The second contribution in (D.1) is simpler to calculate in terms of
differential forms [156],

$\displaystyle\delta\int\tilde{\mathcal{R}}$ $\displaystyle e_{a}De^{a}$ (D.3)
$\displaystyle=$ $\displaystyle\int_{M}\tilde{\mathcal{R}}\left(\delta
e^{a}T_{a}+e^{a}\mbox{d}\delta
e_{a}+e^{a}\delta\omega_{ab}e^{b}+e^{a}\omega_{ab}\delta e^{b}\right)$
$\displaystyle=$ $\displaystyle\int_{M}\left(\tilde{\mathcal{R}}2\delta
e_{a}T^{a}-\tilde{\mathcal{R}}\delta\omega_{ab}e^{a}e^{b}-\delta
e_{a}e^{a}\mbox{d}\tilde{\mathcal{R}}\right)+\int_{\partial M}\delta
e_{a}\tilde{\mathcal{R}}e^{a},$

which implies

$\displaystyle*$
$\displaystyle\mathsf{J}^{a}=-\beta_{1}\left(2\tilde{\mathcal{R}}T^{a}-e^{a}\mbox{d}\tilde{\mathcal{R}}\right),\;*\mathsf{J}^{ab}=-\beta_{1}\left(-\tilde{\mathcal{R}}e^{a}e^{b}\right),$
$\displaystyle*$
$\displaystyle\mathsf{j}^{a}=-\beta_{1}\tilde{\mathcal{R}}e^{a},\;*\mathsf{j}^{ab}=0.$
(D.4)

Here we kept track of boundary terms and calculated the contributions to
boundary currents
$\mathsf{j}^{a}=\mathsf{j}_{\mu}^{\;a}\mbox{d}x^{\mu},\;\mathsf{j}^{ab}=\mathsf{j}_{\;\;\mu}^{ab}\text{d}x^{\mu}$,
which are relevant for our discussion in section 2.5.4.2. Collecting all of
the bulk contributions one finds (2.111).

In section 2.5.4.2 we wrote down (2.111) for a product geometry with respect
to the coordinate $z$, and assumed torsion vanishes. Here we generalize to
non-zero torsion. With non zero torsion, (2.113) generalizes to

$\displaystyle\left\langle\mathsf{J}^{\alpha z}\right\rangle_{\text{gpCS}}=$
$\displaystyle-\beta_{1}\frac{1}{\left|e\right|}\varepsilon^{z\alpha\beta}\partial_{\beta}\tilde{\mathcal{R}},$
(D.5)
$\displaystyle\left\langle\mathsf{J}^{z\alpha}\right\rangle_{\text{gpCS}}=$
$\displaystyle\beta_{1}\left[\frac{1}{\left|e\right|}\varepsilon^{z\alpha\beta}\partial_{\beta}\tilde{\mathcal{R}}+\frac{1}{\left|e\right|}\varepsilon^{z\beta\gamma}C_{\beta\gamma}^{\;\;\alpha}\tilde{\mathcal{R}}\right].$

For $z=t$, which describes a time independent situation, we find

$\displaystyle\left\langle J_{E}^{i}\right\rangle_{\text{gpCS}}=$
$\displaystyle\beta_{1}\varepsilon^{ij}\partial_{j}\tilde{\mathcal{R}},$ (D.6)
$\displaystyle\left\langle P_{i}\right\rangle_{\text{gpCS}}=$
$\displaystyle-\beta_{1}\left[g_{ik}\varepsilon^{kj}\partial_{j}\tilde{\mathcal{R}}+2o\left|e\right|\tilde{\mathcal{R}}C_{12i}\right],$

which generalizes (2.114). Explicit expressions for the contorsion $C_{12i}$
are given in appendix B.4. Equation (D.6) is compatible with the operator
equation (2.61), and the density response (2.112).

In the case $z=y$, the inflow (2.116) generalizes to

$\displaystyle\left\langle
t_{\text{cov}\;\alpha}^{y}\right\rangle_{\text{gpCS}}$
$\displaystyle=-\left|e\right|\left\langle\mathsf{J}_{\;\alpha}^{y}\right\rangle_{\text{gpCS}}$
(D.7) $\displaystyle=-\beta_{1}\left[g_{\alpha\beta}\varepsilon^{\beta\gamma
y}\partial_{\gamma}\tilde{\mathcal{R}}+2o\left|e\right|C_{01\alpha}\tilde{\mathcal{R}}\right].$

For the order parameter
$\Delta=\Delta_{0}e^{i\theta\left(t,x\right)}\left(1+f\left(t,x\right),\pm
i\right)$ that we consider in this case, we find using appendix B.4 that
$C_{01t}=0,\;C_{01x}=e_{1}^{\;i}\frac{1}{2}\partial_{t}g_{ij}$. The boundary
current (2.117) is unchanged, but the bulk+boundary conservation equation
(2.118) is generalized to

$\displaystyle\tilde{\nabla}_{\alpha}\left\langle\mathsf{j}_{\;\beta}^{\alpha}\right\rangle_{\text{gpCS}}-C_{ab\beta}\left\langle\mathsf{j}^{[ab]}\right\rangle_{\text{gpCS}}=\left\langle\mathsf{J}^{y\beta}\right\rangle_{\text{gpCS}},$
(D.8)

so that bulk+boundary conservation still holds for the current from gpCS, in
the presence of torsion.

###  Appendix E Perturbative calculation of the relativistic effective action

Here we present a perturbative calculation of the effective action for the RC
background fields $e,\omega$ induced by a Majorana spinor in 2+1 dimensions. A
perturbative calculation requires three types of input: free propagators,
interaction vertices, and a renormalization scheme to handle UV divergences.
In our case the propagator and vertices are standard in the context of the
coupling of relativistic fermions to gravity, but the renormalization scheme
will not be standard in this context.

The standard renormalization schemes used in the literature are aimed at
preserving Lorentz symmetry, obtaining properly quantized coefficients for CS
terms, and obtaining finite results that do not depend on a regulator [156,
157]. This is usually done as follows. First, one introduces a Lorentz
invariant regulator, such as a frequency and wave-vector cutoff
$\Lambda_{\text{rel}}$, then one introduces Pauli-Villars regulators, and
tunes their masses such that the limit $\Lambda_{\text{rel}}\rightarrow\infty$
produces finite results and properly quantized CS coefficients.

In contrast, we take the lattice model (2.15) as a microscopic description of
the $p$-wave SC, and the relativistic continuum limit as an approximation of
it. As we obtained naturally in sections 2.2.1 and 2.2.3, this means that
there are four Majorana spinors, with different orientations and masses, and a
wave-vector cutoff $\Lambda_{UV}\sim 1/a$, but no frequency cutoff, as
dictated by the lattice model. Note that these multiple Majorana spinors are
not Pauli-Villars regulators, simply because they are all fermions. None of
them has the “wrong statistics”. The cutoff $\Lambda_{UV}$ is a physical
parameter of the model and we do not wish to take it to infinity. Thus wave-
vector integrals cannot diverge. In contrast, since time is continuous, there
is no physical frequency cutoff, and divergences in frequency integrals do
appear. These divergences are unphysical, and can be viewed as a byproduct of
the construction of the path integral by time discretization. These
divergences need to be renormalized in the usual sense, and we do this by
minimal subtraction.

