# Exploiting oriented field projectors to open topological gaps
in plasmonic nanoparticle arrays

Álvaro Buendía<EMAIL_ADDRESS>Instituto de Estructura de la Materia,
Consejo Superior de Investigaciones Científicas, Serrano 121, 28006 Madrid,
Spain  José 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  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

###### Abstract

In the last years there have been multiple proposals in nanophotonics to mimic
topological condensed matter systems. However, nanoparticles have degrees of
freedom that atoms lack of, like dimensions or shape, which can be exploited
to explore topology beyond electronics. Elongated nanoparticles can act like
projectors of the electric field in the direction of the major axis. Then, by
orienting them in an array the coupling between them can be tuned, allowing to
open a gap in an otherwise gapless system. As a proof of the potential of the
use of orientation of nanoparticles for topology, we study 1D chains of
prolate spheroidal silver nanoparticles. We show that in these arrays spatial
modulation of the polarization allows to open gaps, engineer hidden
crystalline symmetries and to switch on/off or left/right edge states
depending on the polarization of the incident electric field. This opens a
path towards exploiting features of nanoparticles for topology to go beyond
analogues of condensed matter systems.

††preprint: APS/123-QED

##  I Introduction

The exciting discovery of the topological phase of matter systems has inspired
many new fields in physics, particularly in photonics [1, 2, 3] ; in fact, in
recent years, we have witnessed an exponential growth of interests in that
direction. Mimicking the phenomenology of topological insulators has been the
driving force until now. However, it is becoming clear that a further step
needs to be taken, i.e., to push forward new topological photonic
phenomenology that does not have a material counterpart.

Topological insulators are possible thanks to the fermionic nature of
electrons [4], but photons cannot take advantage of such symmetry. Initial
solutions have been proposed based on gyromagnetic photonic crystals [5],
bianisotropic materials [6], and coupled waveguides and resonators [7, 8].

All the previous systems use some kind of time-reversal property not present
in simple photonics systems without magnetic response. In addition, there is
always a strong interest in achieving very small and faster devices for
nanotechnological applications. Typical examples are microprocessors, but
light interacts weakly with the material at the nanoscale. Moreover, one would
like to have such photonic properties in the visible, where most of the
molecular electronic transitions happens, making such zone relevant in light-
matter interaction. With these goals and restriction in mind, metal
nanoparticles using plasmonic resonances are probably the best candidates.
This has made it possible for many researchers to look at what we can call
topological nanoparticle photonics [9, 10].

Plasmonic nanoparticles provide an excellent platform for light-matter
interaction, but being not simple to break time-reversal symmetry in the
visible range, a typical approach is using crystal symmetries [11, 12, 13, 14,
15, 16]. Such approach has also been explored for radiative heat transfer with
interesting results [17].

In addition, particular care needs to be taken due to the long-range nature of
these interactions and the radiative corrections [18, 19, 10], which can spoil
the topological protection of the system. Here, instead of focusing on such
loss of protection, we explore degrees of freedom of the nanoparticles which
could be exploited for topology beyond condensed matter systems.

This paper is organized as follows. In section II we introduce the simplest
topological system, a dimer chain known as SSH model, and its extensions for
larger unit cells. In section III we study and compare different plasmonic
counterparts of these chains. Finally, in section IV we discuss the excitation
and switching of the edge states of the plasmonic chains of elongated
nanoparticles by an incoming electric field.

##  II SSH model and topological phases

The Su-Schrieffer-Heeger (SSH) model is the simplest system with topological
protection. It was firstly proposed in Ref. [20] to describe the physics of
the polyacetylene chain, which alternates double and simple (or strong and
weak) bonds between adjacent carbon atoms. In FIG. 1(a) we show a scheme of
this model, which consists of a one-dimensional diatomic chain with two
staggered hopping amplitudes between nearest neighbors, namely $v$ and $w$.
Its tight-binding Hamiltonian is:

$\displaystyle\mathcal{H}(q)=\begin{pmatrix}0&v+we^{-iqd}\\\
v+we^{iqd}&0\end{pmatrix},$ (1)

and satisfies the Schrödinger equation:

$\displaystyle\mathcal{H}(q)\ket{u_{n}(q)}=E_{n}(q)\ket{u_{n}(q)}$ (2)

This system has two different distinct topological phases: it is trivial when
the bond between particles in adjacent unit cells is weaker than the one
between particles within a unit cell ($|w|<|v|$) and topological when it is
stronger ($|w|>|v|$). When we chop the periodic chain commensurately with the
topological unit cell, it hosts strongly localized zero-energy states at both
ends. These edge states are robust to disorder and perturbations that respect
its symmetries: sublattice symmetry (also known as chirality) and
mirror/inversion symmetries.

Figure 1: (a) The simplest model with topological protection, the SSH chain,
consisting of a diatomic chain with staggered hoppings $v$ and $w$. (b)
Extended 3-particle SSH chain (or SSH3), alternating hoppings $u,v$ and $w$.
(c) Extended 4-particle SSH chain (or SSH4), alternating hoppings $t,u,v$ and
$w$.

Sublattice symmetry stems from the existence of two sublattices ($A$ and $B$)
with bonds between sublattices but not within a sublattice. This implies the
Hamiltonian is anti-block-diagonal, i.e.:

$\mathcal{H}(q)=\begin{pmatrix}0&h(q)\\\ h^{\dagger}(q)&0\end{pmatrix}.$ (3)

Chirality makes the spectrum symmetric around zero energy and fixes the energy
of the edge states at zero, isolating them from the bulk states. It also makes
each edge state be localized in just one of the sublattices.

$\Gamma\mathcal{H}(q)\Gamma=-\mathcal{H}(q)\qquad\Gamma=\begin{pmatrix}I&0\\\
0&-I\end{pmatrix}$ (4)

The SSH model is mirror-symmetric because the system remains invariant under
spatial inversion in the $x$-axis, i.e., under the operation of $\Pi$:

$\Pi\mathcal{H}(q)\Pi=\mathcal{H}^{T}(-q)=\mathcal{H}(q)$ (5)

The SSH chain is also inversion-symmetric, as it remains invariant under the
subsequent spatial inversion in all axes. Mirror or inversion symmetries lead
to the double degeneracy of the edge states, even when the sublattice symmetry
is broken.

As long as chirality and/or mirror symmetries are respected, the Zak phase
[21] $\gamma$, a bulk property, is a topological invariant and predicts the
existence of edge states in the terminated system [22]. This is known as bulk-
boundary correspondence. The Zak phase for each band and for the gap is:

$\displaystyle\gamma_{n}$ $\displaystyle=\int_{\mathrm{BZ}}\langle
u_{n}(q)|\frac{\partial}{\partial_{q}}|u_{n}(q)\rangle dq$
$\displaystyle\gamma$ $\displaystyle=\sum_{\textnormal{below gap}}\gamma_{n}$
(6)

In electronic systems, when the system is neutral, only half of the bands are
below the Fermi level, so in the SSH model only the lower band contributes to
the Zak phase.

In the following subsection we introduce some extensions of the SSH model.

###  II.1 Extended unit cell SSH models

Due to the simplicity of the system, several generalizations of the SSH model
have been made, for example by adding hoppings between further neighbors [23],
or by extending to two dimensions in a square array [24]. This model can also
be generalized to one-dimensional chains with larger linear unit cells [25,
26, 27, 28] or rhombus unit cells [29]. These systems are topologically more
complex than the SSH model, featuring several gaps and non-zero edge states.
They can also exhibit other kinds of topological protection like square-root
topology [30].

