# Chirality-induced emergent spin-orbit coupling with photons

Jonah S. Peter<EMAIL_ADDRESS>Department of Physics, Harvard
University, Cambridge, Massachusetts 02138, USA  Stefan Ostermann  Department
of Physics, Harvard University, Cambridge, Massachusetts 02138, USA  Susanne
F. Yelin  Department of Physics, Harvard University, Cambridge, Massachusetts
02138, USA

###### Abstract

Chirality, or handedness, is a geometrical property denoting a lack of mirror
symmetry. Chiral systems are ubiquitous in nature and are associated with the
non-reciprocal interactions observed in topological materials and complex
biomolecules, among other phenomena. In this paper, we demonstrate that chiral
arrangements of dipole-coupled quantum emitters can facilitate the
unidirectional transport of photonic spin excitations without breaking time-
reversal symmetry. We show that this spin-selective transport stems from a
universal emergent spin-orbit coupling induced by the chiral geometry, which
in turn, results in a nontrivial topology. We also examine the effects of
collective dissipation on the dynamics and find that many-body coherences lead
to helicity-dependent photon emission: an effect we call helical
superradiance. Applications to emerging quantum technologies as well as the
potential role of spin-selective photon transport in nature are discussed.

Chirality (or handedness) is ubiquitous in nature and can be observed in
macromolecular structures like DNA, down to the single particle level as in
Weyl fermions or photons with circular polarization. An object is chiral if it
cannot be superimposed on its mirror image by a rotation. This property allows
chiral systems to unidirectionally propagate photons or spin excitations
without breaking time-reversal ($\mathcal{T}$) symmetry. Platforms that admit
such chiral (or helical) modes, including photonic nanostructures [1, 2] and
topological insulators [3, 4, 5], are at the forefront of recent innovations
in both fundamental physics and applied quantum technologies.

Fig. 1: Overview of the model and results described in this work. (a) Photons
with opposite helicity preferentially interact with structures of opposite
chirality. For a given propagation direction (black arrows), collective
dissipation results in the superradiant emission of light with polarization
determined by the chirality of the structure (red and blue halos). (b) The
long-range hopping model used to describe the transport dynamics. Purple
spheres denote emitters with a V-type level structure consisting of two
degenerate hyperfine orbitals $\ket{\uparrow}$ and $\ket{\downarrow}$ excited
by left ($\sigma^{+}$; blue) and right ($\sigma^{-}$; red) circularly
polarized light, respectively. Decay to the ground state $\ket{g}$ occurs with
spontaneous emission rate $\Gamma_{0}$ from each excited state. Emitters
arranged in a helix with radius $r_{0}$, pitch $a$, and $\mathcal{N}$ atoms
per $2\pi$ turn are coupled through long-range dipole-dipole interactions with
coherent and dissipative hopping rates $J_{ij}^{\sigma\sigma^{\prime}}$ and
$\Gamma_{ij}^{\sigma\sigma^{\prime}}$, respectively. Initialization at one end
of the helix results in spin-selective transport. (c) Fundamental processes
governing the transport dynamics. Flip-flop interactions between emitters $i$
and $j$ are mediated through photon exchanges that excite orbitals with either
the same (top) or opposite (bottom) polarization. When the photon spin is
changed, conservation of angular momentum requires that the interaction picks
up an additional position-dependent phase (see also Eq. (1) and Methods). (d)
Population dynamics for left-handed (left-hand panels) and right-handed
(right-hand panels) helices after initialization in an unpolarized mixed state
at either the bottom (top two panels) or top (bottom two panels) of the helix.
Blue (red) curves denote the total population in the $\uparrow$ ($\downarrow$)
manifold as a function of time. Black arrows denote the propagation direction
of the initial wave packet. Dashed lines indicate the time $t=7.9/\Gamma_{0}$
at which the wave packet reaches the opposite end of the helix (see also panel
e and Fig. 3b). (e) Snapshots of the individual emitter populations for the
left-handed (left-hand panels) and right-handed (right-hand panels) helices at
time $t=7.9/\Gamma_{0}$. Blue (red) bars denote the spin $\uparrow$
($\downarrow$) populations of each emitter along the longitudinal axis of the
helix.

Over the last two decades, several experiments have demonstrated that chiral
molecules can act as efficient spin filters for low energy helical electrons
in a phenomenon known as the Chiral-Induced Spin Selectivity (CISS) effect [6,
7]. The discovery of a spin-selective electron transport mechanism based on
organic molecules has implicated the CISS effect in numerous potential
applications including enantioseparation [8], electrochemistry [9], and the
development of spintronics devices [10, 11]. Recent proposals have also
suggested that CISS may be involved in biological electron transport [12, 13]
and could have contributed to the emergence of “biological homochirality” (the
observation that DNA and other biomolecules exist almost exclusively with one
handedness) during the origin of life [14, 15].

Of course in addition to charged particles, the other fundamental component of
the electromagnetic interaction is light. Photons can also carry spin angular
momentum, which is encoded in their two orthogonal polarizations. The coupling
of photons to dipolar quantum emitters [16, 17] can result in efficient photon
transport [18, 19, 20] and cooperative phenomena including the superradiant
and subradiant emission of light [21, 22]. A central question is then whether
geometrical chirality can also facilitate the spin-selective transport of
helical modes in photonic systems, while preserving $\mathcal{T}$ symmetry.

In this work, we show that helical excitations induced by circularly polarized
photons do indeed propagate selectively through chiral emitter geometries.
This phenomenon—which occurs at zero magnetic field and without breaking
$\mathcal{T}$ symmetry—results from an emergent spin-orbit coupling (SOC) that
is universal to all chiral geometries. The dynamics are further enriched at
subwavelength scales where collective dissipation results in helicity-
dependent photon emission: an effect we call helical superradiance (Fig. 1a).
We show that spin-selective chiral transport can be attributed to a non-
Abelian gauge field [23] that results in an excitation band structure with
nontrivial topology. We conclude by discussing potential experimental
implementations of these effects and highlighting the possible implications of
their occurrence in nature.

##  I Spin-Selective Dynamics

Excitation transport between atoms or molecules can be modeled as a collection
of quantum emitters interacting with a radiation field. As a minimal model to
unveil the effects described above, we focus on a helix as the archetypal
chiral geometry—though our results are universal to all chiral structures (see
Methods). We consider a helical array of V-type quantum emitters, each with
two hyperfine transitions excited by left $(\sigma^{+})$ and right
$(\sigma^{-})$ circularly polarized light, respectively, and resonance
frequency $\omega_{0}=2\pi c/\lambda$ (Fig. 1b). No Zeeman splitting is
imposed between the hyperfine states. Long-range dipole-dipole interactions
between emitters $i$ and $j$ are mediated through real and virtual photon
exchanges that couple orbitals $\ket{\sigma_{i}}$ and
$\ket{\sigma^{\prime}_{j}}$ with either the same or opposite polarization (see
Fig. 1c and Methods). After tracing out the field degrees of freedom in the
Born and Markov approximations [21], this system can be described as a
collection of pseudospin-$1/2$ hard-core bosons via the Hamiltonian [24],

$H=\sum_{i=1}^{N}\sum_{\sigma}\omega_{0}b^{{\dagger}}_{i\sigma}b_{i\sigma}+\sum^{N}_{i,j\neq
i=1}\sum_{\sigma,\sigma^{\prime}}J_{ij}^{\sigma\sigma^{\prime}}b^{{\dagger}}_{i\sigma}b_{j\sigma^{\prime}}$
(1)

(we set $\hbar\equiv 1$ here and throughout this work). Here,
$b^{{\dagger}}_{i\sigma}$ $(b_{i\sigma})$ creates (annihilates) an excitation
at site $i$ with spin $\sigma\in\\{\uparrow,\downarrow\\}$, $N$ is the total
number of emitters, and $J_{ij}^{\sigma\sigma^{\prime}}$ are the spin-
dependent hopping rates between emitters located at positions $\mathbf{r}_{i}$
and $\mathbf{r}_{j}$. The hopping rates are determined by the real part of the
free space electromagnetic Green’s tensor evaluated at the relative coordinate
$\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}$ and include an explicit phase
dependence related to the azimuthal positions of the emitters (see Methods).

In addition to the coherent long-range couplings described by the Hermitian
Hamiltonian (1), we also consider collective dissipation that arises through
coupling to the electromagnetic vacuum at zero temperature. The full open
system dynamics include non-unitary evolution with the Lindbladian
superoperator,

$\mathcal{L}[\rho]=\sum^{N}_{i,j=1}\frac{\Gamma_{ij}^{\sigma\sigma^{\prime}}}{2}\left(2b_{j\sigma^{\prime}}\rho
b^{{\dagger}}_{i\sigma}-\left\\{b^{{\dagger}}_{i\sigma}b_{j\sigma^{\prime}},\rho\right\\}\right)$
(2)

that describes cooperative decay to the vacuum with rates
$\Gamma_{ij}^{\sigma\sigma^{\prime}}$ and single-emitter spontaneous emission
with rate $\Gamma_{0}\equiv\Gamma^{\sigma\sigma}_{ii}$, determined by the
imaginary part of the electromagnetic Green’s tensor. The time evolution of an
initial state $\rho_{0}\equiv\rho(t=0)$ is then governed by the quantum
optical master equation
$\dot{\rho}=-i\left[H,\,\rho\right]+\mathcal{L}[\rho]$. Throughout this work,
we focus on the single-excitation subspace which is sufficient to observe
spin-selective chiral transport.