To set up the perturbative calculation, we write the action $S_{\text{RC}}$ in
terms of the spinor densities $\Psi=\left|e\right|^{1/2}\chi$, and using the
explicit form (B.3),

$\displaystyle S_{\text{RC}}=\frac{1}{2}\int\mbox{d}^{3}x\overline{\Psi}$
$\displaystyle\left[\frac{1}{2}ie_{a}^{\;\mu}\gamma^{a}\overleftrightarrow{\partial_{\mu}}-\frac{1}{4}\omega_{abc}\varepsilon^{abc}-m\right]\Psi$
(E.1) $\displaystyle=\frac{1}{2}\int\mbox{d}^{3}x\overline{\Psi}$
$\displaystyle\left[ie_{a}^{\;\mu}\gamma^{a}\partial_{\mu}+\frac{i}{2}\left(\partial_{\mu}e_{a}^{\;\mu}\right)\gamma^{a}\right.\left.-\frac{1}{4}\omega_{abc}\varepsilon^{abc}-m\right]\Psi.$

Assuming for now that the vielbein has a positive orientation, we insert
$e_{a}^{\;\mu}=\delta_{a}^{\mu}+h_{a}^{\;\mu}$ with small $h$, and split the
action into an inverse propagator $G^{-1}$ and vertices $V$,

$\displaystyle S_{\text{RC}}=$
$\displaystyle\frac{1}{2}\int\mbox{d}^{3}x\Psi^{\dagger}\gamma^{0}\left[G^{-1}+V\right]\Psi,$
(E.2) $\displaystyle G^{-1}=$ $\displaystyle
i\delta_{a}^{\;\mu}\gamma^{a}\partial_{\mu}-m,\;V=V_{1}+V_{2},$ $\displaystyle
V_{1}=$ $\displaystyle
i\gamma^{a}h_{a}^{\;\mu}\partial_{\mu}+\frac{i}{2}\gamma^{a}\left(\partial_{\mu}h_{a}^{\;\mu}\right),\;V_{2}=-\frac{1}{4}\omega_{abc}\varepsilon^{abc}.$

The vertex $V_{1}$ is first order in the perturbation $h$. The vertex $V_{2}$
is given explicitly by

$\displaystyle
V_{2}=-\frac{1}{4}\omega_{ab\mu}e_{c}^{\;\mu}\varepsilon^{abc}=-\frac{1}{4}\omega_{ab\mu}\delta_{c}^{\mu}\varepsilon^{abc}-\frac{1}{4}\omega_{ab\mu}h_{c}^{\;\mu}\varepsilon^{abc},$
(E.3)

and therefore contains a term of order $\omega$ and a term quadratic in the
perturbations, of order $h\omega$. Terms in vertices which are nonlinear in
perturbations are sometimes called contact terms, and the above contribution
to $V_{2}$ is the only contact term in our scheme. Note that there is no
vertex related to the volume element $\left|e\right|$, because the fundamental
fermionic degree of freedom is the spinor density $\Psi$, see appendix B.3. In
expressions written in terms of $h_{a}^{\;\mu}$ we use $\eta_{\mu\nu}$ to
raise and lower coordinate indices and $\delta_{a}^{\mu}$ to map internal
indices to coordinate indices, so in practice there is no difference between
these indices in such expressions.

The perturbative expansion of the effective action is given by

$\displaystyle 2W_{\text{RC}}=$
$\displaystyle-2i\log\mbox{Pf}\left(i\gamma^{0}\left(G^{-1}+V\right)\right)$
(E.4) $\displaystyle=$ $\displaystyle-i\mbox{Tr}\left(\log
i\gamma^{0}G^{-1}\right)$
$\displaystyle-i\mbox{Tr}\left(GV\right)+\frac{i}{2}\mbox{Tr}\left(GV\right)^{2}+O\left(V^{3}\right),$

which, apart from the first term, is a sum over Feynman diagrams with a
fermion loop and any number of vertices $V$. We will be interested in
$W_{\text{RC}}$ to second order in the perturbations $h$ and $\omega$ and up
to third order in derivatives. Terms of first oder in $h,\omega$ correspond to
properties of the unperturbed ground state, or vacuum, while terms of second
order correspond to linear response coefficients. The first term is
independent of $h,\omega$ and corresponds to the ground state energy of the
unperturbed system. This information can also be obtained from the term linear
in $h$, and we therefore ignore $\mbox{Tr}\log i\gamma^{0}G^{-1}$ in the
following. Expanding the vertices,

$\displaystyle 2W_{\text{RC}}=$
$\displaystyle-i\mbox{Tr}\left(GV_{1}\right)-i\mbox{Tr}\left(GV_{2}\right)+\frac{i}{2}\mbox{Tr}\left(GV_{1}\right)^{2}$
(E.5)
$\displaystyle+\frac{i}{2}\mbox{Tr}\left(GV_{2}\right)^{2}+i\mbox{Tr}\left(GV_{1}GV_{2}\right)+O\left(V^{3}\right).$

These functional traces can now be written as integrals over Fourier
components and traces over spinor indices,

$\displaystyle\mbox{Tr}\left(GV_{1}\right)=$ $\displaystyle-
h_{a}^{\;\mu}\left(p=0\right)\int_{q}q_{\mu}\mbox{tr}\left(\gamma^{a}G_{q}\right),$
(E.6) $\displaystyle\mbox{Tr}\left(GV_{2}\right)=$
$\displaystyle\omega\left(p=0\right)\int_{q}\mbox{tr}\left(G_{q}\right),$
$\displaystyle\mbox{Tr}\left(GV_{1}\right)^{2}=$
$\displaystyle\int_{p}h_{a}^{\;\mu}\left(p\right)h_{b}^{\;\nu}\left(-p\right)\int_{q}\left(q+\frac{1}{2}p\right)_{\mu}\left(q+\frac{1}{2}p\right)_{\nu}\text{tr}\left(\gamma^{a}G_{q}\gamma^{b}G_{p+q}\right),$
$\displaystyle\mbox{Tr}\left(GV_{2}\right)^{2}=$
$\displaystyle\int_{p}\omega\left(p\right)\omega\left(-p\right)\int_{q}\text{tr}\left(G_{q}G_{p+q}\right),$
$\displaystyle\mbox{Tr}\left(GV_{1}GV_{2}\right)=$
$\displaystyle-\int_{p}h_{a}^{\;\mu}\left(p\right)\omega\left(-p\right)\int_{q}\left(q+\frac{1}{2}p\right)_{\mu}\text{tr}\left(\gamma^{a}G_{q}G_{p+q}\right),$

where $\omega=-\frac{1}{4}\omega_{ab\mu}e_{c}^{\;\mu}\varepsilon^{abc}$, and
$\int_{p}=\int\frac{\text{d}^{3}p}{\left(2\pi\right)^{3}}$. Our conventions
for the Fourier transform of a function $f$ is
$f\left(x\right)=\int_{q}e^{iq_{\mu}x^{\mu}}f\left(q\right)$. The Fourier
transform of the Greens function is then
$G_{q}=-\frac{1}{\cancel{q}+m}=-\frac{\cancel{q}-m}{q^{2}-m^{2}}$, where
$\cancel{q}=q_{a}\gamma^{a}$. The spinor traces are evaluated using the usual
identities for gamma matrices in 2+1 dimensions,

tr
$\displaystyle\left(\gamma^{a}\right)=0,\;\text{tr}\left(\gamma^{a}\gamma^{b}\right)=2\eta^{ab},\;\text{tr}\left(\gamma^{a}\gamma^{b}\gamma^{c}\right)=\pm
2i\varepsilon^{abc},$ tr
$\displaystyle\left(\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d}\right)=2\left(\eta^{ab}\eta^{cd}-\eta^{ac}\eta^{bd}+\eta^{ad}\eta^{bc}\right).$
(E.7)

