# Geometric Induction in Chiral Superfluids

Qing-Dong Jiang12<EMAIL_ADDRESS>A. Balatsky34
1 Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240,
China
2School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China
3Nordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullbacken 23, SE-106 91 Stockholm, Sweden
4 UCONN, Department of Physics, Storrs, CT 06269, USA

###### Abstract

We explore the properties of chiral superfluid thin films coating a curved
surface. Due to the vector nature of the order parameter, a geometric gauge
field emerges and leads to a number of observable effects such as anomalous
vortex-geometric interaction and curvature-induced mass/spin supercurrents. We
apply our theory to several well-known phases of chiral superfluid
$\rm{}^{3}He$ and derive experimentally observable signatures. We further
discuss the cases of flexible geometries where a soft surface can adapt itself
to compensate for the strain from the chiral superfluid. The proposed
interplay between geometry and chiral superfluid order provides a fascinating
avenue to control and manipulate quantum states with strain.

Geometric phases, rooted in the concept of parallel transport and related to
topology, figure prominently in a startling variety of physical contexts,
ranging from optics and hydrodynamics to quantum field theory and condensed
matter physics ShapereW . In classical systems, for example, the geometric
phase shift of the Foucault pendulum is equal to the enclosed solid angle
subtended at the earth’s center foucault . Other classical examples of
geometric phases include the motion of deformable bodies ShapereW2  and
tangent-plane order on a curved substrate ParkL ; ATuner . In quantum
mechanics, the geometric phases arise from slowly transporting an eigenstate
round a circuit C by varying parameters $\mathbb{R}$ in its Hamiltonian
$\hat{\mathbb{H}}(\mathbb{R})$ VinitskiiD . For example, the geometric phase
of a single-electron Bloch wave function in the Brillouin zone is essential
for topological states of matter such as the quantum Hall effect and
topological insulators XiaoCN .

Beyond the single-electron picture, the concept of geometric phase has become
a defining property of topological superconductors, where Cooper pairs can
directly inherit their geometric phases from the two paired electrons QiZ .
Chiral superconductors, a particularly interesting class of topological
superconductors Maeno , have received great attention due to their promise of
hosting Majorana zero modes in vortex cores and at edges, which are central to
several proposals for topological quantum computation Kopnin ; Volovik .

In a chiral $p$-wave superconductor, the Cooper pairs carry orbital angular
momentum (OAM) of $\hbar$, and the order parameter is a complex vector defined
in the tangent plane of a two-dimensional (2D) surface
$|\Psi\rangle=\psi~{}\left(\mathbb{\hat{e}_{1}}\pm
i\,\mathbb{\hat{e}_{2}}\right)/\sqrt{2}$ with $\mathbb{\hat{e}_{1}}$ and
$\mathbb{\hat{e}_{2}}$ the local orthogonal basis and $\psi$ the complex
amplitude Volovik ; JSauls ; Volovik1 . Here $\pm$ sign denotes the chirality
and the direction of the OAM. When such an order parameter with positive
chirality evolves in a circuit on a curved 2D surface (Fig. 1 as an
illustration), a geometric phase arises according to the formula
$\frac{1}{\langle\Psi|\Psi\rangle}\oint_{C}{\langle\Psi|i\partial_{\mu}|\Psi\rangle}dl^{\mu}=\oint_{C}\omega_{\mu}dl^{\mu}$.
Here
$\omega_{\mu}=\mathbf{\hat{e}_{1}}\cdot\partial_{\mu}\mathbf{\hat{e}_{2}}$ is
the geometric connection whose curl is Gaussian curvature (see Sec. S-I in the
Supplemental Material, where we present the mathematical foundation of
geometric connection Append ). Generalization to a chiral $\ell$-wave order
parameter, describing a condensate of Cooper pairs with orbital angular
momentum $\ell\hbar$, yields a geometric phase
$\ell\oint_{C}\omega_{\mu}dl^{\mu}$ Append .

Figure 1: Schematic illustration of transporting a vectorial order on a curved
surface. The height $h(\mathbf{r},t)$ measures the deviation of a curved
surface from a plane.

The geometric connection $\omega_{\mu}$ may lead to a number of intriguing
effects, such as the geo-Meissner effect KvorningH  and the geometric
Josephson effect JiangHW , which serve as definitive signatures of chiral
superconductivity.

In this Letter, we study the interplay between chiral superfluidity and
geometry. We are motivated by the following observations: i) Chiral
superfluids are charge-neutral condensates. Therefore, the corresponding
electromagnetic signature must be qualitatively different from that of
superconductors. ii) Unlike chiral superconductors, chiral superfluids are
observed in nature (3He-A phase) Vollhardt  and provide a testbed for our
proposed geometric induction theory. iii) The study of interactions between
chiral-superfluid vortices and geometry, while experimentally feasible, is
still lacking in the literature. iv) Geometry may provide a practical knob to
manipulate novel quantum states, such as the Majorana zero mode in a vortex.
Thus it may offer a unique route to quantum manipulation including braiding -
central to topological quantum computation Nayak ; chung .

The paper is organized as follows: We first develop the necessary formalism
for 2D chiral superfluids covering a curved surface. We then study the
interaction between vortices and geometry, aiming at controlling quantum
states with geometry. Next, we derive mass current and spin current induced by
Gaussian curvature in several well-known phases of chiral superfluid 3He, and
we obtain the associated electromagnetic signatures. Finally, we study the
quantum backaction of a chiral superfluid on a flexible surface.

Emergent geometric gauge fields.— The order parameter of a chiral $\ell$-wave
superfluid can be generically written as a rank-$\ell$ tensor, i.e.,

$\Psi=\psi~{}\underbrace{\epsilon_{\pm}\otimes\epsilon_{\pm}\dots\otimes\epsilon_{\pm}}_{\text{$\ell$
times}},$ (1)

where $\epsilon_{\pm}=\frac{1}{\sqrt{2}}\left(\mathbf{\hat{e}_{1}}\pm
i\,\mathbf{\hat{e}_{2}}\right)$ denote chiral basis, and
$\psi=\sqrt{\rho}e^{i\theta}$ is the complex amplitude in terms of the
superfluid density $\rho$ and phase $\theta$. $\ell=1$ ($\ell=2...$)
corresponds to the order parameter of chiral p-wave (d-wave…) superfluids. In
this paper, we consider the positive chirality. The negative chirality cases
can be obtained by reversing the sign of $\ell$ in our formulas.

On a curved surface (substrate), the minimal Lagrangian of a chiral
$\ell$-wave superfluid reads

$\mathcal{L}_{\text{sf}}=i\hbar\psi^{*}D_{t}\psi-\frac{\hbar^{2}g^{ij}}{2m}(D_{i}\psi)^{*}(D_{j}\psi)-V(|\psi|),$
(2)

where $g^{ij}$ is inverse the metric tensor $g_{ij}$, $m$ is the mass of a
Cooper pair, and $V(|\psi|)$ is a symmetry-breaking potential. Since the order
parameter $\psi(t,\mathbb{r})$ depends on the choice of orthonormal basis
$\mathbf{\hat{e}_{1}}$ and $\mathbf{\hat{e}_{2}}$, one needs to use the
covariant derivatives $D_{\mu}$ $(\mu=0,1,2)$ defined by

$D_{\mu}=\partial_{\mu}+i\ell\,\omega_{\mu},$ (3)

where
$\omega_{\mu}=\mathbf{\hat{e}_{1}}\cdot\mathbf{\partial_{\mu}\hat{e}_{2}}$ is
the geometric connection originating from parallel transport of a vector on a
curved surface Append . The geometric connection $\omega_{\mu}$ is a geometric
gauge field akin to the electromagnetic vector potential, with the Gaussian
curvature playing the role of a magnetic field. It was shown that a similiar
Lagrangian can induce Hall viscosity Hoyos  and thermal Hall effect GolanS .
From the covariant derivatives, we can obtain the total field strength tensor
$T_{\mu\nu}=i\,\left[D_{\mu},D_{\nu}\right]=-\ell\,G_{\mu\nu},$ where
$G_{\mu\nu}=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}$ is the
geometric field tensor, and correspondingly, we define the electric- and
magneticlike field strength:

${\mathcal{E}}^{i}=\frac{1}{2}\frac{\epsilon^{i\mu\nu}}{\sqrt{g}}G_{\mu\nu},~{}~{}\mathcal{B}=\frac{1}{2}\frac{\epsilon^{0ij}}{\sqrt{g}}G_{ij}.$
(4)

with $i,j$ taking values $1$ or $2$. Physically, $\mathcal{B}$ is the Gaussian
curvature of a curved surface, and the meaning of $\boldsymbol{\mathcal{E}}$
will be clear later. In what fellows, we will discuss a number of effects that
originate from the geometric gauge field.

Anomalous vortex-geometry interaction.— To discuss vortex physics, we rewrite
Eq.(2) in terms of superfluid density $\rho$ and phase $\theta$, i.e., set
$\Psi=\sqrt{\rho}e^{i\theta}$ to get

$\displaystyle\mathcal{L}_{\text{sf}}$ $\displaystyle=$ $\displaystyle
i\hbar\rho\left(\partial_{0}\theta+\ell\omega_{0}\right)-\frac{\hbar^{2}\rho
g^{ij}}{2m}(\partial_{i}\theta+\ell\omega_{i})(\partial_{j}\theta+\ell\omega_{j})$
(5) $\displaystyle-V(\rho),$

where the potential $V(\rho)=A\left(\rho-\bar{\rho}\right)^{2}$ guarantees
that the superfluid acquires a finite average density $\bar{\rho}$. Upon
integrating out the fluctuations of density, one obtains

$\displaystyle\mathcal{L}_{\text{sf}}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{0}}{2}\left(\partial_{0}\theta+\ell\,\omega_{0}\right)^{2}-\frac{\gamma_{s}}{2}(\boldsymbol{\nabla}\theta+\ell\,\boldsymbol{\omega})^{2},$
(6)

where $\gamma_{0}={\hbar^{2}}/{2A}$ indicates fluctuation strength and
$\gamma_{s}={\hbar^{2}{\bar{\rho}}}/{m}$ denotes the superfluid stiffness.
Upon rescaling temporal and spatial coordinates, we arrive at an effective
Lagrangian density of the Lorentz-invariant form:

$\mathcal{L}_{\text{eff}}=\frac{\gamma}{2}\left(\partial_{\mu}\theta+\ell\,\omega_{\mu}\right)^{2}.$
(7)

To discuss vortex interactions and dynamics, we introduce the alternative form

$\mathcal{L}_{\text{eff}}=-\frac{1}{2\gamma}\xi_{\mu}^{2}+\xi^{\mu}\left(\partial_{\mu}\theta+\ell\,\omega_{\mu}\right),$
(8)

which gives Eq.(7) after integrating out the auxiliary field $\xi^{\mu}$.
Without loss of generality, one can take the phase $\theta$ as a smoothly
fluctuating field, except that at vortices where it winds around $2\pi$ Zee .
Therefore, one can write
$\partial_{\mu}\theta=\partial_{\mu}\theta_{\text{smooth}}+\partial_{\mu}\theta_{\text{vortex}}$,
and plug it into the Eq.(8), yielding

$\mathcal{L}_{\text{eff}}=-\frac{1}{2\gamma}\xi_{\mu}^{2}+\xi^{\mu}\left(\partial_{\mu}\theta_{\text{smooth}}+\partial_{\mu}\theta_{\text{vortex}}+\ell\,\omega_{\mu}\right).$
(9)

Integrating out $\theta_{\text{smooth}}$, we get the constraint for
$\partial_{\mu}\xi^{\mu}=0$, which can be automatically satisfied by the
substitution
$\xi^{\mu}\equiv\varepsilon^{\mu\nu\lambda}\partial_{\nu}a_{\lambda}$. Notice
that, on a curved surface,
$\varepsilon^{\mu\nu\lambda}\equiv\epsilon^{\mu\nu\lambda}/\sqrt{g}$, and
$a_{\mu}$ can be understood as a gauge field, because the change
$a_{\mu}\rightarrow a_{\mu}+\partial_{\mu}\Gamma$ does not change $\xi^{\mu}$.
With this substitution, we can write the action in terms of $a_{\mu}$:

$\displaystyle\mathcal{S}_{\text{eff}}=\int
dtd^{2}r\sqrt{g}\left[-\frac{f_{\mu\nu}^{2}}{4\gamma}+a_{\lambda}\varepsilon^{\lambda\nu\mu}\partial_{\nu}\left(\partial_{\mu}\theta_{\text{vortex}}+\ell\,\omega_{\mu}\right)\right]$

where $f_{\mu\nu}=\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}$ is the strength
tensor of the $a_{\mu}$ field. We reveal the physical meaning of the second
term of the above equation. Integrating the zero component
$\varepsilon^{0\mu\nu}\partial_{\nu}\partial_{\mu}\theta_{\text{vortex}}$ over
a region containing a vortex yields $\int
d^{2}r\sqrt{g}\,\varepsilon^{0\mu\nu}\partial_{\mu}\partial_{\nu}\theta_{\text{vortex}}=\oint
d\mathbb{r}\cdot\boldsymbol{\nabla}\theta_{\text{vortex}}=2\pi$. We thus
recognize
$\varepsilon^{0\mu\nu}\partial_{\mu}\partial_{\nu}\theta_{\text{vortex}}$ as
the density of vortices, i.e., the time component of a vortex current density

$\displaystyle j_{\text{vor}}^{\lambda}$ $\displaystyle=$
$\displaystyle\varepsilon^{\lambda\mu\nu}\partial_{\mu}\partial_{\nu}\theta_{\text{vortex}}.$
(10)

One the other hand, we realize that
$\varepsilon^{0\mu\nu}\partial_{\mu}\omega_{\nu}=\mathcal{B}$ and
$\varepsilon^{i\mu\nu}\partial_{\mu}\omega_{\nu}=\mathcal{E}^{i}$ are the
geometric field strength defined in Eq.(4). Therefore, we also identify a
geometric current

$\displaystyle j_{\text{geo}}^{\lambda}$ $\displaystyle=$
$\displaystyle\varepsilon^{\lambda\mu\nu}\partial_{\mu}\omega_{\nu}=\left(\mathcal{B},\mathcal{E}^{1},\mathcal{E}^{2}\right).$
(11)

Substituting vortex current and geometric current into the effective action,
we obtain the effective Lagrangian density for vortices and geometry

$\mathcal{L}_{\text{vor-
geo}}=-\frac{1}{4\gamma}f_{\mu\nu}^{2}+a_{\lambda}\left(j_{\text{vor}}^{\lambda}+j_{\text{geo}}^{\lambda}\right).$
(12)

This central equation governs the dynamics and interactions of vortices and
geometry in a chiral superfluid covering a curved surface. There are three
types of interactions mediated by the gauge field $a_{\mu}$, namely vortex-
vortex interaction, geometry-geometry interaction, and vortex-geometry
interaction. The vortex-geometry interaction resembles the quasiparticle-
geometry coupling (the Wen-Zee term WenZee1992 ) of quantum Hall (QH) liquids.
In SM Append , we derive an alternative form of Eq.(12), revealing the
similarity and differences between chiral superfluidity and QH physics. While
the analogy has been realized in the literature previously StoneRoy2004 ;
GolkarMoroz , the field theory of chiral superfluidity has two key differences
with QHE: the gauge field action is Maxwell-like instead of Chern-Simons
action and we have Aharonov-Casher gauge potential term absent in QHE. These
differences lead to qualitatively different electromagnetic responses.

