS match with the spatial positions of the nanoparticles. The absolute
maxima are at the edge, as expected. (c) Square of the free-space Green
function of two molecules over adjacent edges (orange line) versus square of
the effective Green function (purple line), mediated by coupling to the
plasmonic array. Black line represents the enhancement of the energy transfer.
Energy transfer is enhanced for almost all frequencies, but especially near
the frequency of the bright edge state. (d) Energy transfer enhancement at the
same frequency for donor fixed at $\mathbf{r}_{\mathrm{D}}$ and mapping over
the position of acceptor $\mathbf{r}_{\mathrm{A}}$. The energy transfer
between donor and acceptor is maximally enhanced along the edge due to
coupling with the edge state.

In Fig. 2(a), red, green and blue lines represent the LDOS over a nanoparticle
at the corner, edge and bulk, at a height $z=0.2d$. We can see the peaks match
with the frequencies of the normal modes, plotted under the LDOS. We see the
maximum LDOS is at the frequency of the dark edge states. In panel (b) we plot
the LDOS at the frequency of the bright edge state
$(\omega_{mol}=1.018\omega_{sp})$, mapping the position of the probe
$\textbf{r}_{0}$ at constant height $z=0.2d$. We see the maxima of the LDOS
are over the positions of the nanoparticles on the edge of the array, as
expected.

##  3 Enhanced molecular energy transfer in the presence of topological edge
states

In this section, we discuss the effect of the topological plasmonic array on
the energy transfer between two molecules located near the array. Molecular
energy transfer mediated by the dipole-dipole interaction, known as Förster
resonance energy transfer (FRET), is proportional to the square of the Green’s
dyadic 54, i.e., at short distances, it decays like $R^{-6}$. In the medium
and long-range energy transfer decays like $R^{-4}$ and $R^{-2}$,
respectively, but in these medium and long-range regimes energy transfer is
not experimentally significant. To extend the range of the energy transfer for
molecular spectroscopy or the catalysis of chemical reactions, the molecules
can be coupled to plasmonic modes, due to their long-range nature 18.

In Section 3.1 we study the system of a single donor molecule and a single
acceptor molecule, spatially separated and located above the array. We
demonstrate that by choosing array parameters such that the donor and acceptor
molecules have resonant frequency commensurate with the topological edge
state, molecular energy transfer is both greatly enhanced with respect to
free-space energy transfer, and also highly directional along the edge of the
system. In Section 3.2 we move on to study the more realistic system of a
molecular ensemble on top of the nanoparticle array. We demonstrate that this
system may enter the strong coupling regime, in which the enhancement of
molecular energy transfer via coupling to a topological edge state may be
increased even further via the process known as polariton assisted resonance
energy transfer (PARET) 10.

###  3.1 Energy transfer enhancement via weak coupling between molecules and
topological edge states

From now on, we consider the molecules as electric dipoles. Interaction
involving higher multipoles are very short-ranged so the main contribution to
energy transfer will be the dipole-dipole contribution. The power transmitted
from the donor molecule dipole at position $\mathbf{r}_{\mathrm{D}}$ to an
acceptor molecule dipole at position $\mathbf{r}_{\mathrm{A}}$ is given by 54:

$P_{\mathrm{D}\rightarrow\mathrm{A}}=\frac{\omega}{2}\Im(\alpha_{\mathrm{mol}}(\omega))\Big{|}\textbf{u}_{\mathrm{A}}\cdot\textbf{E}_{\mathrm{D}}(\textbf{r}_{\mathrm{A}})\Big{|}^{2},$
(13)

where $\textbf{u}_{\mathrm{A}}$ is a unitary vector in the direction of the
acceptor dipole, $\alpha_{\mathrm{mol}}(\omega)$ is the polarizability of the
molecules, which we model as a Lorentzian centered at frequency
$\omega_{\mathrm{mol}}$ and width $\Gamma$:

$\alpha_{\mathrm{mol}}(\omega)=-\frac{6\pi\epsilon_{0}}{k^{3}}\frac{\Gamma/2}{(\omega-\omega_{\mathrm{mol}})-i\Gamma/2},$
(14)

and $\textbf{E}_{\mathrm{D}}(\mathbf{r}_{\mathrm{A}})$ is the electric field
created by the donor dipole at position $\textbf{r}_{\mathrm{A}}$:

$\displaystyle\mathbf{E}_{\mathrm{D}}(\mathbf{r}_{\mathrm{A}})$
$\displaystyle=\frac{k^{2}}{\epsilon_{0}\epsilon_{\mathrm{B}}}\overleftrightarrow{\mathbf{G}}(\omega,\mathbf{r}_{\mathrm{A}},\mathbf{r}_{\mathrm{D}})\mathbf{p}_{\mathrm{D}}.$
(15)

In the presence of the plasmonic array, the power transferred via excitation
of the array is given by:

$P_{\mathrm{D}\rightarrow\mathrm{A}}=\frac{\omega_{mol}}{2}\Im(\alpha_{\mathrm{mol}}(\omega_{mol}))\Big{|}\textbf{u}_{\mathrm{A}}\cdot\overleftrightarrow{\textbf{G}}_{\mathrm{eff}}(\omega_{mol},\textbf{r}_{\mathrm{A}},\textbf{r}_{\mathrm{D}})\textbf{p}_{\mathrm{D}}\Big{|}^{2}$
(16)

We define an enhancement factor of the power transmitted from donor to
acceptor by the presence of the array:

$\eta(\omega_{mol},\textbf{r}_{\mathrm{D}},\textbf{r}_{\mathrm{A}})=\frac{\Big{|}\mathbf{u}_{\mathrm{A}}\cdot\overleftrightarrow{\mathbf{G}}_{\mathrm{eff}}(\omega_{\mathrm{mol}},\mathbf{r_{\mathrm{A}}},\mathbf{r}_{\mathrm{D}})\mathbf{u}_{\mathrm{D}}\Big{|}^{2}}{\Big{|}\mathbf{u}_{\mathrm{A}}\cdot\overleftrightarrow{\mathbf{G}}(\omega_{\mathrm{mol}},\mathbf{r}_{\mathrm{A}},\mathbf{r}_{\mathrm{D}})\mathbf{u}_{\mathrm{D}}\Big{|}^{2}},$
(17)

where $\textbf{u}_{D}$ is a unitary vector in the direction of the donor
dipole. In Fig. 2(c) we plot the square of the effective Green function via
the array (purple line) versus the square of the free-space dipole-dipole
Green function (orange line) for the donor and acceptor molecules positioned
over adjacent edges. We also plot the enhancement factor, which is the
quotient of the two. We see that energy transfer is enhanced for the entirety
of the interval $\omega_{mol}\in(0.96\omega_{sp},1.06\omega_{sp})$, but that
enhancement is maximum when the molecular resonance is close to the
frequencies of the edge states. There are two dips in the energy transfer
enhancement around $0.995\omega_{sp}$ and $1.035\omega_{sp}$, matching with
resonance frequencies of bulk states, which might prevent coupling to edge
states.

Even though the dark edge states exhibit a larger LDOS, the energy transfer
enhancement via the bright edge states is larger by almost a factor $10^{2}$.
This may be due to destructive interference in the electric field radiated by
the dark state. However, the peak in the energy transfer for the dark edge
state is narrower, as dark states are lower in loss and less screened by
nearby bulk and corner states.

In Fig. 2(d) we map the energy transfer enhancement at the frequency of the
bright edge state $(\omega=1.018\omega_{\mathrm{sp}})$. We fix the position of
the donor over a hotspot, or a maximum of the LDOS, and vary the position of
the acceptor. We see that the energy transfer is strongly enhanced along the
edge.

###  3.2 Energy transfer enhancement via strong coupling between molecules and
topological edge states

In a realistic molecule-nanophotonic system, the molecular material will
generally contain a great number of molecules, $N_{\mathrm{mol}}$, instead of
the simple picture of two isolated donor and acceptor molecules. In fact, it
is the behaviour of molecular ensembles with $N_{\mathrm{mol}}$ large enough
to enter the strong coupling regime that are at the centre of the burgeoning
fields of molecular polaritonics and polaritonic chemistry 55, 5. While in
Section 3.1 we demonstrated the ability of topological edge states to enhance
molecular energy transfer via weak coupling, we now describe two complementary
methods to study a system of molecules strongly coupled to modes of the
topological array.