First, as we show in FIG. 1(b), we consider a linear chain with three
alternating hopping amplitudes $u,v$ and $w$ This lattice has three different
sublattices $A,B$ and $C$. The topology of this system has been discussed in
Refs. [31, 28, 32]

$\displaystyle\mathcal{H}(q)=\begin{pmatrix}0&u&we^{-iqd}\\\ u&0&v\\\
we^{iqd}&v&0\end{pmatrix}.$ (7)

A three-way generalization of the sublattice symmetry can be made for this
chain:

$\displaystyle\Gamma_{3}^{-1}\mathcal{H}(q)\Gamma_{3}+\Gamma_{3}^{-2}\mathcal{H}(q)\Gamma_{3}^{2}=-\mathcal{H}(q),$
$\displaystyle\Gamma_{3}=\begin{pmatrix}1&0&0\\\ 0&e^{2i\pi/3}&0\\\
0&0&e^{-2i\pi/3}\end{pmatrix},\qquad\Gamma_{3}^{3}=I.$ (8)

This symmetry is the same that features the breathing Kagome lattice [33].
However, due to the absence of the $C_{3}$ rotational symmetry also present in
the Kagome lattice, in this 1D system there are no three degenerate zero-
states but non-zero edge states. Additionally, the edge states are not
localized in just one of the sublattices but the two closer to the edge. When
the chain is mirror symmetric, i.e $|u|=|v|$, the edge states come in two
degenerate pairs at energies $-E$ and $E$ when $|w|>|u|$. Each gap has a
distinct Zak phase, that is quantized by mirror symmetry. Due to the three-way
chirality they are not independent, but the same.

Similarly, we can consider a chain with a 4-particle unit cell, shown in FIG.
1(c), with hoppings $t,u,v$ and $w$. The tight-binding Hamiltonian (in the
base $A,C,B,D$) is:

$\displaystyle\mathcal{H}(q)=\begin{pmatrix}0&0&t&we^{-iqd}\\\ 0&0&u&v\\\
t&u&0&0\\\ we^{iqd}&v&0&0\end{pmatrix}$ (9)

Due to the even number of particles in the unit cell, this chain recovers the
sublattice symmetry from the SSH model. The transition for the central gap
occurs when $|tv|=|uw|$. When $|tv|>|uw|$ the system is in the trivial phase,
whereas for $|tv|<|uw|$ the system has symmetry-protected zero-energy states
that localize exclusively in even or odd sublattices.

However, the system also features non-zero energy states in the lower/upper
gaps. These states inherit properties from the four-way generalized chirality,
which is given by:

$\displaystyle\Gamma_{4}^{-1}\mathcal{H}(q)\Gamma_{4}+\Gamma_{4}^{-2}\mathcal{H}(q)\Gamma_{4}^{2}+\Gamma_{4}^{-3}\mathcal{H}(q)\Gamma_{4}^{3}=-\mathcal{H}(q),$
$\displaystyle\Gamma_{4}=\begin{pmatrix}1&0&0&0\\\ 0&i&0&0\\\ 0&0&-1&0\\\
0&0&0&-i\end{pmatrix},\qquad\Gamma_{4}^{4}=I.$ (10)

This symmetry makes the non-zero edge states be localized in three out of the
four sublattices. However, this symmetry doesn’t close lower/upper gaps or
quantize their Zak phases, but spatial symmetries do. The system is
mirror/inversion symmetric when $|t|=|v|$. Mirror symmetric non-zero energy
edge states appear for $|w|>|u|$.

Now we focus in arrays of plasmonic nanoparticles. Previously, optical
response of metallic nanoparticles has been used to mimic topological
condensed matter systems, as in zigzag chains [34, 35], diatomic chains of
nanospheres [36], or breathing Kagome [37] and breathing honeycomb plasmonic
metasurfaces [38, 39]. However, in these systems it has been shown that long-
range interactions between nanoparticles must be considered, which have a
striking effect on the topology of the system [18, 19].

Electric fields produce localized surface plasmon resonances (LSPR) in
metallic nanoparticles. A small single nanoparticle with $a\ll\lambda$ (where
$a$ is the particle radius and $\lambda$ is the wavelength of incoming light),
such that $a>3-4$ nm to avoid quantum effects, scatters an incident electric
field $\textbf{E}_{inc}$ approximately like a dipole p:

$\displaystyle\mathbf{p}=\epsilon_{\mathrm{B}}\overleftrightarrow{\alpha}(\omega)\mathbf{E}_{inc},$
(11)

where $\epsilon_{B}$ is the permittivity of the background medium and
$\overleftrightarrow{\alpha}(\omega)$ is the polarizability tensor.

The dipolar approximation still holds for an array of nanoparticles, if they
are separated a distance of at least $3a$.

Then, each dipole in the array is determined by both the incident electric
field and the scattered electric field by the rest of the dipoles:

$\displaystyle\mathbf{p}_{n}$
$\displaystyle=\overleftrightarrow{\alpha_{n}}(\omega)\left(\textbf{E}_{inc}+\frac{k^{2}}{\epsilon_{0}}\sum_{m\neq
n}\overleftrightarrow{\mathbf{G}}(\mathbf{r}_{n},\mathbf{r}_{m},\omega)\mathbf{p}_{m}\right),$
(12)

where $n,m$ are sites in the array, $\mathbf{p}_{n,m}$ are the dipoles in
positions $\mathbf{r}_{n,m}$ and
$\overleftrightarrow{\mathbf{G}}(\mathbf{r}_{n},\mathbf{r}_{m},\omega)$ is the
Green’s dyadic function, which in the quasi-static regime $kR\gg 1$ is given
by:

$\displaystyle{\overleftrightarrow{{\bf
G}}}(\mathbf{r}_{n},\mathbf{r}_{m},\omega)$ $\displaystyle=\frac{1}{4\pi
k^{2}R^{3}}\left[-I+3\frac{\mathbf{R}\otimes\mathbf{R}}{R^{2}}\right],$ (13)

where $\mathbf{R}=\mathbf{r}_{n}-\mathbf{r}_{m}$, $R=|\mathbf{R}|$, and
$k=\sqrt{\epsilon_{B}}\omega/c$ is the wavevector.

In the next subsections we’ll consider examples of topological arrays of
nanoparticles.

##  III Topological plasmonic chains

###  III.1 Chain of nanospheres

First, we consider a single spherical metallic nanoparticle. A nanosphere has
spherical symmetry, so its tensor polarizability behaves like a scalar,
$\alpha(\omega)$, which in the quasi-static limit $a\ll\lambda$ is:

$\displaystyle\alpha(\omega)=4\pi
a^{3}\epsilon_{0}\frac{\epsilon(\omega)-\epsilon_{\mathrm{B}}}{\epsilon(\omega)+2\epsilon_{\mathrm{B}}}.$
(14)

In figure 2(a) we see the optical response of a single silver nanosphere of
radius $a=12.5$ nm to a linear-polarized electric field (blue curve), that
shows a resonance for $\hbar\omega_{sp}\sim 2.75$ eV.