In the static limit, Eq.(12) can be understood by analogy to the Coulomb gas
model: the Gaussian curvature $\mathcal{B}(\mathbb{r})$ plays the role of a
non-uniform background charge distribution and the vortices appear as point-
like sources with electrostatic charges equal to their winding number. As a
result, the vortices tend to position themselves so that the Gaussian
curvature is screened: the negative ones on maximum or minimum while the
positive ones on the saddles of a surface.

Figure 2: (a) shows the spatial-dependent Gaussian curvature of a Gaussian
bump of three different aspect ratios $\alpha=1,\,0.8$ and $0.6$. (b) shows
the corresponding spatial-dependent geometric potential, and $E_{\rm
core}\approx\hbar^{2}\rho_{s}/m$ is a typical 2D vortex core energyMmondal .

Let us quantify the strength of vortex-geometric interaction by considering a
vortex in a rotational symmetric 2D surface specified by a three-dimensional
vector
$\mathbb{R}(r,\varphi)=\left(r\cos\varphi,r\sin\varphi,h_{0}\exp{\left(-r^{2}/2r_{0}^{2}\right)}\right)$,
where $r$ and $\varphi$ are plane polar coordinates. Clearly,
$\mathbb{R}(r,\varphi)$ describes a static Gaussian bump with a maximum height
$h_{0}$ and spatial extent $\sim r_{0}$. It is useful to characterize the
deviation of the bump from a plane in terms of a dimensionless aspect ratio
$\alpha\equiv h_{0}/r_{0}$. We can define local orthonormalized basis vectors
$\mathbf{\hat{e}_{r}}$ and $\boldsymbol{\hat{e}_{\varphi}}$ by normalizing two
orthogonal tangent vectors $\mathbb{t}_{r}=\partial\mathbb{R}/\partial r$ and
$\mathbb{t}_{\varphi}=\partial\mathbb{R}/\partial\varphi$. The components of
the geometric gauge field introduced in Eq.(3) are given by
$\omega_{i}=\mathbf{\hat{e}_{r}}\cdot\partial_{i}\boldsymbol{\hat{e}_{\varphi}}$,
i.e., $\omega_{r}=0$ and $\omega_{\varphi}=-1/\sqrt{c(r)}$ with $c(r)\equiv
1+\frac{\alpha^{2}r^{2}}{r_{0}^{2}}\exp\left(-\frac{r^{2}}{r_{0}^{2}}\right)$.
Consequently, the Gaussian curvature of the bump can be obtained
$\mathcal{B}(r)=\frac{\alpha^{2}}{r_{0}^{2}c(r)^{2}}\left(1-\frac{r^{2}}{r_{0}^{2}}\right)\exp{(-r^{2}/r_{0}^{2})}$,
which generates a geometric potential

$\displaystyle V_{\text{geo}}(\mathbb{r})=\int
d^{2}r^{\prime}\sqrt{g(\mathbb{r}^{\prime})}\,\mathcal{B}(\mathbb{r}^{\prime})\,\Gamma(\mathbb{r},\mathbb{r}^{\prime})$
(13)

via the propagator $\Gamma(\mathbb{r}^{\prime},\mathbb{r})$ of the gauge field
$a_{\mu}$. Here $g(\mathbb{r}^{\prime})=c(r^{\prime})$ is the determinant of
the metric. One can employ a conformal transformation to obtain the propagator
$\Gamma(\mathbb{r}^{\prime},\mathbb{r})$ and then the geometric potential
Append

$V_{\text{geo}}(r)={\frac{\hbar^{2}\rho_{s}}{m}}\int_{r}^{\infty}dr^{\prime}\frac{\sqrt{c(r^{\prime})}-1}{r^{\prime}}.$
(14)

Note that we have viewed a vortex as a point defect and ignored its self-
energy. When considering the self-energy of a vortex, there exists an
additional geometric interaction, which is always smaller than the geometric
interaction we considered (See details in SM Append ).

The vortex-geometry interaction provides a unique route to control the
position of a vortex. And since a localized Majorana mode is associated with a
vortex in a chiral superfluid, one can adiabatically braid Majorana modes by
mechanically engineering geometric curvature, as is illustrated in Fig. 3 (a).
We plot the geometric potential (for vortices) generated by two valleys in
Fig. 3 (b). It shows that the geometric potential is comparable to the self
energy of a vortex. Therefore, the vortex-geometry interaction offers a
promising route to perform topological quantum computing in the future.

Figure 3: (a) Schematic demonstration of quantum braiding by engineering
geometric curvature. (b) shows the the geometric potential versus distance
with aspect ratio $\alpha=0.8$ for each valley.

Anomalous mass and spin supercurrent in 3He superfluid thin film.— We apply
geometric induction theory in chiral superfluid 3He film. While both 3He and
4He are superfluids at sufficiently low temperature, the superfluidity in 3He
is more closely resembles superconductivity than the superfluid 4He. Because,
unlike 4He, 3He atoms are fermions that have to be paired to become
superfluid. In 3He the strong repulsive force exerted by the atomic cores
prevents s-wave pairing: instead, the pairs form an orbital p-wave state, with
$L$ and $S$ are both equal to $\hbar$. We will consider the 3He-A phase where
Cooper pairs possess finite angular momentum in the z-direction $L_{z}$. Near
a surface, surface scattering favors the orbital angular momentum $L_{z}$
perpendicular to the surface IkegamiTK . As a result, our geometric induction
theory applies.

In 3He-$\rm A$ ($\rm A_{1}$, $\rm A_{2}$) phase the spin-up and spin-down
components have the same chirality, and the corresponding order parameter
reads Vollhardt

$\Psi_{A}=\frac{1}{\sqrt{2}}\left(\mathbf{\hat{e}_{x}}+i\,\mathbf{\hat{e}_{y}}\right)\left(\sqrt{\rho_{\uparrow}}e^{i\theta_{\uparrow}}|\uparrow\rangle+\sqrt{\rho_{\downarrow}}e^{i\theta_{\downarrow}}|\downarrow\rangle\right)$
(15)

where $\rho_{\uparrow/\downarrow}$ and $\theta_{\uparrow/\downarrow}$ are the
superfluid density and phase of the spin-up/down component, respectively.
Depending on the relative magnitude of $\rho_{\uparrow}$ and
$\rho_{\downarrow}$, this order parameter can describe 3He-A phase
($\rho_{\uparrow}=\rho_{\downarrow}$), $\rm A_{1}$ phase (either
$\rho_{\uparrow}$ or $\rho_{\downarrow}$ vanishes), or $\rm A_{2}$ phase
($\rho_{\uparrow}\neq\rho_{\downarrow}$). Assuming constant superfluid
density, we can obtain the Ginzburg-Landau (GL) Lagrangian density for 3He
superfluid thin film embedded on a curved surface

$\displaystyle\mathcal{L}_{A}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{\uparrow}}{2}\left(\partial_{\mu}\theta_{\uparrow}+{\omega_{\mu}}+\mathcal{A}^{ac}_{\mu}\right)^{2}+\frac{\gamma_{\downarrow}}{2}\left(\partial_{\mu}\theta_{\downarrow}+{\omega_{\mu}}-\mathcal{A}^{ac}_{\mu}\right)^{2}$
(16) $\displaystyle+\text{interacting terms + potential terms...}$

where $\gamma_{\uparrow/\downarrow}=\frac{\rho_{\uparrow/\downarrow}}{m}$
denotes the stiffness for spin-up/down component;
$\left(\mathcal{A}^{ac}_{0},{\mathcal{A}^{ac}_{k}}\right)=\left(\mu_{i}B_{i},\varepsilon_{ijk}E^{i}\mu^{j}\right)$
is the Aharonov-Casher (AC) gauge field arising due to a magnetic moment
$\boldsymbol{\mu}$ moving in an electromagnetic field
$\left(\mathbf{E},\mathbf{B}\right)$ AharonovC ; Shen .

One can obtain the current density of the spin-up and spin-down components
from the Lagrangian density $\mathcal{L}_{A}$
$j_{\mu}^{\uparrow/\downarrow}=\gamma_{\uparrow/\downarrow}\left[\partial_{\mu}\theta_{\uparrow/\downarrow}+\omega_{\mu}\pm{\mathcal{A}^{ac}_{\mu}}\right]$.
Defining a total mass current
$j_{\mu}^{\text{m}}=j_{\mu}^{\uparrow}+j_{\mu}^{\downarrow}$ and a total spin
current $j_{\mu}^{\text{s}}=j_{\mu}^{\uparrow}-j_{\mu}^{\downarrow}$ yields
the matrix formula:

$\displaystyle\left(\begin{array}[]{cc}j_{\mu}^{\text{m}}\\\
j_{\mu}^{\text{s}}\end{array}\right)=\left(\begin{array}[]{cc}\gamma^{\text{m}}&\gamma^{\text{s}}\\\
\gamma^{\text{s}}&\gamma^{\text{m}}\end{array}\right)\cdot\left(\begin{array}[]{cc}\omega_{\mu}\\\
{\mathcal{A}^{ac}_{\mu}}\end{array}\right),$ (23)

where $\gamma^{\text{m/s}}\equiv\gamma_{\uparrow}\pm\gamma_{\downarrow}$, and
the phase gradient term is absorbed into the $\omega_{\mu}$ and
$\mathcal{A}_{\mu}^{ac}$ by a gauge transformation. One can immediately make
several useful predictions from Eq.(23). In $\rm{}^{3}He$-A phase
$\gamma^{\text{s}}=0$ indicates that Gaussian curvature drives a mass current
whereas the AC gauge field drives a spin current. In $\rm{}^{3}He$-$\rm A_{1}$
or $\rm A_{2}$ phase, however, $\gamma^{\text{s}}$ is finite so that either
Gaussian curvature or an AC gauge field can drive both mass current and spin
current, simultaneously. Generally, symmetry should allow a spin-spin
interaction term Leggett1968  such as $j_{\mu}^{\uparrow}\,j^{\mu\downarrow}$.
As is discussed in detail in SM Append , the spin-spin interaction effectively
shifts the strength of mass or spin stiffness.

Electromagnetic signature.— We obtain the electromagnetic signature of chiral
superfluids induced by geometric gauge fields, and for definiteness we take
$\rm{}^{3}He$-A phase as an example. Minimization of GL action with respect to
the four-vector potential $A_{\mu}=\left(\phi,\mathbf{A}\right)$ leads to the
effective electric charge and electric current density Append :

$\displaystyle\sigma_{c}=-\gamma^{\rm
s}\,\mu\,\mathcal{B}(\mathbf{r}),~{}~{}~{}\mathbb{J}_{c}=\gamma^{\rm
s}\,\boldsymbol{\mu}\times\boldsymbol{\mathcal{E}}(\mathbf{r})$ (24)

where $\mathcal{B}(\mathbf{r})$ and $\boldsymbol{\mathcal{E}}(\mathbf{r})$ are
the magnetic-like and electric-like geometric field strength in Eq.(4);
$\boldsymbol{\mu}=\mu\mathbb{\hat{e}_{3}}$ is the magnetic moment
perpendicular to the surface. The definition of the geometric field strength
leads to the the Maxwell-like equation
$\boldsymbol{\nabla}\times\boldsymbol{\mathcal{E}}=\partial_{t}\mathcal{B}$,
which further guarantees the current conservation
$\partial_{t}\sigma_{c}+\boldsymbol{\nabla}\cdot\mathbf{J}_{c}=0$. One
observes that effective charge density and electric current density can emerge
when there is a stiffness difference between the spin-up component and the
spin-down component. Similar reasoning enables us to obtain the effective
electric charge density and current density for several other chiral phases of
3He Append . We assume a superfluid density $\rho\approx 10^{22}/{\rm m^{2}}$
and a Gaussian curvature $\mathcal{B}\approx 1/(100{\rm\mu m})^{2}$. The
effective charge density can induce an electric field $E\approx 10^{-3}{\rm
V/m}$.