Fig. 2: Spin polarization as a function of helix radius and pitch. Colors show
the spin polarization, $\Delta$, calculated for left-handed (left) and right-
handed (right) helices. Black crosses denote the geometrical parameters used
in Figs. 1d,e. The spin polarization for a given radius and pitch is reversed
between the two chiralities, though the magnitude and sign depend nontrivially
on the geometry through the electromagnetic Green’s tensor. Fig. 3: Band
structure and field emissions for a chiral lattice geometry. (a) Collective
excitation band structure for an infinite left-handed helix plotted in the
first Brillouin zone. The geometry is the same as in Figs. 1d,e. The energies
$\varepsilon_{nk}$ of each Bloch mode are shifted by the coherent couplings
$J_{ij}^{\sigma\sigma^{\prime}}$ (without loss of generality, we subtract off
the constant energy contribution $\omega_{0}$). Filled squares (diamonds)
denote modes that are predominantly spin $\uparrow$ ($\downarrow$). Open
circles indicate equal superposition modes at the $\mathcal{T}$ invariant
points $k=0,\pm\pi/a$. The anti-symmetric spin structure of each band results
from the emergent SOC induced by the chiral geometry. The coloring of each
symbol denotes the collective decay rate $\Gamma_{nk}$ of each mode. Bright
colors are superradiant and dark colors are subradiant. Dotted lines indicate
the light cone at $k=k_{0}$. (b) Snapshots of the emitted light field
intensity at $t=1/\Gamma_{0}$ (top two panels) and $t=7.9/\Gamma_{0}$ (bottom
two panels). Field intensities are shown in the $y$-$z$ plane at a distance
$x=10r_{0}$ from the helical axis so as to not include contributions from the
evanescent fields of the emitters. A projection of the helix geometry onto the
$y$-$z$ plane is shown in white. The left (right) two panels show the
contribution from spin $\uparrow$ ($\downarrow$) photons to the field
intensity. Bright colors demonstrate that helical superradiance is enhanced
when the wave packet is localized at either end of the helix. The maximum
intensities from each polarization are indicated in the color bar to the
right.

To illustrate the dynamics, we consider left- ($\xi=1)$ and right- ($\xi=-1$)
handed helices oriented along the $z$-axis with radius $r_{0}=0.05\lambda$,
pitch $a=0.175\lambda$, $\mathcal{N}=3$ emitters per $2\pi$ turn, and
$M=N/\mathcal{N}=20$ turns total. Fig. 1d shows the total population summed
across all emitters in each spin manifold,
$\sum_{i}\braket{b^{{\dagger}}_{i\sigma}b_{i\sigma}}$, as a function of time
for initial states
$\rho_{0}=\frac{1}{2}\left(\ket{\uparrow_{i}}\bra{\uparrow_{i}}+\ket{\downarrow_{i}}\bra{\downarrow_{i}}\right)$,
corresponding to unpolarized statistical mixtures of up and down spins
localized at a single emitter $i$. The top (bottom) two panels show the
dynamics when the starting emitter is located at the bottom (top) of the
helix. The spin character of each state is given by the expectation value
$\braket{\mathcal{S}_{z}}$ where
$\mathcal{S}_{z}=\openone_{N}\otimes\sigma_{z}=\sum_{i}(b^{{\dagger}}_{i\uparrow}b_{i\uparrow}-b^{{\dagger}}_{i\downarrow}b_{i\downarrow})$,
$\openone_{N}$ is the $N\times N$ identity matrix, and $\sigma_{z}$ is the
third Pauli operator. As the initial spin wave propagates through the helix
with group velocity $\mathbf{v}=v\hat{\mathbf{z}}$ (black arrows), it acquires
a helicity $\eta=\braket{\mathcal{S}_{z}}v/\lvert v\rvert$. For the left-
handed chirality (left-hand panels of Fig. 1d), the positive helicity states,
i.e., spin $\uparrow$($\downarrow$) propagating upwards (downwards), propagate
with enhanced dissipation. By contrast for the right-handed chirality (right-
hand panels of Fig. 1d), the negative helicity states, i.e., spin $\downarrow$
($\uparrow$) propagating upwards (downwards), are preferentially scattered. In
both cases, the product $\chi\equiv\xi\ \mathrm{sgn}(\eta)=-1$ describes
configurations that are robust to population loss.

Fig. 1e shows a snapshot of the individual emitter populations when the
initial upward-moving wave packet reaches the top end of the helix
($t=7.9/\Gamma_{0}$; denoted by dashed lines in Fig. 1d). For the left-handed
helix, spin $\downarrow$ excitations exhibit longer lifetimes and are more
efficiently transported to the opposite end. The same initial condition
results in the preferential transport of spin $\uparrow$ excitations for the
right-handed helix. For the more general case when the initial state is a
_polarized_ statistical mixture of spins, the same transport dynamics result
in the predominant excitation of only one chirality (Extended Data Fig. 1).

The magnitude of this chiral transport can be quantified by the spin
polarization,

$\Delta=\int_{0}^{\tau}dt\sum_{i}\frac{\left(\braket{b^{{\dagger}}_{i\uparrow}b_{i\uparrow}}-\braket{b^{{\dagger}}_{i\downarrow}b_{i\downarrow}}\right)}{\left(\braket{b^{{\dagger}}_{i\uparrow}b_{i\uparrow}}+\braket{b^{{\dagger}}_{i\downarrow}b_{i\downarrow}}\right)},$
(3)

defined as the difference in population between the spin manifolds integrated
until the wave packet reaches the opposite end of the helix at time $\tau$.
Fig. 2 shows the value of $\Delta$ for left- and right-handed helices of
varying radius and pitch. For any given geometry, the spin polarization is
equal and opposite between the left- and right-handed helices. However, for
fixed chirality $\xi$, both the magnitude and sign of the spin polarization
depend nontrivially on the geometric proportions of the helix. This fact may
be traced back to the nontrivial positional dependence of the electromagnetic
Green’s tensor and to the presence of long-range all-to-all couplings that
introduce multiple frequency scales to the dynamics (see Extended Data Figs. 2
and 3).

##  II Emergent Spin-Orbit Coupling and Helical Superradiance

The spin-selective dynamics can be understood by examining the band structure
of the collective helical modes. A helix with $N$ commensurate, evenly
distributed emitters is a non-Bravais lattice with $\mathcal{N}$ sublattices
and $M$ unit cells. This geometry is periodic along the $z$-axis and the
Hamiltonian (1) can be expressed in momentum space by performing the discrete
Fourier transform
$b_{i\mu\sigma}=(1/\sqrt{N})\sum_{\mathbf{k}}\exp{(i\mathbf{k}\cdot\mathbf{r}^{\mu}_{i})}b_{k\mu\sigma}$,
where $\mu$ is the sublattice index denoting the $M$ emitters along one
sublattice and $\mathbf{k}=k\hat{\mathbf{z}}$ is the lattice quasimomentum.
The eigenstates of the resultant $k$-space Hamiltonian are collective Bloch
modes of the form $\ket{\psi_{nk}}=\mathrm{e}^{ikz}\ket{u_{nk}}$ with energy
$\varepsilon_{nk}$ and band index $n$.

After a local gauge transformation (see Methods), the set of
$\varepsilon_{nk}$ describe $2\mathcal{N}$ energy bands that are periodic in
the first Brillouin zone, $k\in[-\pi/a,\pi/a]$, and become continuous as
$M\rightarrow\infty$ [25]. Fig. 3a shows an exemplary band structure for an
infinite left-handed helix with the same parameters used in Figs. 1d,e. Filled
squares denote modes that are predominantly spin $\uparrow$ with
$\braket{\mathcal{S}_{z}}>0$, whereas filled diamonds represent modes that are
predominantly spin $\downarrow$ with $\braket{\mathcal{S}_{z}}<0$. Open
circles correspond to equal superposition states with
$\braket{\mathcal{S}_{z}}=0$.

The qualitative features of the band structure shown in Fig. 3a can be
understood in terms of the symmetry properties of the Hamiltonian (1). This
Hamiltonian is $\mathcal{T}$ symmetric but breaks spin rotation symmetry
through the spin-flip term $\propto b_{i,\uparrow}^{\dagger}b_{j,\downarrow}$.
$\mathcal{T}$ symmetry transforms $\mathcal{S}_{z}\to-\mathcal{S}_{z}$ and
therefore requires modes with quasimomentum $\pm k$ to have equal energy but
opposite spin and group velocity $v=d\varepsilon_{k}/dk$. Besides the
$\mathcal{T}$ invariant points at $k=0,\pm\pi/a$ (for which $\mathcal{T}$
symmetry requires $v,\braket{\mathcal{S}_{z}}=0$), broken spin rotation
symmetry allows each mode to experience spin mixing. The spin character of
each Bloch mode is guaranteed to be nonzero at all $k\neq 0,\pm\pi/a$ when the
geometry is chiral (see Methods). This property allows modes with finite
dispersion ($v\neq 0$) to experience spin-momentum locking. The resultant SOC
imposes an energy penalty on helical modes in the $\chi=1$ configuration and
results in reduced population and less efficient spin transport (see Figs.
1d,e and 2). As shown in the Methods, this emergent SOC is a universal feature
of all chiral geometries.