When a plasmonic mode couples to an emitter, two hybridized bands form, lower
($\omega_{\mathrm{{LP}}}<\omega_{\mathrm{mol}}$) and upper
($\omega_{\mathrm{UP}}>\omega_{\mathrm{mol}}$) polaritons, separated by a
spectral gap ($\Omega=\omega_{\mathrm{UP}}-\omega_{\mathrm{LP}}$) which is
known as Rabi splitting. When this gap exceeds the plasmonic and molecular
losses of the original uncoupled modes, the system is strongly coupled.

We first demonstrate that strong coupling to a topological edge state can be
achieved by studying a moderate number of molecules placed in the field maxima
above the nanoparticles at the system edge. Using this model, we demonstrate
strong coupling to a topological edge state and subsequent enhancement of
molecular energy transfer. Then, in order to consider a system of realistic
molecular density and spatial positioning, we treat the molecular system as an
effective continuous medium in which the nanoparticle array is embedded. Using
this effective medium model we then study the effect of strong coupling on
molecular energy transfer. We show the results of the two method give good
qualitative agreement.

####  3.2.1 Demonstration of strong coupling to a topological edge state with
a finite array of molecules

The effective polarizability of the molecular array (analogous to Equation (8)
for the nanoparticle array) is:

$\displaystyle[\overleftrightarrow{\alpha}_{\mathrm{eff}}^{\mathrm{mol}}(\omega)]^{-1}_{3n-2;3n;3m-2;3m}=\begin{cases}\overleftrightarrow{\alpha}_{\mathrm{mol}}^{-1}(\omega)&\mbox{if
}n=m,\\\
-\frac{k^{2}}{\epsilon_{0}}\overleftrightarrow{\textbf{G}}(\omega,\textbf{r}_{n},\textbf{r}_{m})&\mbox{if
}n\neq m.\end{cases}$ (18)

From Equation 8 and 18, we obtain the effective polarizability of the whole
system as:

$\displaystyle[\overleftrightarrow{\alpha}_{\mathrm{eff}}^{\mathrm{NP}\mbox{-}\mathrm{mol}}(\omega)]^{-1}=\begin{pmatrix}\overleftrightarrow{\alpha}_{\mathrm{eff}}^{-1}(\omega)&&-\frac{k^{2}}{\epsilon_{0}}\mathcal{G}^{\mathrm{NP}\mbox{-}\mathrm{mol}}(\omega)\\\
-\frac{k^{2}}{\epsilon_{0}}\left(\mathcal{G}^{\mathrm{NP}\mbox{-}\mathrm{mol}}(\omega)\right)^{T}&&\overleftrightarrow{\alpha}_{\mathrm{eff}}^{\mathrm{mol}-1}(\omega)\end{pmatrix}$
(19)

where $\mathcal{G}^{\mathrm{NP}\mbox{-}\mathrm{mol}}(\omega)$ is a $N\times
N_{\mathrm{mol}}$ matrix whose blocks are given by the Green dyadic between
nanoparticles and molecules.

From Equation 19 we can understand the coupling between molecular and
plasmonic arrays. In the limit where the nanoparticle and molecular array
polarizabilities do not spectrally overlap, both arrays are fully uncoupled.
When the polarizabilities have spectral overlap and the Green function between
molecules and nanoparticles is large enough, the system will be strongly
coupled.

We expect the coupling to be optimal when the spectral and spatial overlap of
the molecules with the edge states is maximal. In order to demonstrate strong
coupling to edge states, we consider an array of molecules placed at a height
$z$ above the nanoparticles at the edge of the system, coinciding with the
spatial maxima of the LDOS as shown in Fig. 2(b).