Now we consider a plasmonic analogue of the SSH model, i.e., a chain of
nanospheres with two alternate distances: the intracell distance
$\beta\frac{d}{2}$ and the intercell distance
$\left(2-\beta\right)\frac{d}{2}$, $d$ being the size of the unit cell (see
scheme on FIG. 2(b)). For any array of nanospheres and in absence of incident
electric field we can rewrite EQ. 12 as:

$\frac{1}{\alpha(\omega)}\textbf{p}_{n}=\frac{k^{2}}{\epsilon_{0}}\sum_{m\neq
n}\overleftrightarrow{\textbf{G}}(\textbf{r}_{n},\textbf{r}_{m},\omega)\cdot\textbf{p}_{m}.$
(15)

After Bloch, coupled-dipole equations can be compacted in a matrix equation:

$\mathcal{G}(\textbf{q})\textbf{P}=\frac{1}{\alpha(\omega)}\textbf{P},$ (16)

where $\textbf{P}=(p_{1x},p_{2x},p_{1y},p_{2y},p_{1z},p_{2z})$ [18]. As we
see, this is equivalent to the Schrödinger equation in EQ. 2, where the dipole
vector, the inverse of the polarizability, and the Green’s matrix
$\mathcal{G}(q)$ take the roles of, respectively, eigenvectors, eigenvalues,
and the Bloch Hamiltonian $\mathcal{H}(q)$. Explicitly, $\mathcal{G}(q)$ terms
are:

Figure 2: Arrays of plasmonic nanoparticles (a) Extinction cross sections of a
silver nanosphere of radius $a=12.5$ nm (blue curve) and prolate spheroidal
($a=12.5$ nm, $b=c=5$ nm) silver nanoparticles (red curve for the major axis
polarization and green line for any polarization in the perpendicular plane)
embedded in glass ($\epsilon_{B}=2.25$); Ag dielectric function from [40].
(b-c) Schematic of the SSH analogue consisting of plasmonic chains with (b)
alternating distances between nanospheres and (c) equidistant nanospheroids
alternating orientations. (d) Interaction between nanospheroids depending on
their orientations. Projection of the Green dyadic’s function on the
directions of the nanoparticles normalized by $\frac{1}{4\pi k^{2}R^{3}}$.

$\displaystyle\mathcal{G}_{\mu\nu}=\delta_{\mu\nu}\frac{2m_{\nu}}{\pi
d^{3}}\begin{pmatrix}0&\frac{1}{\beta^{3}}+\frac{1}{(2-\beta)^{3}}e^{-iqd}\\\
\frac{1}{\beta^{3}}+\frac{1}{(2-\beta)^{3}}e^{iqd}&0\end{pmatrix},$ (17)

###  III.2 Coupled-dipole equations

where $\mu,\nu=x,y,z$ are the polarizations of each pair of dipoles, and
$m_{v}$ is $m_{x}=2$ for the longitudinal polarization and $m_{y,z}=-1$ for
the transversal polarizations. This is, the plasmonic dimer chain of
nanospheres is equivalent to three independent copies of the SSH (EQ. (1)),
one per polarization, with $v=\frac{2m_{\nu}}{\pi\epsilon_{0}\beta^{3}d^{3}}$
and $w=\frac{2m_{\nu}}{\pi\epsilon_{0}(2-\beta)^{3}d^{3}}$. The dispersion
bands $\omega(q)$ can be calculated from EQ. 16, searching for the solutions
of:

$\lambda_{n}-\frac{1}{\alpha(\omega)}=0.$ (18)

$\lambda_{n}$ being the $n$th eigenvalue of $\mathcal{G}(q)$. The zero-energy
modes typical of the finite SSH model translate in this system to six (two per
polarization) resonant modes localized at the edges of the chain at the
surface plasmon frequency of a single nanosphere $\omega_{sp}$.

Apart from the plasmonic diatomic 1D chain, the zigzag chain [34, 35], which
alternates angles between the nanoparticles, has also been proposed to mimic
the topology of the SSH model. This system exploits the polarization asymmetry
between longitudinal and transversal modes and allows to select edge modes by
changing the polarization of the incoming electric field. In the next
subsection we will get advantage of this same anisotropy not in the geometry
of the array but in the shape of the nanoparticles, adding a new degree of
freedom to the system.

###  III.3 Chain of nanospheroids

When the nanoparticles are not spherical,
$\overleftrightarrow{\alpha_{i}}(\omega)$ are not proportional to the identity
matrix anymore, so they affect the polarization of the dipoles. This asymmetry
has no analogy in tight binding models and can be exploited to explore new
topological systems. For example, if we replace the nanospheres in the
previous chain by parallel nanorods, we can filter modes by in-plane or out-
of-plane polarizations. However, by orienting the nanoparticles in alternating
directions, we can force modes beyond the plasmonic nanosphere chain.

Previously, gradual change of orientation in arrays of anisotropic
nanoparticles or nanoholes has been exploited in Pancharatnam-Berry
metasurfaces (also known as geometric phase metasurfaces) for example to
enable polarization-dependent control of light [41] or to create vortex beams
[42].

Here we use orientation of elongated nanoparticles not to build a geometric
phase in polarization, but to tune the interactions between nanoparticles in
order to open a topological gap. However, the spatial modulation of
polarization in our system plays a role in the control of edge states,
allowing to switch them off by changing the polarization of the incident
electric field, as we’ll see in last section.

The strategy pursued in this paper to open a topological gap is similar to in
Ref. [43], where it is opened by orientation in the transversal plane of
bianisotropic particles in an equidistant array. However, our study is more
general and in the visible range.

Let us now assume our elongated nanoparticles are prolate spheroidal
nanoparticles with axis half-lengths $a>b=c$ with major axis pointing in the z
direction. The results will be qualitatively equivalent for any other
elongated shape, so we’re not losing generality by making this choice. In this
case the polarizability tensor is:

$\overleftrightarrow{\alpha}(\omega)=\begin{pmatrix}\alpha_{c}(\omega)&0&0\\\
0&\alpha_{c}(\omega)&0\\\ 0&0&\alpha_{a}(\omega)\end{pmatrix},$ (19)

where the quasistatic polarizabilities $\alpha_{l}$ with $l\in[a,b,c]$ are
[44]:

$\alpha_{l}(\omega)=V\frac{\epsilon(\omega)-\epsilon_{b}}{\epsilon_{b}+L_{l}(\epsilon(\omega)-\epsilon_{b})},$
(20)

$V$ being the volume of the spheroid, $V=\frac{4}{3}\pi ac^{2}$,and $L_{l}$
are geometric factors (see appendix A).

In FIG. 2(a) we can see the extinction cross section (see A for details) of a
single silver nanospheroid with major axis $a=12.5$ nm and minor axes
$b=c=0.4a=5$ nm. Red curve represents the response to a field polarized
parallel to the major axis, while for the green curve the field is polarized
normal to the major axis. As we see, the resonance wavelengths are separate
enough ($\hbar\omega_{spc}\sim 2.96$ eV, whereas $\hbar\omega_{spa}\sim 2.0$
eV), that in the proximity of the major axis resonance,
$\alpha_{c}(\omega\simeq\omega_{spa})\simeq 0$, so we can approximate the
tensor polarizability to:

$\overleftrightarrow{\alpha}(\omega\simeq\omega_{spa})\simeq\begin{pmatrix}0&0&0\\\
0&0&0\\\ 0&0&\alpha_{a}(\omega)\end{pmatrix}.$ (21)

This means the polarizability tensor acts like a projection operator [45],
projecting the dipole in the direction of the major axis.

Now we consider an array of nanospheroids, with major axis oriented in the
directions $\textbf{u}_{n}=(\cos{\varphi_{n}},\sin{\varphi_{n}},0)^{T}$, where
$\varphi_{n}$ is the angle of the particle $n$ with respect to the $x$ axis.
We assume all the particles are oriented in the plane $xy$ for the sake of
simplicity. As the $y$ and $z$ axes are indistinguishable, all results will be
the same for the $xz$ plane. The equations for the general case, with
nanoparticles oriented in any direction, are in appendices A,B.