Fig. 4: Demonstration of nontrivial topology for chiral geometries. Each plot
shows the collective band structure for a left-handed helix constructed with
$\mathcal{N}$ emitters per $2\pi$ turn. The radius and pitch are the same as
in Figs. 1d,e, and 3. A topological phase transition occurs at
$\mathcal{N}=3$, characterized by the opening of an energy band gap (gray
shaded regions) and a nontrivial Zak phase. Colors show the spin character
$\braket{\mathcal{S}_{z}}$ of each mode and demonstrate the emergence of a
finite SOC in the topologically nontrivial regime.

The energy and spin character of each mode are dictated solely by the
Hermitian Hamiltonian (1), which is sufficient to explain the spin-selective
chiral transport (see Methods and Extended Data Fig. 4). Nevertheless,
inclusion of the Lindbladian (2) enhances this dynamical effect due to
collective dissipation at subwavelength scales. In the single-excitation
subspace, the dynamics generated by Eqs. (1) and (2) are equivalent to those
evoked by the non-Hermitian effective Hamiltonian,

$H_{\mathrm{eff}}=H-i\sum_{i,j=1}^{N}\frac{\Gamma_{ij}^{\sigma\sigma^{\prime}}}{2}b_{i\sigma}^{\dagger}b_{j\sigma^{\prime}}.$
(4)

This effective Hamiltonian has complex eigenvalues
$\tilde{\varepsilon}_{nk}=\varepsilon_{nk}+i\Gamma_{nk}$, where $\Gamma_{nk}$
is the vacuum decay rate of each Bloch mode (indicated by the color coding of
symbols in Fig. 3a). At subwavelength distances, cooperative resonances can
enhance or suppress the decay rate of a given mode away from the bare emitter
decay rate, $\Gamma_{0}$, depending on whether that mode lies inside or
outside the light cone (dotted lines in Fig. 3a). Bright modes (those inside
the light cone) are superradiant with $\Gamma_{nk}>\Gamma_{0}$, whereas dark
modes (outside the light cone) experience a momentum mismatch with the
available vacuum modes and are subradiant with $\Gamma_{nk}\approx 0$.

The effect of this Lindbladian is to passively break time-reversal symmetry
[26] by adding only an overall decay to each mode without influencing the
energetics or spin properties of the band structure described above. These
decay terms may be interpreted as imaginary energies resulting from evolution
with $H_{\mathrm{eff}}$. As for the real energies $\varepsilon_{nk}$,
$\mathcal{T}$ symmetry requires that the decay rates $\Gamma_{nk}$ be equal
for modes with $\pm k$. Because these modes necessarily have opposite spin and
group velocity, they must have identical helicity. As such, the two maximally
superradiant modes always enhance the population loss for the same helicity.
Because space-inversion transforms $\xi\to-\xi$ and $\eta\to-\eta$, the
superradiant helicity is reversed between the left- and right-handed
chiralities and amplifies the chiral transport. As shown in Figs. 1d,e, this
dynamical effect results in the preferential population of a given spin
manifold determined by the propagation direction and chirality of the
geometry.

Fig. 3b demonstrates this helical superradiance explicitly for the finite
left-handed helix used in Figs. 1d,e. The left (right) panels of Fig. 3b show
the intensity of the emitted dipole field
$\braket{\mathbf{E}^{-}_{\sigma}(\mathbf{r},t)\cdot\mathbf{E}^{+}_{\sigma}(\mathbf{r},t)}$
radiated by $\uparrow$ ($\downarrow$) excitations. The top two panels show the
initial superradiant burst at $t=1/\Gamma_{0}$ after initialization with an
unpolarized mixed state at the bottom end of the helix (see also top left
panel of Fig. 1d). The intense emission of $\uparrow$-polarized photons
results from dynamically generated overlaps between the evolving state,
$\rho(t)$, and the positive helicity superradiant modes of Fig. 3a. At
intermediate times, the wave packet enters subradiant guided modes [21] and
radiation into the vacuum via the bulk emitters is suppressed. Strong
superradiant emission resumes at $t=7.9/\Gamma_{0}$ out of the opposite end of
the helix (bottom two panels of Fig 3b).

##  III Topological properties

The SOC identified above suggests a nontrivial topology for the energy bands
of chiral geometries and the existence of topologically protected edge modes
[27, 28, 4]. Conservation of angular momentum requires that the two-body spin-
flip process in Eq. (1) (see also Fig. 1c) described by
$J_{ij}^{\uparrow\downarrow}$ picks up a complex phase,
$\exp{(-2i\phi_{ij})}$, related to the azimuthal positions of the two emitters
(see Methods) in order to compensate for the changing photon polarization.
This phase dependence gives rise to an emergent gauge field that results in a
nontrivial topology. In 1D, the topological properties of the band structure
are characterized by the Zak phase [29],

$\varphi=\oint_{\mathcal{C}}\operatorname{Tr}{[\mathbf{A}(k)]}dk,$ (5)

where $A_{mn}(k)=i\braket{u_{mk}}{\partial_{k}u_{nk}}$ is the non-Abelian
$U(\mathcal{N})$ gauge field [23] and $\mathcal{C}$ defines a closed loop in
reciprocal space around the Brillouin zone torus. The Zak phase is defined
modulo $2\pi$ and can be either $0$ (trivial) or $\pi$ (nontrivial).

Fig. 4 demonstrates a transition to topologically nontrivial phases as a
function of $\mathcal{N}$. When $\mathcal{N}=1$, the lattice is a simple chain
of transversely polarized emitters with lattice spacing $a$. In this case, the
two spin manifolds are uncoupled with $J_{ij}^{\uparrow\downarrow}=0$ and the
band structure consists of two degenerate copies of the band structure for a
chain of linearly polarized two-level emitters [21]. Spin rotation symmetry
requires that the two degenerate states at each $k$ have
$\braket{\mathcal{S}_{z}}=\pm 1$, which averages to zero at every point in the
Brillouin zone. For $\mathcal{N}=2$, the geometry is that of a staggered chain
with longitudinal separation $a$ and transverse separation $2r_{0}$. Here,
uniform spin mixing occurs at each mode through
$J_{ij}^{\uparrow\downarrow}\neq 0$, and the band structure resembles that of
two 1D chains separated by a finite interaction energy. The presence of
reflection symmetry for $\mathcal{N}=1,2$ prevents the emergence of a finite
SOC (see Methods). The minimal chiral geometry is realized for $\mathcal{N}=3$
and is characterized by a finite energy gap and the existence of emergent SOC.
The Zak phases calculated for the separate upper and lower bundles of
$\mathcal{N}$ bands on each side of this energy gap are nontrivial, indicating
the transition to a topologically distinct phase. The edge modes for the
$\mathcal{N}=3$ geometry are shown in Fig. 5. This nontrivial topology
persists for $\mathcal{N}\geq 3$ as long as the geometry remains chiral.

These topological considerations might also be relevant to the spin-selective
transport of helical electrons observed in the CISS effect. In the independent
electron approximation, there is no spin-exchange contribution to the hopping
amplitudes and the Hamiltonian (1) is also applicable to spin-1/2 fermions,
albeit with different behavior under $\mathcal{T}$ (see Methods).
Nevertheless, the influence of the gauge field in Eq. (5) has not been
considered in any theoretical analysis of electron spin polarization in chiral
molecules thus far. Accounting for the emergent SOC induced by particle motion
along the chiral geometry might resolve the discrepancies between theoretical
models of CISS and experimentally observed spin polarizations.

Fig. 5: A topologically nontrivial band structure leads to topologically
protected edge modes. Purple bars denote emitter populations in the edge modes
realized for a finite helix of $\mathcal{N}=3$ and $M=20$ unit cells with the
same geometry used in Figs. 1d,e, 3, and 4. The edge modes are strongly
localized at the top and bottom ends of the helix with minimal extension into
the bulk. Both modes experience uniform spin-mixing. Topological protection
guarantees that the existence of these modes is robust to the presence of
disordering perturbations that do not close the energy gap.

##  IV Towards Experimental Implementations

The spin-selective chiral transport and emergent SOC presented in this work
could be observed with a variety of different platforms. A potential
implementation based on ultra-cold atoms trapped in an optical potential built
from Laguerre-Gauss modes of light is particularly intriguing and could be
realized with readily available techniques [30, 31, 32]. Laguerre-Gauss modes
are the eigenmodes of the paraxial wave equation and characterize states with
orbital angular momentum quantum number, $l$. The $l=1$ state exhibits a
cylindrical geometry with a phase advance that winds once per wavelength.
Interfering this mode with an orthogonally polarized plane wave shapes the
field intensity into a left-handed helix. A similar setup using the $l=-1$
mode yields a right-handed helix of the same geometrical proportions.
Intersecting this field with a cloud of red-detuned atoms would trap some of
these atoms at intensity maxima determined by the helical potential. An
additional long-range interaction potential (which could, for example, be
imposed by Rydberg dressing the atoms via an additional laser [33, 34]) would
result in a periodic arrangement of atoms along the helix.

