# Spin-polarized antichiral exciton-polariton edge states

Ruiqi Bao  S. Mandal<EMAIL_ADDRESS>Huawen Xu  Xingran Xu  R.
Banerjee  Timothy C. H. Liew<EMAIL_ADDRESS>Division of Physics and
Applied Physics, School of Physical and Mathematical Sciences, Nanyang
Technological University, Singapore 637371, Singapore

###### Abstract

We consider theoretically a system of exciton-polariton micropillars arranged
in a honeycomb lattice. The naturally present TE-TM splitting and an
alternating Zeeman splitting, where the different sublattices experience
opposite Zeeman splitting, shifts the Dirac points in energy, giving rise to
antichiral behavior. In a strip geometry having zigzag edges, two pairs of
edge states exist and propagate in the same direction (including the states at
the opposite edges). The edge modes localized at the opposite edges have
opposite spins (circular polarizations), which leads to co-propagating
$``\pm"$ spin channels. The antichiral edge states are protected by non-zero
winding numbers and can propagate around a 60 degree bend without being
reflected. We further compare the transport properties of these edge states
with chiral edge modes and propose a scheme to realize them experimentally.

##  I Introduction

The quantum Hall effect Klitzing  _et al._ (1980), where free electron gas
subjected to a perpendicular magnetic field shows quantized Hall conductance,
has laid the platform for topological phases. A few years later, the Haldane
model was proposed Haldane (1988), where the quantum Hall effect was shown
without any net magnetic field. The Haldane model has gone onto become the
backbone of many noble topological systems such as anomalous Chern insulators
Jotzu  _et al._ (2014); Zhao  _et al._ (2020), quantum spin-Hall insulators
Kane and Mele (2005), etc.

Recently, a new type of topological phase, which shows antichiral edge states,
has been introduced based on a modified-Haldane model Colomés and Franz
(2018). In contrast to previous topological phases, systems with antichiral
edge states do not host a bulk band gap. Instead, in a strip geometry both the
edge states propagate in the same direction and counter-propagating states
within the same energy window lie in the bulk. Although, the antichiral edge
states reside along with the bulk modes, they were shown to be reasonably
robust against disorder Colomés and Franz (2018). Because of their peculiar
properties, the antichiral edge states have been under intense investigation
in various system including in exciton-polaritonics Mandal  _et al._ (2019a),
photonics Chen  _et al._ (2020); Zhou  _et al._ (2020); Chen and Li (2022),
acoustics Yu  _et al._ (2021); Wu  _et al._ (2020), electric circuits Yang
_et al._ (2021) and others Bhowmick and Sengupta (2020); Mannaï and Haddad
(2020); Breitkreiz (2020); Mizoguchi and Koma (2021); Cheng  _et al._ (2021).

Figure 1: Schematic of a honeycomb lattice consisting exciton-polariton
micropillars. “A” and “B” correspond to the different sublattices of the
system, which experience opposite Zeeman splittings. In presence of the
sublattice dependent Zeeman splitting and the TE-TM splitting, spin-polarized
antichiral edge states appear, where opposite edge attain opposite spins and
propagate in the same direction.

Although, the antichiral edge states have been explored over a variety of
systems, the spin degree of freedom has not been associated with them till
now. In this work, we propose theoretically spin-polarized antichiral edge
states in an exciton-polariton system. Exciton-polaritons are quasi particles
that arise due to the strong coupling between quantum well excitons and
microcavity photons Deng  _et al._ (2010); Carusotto and Ciuti (2013); Byrnes
_et al._ (2014). Due to their excitonic component, polaritons exhibit strong
nonlinearity, which has lead to a variety of optical devices Sanvitto and
Kéna-Cohen (2016); Feng  _et al._ (2021), optical computation Kyriienko  _et
al._ (2019); Xu  _et al._ (2020); Banerjee and Liew (2020); Harrison  _et al._
(2022), and others Xu  _et al._ (2021a); Zvyagintseva  _et al._ (2022). The
finite lifetime of the photons gives rise to interesting non-Hermitian
physics. Circularly polarized photons provide a spin degree of freedom to the
polaritons, which show many rich effects. For example, the naturally present
energy splitting between the transverse electric (TE) and transverse magnetic
(TM) modes (also known as the TE-TM splitting) acts as an effective magnetic
field Kavokin  _et al._ (2005), which leads to the realization of the optical
spin-Hall effect both in the linear Leyder  _et al._ (2007) and nonlinear
Kammann  _et al._ (2012) regimes. The excitonic component along with the spin
degree of freedom leads to the Zeeman splitting in presence of perpendicular
magnetic field Larionov  _et al._ (2010). These fascinating properties of
polaritons make them ideal for studying topological phases in polariton
lattices.

Polariton Chern insulators Karzig  _et al._ (2015); Bardyn  _et al._ (2015);
Nalitov  _et al._ (2015); Banerjee  _et al._ (2018); Li  _et al._ (2018); Sun
_et al._ (2019) and related phases Solnyshkov  _et al._ (2018); Mandal  _et
al._ (2019b) have been explored theoretically and later realized
experimentally Klembt  _et al._ (2018). Strong nonlinear polariton-polariton
interaction gives rise to different kinds of nonlinear topological behavior
including topological solitons Kartashov and Skryabin (2016); Gulevich  _et
al._ (2017); Pernet  _et al._ (2022); Solnyshkov  _et al._ (2017), interaction
induced chiral edge states Bleu  _et al._ (2017), bistable topological
insulators Kartashov and Skryabin (2017), nonlinear higher-order topological
insulators Banerjee  _et al._ (2020), etc. In some cases, nonlinearity can
alone induce topological phase transition Bardyn  _et al._ (2016); Sigurdsson
_et al._ (2017); Banerjee  _et al._ (2021). The inherent non-Hermiticity of
the polaritons has also lead to a variety of non-Hermitian topological
behavior such as exceptional points Gao  _et al._ (2015a, 2018); Khurgin
(2020); Comaron  _et al._ (2020); Su  _et al._ (2021); Liao  _et al._ (2021);
Krol  _et al._ (2022), non-reciprocal transport Mandal  _et al._ (2020); Xu
_et al._ (2021b), skin effect Mandal  _et al._ (2022), double-sided skin
effect Xu  _et al._ (2021c), non-Hermitian topological edge-modes Pickup  _et
al._ (2020); Comaron  _et al._ (2020), etc. However, none of the above works
can show topologically protected spin transport and the search for a system
exhibiting robust spin transport is still an open question.

Here, we consider exciton-polaritons in a honeycomb lattice in the presence of
the TE-TM splitting and subjected to an alternating Zeeman splitting, where
all the “A” (“B”) sublattices are considered to have positive (negative)
Zeeman splitting. The same system with a uniform Zeeman splitting is known to
give rise to a Chern insulating phase, where in a strip geometry, the bulk
band gap hosts two pairs of counter-propagating edge states Nalitov  _et al._
(2015); Bardyn  _et al._ (2015); Klembt  _et al._ (2018). Unlike the Chern
insulator case, our present system does not show a bulk bandgap. Instead, the
Dirac points split in energy. The non-trivial topology of the system connects
the Dirac points with two pairs of zigzag edge modes with each pair residing
on the opposite edges of a finite strip. Consequently, the edge modes
including the opposite edges obtain the same sign of the group velocity.
Unlike the Chern insulator case, here the system becomes antichiral, where all
the edge modes propagate in the same direction. We further find that the edge
states residing at the opposite edges of the strips have opposite circular
polarization, which enables robust spin transport along the edges of the
system. However, due to the fact that the edge modes reside with the counter-
propagating bulk modes, the system is less robust compared to their Chern
insulator counterpart. Our simulation shows that the system shows fair
robustness against lattices defects and the efficiency of pulse propagation
remains lower than but close to the Chern insulators.