Geometric induction in a flexible superfluid thin film.— We consider the
geometric induction theory of a chiral superfluid embedded on a flexible
surface. The flexibility of the surface provides additional degrees of freedom
to minimize the total GL action:

$\displaystyle S_{\text{tot}}=\int dtd^{2}r\sqrt{g}\,\,$
$\displaystyle\left\\{\frac{\gamma}{2}\right.\left(\partial_{\mu}\theta+\ell\,\omega_{\mu}\right)^{2}$
(25)
$\displaystyle\left.+\left[\frac{\kappa_{0}}{2}\left(\partial_{t}h\right)^{2}-\frac{\kappa_{r}}{2}\left(\nabla^{2}h\right)^{2}\right]\right\\}$

where the first term and the second term represent the the Lagrangian of
chiral superfluid and geometry, respectively. To describe a flexible surface,
we use height $h(x,y,t)$ \- the deviation of a curved surface from a plane -
to parametrize a 2D surface. The geometric stiffness $\kappa_{0}$ and
$\kappa_{r}$ measure the softness of the surface Nelson . The geometric
connection
$\omega_{\mu}=\frac{1}{2}\varepsilon^{0\beta\gamma}\partial_{\gamma}\left(\partial_{\beta}h\partial_{\mu}h\right)$
embodies the essential interaction between a chiral superfluid and geometry.
Minimizing the GL action with respect to the $h$, one obtains the equation of
motion for geometry to linear order in height and supercurrent density
$j_{\mu}$:

$\displaystyle\kappa_{0}\partial^{2}_{t}h-\kappa_{r}\nabla^{4}h=\ell\,\left(\partial_{\mu}\partial_{\beta}h\right)\varepsilon^{0\beta\gamma}\partial_{\gamma}j^{\mu}.$
(26)

Figure 4: The energy-momentum dispersion of $h$ is shown for three stiffness
$\kappa_{r}$. For comparison, a suspended graphene has a stiffness
$k_{r}=1{\rm eV}$. Numerically, we have assumed a reasonable superfluid
density $\rho\approx 10^{22}/{\rm m^{2}}$ and superfluid current gradient
$\Gamma=10^{-7}{\rm J/m^{2}}$. We set $\ell=1$ and $k_{y}=0$ in the plot.

The dynamic of geometry is qualitatively modified due to the presence of
chiral superfluid. To quantify the influence of chiral superfluidity on
geometry, we study the energy-momentum dispersion of $h$ (usually called
flexural modes) by assuming a supercurrent in the x-direction with a gradient
$\Gamma\equiv\partial_{y}j^{x}$ in the y-direction. We obtain the modified
dispersion relation due to the backaction of the chiral superfluid:

$\displaystyle
E_{\pm}=\pm\sqrt{\frac{\kappa_{r}}{\kappa_{0}}\left(k_{x}^{2}+k_{y}^{2}\right)^{2}+\frac{\ell\,\Gamma}{\kappa_{0}}k_{x}^{2}}.$
(27)

In Fig. 4 we see that the energy-momentum dispersion in the x-direction
becomes Dirac type at small momentum, i.e., $k_{x}\ll
k_{c}\equiv\sqrt{|{\ell\Gamma}/{\kappa_{r}}|}$, with the critical speed
$v_{c}=\sqrt{\ell\,\Gamma/\kappa_{0}}$. Given $\ell=1$, $\Gamma=10^{-7}{\rm
J/m^{2}}$, and the geometric stiffness $\kappa_{0}\approx 7.6\times
10^{-8}{\rm g/cm^{2}}$ (values taken from grapheneflxuralGraphene ), we can
estimate the emergent critical speed $v_{c}\approx 0.36{\rm m/s}$.

Summary.— We have studied the intriguing interplay between chiral
superfluidity and geometry. Due to the chiral order parameter, a geometric
gauge field emerges and induces anomalous dynamics and interactions in chiral
superfluids. Based on the anomalous interaction between vortices and geometry,
we proposed a mechanical approach to control the positions of vortices, which
creates a new route for quantum braiding. We further show that both mass
supercurrent and spin supercurrent can be driven by a Gaussian curvature. And
we also obtained the geometry-induced electromagnetic signatures. Finally, we
study the backaction of chiral superfluidity on geometry. We find that the
dispersion of geometry shifts from quadratic to linear due to the presence of
chiral superfluidity.

Several proposed effects illustrate the opportunities of controlling quantum
states with strain, e.g., pseudo-electromagnetic fields in 3D topological
semimetals Rllan , uniaxial pressure control of competing orders in a high-
temperature superconductor HKim , strain and ferroelectric soft-mode induced
superconductivity in strontium titanate KDunnett . While it is known that
strain can affect the superconducting state, this work highlights the
opportunities to induce and modify spin and mass currents in superfluids.

Acknowledgement: We are grateful for useful discussions and previous
collaborations with T. H. Hansson and F. Wilczek. Q.-D. Jiang was sponsored by
Pujiang Talent Program 21PJ1405400 and TDLI starting up grant. AB was
supported by the European Research Council ERC HERO-810451 grant, University
of Connecticut, and the Swedish Research Council (VR).

## References

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Supplemental Materials

S-I. Mathematical description of geometric connection.

In this section, we deal with tangent-plane order on a curved surface by
reviewing relevant concepts in differential geometry, mostly to establish
notations. In standard literature, what we called geometric connection is
actually called spin connection although it has nothing to do with real spin.
To avoid confusion, we only call it geometric connection in the main text.
However, we recover its standard name “spin connection” in the supplementary
for the sake of mathematical consistence.

Differential geometry of a two-dimensional surface.— A two-dimensional surface
embedded in three-dimensional Euclidean space can be parametrized by a three-
dimensional vector
$\mathbb{R}(\mathbb{r})=(R_{1}(\mathbb{r}),\,R_{2}(\mathbb{r}),\,R_{3}(\mathbb{r}))$,
as a function of a two-dimensional parameter $\mathbb{r}=(x^{1},x^{2})$.
Covariant tangent-plane vectors are defined as

$\displaystyle\mathbb{t}_{\alpha}=\partial_{\alpha}\mathbb{R},~{}\alpha=1,2,$
(28)

where $\partial_{\alpha}=\partial/\partial x^{\alpha}$ with
$\alpha,\,\beta...$ to denote components of vectors and tensors written in the
local coordinates. The metric tensor is
$g_{\alpha}=\mathbb{t}_{\alpha}\cdot\mathbb{t}_{\beta}$, and the square root
of the determinant of $g_{\alpha\beta}$, $\sqrt{g}=\sqrt{{\rm
det}\,g_{\alpha\beta}}$, is useful for constructing invariant area.
$g^{\alpha\beta}$, the inverse of $g_{\alpha\beta}$, is defined as
$g^{\alpha\beta}g_{\beta\gamma}=\delta_{\gamma}^{\alpha}$. One can define
contravariant tangent-plane vectors
$\mathbb{t}^{\alpha}=g^{\alpha\beta}\mathbb{t}_{\beta}$ satisfying
$\mathbb{t}^{\alpha}\cdot\mathbb{t}_{\beta}=\delta^{\alpha}_{\beta}$. Any
vector $\mathbb{V}$ in the tangent plane can be expressed as
$\mathbb{V}=V^{\alpha}\mathbb{t}_{\alpha}=V_{\alpha}\mathbb{t}^{\alpha}$,
where $V_{\alpha}=\mathbb{V}\cdot t_{\alpha}$ and
$V^{\alpha}=\mathbb{V}\cdot\mathbb{t}^{\alpha}=g^{\alpha\beta}V_{\beta}$ are
the covariant and contravariant components of $\mathbb{V}$.

A unit $\mathbb{\hat{n}}$ normal to the surface can be constructed from
$\mathbb{t}_{1}$ and $\mathbb{t}_{2}$, namely
$\mathbb{\hat{n}}=\frac{\mathbb{t}_{1}\times\mathbb{t}_{2}}{|\mathbb{t}_{1}\times\mathbb{t}_{2}|}$.
And the curvature tensor can be obtained via the formula

$\displaystyle
K_{\alpha\beta}=\mathbb{\hat{n}}\cdot\partial_{\alpha}\partial_{\beta}\mathbb{R}.$
(29)

which is very useful for calculating the mean (extrinsic) curvature
$H=\frac{1}{2}K_{\alpha}^{\alpha}$ and the Gaussian (intrinsic) curvature
$\mathcal{B}={\rm det}\,K_{\beta}^{\alpha}={\rm
det}\,(g^{\alpha\gamma}K_{\gamma\beta})={\rm det}\,K_{\gamma\beta}\,{\rm
det}g^{\alpha\gamma}$.

To be specific, let us calculate the Gaussian curvature of a curved surface
parametrized in the Monge representation, i.e., $\mathbb{r}=(x^{1},x^{2})$ and
$\mathbb{R}(\mathbb{r})=(\mathbb{r},\,h(\mathbb{r}))$. In the Monge
representation, one can obtain the metric tensor $g_{\alpha\beta}$ and the
inverse metric tensor $g^{\alpha\beta}$:

$\displaystyle
g_{\alpha\beta}=\partial_{\alpha}\mathbb{R}\cdot\partial_{\beta}\mathbb{R}=\left(\begin{array}[]{cc}1+(\partial_{1}h)^{2}&\partial_{1}h\partial_{2}h\\\
\partial_{1}h\partial_{2}h&1+(\partial_{2}h)^{2}\end{array}\right),$ (32)
$\displaystyle g^{\alpha\beta}=g_{\alpha\beta}^{-1}=\frac{1}{1+(\nabla
h)^{2}}\left(\begin{array}[]{cc}1+(\partial_{2}h)^{2}&-\partial_{1}h\partial_{2}h\\\
-\partial_{1}h\partial_{2}h&1+(\partial_{1}h)^{2}\end{array}\right),$ (35)

where $(\nabla h)^{2}=(\partial_{1}h)^{2}+(\partial_{2}h)^{2}$. Based on the
matrix form of metric tensor, one can conveniently write
$g_{\alpha\beta}=\delta_{\alpha\beta}+(\partial_{\alpha}h)(\partial_{\beta}h)$
and
$g^{\alpha\beta}=\delta_{\alpha\beta}-(\partial_{\alpha}h)(\partial_{\beta}h)$
to the second order approximation. The curvature tensor $K_{\alpha\beta}$ is

$\displaystyle
K_{\alpha\beta}=\mathbb{n}\cdot\partial_{\alpha}\partial_{\beta}\mathbb{R}=\frac{1}{\sqrt{1+(\nabla
h)^{2}}}\left(\begin{array}[]{cc}\partial_{1}^{2}h&\partial_{1}\partial_{2}h\\\
\partial_{1}\partial_{2}h&\partial_{2}^{2}h\end{array}\right).$ (38)

Consequently, the Gaussian curvature and mean curvature can be derived via the
following formulas:

$\displaystyle\mathcal{B}={\rm det}K_{\alpha\beta}\,{\rm
det}\,g^{\beta\gamma}=\frac{|K_{\alpha\beta}|}{|g_{\beta\gamma}|}=\frac{1}{[1+(\nabla
h)^{2}]^{2}}\left[\partial_{1}^{2}h\partial_{2}^{2}h-(\partial_{1}\partial_{2}h)^{2}\right].$
(39) $\displaystyle H=K_{\alpha\beta}g^{\beta\alpha}=\frac{1}{[1+(\nabla
h)^{2}]^{\frac{3}{2}}}\left\\{[1+(\partial_{2}h)^{2}]\partial_{1}^{2}h+[1+(\partial_{1}h)^{2}]\partial_{2}^{2}h-2(\partial_{1}\partial_{2}h)(\partial_{1}h\partial_{2}h)\right\\}.$
(40)

For later convenience, we need to define the antisymmetric tensor
$\varepsilon_{\alpha\beta}$ via

$\displaystyle\varepsilon_{\alpha\beta}=\mathbb{\hat{n}}\cdot(\mathbb{t}_{\alpha}\times\mathbb{t}_{\beta})=\sqrt{g}\epsilon_{\alpha\beta},$
(41)

where $g=det\,g_{\alpha\beta}$ and $\epsilon_{\alpha\beta}$ is the
antisymmetric tensor with $\epsilon_{12}=-\epsilon_{21}=1$. The contravariant
tensor is

$\displaystyle\varepsilon^{\alpha\beta}=\mathbb{\hat{n}}\cdot(\mathbb{t}^{\alpha}\times\mathbb{t}^{\beta})=\epsilon_{\alpha\beta}/\sqrt{g},$
(42)

and satisfies
$\varepsilon^{\alpha\beta}\varepsilon_{\beta\gamma}=-\delta^{\alpha}_{\gamma}$.
Finally, the mixed tensor
$\varepsilon^{\alpha}_{\beta}=g^{\alpha\gamma}\varepsilon_{\gamma\beta}$
rotates a vector by $\pi/2$, because
$V_{\alpha}\varepsilon^{\alpha}_{\beta}V^{\beta}=\varepsilon_{\alpha\beta}V^{\alpha}V^{\beta}=0$
and
$\varepsilon^{\alpha}_{\beta}V^{\beta}\varepsilon_{\alpha}^{\gamma}V_{\gamma}=V^{\alpha}V_{\alpha}$.