An alternative approach could be realized using a 3D optical tweezer array.
Coherent oscillations between atoms with V-type level structures have been
demonstrated, e.g., by isolating the 60S1/2 and 60P3/2 Rydberg manifolds of
87Rb [35, 36]. The 17.2 GHz transition frequency between these manifolds
allows for subwavelength dynamics at $\mu$m-scale tweezer separations.
Although helical superradiance may not be observable with this platform due to
long Rydberg excitation lifetimes, the spin-selective transport and
topological properties of the system could still be achieved. Helical
superradiance could, however, potentially be observed by realizing the V-level
structure with the 3P0 and 3D1 manifolds of 88Sr [18].

##  V Outlook

Here we have demonstrated spin-selective photonic excitation transport where
the internal emitter degrees of freedom are coupled to an underlying chiral
geometry. This phenomenon has a complete description within the electric
dipole approximation, placing it within the new class of nonmagnetic chiral
interactions exhibiting very strong optical responses [37, 38]. Precise
control over the transport and superradiant or subradiant emission of photons
is a fundamental goal of quantum information science and could contribute to
the development of new quantum technologies. The exploitation of excitation
helicity as an additional degree of freedom might also allow for new types of
chiral-selective chemistry driven by circularly polarized light, without
reliance on higher order multipole moments.

The results of this work may also play a role in fundamental processes in
nature. The well-documented importance of photoexcitation dynamics to the
stability and reactivity of prebiotic molecules (e.g., RNA, DNA) [39, 40, 41]
evokes the question of whether the effects presented here might have
contributed to the emergence of biological homochirality. Although the
mechanisms responsible for the homochirality of life are unknown, recent
studies have suggested that chiral-selective processes during prebiotic
synthesis could have seeded an initial imbalance that was amplified by
subsequent chemical reactions [42, 43]. In particular, a bias in the net flux
of circularly polarized photons—as has been observed in nearby star forming
regions [44, 45, 46]—has long been suggested as a potential symmetry breaking
agent [47, 48, 49]. Previous proposals for translating this bias into
homochiral chemistry typically invoke traditional chiroptical phenomena (e.g.,
asymmetric photolysis) that result in enantiomeric excesses of only a few
percent [50, 51, 52, 53]. The helical superradiance identified in this work
provides a new mechanism for chiral amplification that does not rely on weak
magnetic interactions and could potentially lead to a much larger chiral bias.
In this scenario, the predominant circular polarization incident along a given
propagation direction would excite one chirality into short-lived superradiant
modes and the other chirality into long-lived subradiant modes. These long-
lived modes would be preferentially available for photochemical reactions,
resulting in different chemistry for right- and left-handed molecules. Such a
process could contribute to the enantiomeric excesses observed in carbonaceous
chondrites [54] and may have provided an exogenous source of homochiral
material to the early Earth.

##  VI Methods

###  VI.1 Dipole-dipole interactions

The free space electromagnetic Green’s tensor is the solution to the wave
equation,

$\nabla^{2}\mathbf{G}(\mathbf{r},\mathbf{r^{\prime}},k)-k^{2}\mathbf{G}(\mathbf{r},\mathbf{r^{\prime}},k)=\delta(\mathbf{r}-\mathbf{r^{\prime}})\openone$
(6)

for observational coordinates $\mathbf{r}$ and source coordinates
$\mathbf{r}^{\prime}$. In the case where the emitters located at positions
$\mathbf{r}_{j}$ are well approximated by point electric dipoles, the Green’s
tensor between emitters $i$ and $j$ depends only on the relative coordinate
$\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}$. In the Born and Markov
approximations, the Green’s tensor may further be regarded as dispersionless
and is given by [21],

$\mathbf{G}(\mathbf{r}_{ij},k_{0})=\frac{\mathrm{e}^{ik_{0}r}}{4\pi
k_{0}^{2}r^{3}}\Big{[}(k_{0}^{2}r^{2}+ik_{0}r-1)\openone\\\
-(k_{0}^{2}r^{2}+3ik_{0}r-3)\frac{\mathbf{r}_{ij}\otimes\mathbf{r}_{ij}}{r^{2}}\Big{]}.$
(7)

The $\sigma$ component of the positive frequency electric field calculated at
position $\mathbf{r}$ is then calculated (in the time domain) as [21]

$\mathbf{E}^{+}_{\sigma}(\mathbf{r})=\mu_{0}\omega_{0}^{2}\sum_{j=1}^{N}\mathbf{G}(\mathbf{r}-\mathbf{r}_{j},k_{0})\cdot\boldsymbol{\wp}_{j\sigma}b_{j\sigma},$
(8)

where $\boldsymbol{\wp}_{i\sigma}$ is the transition dipole matrix element
vector for orbital $\sigma\in\\{\uparrow,\downarrow\\}$ on emitter $j$,
$b_{j\sigma}$ is the bosonic lowering operator defined in the main text, and
$\mu_{0}$ is the vacuum permeability. Consequently, dipole-dipole interactions
between pairs of emitters are characterized by
$-\boldsymbol{\wp}^{\dagger}_{i\sigma}\cdot\mathbf{E}^{+}_{j\sigma^{\prime}}(\mathbf{r}_{i})=J^{\sigma\sigma^{\prime}}_{ij}-(i/2)\Gamma^{\sigma\sigma^{\prime}}_{ij}$,
where

$\displaystyle J^{\sigma\sigma^{\prime}}_{ij}$
$\displaystyle=-\mu_{0}\omega_{0}^{2}\boldsymbol{\wp}^{\dagger}_{i\sigma}\cdot\operatorname{Re}\mathbf{G}(\mathbf{r}_{ij},k_{0})\cdot\boldsymbol{\wp}_{j\sigma^{\prime}}$
(9) $\displaystyle\Gamma^{\sigma\sigma^{\prime}}_{ij}$
$\displaystyle=2\mu_{0}\omega_{0}^{2}\boldsymbol{\wp}^{\dagger}_{i\sigma}\cdot\operatorname{Im}\mathbf{G}(\mathbf{r}_{ij},k_{0})\cdot\boldsymbol{\wp}_{j\sigma^{\prime}}$
(10)

describe the coherent and dissipative parts of the interaction, respectively,
and
$\Gamma_{0}\equiv\Gamma_{ii}^{\sigma\sigma}=\omega_{0}^{3}\lvert\boldsymbol{\wp}_{i\sigma}\rvert^{2}/3\pi\hbar\epsilon_{0}c^{3}$
is the spontaneous emission rate of each orbital.

For a helix periodic along the $z$-axis with radius $r_{0}$, pitch $a$, and
$\mathcal{N}$ emitters per $2\pi$ turn, the relative coordinate between
emitters $i$ and $j$ can be written as
$\mathbf{r}_{ij}=x_{ij}\hat{\mathbf{x}}+y_{ij}\hat{\mathbf{y}}+z_{ij}\hat{\mathbf{z}}$
for

$\displaystyle x_{ij}$
$\displaystyle=2r_{0}\sin\left[(\mu-\nu)\frac{\pi}{\mathcal{N}}\right]\cos\left[(\mu+\nu)\frac{\pi}{\mathcal{N}}\right]$
$\displaystyle y_{ij}$
$\displaystyle=2r_{0}\sin\left[(\mu-\nu)\frac{\pi}{\mathcal{N}}\right]\sin\left[(\mu+\nu)\frac{\pi}{\mathcal{N}}\right]$
$\displaystyle z_{ij}$
$\displaystyle=\left[(\mu-\nu)\frac{a}{\mathcal{N}}+al\right],$ (11)

where $\mu\ (\nu)=i\ (j)\ \mathrm{mod}\ \mathcal{N}$, and $l$ denotes the
number of unit cells between $i$ and $j$. Plugging
$r_{ij}=\lvert\mathbf{r}_{ij}\rvert$ and $\Gamma_{0}$ into Eq. (9), the spin-
flip interaction is found to be

$J_{ij}^{\uparrow\downarrow}=\frac{3r_{0}^{2}\lambda\Gamma_{0}}{4\pi
k_{0}^{2}r_{ij}^{5}}\sin^{2}\left(\frac{\phi_{i}-\phi_{j}}{2}\right)\mathrm{e}^{-2i\phi_{ij}}\times\\\
\left[(3-k_{0}^{2}r_{ij}^{2})\cos(k_{0}r_{ij})+3k_{0}r_{ij}\sin(k_{0}r_{ij})\right]$
(12)

for $\phi_{i}=\tan^{-1}(y_{i}/x_{i})$ and
$\phi_{ij}\equiv(\phi_{i}+\phi_{j})/2$.