The sign $\pm$ distinguishes the two inequivalent representations of gamma
matrices in 2+1 dimensions, and with our chosen representation,
$\text{tr}\left(\gamma^{a}\gamma^{b}\gamma^{c}\right)=2i\varepsilon^{abc}$.
Using these identities yields for the single vertex diagrams

$\displaystyle\mbox{Tr}\left(GV_{1}\right)=2\eta^{ab}h_{a}^{\;\mu}\left(p=0\right)\int_{q}\frac{q_{\mu}q_{b}}{q^{2}-m^{2}},$
(E.8)
$\displaystyle\mbox{Tr}\left(GV_{2}\right)=2m\omega\left(p=0\right)\int_{q}\frac{1}{q^{2}-m^{2}}.$

The expressions for the diagrams with two vertices are more complicated, so
let us start by analyzing the single vertex diagrams. This will suffice to
demonstrate our renormalization scheme and compare it to direct calculations
within the lattice model and to renormalizations which are more natural in the
context of relativistic QFT.

####  E.1 Single vertex diagrams

From (E.5) and (E.8) it follows that

$\displaystyle
W_{\text{RC}}=\Lambda_{\;\mu}^{a}\int\text{d}^{3}xh_{a}^{\;\mu}+s\int\text{d}^{3}x\omega+O\left(V^{2}\right),$
(E.9)

where

$\displaystyle\Lambda_{\;\mu}^{a}=-i\eta^{ab}\int\frac{\text{d}^{3}q}{\left(2\pi\right)^{3}}\frac{q_{\mu}q_{b}}{q^{2}-m^{2}},\;s=-i\int\frac{\text{d}^{3}q}{\left(2\pi\right)^{3}}\frac{m}{q^{2}-m^{2}},$
(E.10)

can now be recognized as the energy-momentum tensor and spin density of the
unperturbed ground state,

$\displaystyle\left\langle\mathsf{J}_{\;\mu}^{a}\right\rangle=$
$\displaystyle-\Lambda_{\;\mu}^{a},$ (E.11)
$\displaystyle\left\langle\mathsf{J}^{ab\mu}\right\rangle=$
$\displaystyle-\frac{1}{4}\frac{1}{2}\left\langle\overline{\chi}\chi\right\rangle\delta_{c}^{\mu}\varepsilon^{abc}=-\frac{1}{4}s\delta_{c}^{\mu}\varepsilon^{abc}.$

Preforming a Wick rotation $q_{0}\mapsto iq_{0}$,

$\displaystyle\Lambda_{\;\mu}^{a}=\delta^{ab}\int\frac{\text{d}^{3}q}{\left(2\pi\right)^{3}}\frac{q_{\mu}q_{b}}{\left|q\right|^{2}+m^{2}},\;s=-\int\frac{\text{d}^{3}q}{\left(2\pi\right)^{3}}\frac{m}{\left|q\right|^{2}+m^{2}},$
(E.12)

where $\left|\cdot\right|$ is the euclidian norm. We start by calculating $s$
in our lattice motivated renormalization scheme. In this scheme the integral
reads

$\displaystyle
s=-\int_{\left|\boldsymbol{q}\right|<\Lambda_{UV}}\frac{\text{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\int_{-\infty}^{\infty}\frac{\text{d}q_{0}}{2\pi}\frac{m}{q_{0}^{2}+\left|\boldsymbol{q}\right|^{2}+m^{2}},$
(E.13)

where $\Lambda_{UV}$ is a physical cutoff related to the lattice spacing by
$\Lambda_{UV}\sim a^{-1}$. The $q_{0}$ integral converges, and does not
require renormalization. It yields the result within the lattice motivated
scheme,

$\displaystyle
s=-\frac{1}{2}\int_{\left|\boldsymbol{q}\right|<\Lambda_{UV}}\frac{\text{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\frac{m}{\sqrt{\left|\boldsymbol{q}\right|^{2}+m^{2}}},$
(E.14)

and adding the operator ordering correction gives the ground state charge
density

$\displaystyle\rho=\left\langle
J^{t}\right\rangle=-\frac{1}{2}\int_{\left|\boldsymbol{q}\right|<\Lambda_{UV}}\frac{\text{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\left(1-\frac{m}{\sqrt{\left|\boldsymbol{q}\right|^{2}+m^{2}}}\right).$
(E.15)

After summing over low energy Majorana spinors and restoring units, this
coincides with the relativistic limit of the exact ground state charge density
of the lattice model [15],

$\displaystyle\rho=-\frac{1}{2}\int_{BZ}\frac{\mbox{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\left(1-\frac{h_{\boldsymbol{q}}}{\sqrt{\left|\Delta_{\boldsymbol{q}}\right|^{2}+h_{\boldsymbol{q}}^{2}}}\right),$
(E.16)

where $h_{\boldsymbol{q}},\Delta_{\boldsymbol{q}}$ were defined in section
2.2.1.

For comparison we calculate the $s$ integral in a standard renormalization
scheme of relativistic QFT. In this approach the integral does not converge.
We introduce a frequency and wave-vector cutoff $\Lambda_{\text{rel}}$, and
restrict the integration to $\left|q\right|<\Lambda_{\text{rel}}$. This yields

$\displaystyle
s=-\int_{\left|q\right|<\Lambda_{\text{rel}}}\frac{\text{d}^{3}q}{\left(2\pi\right)^{3}}\frac{m}{\left|q\right|^{2}+m^{2}}=-\frac{\Lambda_{\text{rel}}m}{2\pi^{2}}+\frac{m^{2}\text{sgn}m}{4\pi}+O\left(\frac{m}{\Lambda_{\text{rel}}}\right).$
(E.17)

A simple way to proceed is to preform minimal subtraction, which means we
remove the diverging piece, and take
$\Lambda_{\text{rel}}/m\rightarrow\infty$. This gives the fully relativistic
result

$\displaystyle s=\frac{m^{2}\text{sgn}m}{4\pi}.$ (E.18)

Comparing with (2.5.2.2) we find $\zeta_{H}=s=\frac{m^{2}\text{sgn}m}{4\pi}$
for a positive orientation which is essentially the torsional Hall viscosity
of [156]424242It is not exactly the same result because we did not use the
same relativistic renormalization scheme.. The relativistic result can also be
obtained by expanding the lattice result (E.14) in $\Lambda_{UV}$ and keeping
the $O\left(1\right)$ piece. This is a general feature, the $O\left(1\right)$
piece of any coefficient in the effective action is always relativistic.