Figure 2: (a)Honeycomb lattice consisting of two types of sublattices, “A” and
“B”. $a_{1}$ and $a_{2}$ are the primitive lattice vectors. (b) Lines
connecting the high symmetry points of the 2D Brillouin zone, along which the
band structures are calculated. (c) Band structure for a trivial Honeycomb
lattice ($\Delta_{T}=\Delta_{A}=\Delta_{B}=0$). (d) Effect of Zeeman splitting
on the honeycomb lattice band structure
($\Delta_{T}=0,~{}\Delta_{A}=\Delta_{B}=0.3J$). (e) Effect of the TE-TM
splitting on the band structure
($\Delta_{T}=0.3J,~{}\Delta_{A}=\Delta_{B}=0$). (f) Chern insulating band
structure with a non-trivial bulk bandgap
($\Delta_{T}=\Delta_{A}=\Delta_{B}=0.3J$). (g) Antichiral band structure with
the two valleys shifted in energy ($\Delta_{T}=\Delta_{A}=-\Delta_{B}=0.3J$).
(h) Winding numbers as a function of $k_{1}$ for the antichiral bands, which
become nontrivial for $k_{1}>|k_{D}|$, predicting the existence of two pairs
of antichiral edge states in that momentum window.

##  II The model

We start by considering two coupled micropillars “A” and “B” in presence of
the Zeeman splitting and the TE-TM splitting. The corresponding Hamiltonian
using the bispinor basis
$\Psi=[\Psi_{A}^{+},\Psi_{A}^{-},\Psi_{B}^{+},\Psi_{B}^{-}]^{T}$ can be
expressed in the tight-binding limit as

$\displaystyle\mathcal{H}_{AB}=\begin{bmatrix}\Delta_{A}&0&J&\Delta_{T}e^{-2i\theta}\\\
0&-\Delta_{A}&\Delta_{T}e^{2i\theta}&J\\\
J&\Delta_{T}e^{-2i\theta}&\Delta_{B}&0\\\
\Delta_{T}e^{2i\theta}&J&0&-\Delta_{B}\end{bmatrix}.$ (1)

Here $J$ is the coupling strength between the pillars, $\Delta_{T}$ is the TE-
TM splitting, $\theta$ is the orientation angle connecting the two pillars
with respect to a reference axis, and $\Delta_{A(B)}$ is the Zeeman splitting
corresponding to the pillar “A”(“B”). We stress that while staggered
potentials were considered in previous work predicting a valley-Hall effect
Bleu  _et al._ (2018), here it is the Zeeman splitting that is staggered. We
discuss explicit ways that it can be realized in section VI. The difference
between staggered potentials and staggered Zeeman splitting is that in the
case that a uniform magnetic field is applied together with a staggered
potential, we would generally obtain the chiral edge states of a Chern
insulator rather than antichiral ones. However, by properly tunning the
parameters it is possible to completely gap out one of the valleys, leaving
the other one gapped. Although, the system no longer remains a Chern
insulator, it can act as a robust spin filter, where only one of the two spins
are allowed to propagate along one of the edges. Such a regime is discussed in
details in Ref. Mandal  _et al._ (2019b).

Next, we arrange such pillar pairs in a honeycomb lattice structure and write
down its corresponding Hamiltonian in the reciprocal space by taking periodic
boundaries in both the directions. This can be achieved by writing
$\Psi^{\pm}_{A,B,n}=\Psi^{\pm}_{A,B}e^{i\mathbf{k}.{\mathbf{x}}_{n}}$ where
${\mathbf{x}}_{n}$ is the position of each site and reducing the basis back to
that of $[\Psi_{A}^{+},\Psi_{A}^{-},\Psi_{B}^{+},\Psi_{B}^{-}]^{T}$.

$\displaystyle\mathcal{H}_{\mathbf{k}}=\begin{bmatrix}\Delta_{A}&0&-g_{\mathbf{k}}J&-g^{+}_{\mathbf{k}}\Delta_{T}\\\
0&-\Delta_{A}&-g^{-}_{\mathbf{k}}\Delta_{T}&-g_{\mathbf{k}}J\\\
-g^{*}_{\mathbf{k}}J&-{g^{-}_{\mathbf{k}}}^{*}\Delta_{T}&\Delta_{B}&0\\\
-{g^{+}_{\mathbf{k}}}^{*}\Delta_{T}&-g^{*}_{\mathbf{k}}J&0&-\Delta_{B}\end{bmatrix},$
(2)

where

$g_{\mathbf{k}}=\sum_{n=1}^{3}\exp\left({-i\mathbf{k}.{\mathbf{r}}_{n}}\right)$

and

$g^{\pm}_{\mathbf{k}}=\sum_{n=1}^{3}\exp\left({-i\left[\mathbf{k}.{\mathbf{r}}_{n}\mp
2\theta_{n}\right]}\right).$

Here $\mathbf{r}_{n}$ represent the vectors connecting the three nearest “B”
sites from a single “A” site (Fig. 2(a)) and $\theta_{n}=2\pi(n-1)/3$ are the
angles of those vectors with respect to one of them (say $\mathbf{r}_{1}$).

To investigate the effect of $\Delta_{T}$, $\Delta_{A}$, and $\Delta_{B}$, we
plot the band structure of the system along a line connecting the high
symmetry points in the 2D Brillouin zone. Such a line is shown in Fig. 2(b).
Before proceeding, we define the degree of circular polarization corresponding
to a state as

$\displaystyle
S_{z}=\frac{\left|\Psi^{+}_{A}\right|^{2}+\left|\Psi^{+}_{B}\right|^{2}-\left|\Psi^{-}_{A}\right|^{2}-\left|\Psi^{-}_{B}\right|^{2}}{\left|\Psi^{+}_{A}\right|^{2}+\left|\Psi^{+}_{B}\right|^{2}+\left|\Psi^{-}_{A}\right|^{2}+\left|\Psi^{-}_{B}\right|^{2}}.$
(3)

The Hamiltonian of the system for $\Delta_{T}=\Delta_{A}=\Delta_{B}=0$
corresponds to trivial Graphene. Consequently, we obtain the Graphene band
structure by diagonalizing Eq. (2), where the lower and upper band touch each
other at the $K$ and $K^{\prime}$ valleys forming the Dirac cones (see Fig.
2(c)). In this case, the bands corresponding to the “$\pm$” spins are
degenerate or equivalently they are linearly polarized with $S_{z}=0$. This
degeneracy can be lifted by making $\Delta_{A}=\Delta_{B}=\Delta$, which
breaks the time-reversal symmetry (TR) and shifts the bands corresponding to
the “$\pm$” spins in energy by $2\Delta$ (see Fig. 2(d)). Next, we make
$\Delta_{T}\neq 0$ keeping $\Delta_{A}=\Delta_{B}=0$. In this case, the TR
symmetry is restored and the bands become linearly polarized. One significant
difference from the previous two cases is that the band structure is no longer
linear near the $K$ and $K^{\prime}$ valleys (see Fig. 2(e)). Making
$\Delta_{T}\neq 0$ and $\Delta_{A}=\Delta_{B}\neq 0$ gives rise to the non-
trivial topological band structure with the system transiting to a Chern
insulating phase having Chern number 2 Nalitov  _et al._ (2015); Bardyn  _et
al._ (2015); Klembt  _et al._ (2018). A nontrivial bulk band gap opens up due
to the band inversion, which can be identified by the opposite values of
$S_{z}$ of the second and third bands near the valleys (see Fig. 2(f)).
Finally, we introduce the antichiral band structure in Fig. 2(g) by choosing
$\Delta_{T}\neq 0$ and $\Delta_{A}=-\Delta_{B}\neq 0$. Unlike the Chern
insulator band structure, the bulk bands are not gapped here. Instead the
states near the $K$ and $K^{\prime}$ prime valleys shift in energy. The
introduction of $\Delta_{A(B)}$ breaks the TR symmetry. However, unlike Figs.
2(d,f) the bulk bands do not have any preferred spins and stay linearly
polarized.