###  VI.2 Mapping to a pseudospin formalism

In the Cartesian basis, the transition dipole matrix element vectors for left
and right circularly polarized excitations are given by

$\boldsymbol{\wp}_{\pm}=\frac{\lvert\boldsymbol{\wp}\pm\rvert}{\sqrt{2}}\begin{pmatrix}1\\\
\pm i\\\ 0\end{pmatrix}.$ (13)

In the single excitation manifold, the ground state, $\ket{g}$, of each
emitter may be traced out of the Hilbert space and the excited states,
$\ket{\sigma}$, are represented succinctly using the basis vector mapping
$\ket{\uparrow}=\ket{+}=\begin{pmatrix}1,0\end{pmatrix}^{T}$,
$\ket{\downarrow}=\ket{-}=\begin{pmatrix}0,1\end{pmatrix}^{T}$ such that
$b^{\dagger}_{\uparrow}b_{\uparrow}-b^{\dagger}_{\downarrow}b_{\downarrow}=\ket{\uparrow}\bra{\uparrow}-\ket{\downarrow}\bra{\downarrow}=\sigma_{z}$.
The dipole-dipole Green’s tensor (7) neglects electronic exchange
interactions, which is a good approximation when the orbitals on adjacent
emitters are well-separated. This allows for the circularly polarized
excitations to be described using bosonic statistics with a pseudospin-1/2
degree of freedom characterized by $\mathcal{S}_{z}$. This bosonization,
however, implies a deviation from the usual fermionic time-reversal
properties, as discussed below.

It is familiar from traditional band theory that spin-1/2 excitations obeying
fermionic statistics exist as Kramers’ pairs when $\mathcal{T}$ symmetry is
preserved. This constraint requires at least a two-fold degeneracy at every
point in the Brillouin zone when inversion symmetry is present. If inversion
symmetry is broken (i.e., the lattice geometry is chiral), $\mathcal{T}$
symmetry still requires this degeneracy be preserved at all $\mathcal{T}$
invariant quasimomenta. In either case, the spin of each excitation may be
interpreted as a vector on the Bloch sphere.

In the bosonic case, no such degeneracy is required and $\mathcal{T}$ may be
written as $\mathcal{T}=\mathcal{K}\mathcal{S}_{x}$ where $\mathcal{K}$ is the
complex conjugation operator. The polarization of each excitation is instead
given by a vector on the Poincare sphere [55, 56], with left and right
circular polarizations residing at the north and south poles, respectively. In
contrast to the fermionic case (where an equal superposition of $\uparrow$ and
$\downarrow$ spins results in an equal magnitude spin-1/2 excitation pointing
in the orthogonal plane), vectors residing along the equator of the Poincare
sphere do not carry angular momentum. That is, an equal superposition of left
and right circularly polarized excitations yields a linearly polarized
excitation of pseudospin-0 (see Eq. (13)). Whereas the $\mathcal{T}$ operator
acting on a fermionic system corresponds to a complete inversion of the Bloch
vector through the origin, the bosonic $\mathcal{T}$ results in a reflection
through the Poincare sphere equatorial plane and leaves vectors residing in
this plane unchanged.

###  VI.3 Band structure calculations

Accounting for a potential Zeeman field, $\mathbf{B}$, the general form of the
Hamiltonian for a collection of V-type quantum emitters interacting via the
Green’s tensor formalism is

$H=\sum_{i=1}^{N}\sum_{\sigma}\omega_{0}b^{{\dagger}}_{i\sigma}b_{i\sigma}+\sum_{i=1}^{N}\boldsymbol{\mu}\cdot\mathbf{B}(b^{{\dagger}}_{i\uparrow}b_{i\uparrow}-b^{{\dagger}}_{i\downarrow}b_{i\downarrow})\\\
+\sum^{N}_{i,j\neq
i=1}\sum_{\sigma,\sigma^{\prime}}J_{ij}^{\sigma\sigma^{\prime}}b^{{\dagger}}_{i\sigma}b_{j\sigma^{\prime}}.$
(14)

For an infinite, periodic, non-Bravais lattice, the substitution
$b_{j\mu\sigma}=(1/\sqrt{N})\sum_{k}\exp{(i\mathbf{k}\cdot\mathbf{r}^{\mu}_{j})}b_{k\mu\sigma}$
for sublattice index $\mu$ and quasimomentum $\mathbf{k}$ into Eq. (14) yields
the $\mathbf{k}$-space Hamiltonian [25, 57],

$H=\sum_{\mathbf{k}}\sum_{\mu,\nu}\sum_{\sigma,\sigma^{\prime}}h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})b^{{\dagger}}_{\mathbf{k}\mu\sigma}b_{\mathbf{k}\nu\sigma^{\prime}}$
(15)

with matrix elements

$h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})=\omega_{0}\delta_{\mu\nu}\delta_{\sigma\sigma^{\prime}}+\zeta_{\mu\sigma,\nu\sigma^{\prime}}+\chi^{I}_{\mu\sigma,\nu\sigma^{\prime}}+\chi^{II}_{\mu\sigma,\nu\sigma^{\prime}},$
(16)

where

$\displaystyle\zeta_{\mu\sigma,\nu\sigma^{\prime}}$
$\displaystyle=\boldsymbol{\mu}\cdot\mathbf{B}(\delta_{\sigma\uparrow}\delta_{\sigma^{\prime}\uparrow}-\delta_{\sigma\downarrow}\delta_{\sigma^{\prime}\downarrow})\delta_{\mu\nu}$
(17) $\displaystyle\chi^{I}_{\mu\sigma,\nu\sigma^{\prime}}$
$\displaystyle=\sum_{\mathbf{R}_{\mu}\neq
0}\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{R}_{\mu}}J_{\sigma\sigma^{\prime}}(\mathbf{R}_{\mu})\delta_{\mu\nu}$
(18) $\displaystyle\chi^{II}_{\mu\sigma,\nu\sigma^{\prime}}$
$\displaystyle=\sum_{\mathbf{R}_{\mu}}\mathrm{e}^{-i\mathbf{k}\cdot(\mathbf{R}_{\mu}+\mathbf{n}_{\mu\nu})}J_{\sigma\sigma^{\prime}}(\mathbf{R}_{\mu}+\mathbf{n}_{\mu\nu})(1-\delta_{\mu\nu}).$
(19)

Here, the (infinite) set of $\mathbf{R}_{\mu}$ denote vectors of the
underlying Bravais lattice and
$\mathbf{n}_{\mu\nu}=\mathbf{n}_{\mu}-\mathbf{n}_{\nu}$ is the vector pointing
from sublattice $\nu$ to $\mu$ within a given unit cell. $\zeta$ describes the
Zeeman splitting between $\uparrow$ and $\downarrow$ orbitals with magnetic
moment $\boldsymbol{\mu}$ due to an external magnetic field $\mathbf{B}$.
Finally, $\chi^{I}$ describes interactions between emitters on the same
sublattice and $\chi^{II}$ describes interactions between different
sublattices.

To enforce periodicity of the Brillouin zone, it is necessary to apply the
local gauge transformation [58]
$b_{\mathbf{k}\mu\sigma}\to\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{n}_{\mu}}b_{\mathbf{k}\mu\sigma}$.
A redefinition of $h(\mathbf{k})$ to include this $\mathbf{k}$-dependent phase
transforms

$\chi^{II}_{\mu\sigma,\nu\sigma^{\prime}}\to\sum_{\mathbf{R}_{\mu}}\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{R}_{\mu}}J_{\sigma\sigma^{\prime}}(\mathbf{R}_{\mu}+\mathbf{n}_{\mu\nu})(1-\delta_{\mu\nu})$
(20)

and ensures that $h(\mathbf{k}+\mathbf{G}_{\mu})=h(\mathbf{k})$ for any
reciprocal lattice vector $\mathbf{G}_{\mu}\cdot\mathbf{R}_{\mu}=2\pi$.

For the quasi-1D helical geometries used in this work, the lattice vectors are
given by $\mathbf{R}_{\mu}=al\hat{\mathbf{z}}$ for lattice constant $a$ and
$l\in\mathbb{Z}$. The infinite sums contained in Eqs. (18) and (19) are easily
computed numerically and typically converge for $\lvert l\rvert\lesssim 50$.

###  VI.4 Topological classification

For a non-Bravais lattice with $\mathcal{N}$ sublattices, the Hamiltonian (15)
gives rise to $2\mathcal{N}$ Bloch modes of the form
$\ket{\psi_{n\mathbf{k}}}=\mathrm{e}^{i\mathbf{k}\cdot\mathbf{r}}\ket{u_{n\mathbf{k}}}$.
The $\mathcal{N}$ bands on either side of the energy gap obey a
$U(\mathcal{N})$ gauge freedom,

$\ket{u_{n\mathbf{k}}}\to\sum_{m}U_{mn}(\mathbf{k})\ket{u_{n\mathbf{k}}},$
(21)

where the $\mathcal{N}\times\mathcal{N}$ unitary matrix $U_{mn}$ describes an
equivalence class of physically identical Bloch manifolds. The non-Abelian
Berry connection for each isolated manifold then follows as [23, 59]

$\mathbf{A}_{mn}(\mathbf{k})=i\braket{u_{m\mathbf{k}}}{\nabla_{\mathbf{k}}u_{n\mathbf{k}}}.$
(22)

For a closed loop $\mathcal{C}$ around the first Brillouin zone, the Berry
phase is given by

$\varphi=\oint_{\mathcal{C}}\operatorname{Tr}{[\mathbf{A}(\mathbf{k})]}\cdot
d\mathbf{k}=-\operatorname{Im}\left\\{\ln\left[\det\left(\mathcal{W}_{\mathcal{C}}\right)\right]\right\\},$
(23)

where $\mathcal{W}_{\mathcal{C}}$ is the Wilson loop for the path
$\mathcal{C}$ traversed in reciprocal space. In 1D, for discretized
$k=k_{0},...,k_{L}$, the Wilson loop may be written as [60, 61]