Let us now turn to the calculation of the ground state energy-momentum tensor
$\Lambda_{\;\mu}^{a}$. With a relativistic regulator $\Lambda_{\;\mu}^{a}$ is
$O\left(3\right)$ invariant and must therefore be proportional to the
identity,

$\displaystyle\Lambda_{\;\mu}^{a}=\delta^{ab}\int_{\left|q\right|<\Lambda_{\text{rel}}}\frac{\text{d}^{3}q}{\left(2\pi\right)^{3}}\frac{q_{\mu}q_{b}}{\left|q\right|^{2}+m^{2}}=\delta_{\mu}^{a}\frac{\Lambda}{2\kappa_{N}},$
(E.19)

with the cosmological constant

$\displaystyle\frac{\Lambda}{2\kappa_{N}}=\frac{1}{3}\int_{\left|q\right|<\Lambda_{\text{rel}}}\frac{\text{d}^{3}q}{\left(2\pi\right)^{3}}\frac{\left|q\right|^{2}}{\left|q\right|^{2}+m^{2}}=\frac{1}{3}\left[\frac{\Lambda_{\text{rel}}^{3}}{6\pi^{2}}-\frac{\Lambda_{\text{rel}}m^{2}}{2\pi^{2}}+\frac{\left|m\right|^{3}}{4\pi}+O\left(\frac{m}{\Lambda_{\text{rel}}}\right)\right].$
(E.20)

Keeping the $O\left(1\right)$ piece we find the relativistic expression

$\displaystyle\Lambda_{\;\mu}^{a}=\delta_{\mu}^{a}\frac{\Lambda}{2\kappa_{N}}=\delta_{\mu}^{a}\frac{\left|m\right|^{3}}{6\pi},$
(E.21)

which again, is essentially the result of [156]. With the lattice motivated
renormalization scheme,

$\displaystyle\Lambda_{\;\mu}^{a}=\delta^{ab}\int_{\left|\boldsymbol{q}\right|<\Lambda_{UV}}\frac{\text{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\int_{-\infty}^{\infty}\frac{\text{d}q_{0}}{2\pi}\frac{q_{\mu}q_{b}}{q_{0}^{2}+\left|\boldsymbol{q}\right|^{2}+m^{2}}.$
(E.22)

Here the $q_{0}$ integral for $\Lambda_{\;t}^{0}$ does not converge, and needs
to be regularized. We do this by introducing a frequency cutoff $\Lambda_{0}$,

$\displaystyle\Lambda_{\;\mu}^{a}=\delta^{ab}\int_{\left|\boldsymbol{q}\right|<\Lambda_{UV}}\frac{\text{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\int_{-\Lambda_{0}}^{\Lambda_{0}}\frac{\text{d}q_{0}}{2\pi}\frac{q_{\mu}q_{b}}{q_{0}^{2}+\left|\boldsymbol{q}\right|^{2}+m^{2}}.$
(E.23)

Unlike $\Lambda_{UV}$ which is a physical parameter of the model,
$\Lambda_{0}$ is a fictitious cutoff which we take to infinity at the end of
the calculation. The $q_{0}$ divergence can be interpreted as an artifact of
time discretization [130]. At this point the domain of integration is not a
ball in Euclidian Fourier space but a cylinder, so it is not invariant under
$O\left(3\right)$, only under $O\left(2\right)$434343More accurately, the
domain of integration for each lattice fermion is not the disk
$\left\\{\left|\boldsymbol{q}\right|<\Lambda_{UV}\right\\}$ but the square
$\left[-\Lambda_{UV}/2,\Lambda_{UV}/2\right]^{2}$ with $\Lambda_{UV}=\pi/a$,
which is a quarter of the Brillouin zone $BZ$, see section 2.2.3. The symmetry
group of this domain is not $O\left(2\right)$ but the point group symmetry of
the lattice $D_{4}\subset O\left(2\right)$. This subtlety has no effect on the
following.  and the reflection $q\mapsto-q$. This implies that the tensor
$\Lambda_{\;\mu}^{a}$ takes the form

$\displaystyle\Lambda_{\;t}^{0}=$
$\displaystyle\int_{\left|\boldsymbol{q}\right|<\Lambda_{UV}}\frac{\text{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\int_{-\Lambda_{0}}^{\Lambda_{0}}\frac{\text{d}q_{0}}{2\pi}\frac{q_{0}^{2}}{q_{0}^{2}+\left|\boldsymbol{q}\right|^{2}+m^{2}},$
(E.24) $\displaystyle\Lambda_{\;j}^{A}=$
$\displaystyle\frac{1}{2}\delta_{j}^{A}\int_{\left|\boldsymbol{q}\right|<\Lambda_{UV}}\frac{\text{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\int_{-\Lambda_{0}}^{\Lambda_{0}}\frac{\text{d}q_{0}}{2\pi}\frac{\left|\boldsymbol{q}\right|^{2}}{q_{0}^{2}+\left|\boldsymbol{q}\right|^{2}+m^{2}},$

with all other components vanishing. The $q_{0}$ integral for the energy
density $\Lambda_{\;t}^{0}$ gives

$\displaystyle\Lambda_{\;t}^{0}=\frac{1}{2}\int_{\left|\boldsymbol{q}\right|<\Lambda_{UV}}\frac{\mbox{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\left[\frac{2\Lambda_{0}}{\pi}-\sqrt{\left|\boldsymbol{q}\right|^{2}+m^{2}}+O\left(\frac{m}{\Lambda_{0}}\right)\right].$
(E.25)

Keeping the $O\left(1\right)$ piece we find, within the lattice motivated
scheme,

$\displaystyle\Lambda_{\;t}^{0}=-\frac{1}{2}\int_{\left|\boldsymbol{q}\right|<\Lambda_{UV}}\frac{\mbox{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\sqrt{\left|\boldsymbol{q}\right|^{2}+m^{2}},$
(E.26)

which is the familiar expression for the ground state energy of a single
Majorana fermion, which is half the energy of a filled Dirac sea. The $q_{0}$
integral for $\Lambda_{\;j}^{A}$ converges and gives the pressure

$\displaystyle\Lambda_{\;j}^{A}=\frac{1}{2}\delta_{j}^{A}\frac{1}{2}\int_{\left|\boldsymbol{q}\right|<\Lambda_{UV}}\frac{\mbox{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\frac{\left|\boldsymbol{q}\right|^{2}}{\sqrt{\left|\boldsymbol{q}\right|^{2}+m^{2}}}.$
(E.27)

We see that the ground state energy density and pressure are no longer equal.
In other words the ground state energy-momentum tensor is not Lorentz
invariant, due to the lattice renormalization scheme. It may be surprising
that the expression (E.26) for the energy density is part of an energy
momentum tensor which is not Lorentz invariant. This has been discussed in the
literature in the context of the cosmological constant problem [312, 313,
314].

Let us now compare the above with the lattice model. For the energy density we
need to add the operator ordering correction,

$\displaystyle\varepsilon=\left\langle
t_{\text{cov}\;t}^{t}\right\rangle=\frac{1}{2}\int_{\left|\boldsymbol{q}\right|<\Lambda_{UV}}\frac{\mbox{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\left(m-\sqrt{\left|\boldsymbol{q}\right|^{2}+m^{2}}\right).$
(E.28)

Restoring units and summing over Dirac points, we recognize the above as the
relativistic approximation of the ground state energy density of the lattice
model [27],

$\displaystyle\varepsilon=\frac{1}{2}\int_{BZ}\frac{\text{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\left(h_{\boldsymbol{q}}-E_{\boldsymbol{q}}\right).$
(E.29)

The above calculations of simple ground state properties serve as consistency
checks. We have seen explicitly that these quantities are UV sensitive. With
the lattice motivated renormalization scheme the effective action produces
physical quantities that approximate those of the lattice model, which are
distinct from those obtained with a relativistic scheme. In the following we
will focus on UV insensitive terms. In doing so we will also ignore operator
ordering corrections, because these always contain
$\delta^{2}\left(0\right)\sim\int_{\left|\boldsymbol{q}\right|<\Lambda_{UV}}\frac{\text{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\sim\Lambda_{UV}^{2}$
and are therefore UV sensitive.