Description of the tangent-plane order.— We focus on the situation where the
tangent-plane order has a fixed magnitude. Therefore, it is useful to
introduce a set of orthonormal tangent-plane basis vectors $\mathbb{e}_{1}$
and $\mathbb{e}_{2}$ satisfying

$\displaystyle\mathbb{\hat{e}_{a}}\cdot\mathbb{\hat{e}_{b}}=\delta_{ab},~{}\mathbb{\hat{n}}\cdot\mathbb{\hat{e}_{a}}=0;~{}(a=1,2)$
(43)

A tangent vector $\mathbb{V}$ can be expressed in the basis
$\\{\mathbb{\hat{e}_{1}},\,\mathbb{\hat{e}_{2}}\\}$ as well as that defined by
the covariant or contravariant vectors: $\mathbb{V}=V_{a}\mathbb{\hat{e}_{a}}$
where $V_{a}=\mathbb{\hat{e}_{a}}\cdot\mathbb{V}$. Notice that the covariant
derivatives are the derivatives projected into the tangent plane. Components
of the covariant derivative of a vector $\mathbb{V}$ relative to the
orthonormal basis are

$\displaystyle
D_{\alpha}V_{a}\equiv\mathbb{\hat{e}_{a}}\cdot(\partial_{\alpha}\mathbb{V})=\partial_{\alpha}V_{a}+\mathbb{\hat{e}_{a}}\cdot\partial_{\alpha}\mathbb{\hat{e}_{b}}\,V_{b}=\partial_{\alpha}V_{a}+\epsilon_{ab}\omega_{\alpha}\,V_{b},$
(44)

where
$\omega_{\alpha}=\mathbb{\hat{e}_{1}}\cdot\partial_{\alpha}\mathbb{\hat{e}_{2}}$
is the spin-connection whose curl is the Gaussian curvature,
i.e.,$\varepsilon^{\alpha\beta}\partial_{\alpha}\omega_{\beta}=\mathcal{B}$.
Note that it is $\varepsilon$ instead of $\epsilon$ in the expression of the
above formula. We need to use $\varepsilon$ because we want to calculate the
curl in the normal direction of the surface, not the curl in the z-direction.

Spin connection and affine connection.— It is instructive to see how to obtain
$\boldsymbol{\omega}$ MBowick2009a . To describe a tangent-plane order, one
need to define a local tangent-plane orthonormal coordinate frame (a tetrad,
or a vielbein) satisfying

$\displaystyle\mathbb{\hat{e}_{a}}=e_{a}^{\alpha}\mathbb{t}_{\alpha};~{}\mathbb{e}_{b}=e_{b}^{\beta}\mathbb{t}_{\beta},$
(45)

where $\mathbb{t}_{\alpha}$ and $\mathbb{t}_{\beta}$ are the local tangent-
plane vectors defined before. (Note that $\mathbb{t}_{\alpha}$ and
$\mathbb{t}_{\beta}$ are generally not unit vectors.) We require
$\mathbb{e}_{a}$ and $\mathbb{e}_{b}$ to be orthonormal, which means
$\mathbb{e}_{a}\cdot\mathbb{e}_{b}=\delta_{ab}$. This equally indicates

$\displaystyle\delta_{ab}$ $\displaystyle=$
$\displaystyle\mathbb{\hat{e}_{a}}\cdot\mathbb{\hat{e}_{b}}$ (46)
$\displaystyle=$
$\displaystyle(e_{a}^{\alpha}\mathbb{t}_{\alpha})\cdot(e_{b}^{\beta}\mathbb{t}_{\beta})=e_{a}^{\alpha}\,e_{b}^{\beta}\,g_{\alpha\beta}.$

The spin connection is then obtained from

$\displaystyle\mathbb{\hat{e}_{a}}\cdot\partial_{\alpha}\mathbb{\hat{e}_{b}}$
$\displaystyle=$
$\displaystyle(e_{a}^{\beta}\mathbb{t}_{\beta})\cdot\partial_{\alpha}\,(e_{b}^{\gamma}\mathbb{t}_{\gamma})$
(47) $\displaystyle=$
$\displaystyle(e_{a}^{\beta}\partial_{\alpha}e_{b}^{\gamma})\,g_{\beta\gamma}+e_{a}^{\beta}e_{b}^{\gamma}\,(\partial_{\alpha}\,\mathbb{t}_{\gamma}\cdot\mathbb{t}_{\beta})$
$\displaystyle=$
$\displaystyle(e_{a}^{\beta}\partial_{\alpha}e_{b}^{\gamma})\,g_{\beta\gamma}+e_{a}^{\beta}e_{b}^{\gamma}\,\Gamma_{\beta\gamma\alpha},$

where $\Gamma_{\beta\alpha\gamma}$ is the Christoffel symbols (affine
connection) of the first kind. The Christoffel symbols
$\Gamma_{\nu\lambda}^{\mu}$ are used to specify the parallel transport in a
tetrad-free language: $V^{\beta}\rightarrow
V^{\beta}-\Gamma^{\beta}_{\alpha\gamma}dx^{\gamma}V^{\alpha}$
FrancescoMathieua . In fact, $\Gamma_{\beta\alpha\gamma}$ can be obtained
directly from the metric alone, via

$\displaystyle\Gamma_{\beta\alpha\gamma}=\frac{1}{2}\left(\frac{\partial
g_{\beta\alpha}}{\partial x^{\gamma}}+\frac{\partial g_{\beta\gamma}}{\partial
x^{\alpha}}-\frac{\partial g_{\alpha\gamma}}{\partial x^{\beta}}\right).$ (48)

One should notice that the above procedures still cannot uniquely determine
$\boldsymbol{\omega}$. One can decide a particular form of
$\boldsymbol{\omega}$ only after choosing a particular set of
$\\{e_{a}^{\alpha},\,e_{b}^{\beta}\\}$ satisfying Eq.(46). (The reason is
quite physically intuitive. Affine connection is a connection for close
tangent planes; while spin connection here is a connection for vectors in
tangent planes. Even tangent plane is fixed, one still has the freedom to
choose the local coordinate frame.) However, the Gaussian curvature is a
gauge-invariant quantity that doesn’t depend on what particular frame you
choose. With the above procedures, we can, for sure, get one form of spin
connection $\boldsymbol{\omega}$ in the Monge representation.

Let’s work out the spin connection in the Monge representation. Due to the
expression of metric $g_{ij}$, we can derive the Christoffel symbols

$\displaystyle\Gamma_{\beta\alpha\gamma}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\partial_{\gamma}g_{\beta\alpha}+\partial_{\alpha}g_{\beta\gamma}-\partial_{\beta}g_{\alpha\gamma}\right)$
(49) $\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{\partial_{\gamma}[(\partial_{\beta}h)(\partial_{\alpha}h)]+\partial_{\alpha}[(\partial_{\beta}h)(\partial_{\gamma}h)]-\partial_{\beta}[(\partial_{\alpha}h)(\partial_{\gamma}h)]\right\\}$
$\displaystyle=$
$\displaystyle(\partial_{\beta}h)\,(\partial_{\alpha}\partial_{\gamma}h)$

Next, we choose vielbein that satisfies equation (46). If we write (46) in an
explicit manner

$\displaystyle\left(\begin{array}[]{cc}e_{1}^{1}&e_{1}^{2}\\\
e_{2}^{1}&e_{2}^{2}\end{array}\right)\left(\begin{array}[]{cc}1+(\partial_{1}h)^{2}&(\partial_{1}h)(\partial_{2}h)\\\
(\partial_{1}h)(\partial_{2}h)&(1+\partial_{2}h)^{2}\end{array}\right)\left(\begin{array}[]{cc}e_{1}^{1}&e_{2}^{1}\\\
e_{1}^{2}&e_{2}^{2}\end{array}\right)=\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)$ (58)

Notice that there are in total 4 unknowns, $e_{1}^{1}$, $e_{1}^{2}$,
$e_{2}^{1}$, $e_{2}^{2}$, while only three equations. (It looks like that we
have 4 equations, but two of them are identical.)

$\displaystyle
e_{1}^{1}e_{1}^{1}[1+(\partial_{1}h)^{2}]+e_{1}^{1}e_{1}^{2}[(\partial_{1}h)(\partial_{2}h)]+e_{1}^{1}e_{1}^{2}[(\partial_{1}h)(\partial_{2}h)]+e_{1}^{2}e_{1}^{2}[1+(\partial_{2}h)^{2}]=1$
(59a) $\displaystyle
e_{1}^{1}e_{2}^{1}[1+(\partial_{1}h)^{2}]+e_{1}^{2}e_{2}^{1}[(\partial_{1}h)(\partial_{2}h)]+e_{2}^{2}e_{1}^{1}[(\partial_{1}h)(\partial_{2}h)]+e_{1}^{2}e_{2}^{2}[1+(\partial_{2}h)^{2}]=0$
(59b) $\displaystyle
e_{2}^{1}e_{2}^{1}[1+(\partial_{1}h)^{2}]+e_{2}^{1}e_{2}^{2}[(\partial_{1}h)(\partial_{2}h)]+e_{2}^{2}e_{2}^{1}[(\partial_{1}h)(\partial_{2}h)]+e_{2}^{2}e_{2}^{2}[1+(\partial_{2}h)^{2}]=1$
(59c)

If we choose the gauge $e_{2}^{1}=e_{1}^{2}$, then solving the above equations
yields

$\displaystyle
e_{1}^{1}=1-\frac{1}{2}(\partial_{1}h)^{2};~{}e_{2}^{2}=1-\frac{1}{2}(\partial_{2}h)^{2};~{}e_{1}^{2}=e_{2}^{1}=-\frac{1}{2}(\partial_{1}h)(\partial_{2}h).$
(60)

Substitute Eqs.(49) and (60) into Eq.(47), we can obtain the spin connection

$\displaystyle\mathcal{W}_{\alpha}$ $\displaystyle=$
$\displaystyle\mathbb{e}_{1}\cdot\partial_{\alpha}\mathbb{e}_{2}$ (61)
$\displaystyle=$ $\displaystyle
e_{1}^{\beta}\partial_{\alpha}e_{2}^{\gamma}g_{\beta\gamma}+e_{1}^{\beta}e_{2}^{\gamma}\Gamma_{\beta\gamma\alpha}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}[(\partial_{1}\partial_{\alpha}h)\partial_{2}h+\partial_{1}h(\partial_{2}\partial_{\alpha}h)]+\partial_{1}h\,(\partial_{2}\partial_{\alpha}h)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon^{\beta\gamma}[(\partial_{\beta}h)(\partial_{\gamma}\partial_{\alpha}h)]=\frac{1}{2}\epsilon^{\beta\gamma}\partial_{\gamma}[(\partial_{\beta}h)\,(\partial_{\alpha}h)],$

where we have neglected terms of order higher than $(\nabla h)^{2}$. Note that
this is consistent with the result in Ref. SeungNelson .

Spin connection in chiral basis.— We will also find it useful to use a
circular basis defined by the vectors

$\displaystyle\boldsymbol{\epsilon}_{\pm}=\frac{1}{\sqrt{2}}(\mathbb{\hat{e}_{1}}\pm
i\mathbb{\hat{e}_{2}})=\boldsymbol{\epsilon}_{\mp}^{*}$ (62)

satisfying
$\boldsymbol{\epsilon}_{a}\cdot\boldsymbol{\epsilon}_{b}^{*}=\delta_{ab}$ with
$a,\,b=\pm$. In this basis,
$\mathbb{V}=\tilde{V}_{a}\boldsymbol{\epsilon}_{a}^{*}$, and the covariant
derivative,

$\displaystyle D_{\alpha}\tilde{V}_{\pm}\equiv$
$\displaystyle\boldsymbol{\epsilon}_{\pm}\cdot\partial_{\alpha}\mathbb{V}=\partial_{\alpha}\tilde{V}_{\pm}+\boldsymbol{\epsilon}_{\pm}\cdot\partial_{\alpha}\boldsymbol{\epsilon}_{a}^{*}\,V_{a}$
$\displaystyle=$ $\displaystyle\partial_{\alpha}\tilde{V}_{\pm}\mp
i\omega_{\alpha}\tilde{V}_{\pm}=(\partial_{\alpha}\mp
i\omega_{\alpha})\tilde{V}_{\pm}$ (63)

has a particular simple form Park1996 .

The physical meaning of spin connection and affine connection.— We have shown
that spin connection can be expressed in terms of affine connection. Let us
again see the difference between affine connection (Christoffel symbols) and
spin connection in terms of their physical meanings. One can write a vector
either in terms of tangent-plane coordinates
($\mathbb{t}_{\alpha},\mathbb{t}_{\beta}$) or in terms of local coordinates
($\mathbb{\hat{e}_{a}},\mathbb{\hat{e}_{b}}$):

i) If we choose to express a vector in terms of tangent-plane coordinates
(non-orthogonal and non-unit), we can get

$\displaystyle\partial_{\alpha}\mathbb{V}=$
$\displaystyle\partial_{\alpha}[V^{\beta}\mathbb{t}_{\beta}]$ (64)
$\displaystyle=$
$\displaystyle[\partial_{\alpha}V^{\beta}]\mathbb{t}_{\beta}+V^{\beta}\partial_{\alpha}\mathbb{t}_{\beta}$
$\displaystyle=$
$\displaystyle[\partial_{\alpha}V^{\beta}]\mathbb{t}_{\beta}+V^{\beta}\Gamma_{\beta\alpha}^{\gamma}\mathbb{t}_{\gamma}+V^{\beta}N_{\beta\alpha}\hat{n}$

In the second line, the second term expresses the effect that the basis
vectors themselves vary as we move about. One should notice that
$\partial_{\alpha}\mathbb{V}$ contains a component along $\hat{n}$, the normal
to the surface. The Christoffel symbols only include the information of basis
vector variation in the tangent plane. The definition of covariant derivative
on components defined in global coordinate is

$\displaystyle D_{\alpha}V^{\beta}\equiv$
$\displaystyle\mathbb{t}^{\beta}\cdot\partial_{\alpha}\mathbb{V}=\mathbb{t}^{\beta}\cdot\partial_{\alpha}(V^{\gamma}\mathbb{t}_{\gamma})$
(65) $\displaystyle=$
$\displaystyle\partial_{\alpha}V^{\beta}+\Gamma_{\gamma\alpha}^{\beta}V^{\gamma},$

where the out-of-tangent-plane component is projected out.

ii) If we choose to express a vector in terms of local orthonormal
coordinates, we get

$\displaystyle\partial_{\alpha}\mathbb{V}=$
$\displaystyle\partial_{\alpha}[V^{a}\mathbb{\hat{e}_{a}}]=[\partial_{\alpha}V^{a}]\hat{e}_{a}+V^{a}\partial_{\alpha}\mathbb{\hat{e}_{a}}$
(66) $\displaystyle=$
$\displaystyle[\partial_{\alpha}V^{a}]\hat{e}_{a}+V^{c}\omega_{c\alpha}^{b}\mathbb{\hat{e}_{b}}.$

The definition of covariant derivative on components defined in the local
frame is

$D_{\alpha}V^{a}\equiv\mathbb{\hat{e}_{a}}\cdot\partial_{\alpha}\mathbb{V}=\partial_{\alpha}V^{a}+\omega_{c\alpha}^{a}V^{c},$
(67)

where $\omega_{c\alpha}^{a}$ is the spin connection. That is the spin
connection connects vectors defined in local orthomormal coordinates
$\mathbb{\hat{e}_{\alpha}}$ while affine connection connects vectors defined
in terms of $\mathbb{t}_{\alpha}$.