$\displaystyle\mathcal{W}_{\mathcal{C}}$
$\displaystyle=\mathcal{P}\exp{\bigg{\\{}\oint_{\mathcal{C}}-iA_{mn}(k)dk\bigg{\\}}}$
$\displaystyle=\prod_{i=0}^{L-1}\exp{\\{-iA^{(k_{i},k_{i+1})}_{mn}dk\\}},$
(24)

where $\mathcal{P}$ is the path-ordering operator and

$\displaystyle\exp{\\{-iA^{(k_{i},k_{i+1})}_{mn}dk\\}}$
$\displaystyle\approx\delta_{mn}-iA^{(k_{i},k_{i+1})}_{mn}dk$
$\displaystyle=\delta_{mn}+\bra{u_{mk_{i}}}(\ket{u_{nk_{i+1}}}-\ket{u_{nk_{i}}})$
$\displaystyle=M^{(k_{i},k_{i+1})}_{mn}$ (25)

for overlap matrix
$M^{(k_{i},k_{i+1})}_{mn}=\braket{u_{mk_{i}}}{u_{nk_{i+1}}}$, provided the
cell-periodic functions are specified in the periodic gauge where
$\ket{u_{nk_{0}}}=\ket{u_{nk_{L}}}$. In this case, the Wilson loop is easily
computed as

$\mathcal{W}_{\mathcal{C}}=\prod_{i}M^{(k_{i},k_{i+1})}_{mn}.$ (26)

Importantly, while $A_{mn}(k)$ is gauge-dependent, the 1D Berry phase (or Zak
phase) is gauge invariant modulo $2\pi$.

###  VI.5 Universality to all chiral geometries

In this section, we show that chirality is both a necessary and sufficient
condition for the existence of emergent SOC in pseudospin-1/2 bosons.

The parity, or space-inversion, operator $\mathcal{I}$ leaves the total
Hamiltonian (14) unchanged but can act non-trivially on the matrix elements
$h_{\mu\sigma,\nu,\sigma^{\prime}}(\mathbf{k})$ for particular values of
$\mathbf{k}$. $\mathcal{I}$ transforms $\mathbf{k}\to-\mathbf{k}$,
$\mathbf{R}_{\mu}\to-\mathbf{R}_{\mu}$, and
$\mathbf{n}_{\mu\nu}\to-\mathbf{n}_{\mu\nu}$ such that

$\mathcal{I}H\mathcal{I}^{-1}=\sum_{\mathbf{k}}\sum_{\mu,\nu}\sum_{\sigma,\sigma^{\prime}}h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})b^{{\dagger}}_{\mathbf{-k}\mu\sigma}b_{\mathbf{-k}\nu\sigma^{\prime}}$
(27)

(the coupling strengths $J_{\sigma\sigma^{\prime}}(\mathbf{R}_{\mu})$ remain
unchanged; see Eq. (7)). A redefinition of $\mathbf{k}\to-\mathbf{k}$ (which
leaves the total Hamiltonian invariant due to the sum over all $\mathbf{k}$)
transforms $b_{\mathbf{-k}\mu\sigma}\to b_{\mathbf{k}\mu\sigma}$ and

$\displaystyle\chi^{I}_{\mu\sigma,\nu\sigma^{\prime}}$
$\displaystyle\to\sum_{\mathbf{R}_{\mu}\neq
0}\mathrm{e}^{i\mathbf{k}\cdot\mathbf{R}_{\mu}}J_{\sigma\sigma^{\prime}}(\mathbf{R}_{\mu})\delta_{\mu\nu}$
(28) $\displaystyle\chi^{II}_{\mu\sigma,\nu\sigma^{\prime}}$
$\displaystyle\to\sum_{\mathbf{R}_{\mu}}\mathrm{e}^{i\mathbf{k}\cdot\mathbf{R}_{\mu}}J_{\sigma\sigma^{\prime}}(\mathbf{R}_{\mu}+\mathbf{n}_{\mu\nu})(1-\delta_{\mu\nu})$
(29)

while leaving the other terms in
$h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})$ unchanged. The sum over
$\mathbf{R}_{\mu}$ allows for a redefinition of
$\mathbf{R}_{\mu}\to-\mathbf{R}_{\mu}$, which (again noting that
$J_{\sigma\sigma^{\prime}}(-\mathbf{R}_{\mu})=J_{\sigma\sigma^{\prime}}(\mathbf{R}_{\mu}))$
further transforms

$\displaystyle\chi^{I}_{\mu\sigma,\nu\sigma^{\prime}}$
$\displaystyle\to\sum_{\mathbf{R}_{\mu}\neq
0}\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{R}_{\mu}}J_{\sigma\sigma^{\prime}}(\mathbf{R}_{\mu})\delta_{\mu\nu}$
(30) $\displaystyle\chi^{II}_{\mu\sigma,\nu\sigma^{\prime}}$
$\displaystyle\to\sum_{\mathbf{R}_{\mu}}\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{R}_{\mu}}J_{\sigma\sigma^{\prime}}(\mathbf{R}_{\mu}-\mathbf{n}_{\mu\nu})(1-\delta_{\mu\nu}).$
(31)

The net effect of spatial inversion is then to leave the total Hamiltonian
invariant but to take $\mathbf{n}_{\mu\nu}\to-\mathbf{n}_{\mu\nu}$ in
$h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})$, which is equivalent to
transforming the real-space non-Bravais lattice to its mirror image. When the
geometry is chiral,
$J_{\sigma\sigma^{\prime}}(\mathbf{R}_{\mu}+\mathbf{n}_{\mu\nu})\neq
J_{\sigma\sigma^{\prime}}(\mathbf{R}_{\mu}-\mathbf{n}_{\mu\nu})$ leading to
profound effects on the band structure.

We begin by demonstrating that chirality is a necessary condition to observe
the emergent SOC described in the main text.

######  Definition 1.

Let $\mathcal{S}_{i}=\openone_{\mathcal{N}}\otimes\sigma_{i}$ with
$\openone_{\mathcal{N}}$ the $\mathcal{N}\times\mathcal{N}$ identity matrix
and $\sigma_{i}$ the Pauli operators for $i\in\\{x,y,z\\}$. $\mathcal{S}_{i}$
has the following properties:

  1. 1.

Unitary: $\mathcal{S}^{\dagger}_{i}=\mathcal{S}^{-1}_{i}$

  2. 2.

Hermitian: $\mathcal{S}^{*}_{i}=\mathcal{S}_{i}^{T}$.

  3. 3.

Involutory: $\mathcal{S}_{i}=\mathcal{S}_{i}^{-1}$.

######  Lemma 1.

Let $M$ be a $2\mathcal{N}\times 2\mathcal{N}$ Hermitian matrix satisfying
$\mathcal{S}_{x}M\mathcal{S}^{-1}_{x}=M^{*}$. If $\ket{\lambda}$ is an
eigenstate of $M$ satisfying $M\ket{\lambda}=\lambda\ket{\lambda}$ for
eigenvalue $\lambda$, then $\mathcal{S}_{x}\ket{\lambda}$ is an eigenstate of
$M^{*}$ with the same eigenvalue $\lambda$.

###### Proof.

It follows from the involutory nature of $\mathcal{S}_{x}$ that
$\mathcal{S}_{x}M=\mathcal{S}_{x}M\mathcal{S}^{-1}_{x}\mathcal{S}_{x}=M^{*}\mathcal{S}_{x}$.
Acting with $\mathcal{S}_{x}$ on the eigenvalue equation for $M$ then gives
$\mathcal{S}_{x}M\ket{\lambda}=M^{*}\mathcal{S}_{x}\ket{\lambda}=\lambda\mathcal{S}_{x}\ket{\lambda}$.
∎

######  Lemma 2.

Let $M$ be a $2\mathcal{N}\times 2\mathcal{N}$ Hermitian matrix satisfying
$\mathcal{S}_{x}M\mathcal{S}^{-1}_{x}=M^{*}$. If $\ket{\lambda}$ is an
eigenstate of $M$ satisfying $M\ket{\lambda}=\lambda\ket{\lambda}$ for
eigenvalue $\lambda$, then the states $\ket{\lambda^{*}}$ and
$\mathcal{S}_{x}\ket{\lambda}$ are both eigenstates of $M^{*}$ with the same
eigenvalue $\lambda$.