Figure 3: Energy transfer mediated by strong coupling between molecules and
topological bright edge states: (a) Strong coupling between array of molecules
and plasmonic array of $30\times 30$ silver nanoparticles, with $a=10nm$,
$d=150nm$, intra-cell distance $\beta d/2=0.8a$ and $\hbar\omega_{sp}=2.50eV$
and losses $\hbar\gamma=1meV$, depending on the molecular frequency
$\omega_{mol}$. The molecules (black dots) are at the LDOS hotspots, over the
nanoparticles on the edge. Red, green and blue dotted lines are corner, edge
and bulk states. Black dashed line is $\omega=\omega_{mol}$. White dots are
the donor/acceptor molecules (b) Spectrum of the strongly coupled system and
energy transfer between molecules resonant with bright edge state
($\omega_{mol}=1.018\omega_{sp},\hbar\Gamma=0.1meV$) on adjacent edges
depending on the distance between the molecules and the plasmonic array $z/d$
overlaid with the energy transfer enhancement between molecules on adjacent
edges. (c) Strong coupling spectrum of the same plasmonic array embedded in an
effective continuous molecular medium depending on its resonance frequency
$\omega_{mol}$. (d) Spectrum of the strongly coupled system plasmonic array-
effective medium and energy transfer between molecules resonant with bright
edge state ($\omega_{mol}=1.018\omega_{sp},\hbar\Gamma=0.1meV$) on adjacent
edges depending on the distance between the molecules and the plasmonic array
$z/d$.

In Fig. 3(a) we plot the spectrum of the coupled system for $z=0.15d$ for
varying $\omega_{\mathrm{mol}}$. We see each plasmonic edge band hybridize
with the molecules and split into two polaritonic bands
($\omega_{UP}>\omega_{\mathrm{mol}}$ and $\omega_{LP}<\omega_{\mathrm{mol}}$).
The corner and bulk modes however, only couple weakly to the molecules, due to
their spatial distribution.

To compute the energy transfer enhancement, we define an effective Green
function between any two molecules $n_{D},n_{A}\in[N+1,N+N_{mol}]$ in the
molecular array via strong coupling to the plasmonic array. From
$\textbf{p}=\overleftrightarrow{\alpha}(\omega)\textbf{E}$ and Equation (15),
we derive (see Appendix B):

$\displaystyle\overleftrightarrow{\textbf{G}}_{\mathrm{eff}}^{\mathrm{SC}}(\omega,\textbf{r}_{\mathrm{A}},\textbf{r}_{\mathrm{D}})=\frac{\epsilon_{0}}{k^{2}}\overleftrightarrow{\alpha}_{\mathrm{mol}}^{-1}(\omega)\cdot\left[\overleftrightarrow{\alpha}_{\mathrm{eff}}^{\mathrm{NP}\mbox{-}\mathrm{mol}}(\omega)\right]_{3n_{A}-2;3n_{A},3n_{D}-2;3n_{D}}\overleftrightarrow{\alpha}_{\mathrm{mol}}^{-1}(\omega)$
(20)

When $z\rightarrow\infty$ this effective Green function simply reduces to the
free-space dipole-dipole Green function (see Appendix B).

When the molecules are strongly coupled to the system, molecular energy
transfer occurs via polariton modes - the molecular component of polaritons
varies with frequency, resulting in a frequency-dependent energy transfer
enhancement factor:

$\eta(\omega,\omega_{\mathrm{mol}},\textbf{r}_{\mathrm{D}},\textbf{r}_{\mathrm{A}})=\frac{\Big{|}\mathbf{u}_{\mathrm{A}}\cdot\overleftrightarrow{\mathbf{G}}_{\mathrm{eff}}^{\mathrm{SC}}(\omega,\mathbf{r}_{\mathrm{A}},\mathbf{r}_{\mathrm{D}})\mathbf{u}_{\mathrm{D}}\Big{|}^{2}}{\Big{|}\mathbf{u}_{\mathrm{A}}\cdot\overleftrightarrow{\mathbf{G}}(\omega_{\mathrm{mol}},\mathbf{r}_{\mathrm{A}},\mathbf{r}_{\mathrm{D}})\mathbf{u}_{\mathrm{D}}\Big{|}^{2}}.$
(21)

In Fig. 3(b) we plot the spectrum and energy transfer enhancement factor,
varying the distance between nanoparticles and molecules from $z=0.3d$ to
$z=0.1d$. Red, green and blue dots represent the frequencies of the corner,
edge and bulk modes.