####  E.2 Two vertex diagrams

Let us now turn to the calculation of the more interesting second order terms,
which correspond to linear responses. After preforming the traces over gamma
matrices one finds

$\displaystyle\mbox{Tr}\left(GV_{1}\right)^{2}=$
$\displaystyle-2im\varepsilon^{abc}\int_{p}h_{a}^{\;\mu}\left(p\right)h_{b}^{\;\nu}\left(-p\right)p_{c}\int_{q}\frac{\left(q+\frac{1}{2}p\right)_{\mu}\left(q+\frac{1}{2}p\right)_{\nu}}{\left(q^{2}-m^{2}\right)\left(\left(p+q\right)^{2}-m^{2}\right)}$
(E.30)
$\displaystyle+2m^{2}\eta^{ab}\int_{p}h_{a}^{\;\mu}\left(p\right)h_{b}^{\;\nu}\left(-p\right)\int_{q}\frac{\left(q+\frac{1}{2}p\right)_{\mu}\left(q+\frac{1}{2}p\right)_{\nu}}{\left(q^{2}-m^{2}\right)\left(\left(p+q\right)^{2}-m^{2}\right)}$
$\displaystyle+2\left(\eta^{ac}\eta^{bd}-\eta^{ab}\eta^{cd}+\eta^{ad}\eta^{cb}\right)\int_{p}h_{a}^{\;\mu}\left(p\right)h_{b}^{\;\nu}\left(-p\right)\int_{q}\frac{\left(q+\frac{1}{2}p\right)_{\mu}\left(q+\frac{1}{2}p\right)_{\nu}q_{c}\left(p+q\right)_{d}}{\left(q^{2}-m^{2}\right)\left(\left(p+q\right)^{2}-m^{2}\right)},$
$\displaystyle\mbox{Tr}\left(GV_{2}\right)^{2}=$
$\displaystyle\int_{p}\omega\left(p\right)\omega\left(-p\right)\int_{q}\frac{2\eta^{ab}q_{a}\left(p+q\right)_{b}+2m^{2}}{\left(q^{2}-m^{2}\right)\left(\left(p+q\right)^{2}-m^{2}\right)},$
(E.31) $\displaystyle\mbox{Tr}\left(GV_{1}GV_{2}\right)=$
$\displaystyle-2i\varepsilon^{abc}\int_{p}h_{a}^{\;\mu}\left(p\right)\omega\left(-p\right)p_{c}\int_{q}\frac{\left(q+\frac{1}{2}p\right)_{\mu}q_{b}}{\left(q^{2}-m^{2}\right)\left(\left(p+q\right)^{2}-m^{2}\right)}$
(E.32)
$\displaystyle+2m\eta^{ab}\int_{p}h_{a}^{\;\mu}\left(p\right)\omega\left(-p\right)\int_{q}\frac{\left(q+\frac{1}{2}p\right)_{\mu}\left(2q+p\right)_{b}}{\left(q^{2}-m^{2}\right)\left(\left(p+q\right)^{2}-m^{2}\right)}.$

One is then left with the calculation of the integrals over the loop momenta
$q$ in the above equations. The first step in doing so is Wick rotating to
euclidian signature by changing $q_{0}\mapsto iq_{0},\;p_{0}\mapsto ip_{0}$ in
the $q$ integrals.

At this point one can use Feynman parameters to simplify the form of the
integrands, but since we are only interested in the effective action to low
orders in derivatives of the background fields, we find it simpler to expand
the integrands in powers of $p/m$.

We start with the first integral in (E.30), which contains the gCS term.
Expanding the integrand in $p/m$ we find

$\displaystyle\frac{\left(q+\frac{1}{2}p\right)_{\mu}\left(q+\frac{1}{2}p\right)_{\nu}}{\left(q^{2}+m^{2}\right)\left(\left(p+q\right)^{2}+m^{2}\right)}=$
$\displaystyle\frac{q_{\mu}q_{\nu}}{\left(m^{2}+q^{2}\right)^{2}}+\left[\frac{p_{(\mu}q_{\nu)}}{\left(m^{2}+q^{2}\right)^{2}}-2\frac{q_{\mu}q_{\nu}p\cdot
q}{\left(m^{2}+q^{2}\right)^{3}}\right]$ (E.33)
$\displaystyle+\left[\frac{p_{\mu}p_{\nu}}{4\left(m^{2}+q^{2}\right)^{2}}-\frac{p^{2}q_{\mu}q_{\nu}}{\left(m^{2}+q^{2}\right)^{3}}-\frac{2p_{(\mu}q_{\nu)}p\cdot
q}{\left(m^{2}+q^{2}\right)^{3}}+\frac{4q_{\mu}q_{\nu}\left(p\cdot
q\right)^{2}}{\left(m^{2}+q^{2}\right)^{4}}\right]+O\left(p^{3}\right)$

where terms are grouped according to their order in $p/m$.The $q$ integral
over the $O\left(1\right)$ terms diverges, and therefore produces a UV
sensitive term in the effective action. With a relativistic renormalization we
find

$\displaystyle
2W_{\text{RC}}=\frac{m^{2}\text{sgn}\left(m\right)}{4\pi}\int\text{d}^{3}xh_{a}^{\;\mu}\eta_{\mu\nu}\varepsilon^{abc}\partial_{b}h_{c}^{\;\nu}+\cdots$
(E.34)

Comparing with (2.5.2.2) and using
$e_{a}De^{a}=\varepsilon^{abc}h_{a\nu}\partial_{b}h_{c}^{\;\nu}\text{d}^{3}x+\cdots$
we find again the torsional Hall viscosity
$\zeta_{H}=\frac{m^{2}\text{sgn}\left(m\right)}{4\pi}$, for positive
orientation. With the lattice renormalization the $\eta_{\mu\nu}$ in the above
is replaced by a non Lorentz invariant tensor, but in Sec.2 we are only
interested in UV insensitive responses, so we will not discuss it here further
(see Appendix H.1).

The $q$ integral over the $O\left(p/m\right)$ terms vanishes because it is odd
under the reflection $q\mapsto-q$.

The $O\left(p^{2}/m^{2}\right)$ contributions are most interesting for us. The
$q$ integral over these converges, and therefore produces UV insensitive terms
in the effective action. Instead of calculating the integral with the finite
physical $\Lambda_{UV}$, we can calculate it with $\Lambda_{UV}/m=\infty$ at
the expense of producing small $O\left(m/\Lambda_{UV}\right)$ corrections.
Then the calculation reduces to a standard calculation within relativistic QFT
which has appeared a few times in the literature with slightly different
conventions [257, 258, 259, 260], and which is done below for completeness.
See also [261] for a recent heat kernel calculation and review of the
literature, and [315] for similar computations in 4+1 dimensions. With
$\Lambda_{UV}/m=\infty$ the integral is Lorentz invariant and this implies the
standard reductions to radial functions such as
$q_{\mu}q_{\nu}\mapsto\frac{1}{3}\eta_{\mu\nu}q^{2}$. The
$O\left(p^{2}/m^{2}\right)$ contributions in (E.33) then reduce to