S-II. Spin connection as a gauge field.

In the main text, we have shown the emergence of a geometric phase when
parallel transporting a complex vector. Here, we give the mathematical
derivations of the geometric gauge field arising from transporting a complex
tensor.

Method 1.— One may first understand the appearance of geometric gauge field
through a hand-waving argument as follows: With the assumption of constant
superfluid density $\rho$, the chiral order parameter is totally determined by
its local phase, i.e., $\Psi(\mathbb{r})=(\psi_{x}\pm
i\psi_{y})^{\ell}=\sqrt{\rho}\langle exp[\pm i\ell\theta(\mathbb{r})]\rangle$,
where $\ell\hbar$ is the angular momentum of a Cooper pair. Note that since
the local $U(1)$ phase $\theta(\mathbb{r})$ depends on the choice of
orthonormal vectors $\mathbb{\hat{e}_{1}}$ and $\mathbb{\hat{e}_{2}}$, so does
the order parameter $\Psi(\mathbb{r})$. This means that any spatial
derivatives for $\Psi$ must be covariant derivatives, namely,
$\partial_{\mu}\rightarrow D_{\mu}=\partial_{\mu}+i\ell\omega_{\mu}$. Here,
$\omega_{\mu}=\mathbb{\hat{e}_{1}}\cdot\partial_{\mu}\mathbb{\hat{e}_{2}}$ is
the spin connection that originates from parallel transporting the position-
dependent orthonomal vectors.

Method 2.— The order parameter for a chiral $\ell-$wave superfluid can be
generically written as

$\displaystyle\Psi=\sqrt{\frac{\rho_{+}}{2^{\ell}}}e^{i\theta_{+}}\left(\mathbb{\hat{e}_{1}}+i\mathbb{\hat{e}_{2}}\right)^{\ell}+\sqrt{\frac{\rho_{-}}{2^{\ell}}}e^{i\theta_{-}}\left(\mathbb{\hat{e}_{1}}-i\mathbb{\hat{e}_{2}}\right)^{\ell}.$
(68)

where $\rho_{\pm}$ represents the superconducting-carrier density for $\pm$
pairing parity, respectively. Here $\Psi$ is a rank-$\ell$ tensor with its
magnitude $|\Psi|=\Psi\cdot\Psi^{*}$, where the dot means the inner product of
two tensors.

The action for a $\ell-$wave chiral superfluid is

$S_{sc}=\int dt\int
d^{2}r\sqrt{g}\left\\{i\hbar\Psi^{*}\cdot\partial_{0}\Psi-\frac{\hbar^{2}g^{ij}}{2m}(\partial_{i}\Psi)^{*}\cdot(D_{j}\Psi)+V(|\Psi|)\right\\}.$
(69)

There is no spin connection term in the expression because the spin connection
is associated with the components, not the vector itself. In the following, we
choose the potential term $V(|\Psi|)$ that only favors the $``+"$ chirality,
i.e., $\rho_{+}\neq 0$ and $\rho_{-}=0$. Our results can be easily generalized
to the $``-"$ chirality case, straightforwardly.

Let us first examine the time-dependent part, $\Psi^{*}\cdot\partial_{0}\Psi$:

$\displaystyle\Psi^{*}\cdot D_{0}\Psi=$
$\displaystyle\rho_{+}\left(i\partial_{0}\theta\right)+\frac{\rho_{+}}{2^{\ell}}(\mathbb{\hat{e}_{1}}-i\mathbb{\hat{e}_{2}})^{\ell}\cdot\left[\partial_{0}(\mathbb{\hat{e}_{1}}+i\mathbb{\hat{e}_{2}})^{\ell}\right]$
(70) $\displaystyle=$
$\displaystyle\rho_{+}\left(i\partial_{0}\theta\right)+\frac{\rho_{+}}{2^{\ell}}\ell(\mathbb{\hat{e}_{1}}-i\mathbb{\hat{e}_{2}})^{\ell}\cdot(\mathbb{\hat{e}_{1}}+i\mathbb{\hat{e}_{2}})^{\ell-1}\partial_{0}(\mathbb{\hat{e}_{1}}+i\mathbb{\hat{e}_{2}})$
$\displaystyle=$
$\displaystyle\rho_{+}\left(i\partial_{0}\theta\right)+\frac{\rho_{+}}{2}\ell(\mathbb{\hat{e}_{1}}-i\mathbb{\hat{e}_{2}})\cdot\partial_{0}(\mathbb{\hat{e}_{1}}+i\mathbb{\hat{e}_{2}})$
$\displaystyle=$
$\displaystyle\rho_{+}i\left(\partial_{0}\theta+\ell\omega_{0}\right)$

where $\omega_{0}=\mathbb{\hat{e}_{1}}\cdot\partial_{0}\mathbb{\hat{e}_{2}}$
is the temporal part of the spin connection.

We then examine the space-dependent part,
$(\partial_{i}\Psi)^{*}\cdot(\partial_{j}\Psi)$:

$\displaystyle(\partial_{i}\Psi)^{*}\cdot(\partial_{j}\Psi)=$
$\displaystyle\frac{\rho_{+}}{2^{\ell}}\partial_{i}\left[e^{-i\theta_{+}}(\mathbb{\hat{e}_{1}}-i\mathbb{\hat{e}_{2}})^{\ell}\right]\cdot\partial_{j}\left[e^{i\theta_{+}}(\mathbb{\hat{e}_{1}}+i\mathbb{\hat{e}_{2}})^{\ell}\right]$
(71) $\displaystyle=$
$\displaystyle\rho_{+}\left(\partial_{i}\theta_{+}\right)\left(\partial_{j}\theta_{+}\right)+\frac{\rho_{+}}{2^{\ell}}\left(-i\partial_{i}\theta_{+}\right)(\mathbb{\hat{e}_{1}}-i\mathbb{\hat{e}_{2}})^{\ell}\cdot\partial_{j}(\mathbb{\hat{e}_{1}}+i\mathbb{\hat{e}_{2}})^{\ell}$
$\displaystyle+\frac{\rho_{+}}{2^{\ell}}\left(i\partial_{j}\theta_{+}\right)(\mathbb{\hat{e}_{1}}+i\mathbb{\hat{e}_{2}})^{\ell}\cdot\partial_{i}(\mathbb{\hat{e}_{1}}-i\mathbb{\hat{e}_{2}})^{\ell}+\frac{\rho_{+}}{2^{\ell}}\partial_{i}(\mathbb{\hat{e}_{1}}-i\mathbb{\hat{e}_{2}})^{\ell}\cdot\partial_{j}(\mathbb{\hat{e}_{1}}+i\mathbb{\hat{e}_{2}})^{\ell}$
$\displaystyle=$
$\displaystyle\rho_{+}\left(\partial_{i}\theta_{+}\right)\left(\partial_{j}\theta_{+}\right)+\rho_{+}\ell(-i\partial_{i}\theta_{+})\left(i\omega_{j}\right)+\rho_{+}\ell(i\partial_{j}\theta_{+})\left(-i\omega_{i}\right)+\rho_{+}\ell^{2}\left(\omega_{i}\omega_{j}\right)$
$\displaystyle=$
$\displaystyle\rho_{+}\left(\partial_{i}\theta_{+}+\ell\omega_{i}\right)\left(\partial_{j}\theta_{+}+\ell\omega_{j}\right)$

Here $\omega_{i}=\mathbb{\hat{e}_{1}}\cdot\partial_{i}\mathbb{\hat{e}_{2}}$ is
the spatial part of the spin connection. A key step to accomplish the
derivation is that one needs to verify that
$\partial_{i}(\mathbb{\hat{e}_{1}}+i\mathbb{\hat{e}_{2}})\cdot\partial_{j}(\mathbb{\hat{e}_{1}}-i\mathbb{\hat{e}_{2}})=2\omega_{i}\omega_{j}$.
To prove this, one could insert a unit tensor between two vectors, i.e.,
$\partial_{i}(\mathbb{\hat{e}_{1}}+i\mathbb{\hat{e}_{2}})\cdot(\mathbb{\hat{e}_{1}}\mathbb{\hat{e}_{1}}+\mathbb{\hat{e}_{2}}\mathbb{\hat{e}_{2}})\cdot\partial_{j}(\mathbb{\hat{e}_{1}}-i\mathbb{\hat{e}_{2}})=(i\mathbb{\hat{e}_{1}}\cdot\partial_{i}\mathbb{\hat{e}_{2}})(-i\mathbb{\hat{e}_{1}}\cdot\partial_{j}\mathbb{\hat{e}_{2}})+(\mathbb{\hat{e}_{2}}\cdot\partial_{i}\mathbb{\hat{e}_{1}})(\mathbb{\hat{e}_{2}}\cdot\partial_{j}\mathbb{\hat{e}_{1}})=2\omega_{i}\omega_{j}$.

S-III. Comparison with the topological field theory of (fractional) quantum
Hall effect.

We do two things in this section. First, we review the deep analogy between
superfluidity and (fractional) quantum Hall (QH) effect, mainly to refresh the
reader’s memory and to make this section self-contain. Second, we compare the
similarity and differences between chiral superfluidity and the QH effect. QH
effect shares many similar properties with superfluidity such as no
dissipation and vortex soliton excitations ZHK1989app . This observation
indicates that there might exist a universal theoretical understanding of the
two very different phenomena. Indeed, according to composite-boson theory, the
QH effect can be understood as the Bose-Einstein condensation of charged
bosons ZHK1989app . Each electron is viewed as a composition of a charged
boson and an odd number of fundamental flux units $\phi_{0}=h/e$. This can be
accomplished by introducing a statistical gauge field $\alpha_{\mu}$ which is
determined by the particle density $\rho(\mathbb{r})$ of charged bosons:

$\nabla\times\boldsymbol{\alpha}(\mathbb{r})=(2k+1)\phi_{0}\rho(\mathbb{r}).~{}(\text{$k$
is an integer}).$ (72)

This formula means that a unit charge induces a statistical flux
$\oint\boldsymbol{\alpha}\cdot d\mathbb{l}=(2k+1)\phi_{0}$. As one
interchanges two of these boson-flux composites, a Aharonov-Bohm phase factor
of $\exp\left\\{i\frac{e}{\hbar}\int_{0}^{\pi}\boldsymbol{\alpha}\cdot
d\mathbb{l}\right\\}=-1$ is obtained, which correctly reproduces the fermionic
statistic of electrons.

According to the composite-boson theory, fractional filling (odd denominator,
in particular) of QH effect can be understood as follows: Zero longitudinal
resistivity indicates that the system forms a BEC condensate of charged
bosons. According to the Meissner effect, the total gauge field seen by
charged bosons (i.e., EM field $A_{\mu}$ and statistic gauge field
$\alpha_{\mu}$) must be canceled out in the bulk. Therefore, a unit bosonic
charge has to associate with $(2k+1)\phi_{0}$ statistic flux and an opposite
$-(2k+1)\phi_{0}$ magnetic flux. The filling factor is defined as the ratio
between electron number and magnetic flux number, i.e.,
$\nu=\frac{N_{e}}{N_{\phi_{B}}}=\frac{1}{2k+1}$.

Beyond the fractional filling factor, the superfluid analogy also provides a
useful picture to understand other behaviors in the QH effect. For example,
the stability of the Hall plateau may be due to the pinning of statistical
vortices induced by adding additional charges into the system stone1990app .

Having reviewed the background, we are ready to compare chiral superfluidity
with QH effect. Chiral superfluidity has additional similarities with QH
effect because they both couple to geometry. Some of the similarities has been
noticed in the literature StoneRoy2004 ; MorozHoyos . To make the comparison
concrete, let us start from the effective action that describes the long-range
physics of QH liquids on a curved surface WenZee1992app :

$\mathcal{S}_{\text{QH}}=\int
dtd^{2}r\sqrt{g}\left[\frac{1}{4\pi}\left(\varepsilon^{\mu\nu\rho}\alpha_{\mu}\partial_{\nu}a_{\rho}+2A_{\mu}\varepsilon^{\mu\nu\rho}\partial_{\nu}\alpha_{\rho}+2s\,\omega_{\mu}\varepsilon^{\mu\nu\rho}\partial_{\nu}\alpha_{\rho}\right)+\alpha_{\mu}j^{\mu}\right]$
(73)

The action contains several pieces: The first term is the well-known Chern-
Simons interaction, describing the topological dynamics of the Hall fluid. The
second and third terms describe the coupling of the dynamics to the
electromagnetic field and the curvature of the space, respectively. Finally,
the last term accounts for the coupling between the gauge potential $a_{\mu}$
and $j_{\mu}$, the current of the quasiparticles (or vortices) in QH fluid.
The parameter $s$ in the third term is known as the Wen-Zee shift describing
the coupling between QH liquid and geometry. Physically, this geometric
coupling comes about because an electron in a QH liquid carries a “orbital
spin” due to its cyclotron motion, and acquires an additional Berry phase when
moving on a curved surface.