Due to the projection of the polarizability in the directions of the major
axes of the particles, the vectorial coupled-dipole equations in an array of
nanospheroids can be reduced to scalar equations:

$\frac{1}{\alpha_{a}(\omega)}p_{n}=\frac{k^{2}}{\epsilon_{0}}\sum_{m\neq
n}G_{\textbf{u}_{m},\textbf{u}_{n}}p_{m},$ (22)

where $G_{\textbf{u}_{m},\textbf{u}_{n}}=({\overleftrightarrow{{\bf
G}}}(\textbf{r}_{m},\textbf{r}_{n},\omega)\cdot\textbf{u}_{m})\cdot\textbf{u}_{n}$
is the projection of the polarizability tensor in the directions
$\textbf{u}_{m}$,$\textbf{u}_{n}$.

The coupled-dipole equations can again be rewritten in a matrix form:

$\mathcal{G}_{a}(\textbf{k})\textbf{P}_{a}=\frac{1}{\alpha_{a}(\omega)}\textbf{P}_{a},$
(23)

where in this case $\mathcal{G}_{a}(\textbf{k})$ is a $N\times N$ matrix with
elements given by $G_{\textbf{u}_{m},\textbf{u}_{n}}$ and
$\textbf{P}_{a}=(p_{1},...,p_{N})$ is a vector containing the module of the
dipoles.

First, let us consider a 1D chain of equidistant nanoparticles, separated by a
distance $R$. We can see an scheme of this chain in FIG. 2(c). As all the
particles are in the $x$ axis, the Green dyadic’s function reduces to:

$\overleftrightarrow{\textbf{G}}(\textbf{r}_{m},\textbf{r}_{n})=\frac{1}{4\pi
k^{2}R^{3}}\begin{pmatrix}2&0&0\\\ 0&-1&0\\\ 0&0&-1\end{pmatrix},$ (24)

and its projection onto the $\textbf{u}_{i}$ and $\textbf{u}_{j}$ axes is:

$G_{\textbf{u}_{m},\textbf{u}_{n}}=\frac{2\cos{\varphi_{m}}\cos{\varphi_{n}}-\sin{\varphi_{m}}\sin{\varphi_{n}}}{4\pi
k^{2}R^{3}}.$ (25)

_This means that we can tune the coupling between nanoparticles by rotating
them._ In FIG. 2(d) we plot the interaction
$G_{\textbf{u}_{m},\textbf{u}_{n}}$ multiplied by $4\pi k^{2}R^{3}$ depending
on the orientation angles $\varphi_{m}$ and $\varphi_{n}$. This interaction
ranges from $2$ (parallel dipoles in the longitudinal direction) to $-2$
(antiparallel and in the longitudinal direction), passing by $-1$ (parallel
dipoles in the transversal polarization). For any pair of angles in the zero
contour line the interaction is suppressed. For example, orthogonal
nanospheroids oriented in $x$ and $y$ directions don’t interact between them,
as we see for $\varphi=0$ in the colormap.

This zero interaction was impossible to achieve in the nanosphere chain and it
could only be approximated by separating the particles a long distance (see
EQ. 17). This could be interesting for switching off some even neighbor
interactions that can break sublattice symmetry, but here we will restrict to
the first-neighbor approximation. This approximation is accurate only in the
quasi-static regime, that is, when the nanoparticles and the distances between
them are small, so $kR\ll 1$.

In the next subsections we will consider linear arrays of nanospheroids. With
nanospheres, a linear chain where the nanoparticles were equidistant would be
gapless and equivalent to the $v=w$ case in the SSH model. However, by
substituting the nanospheres by nanospheroids and adding the orientation as a
degree of freedom a gap can be opened.

####  III.3.1 2 particles/unit cell

First we consider the simplest system which may have a topological gap. This
is a linear array with two particles/unit cell and with a distance between
adjacent particles $R=\frac{d}{2}$, being $d$ the size of the unit cell. For
nanospheroids, the Green matrix of the system would be:

$\mathcal{G}(q)=G_{\textbf{u}_{A},\textbf{u}_{B}}\begin{pmatrix}0&1+e^{-iqd}\\\
1+e^{iqd}&0\end{pmatrix}.$ (26)

This system is equivalent to the SSH model with $v=w=G_{u_{A},u_{B}}$, due to
the reciprocity $G_{u_{A},u_{B}}=G_{u_{B},u_{A}}$. This means that by rotating
the spheroidal nanoparticles in a 2-particle unit cell, we can change the
amplitude of the bands but not open a gap.

####  III.3.2 3 particles/unit cell

Figure 3: Plasmonic analogue of the mirror-symmetric SSH3 model: periodic
chain of 3 prolate silver nanospheroids per unit cell, with long spheroidal
axes forming angles with the chain direction
$\varphi_{A}=\varphi_{B}-\frac{\pi}{2}=\varphi_{C}=\varphi$. The dimensions of
the nanoparticles are $a=12.5\text{ nm}$ and $b=c=0.4a=5$ nm. (a) Unit cells
for $\varphi=-\frac{\pi}{4},$ $\varphi=0,$ $\varphi=\frac{\pi}{4}$. For any
$\varphi$ the unit cell is inversion symmetric. (b) Equivalent tight binding
unit cells. Solid and dashed black lines strong and weak hoppings. (c)
Plasmonic bands of the periodic system. Solid lines represent the bands for
$\varphi=0$, dashed ones represent the bands for
$\varphi=-\frac{\pi}{4},\frac{\pi}{4}$ and dotted lines are
$\varphi=-\varphi_{t},\varphi_{t}$, with $\varphi_{t}\sim 0.16\pi$, where the
lower and upper gaps close and topological transitions occur. (d) Plasmonic
spectrum of a finite chain of 99 nanoparticles (33 unit cells). Bulk states
are represented by blue lines, while green curves represent the two pairs of
three-way-chiral edge states, that appear between $-\varphi_{t}$ and
$\varphi_{t}$. (e) Zak phase of lower/upper gaps, which matches with the
existence of edge states in panel (d). (f) Edge state for
$\varphi=-\frac{\pi}{30}$. Due to the three-way chirality the edge state is
localized at the two sublattices closer to each edge.

However, if we enlarge the unit cell, we can open a gap in an array of
equidistant particles. Let us consider a 3-particle unit cell with
$R=\frac{d}{3}$. Then the Green dyadic’s matrix is:

$\displaystyle\mathcal{G}(q)=\begin{pmatrix}0&G_{\textbf{u}_{A},\textbf{u}_{B}}&G_{\textbf{u}_{A},\textbf{u}_{C}}e^{-iqd}\\\
G_{\textbf{u}_{A},\textbf{u}_{B}}&0&G_{\textbf{u}_{B},\textbf{u}_{C}}\\\
G_{\textbf{u}_{A},\textbf{u}_{C}}e^{iqd}&G_{\textbf{u}_{B},\textbf{u}_{C}}&0\end{pmatrix},$
(27)

which is equivalent to the Bloch Hamiltonian of the SSH3 model (EQ. 7). The
condition for the unit cell to be mirror-symmetric is
$\varphi_{A}=-\varphi_{C}$ and $\varphi_{B}=0,\frac{\pi}{2}$. The condition
for the unit cell to be inversion-symmetric is, however, less restrictive. As
a single nanospheroid is inversion-symmetric, the only condition is that
particles $A$ and $C$ are inversion-symmetric with respect to each other, that
is $\varphi_{A}=\varphi_{C}$.