The antichiral system does not have a bulk band gap. Consequently, it is not
possible to assign a Chern number to characterize the topology of the system.
It is possible to calculate a relevant winding number. However, unlike all the
previous antichiral systems, instead of one, two bands are involved here.
Consequently, one needs to define the non-abelian Berry connection in order to
calculate the winding number (see appendix). This is defined as Qi  _et al._
(2008)

$\displaystyle\mathcal{F}^{m,n}_{k_{1}}=i\langle
u^{m}_{k_{1}}(k_{2})\left|\frac{\partial}{\partial
k_{2}}\right|u^{n}_{k_{1}}(k_{2})\rangle,$ (4)

where $m,n=1,2$ corresponds to the lowest two bands as indicated in Fig. 2(g).
$u^{n}_{k_{1}}(k_{2})$ is the Bloch wave function. $k_{1}$ and $k_{2}$ are the
wave vectors along the $a_{1}$ and $a_{2}$ directions, respectively (see Fig.
2(a)). The winding number is given as

$\displaystyle\nu_{k_{1}}=\nu^{+}_{k_{1}}+\nu^{-}_{k_{1}}=\frac{1}{\pi}\oint_{\text{BZ}}dk_{2}\text{Tr}[\mathcal{F}_{k_{1}}],$
(5)

where $\nu_{k_{1}}$ is the total winding number of the system,
$\nu^{\pm}_{k_{1}}$ correspond to the contribution from the two bands, and the
integration is taken over the Brillouin zone along the ${k_{2}}$ direction for
each value of $k_{1}$. In Fig. 2(h) the winding numbers are plotted as a
function of $k_{1}$, which shows

$\displaystyle\nu^{+}_{k_{1}}=-\nu^{-}_{k_{1}}\approx\left\\{\begin{matrix}0,~{}\text{for}\left|k_{1}\right|<k_{D}\\\
1,~{}\text{for}\left|k_{1}\right|>k_{D}\end{matrix}\right.$ (6)

where $k_{D}$ is the position of the Dirac point. The total winding number of
the system is always 0. However, $\nu^{\pm}_{k_{1}}$ deviates from 0 and
becomes close to $\pm 1$ as shown in the shaded region in Fig. 2(h). Non-zero
winding corresponds to the non-trivial topology and predicts the existence of
antichiral edge states Colomés and Franz (2018). Two non-trivial branches in
the winding number predicts two pairs of antichiral edge modes. This is
different from previously studied systems, where only a pair of antichiral
edge modes are obtained.

Figure 3: Topological band structures for antichiral (a-b) and Chern insulator
(c-d) cases. The states are color coded with respect to the contribution from
the edges in (a,c) and with respect to the contribution from the spins in
(b,d). Antichiral edge states are co-propagating having almost pure spins in a
certain energy window. The chiral edge states in the Chern insulator case are
counter-propagating and have mixed spins. Here $a$ is the periodicity along
the $x$ direction, $\langle y\rangle$ represents of the mean position of the
states along the $y$ axis with $-L$ and $+L$ being the boundaries. $S_{z}$ is
defined in Eq. (3). All the parameters are kept the same as those in Fig. 2.

##  III Topological band structure

In this section, we provide topological band structures by considering the
system periodic along the $x$ direction and truncated along the $y$ direction
with the zigzag boundaries. In Figs. 3(a-b) the numerically calculated band
structure for the antichiral system $(\Delta_{T}=\Delta_{A}=-\Delta_{B})$ is
shown. As predicted from the non-zero winding numbers, indeed two pairs of
antichiral edge modes connecting the Dirac points appear and co-exist with the
bulk modes. Generally, the edge modes localized at the opposite edges are
degenerate. However, for better visualization, we have lifted the degeneracy
of the edge modes by adding a inversion symmetry breaking term
$m\sigma_{z}\otimes\sigma_{o}$ to the Hamiltonian in Eq. (2). Here
$\sigma_{z}$ is the $z$ component of the Pauli matrix, $\sigma_{o}$ is a
$2\times 2$ identity matrix, and $\otimes$ represents the tensor product. We
have chosen a small $m=10^{-2}$ for Fig. 3(a-b), which does not alter any
property of the system. This is necessary for visualizing the properties of
the edge modes from the band structure, otherwise the opposite edge modes in
each pair would overlap with each other making it difficult to distinguish
them.

To study the properties of the edge modes, we color code the band structure as
a contribution of localization in Fig. 3(a) and as a contribution of $S_{z}$
in Fig. 3(b). All the edge modes have the same group velocity, which makes
them propagate in the same direction. Such one directional propagation at the
edges is balanced by the counter propagating bulk modes. Although, the bulk
remains linearly polarized, the antichiral edge modes at the two opposite
edges have opposite values of $S_{z}$ and in a particular energy window
$S_{z}\rightarrow\pm 1$. As we show in the next section, this leads to
different spin channels in the same sample, where opposite spins propagate
along the different edges of a finite sample in the same direction.

In Figs. 3(c-d), we provide the Chern insulating band structure
$(\Delta_{T}=\Delta_{A}=\Delta_{B})$ for comparison. Unlike the antichiral
system, here the bulk bands are gapped and two pairs of edge modes appear.
Since the edge states at the different edges have opposite group velocities,
they propagate in the opposite directions. Here the bulk bands are not
linearly polarized and have some preferred value of $S_{z}$. Near the Dirac
points, $S_{z}\rightarrow\pm 1$. However, the edge modes deviate from the pure
spins and acquire some mixed spins.

##  IV Spin propagation

Figure 4: (a-c) Polariton propagation under the effect of two pulses. The
arrows in (b) represents the positions of two defects deliberately introduced
by removing a site at each edge. Polaritons can propagate around the defect
without significant backscattering. (d) Polariton propagation under continuous
coherent pumps. Polaritons with “$-$” spin propagate along the upper edge,
while the polaritons with “$+$” spin propagate along the lower edge. The
yellow stars correspond to the positions of linearly polarized coherent pulses
(a-c) and pumps (d). Parameters: $F_{0}=J$, $\Gamma=0.002J$, $\sigma=3$,
$\tau_{0}/\hbar=50/J$, $\hbar\omega_{p}=-0.207J$, $t_{u}=\hbar/J$ and
$ak_{p}=3.42$. All other parameters are kept the same as those in Figs.
3(a-b). Figure 5: Schematic diagram of antichiral system arranged in a
triangle (a) and a parallelogram shape (d). The triangle shaped system has
only one type of edge. (b-c) Only spin “+” polaritons can propagate around the
bending without being backscattered. (e-f) The parallelogram consists of two
types of zigzag edges and “$\pm$” spins propagate along those edges,
separately. (g) At the top left corner, where both edges meet, polaritons
scatter into the bulk. (h) For the Chern insulator system, polaritons can
propagate along the edges and do not scatter into the bulk. Each arm consists
of 15 sites in (a-c) and 25 sites in (d-h). The yellow stars represent
linearly polarized coherent pumps. Parameters: (a-g) All other parameters are
kept the same as those in Fig. 4. (h) $\Delta_{T}=\Delta_{A}=\Delta_{B}=0.3J$,
$\hbar\omega_{p}=0.07J$, and $ak_{x}=2.89$.

In this section, we investigate the polariton propagation in the antichiral
system and find out their robustness by removing one site from each edge. We
consider a stripe lattice formed by 12 unit cells along the $y$ direction and
20 unit cells along the $x$ direction. The upper (lower) edge of the system is
“A” (“B”) type zigzag edge (see Fig. 1). The evolution of the polaritons under
the coherent excitation is governed by the following time dependent
Schrödinger equation

$i\hbar\frac{\partial\Psi_{\pm}}{\partial
t}=H\Psi_{\pm}-i\Gamma\Psi_{\pm}+F_{\pm}(x,y,t)\exp{\left[i\left(k_{p}x-\omega_{p}t\right)\right]},$
(7)

where

$\displaystyle
F_{\pm}=F_{0}\left\\{\begin{matrix}\exp\left[{-\frac{[(x-x_{0})^{2}+(y-y_{0})^{2}]}{2\sigma^{2}}-\frac{t^{2}}{2\tau_{0}^{2}}}\right],~{}\text{for
pulse,}\\\
\exp\left[{-\frac{[(x-x_{0})^{2}+(y-y_{0})^{2}]}{2\sigma^{2}}}\right],~{}\text{for
pump}\end{matrix}\right..$ (8)

Here $H$ is the Hamiltonian representing the system which can be written by
repeating the Hamiltonian $H_{AB}$, $F_{\pm}$ represents the circularly
polarized components of Gaussian shaped coherent pulses or pumps having
amplitude $F_{0}$, energy $\hbar\omega_{p}$, and momentum $k_{p}$; and
$\Gamma$ is the decay rate of polaritons. The expression in Eq. (8) represents
the profiles of $F_{\pm}$ in space and in time. $(x_{0},y_{0})$ represent the
position of $F_{\pm}$ having widths $\sigma$ in space and $\tau_{0}$ in time.