###### Proof.

The Hermiticity of $M$ implies that the eigenvalues $\lambda$ are real and
that $M^{*}\ket{\lambda^{*}}=\lambda\ket{\lambda^{*}}$. In conjunction with
Lemma (1), this implies that $\ket{\lambda^{*}}$ and
$\mathcal{S}_{x}\ket{\lambda}$ are both eigenstates of $M^{*}$ with eigenvalue
$\lambda$. ∎

######  Theorem 1.

Let $M$ be a $2\mathcal{N}\times 2\mathcal{N}$ Hermitian matrix with basis
vectors $\ket{\mu}\otimes\ket{\sigma}$ such that
$\mathrm{dim}(\ket{\mu})=\mathcal{N}$ and $\mathrm{dim}(\ket{\sigma})=2$. Let
$\ket{\bar{\sigma}}=\sigma_{x}\ket{\sigma}$ denote the exchanged version of
$\ket{\sigma}$. Let
$\ket{u_{n}}=\sum_{\mu}\sum_{\sigma}c^{n}_{\mu\sigma}\ket{\mu}\otimes\ket{\sigma}$
be a general eigenstate of $M$ satisfying the eigenvalue equation
$M\ket{u_{n}}=\varepsilon_{n}\ket{u_{n}}$ where $n=1,...,2\mathcal{N}$. If
$\mathcal{S}_{x}M\mathcal{S}^{-1}_{x}=M^{*}$, then the eigenstates
$\ket{u_{n}}$ and $\ket{u_{m}}$ satisfy
$c^{n}_{\mu^{\prime}\bar{\sigma}^{\prime}}=(c^{m}_{\mu^{\prime}\sigma^{\prime}})^{*}$
for $\varepsilon_{n}=\varepsilon_{m}$.

###### Proof.

Left-multiplying both sides of the eigenvalue equation for $M$ by
$\mathcal{S}_{x}$ yields

$\displaystyle\mathcal{S}_{x}M\ket{u_{n}}$
$\displaystyle=\varepsilon_{n}\mathcal{S}_{x}\ket{u_{n}}$
$\displaystyle=\varepsilon_{n}\sum_{\mu}\sum_{\sigma}c^{n}_{\sigma\mu}\ket{\mu}\otimes\sigma_{x}\ket{\sigma}.$
(32)

Acting on Eq. (VI.5) with the arbitrary basis vector
$\bra{\mu^{\prime}}\otimes\bra{\sigma^{\prime}}$ produces

$\displaystyle(\bra{\mu^{\prime}}\otimes\bra{\sigma^{\prime}})\mathcal{S}_{x}M\ket{u_{n}}$
$\displaystyle=\varepsilon_{n}\sum_{\mu}\sum_{\sigma}c^{n}_{\mu\sigma}\braket{\mu^{\prime}}{\mu}\otimes\bra{\sigma^{\prime}}\sigma_{x}\ket{\sigma}$
$\displaystyle=\varepsilon_{n}\sum_{\mu}\sum_{\sigma}c^{n}_{\mu\sigma}\delta_{\mu^{\prime}\mu}\delta_{\bar{\sigma}^{\prime}\sigma}$
$\displaystyle=\varepsilon_{n}c^{n}_{\mu^{\prime}\bar{\sigma}^{\prime}},$ (33)

where in the second line we have used the Hermiticity of $\sigma_{x}$ applied
to $\bra{\sigma^{\prime}}\sigma_{x}^{\dagger}$. Considering now the left-hand
side of Eq. (VI.5), we have

$\displaystyle(\bra{\mu^{\prime}}\otimes\bra{\sigma^{\prime}})\mathcal{S}_{x}M\ket{u_{n}}$
$\displaystyle=(\bra{\mu^{\prime}}\otimes\bra{\sigma^{\prime}})M^{*}\mathcal{S}_{x}\ket{u_{n}}$
$\displaystyle=(\bra{\mu^{\prime}}\otimes\bra{\sigma^{\prime}})\varepsilon_{n}\mathcal{S}_{x}\ket{u_{n}},$
(34)

where the second line follows from Lemma (1). Applying Lemma (2), we may
define $\ket{u^{*}_{m}}\equiv\mathcal{S}_{x}\ket{u_{n}}$ as an eigenstate of
$M^{*}$, such that the right-hand side of Eq. (VI.5) simplifies to

$\displaystyle\varepsilon_{n}(\bra{\mu^{\prime}}\otimes\bra{\sigma^{\prime}})\ket{u^{*}_{m}}$
$\displaystyle=\varepsilon_{n}\sum_{\mu}\sum_{\sigma}(c^{m}_{\mu\sigma})^{*}\braket{\mu^{\prime}}{\mu}\otimes\braket{\sigma^{\prime}}{\sigma}$
$\displaystyle=\varepsilon_{n}(c^{m}_{\mu^{\prime}\sigma^{\prime}})^{*}$ (35)

in analogy with Eq. (VI.5). It follows from Eqs. (VI.5) and (VI.5) that
$c^{n}_{\mu^{\prime}\bar{\sigma}^{\prime}}=(c^{m}_{\mu^{\prime}\sigma^{\prime}})^{*}$.
∎

######  Corollary 1.

Let $M$ and $\ket{u_{n}}$ be defined as in Theorem (1). If the eigenstate
coefficients satisfy
$c^{n}_{\mu^{\prime}\bar{\sigma}^{\prime}}=(c^{m}_{\mu^{\prime}\sigma^{\prime}})^{*}$,
then $\mathcal{S}_{x}M\mathcal{S}^{-1}_{x}\neq M^{*}$.

###### Proof.

Theorem (1) states that if $\mathcal{S}_{x}M\mathcal{S}^{-1}_{x}=M^{*}$, then
$c^{n}_{\mu^{\prime}\bar{\sigma}^{\prime}}=(c^{m}_{\mu^{\prime}\sigma^{\prime}})^{*}$.
By modus tollens, this statement is logically equivalent to its
contrapositive. Thus if
$c^{n}_{\mu^{\prime}\bar{\sigma}^{\prime}}=(c^{m}_{\mu^{\prime}\sigma^{\prime}})^{*}$,
we must have $\mathcal{S}_{x}M\mathcal{S}^{-1}_{x}\neq M^{*}$. ∎

######  Theorem 2.

Let $H$ be a $\mathcal{T}$ symmetric, Hermitian Hamiltonian describing an
infinite non-Bravais lattice with $\mathcal{N}$ sublattices $\\{\mu,\nu\\}$,
bosonic pseudospin-1/2 degree of freedom $\\{\sigma,\sigma^{\prime}\\}$, and
reciprocal space matrix elements
$h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})$. Let
$\ket{\psi_{n\mathbf{k}}}=\mathrm{e}^{i\mathbf{k}\cdot\mathbf{r}}\ket{u_{n\mathbf{k}}}$
denote a general Bloch mode of $H$ with $\ket{u_{n\mathbf{k}}}$ the
corresponding cell-periodic eigenstate satisfying
$h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})\ket{u_{n\mathbf{k}}}=\varepsilon_{n\mathbf{k}}\ket{u_{n\mathbf{k}}}$.
If the lattice geometry is not chiral, then $\braket{\mathcal{S}_{z}}=0$ at
every point in the Brillouin zone.

###### Proof.

Without loss of generality, $H$ may be given by Eqs. (14-20) with
$\mathbf{B}=0$ and $J_{\sigma\sigma^{\prime}}=J^{*}_{\sigma^{\prime}\sigma}$.
$\mathcal{T}$ symmetry implies that

$\mathcal{T}h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})\mathcal{T}^{-1}=h^{*}_{\mu\bar{\sigma},\nu\bar{\sigma}^{\prime}}(\mathbf{-k})=h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k}).$
(36)

For an achiral lattice geometry, inversion symmetry further requires
$J_{\sigma\sigma^{\prime}}(\mathbf{R}_{\mu}+\mathbf{n}_{\mu\nu})=J_{\sigma\sigma^{\prime}}(\mathbf{R}_{\mu}-\mathbf{n}_{\mu\nu})$
such that

$\mathcal{I}h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})\mathcal{I}^{-1}=h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{-k})=h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})$
(37)

for all $\mathbf{k}$ (see the discussion at the beginning of this section).
Thus, from Eqs. (36) and (37), we have

$h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})=h^{*}_{\mu\bar{\sigma},\nu\bar{\sigma}^{\prime}}(\mathbf{k})=\mathcal{S}_{x}h^{*}_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})\mathcal{S}^{-1}_{x},$
(38)

which, by Theorem (1), implies $\lvert
c^{n}_{\mu\bar{\sigma}}(\mathbf{k})\rvert^{2}=\lvert
c^{m}_{\mu\sigma}(\mathbf{k})\rvert^{2}$ for all eigenstates
$\ket{u_{n\mathbf{k}}}$ and $\ket{u_{m\mathbf{k}}}$ satisfying
$\varepsilon_{n}(\mathbf{k})=\varepsilon_{m}(\mathbf{k})$.

When $n=m$, $\ket{u_{n\mathbf{k}}}$ and $\ket{u_{m\mathbf{k}}}$ represent the
same state up to a $U(1)$ gauge ambiguity and
$c^{n}_{\mu\sigma}(\mathbf{k})=\mathrm{e}^{i\alpha}c^{m}_{\mu\sigma}(\mathbf{k})$
for $\alpha\in\mathbb{R}$. Noting that

$\displaystyle\sum_{\mu}\lvert
c^{n}_{\mu\bar{\sigma}}(\mathbf{k})\rvert^{2}-\sum_{\mu}\lvert
c^{m}_{\mu\sigma}(\mathbf{k})\rvert^{2}$ $\displaystyle=\sum_{\mu}\lvert
c^{n}_{\mu\bar{\sigma}}(\mathbf{k})\rvert^{2}-\sum_{\mu}\lvert
c^{n}_{\mu\sigma}(\mathbf{k})\rvert^{2}$
$\displaystyle=\bra{u_{n\mathbf{k}}}\mathcal{S}_{z}\ket{u_{n\mathbf{k}}}$ (39)

proves the theorem for $n=m$.