The condition for $\mathcal{G}(q)$ to be equivalent to the mirror-symmetric
SSH3 is
$|G_{\textbf{u}_{A},\textbf{u}_{B}}|=|G_{\textbf{u}_{2},\textbf{u}_{3}}|$.
This is satisfied by any two pair of angles $\varphi_{A},\varphi_{B}$ and
$\varphi_{B},\varphi_{C}$ that lay on the same or opposite contour line in
Fig. 2(d). Explicitly, this occurs for
$\varphi_{B}=\arctan\left(\frac{2\left(\cos{\varphi_{A}}\pm\cos{\varphi}_{C}\right)}{\left(\sin{\varphi_{A}}\pm\sin{\varphi_{C}}\right)}\right)$.
This includes mirror symmetric and inversion symmetric previous conditions.
However, it goes beyond them. For example, if we fix
$\varphi_{A}=0,\varphi_{C}=\frac{\pi}{4}$, then the Green dyadic is
accidentally mirror symmetric for $\varphi_{B}\sim 0.43\pi$ and
$\varphi_{B}\sim 0.78\pi$.

This accidental symmetry stems from the fact that we are ignoring the
orientation of the nanoparticles in the equations, so it’s a symmetry of the
strength of the interaction between particles. Due to the anisotropy between
longitudinal and transversal modes, these symmetries happen for apparently
random values of the orientations. However, this accidental symmetry is enough
to quantize the Zak phase, as in the true mirror symmetric and inversion
symmetric cases.

In FIG. 3 we show the topological transition in this plasmonic analogue of the
mirror-symmetric SSH3 model with $a=12.5$nm, $d=15a$ and $R=\frac{d}{3}=5a$.
By orienting the nanoparticles at
$\varphi_{A}=\varphi_{B}+\frac{\pi}{2}=\varphi_{C}=\varphi$, upper and lower
gaps close when $|G_{u_{C},u_{A}}|=|G_{u_{A},u_{B}}|=|G_{u_{B},u_{C}}|$, that
is, for $\varphi=\pm\varphi_{t}\sim\pm 0.16\pi$. For $|\varphi|>\varphi_{t}$
and $|G_{u_{C},u_{A}}|<|G_{u_{A},u_{B}}|$, the system is in the trivial phase,
while for $|\varphi|<\varphi_{t}$ and $|G_{u_{C},u_{A}}|>|G_{u_{A},u_{B}}|$
there are non-zero edge states in the upper/lower gaps. We can see the gaps
closing and reopening for the bands of the periodic chain in panel (c) and for
the finite chain (d). After the closing, double degenerate topological edge
states (red solid lines in panel (d)) arise at the edges of the chain,
localizing in odd sublattices at left edges and even sublattices at right
edges due to sublattice symmetry. The appearance of these edge states matches
the steps in the Zak phase, shown in panel (e). We also show one of the edge
states of the lower gap for $\varphi_{D}=-\varphi_{B}=\frac{\pi}{4}$ in FIG.
3(f), which as we see inherits the symmetry of the three-way chirality,
localizing in two of the sublattices.

Interestingly, the asymmetry of transversal and longitudinal Green dyadic’s
functions can be exploited not just to open a gap but also to engineer
accidental spatial symmetries or to suppress interaction between particles.
This system is therefore more flexible than the linear and zigzag chains of
nanospheres, allowing to play with symmetries, which yield a response of the
edge states tunable by the external electric field, to be studied in last
section.

####  III.3.3 4 particles/unit cell

Figure 4: Plasmonic analogue of the SSH4 model: periodic chain of 4 prolate
silver nanospheroids per unit cell, with major axes forming angles with the
chain direction
$\varphi_{A}=\varphi_{B}-\frac{\pi}{2}=\varphi_{C}-\frac{\pi}{2}=\varphi_{D}=\varphi$.
The dimensions of the nanoparticles are $a=12.5\text{ nm}$ and $b=c=0.4a=5$
nm. (a) Unit cells for $\varphi=0,\varphi=\frac{\pi}{4}$ and
$\varphi=\frac{\pi}{2}$. For all values of $\varphi$, the unit cell is mirror-
symmetric, but for $\varphi=0,\frac{\pi}{2}$ it is also inversion-symmetric.
(b) Equivalent tight binding unit cells. Solid and dashed black lines
represent weak and strong hoppings. (c) Plasmonic bands of the periodic
system. Solid curves represent the bands for $\varphi=0,-\frac{\pi}{2}$,
whereas dash ones represent the bands for $\varphi_{B}=0$, at the gap closing.
(d) Plasmonic spectrum of a finite chain of 100 nanoparticles (25 unit cells).
Bulk states are represented by blue lines, while red lines represent the pair
of edge states, that appear after the gap closing at $\varphi_{B}=0$. (e) Zak
phases of central gap (solid red line) and lower/upper gaps (dashed green
line), which match with the number of edge states in panel (d). The Zak phase
of the central gap is quantized by chiral and mirror/inversion symmetries,
while the lower and upper gap Zak phases are quantized just by
mirror/inversion symmetries. (f) Edge state for
$\varphi_{D}=-\varphi_{B}=\frac{\pi}{4}$. Due to sublattice symmetry left edge
chiral states (red arrows) localize at odd sublattices ($A$ and $C$) while
right edge state localize at even sublattices ($B$ and $D$), while four-way
chiral edge states (green arrows) are localized in the closest three
sublattices to the edge.

Next, we consider a larger linear unit cell of 4 nanoparticles separated by a
distance $R=\frac{d}{4}$. For this system, generally, $\mathcal{G}(q)$ in the
base ($A,C,B,D$) is:

$\begin{pmatrix}0&0&G_{\textbf{u}_{A},\textbf{u}_{B}}&G_{\textbf{u}_{A},\textbf{u}_{D}}e^{-iqd}\\\
0&0&G_{\textbf{u}_{B},\textbf{u}_{C}}&G_{\textbf{u}_{C},\textbf{u}_{D}}\\\
G_{\textbf{u}_{A},\textbf{u}_{B}}&G_{\textbf{u}_{B},\textbf{u}_{C}}&0&0\\\
G_{\textbf{u}_{A},\textbf{u}_{D}}e^{iqd}&G_{\textbf{u}_{C},\textbf{u}_{D}}&0&0\end{pmatrix}.$
(28)

This is equivalent to the Hamiltonian of the SSH4 model (EQ. 9) with hoppings
given by EQ. 25. As we see, this matrix is block-antidiagonal, i.e. sublattice
symmetric. This is because there are two sublattices (odd and even sites) with
only inter-sublattice connections.

The conditions for this system to be geometrically mirror (inversion)
symmetric are $\varphi_{1}=\mp\varphi_{4}$ and $\varphi_{2}=\mp\varphi_{3}$.
However, the only condition for $\mathcal{G}(q)$ to be effectively mirror-
symmetric is
$|G_{\textbf{u}_{A},\textbf{u}_{B}}|=|G_{\textbf{u}_{C},\textbf{u}_{D}}|$.
This is satisfied by any two pairs of $\varphi_{A},\varphi_{B}$ and
$\varphi_{C},\varphi_{D}$ that lie in the same or opposite contour line. True
or hidden mirror/inversion symmetries quantize the Zak phase of all gaps.