We choose the molecules to be resonant with the bright edge states, such that
$\omega_{\mathrm{mol}}=1.018\omega_{\mathrm{sp}}$, as these showed larger
enhancement factors than the dark states in the weak coupling regime (as
demonstrated in Section 3.1). In Appendix C we study coupling to dark edge
states. When the molecules are very close to the nanoparticle array
($z/d\lesssim 0.2$), they not only strongly couple with the edge state on
resonance with the molecules, but also with the off-resonance edge states.
This is due to plasmonic losses, meaning that the spectral overlap between
edge states is non-zero.

The colormap in Fig. 3(b) represents the enhancement factor between two
molecules over two particles in adjacent edges of the array. As we see, the
maxima of the enhancement matches with the frequencies of the strongly coupled
edge states. We see the enhancement factor grows when $z$ decreases, i.e. it
is proportional to the coupling strength. We see that for strongly coupled
molecules we get factors up to $10^{15}$, contrasting with the maximum factors
around $10^{6}$ in the weak coupling regime.

####  3.2.2 Strong coupling using a continuous effective medium model

In a molecule-nanophotonic system made with straightforward fabrication
techniques, molecules will not only be located at the hotspots above
nanoparticles, nor only present over nanoparticles at the edge of the system.
A more realistic system would entail a randomly distributed layer of molecules
over the nanoparticle array, with a large number of molecules per unit cell.
Modelling this molecular system via the same coupled-dipole approach as the
previous section would be computationally exhaustive. In the limit that the
average distance between molecules is much smaller than the distance between
nanoparticles, spatial disorder will be negligible and we can treat the
molecular system as an effective continuous medium in which the nanoparticles
are embedded (see Fig. $\ref{fig:StrongCoup}(c))$.

In Equation (1) we substitute the constant background permittivity
$\epsilon_{B}$ by a Lorentz oscillator model for the material:

$\epsilon_{B}(\omega,\omega_{\mathrm{mol}})=\epsilon_{\infty}-\frac{\omega_{\mathrm{mol}}^{2}f_{\mathrm{mol}}}{\omega_{\mathrm{mol}}^{2}-\omega^{2}-i\Gamma_{m}\omega},$
(22)

where $f_{\mathrm{mol}}$ is the dimensionless oscillator strength of the
molecular material and $\Gamma_{m}$ the optical loss of the effective medium.
In Fig. 3(c) we calculate the spectrum for the arrray embedded in a effective
medium with $f_{\mathrm{mol}}=10^{-3}$, $\Gamma_{m}=0.1$ meV and
$\omega_{\mathrm{mol}}$ ranging from $0.9\omega_{\mathrm{sp}}$ to
$1.1\omega_{\mathrm{sp}}$.

Contrasting with the case with the molecules over the edge in Fig. 3(a), here
we see large polaritonic splittings for bulk and corner states too. This is
due to the molecules being extended all over space and not only placed over
the edges.

Finally, we study the energy transfer in the effective medium description by
adding two extra dipoles (donor and acceptor) and using the strongly-coupled
dyadic Green function. In Fig. 3(d) we plot the spectrum overlaid on top of
the energy transfer enhancement. From the spectrum, we see that the energy
level splitting of the molecule-edge state polaritons is reduced due to level
pushing from the bulk and corner mixed states, due to the spatial overlap of
the molecular system and all of the edge, bulk and corner states. However, the
energy transfer enhancement via the edge states at the frequency of the
polaritons is still clearly preserved.

##  4 Conclusions

Molecular energy transfer in free space is short-ranged, while coupling to
photonic modes can substantially extend the distance over which molecules may
interact, with promising applications. In this paper we have illustrated how
both weak and strong coupling between molecules and the topological edge
states of a 2D SSH metallic nanoparticle array can provide energy transfer
enhancement using a coupled-dipole model formalism. However, we note that this
concept is not limited to the SSH2D model, nor to plasmonic systems. This
formalism can be used to study coupling in any finite heterogeneous array of
couple dipoles, such as dielectric nanoparticle systems, quantum dots, etc.