In a chiral superfluid, such as 3He, while the carriers are charge-neutral,
they could have finite magnetic dipoles. And an Aharonov-Casher gauge field
must be taken into account, as was shown in the main text. Adding up the
Aharonov-Casher term, the effective action for spin-up and spin-down
components reads (see the part above Eq.(10) in the main text)

$\displaystyle\mathcal{S}^{\uparrow/\downarrow}_{\text{eff}}=\int
dtd^{2}r\sqrt{g}\left[-\frac{\left(f_{\mu\nu}^{\uparrow/\downarrow}\right)^{2}}{4\gamma_{\uparrow/\downarrow}}+a^{\uparrow/\downarrow}_{\lambda}\varepsilon^{\lambda\nu\mu}\partial_{\nu}\left(\partial_{\mu}\theta^{\uparrow/\downarrow}_{\text{vortex}}\pm\mathcal{A}_{\mu}^{ac}+\ell\,\omega_{\mu}\right)\right]$
(74)

where
$f_{\mu\nu}^{\uparrow/\downarrow}=\partial_{\mu}a^{\uparrow/\downarrow}_{\nu}-\partial_{\nu}a^{\uparrow/\downarrow}_{\mu}$
is the strength tensor of the $a_{\mu}^{\uparrow/\downarrow}$ field. Note that
we have assumed that the spin-up component and the spin-down component have
the same chirality (e.g. 3He-A phase). We can derive the effective field
theory in terms of mass current and spin current:

$\displaystyle\begin{aligned}
\mathcal{S}_{\text{eff}}=&\mathcal{S}_{\text{eff}}^{\uparrow}+\mathcal{S}_{\text{eff}}^{\downarrow}\\\
=&\int
dtd^{2}r\sqrt{g}\left[-\frac{{\gamma^{m}}}{2\left({\gamma^{m}}^{2}-{\gamma^{s}}^{2}\right)}\left(f_{\mu\nu}^{m}\right)^{2}+a_{\lambda}^{m}\varepsilon^{\lambda\nu\mu}\partial_{\nu}\left(\partial_{\mu}\theta_{\text{vortex}}^{m}+\ell\,\omega_{\mu}\right)\right.\\\
&-\frac{{\gamma^{m}}}{2\left({\gamma^{m}}^{2}-{\gamma^{s}}^{2}\right)}\left(f_{\mu\nu}^{s}\right)^{2}+a_{\lambda}^{s}\varepsilon^{\lambda\nu\mu}\partial_{\nu}\left(\partial_{\mu}\theta_{\text{vortex}}^{s}+\mathcal{A}_{\mu}^{ac}\right)+\left.\frac{{\gamma^{s}}}{2\left({\gamma^{m}}^{2}-{\gamma^{s}}^{2}\right)}f_{\mu\nu}^{m}\left(f^{\mu\nu}\right)^{s}\right]\end{aligned}$
(75)

with the following mass-related and spin-related definitions
$\lambda_{m}=\lambda_{\uparrow}+\lambda_{\downarrow}$,
$\lambda_{s}=\lambda_{\uparrow}-\lambda_{\downarrow}$,
$\theta^{m}=\left(\theta_{\uparrow}+\theta_{\downarrow}\right)/2$,
$\theta^{s}=\left(\theta_{\uparrow}-\theta_{\downarrow}\right)/2$,
$a^{m}_{\mu}=a^{\uparrow}_{\mu}+a^{\downarrow}_{\mu}$,
$a^{s}_{\mu}=a^{\uparrow}_{\mu}-a^{\downarrow}_{\mu}$,
$f_{\mu\nu}^{m}=\partial_{\mu}a_{\nu}^{m}-\partial_{\nu}a_{\mu}^{m}$, and
$f_{\mu\nu}^{s}=\partial_{\mu}a_{\nu}^{s}-\partial_{\nu}a_{\mu}^{s}$. In terms
of mass vortex current
$j_{m,\text{vor}}^{\lambda}=\varepsilon^{\lambda\mu\nu}\partial_{\mu}\partial_{\nu}\theta^{m}_{\text{vortex}}$
and spin vortex current
$j_{s,\text{vor}}^{\lambda}=\varepsilon^{\lambda\mu\nu}\partial_{\mu}\partial_{\nu}\theta^{s}_{\text{vortex}}$,
this effective action can be rewritten as

$\displaystyle\begin{aligned} \mathcal{S}_{\text{eff}}=\int
dtd^{2}r\sqrt{g}&\left[-\frac{{\gamma^{m}}}{2\left({\gamma^{m}}^{2}-{\gamma^{s}}^{2}\right)}\left(f_{\mu\nu}^{m}\right)^{2}+\ell\,\omega_{\mu}\varepsilon^{\mu\nu\rho}\partial_{\nu}a_{\rho}^{m}+a_{\mu}^{m}j_{m,\text{vor}}^{\mu}\right.\\\
&\,\left.-\frac{{\gamma^{m}}}{2\left({\gamma^{m}}^{2}-{\gamma^{s}}^{2}\right)}\left(f_{\mu\nu}^{s}\right)^{2}+\mathcal{A}_{\mu}^{ac}\varepsilon^{\mu\nu\rho}\partial_{\nu}a_{\rho}^{s}+a_{\mu}^{s}j_{s,\text{vor}}^{\mu}+\frac{{\gamma^{s}}}{2\left({\gamma^{m}}^{2}-{\gamma^{s}}^{2}\right)}f_{\mu\nu}^{m}\left(f^{s}\right)^{\mu\nu}\right]\end{aligned}$
(76)

where the last term represents the interaction between mass gauge field
$a^{m}$ and spin gauge field $a^{s}$. We emphasize that the last term is
important because it embodies curvature-induced spin current and
electromagnetic signatures. Note that if spin-up component and spin-down
component have different chiralities (e.g. $\rm{}^{3}He$ planar phase), the
spin connection will couple to spin current, and one can obtain the term
$\left(\ell\,\omega_{\mu}+\mathcal{A}_{\mu}^{ac}\right)\varepsilon^{\mu\nu\rho}\partial_{\nu}a_{\rho}^{s}$.

While this effective action also contains a Wen-Zee-like term (as was noticed
previously MorozHoyos ), there are two essential differences. First, the gauge
field $a_{\mu}$ is described by the Maxwell-like action $f^{\mu\nu}f_{\mu\nu}$
instead of a Chern-Simons-like action. Therefore, integrating out the
auxiliary gauge field leads to very different physical responses. Second, the
action contains an Aharonov-Casher gauge field $\mathcal{A}_{\mu}^{ac}$
instead of an electromagnetic vector potential $A_{\mu}$. This leads to a very
different electromagnetic response, which we have demonstrated in the main
text.

S-IV. Conformal transformation and the propagator of $a_{\mu}$ on a curved 2D
surface.

To determine the propagator of a gauge field on a curved surface, it is often
convenient to employ a conformal transformation. Under a conformal
transformation, a metric of a curve surface can be written as

$g_{ij}(\mathbb{r})=e^{s(\mathbb{r})}\delta_{ij},$ (77)

which differs from the flat space one only by a conformal factor
$e^{s}(\mathbb{r})$. The conformal factor includes all the information of the
curved surface, and the propagator of a gauge field on a curved surface will
look much neater. To demonstrate the whole mathematical procedure, let us
assume a curved surface with finite Gaussian curvature described by a three-
dimensional vector
$\mathbb{R}(r,\varphi)=\left(r\cos\varphi,r\sin\varphi,h_{0}\exp{\left(-r^{2}/2r_{0}^{2}\right)}\right)$,
where $r$ and $\varphi$ are plane polar coordinates. According to differential
geometry, the metric is defined as
$g_{\mu\nu}=\mathbb{t}_{\mu}\cdot\mathbb{t}_{\nu}$ with
$\mathbb{t}_{\mu}\equiv\partial_{\mu}\mathbb{R}$. Therefore, in polar
coordinates, the metric
$g_{rr}=\mathbb{t}_{r}\cdot\mathbb{t}_{r}=1+\frac{\alpha^{2}r^{2}}{r_{0}^{2}}e^{-r^{2}/r_{0}^{2}}$
with $\alpha=h_{0}/r_{0}$, $g_{\phi\phi}=1$, and $g_{r\phi}=g_{\phi r}=0$.
Therefore, one can express the area element in polar coordinates, i.e.,

$ds^{2}=\left(1+\frac{\alpha^{2}r^{2}}{r_{0}^{2}}e^{-\frac{r^{2}}{r_{0}^{2}}}\right)dr^{2}+r^{2}d\phi^{2}.$

One could stretch the radial part (i.e., $r\rightarrow\mathcal{R}(r)$) and
assume the new metric to be of the form

$ds^{2}=e^{s(\mathbb{r})}\left(d\mathcal{R}^{2}+\mathcal{R}^{2}d\phi^{2}\right).$

To ensure the above two metrics describe the same geometry, we require

$\frac{d\mathcal{R}}{\mathcal{R}}=\sqrt{c(r)}\,\frac{dr}{r}~{}~{}~{}~{}~{}~{}e^{s(r)}=\frac{r^{2}}{\mathcal{R}^{2}}$

with $c(r)\equiv
1+\frac{\alpha^{2}r^{2}}{r_{0}^{2}}e^{-\frac{r^{2}}{r_{0}^{2}}}$. The solution
reads

$\mathcal{R}=re^{-\int_{r}^{\infty}\frac{dr^{\prime}}{r^{\prime}}\left(\sqrt{c(r^{\prime})}-1\right)}$
(78)

and

$s(r)=2\int_{r}^{\infty}\frac{dr^{\prime}}{r^{\prime}}\left(\sqrt{c(r^{\prime})}-1\right).$
(79)

This particular solution leaves the origin and the point at the infinity
invariant. The propagator $\Gamma(\mathbb{r},\mathbb{r}^{\prime})$ can be
obtained from solving the following Laplacian equation on a curved surface

$D^{i}D_{i}\Gamma(\mathbb{r},\mathbb{r}^{\prime})=-\frac{\delta(\mathbb{r},\mathbb{r}^{\prime})}{\sqrt{g}}$
(80)

where
$D^{i}D_{i}\equiv\left(1/\sqrt{g}\right)\partial_{i}\left(\sqrt{g}g^{ij}\partial_{j}\right)$.
Under the conformal transformation $\sqrt{g(r,\phi)}\rightarrow
e^{s(r)}\sqrt{g(\mathcal{R},\phi)}$ and $g^{ij}(r,\phi)\rightarrow
e^{-s(r)}g^{ij}(\mathcal{R},\phi)$. Then the exponential factors cancel out,
leading to a Laplacian of a flat plane in terms of coordinates
$\left(\mathcal{R}(r),\phi\right)$. As a result, the propagator reads

$\Gamma(\mathbb{r},\mathbb{r}^{\prime})=-\frac{1}{4\pi}\ln\left[\mathcal{R}(r)^{2}+\mathcal{R}(r^{\prime})^{2}-2\mathcal{R}(r)\mathcal{R}(r^{\prime})\cos\left(\phi-\phi^{\prime}\right)\right]$
(81)

Note that $\Gamma(\mathbb{r},\mathbb{r}^{\prime})$ differs from the real flat
space expression only be a stretch of the radial part. The geometric potential
is defined as

$V_{\text{geo}}=\int
d^{2}r^{\prime}\sqrt{g}\mathcal{B}(\mathbb{r}^{\prime})\Gamma(\mathbb{r}^{\prime},\mathbb{r}).$
(82)

Act on both sides of the above equation with the covariant Laplacian and using
the definition of the propagator, we can obtain

$D_{i}D^{i}V_{\text{geo}}(\mathbb{r})=\mathcal{B}(\mathbb{r}).$ (83)

The Gaussian curvature can be expressed in the conformal coordinates:

$\mathcal{B}(x,y)=-e^{-s(r)}\left(\partial_{x}^{2}+\partial_{y}^{2}\right)\frac{s(x,y)}{2},$
(84)

and so does $D_{i}D^{i}V_{\text{geo}}(\mathbb{r})$:

$D_{i}D^{i}V_{\text{geo}}(\mathbb{r})=e^{-s(x,y)}\left(\partial_{x}^{2}+\partial_{y}^{2}\right)V_{\text{geo}}(x,y).$
(85)

Substitute Eqs.(84) and (85) into Eq.(83), we identify the geometric potential
as

$V_{\text{geo}}(r)=\frac{s(x,y)}{2}=\int_{r}^{\infty}dr^{\prime}\frac{\sqrt{c(r^{\prime})}-1}{r^{\prime}}.$
(86)

Thus we have proved the Eq.(14) in the main text by adding back the
coefficient $\hbar^{2}\rho_{s}/m$, which we have dropped in writing the
propagator. In the case of two localized bumps on a surface, the total
geometric potential adds up via the formula Eq.(82).

S-V. Vortex density and Gaussian curvature attachment.

In a series of delicate works (see Refs.[24a,24b,29a,29b] in the main text),
the authors derived the vorticity and Gaussian curvature attachment by
minimizing the following Lagrangian density for chiral superfluids

$\displaystyle\mathcal{L}_{\text{ch}}=\frac{J_{i}J^{i}}{2\rho}-E(\rho)-\ell\,\omega_{\mu}J^{\mu},$
(87)

where $J_{\mu}$ represents supercurrent density, $E(\rho)$ is the internal
energy density, and $\ell$ is the orbital spin. By minimizing this Lagrangian,
one can obtain the vorticity equation for charge-neutral superfluids:

$\displaystyle j_{\text{vor}}^{0}=\ell\,\mathcal{B},$ (88)

where $j_{\text{vor}}^{0}=\frac{\varepsilon^{ij}\partial_{i}J_{j}}{\rho}$
represent vorticity density, and $\mathcal{B}$ is Gaussian curvature.