Figure 5: Plasmonic analogue of the SSH model: periodic chain of 4 prolate
silver nanospheroids per unit cell, with long spheroidal axes forming angles
with the chain direction $\varphi_{A}=\varphi_{C}=\frac{\pi}{4}$ and
$\varphi_{B}=-\varphi_{D}$ let free. The dimensions of the nanoparticles are
$a=12.5\text{ nm}$ and $b=c=0.4a=5$ nm. (a) Unit cells for
$\varphi_{D}=-\pi/4,0,\pi/4$. (b) Equivalent tight binding unit cells; the
effective unit cells are diatomic, as in the SSH model. Solid and dashed black
lines represent strong and weak bonds. (c) Plasmonic bands of the periodic
system. Solid curves represent the bands for
$\varphi_{B}=-\frac{\pi}{4},-\frac{\pi}{4}$, while dashed ones represent the
bands for $\varphi_{B}=0$, at the gap closing. (d) Plasmonic spectrum of a
finite chain of 100 nanoparticles (25 unit cells). Bulk states are represented
by blue lines, while red lines represent the pair of edge states, that appear
after the gap closing at $\varphi_{B}=0$. (e) Zak phase of central gap, which
matches the number of edge states in panel (d). (f) Edge state for
$\varphi_{D}=-\varphi_{B}=\frac{\pi}{4}$. Due to sublattice symmetry, the left
edge state is localized at odd sublattices ($A$ and $C$) while the right edge
state is localized at even sublattices ($B$ and $D$).

Figure 6: Excitation of edge states of different plasmonic chains by an
incoming linearly polarized electric field at normal incidence, depending on
its angle of polarization $\eta$. We plot the square of the module of dipolar
moments at each site $\textbf{p}_{i}$, normalized by its maximum value for all
polarizations and sites. We use red and green gradients for chiral (central
gap) in panels (a-d) and generalized 4-way chiral (lower gap) edge states in
panels (e-h). As insets we plot the spectra of the chains, where blue, red and
green dots represent bulk states, central gap edge states, and lower/upper gap
edge states. (a) Mirror symmetric unit cell (see Fig
5),$\varphi_{n}=\pi/4[-1,-1,1,1]$. The chain breaks inversion symmetry, while
incident electric field breaks mirror symmetry (except for $\eta=0$ and
$\eta=\pi$), allowing to switch off left or right edge states separately for
$\eta\sim\frac{\pi}{7}$ and $\eta\sim\frac{6\pi}{7}$. Then, when applying a
circular incident electric field, edge states bounce back and forth between
the edges. (b) Inversion symmetric unit cell. Electric field doesn’t break
inversion symmetry, but allows to switch on and off both edge states
simultaneously. (c) Accidentally mirror symmetric unit cell. As both true
mirror and inversion symmetries are broken, both the amplitude and phase of
the edge states differ. (d) Non-spatial-symmetric unit cell. As left-right
edge states are still degenerate due to chiral symmetry, we have a response
for both edges similar to case (c). (e) Mirror symmetric unit cell. (f,g)
Accidentally mirror symmetric unit cells. (h) Non-spatial-symmetric unit cell.
Due to the absence of symmetries, left and right edge states are not
degenerate and can be excited separately.

In FIG. 4 we show a plasmonic analogue of the SSH4 model with $a=12.5$nm,
$d=15a$ and $R=\frac{d}{4}=3.75a$. By starting from the unit cell with
$\varphi_{A}=\varphi_{B}-\pi/2=\varphi_{C}-\pi/2=\varphi_{D}=0$ (left unit
cell in panel (a)) and rotating all nanoparticles an angle $\varphi$, two
different topological transitions are crossed. As we see in panels (c) and
(d), the first one occurs at $\varphi=\varphi_{t}\sim 0.12\pi$
($\varphi=\frac{\pi}{2}-\varphi_{t}$), where
$|G_{\textbf{u}_{A},\textbf{u}_{B}}G_{\textbf{u}_{C},\textbf{u}_{D}}|=|G_{\textbf{u}_{B},\textbf{u}_{C}}G_{\textbf{u}_{D},\textbf{u}_{A}}|$
and the central gap closes. After reopening, the edge states (red solid line
in panel (d)) disappear (appear). For $\varphi=0$,
$G_{\textbf{u}_{D},\textbf{u}_{A}}=G_{\textbf{u}_{B},\textbf{u}_{C}}$ and
$G_{\textbf{u}_{A},\textbf{u}_{B}}=G_{\textbf{u}_{C},\textbf{u}_{D}}$, so
upper and lower gaps close and its edge states (green solid line in panel (d))
disappear after the closing, when $G_{u_{D},u_{A}}<G_{u_{B},u_{C}}$. In panel
(e) we show the Zak phase for all the gaps, which represent the existence of
edge states in each gap. In panel (f) we plot the dipoles for the edge states
in the central (red arrows) and lower (green arrows) gaps. As we see, the
former respects the sublattice symmetry and is localized only in odd or even
sublattices, while the latter has only zero weight in one of the sublattices.

Interestingly, due to accidental symmetries, in this system we can also
recover the topology of the SSH model. When
$G_{\textbf{u}_{D},\textbf{u}_{A}}=G_{\textbf{u}_{B},\textbf{u}_{C}}$ and
$G_{\textbf{u}_{A},\textbf{u}_{B}}=G_{\textbf{u}_{C},\textbf{u}_{D}}$ (or
equivalently, $t=v$ and $u=w$ in FIG. 1(c)), upper and lower gaps close and we
have an analogue of the SSH model. Even when the period of the real unit cell
is $4R$, the effective tight binding unit cell has a period of $2R$.

In FIG. 5 we see a possible realization of this analogue of the SSH for
$a=12.5$nm, $d=15a$ and $R=\frac{d}{4}=3.75a$. By fixing the direction of the
nanoparticles in sites $A$ and $C$ at
$\varphi_{A}=-\varphi_{C}=-\frac{\pi}{4}$ and rotating
$\varphi_{B}=-\varphi_{D}$ from $-\frac{\pi}{4}$ (left unit cell in FIG. 5(a))
to $\frac{\pi}{4}$ (right unit cell), a topological transition occurs at
$\varphi_{B}=-\varphi_{D}=-\frac{\pi}{4}=0$ (middle unit cell), where the gap
closes due to all $G_{\textbf{u}_{i},\textbf{u}_{j}}$ being equivalent, as in
the SSH model for $v=w$ (middle unit cell in panel (b)). In this system
there’s also a transition from an inversion symmetric unit cell
$(\varphi_{B}=-\frac{\pi}{4})$ to an accidentally mirror symmetric one
$(-\frac{\pi}{4}<\varphi_{B}<\frac{\pi}{4})$ to a mirror symmetric unit cell
$(\varphi_{B}=\frac{\pi}{4})$, so the equivalent tight binding unit cells
(panel (b)) always remain mirror symmetric, as in the SSH. We can see the gap
closing and reopening for the bands of the periodic chain. (panel (c)) and for
the finite chain (panel (d)). We also plot the Zak phase (panel (e)), that
compared to that of panel (d), we see it represents the existence of edge
states (red line). After the closing, double degenerate topological edge
states arise at the edges of the chain, localizing in odd sublattices at the
left edge and in even sublattices at the right edge due to sublattice
symmetry. We show one of the edge states for
$\varphi_{D}=-\varphi_{B}=\frac{\pi}{4}$ in panel (f).

In the next section we will study how these edge states are excited by an
incident electric field, depending on its polarization.