In Fig. 4 the polariton propagation under two linearly polarized
($F_{+}=F_{-}$) pulses positioned at the two edges (represented by the yellow
stars) are shown. As it can be seen from the band structure in Fig. 3(a-b) the
edge modes have negative group velocity. Consequently, polaritons propagate
from right to left along both the edges. To check the robustness, we have
removed one site at each edge as indicated by the arrows in 4(b). As it can be
seen in Figs. 4(a-c) the polaritons can propagate around the defect without
significant backscattering. However, the propagation around the defect is not
perfect and in the next section, we have shown that the backscattering is
around 10%. Because of the presence of the decay term $\Gamma$, the intensity
created by the pulse decreases exponentially. Consequently, for visulization
purpose the intensity is renormalized at each time step in Figs. 4(a-c). Next,
to show that the edges are circularly polarized, we use continuous pump
instead of pulses. It should be noted that we use linearly polarized pumps to
inject both $\pm$ polaritons. However, the system only allows $``+"$
polaritons to propagate along the lower edge and $``-"$ polaritons to
propagate along the upper edge, creating two spin channels.

The dynamics of the polaritons become more interesting if instead of a strip
geometry, triangular and parallelogram shaped geometries are considered. In
Fig. 5(a), (d) schematics of triangular and parallelogram shaped geometries
are shown, respectively. For triangular geometry, one of the edges of the
system can be considered as periodic while the other edge does not appear in
the structure. As a result, under a linearly polarized coherent pump, spin “+”
polaritons can propagate along the edges without being backscattered into the
bulk (see Figs. 5(b-c)). This scenario changes for parallelogram shaped
geometry as it consists of both type of edges. Consequently, $``\pm"$
polaritons propagate along different edges (see Figs. 5(e-f)). Due to the
presence of $\Gamma$ the intensity of the polaritons decreases as they
propagate away from the excitation spot. It should be noted that in both the
structures, polaritons can go around the 60 degree bends perfectly. At 120
degree bends the two types of edge states meet (see the top left corner in
Fig. 5(d)). Since both the edge states propagate in the same direction,
polaritons can not continue propagating along the edges. Instead they scatter
into the bulk. Such a situation is shown in Fig. 5(g), where the pump is
positioned at the top left corner, where the opposite edges meet. For
comparison, in Fig. 5(h) we have considered the case for the Chern insulator,
which understandably does not scatter into the bulk and continues propagating
along the edges.

##  V Efficiency comparison

To estimate the backscattering while going around the defect, we consider the
similar geometry as shown in Figs. 4(a-c), where the edge contains a defect.
Next, we position a linearly polarized coherent pulse at the right end of the
strip and define the efficiency of the system as

$\eta=\frac{I_{l}-I_{r}}{I_{l}+I_{r}}$ (9)

where $I_{l}$ is the total intensity of the sites situated at the left of the
defect (from both edge and bulk) and $I_{r}$ is the total intensity of sites
situated at the right of the defect (from both edge and bulk). The intensities
are considered at longer times when the polaritons pass the defect. In Fig. 6
the efficiency of the antichiral system is plotted in blue as a function of
different system size $N$ as defined in the inset. For larger $N$, the
efficiency stays near the 90%, but never reaches 100%, which is an indication
that there is always around 10% backscattering when the antichiral edge states
go around a defect. This is understandable, as the antichiral edge modes
reside together with the bulk modes in the same energy window, a defect can
couple the edge and bulk modes. However, Since the edge modes are spatially
separated from the bulk modes, the system still shows fair robustness and the
efficiency remains around 90 %. The efficiency decreases rapidly as $N$
decreases.

For comparison with the Chern insulator edge states, we also perform the same
steps and obtain the intensity as shown in red. As expected, the Chiral edge
modes of the Chern insulators show near perfect transmission around a defect
and the efficiency remains around 100 %. Similar to antichiral system the
efficiency decreases rapidly for lower $N$. The low $\eta$ for lower $N$ is
understandable from the fact that, being topological in nature, the robustness
of the edge modes in both the systems is associated with the bulk properties.
However, if sufficient bulk is not provided to the system, the system can not
show the expected robustness. We should notice that the nature of
backscattering is different for the antichiral and Chern insulator case. For
the antichiral system, the drop in efficiency arises due to the coupling of
co-propagating antichiral edge states with the counter propagating bulk
states. However, for the Chern insulator case, when $N$ is small, a defect
couples the two counter-propagating edge states, reducing the efficiency.

Figure 6: Transmission efficiency comparison between anti-chiral edge states
(in blue) and the Chern insulator edge states (in red) with different number
of layers $N$, as defined in the inset. The Chern insulator can reach almost
perfect transmission without backscattering when number of layers is larger
than 10. Understandably, the efficiency of antichiral edge states remains
lower than that of the Chern insulator, but still reaches around $90\%$. Both
types of system has a site missing at the edge.

##  VI Experimental proposal

Figure 7: (a) Spatial profile of $P_{+}$ with peaks occurring in only one type
of sublattices. The green circles represent the micropillars. (b-e) Band
structure of the system. All four of them corresponds to the same system. For
presentation purpose, upper edge states are omitted in (b-c) and lower edge
states are omitted in (d-e). The bulk is linearly polarized, while the
opposite edge states have opposite spins.

In this section, we discuss and propose a scheme to realize the spin-polarized
antichiral edge states in the experiments. The key elements in our scheme are
1) honeycomb lattice potential for the polaritons, 2) TE-TM splitting, and 3)
sublattice dependent Zeeman splitting. Arranging exciton-polariton
micropillars in a variety of periodic lattices is a routine task in
experiments. The TE-TM splitting is known to occur naturally in microcavities
due to the polarization dependent reflection from the cavity mirrors Panzarini
_et al._ (1999). A Chern insulator was realized based on these ingredients,
when a honeycomb lattice was subjected to a strong perpendicular magnetic
field, which induced a uniform Zeeman splitting Klembt  _et al._ (2018). In
order to realize the antichiral system, instead of a uniform Zeeman splitting,
the sign of the Zeeman splitting needs to be opposite for different
sublattices. For this reason, the set up with real magnetic field used for the
realization of Chern insulators seems challenging and requires an alternative
way. In Ref. Sun  _et al._ (2019), an additional layer of ferromagnetic
material was proposed to induced the Zeeman splitting. Instead of one, two
ferromagnetic material layers with opposite predefined magnetic momentums may
be used and etched to make the Zeeman splitting opposite in different
sublattices.

However, using the inherent nonlinearity of the polaritons seems to be the
most straight-forward way. In a honeycomb lattice under the nonpolarized
incoherent pump, it is possible to form polariton condensate in an
antiferromagnetic configuration, where different types of sublattices attain
opposite circular polarization Sigurdsson  _et al._ (2019). In such a
scenario, due to the nonlinear interactions, polaritons with opposite spins
will experience blueshifts in different sublattices, giving rise to the
necessary sublattice dependent Zeeman splitting in our scheme.