$\displaystyle\frac{1}{4}$
$\displaystyle\frac{p_{\mu}p_{\nu}}{\left(m^{2}+q^{2}\right)^{2}}-\frac{1}{3}\frac{p^{2}\eta_{\mu\nu}q^{2}}{\left(m^{2}+q^{2}\right)^{3}}-\frac{2}{3}\frac{p_{\mu}p_{\nu}q^{2}}{\left(m^{2}+q^{2}\right)^{3}}$
$\displaystyle+\frac{4}{15}\frac{p^{2}\eta_{\mu\nu}+2p_{\mu}p_{\nu}}{\left(m^{2}+q^{2}\right)^{4}}q^{4},$
(E.35)

and preforming the $q$ integral yields

$\displaystyle\frac{1}{96\pi\left|m\right|}\left(p_{\mu}p_{\nu}-p^{2}\eta_{\mu\nu}\right).$
(E.36)

This corresponds to the following term in the effective action

$\displaystyle 2W_{\text{RC}}=$
$\displaystyle\frac{\text{sgn}\left(m\right)}{2}\frac{1}{96\pi}2\int\text{d}^{3}xh_{a}^{\;\mu}\varepsilon^{abc}\partial_{c}\left(\partial_{\mu}\partial_{\nu}-\partial^{2}\eta_{\mu\nu}\right)h_{b}^{\;\nu}$
$\displaystyle+\cdots$ (E.37)

To identify this term it is easiest to fix a Lorentz gauge where
$h_{\left[\mu\nu\right]}=0$. In terms of the $p$-wave SC this corresponds to
$U\left(1\right)$ gauge fixing the phase $\theta$ of the order parameter to
$0$, along with an additional boost which is only a symmetry in the
relativistic limit. Then $h$ corresponds also to the first order metric
perturbation,
$g_{\mu\nu}=\eta_{\mu\nu}-2h_{(\mu\nu)}=\eta_{\mu\nu}-2h_{\mu\nu}$, and the
above corresponds to the expansion of the gCS term

$\displaystyle
2W_{\text{RC}}=\frac{\text{sgn}\left(m\right)}{2}\frac{1}{96\pi}\int
Q_{3}\left(\tilde{\Gamma}\right)+\cdots$ (E.38)

to second order in $h$. In preforming such expansions we found the Mathematica
package xAct very useful [316, 317]. Equation (E.38) corresponds to
$\kappa_{H}=\frac{1}{48\pi}\frac{\text{sgn}\left(m\right)}{2}$. We note that
within the perturbative calculation there is no difference between $\int
Q_{3}\left(\tilde{\Gamma}\right)$ and $\int Q_{3}\left(\tilde{\omega}\right)$,
see (LABEL:eq:70).

The above result is valid for a vielbein
$e_{a}^{\;\mu}=\delta_{a}^{\mu}+h_{a}^{\;\mu}$ which has a positive
orientation. A vielbein with a negative orientation can be written as
$e_{a}^{\;\mu}=L_{\;a}^{b}\left(\delta_{b}^{\mu}+h_{b}^{\;\mu}\right)$ where
$L$ is a Lorentz transformation with $\text{det}L=-1$. We can deal with such
vielbeins by absorbing $L$ into the gamma matrices, $\gamma^{a}\mapsto
L_{\;a}^{b}\gamma^{a}$. The only effect that this change has on the traces
(E.7) is changing
$\text{tr}\left(\gamma^{a}\gamma^{b}\gamma^{c}\right)=2i\varepsilon^{abc}$ to
$\text{tr}\left(\gamma^{a}\gamma^{b}\gamma^{c}\right)=-2i\varepsilon^{abc}$.
The metric is independent of the orientation and so is $\tilde{\Gamma}$, so
the result valid for both orientations is

$\displaystyle
2W_{\text{RC}}=\frac{\text{sgn}\left(m\right)o}{2}\frac{1}{96\pi}\int
Q_{3}\left(\tilde{\Gamma}\right)+\cdots$ (E.39)

where $o=\text{sgn}\left(\text{det}e\right)$ is the orientation. The second
and third lines in (E.30) correspond, with a relativistic regulator, to
$O\left(h^{2}\right)$ contributions to the cosmological constant and E-H term
which are UV sensitive.

One can compute the other traces in the same manner. The only additional UV
insensitive contribution comes from the second integral in (E.32). It is given
by

$\displaystyle 2W_{\text{RC}}=$
$\displaystyle\frac{\text{sgn}\left(m\right)}{2}\frac{1}{96\pi}\int\text{d}^{3}x4\omega\left(\partial_{\mu}\partial_{\nu}-\partial^{2}\eta_{\mu\nu}\right)2h^{\mu\nu}$
$\displaystyle+\cdots$ (E.40)

This corresponds to the expansion of the gpCS term to second order in the
vertices,

$\displaystyle
2W_{\text{RC}}=\frac{\text{sgn}\left(m\right)}{2}\frac{1}{96\pi}\int\text{d}^{3}x\left|e\right|\tilde{\mathcal{R}}c+\cdots$
(E.41)

where we have used (2.82), the expansion of the curvature
$\tilde{\mathcal{R}}=-2\left(\partial_{a}\partial_{b}-\partial^{2}\eta_{ab}\right)h^{ab}+O\left(h^{2}\right)$,
the definition
$c=\varepsilon^{abc}\left(\omega_{abc}-\tilde{\omega}_{abc}\right)$, and the
expansion
$\tilde{\omega}_{abc}\varepsilon^{abc}=-\varepsilon^{abc}\partial_{a}h_{bc}+O\left(h^{2}\right)$
of the LC spin connection. Note that in the Lorentz gauge
$h_{\left[ab\right]}=0$, $\tilde{\omega}_{abc}\varepsilon^{abc}$ vanishes to
first order. This completes the calculation of the UV insensitive terms in the
effective action which we have studied in Sec.2.

###  Appendix F Non-relativistic geometric quantities and their perturbative
expansion

We write $E_{A}^{\;i}=\delta_{A}^{\;i}+H_{A}^{\;i}$ for the inverse vielbein,
and expand the relevant geometric quantities in $H$. For the inverse metric
$G^{ij}=E_{A}^{\;i}\delta^{AB}E_{B}^{\;j}$ and volume element
$\sqrt{G}=\left|E\right|=\left|\text{det}\left(E_{A}^{\;i}\right)\right|$ we
find

$\displaystyle G^{ij}=$ $\displaystyle\delta^{ij}+2H^{(ij)}+H_{A}^{\;i}H^{Aj}$
(F.1) $\displaystyle=$ $\displaystyle\delta^{ij}+\delta G^{ij},$
$\displaystyle\sqrt{G}=$ $\displaystyle
1-H_{A}^{\;A}+\frac{1}{2}H_{A}^{\;A}H_{B}^{\;B}+\frac{1}{2}H_{A}^{\;B}H_{B}^{\;A}+O\left(H^{3}\right),$
$\displaystyle\log\sqrt{G}=$ $\displaystyle-
H_{A}^{\;A}+\frac{1}{2}H_{A}^{\;B}H_{B}^{\;A}+O\left(H^{3}\right),$

where, in expanded expressions, all index manipulations are trivial, and in
particular, there is no difference between coordinate indices $i,j$ and
$SO\left(2\right)_{L}$ indices $A,B$. Note that the strain used in Sec.3 is
given by
$u_{ij}=\left(G_{ij}-\delta_{ij}\right)/2=-H_{(ij)}+O\left(H^{2}\right)$. We
use the notation $\varepsilon^{\mu\nu\rho}$ for the totally anti-symmetric
(pseudo) tensor, normalized such that $\varepsilon^{xyt}=1/\sqrt{G}$, as well
as $\varepsilon^{ij}=\varepsilon^{ijt}$. The non-relativistic spin connection
used in Sec.3 is the $SO\left(2\right)_{L}$ connection