##  IV Switching edge states by incoming electric field

A difference between electronics and plasmonics is that plasmons are not
fermions, so the bands are not naturally half-filled. In photonics, we need an
incident field that overlaps spatially with the eigensolutions of the array.
Once the incident field is fixed, by inverting EQ. 12, the dipoles of the
chain are given by:

$\displaystyle\textbf{P}=\left(\mathcal{G}(\omega)-\frac{1}{\alpha(\omega)}I\right)^{-1}\textbf{E}_{inc}(\omega,t),$
(29)

$\textbf{E}_{inc}$ being a $3\times N$ vector which contains the field
$E_{inc}(t)$ evaluated at the position of each nanoparticle. For a chain of
nanospheroids, the equation reduces to:

$\displaystyle\textbf{P}_{a}=\left(\mathcal{G}_{a}(\omega)-\frac{1}{\alpha_{a}(\omega)}I\right)^{-1}\textbf{E}^{a}_{inc}(\omega,t),$
(30)

where $\textbf{E}_{inc}^{a}$ is a vector of the projections of the electric
field in the directions of the major axes of the nanoparticles. Such
projections are key in order to excite or not protected states. When all the
particles in the array are spherical or oriented in the same direction, the
electric fields affect almost equally all the nanoparticles. However, when
nanoparticles are oriented in different directions, the external electric
field can couple to real and hidden spatial symmetries.

Let’s analyze what happens when we excite the edge states of the plasmonic
chains. In FIG. 6 we plot the dipolar response to a linearly polarized
electric field at normal incidence, depending on its polarization:

$\textbf{E}_{inc}(\omega,t)\propto\begin{pmatrix}\cos\eta\\\ \sin\eta\\\
0\end{pmatrix},$ (31)

where $\eta$ is the angle of polarization of the electric field with respect
to the $x$ axis. In order to the field to resonate with the nanoparticles and
with the edge state mode, due to the losses of the nanoparticles, we need the
incoming electric field to have the frequency of the edge state and a finite
lifetime, that is, a pulse.

In FIG. 6(a-d) we analize the edge states of the central gap, which are
protected by chiral symmetry. In panels (e-h) we excite the edge states in the
lower gap of the SSH4 chain, which are 4-way-chiral-symmetric. For both types
we consider chains with mirror, inversion, accidental mirror, and no spatial
symmetries to see how this affects the optical response.

In FIG. 6(a) we see the response of the SSH nanospheroid chain with mirror
symmetry and $-\varphi_{A}=-\varphi_{B}=\varphi_{C}=\varphi_{D}=\frac{\pi}{4}$
to a linearly polarized electric field at the frequency of the surface plasmon
$\omega_{spa}$, depending on its polarization. Since all the nanoparticles are
oriented at diagonals when the field is polarized in $x$ or $y$ directions
$(\eta=0,\frac{\pi}{2})$, all the particles are equally perturbed so mirror
symmetry holds, and left and right edge states are identical.

However, when we apply an electric field oriented at $\eta\neq
0,\frac{\pi}{2}$, the interaction with the external field depends on the
nanoparticle. The sublattice symmetry is still preserved, as we see in Fig 6
(a). However, the external field breaks the mirror symmetry, allowing to have
a different response at left and right edges. For $\eta\simeq\pi/7$
($\eta\simeq\frac{6\pi}{7}$), the dipolar response is localized only at the
left (right) edge. Then, by changing the polarization of the field we can
select left, right or both edge states with the same or different weight.

Now if we apply a circularly polarized electric field at normal incidence,
this is:

$\textbf{E}_{inc}(\omega,t)\propto\begin{pmatrix}\cos(\eta(t))\\\
\sin(\eta(t))\\\ 0\end{pmatrix}.$ (32)

The nanospheroids convert the circular polarization of the incoming field to
linear polarization. Then, the oscillations in left and right edges are not in
phase, so the edge states ”bounce back and forth” between left and right
edges. Over a period T, the response of the chain loops two times over the
$\eta$ axis in FIG. 6(a).

However, when the chain is inversion-symmetric, for example the one in panel
(c) ($\varphi_{n}=\pi/8+[0,\pi/2,\pi/2,0]$), the electric field preserves this
symmetry, so the response in both edges is the same. We can switch on/off both
edge states simultaneously. If the electric field is circularly polarized,
then the oscillations in both edges are in phase.

When the chain is accidentally mirror-symmetric (panel (c)), or has no spatial
symmetries, the field couples more intensely to one of the edges. If we apply
a circularly polarized electric field, the oscillations in the edges would not
be just dephased as the bouncing states in the mirror symmetry chain, but they
would differ also in amplitude.

For the edge states in lower (or upper) gaps, however, we find a different
scenario. In a mirror symmetric SSH4 chain (panel (e),
$\varphi_{n}=[\pi/4,\pi/2,\pi/2,\pi/4]$), the external field doesn’t appear to
break the symmetry between edges. This may be due to the coexistence of
generalized chiral symmetry and spatial symmetries. This means that we cannot
select right or left edges. If the field is circularly polarized, then the
oscillations in both edges are in phase. The same occurs for an inversion-
symmetric unit cell (panel (f), $\varphi_{n}=[0,\pi/4,\pi/4,\pi/4,0]$) and in
the accidentally symmetric case (panel (g),
$\varphi_{n}=[0,\pi/4,\pi/4,0.71\pi])$).

Finally, if the SSH4 unit cell has no spatial symmetries (panel (h),
$\varphi_{n}=[0,\pi/4,\pi/4,0.8\pi]$), left and right lower/upper gap edge
states are no longer degenerate, so we can excite them separately at different
frequencies. However, these edge states don’t have any kind of topological
protection and can be pushed out of the gap by disorder and hybridize with
bulk states.

As we see, by using elongated nanoparticles we have gained control in edge
states, making possible to switch them off, select left, right, both, or
bouncing edge states. This wasn’t feasible in the nanosphere chain, as all the
particles were identical and have isotropic response for all the polarizations
of the electric field.

##  V Conclusions

In previous years, there have been several proposals to mimic topological
electronic systems in photonics. Periodic arrays of metallic nanoparticles are
an interesting platform to study topology in nanophotonics due to their
plasmonic resonances in the visible range and their tunability. Here we have
proposed means to open a topological gap not by rearranging the particles in
an array as in crystalline topological electronic systems, but by orientating
elongated particles. By adding this degree of freedom, we can mimic
topological chains as the SSH model or its greater unit cell extensions in an
equidistant array. The spatial polarization modulation allows also to switch
on/off or select right, left or bouncing edges states, by changing the
polarization of the incoming electric field, as in the zigzag plasmonic chain.
However, this system is even more flexible, making possible also to suppress
interaction between nanoparticles or to engineer accidental spatial
symmetries. In this paper we proved that orientation of nanoparticles can open
a topological gap in an otherwise gapless system (which could also be
exploited in gapped $1D$ and $2D$ arrays), allowing to filter modes or play
with symmetries and polarizations. This opens a path towards exploiting
features of nanoparticles for topology without a counterpart in condensed
matter systems.

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##  Appendix A Polarizability of metallic nanospheres and nanospheroids

Generally, the electric polarizability $\overleftrightarrow{\alpha}(\omega)$
acts like a tensor:

$\displaystyle\overleftrightarrow{\alpha}(\omega)=\begin{pmatrix}\alpha_{xx}&\alpha_{xy}&\alpha_{xz}\\\
\alpha_{xy}&\alpha_{yy}&\alpha_{yz}\\\
\alpha_{xz}&\alpha_{yz}&\alpha_{zz}\end{pmatrix}.$ (33)

For a nanosphere, due to the spherical symmetry, the polarizability is
proportional to the identity matrix,
$\overleftrightarrow{\alpha}(\omega)=\alpha(\omega)I$. When we are working in
the limit $a\gg\lambda$ we can take the quasi-static approximation, assuming
only the first Mie coefficient contributes to the polarizability, so
$\alpha(\omega)$ is:

$\displaystyle\alpha(\omega)=4\pi
a^{3}\epsilon_{0}\frac{\epsilon(\omega)-\epsilon_{\mathrm{B}}}{\epsilon(\omega)+2\epsilon_{\mathrm{B}}}.$
(34)

The permittivity of the medium, $\epsilon(\omega)$ can be approximated by a
Drude-Lorentz model [40]:

$\epsilon(\omega)=\epsilon_{r}-\sum_{j}\frac{\omega^{2}_{\mathrm{P},j}}{\omega(\omega+i\gamma_{j})}-\sum_{j}\frac{\Delta\epsilon_{j}\Omega^{2}_{j}}{\omega^{2}-\Omega^{2}_{j}+i\omega\Gamma_{j}},$
(35)

where $\epsilon_{r}$ is the static dielectric constant,
$\omega_{\mathrm{P},j}$ are plasma frequencies, $\gamma_{j}$ and $\Gamma_{j}$
are the damping constants, $\Omega_{j}$ are resonant frequencies, and
$\Delta\epsilon_{j}$ are related to the oscillator strengths. In our work we
use the parameters for silver:
$\epsilon_{r}=4.6,\;\omega_{\mathrm{P},0}=9.0,\;\gamma_{0}=0.07,\;\Gamma_{0}=1.2,\;\Omega_{0}=4.9,\;\Delta\epsilon_{0}=1.10$
[40].