Another approach could be using circularly polarized incoherent pumps. It was
recently verified experimentally that in micropillar structures, a circularly
polarized incoherent pump can give rise to optically induced Zeeman splitting
up to 0.2 meV Real  _et al._ (2021). In what follows, we implement this recent
experimental finding in our scheme and realize the spin-polarized antichiral
edge states. Polaritons under the effect of external incoherent pump can be
described using the following Gross-Pitaevskii equation (GPE) Carusotto and
Ciuti (2013)

$\displaystyle i\hbar\frac{\partial\psi_{\pm}}{\partial t}=$
$\displaystyle\left[-\frac{\hbar^{2}\nabla^{2}}{2m_{o}}+V(x,y)+gP_{\pm}(x,y)\right]\psi_{\pm}$
$\displaystyle+\left(\alpha_{1}|\psi_{\pm}|^{2}+\alpha_{2}|\psi_{\mp}|^{2}-i\alpha_{NL}|\psi_{\pm}|^{2}\right)\psi_{\pm}$
$\displaystyle+i\left[P_{\pm}(x,y)-\Gamma\right]\psi_{\pm}+\Delta_{T}\left(i\frac{\partial}{\partial
x}\pm\frac{\partial}{\partial y}\right)^{2}\psi_{\mp}.$ (10)

Here $\psi_{\pm}$ is the polariton wave function corresponding to the “$\pm$”
spins, $m_{o}$ is the polartion effective mass, and $V(x,y)$ is the underlying
honeycomb lattice potential. $gP_{\pm}$ is the spin dependent potential
experienced by the polaritons due to interaction with the excitonic reservoir
created by the circularly polarized incoherent pumps $P_{\pm}$ with $g$ being
a dimensionless parameter. $\Delta_{T}$ is the TE-TM splitting.
$\alpha_{1(2)}$ is the polariton-polariton interaction coefficient between
same (opposite) spins. $\alpha_{NL}$ is the nonlinear loss and $\Gamma$ is the
linear decay.

The incoherent pump can be patterned into a lattice shape. One way is to use a
spatial light-modulator, which has been used to make lattice potentials in a
variety of works Ohadi  _et al._ (2017, 2018); Töpfer  _et al._ (2021); Opala
_et al._ (2022). An alternative technique is to interference plane waves with
different in-plane wavevector components which has been demonstrated
experimentally in Luo  _et al._ (2020). Here in this work, we use
superposition of four waves to prepare the incoherent pumps, such that the
resultant wave has maxima at one type of sublattice, while minima at the other
type of sublattices. Here we emphasize that the four waves we are using here
come from the same laser by using additional external optics devices. In this
case, $P_{+}$ can be expressed as

$\displaystyle P_{+}(x,y)=P_{0}f(x,y),$ (11)

where

$\displaystyle f(x,y)=$ $\displaystyle\cos\left(K_{1}\cdot
R\right)+\cos\left(K_{2}\cdot R\right)+\cos\left(K_{3}\cdot R\right)$
$\displaystyle+\cos\left(K_{4}\cdot R\right)$ (12)

with $K_{1}=4\pi(0,1/\sqrt{3})/a$, $K_{2,3}=2\pi(1,\pm 1/\sqrt{3})/a$,
$K_{4}=(0,0)$. In the above $a$ is the periodicity of the lattice along the
$x$ direction, $P_{0}$ controls the amplitude of the pump, $f$ is the
superposition of four waves that creates the maxima at a particular type of
sublattices, and the last term is a wave which has zero in-plane wave vector
that fixes the minima of $P_{+}$ to zeros. In Fig. 7(a) the spatial profile of
$P_{+}$ is plotted, which shows that indeed the maxima of $P_{+}$ coincide
with only one type of sublattices. $P_{-}$ also has a similar spatial form,
however, one needs to shift $f$ spatially such that the maxima now coincide
with the other type of sublattice.

We choose the pillar diameter to be $2~{}\mu$m, the lattice periodicity around
$a=2.95~{}\mu$m Whittaker  _et al._ (2018); Su  _et al._ (2020); Jacqmin  _et
al._ (2014), effective mass $m_{0}=5\times 10^{-5}m_{e}$, where $m_{e}$ is the
free electron mass, and the potential depth of the micropillars to be around 4
meV with 0 meV being the minimum. We also fix $\Delta_{T}=0.12$ meV.$\mu$m2
Klembt  _et al._ (2018); Dufferwiel  _et al._ (2015); Sedov  _et al._ (2019)
and $\Gamma=5.9~{}\mu$eV Nelsen  _et al._ (2013), which corresponds to a
polariton lifetime around 55 ps. By fixing $P_{0}=2.3~{}\mu$eV we stay near
the condensation threshold, such that $|\psi_{\pm}|^{2}\approx$0\. This allows
us to be in the linear limit and calculate the band structure of the system
using the Bloch theorem. We also set $g=2.3$, which induces optical Zeeman
shift of the lower energy modes around 0.2 meV, which is consistent with
experiment Real  _et al._ (2021).

Now that we have all the ingredients, by applying the Bloch theorem to the
linear Hamiltonian corresponding to the Eq. (VI), we calculate the band
structure of the system corresponding to a strip geometry with zigzag edges as
shown in Figs. 7(b-e). Similar to the tight binding model, our continuous
model also predicts the antichiral band structure, where the Dirac points
shifts in energy and two pairs of edge states appear. As already discussed,
the opposite edge states in our system are degenerate. Consequently, to
clearly show that opposite edge states have opposite spins, we plot the same
band structure four times. In Figs. 7(b-c) we have omitted the edge states
located at the upper edge of the strip ($\langle y\rangle=+L$), while in Figs.
7(d-e) we have omitted the edge states located at the lower edge of the strip
($\langle y\rangle=-L$). Indeed, the opposite edge states attain opposite
spins, while the bulk remains linearly polarized. Although, the topological
properties of the continuous model band structure is the same with those from
the tight-binding band structure, there are a few differences. For example,
the upper of pair edge states does not have similar dispersion as the lower
pair. This may arise due to the next nearest neighbour hopping, which is
naturally present in the continuous model as well as due to the continuous
nature of the effective Zeeman and TE-TM splittings.

The parameters that we have used are flexible. However, the TE-TM splitting
and effective sublattice dependent Zeeman splitting are the crucial
parameters. For the present parameters, the energy window, where the
antichiral edge states resides is around 0.12 meV. This is similar to the
topological bandgap obtained experimentally for the polariton Chern insulator
Klembt  _et al._ (2018). Being topological in nature, a slight change in the
parameters will not hamper the antichiral modes. However, the case where the
Zeeman splitting decreases will result in a decrease in the energy window of
interest.

For simplicity, we have only considered only the necessary terms in the GPE in
Eq. (VI) to obtain the desired effect. For example, we have ignored the spin
relaxation of the excitonic reservoir and assumed it to preserve their
circular polarization perfectly. Although this is not entirely true,
experiment has shown that under circularly polarized incoherent pump the
reservoir can have up to 17% preferred $S_{z}$ C. Zambon  _et al._ (2019),
which should be enough as long as the blueshift difference between opposite
circularly polarized polariton modes in a pillar is around 0.2 meV.

##  VII Discussion and conclusion

The spin degree of freedom of exciton-polaritons makes them suitable for
optical spintronic devices. Devices based on polariton spins have been
realized experimentally, which includes spin switches Amo  _et al._ (2010),
logic gates Gao  _et al._ (2015b), etc. However, for the realization of a
complete optical network, it is important to be able to communicate among
different elements without losing the spin information. Unfortunately, the
Chiral polaritonic edge modes do not preserve spins while propagating. Our
proposed antichiral edge states could be useful in this purpose. Besides all
the edge modes in our system propagate in one direction, which may be helpful
in transferring oneway information through both the edges, unlike in the Chern
insulator where only one edge can be used.

We note that antichiral edge states for exciton-polaritons were proposed in a
previous work Mandal  _et al._ (2019a). The present work is significantly
different from the previous one in many ways. First, in the previous work the
antichiral edge modes were not coherent polaritonic states. Instead, those
were fluctuations on top of a steady state. Second, in Ref. Mandal  _et al._
(2019a) the antichiral edge states located at opposite edges did not have
opposite spins. Last, in the previous work, the antichiral edge states were
shown to propagate only in a straight strip, while their propagation around
different types of bendings was not discussed.