When $n\neq m$, $\ket{u_{n\mathbf{k}}}$ and $\ket{u_{m\mathbf{k}}}$ are
degenerate eigenstates such that the superpositions
$\ket{u_{\pm}}\equiv(1/\sqrt{2})(\ket{u_{n\mathbf{k}}}\pm\ket{u_{m\mathbf{k}}})$
are also eigenstates of $h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})$ with
the same eigenvalue. The coefficients of these eigenstates in the
superposition basis are
$c^{\pm}_{\mu\sigma}(\mathbf{k})=\frac{1}{\sqrt{2}}\left(c^{n}_{\mu\sigma}(\mathbf{k})\pm
c^{m}_{\mu\sigma}(\mathbf{k})\right)$. It then follows from Theorem (1) that

$\displaystyle\lvert c^{\pm}_{\mu\sigma}(\mathbf{k})\rvert^{2}$
$\displaystyle=\frac{1}{2}\Big{(}\lvert
c^{n}_{\mu\sigma}(\mathbf{k})\rvert^{2}+\lvert
c^{m}_{\mu\sigma}(\mathbf{k})\rvert^{2}$ $\displaystyle\quad\,\pm
c^{n}_{\mu\sigma}(\mathbf{k})(c^{m}_{\mu\sigma}(\mathbf{k}))^{*}\pm(c^{m}_{\mu\sigma}(\mathbf{k}))^{*}c^{n}_{\mu\sigma}(\mathbf{k})\Big{)}$
$\displaystyle=\frac{1}{2}\Big{(}\lvert
c^{m}_{\mu\bar{\sigma}}(\mathbf{k})\rvert^{2}+\lvert
c^{n\mathbf{k}}_{\mu\bar{\sigma}}\rvert^{2}$
$\displaystyle\quad\,\pm(c^{m}_{\mu\bar{\sigma}}(\mathbf{k}))^{*}c^{n\mathbf{k}}_{\mu\bar{\sigma}}\pm
c^{n\mathbf{k}}_{\mu\bar{\sigma}}(c^{m}_{\mu\bar{\sigma}}(\mathbf{k}))^{*}\Big{)}$
$\displaystyle=\lvert c^{\pm}_{\mu\bar{\sigma}}(\mathbf{k})\rvert^{2}.$ (40)

In analogy with Eq. (VI.5), performing the summation over $\mu$ concludes the
proof for $n\neq m$. ∎

######  Corollary 2.

Let $\ket{u_{n\mathbf{k}}}$ and $h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})$
be defined as in Theorem (2). If $\braket{\mathcal{S}_{z}}\neq 0$ at any point
in the Brillouin zone, then
$\mathcal{I}h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})\mathcal{I}^{-1}\neq
h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})$ and the lattice geometry must be
chiral.

###### Proof.

The proof follows straightforwardly from Corollary (1), or as the
contrapositive of Theorem (2). ∎

Corollary (2) demonstrates that chirality is a necessary condition for the
generic point $\mathbf{k}$ to have nontrivial spin character at any point in
the Brillouin zone. As discussed in the main text, $\mathcal{T}$ symmetry
implies that all modes with $\braket{\mathcal{S}_{z}}\neq 0$ and finite
dispersion experience spin-momentum locking. It follows that SOC cannot occur
in a bosonic pseudospin-1/2 system unless inversion symmetry is broken.

We now show that broken inversion symmetry is a sufficient condition to
observe emergent SOC. That is, emergent SOC in bosonic pseudospin-1/2 systems
is universal to all chiral geometries.

######  Theorem 3.

Let $\ket{\psi_{n\mathbf{k}}}$, $\ket{u_{n\mathbf{k}}}$, and
$h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})$ be defined as in Theorem (2).
If the lattice geometry is chiral, then $\braket{\mathcal{S}_{z}}\neq 0$ for
all $\mathbf{k}\neq-\mathbf{k}$.

###### Proof.

As discussed above, a chiral lattice geometry implies

$\mathcal{I}h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})\mathcal{I}^{-1}=h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{-k})\neq
h_{\mu\sigma,\nu\sigma^{\prime}}(\mathbf{k})$ (41)

such that

$\displaystyle\mathcal{I}\ket{u_{n\mathbf{k}}}$
$\displaystyle=\mathcal{I}\sum_{\mu}\sum_{\sigma}c^{n}_{\mu\sigma}(\mathbf{k})\ket{\mu}\otimes\ket{\sigma}$
$\displaystyle=\sum_{\mu}\sum_{\sigma}c^{n}_{\mu\sigma}(-\mathbf{k})\ket{\mu}\otimes\ket{\sigma}$
$\displaystyle=\ket{u_{n\mathbf{-k}}}\neq\ket{u_{n\mathbf{k}}}$ (42)

for $\mathbf{k}\neq-\mathbf{k}$. However, $\mathcal{T}$ symmetry requires that
$\ket{\psi_{n\mathbf{k}}}=\mathcal{T}\ket{\psi_{n\mathbf{k}}}$, where

$\displaystyle\mathcal{T}\ket{\psi_{n\mathbf{k}}}$
$\displaystyle=\mathcal{K}\mathcal{S}_{x}\mathrm{e}^{i\mathbf{k}\cdot\mathbf{r}}\sum_{\mu}\sum_{\sigma}c^{n}_{\mu\sigma}(\mathbf{k})\ket{\mu}\otimes\ket{\sigma}$
$\displaystyle=\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{r}}\sum_{\mu}\sum_{\sigma}(c^{n}_{\mu\sigma}(\mathbf{k}))^{*}\ket{\mu}\otimes\sigma_{x}\ket{\sigma}.$
(43)

Noting that $(c^{n}_{\mu\sigma}(\mathbf{k}))^{*}$ are the coefficients for
$\ket{u_{n-\mathbf{k}}}$ with $\sigma\to\bar{\sigma}$, we have
$(c^{n}_{\mu\sigma}(\mathbf{k}))^{*}=c^{n}_{\mu\bar{\sigma}}(-\mathbf{k})$
(see also ref. [62]). It follows similarly from Eq. (VI.5) that
$c^{n}_{\mu^{\prime}\sigma^{\prime}}(\mathbf{k})\neq
c^{n}_{\mu^{\prime}\sigma^{\prime}}(-\mathbf{k})$. Combining these relations
and taking the magnitude squared of both sides results in $\lvert
c^{n}_{\mu^{\prime}\bar{\sigma}^{\prime}}(-\mathbf{k})\rvert^{2}\neq\lvert
c^{n}_{\mu^{\prime}\sigma^{\prime}}(-\mathbf{k})\rvert^{2}$, or equivalently,
$\braket{\mathcal{S}_{z}}\neq 0$. ∎

##  VII Acknowledgements

The authors are grateful to Mikhail D. Lukin and Jonathan Simon for
suggestions on potential experimental implementations. The authors thank
Dimitar D. Sasselov and Rafal Szabla for helpful discussions on chiral
processes in molecules and Ceren B. Dag for useful discussions on topological
systems. S.O. is supported by a postdoctoral fellowship of the Max Planck-
Harvard Research Center for Quantum Optics. All authors acknowledge funding
from the National Science Foundation (NSF) via the Center for Ultracold Atoms
(CUA) Physics Frontiers Centers (PFC) program and via PHY-2207972, as well as
from the Air Force Office of Scientific Research (AFOSR).

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Extended Data Fig. 1: Preferential excitation of one chirality from a
polarized initial state. Panels correspond to the same set-ups as in Fig. 1d
but for initial states $\rho_{0}=\ket{\uparrow_{i}}\bra{\uparrow_{i}}$
corresponding to a polarized spin up excitation localized at the bottom (top
two panels) or top (bottom two panels) of the helix. Extended Data Fig. 2:
Coherent coupling strengths for spin-preserving interactions. Colors denote
the interaction strength $J^{\sigma\sigma}_{ij}/\Gamma_{0}$ with
$\sigma\in\\{\uparrow,\downarrow\\}$ between emitters $i$ and $j$ located on a
helix of radius $r_{0}=0.05\lambda$. The $x$-axis denotes the relative
distance between the emitters in the $\hat{\mathbf{z}}$ direction (along the
longitudinal helical axis). The $y$-axis denotes the relative azimuthal
coordinate between the emitters. Extended Data Fig. 3: Coherent coupling
strengths for spin-flipping interactions. Colors denote the interaction
strength $J^{\sigma\sigma^{\prime}}_{ij}/\Gamma_{0}$ with
$\sigma,\sigma^{\prime}\in\\{\uparrow,\downarrow\\}$ for
$\sigma\neq\sigma^{\prime}$. Geometry and axis labels are the same as in
Extended Data Fig. 2. Extended Data Fig. 4: Spin-selective chiral transport
without dissipation. Panels correspond to the same set-ups as in Fig. 1d but
without the influence of the Lindbladian defined in Eq. (2) (i.e., time
evolution with only the Hamiltonian (1)). SOC resulting from purely unitary
dynamics is sufficient to achieve spin-selective chiral transport (see also
Methods).