$\displaystyle\omega_{t}=$
$\displaystyle\frac{1}{2}\varepsilon^{AB}E_{Ai}\partial_{t}E_{B}^{\;i}$ (F.2)
$\displaystyle=$
$\displaystyle-\frac{1}{2}\partial_{t}\left(\varepsilon^{AB}H_{AB}\right)-\frac{1}{2}\varepsilon^{AB}H_{iA}\partial_{t}H_{B}^{\;i}+O\left(H^{3}\right),$
$\displaystyle\omega_{j}=$
$\displaystyle\frac{1}{2}\left(\varepsilon^{AB}E_{Ai}\partial_{j}E_{B}^{\;i}-\frac{1}{E}\varepsilon^{kl}\partial_{k}G_{lj}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\partial_{j}\left(\varepsilon^{AB}H_{AB}\right)-\partial_{\bot}^{l}H_{(lj)}-\frac{1}{2}\varepsilon^{AB}H_{iA}\partial_{j}H_{B}^{\;i}+O\left(H^{3}\right),$

where $\partial_{\bot}^{l}=\varepsilon^{lk}\partial_{k}$, which is obtained
naturally within Newton-Cartan geometry [164, 44]. This connection is torsion-
full, but has a vanishing “reduced torsion” [41]. In Sec.3, a term $B/2m$ was
implicitly added to $\omega_{t}$, but here we will add it explicitly when
writing expressions for $S_{\text{eff}}$ and $S_{\text{ind}}$. Such a term
appears in the presence of an additional background field $E_{0}^{\;i}$ which
couples to momentum density $P_{i}$ [41, 164], and can be identified with
$G^{ij}A_{j}/m$ in a Galilean invariant system, where $P_{i}=mG_{ij}J^{j}$.
The Ricci scalar is given by

$\displaystyle R=$ $\displaystyle 2\varepsilon^{ij}\partial_{i}\omega_{j}$
(F.3) $\displaystyle=$ $\displaystyle
2\partial_{\bot}^{i}\partial_{\bot}^{j}H_{ij}+O\left(H^{2}\right)$
$\displaystyle=$
$\displaystyle-2\left(\partial^{i}\partial^{j}-\partial^{2}\delta^{ij}\right)H_{ij}+O\left(H^{2}\right).$

###  Appendix G Odd viscosity at non-zero wave-vector: generalities

####  G.1 Definition and $T$ symmetry

We define the viscosity tensor as the linear response of stress to strain rate

$\displaystyle
T^{ij}\left(t,\mathbf{x}\right)=\int\text{d}t\text{d}^{2}\mathbf{x}^{\prime}\eta^{ij,kl}\left(t,\mathbf{x},t^{\prime},\mathbf{x}^{\prime}\right)\partial_{t^{\prime}}H_{kl}\left(t^{\prime},\mathbf{x}^{\prime}\right),$
(G.1)

where

$\displaystyle\eta^{[ij],kl}=0=\eta^{ij,[kl]}.$ (G.2)

In a translationally invariant system we can pass to Fourier components
$T^{ij}\left(\omega,\mathbf{q}\right)=i\omega\eta^{ij,kl}\left(\omega,\mathbf{q}\right)H_{kl}\left(\omega,\mathbf{q}\right)$.
By definition,
$\eta^{ij,kl}\left(t,\mathbf{x},t^{\prime},\mathbf{x}^{\prime}\right)$ is
real, and therefore

$\displaystyle\eta^{ij,kl}\left(\omega,\mathbf{q}\right)=$
$\displaystyle\eta^{ij,kl}\left(-\omega,-\mathbf{q}\right)^{*}.$ (G.3)

Under time reversal $T$,

$\displaystyle\eta^{ij,kl}\left(\omega,\mathbf{q}\right)\mapsto\eta_{T}^{ij,kl}\left(\omega,\mathbf{q}\right)=\eta^{kl,ij}\left(\omega,-\mathbf{q}\right).$
(G.4)

The even and odd viscosities are then defined by
$\eta_{\text{e},\text{o}}=\left(\eta\pm\eta_{T}\right)/2$, and satisfy
$\left(\eta_{\text{e},\text{o}}\right)_{T}=\pm\eta_{\text{e},\text{o}}$. More
explicitly,

$\displaystyle\eta_{\text{e}}^{ij,kl}\left(\omega,\mathbf{q}\right)=+\eta_{\text{e}}^{kl,ij}\left(\omega,-\mathbf{q}\right),$
(G.5)
$\displaystyle\eta_{\text{o}}^{ij,kl}\left(\omega,\mathbf{q}\right)=-\eta_{\text{o}}^{kl,ij}\left(\omega,-\mathbf{q}\right).$

We will see below that in isotropic (or $SO\left(2\right)$ invariant) systems
$\eta$ is even in $\mathbf{q}$, so that

$\displaystyle\eta_{\text{e}}^{ij,kl}\left(\omega,\mathbf{q}\right)=+\eta_{\text{e}}^{kl,ij}\left(\omega,\mathbf{q}\right),$
(G.6)
$\displaystyle\eta_{\text{o}}^{ij,kl}\left(\omega,\mathbf{q}\right)=-\eta_{\text{o}}^{kl,ij}\left(\omega,\mathbf{q}\right),$
(G.7)

which is identical to the definition of $\eta_{\text{e},\text{o}}$ at
$\mathbf{q}=0$ [173, 174, 175, 160, 176].

####  G.2 $SO\left(2\right)$ and $P$ symmetries

Complex tensors satisfying (G.2) and (G.7), in $2$ spatial dimensions, form a
vector space $V\cong\mathbb{C}^{3}$ which can be spanned by [174]

$\displaystyle\sigma^{ab}=2\sigma^{[a}\otimes\sigma^{b]},\;\;a,b=0,x,z,$ (G.8)

where $\sigma^{x},\sigma^{z}$ are the symmetric Pauli matrices, and
$\sigma^{0}$ is the identity matrix. Thus every odd viscosity tensor can be
written as

$\displaystyle\eta_{\text{o}}\left(\omega,\mathbf{q}\right)$
$\displaystyle=\eta_{xz}\left(\omega,\mathbf{q}\right)\sigma^{xz}+\eta_{x0}\left(\omega,\mathbf{q}\right)\sigma^{x0}+\eta_{z0}\left(\omega,\mathbf{q}\right)\sigma^{z0},$
(G.9)

with complex coefficients $\eta_{ab}\left(\omega,\mathbf{q}\right)$. Under a
rotation $R=e^{i\alpha\left(i\sigma^{y}\right)}\in SO\left(2\right)$ the
metric perturbation and stress tensor transform as

$\displaystyle H_{ij}\left(\omega,\mathbf{q}\right)\mapsto
R_{\;i}^{k}R_{\;j}^{l}H_{kl}\left(\omega,R^{-1}\cdot\mathbf{q}\right),$ (G.10)
$\displaystyle T^{ij}\left(\omega,\mathbf{q}\right)\mapsto
R_{\;k}^{i}R_{\;k}^{j}T^{kl}\left(\omega,R^{-1}\cdot\mathbf{q}\right),$

where $\left(R\cdot\mathbf{q}\right)^{i}=R_{\;j}^{i}q^{j}$. The same
transformation rules apply for $R\in O\left(2\right)$, which defines the
parity transformation $P$, in flat space. It follows that