To conclude, we have presented a theoretical scheme to obtain spin-polarized
antichiral edge states for the first time. We use the naturally present TE-TM
splitting and sublattice depend Zeeman splitting in a honeycomb lattice of
exciton-polariton micropillar, where the Dirac points shift in energy and two
pairs of antichiral edge states appear that propagate in the same directions.
While the bulk of the system is linearly polarized, states located at the
opposite edges have opposite spins. Although the antichiral edge states reside
together with the bulk modes, we found that they show fair robustness and can
go around 60 degree bend without being reflected. Our work may be useful in
transferring spins and connecting different spin dependent polariton logic
elements.

##  VIII Acknowledgements

The work was supported by the Ministry of Education, Singapore (Grant No.
MOE2019-T2-1-004).

##  IX Appendix

Here in the appendix we discuss how to calculate the winding number in detail.
Different from the model considered in Colomés and Franz (2018), which has
only one band below the Dirac points, our present model involves two bands.
This makes the Hamiltonian complex and restricts us from calculating the
winding number analytically. Additionally, because of the presence of two
bands the Berry connections becomes non-Abelian (a matrix instead of a scalar)
and one must rely on numerical techniques to calculate the winding number.

To make the integral Gauge invariance, we use the Wilson loop approach
Benalcazar  _et al._ (2017). We first transform the hexagonal BZ to a rhombic
shape with its axes aligned with the $k_{1}$ and $k_{2}$ direction as shown in
Fig. 8.We notice that such a choice of the BZ is not unique, and the
directions $(k_{1},k_{2})$ used in Colomés and Franz (2018) is related to ours
by a rotation of $\pi/12$. However, we choose such a BZ as the Dirac points
are situated at the central of the BZ when projected on any of the two
directions. The Wilson loop operator along $k_{2}$ for a fixed value of
$k_{1}$ can be defined as:

Figure 8: A rhombic shape Brillouin zone with its axis aligned with the
$k_{1}$ and $k_{2}$ direction.

$\displaystyle W(k_{1})=$ $\displaystyle F(k_{1},k_{2}+(N-1)\Delta
k)F(k_{1},k_{2}+(N-2)\Delta k)...$ $\displaystyle F(k_{1},k_{2}+\Delta
k)F(k_{1},k_{2}),$ (13)

where $F$ is a matrix whose elements are given by

$\left[F(k_{1},k_{2})\right]^{mn}=\left\langle u^{m}(k_{1},k_{2}+\Delta
k)|u^{n}(k_{1},k_{2})\right\rangle.$ (14)

Here $\Delta k$ is the grid spacing in the reciprocal space, the integral is
over the unit cell, $u$ is the Bloch state and $m,n=1,2$ corresponds to the
lowest two bands. Next we define a Wannier Hamiltonian $H_{W(k_{1})}$ as

$W(k_{1})=e^{iH_{W(k_{1})}}.$ (15)

The eigenvalues of $H_{W(k_{1})}$ gives the winding numbers for each $k_{1}$
as shown in Fig. 2(h) in the main text.