The extinction cross section is the sum of absorption and scattering cross
sections and for a single nanoparticle as given by [46]:

$\sigma_{\mathrm{ext}}=\sigma_{\mathrm{abs}}+\sigma_{\mathrm{sca}}=\frac{k}{\epsilon_{0}}\textnormal{Im}{(\alpha(\omega)})+\frac{k^{4}}{6\pi\epsilon_{0}^{2}}|\alpha(\omega)|^{2}.$
(36)

Now we consider prolate spheroids, that is, ellipsoids with major axis $a$ and
minor axes $b=c$. The polarizability doesn’t behave like a scalar anymore, but
depends on the polarization of the incoming electric field. The
polarizabilities for the main axes, $\alpha_{l}(\omega)$ with $l\in[a,b,c]$,
are [44]:

$\alpha_{l}(\omega)=V\frac{\epsilon(\omega)-\epsilon_{b}}{\epsilon_{b}+L_{l}(\epsilon(\omega)-\epsilon_{b})},$
(37)

$V$ being the volume of the spheroid, $V=\frac{4}{3}\pi ac^{2}$, and $L_{l}$
are geometric factors given by:

$\displaystyle
L_{a}=\frac{e^{2}}{1-e^{2}}\left(\frac{1}{2\sqrt{1-e^{2}}}\ln\left(\frac{1+\sqrt{1-e^{2}}}{1-\sqrt{{1-e^{2}}}}\right)-1\right),$
$\displaystyle L_{b}=L_{c}=\frac{1-L_{a}}{2},$ (38)

where $e=c/a$ is the eccentricity of the spheroid, ranging from $e=0$ (needle)
to $e=1$ (sphere).

Let us now consider an array of nanoparticles, where the major axis of the
spheroid $n$ is in the direction
$\textbf{u}_{n}=(\sin{\theta_{n}}\cos{\varphi_{n}},\sin{\theta_{n}}\sin{\varphi_{n}},\cos{\theta_{n}})$,
where $\theta_{n}$ and $\varphi_{n}$ are the angles formed by the spheroidal
major axis with respect to the $z$ and $x$ axis. When $q\ll 1$, in the
vicinity of the major axis resonance $\omega_{spa}$,
$\alpha_{c}(\omega\simeq\omega_{spa})\simeq 0$, we can approximate the
polarizability tensor as:

$\overleftrightarrow{\alpha_{n}}(\omega)\simeq\alpha_{a}(\omega)\begin{pmatrix}\sin^{2}\theta_{n}\cos^{2}\varphi_{n}&\sin^{2}\theta_{n}\sin\varphi_{n}\cos\varphi_{n}&\sin\theta_{n}\cos\theta_{n}\cos\varphi_{n}\\\
\sin^{2}\theta_{n}\sin\varphi_{n}\cos\varphi_{n}&\sin^{2}\theta_{n}\sin^{2}\varphi_{n}&\sin\theta_{n}\cos\theta_{n}\sin\varphi_{n}\\\
\sin\theta_{n}\cos\theta_{n}\cos\varphi_{n}&\sin\theta_{n}\cos\theta_{n}\sin\varphi_{n}&\cos^{2}\theta_{n}\end{pmatrix},$
(39)

which projects any vector v in the direction v, i.e.
$\overleftrightarrow{\alpha}(\omega)\textbf{v}=\alpha_{a}(\omega)(\textbf{v}\cdot\textbf{u}_{n})\cdot\textbf{u}_{n}$.

As the direction of the dipoles is fixed, we can project $\textbf{p}_{n}$ in
the direction $u_{n}$, so we get the scalar coupled-dipole equations in EQ.
22.

##  Appendix B Green dyadic’s function projection

As we show in appendix A, near $\omega_{spa}$ the polarizabilities of the
nanospheroids project the dipoles in the directions of the major axes. This
turns in scalar coupled-dipole equations (see EQ. 22). Explicitly, the Green’s
dyadic function projection for two nanospheroids with major axes oriented in
$\textbf{u}_{n}=(\sin{\theta_{n}}\cos{\varphi_{n}},\sin{\theta_{n}}\sin{\varphi_{n}},\cos{\theta_{n}})$
and
$\textbf{u}_{m},(\sin{\theta_{m}}\cos{\varphi_{m}},\sin{\theta_{m}}\sin{\varphi_{m}},\cos{\theta_{m}})$
directions, $G_{\textbf{u}_{m},\textbf{u}_{n}}$, is:

$\begin{split}G_{\textbf{u}_{m},\textbf{u}_{n}}=G_{xx}\cos{\varphi_{m}}\cos{\varphi_{n}}\sin{\theta_{m}}\sin{\theta_{n}}+G_{xy}\sin{\theta_{m}}\sin{\theta_{n}}\sin{(\varphi_{m}+\varphi_{n})}+G_{xz}\left(\cos{\theta_{m}}\cos{\varphi_{n}}\sin{\theta_{n}}+cos{\theta_{n}}\cos{\varphi_{m}}\sin{\theta_{m}}\right)+\\\
+G_{yy}\sin{\varphi_{m}}\sin{\varphi_{n}}\sin{\theta_{m}}\sin{\theta_{n}}+G_{yz}\left(\cos{\theta_{m}}\sin{\varphi_{n}}\sin{\theta_{n}}+cos{\theta_{n}}\sin{\varphi_{m}}\sin{\theta_{m}}\right)+\\\
+G_{zz}\cos{\theta_{m}}\cos{\theta_{n}},\end{split}$ (40)

where $\mu,\nu=x,y,z$ are the polarizations and $G_{\mu\nu}$ are the elements
of the Green dyadic ${\overleftrightarrow{{\bf
G}}}(\textbf{r}_{m},\textbf{r}_{n},\omega)$.

For a linear array of nanoparticles along the $x$ direction, EQ. 40 reduces
to:

$\displaystyle
G_{\textbf{u}_{m},\textbf{u}_{n}}=\frac{(2\cos{\varphi_{m}}\cos{\varphi_{n}}-\sin{\varphi_{m}}\sin{\varphi_{n}})\sin{\theta_{m}}\sin{\theta_{n}}-\cos{\theta_{m}}\cos{\theta_{n}}}{4\pi
k^{2}R^{3}},$ (41)

which for $\theta_{n}=\theta_{m}=\frac{\pi}{2}$ derives in EQ. 25. In this
paper, we restricted the orientations of the dipoles to the $xy$ or $xz$
planes, but by orienting them in the space, we could add another extra degree
of freedom.