In our formulation, the vorticity and Gaussian curvature attachment is
automatically embodied in the geometric potential $V_{\text{geo}}$.
Integrating out $a_{0}$ in Eq.(12) in the main text yielding a Coulomb gas
model with the vortex-geometry interaction $\ell\int
dS\,j^{0}_{\text{vor}}(\mathbb{r})V_{\text{geo}}(\mathbb{r})$. It should be
obvious that vortex density couples to geometric potential in the same manner
as electric charge density couples to electric potential. Therefore, a vortex
sees an effective electric field $E_{i}=D_{i}V_{\text{geo}}(\mathbb{r})$
generated by geometric potential. And according to Gauss law, the total vortex
charge inside a region equals the total effective electric flux that goes
through the boundary

$\displaystyle
q_{\text{vor}}=\ell\oint_{C}dl_{i}D_{i}V_{\text{geo}}(\mathbb{r})=\ell\int
dSD_{i}D^{i}V_{\text{geo}}(\mathbb{r})=\ell\int dS\mathcal{B}(\mathbb{r}),$
(89)

where we have used Eq.(83) in the last step. The differential form of the
above equation is consistent with Eq.(88) for charge-neutral superfluids.

S-VI. Self-energy induced anomalous geometric interactions.

In the main text and above derivations, we have viewed a vortex as a point
defect and ignored its self-energy. When considering the self-energy of a
vortex, an anomalous geometric interaction emerges, as was nicely reviewed in
the literature. This anomalous interaction entails a repulsion of vortices of
either sign from positive curvature and an attraction to negative curvature.
By sharp contrast, the sign of the geometric interaction discussed in the main
text does depend on the sign of vortices: the negative vortices tend to
position themselves on peaks and valleys curvature while the positive ones on
the saddles of a surface.

One may understand the anomalous geometric interaction by analyzing the
streamlines (flow pattern) of a vortex. Streamlines are tangent to the
direction of flow, and their density distribution determines the speed of the
flow. The number of streamlines becomes sparse on a positive curvature.
According to fluid dynamics, sparse streamlines indicate low flowing velocity,
and low velocity indicates high pressure. Therefore, vortices are pushed away
from positive curvature, regardless of the sign of vortex. Another way to
understand this anomalous geometric interaction is by using the electrostatic
analogy. The curvature distorts streamlines in a way similar to that a
metallic boundary distorts the electric field lines of a charge. According to
the image charge method, the interaction is always repulsive between a charge
(either positive or negative) and a metallic boundary. Likewise, the
interaction of a vortex with curvature produces a force whose sign is
independent of the vortex’s sense of rotation.

Beyond understanding the physical picture qualitatively, the conformal mapping
technique (discussed in section S-IV) also allows one to calculate the
anomalous geometric interaction quantitatively. Let us briefly outline the
derivation for normal superfluid vorticesTurnerVitelliNelson . Let us
introduce polar coordinate $r,\varphi$ centered on a vortex of charge
$N_{\text{vor}}$. Near the vortex, the energy density is given by
$\frac{\gamma(N_{\text{vor}})^{2}}{8\pi^{2}r^{2}}$. According to the conformal
mapping, curvature can either stretch or contract the core radius of a vortex
from $a$ to $a^{\prime}=e^{V_{\text{geo}}(\mathbb{r})}a$. On a positive
curvature $V_{\text{geo}}>0$, the core radius is stretched; whereas on a
negative curvature, the core radius is contracted. Therefore, to calculate the
change of the vortex self-energy, one needs to integrate the vortex energy
density over the annulus comprised between $a$ and $a^{\prime}$

$\displaystyle\delta E_{\text{self}}=\int_{a}^{a^{\prime}}2\pi
rdr\frac{\gamma\left(N_{\text{vor}}\right)^{2}}{8\pi^{2}r^{2}}=\frac{(N_{\text{vor}})^{2}}{4\pi}V_{\text{geo}},$
(90)

which is in sharp contrast to the geometric interaction in the main text
$N_{\text{vor}}V_{\text{geo}}$. Including this anomalous geometric interaction
yields a total geometric force
$N_{\text{vor}}\left(1+\frac{N_{\text{vor}}}{4\pi}\right)\boldsymbol{\nabla}V_{\text{geo}}$.
Because a vortex’s self-energy is proportional to the square of its charge, a
vortex of unit charge is therefore energetically favorable. And the anomalous
geometric interaction is always smaller than what we considered in the main
text.

S-VII. Geometric induction in 3He planar phase.

Here we study the anomalous mass current density and spin current density in
3He planar phase, the order parameter of which looks similar to Eq.(91), but
with a key difference Vollhardtappend : The spin-up and spin-down component in
planar phase possess opposite chiralities

$\Psi_{P}=\sqrt{\frac{\rho_{\uparrow}}{2}}\left(\mathbf{\hat{e}_{x}}+i\,\mathbf{\hat{e}_{y}}\right)e^{i\theta_{\uparrow}}|\uparrow\rangle+\sqrt{\frac{\rho_{\downarrow}}{2}}\left(\mathbf{\hat{e}_{x}}-i\,\mathbf{\hat{e}_{y}}\right)e^{i\theta_{\downarrow}}|\downarrow\rangle.$
(91)

The corresponding GL Lagrangian density looks similar to Eq.(16) in the main
text, but the difference is that the spin-up and spin-down component have
opposite chirality

$\displaystyle\mathcal{L}_{P}=\frac{\gamma_{\uparrow}}{2}\left(\partial_{\mu}\theta_{\uparrow}+{\omega_{\mu}}+\mathcal{A}^{ac}_{\mu}\right)^{2}+\frac{\gamma_{\downarrow}}{2}\left(\partial_{\mu}\theta_{\downarrow}-{\omega_{\mu}}-\mathcal{A}^{ac}_{\mu}\right)^{2}$
(92)

Following the same logic, we can obtain the anomalous mass current and spin
current density driven by space curvature and AC gauge field:

$\displaystyle\left(\begin{array}[]{cc}j_{\mu}^{\text{m}}\\\
j_{\mu}^{\text{s}}\end{array}\right)=\left(\begin{array}[]{cc}{\gamma^{m}}&{\gamma^{s}}\\\
{\gamma^{s}}&{\gamma^{m}}\end{array}\right)\cdot\left(\begin{array}[]{cc}\partial_{\mu}\theta^{\text{m}}\\\
\omega_{\mu}+{\mathcal{A}^{ac}_{\mu}}\end{array}\right),$ (99)

where
$\partial_{\mu}\theta^{\text{m}}=\partial_{\mu}\theta_{\uparrow}+\partial_{\mu}\theta_{\downarrow}$
is the total phase gradient. Notice that, in contrast to $\rm{}^{3}He$-A
phase, the planar phase can in principle preserve time-reversal symmetry,
implying $\theta^{m}=0$, $\rho_{s}=0$. Therefore, the above formula indicates
that there is no mass current, but both the curvature and AC phase can drive
spin current.

By minimizing the action, one can also obtain the effective electric charge
density and electric current density for the planar phase. The expressions
have the exact form as Eq.(18) in the main text except that we need to replace
${\gamma^{s}}$ with ${\gamma^{m}}$ since the spin-up and spin-down components
in the planar phase have the same chirality, and therefore the electric
currents add up.

S-VIII. Minimization of the GL action for chiral superfluid

Let us take 3He-A (A1 A2) phase as an example and derive the electromagnetic
response by minimizing GL action with respective to $\\{\phi,\mathbb{A}\\}$.

i) Minimized GL action with respect to electric potential $\phi$.— We can
substitute electric field and magnetic field with gauge potential, i.e.
$\mathbb{E}=-\boldsymbol{\nabla}\phi-\partial_{t}\mathbb{A}$ and
$\mathbb{B}=\boldsymbol{\nabla}\times\mathbb{A}$. In our system, chiral
superfluids flow in 2D, whereas electromagnetic field lives in 3D and magnetic
moment $\boldsymbol{\mu}$ can point to any direction in 3D. Therefore, we use
3D vector notations to accommodate electromagnetic fields and the magnetic
moment in our system. For example, in 2D electron gas, magnetic field is a
scalar $B=\varepsilon^{ij}\partial_{i}A_{j}$, which physically represents a
magnetic field vector
$\mathbb{B}=\varepsilon^{ij3}\partial_{i}A_{j}\mathbb{\hat{e}_{3}}$
perpendicular to the surface. Similarly, the magnetic-like geometric field
strength (i.e., Gaussian curvature) $\mathcal{B}$ is a scalar field, which
corresponds to a vector
$\boldsymbol{\mathcal{B}}=\boldsymbol{\nabla}\times\boldsymbol{\omega}$ that
is perpendicular to the surface. With this preparation, we can express the
free energy in terms of vector notations. We first minimize the Lagrangian
with respect to electric potential $\phi$, yielding

$\displaystyle\begin{aligned}
\delta\mathcal{L}_{A}&=\gamma_{\uparrow}\left[\boldsymbol{\nabla}\theta_{\uparrow}+\boldsymbol{\omega}+\left(\mathbb{E}\times\boldsymbol{\mu}\right)\right]\cdot\delta(\boldsymbol{\nabla}\phi\times\boldsymbol{\mu})-\gamma_{\downarrow}\left[\boldsymbol{\nabla}\theta_{\downarrow}+\boldsymbol{\omega}-(\mathbb{E}\times\boldsymbol{\mu})\right]\cdot\delta(\boldsymbol{\nabla}\phi\times\boldsymbol{\mu})\\\
&={\gamma^{m}}\left[\boldsymbol{\nabla}\theta^{s}+(\mathbb{E}\times\boldsymbol{\mu})\right]\cdot\delta(\boldsymbol{\nabla}\phi\times\boldsymbol{\mu})+{\gamma^{s}}\left[\boldsymbol{\nabla}\theta^{m}+\boldsymbol{\omega}\right]\cdot\delta(\boldsymbol{\nabla}\phi\times\boldsymbol{\mu})\\\
&={\gamma^{m}}\boldsymbol{\nabla}\delta\phi\cdot\left\\{\boldsymbol{\mu}\times\left[\boldsymbol{\nabla}\theta^{s}+(\mathbb{E}\times\boldsymbol{\mu})\right]\right\\}+{\gamma^{s}}\boldsymbol{\nabla}\delta\phi\cdot\left[\boldsymbol{\mu}\times\left(\boldsymbol{\nabla}\theta^{m}+\boldsymbol{\omega}\right)\right]\\\
&\quad\quad(\text{ In this step, we integrated out total derivatives.})\\\
&=-{\gamma^{m}}~{}\delta\phi~{}\boldsymbol{\nabla}\cdot\left\\{\boldsymbol{\mu}\times\left[\boldsymbol{\nabla}\theta^{s}+(\mathbb{E}\times\boldsymbol{\mu})\right]\right\\}-{\gamma^{s}}~{}\delta\phi~{}\boldsymbol{\nabla}\cdot\left[\boldsymbol{\mu}\times\left(\boldsymbol{\nabla}\theta^{m}+\boldsymbol{\omega}\right)\right]\\\
&={\gamma^{m}}~{}\delta\phi~{}\boldsymbol{\mu}\cdot\boldsymbol{\nabla}\times\left[\boldsymbol{\nabla}\theta^{s}+(\mathbb{E}\times\boldsymbol{\mu})\right]+{\gamma^{s}}~{}\delta\phi~{}\boldsymbol{\mu}\cdot\boldsymbol{\nabla}\times\left[\boldsymbol{\nabla}\theta^{m}+\boldsymbol{\omega}\right]\\\
&\approx{\gamma^{s}}~{}\boldsymbol{\mu}\cdot\boldsymbol{\mathcal{B}}\,\delta\phi\quad\text{(Assume
there is no phase singularity.)}\end{aligned}$ (100)

In the above steps, we have used the definitions of mass stiffness
${\gamma^{m}}=\gamma_{\uparrow}+\gamma_{\downarrow}$, spin stiffness
${\gamma^{s}}=\gamma_{\uparrow}-\gamma_{\downarrow}$, mass-current phase
$\theta^{m}=(\theta_{\uparrow}+\theta_{\downarrow})/2$, and spin-current phase
$\theta^{s}=(\theta_{\uparrow}-\theta_{\downarrow})/2$. In the last step, we
ignored the term
${\gamma^{m}}\boldsymbol{\mu}\cdot\boldsymbol{\nabla}\times(\mathbb{E}\times\boldsymbol{\mu})$,
which is a high order contribution compared to the term
${\gamma^{s}}\boldsymbol{\mu}\cdot\boldsymbol{\mathcal{B}}$. This is because,
in the absence of an external electric field, the electric field in the AC
term has to be induced by curvature. Furthermore, the term
${\gamma^{m}}\boldsymbol{\mu}\cdot\boldsymbol{\nabla}\times(\mathbb{E}\times\boldsymbol{\mu})$
is proportional to the square of the magnetic moment, making it even higher
order. Indeed, with a typical radius of curvature $R_{c}=100\rm\mu m$, we can
estimate the ratio between the AC field contribution and the geometric
contribution:
$\frac{\mu^{2}E_{\text{ind}}/R_{c}}{\mu/R_{c}^{2}}=eE_{\text{ind}}R_{c}/m\approx
10^{-15}$, where $\mu=e/m$ is the magnetic moment, and $E_{\text{ind}}$ is the
electric field induced by curvature. Minimization of the total Lagrangian
$\mathcal{L}_{\rm tot}=\mathcal{L}_{A}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ with
respect to $\phi$ leads to the first GL equation that describes charge density
distribution:

$\displaystyle\sigma_{c}=-\frac{\delta\mathcal{L}_{A}}{\delta\phi}=-{\gamma^{s}}\,\mu\,\mathcal{B}.$
(101)

An alternative approach to minimize the Lagrangian is by using the 2D partial
differential form. To do it, we first assume the magnetic moment
$\boldsymbol{\mu}$ perpendicular to the surface, and we can write it as a
scalar. Variation of the Lagrangian with respect to electric potential leads
to

$\displaystyle\begin{aligned}
\delta\mathcal{L}_{A}&=\gamma_{\uparrow}\left(\partial_{i}\theta_{\uparrow}+\omega_{i}+\varepsilon_{ik}E^{k}\mu\right)\varepsilon^{il}\partial_{l}\delta\phi\,\mu-\gamma_{\downarrow}\left(\partial_{i}\theta_{\downarrow}+\omega_{i}-\varepsilon_{ik}E^{k}\mu\right)\varepsilon^{il}\partial_{l}\delta\phi\,\mu\\\
&=\frac{{\gamma^{m}}+{\gamma^{s}}}{2}\left[\partial_{i}\left(\theta^{m}+\theta^{s}\right)+\omega_{i}+\varepsilon_{ik}E^{k}\mu\right]\varepsilon^{il}\partial_{l}\delta\phi\,\mu-\frac{{\gamma^{m}}-{\gamma^{s}}}{2}\left[\partial_{i}\left(\theta^{m}-\theta^{s}\right)+\omega_{i}-\varepsilon_{ik}E^{k}\mu\right]\varepsilon^{il}\partial_{l}\delta\phi\,\mu\\\
&\approx\delta\phi{\gamma^{s}}\mu\varepsilon^{il}\partial_{l}\left(\partial_{i}\theta^{m}+\omega_{i}\right)-\delta\phi{\gamma^{m}}\mu\varepsilon^{il}\partial_{l}\left(\partial_{i}\theta^{s}+\varepsilon_{k}^{j}E^{k}\mu\right)\\\
&\approx{\gamma^{s}}\,\mu\,\mathcal{B}\delta\phi.\end{aligned}$ (102)

Minimization of the total Lagrangian leads to the same first GL equation as
Eq.(101). While the two approaches are equivalent in our case, the first one
may be illuminating for other mixed-dimensional problems.
ii) Minimized GL free energy with respect to vector potential
$\mathbb{A}$.—Here we derive the second GL equation.

$\displaystyle\begin{split}\delta\mathcal{L}_{A}&=\gamma_{\uparrow}\left[\dot{\theta}_{\uparrow}+\omega_{0}-\boldsymbol{\mu}\cdot(\boldsymbol{\nabla}\times\mathbb{A})\right]\left[-\boldsymbol{\mu}\cdot(\boldsymbol{\nabla}\times\delta\mathbb{A})\right]+\gamma_{\downarrow}\left[\dot{\theta}_{\downarrow}+\omega_{0}+\boldsymbol{\mu}\cdot(\boldsymbol{\nabla}\times\mathbb{A})\right]\left[\boldsymbol{\mu}\cdot(\boldsymbol{\nabla}\times\delta\mathbb{A})\right]\\\
&\quad-\gamma_{\uparrow}\left[\boldsymbol{\nabla}\theta_{\uparrow}+\boldsymbol{\omega}+(\mathbb{E}\times\boldsymbol{\mu})\right]\cdot\left[(-\partial_{t}\delta\mathbb{A})\times\boldsymbol{\mu}\right]-\gamma_{\downarrow}\left[\boldsymbol{\nabla}\theta_{\downarrow}+\boldsymbol{\omega}-(\mathbb{E}\times\boldsymbol{\mu})\right]\cdot\left[(\partial_{t}\delta\mathbb{A})\times\boldsymbol{\mu}\right]\\\
&=\gamma_{\uparrow}\delta\mathbb{A}\cdot\boldsymbol{\mu}\times\boldsymbol{\nabla}\left(\dot{\theta}_{\uparrow}+\omega_{0}-\boldsymbol{\mu}\cdot\mathbb{B}\right)-\gamma_{\downarrow}\delta\mathbb{A}\cdot\boldsymbol{\mu}\times\boldsymbol{\nabla}\left(\dot{\theta}_{\downarrow}+\omega_{0}+\boldsymbol{\mu}\cdot\mathbb{B}\right)\\\
&\quad-\gamma_{\uparrow}\delta\mathbb{A}\cdot\boldsymbol{\mu}\times\left[\boldsymbol{\nabla}\dot{\theta}_{\uparrow}+\dot{\boldsymbol{\omega}}+\partial_{t}(\mathbb{E}\times\boldsymbol{\mu})\right]+\gamma_{\downarrow}\delta\mathbb{A}\cdot\boldsymbol{\mu}\times\left[\boldsymbol{\nabla}\dot{\theta}_{\downarrow}+\dot{\boldsymbol{\omega}}-\partial_{t}(\mathbb{E}\times\boldsymbol{\mu})\right]\\\
&=\frac{{\gamma^{m}}+{\gamma^{s}}}{2}\delta\mathbb{A}\cdot\boldsymbol{\mu}\times\boldsymbol{\nabla}\left(\dot{\theta}_{\uparrow}+\omega_{0}-\boldsymbol{\mu}\cdot\mathbb{B}\right)-\frac{{\gamma^{m}}-{\gamma^{s}}}{2}\delta\mathbb{A}\cdot\boldsymbol{\mu}\times\boldsymbol{\nabla}\left(\dot{\theta}_{\downarrow}+\omega_{0}+\boldsymbol{\mu}\cdot\mathbb{B}\right)\\\
&\quad-\frac{({\gamma^{m}}+{\gamma^{s}})}{2}\delta\mathbb{A}\cdot\boldsymbol{\mu}\times\left[\boldsymbol{\nabla}\dot{\theta}_{\uparrow}+\dot{\boldsymbol{\omega}}+\partial_{t}(\mathbb{E}\times\boldsymbol{\mu})\right]+\frac{({\gamma^{m}}-{\gamma^{s}})}{2}\delta\mathbb{A}\cdot\boldsymbol{\mu}\times\left[\boldsymbol{\nabla}\dot{\theta}_{\downarrow}+\dot{\boldsymbol{\omega}}-\partial_{t}(\mathbb{E}\times\boldsymbol{\mu})\right]\\\
&={\gamma^{m}}\delta\mathbb{A}\cdot\boldsymbol{\mu}\times\boldsymbol{\nabla}\left(\dot{\theta}^{s}-\boldsymbol{\mu}\cdot\mathbb{B}\right)+{\gamma^{s}}\delta\mathbb{A}\cdot\boldsymbol{\mu}\times\boldsymbol{\nabla}(\dot{\theta}^{m}+\omega_{0})\\\
&\quad-{\gamma^{m}}\delta\mathbb{A}\cdot\boldsymbol{\mu}\times\left[\boldsymbol{\nabla}\dot{\theta}^{s}+\partial_{t}(\mathbb{E}\times\boldsymbol{\mu})\right]-{\gamma^{s}}\delta\mathbb{A}\cdot\boldsymbol{\mu}\times\left[\boldsymbol{\nabla}\dot{\theta}^{m}+\dot{\boldsymbol{\omega}}\right].\end{split}$
(103)

Minimization of the total Lagrangian $\mathcal{L}_{\rm tot}$ with respect to
$\mathbb{A}$ leads to the second GL equation describing current density:

$\displaystyle\mathbb{J}_{c}=\delta\mathcal{L}_{A}/\delta\mathbb{A}={\gamma^{m}}\boldsymbol{\mu}\times\boldsymbol{\nabla}\left(\dot{\theta}^{s}-\boldsymbol{\mu}\cdot\mathbb{B}\right)+{\gamma^{s}}\boldsymbol{\mu}\times\boldsymbol{\nabla}(\dot{\theta}^{m}+\omega_{0})-{\gamma^{m}}\boldsymbol{\mu}\times\left[\boldsymbol{\nabla}\dot{\theta}^{s}+\partial_{t}(\mathbb{E}\times\boldsymbol{\mu})\right]-{\gamma^{s}}\boldsymbol{\mu}\times\left[\boldsymbol{\nabla}\dot{\theta}^{m}+\dot{\boldsymbol{\omega}}\right]$

To the first order approximation in the absence of electromagnetic field, the
effective charge current is

$\displaystyle\mathbb{J}_{c}={\gamma^{s}}\boldsymbol{\mu}\times\boldsymbol{\nabla}\omega_{0}-{\gamma^{s}}\boldsymbol{\mu}\times\dot{\boldsymbol{\omega}}={\gamma^{s}}\boldsymbol{\mu}\times\boldsymbol{\mathcal{E}}.$
(104)

given the time-independent superfluid phase. In the above formula
$\boldsymbol{\mathcal{E}}=\left(\mathcal{E}_{1},\mathcal{E}_{2}\right)$ where
$\mathcal{E}_{i}\equiv\epsilon_{0ij}\mathcal{E}^{j}$.

S-IX. Introducing spin-spin interactions.

One may further introduce a small spin-spin interaction into the free energy
Leggett1968app . Without spin-flipping terms, spin is still a good quantum
number, and the simplest spin-spin interaction takes the form

$b\left(\partial_{\mu}\theta_{\uparrow}+\omega_{\mu}+A_{\mu}^{ac}\right)\left(\partial_{\mu}\theta_{\downarrow}+\omega_{\mu}-A_{\mu}^{ac}\right),$
(105)

where $b$ measures the spin-spin interaction strength. In what follows, we
show that this spin-spin interaction term does not affect the electromagnetic
response of 3He-A ($A_{1}$, $A_{2}$) phase, but it can affect the
electromagnetic response of 3He planar phase.

To see how the electromagnetic response changes, we again need to minimize the
free energy including the new term. The variation of the free energy with
respect to electric potential $\phi$ yields

$\displaystyle\begin{aligned}
\delta\mathcal{L}_{A}&=\gamma_{\uparrow}\left[\boldsymbol{\nabla}\theta_{\uparrow}+\boldsymbol{\omega}+\left(\mathbb{E}\times\boldsymbol{\mu}\right)\right]\cdot\delta(\boldsymbol{\nabla}\phi\times\boldsymbol{\mu})-\gamma_{\downarrow}\left[\boldsymbol{\nabla}\theta_{\downarrow}+\boldsymbol{\omega}-(\mathbb{E}\times\boldsymbol{\mu})\right]\cdot\delta(\boldsymbol{\nabla}\phi\times\boldsymbol{\mu})\\\
&~{}~{}-b\left[\boldsymbol{\nabla}\theta_{\uparrow}+\boldsymbol{\omega}+(\mathbb{E}\times\boldsymbol{\mu})\right]\cdot\delta(\boldsymbol{\nabla}\phi\times\boldsymbol{\mu})+b\left[\boldsymbol{\nabla}\theta_{\downarrow}+\boldsymbol{\omega}-(\mathbb{E}\times\boldsymbol{\mu})\right]\cdot\delta(\boldsymbol{\nabla}\phi\times\boldsymbol{\mu})\\\
&=\left(\gamma_{\uparrow}-b\right)\left[\boldsymbol{\nabla}\theta_{\uparrow}+\boldsymbol{\omega}+\left(\mathbb{E}\times\boldsymbol{\mu}\right)\right]\cdot\delta(\boldsymbol{\nabla}\phi\times\boldsymbol{\mu})-\left(\gamma_{\downarrow}-b\right)\left[\boldsymbol{\nabla}\theta_{\downarrow}+\boldsymbol{\omega}-(\mathbb{E}\times\boldsymbol{\mu})\right]\cdot\delta(\boldsymbol{\nabla}\phi\times\boldsymbol{\mu}).\end{aligned}$
(106)

One observes that the spin-spin interaction term effectively shifts the
stiffness of spin-up and spin-down components
$\gamma_{\uparrow}^{\prime}=\gamma_{\uparrow}-b$ and
$\gamma_{\downarrow}^{\prime}=\gamma_{\downarrow}-b$. As a result, the new
effective mass stiffness changes to
${\gamma^{m}}^{\prime}=\gamma_{\uparrow}^{\prime}+\gamma_{\downarrow}^{\prime}={\gamma^{m}}-2b$,
while new effective spin stiffness
${\gamma^{s}}^{\prime}=\gamma_{\uparrow}^{\prime}-\gamma_{\downarrow}^{\prime}={\gamma^{s}}$
remains invariant. In this case, spin current will not change but mass current
is reduced. According to Eq.(100) and (101), the electromagnetic signature
will not change. The physical reason is as follows: In a charge-neutral
condensate, all electromagnetic signatures come from the spin current. In
3He-A ($A_{1}$, $A_{2}$) phase, spin current density is determined by spin
stiffness and curvature. And since the spin stiffness does not change,
curvature-induced electromagnetic signatures remain the same. However, in 3He
planar phase, the electromagnetic signature does change. Because, in 3He
planar phase, the spin current density is determined by mass stiffness and
curvature. Therefore, a change of mass stiffness yields a change in
electromagnetic signature. By minimizing the Lagrangian of the 3He planar-
phase, we obtain the new GL equations for curvature induced charge density
$\sigma_{c}=-\left({\gamma^{m}}-2b\right)\mu\,\mathcal{B}$, and electric
current density
$\mathbb{J}_{c}=\left({\gamma^{m}}-2b\right)\,\boldsymbol{\mu}\times\boldsymbol{\mathcal{E}}(\mathbf{r})$.

Supplemental References

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