## References

  * Klitzing  _et al._ (1980) K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance,” Phys. Rev. Lett. 45, 494 (1980).
  * Haldane (1988) F. D. M. Haldane, “Model for a quantum hall effect without landau levels: Condensed-matter realization of the ”parity anomaly”,” Phys. Rev. Lett. 61, 2015 (1988).
  * Jotzu  _et al._ (2014) G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515, 237 (2014).
  * Zhao  _et al._ (2020) Y. Zhao, R. Zhang, R. Mei, L. Zhou, H. Yi, Y. Zhang, J. Yu, R. Xiao, K. Wang, N. Samarth, M. H. W. Chan, C. Liu, and C. Chang, “Tuning the Chern number in quantum anomalous Hall insulators,” Nature 588, 419 (2020).
  * Kane and Mele (2005) C. L. Kane and E. J. Mele, “Quantum spin hall effect in graphene,” Phys. Rev. Lett. 95, 226801 (2005).
  * Colomés and Franz (2018) E. Colomés and M. Franz, “Antichiral edge states in a modified haldane nanoribbon,” Phys. Rev. Lett. 120, 086603 (2018).
  * Mandal  _et al._ (2019a) S. Mandal, R. Ge, and T. C. H. Liew, “Antichiral edge states in an exciton polariton strip,” Phys. Rev. B 99, 115423 (2019a).
  * Chen  _et al._ (2020) J. Chen, W. Liang, and Z. Li, “Antichiral one-way edge states in a gyromagnetic photonic crystal,” Phys. Rev. B 101, 214102 (2020).
  * Zhou  _et al._ (2020) P. Zhou, G. Liu, Y. Yang, Y. Hu, S. Ma, H. Xue, Q. Wang, L. Deng, and B. Zhang, “Observation of photonic antichiral edge states,” Phys. Rev. Lett. 125, 263603 (2020).
  * Chen and Li (2022) J. Chen and Z. Li, “Configurable topological beam splitting via antichiral gyromagnetic photonic crystal,” Opto-Electronic Science 1, 220001 (2022).
  * Yu  _et al._ (2021) L. Yu, H. Xue, and B. Zhang, “Antichiral edge states in an acoustic resonator lattice with staggered air flow,” Journal of Appl. Phys. 129, 235103 (2021).
  * Wu  _et al._ (2020) X. Wu, X. Li, R. Zhang, X. Xiang, J. Tian, Y. Huang, S. Wang, B. Hou, C. T. Chan, and W. Wen, “Deterministic scheme for two-dimensional type-ii dirac points and experimental realization in acoustics,” Phys. Rev. Lett. 124, 075501 (2020).
  * Yang  _et al._ (2021) Y. Yang, D. Zhu, Z. Hang, and Y. Chong, “Observation of antichiral edge states in a circuit lattice,” Science China Physics, Mechanics & Astronomy 64, 257011 (2021).
  * Bhowmick and Sengupta (2020) D. Bhowmick and P. Sengupta, “Antichiral edge states in heisenberg ferromagnet on a honeycomb lattice,” Phys. Rev. B 101, 195133 (2020).
  * Mannaï and Haddad (2020) M. Mannaï and S. Haddad, “Strain tuned topology in the haldane and the modified haldane models,” Journal of Physics: Condensed Matter 32, 225501 (2020).
  * Breitkreiz (2020) M. Breitkreiz, “Parabolic hall effect due to copropagating surface modes,” Phys. Rev. Research 2, 012071 (2020).
  * Mizoguchi and Koma (2021) T. Mizoguchi and T. Koma, “Bulk-edge correspondence in two-dimensional topological semimetals: A transfer matrix study of antichiral edge modes,” Phys. Rev. B 103, 195310 (2021).
  * Cheng  _et al._ (2021) X. Cheng, J. Chen, L. Zhang, L. Xiao, and S. Jia, “Antichiral edge states and hinge states based on the haldane model,” Phys. Rev. B 104, L081401 (2021).
  * Deng  _et al._ (2010) H. Deng, H. Haug, and Y. Yamamoto, “Exciton-polariton bose-einstein condensation,” Rev. Mod. Phys. 82, 1489 (2010).
  * Carusotto and Ciuti (2013) I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299 (2013).
  * Byrnes  _et al._ (2014) T. Byrnes, N. Y. Kim, and Y. Yamamoto, “Exciton–polariton condensates,” Nature Phys. 10, 803 (2014).
  * Sanvitto and Kéna-Cohen (2016) D. Sanvitto and S. Kéna-Cohen, “The road towards polaritonic devices,” Nature Mater. 15, 1061 (2016).
  * Feng  _et al._ (2021) J. Feng, J. Wang, A. Fieramosca, R. Bao, J. Zhao, R. Su, Y. Peng, T. C. H. Liew, D. Sanvitto, and Q. Xiong, “All-optical switching based on interacting exciton polaritons in self-assembled perovskite microwires,” Science Advances 7, eabj6627 (2021).
  * Kyriienko  _et al._ (2019) O. Kyriienko, H. Sigurdsson, and T. C. H. Liew, “Probabilistic solving of $np$-hard problems with bistable nonlinear optical networks,” Phys. Rev. B 99, 195301 (2019).
  * Xu  _et al._ (2020) H. Xu, S. Ghosh, M. Matuszewski, and T. C.H. Liew, “Universal self-correcting computing with disordered exciton-polariton neural networks,” Phys. Rev. Applied 13, 064074 (2020).
  * Banerjee and Liew (2020) R. Banerjee and T. C. H. Liew, “Artificial life in an exciton-polariton lattice,” New Journal of Phys. 22, 103062 (2020).
  * Harrison  _et al._ (2022) S. L. Harrison, H. Sigurdsson, S. Alyatkin, J.D. Töpfer, and P. G. Lagoudakis, “Solving the max-3-cut problem with coherent networks,” Phys. Rev. Applied 17, 024063 (2022).
  * Xu  _et al._ (2021a) H. Xu, T. Krisnanda, W. Verstraelen, T. C. H. Liew, and S. Ghosh, “Superpolynomial quantum enhancement in polaritonic neuromorphic computing,” Phys. Rev. B 103, 195302 (2021a).
  * Zvyagintseva  _et al._ (2022) D. Zvyagintseva, H. Sigurdsson, V. K. Kozin, I. Iorsh, I. A. Shelykh, V. Ulyantsev, and O. Kyriienko, “Machine learning of phase transitions in nonlinear polariton lattices,” Commun. Phys. 5, 8 (2022).
  * Kavokin  _et al._ (2005) A. Kavokin, G. Malpuech, and M. Glazov, “Optical spin hall effect,” Phys. Rev. Lett. 95, 136601 (2005).
  * Leyder  _et al._ (2007) C. Leyder, M. Romanelli, J. Ph. Karr, E. Giacobino, T. C. H. Liew, M. M. Glazov, A. V. Kavokin, G. Malpuech, and A. Bramati, “Observation of the optical spin Hall effect,” Nature Phys. 3, 628 (2007).
  * Kammann  _et al._ (2012) E. Kammann, T. C. H. Liew, H. Ohadi, P. Cilibrizzi, P. Tsotsis, Z. Hatzopoulos, P. G. Savvidis, A. V. Kavokin, and P. G. Lagoudakis, “Nonlinear optical spin hall effect and long-range spin transport in polariton lasers,” Phys. Rev. Lett. 109, 036404 (2012).
  * Larionov  _et al._ (2010) A. V. Larionov, V. D. Kulakovskii, S. Höfling, C. Schneider, L. Worschech, and A. Forchel, “Polarized nonequilibrium bose-einstein condensates of spinor exciton polaritons in a magnetic field,” Phys. Rev. Lett. 105, 256401 (2010).
  * Karzig  _et al._ (2015) T. Karzig, C. E. Bardyn, N. H. Lindner, and G. Refael, “Topological polaritons,” Phys. Rev. X 5, 031001 (2015).
  * Bardyn  _et al._ (2015) C. E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew, “Topological polaritons and excitons in garden-variety systems,” Phys. Rev. B 91, 161413 (2015).
  * Nalitov  _et al._ (2015) A. V. Nalitov, D. D. Solnyshkov, and G. Malpuech, “Polariton $\mathbb{Z}$ topological insulator,” Phys. Rev. Lett. 114, 116401 (2015).
  * Banerjee  _et al._ (2018) R. Banerjee, T. C. H. Liew, and O. Kyriienko, “Realization of hofstadter’s butterfly and a one-way edge mode in a polaritonic system,” Phys. Rev. B 98, 075412 (2018).
  * Li  _et al._ (2018) C. Li, F. Ye, X. Chen, Y. V. Kartashov, A. Ferrando, L. Torner, and D. V. Skryabin, “Lieb polariton topological insulators,” Phys. Rev. B 97, 081103 (2018).
  * Sun  _et al._ (2019) M. Sun, D. Ko, D. Leykam, V. M. Kovalev, and I. G. Savenko, “Exciton-polariton topological insulator with an array of magnetic dots,” Phys. Rev. Applied 12, 064028 (2019).
  * Solnyshkov  _et al._ (2018) D. D. Solnyshkov, O. Bleu, and G. Malpuech, “Topological optical isolator based on polariton graphene,” Appl. Phys. Lett. 112, 031106 (2018).
  * Mandal  _et al._ (2019b) S. Mandal, R. Banerjee, and T. C. H. Liew, “One-way reflection-free exciton-polariton spin-filtering channel,” Phys. Rev. Applied 12, 054058 (2019b).
  * Klembt  _et al._ (2018) S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature 562, 552 (2018).
  * Kartashov and Skryabin (2016) Y. V. Kartashov and D. V. Skryabin, “Modulational instability and solitary waves in polariton topological insulators,” Optica 3, 1228 (2016).
  * Gulevich  _et al._ (2017) D. R. Gulevich, D. Yudin, D. V. Skryabin, I. V. Iorsh, and I. A. Shelykh, “Exploring nonlinear topological states of matter with exciton-polaritons: Edge solitons in kagome lattice,” Scientific Reports 7, 1780 (2017).
  * Pernet  _et al._ (2022) N. Pernet, P. St-Jean, D. D. Solnyshkov, G. Malpuech, N. Carlon Zambon, Q. Fontaine, B. Real, O. Jamadi, A. Lemaître, M. Morassi, L. Le Gratiet, T. Baptiste, A. Harouri, I. Sagnes, A. Amo, S. Ravets, and J. Bloch, “Gap solitons in a one-dimensional driven-dissipative topological lattice,” Nature Physics (2022), 10.1038/s41567-022-01599-8.
  * Solnyshkov  _et al._ (2017) D. D. Solnyshkov, O. Bleu, B. Teklu, and G. Malpuech, “Chirality of topological gap solitons in bosonic dimer chains,” Phys. Rev. Lett. 118, 023901 (2017).