We have given a detailed study of the dispersion and optical properties of the
photonic SSH2D model with all-to-all interactions, and demonstrated that the
weak-coupling of molecules to the topological edge state of this system
already provides enhancement of molecular energy transfer between a single
donor and single acceptor molecule, compared to that in free space. We have
shown that by considering a finite ensemble of molecules at the maxima of the
local density of states at the frequency of the edge states - which correspond
to the positions over the particles on the edge - leads to collective coupling
between the molecules and edge states. When the distance between the molecules
and the array is reduced, the strong coupling regime may be accessed, leading
to polaritonic Rabi splitting of the edge states - even for those edge states
that are off-resonance. This is due to the overlap between two edge states
being small but non-zero, so molecules very close to the surface may couple to
states which are off-resonant. In the strong coupling regime, molecular energy
transfer is enhanced by many orders of magnitude in comparison to the weak-
coupled regime.

We have verified this result at realistic molecular density by considering the
nanoparticle array to be embedded in an effective medium representing the
molecular material. In this case, we have shown that it is harder to couple
exclusively to edge states, due to the spatial overlap of the molecular
material with all photonic states. However, we have shown that coupling to
edge states in this regime still leads to dramatic energy transfer
enhancement. There is a great research effort currently underway to understand
and control the strong coupling of molecules to photonic modes, particularly
in open systems, and in order to engineer long-range molecular energy
transfer. We hope that the results we have reported here will contribute to
both of these efforts via the theoretical demonstration of topological light-
molecule strong coupling, with various potential applications and new research
avenues extending beyond the system studied in the current work.

We thank William Wardley and Felipe Herrera for fruitful discussions. V.G.
thanks the ENSEMBLE3-Centre of Excellence for nanophotonics, advanced
materials and novel crystal growth-based technologies, project (GA No.
MAB/2020/14) carried out within the International Research Agendas program of
the Foundation for Polish Science cofinanced by the European Union under the
European Regional Development Fund.

A.B. and J.A.S.G. acknowledge financial support from the grants BICPLAN6G
(TED2021-131417B-I00) and LIGHTCOMPAS (PID2022-137569NB-C41), funded by
MCIU/AEI/10.13039/501100011033, “ERDF A way of making Europe”, and European
Union NextGenerationEU/PRTR and from MCIU through predoctoral fellowship
PRE2019-090689.

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##  Appendix A Normal modes and symmetries

Even when finite SSH2D arrays break periodicity, when they are commensurate
with the unit cell and respect the spatial symmetries, normal modes still
inherit properties from the periodic system.

The bulk modes respect the symmetries of the bands of the infinite system,
this is the dipolar moments within a single unit cell are either symmetric or
antisymmetric in $x$ and $y$. Depending on parities in $x$ and $y$, we label
the bands as $B_{1}(--),B_{2}(-+),B_{3}(+-),B_{4}(++)$. $B_{4}$ band is the
brightest and highest frequency one. Due to rotational symmetry, $B_{2}$ and
$B_{3}$ bands are degenerate. The $B_{1}$ band is the lowest and darkest band.

The edge modes respect the symmetries of bulk modes in SSH chains. Within
dimers, modes are either symmetric (bright) or antisymmetric (dark). Bright
($E_{1}-E_{2}$) and dark ($E_{3}-E_{4}$) modes are doubly degenerate,
depending on whether they are symmetric or antisymmetric by a 90º rotation. In
Fig. 1(c) we plot the main contributions to $E_{1},E_{3}$ modes at $\Gamma$
point. This corresponds to the modes where the dimers along a single edge are
in-phase.

Corner modes ($C_{1}-C_{4}$) are quadruply degenerated and pinned to
$\omega_{sp}$ due to the generalized sublattice symmetry.