  * Bleu  _et al._ (2017) O. Bleu, D. D. Solnyshkov, and G. Malpuech, “Photonic versus electronic quantum anomalous hall effect,” Phys. Rev. B 95, 115415 (2017).
  * Kartashov and Skryabin (2017) Y. V. Kartashov and D. V. Skryabin, “Bistable topological insulator with exciton-polaritons,” Phys. Rev. Lett. 119, 253904 (2017).
  * Banerjee  _et al._ (2020) R. Banerjee, S. Mandal, and T. C. H. Liew, “Coupling between exciton-polariton corner modes through edge states,” Phys. Rev. Lett. 124, 063901 (2020).
  * Bardyn  _et al._ (2016) C. E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew, “Chiral bogoliubov excitations in nonlinear bosonic systems,” Phys. Rev. B 93, 020502 (2016).
  * Sigurdsson  _et al._ (2017) H. Sigurdsson, G. Li, and T. C. H. Liew, “Spontaneous and superfluid chiral edge states in exciton-polariton condensates,” Phys. Rev. B 96, 115453 (2017).
  * Banerjee  _et al._ (2021) R. Banerjee, S. Mandal, and T. C. H. Liew, “Optically induced topological spin-valley hall effect for exciton polaritons,” Phys. Rev. B 103, L201406 (2021).
  * Gao  _et al._ (2015a) T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Höfling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015a).
  * Gao  _et al._ (2018) W. Gao, X. Li, M. Bamba, and J. Kono, “Continuous transition between weak and ultrastrong coupling through exceptional points in carbon nanotube microcavity exciton–polaritons,” Nature Photonics 12, 362–367 (2018).
  * Khurgin (2020) J. B. Khurgin, “Exceptional points in polaritonic cavities and subthreshold fabry–perot lasers,” Optica 7, 1015–1023 (2020).
  * Comaron  _et al._ (2020) P. Comaron, V. Shahnazaryan, W. Brzezicki, T. Hyart, and M. Matuszewski, “Non-hermitian topological end-mode lasing in polariton systems,” Phys. Rev. Research 2, 022051 (2020).
  * Su  _et al._ (2021) R. Su, E. Estrecho, D. Bieganska, Y. Huang, M. Wurdack, M. Pieczarka, A. G. Truscott, T. C. H. Liew, E. A. Ostrovskaya, and Q. Xiong, “Direct measurement of a non-hermitian topological invariant in a hybrid light-matter system,” Science Advances 7, eabj8905 (2021).
  * Liao  _et al._ (2021) Q. Liao, C. Leblanc, J. Ren, F. Li, Y. Li, D. Solnyshkov, G. Malpuech, J. Yao, and H. Fu, “Experimental measurement of the divergent quantum metric of an exceptional point,” Phys. Rev. Lett. 127, 107402 (2021).
  * Krol  _et al._ (2022) M. Krol, I. Septembre, P. Oliwa, M. Kedziora, K. Lempicka-Mirek, M. Muszyński, R. Mazur, P. Morawiak, W. Piecek, P. Kula, W. Bardyszewski, P. G. Lagoudakis, D. D. Solnyshkov, G. Malpuech, B. Pietka, and J. Szczytko, “Annihilation of exceptional points from different dirac valleys in a 2d photonic system,” Nature Communications 13, 5340 (2022).
  * Mandal  _et al._ (2020) S. Mandal, R. Banerjee, Elena A. Ostrovskaya, and T. C. H. Liew, “Nonreciprocal transport of exciton polaritons in a non-hermitian chain,” Phys. Rev. Lett. 125, 123902 (2020).
  * Xu  _et al._ (2021b) H. Xu, K. Dini, X. Xu, R. Banerjee, S. Mandal, and T. C. H. Liew, “Nonreciprocal exciton-polariton ring lattices,” Phys. Rev. B 104, 195301 (2021b).
  * Mandal  _et al._ (2022) S. Mandal, R. Banerjee, and T. C. H. Liew, “From the topological spin-hall effect to the non-hermitian skin effect in an elliptical micropillar chain,” ACS Photon. 9, 527 (2022).
  * Xu  _et al._ (2021c) X. Xu, H. Xu, S. Mandal, R. Banerjee, S. Ghosh, and T. C. H. Liew, “Interaction-induced double-sided skin effect in an exciton-polariton system,” Phys. Rev. B 103, 235306 (2021c).
  * Pickup  _et al._ (2020) L. Pickup, H. Sigurdsson, J. Ruostekoski, and P. G. Lagoudakis, “Synthetic band-structure engineering in polariton crystals with non-Hermitian topological phases,” Nature Commun. 11, 4431 (2020).
  * Bleu  _et al._ (2018) O. Bleu, G. Malpuech, and D. D. Solnyshkov, “Robust quantum valley hall effect for vortices in an interacting bosonic quantum fluid,” Nature Communications 9, 3991 (2018).
  * Qi  _et al._ (2008) X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B 78, 195424 (2008).
  * Panzarini  _et al._ (1999) G. Panzarini, L. C. Andreani, A. Armitage, D. Baxter, M. S. Skolnick, V. N. Astratov, J. S. Roberts, A. V. Kavokin, M. R. Vladimirova, and M. A. Kaliteevski, “Exciton-light coupling in single and coupled semiconductor microcavities: Polariton dispersion and polarization splitting,” Phys. Rev. B 59, 5082 (1999).
  * Sigurdsson  _et al._ (2019) H. Sigurdsson, Y. S. Krivosenko, I. V. Iorsh, I. A. Shelykh, and A. V. Nalitov, “Spontaneous topological transitions in a honeycomb lattice of exciton-polariton condensates due to spin bifurcations,” Phys. Rev. B 100, 235444 (2019).
  * Real  _et al._ (2021) B. Real, N. Carlon Zambon, P. St-Jean, I. Sagnes, A. Lemaître, L. Le Gratiet, A. Harouri, S. Ravets, J. Bloch, and A. Amo, “Chiral emission induced by optical zeeman effect in polariton micropillars,” Phys. Rev. Research 3, 043161 (2021).
  * Ohadi  _et al._ (2017) H. Ohadi, A. J. Ramsay, H. Sigurdsson, Y. del Valle-Inclan Redondo, S. I. Tsintzos, Z. Hatzopoulos, T. C. H. Liew, I. A. Shelykh, Y. G. Rubo, P. G. Savvidis, and J. J. Baumberg, “Spin order and phase transitions in chains of polariton condensates,” Phys. Rev. Lett. 119, 067401 (2017).
  * Ohadi  _et al._ (2018) H. Ohadi, Y. del Valle-Inclan Redondo, A. J. Ramsay, Z. Hatzopoulos, T. C. H. Liew, P. R. Eastham, P. G. Savvidis, and J. J. Baumberg, “Synchronization crossover of polariton condensates in weakly disordered lattices,” Phys. Rev. B 97, 195109 (2018).
  * Töpfer  _et al._ (2021) J. D. Töpfer, I. Chatzopoulos, H. Sigurdsson, T. Cookson, Y. G. Rubo, and P. G. Lagoudakis, “Engineering spatial coherence in lattices of polariton condensates,” Optica 8, 106–113 (2021).
  * Opala  _et al._ (2022) A. Opala, R. Panico, V. Ardizzone, B. Pietka, J. Szczytko, D. Sanvitto, M. Matuszewski, and D. Ballarini, “Training a neural network with exciton-polariton optical nonlinearity,” Phys. Rev. Applied 18, 024028 (2022).
  * Luo  _et al._ (2020) S. Luo, L. Liao, Z. Zhang, J. Wang, X. Shen, and Z. Chen, “Classical spin chains mimicked by room-temperature polariton condensates,” Phys. Rev. Applied 13, 044052 (2020).
  * Whittaker  _et al._ (2018) C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton polaritons in a two-dimensional lieb lattice with spin-orbit coupling,” Phys. Rev. Lett. 120, 097401 (2018).
  * Su  _et al._ (2020) R. Su, Sanjib G., J. Wang, S. Liu, C. Diederichs, T. C. H. Liew, and Q. Xiong, “Observation of exciton polariton condensation in a perovskite lattice at room temperature,” Nature Physics 16, 301–306 (2020).
  * Jacqmin  _et al._ (2014) T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, “Direct observation of dirac cones and a flatband in a honeycomb lattice for polaritons,” Phys. Rev. Lett. 112, 116402 (2014).
  * Dufferwiel  _et al._ (2015) S. Dufferwiel, Feng Li, E. Cancellieri, L. Giriunas, A. A. P. Trichet, D. M. Whittaker, P. M. Walker, F. Fras, E. Clarke, J. M. Smith, M. S. Skolnick, and D. N. Krizhanovskii, “Spin textures of exciton-polaritons in a tunable microcavity with large te-tm splitting,” Phys. Rev. Lett. 115, 246401 (2015).
  * Sedov  _et al._ (2019) E. S. Sedov, Y. G. Rubo, and A. V. Kavokin, “Polariton polarization rectifier,” Light: Science & Applications 8, 79 (2019).
  * Nelsen  _et al._ (2013) B. Nelsen, G. Liu, M. Steger, D. W. Snoke, R. Balili, K West, and L. Pfeiffer, “Dissipationless flow and sharp threshold of a polariton condensate with long lifetime,” Phys. Rev. X 3, 041015 (2013).
  * C. Zambon  _et al._ (2019) N. C. Zambon, P. St-Jean, M. Milićević, A. Lemaître, A. Harouri, L. Le Gratiet, O. Bleu, D. D. Solnyshkov, G. Malpuech, I. Sagnes, S. Ravets, A. Amo, and J. Bloch, “Optically controlling the emission chirality of microlasers,” Nature Photon. 13, 283 (2019).
  * Amo  _et al._ (2010) A. Amo, T. C. H. Liew, C. Adrados, R. Houdré, E. Giacobino, A. V. Kavokin, and A. Bramati, “Exciton–polariton spin switches,” Nature Photon. 4, 361 (2010).
  * Gao  _et al._ (2015b) T. Gao, C. Antón, T. C. H. Liew, M. D. Martín, Z. Hatzopoulos, L. Viña, P. S. Eldridge, and P. G. Savvidis, “Spin selective filtering of polariton condensate flow,” Appl. Phys. Lett. 107, 011106 (2015b).
  * Benalcazar  _et al._ (2017) W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Quantized electric multipole insulators,” Science 357, 61–66 (2017).