##  Appendix B Strongly coupled effective Green dyadic

In order to derive the strongly-coupled effective Green dyadic, we first
consider an incident field $\hat{\textbf{E}}_{\mathrm{inc}}$ that excites only
the donor molecule $n_{\mathrm{D}}$,
$[\hat{\textbf{E}}_{\mathrm{inc}}]_{3n-2;3n}=\textbf{E}_{\mathrm{D}}\delta_{nn_{\mathrm{D}}}$,
where
$\textbf{E}_{\mathrm{D}}=\alpha_{\mathrm{mol}}^{-1}(\omega)\textbf{p}_{\mathrm{D}}$,
being $\textbf{p}_{\mathrm{D}}$ the electric dipole induced in the donor. The
response of the molecular-plasmonic array to this excitation is given by
$\textbf{P}=\alpha_{\mathrm{eff}}^{\mathrm{NP}-\mathrm{mol}}(\omega)\hat{\textbf{E}}_{\mathrm{inc}}$.
Then, the dipole induced in the acceptor molecule $n_{\mathrm{A}}$ is given
by:

$\displaystyle\textbf{p}_{A}=\left[\alpha_{\mathrm{eff}}^{\mathrm{NP}-\mathrm{mol}}(\omega)\hat{\textbf{E}}_{\mathrm{inc}}\right]_{3n_{\mathrm{A}}-2;3n_{\mathrm{A}}}=\left[\alpha_{\mathrm{eff}}^{\mathrm{NP}-\mathrm{mol}}(\omega)\right]_{3n_{\mathrm{A}}-2;3n_{\mathrm{A}},3n_{\mathrm{D}}-2;3n_{\mathrm{D}}}\overleftrightarrow{\alpha}_{\mathrm{mol}}^{-1}(\omega)\textbf{p}_{\mathrm{D}}$
(B.1)

By analogy with the dipole induced by another dipole in free-space (from Eq.
4)
$\textbf{p}_{A}=\frac{k^{2}}{\epsilon_{0}}\overleftrightarrow{\alpha}_{mol}(\omega)\overleftrightarrow{\textbf{G}}(\omega,\textbf{r}_{A},\textbf{r}_{D})\textbf{p}_{D}$,
we can define an effective Green dyadic via strong coupling to the
nanoparticle-molecular ensemble as:

$\displaystyle\overleftrightarrow{\textbf{G}}_{\mathrm{eff}}^{\mathrm{SC}}(\omega,\textbf{r}_{\mathrm{D}},\textbf{r}_{\mathrm{A}})=\frac{\epsilon_{0}}{k^{2}}\overleftrightarrow{\alpha}_{\mathrm{mol}}^{-1}(\omega)\left[\overleftrightarrow{\alpha}_{\mathrm{eff}}^{\mathrm{NP}\mbox{-}\mathrm{mol}}(\omega)\right]_{3n_{\mathrm{D}}-2;3n_{\mathrm{D}},3n_{\mathrm{A}}-2;3n_{\mathrm{A}}}\overleftrightarrow{\alpha}_{\mathrm{mol}}^{-1}(\omega)$
(B.2)

In the uncoupled limit $z\rightarrow\infty$ and when the distance between
molecules is large enough that $\alpha^{-1}_{\mathrm{mol}}(\omega)\gg
G_{zz}(\omega,\textbf{r}_{\mathrm{D}},\textbf{r}_{\mathrm{A}})$, we recover
the free-space dipole-dipole Green function:

$\displaystyle\lim_{z\rightarrow\infty}[\overleftrightarrow{\textbf{G}}_{\mathrm{eff}}^{\mathrm{SC}}(\omega,\textbf{r}_{\mathrm{D}},\textbf{r}_{\mathrm{A}})]_{zz}=\frac{\epsilon}{k^{2}}\alpha_{\mathrm{mol}}^{-2}(\omega)\left[\begin{pmatrix}\alpha_{\mathrm{mol}}^{-1}(\omega)&-\frac{k^{2}}{\epsilon_{0}}G_{zz}(\omega,\textbf{r}_{\mathrm{D}},\textbf{r}_{\mathrm{A}})\\\
-\frac{k^{2}}{\epsilon_{0}}G_{zz}(\omega,\textbf{r}_{\mathrm{D}},\textbf{r}_{\mathrm{A}})&\alpha_{\mathrm{mol}}^{-1}(\omega)\end{pmatrix}^{-1}\right]_{1,2}=$
$\displaystyle=\frac{\epsilon_{0}}{k^{2}}\frac{\alpha_{\mathrm{mol}}^{-2}(\omega)}{\alpha_{\mathrm{mol}}^{-2}(\omega)-G_{zz}(\omega,\textbf{r}_{\mathrm{D}},\textbf{r}_{\mathrm{A}})^{2}}\left[\begin{pmatrix}\alpha_{\mathrm{mol}}^{-1}(\omega)&\frac{k^{2}}{\epsilon_{0}}G_{zz}(\omega,\textbf{r}_{\mathrm{D}},\textbf{r}_{\mathrm{A}})\\\
\frac{k^{2}}{\epsilon_{0}}G_{zz}(\omega,\textbf{r}_{\mathrm{D}},\textbf{r}_{\mathrm{A}})&\alpha_{\mathrm{mol}}^{-1}(\omega)\end{pmatrix}\right]_{1,2}$
$\displaystyle\simeq
G_{zz}(\omega,\textbf{r}_{\mathrm{D}},\textbf{r}_{\mathrm{A}})$ (B.3)

##  Appendix C Strong coupling to dark edge states

Figure C.1: Energy transfer mediated by strong coupling between molecules and
topological dark edge states: (a) Coupling to an array of $30\times 30$ silver
nanoparticles, with $a=10$ nm, $d=150$ nm, intra-cell distance $\beta
d/2=0.8a$ and $\hbar\omega_{\mathrm{sp}}=2.50$ eV and losses $\hbar\gamma=1$
meV, by placing molecules, resonant with the dark edge states
($\omega_{\mathrm{mol}}=0.983\omega_{\mathrm{sp}}$), over the edge
nanoparticles, at the hotspots of the LDOS. Spectrum overlaid on top of
enhancement of energy transfer between molecules on adjacent edges, varying
the distance between the molecules and the plasmonic array $z/d$. Red, green
and blue dotted lines are corner, edge and bulk states. Black dashed line is
$\omega=\omega_{mol}$. Yellow dots are the donor/acceptor molecules (b)
Coupling to the same array embedded in an effective molecular medium. Spectrum
overlaid on top of enhancement of energy transfer between donor and acceptor
molecules on adjacent edges varying with the vertical distance between the
molecules and the plasmonic array $z/d$.

As we stated in the main text, dark states, despite their name, can be probed
and excited locally by near-field coupling. However, it may be trickier to
excite them, since these modes break the mirror symmetry within the unit cell,
but preserve inversion invariance.

In order to couple to these modes more efficiently, we can break mirror
symmetry in the excitation by placing more than one molecule per plasmonic
unit cell or with one molecule polarized in-plane. Here we restrict to
$z$-polarized molecules. By placing the molecules on every hotspot (over the
particles on the edge), we can couple to both dark and bright edge states.

In Fig. C.1(a) we plot the spectrum and energy transfer enhancement varying
the distance between nanoparticles and molecules from $z=0.3d$ to $z=0.1d$.
Red, green and blue dots represent the frequencies of the corner, edge and
bulk modes.

We choose the molecules to be resonant with the dark edge states, such that
$\omega_{\mathrm{mol}}=0.983\omega_{\mathrm{sp}}$. As with the bright edge
state when the molecules are very close to the metasurface ($z/d\lesssim
0.2$), they do not couple just with the edge state on-resonance but also with
the off-resonance edge states, due to the small spectral overlap between the
edge states.

In Fig. C.1(b) we plot the spectrum and energy transfer enhancement in the
effective medium description, with similars results to the coupling to bright
states. We still find the larger enhancement factor are at the frequency of
the (off-resonance) bright states.

Even when the enhancement of the energy transfer is smaller than for bright
edge states, they will generally be less screened by bulk or corner states, as
bright modes are broader spectrally. This may lead to more confinement in the
edge in the energy transfer than for bright modes.