$\displaystyle\eta^{ij,kl}\left(\omega,\mathbf{q}\right)\mapsto
R_{\;i^{\prime}}^{i}R_{\;j^{\prime}}^{j}R_{\;k^{\prime}}^{k}R_{\;l^{\prime}}^{l}\eta^{i^{\prime}j^{\prime},k^{\prime}l^{\prime}}\left(\omega,R^{-1}\cdot\mathbf{q}\right)$
(G.11)

under $O\left(2\right)$, which is compatible with (G.2), and the decomposition
$\eta=\text{$\eta$}_{\text{o}}+\eta_{\text{e}}$. In particular, equation
(G.11) shows that the viscosity tensor is $P$-even, or more accurately, a
tensor under $P$ rather than a pseudo-tensor. In an
$SO\left(2\right)$-invariant system, the viscosity tensor will also be
$SO\left(2\right)$-invariant

$\displaystyle\eta^{ij,kl}\left(\omega,\mathbf{q}\right)=R_{\;i^{\prime}}^{i}R_{\;j^{\prime}}^{j}R_{\;k^{\prime}}^{k}R_{\;l^{\prime}}^{l}\eta^{i^{\prime}j^{\prime},k^{\prime}l^{\prime}}\left(\omega,R^{-1}\cdot\mathbf{q}\right),\;R\in
SO\left(2\right).$ (G.12)

Note that this holds even when $SO\left(2\right)$ symmetry is spontaneously
broken, as in $\ell$-wave SFs. At $\mathbf{q}=\mathbf{0}$, there is a unique
tensor satisfying (G.12), namely

$\displaystyle\left(\sigma^{xz}\right)^{ij,kl}$
$\displaystyle=-\frac{1}{2}\left(\varepsilon^{ik}\delta^{jl}+\varepsilon^{jk}\delta^{il}+\varepsilon^{il}\delta^{jk}+\varepsilon^{jl}\delta^{ik}\right),$
(G.13)

leaving a single odd viscosity coefficient
$\eta_{xz}\left(\omega\right)=\eta_{\text{o}}^{\left(1\right)}\left(\omega\right)$
[173, 174, 175, 160, 176].

A non-zero $\mathbf{q}$, however, along with the tensors $\delta^{ij}$ and
$\varepsilon^{ij}$, can be used to construct additional
$SO\left(2\right)$-invariant odd viscosity tensors, beyond $\sigma^{xz}$. From
the data $\mathbf{q},\delta^{ij},\varepsilon^{ij}$, three linearly
independent, symmetric, rank-2 tensors can be constructed, which we take to be

$\displaystyle\left(\tau^{0}\right)^{ij}=$ $\displaystyle q^{2}\delta^{ij},$
(G.14) $\displaystyle\left(\tau^{x}\right)^{ij}=$
$\displaystyle-2q_{\bot}^{(i}q^{j)}/q^{2},$
$\displaystyle\left(\tau^{z}\right)^{ij}=$ $\displaystyle
2q^{i}q^{j}/q^{2}-\delta^{ij},$

where $q_{\bot}^{i}=\varepsilon^{ij}q_{j}$. The notation above is due to the
relation

$\displaystyle\begin{pmatrix}\tau^{x}\\\ \tau^{z}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\cos 2\theta&-\sin 2\theta\\\ \sin 2\theta&\cos
2\theta\end{pmatrix}\begin{pmatrix}\sigma^{x}\\\ \sigma^{z}\end{pmatrix}$
(G.15)
$\displaystyle=\frac{1}{q^{2}}\begin{pmatrix}q_{x}^{2}-q_{y}^{2}&-2q_{x}q_{y}\\\
2q_{x}q_{y}&q_{x}^{2}-q_{y}^{2}\end{pmatrix}\begin{pmatrix}\sigma^{x}\\\
\sigma^{z}\end{pmatrix},$

where $\theta=\text{arg}\left(\mathbf{q}\right)$, so that $\tau^{x},\tau^{z}$
are a rotated version of $\sigma^{x},\sigma^{z}$. Moreover, all three $\tau$s
are $SO\left(2\right)$-invariant,
$\tau^{ij}\left(\mathbf{q}\right)=R_{\;i^{\prime}}^{i}R_{\;j^{\prime}}^{j}\tau^{i^{\prime}j^{\prime}}\left(R^{-1}\cdot\mathbf{q}\right)$,
and can therefore be used to construct three $SO\left(2\right)$-invariant odd
viscosity tensors

$\displaystyle\tau^{ab}=2\tau^{[a}\otimes\tau^{b]},\;a,b=0,x,z,$ (G.16)

which form a basis for $V$. Any odd viscosity tensor (at
$\mathbf{q}\neq\mathbf{0}$) can then be written as

$\displaystyle\eta_{\text{o}}\left(\omega,\mathbf{q}\right)$
$\displaystyle=\eta_{\text{o}}^{\left(1\right)}\left(\omega,\mathbf{q}\right)\tau^{xz}+\eta_{\text{o}}^{\left(2\right)}\left(\omega,\mathbf{q}\right)\tau^{0x}+\eta_{\text{o}}^{\left(3\right)}\left(\omega,\mathbf{q}\right)\tau^{0z}.$
(G.17)

Furthermore, for an $SO\left(2\right)$-invariant $\eta_{\text{o}}$, the
coefficients
$\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)},\eta_{\text{o}}^{\left(3\right)}$
depend on $\mathbf{q}$ through its norm, owing to the
$SO\left(2\right)$-invariance of $\tau^{ab}$. We therefore arrive at the
general form of an $SO\left(2\right)$-invariant odd viscosity tensor,

$\displaystyle\eta_{\text{o}}\left(\omega,\mathbf{q}\right)=$
$\displaystyle\eta_{\text{o}}^{\left(1\right)}\left(\omega,q^{2}\right)\tau^{xz}+\eta_{\text{o}}^{\left(2\right)}\left(\omega,q^{2}\right)\tau^{0x}+\eta_{\text{o}}^{\left(3\right)}\left(\omega,q^{2}\right)\tau^{0z}.$
(G.18)

In particular, we see that $\eta_{\text{o}}$ is even in $\mathbf{q}$ (and the
same applies also to the even viscosity $\eta_{\text{e}}$). To determine the
small $\omega,\mathbf{q}$ behavior of the coefficients we change to the
$\mathbf{q}$-independent basis of $\sigma$s,

$\displaystyle\eta_{\text{o}}\left(\omega,\mathbf{q}\right)=$
$\displaystyle\eta_{\text{o}}^{\left(1\right)}\left(\omega,q^{2}\right)\sigma^{xz}$
(G.19)
$\displaystyle+\left[\eta_{\text{o}}^{\left(2\right)}\left(\omega,q^{2}\right)\left(q_{x}^{2}-q_{y}^{2}\right)+\eta_{\text{o}}^{\left(3\right)}\left(\omega,q^{2}\right)\left(2q_{x}q_{y}\right)\right]\sigma^{0x}$
$\displaystyle+\left[\eta_{\text{o}}^{\left(2\right)}\left(\omega,q^{2}\right)\left(-2q_{x}q_{y}\right)+\eta_{\text{o}}^{\left(3\right)}\left(\omega,q^{2}\right)\left(q_{x}^{2}-q_{y}^{2}\right)\right]\sigma^{0z}.$

In gapped systems (such as QH states) $\eta_{\text{o}}$ will be regular around
$\omega=0=q$, and so will the coefficients
$\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)},\eta_{\text{o}}^{\left(3\right)}$.
In gapless systems (such as $\ell$-wave SFs) there will be a singularity at
$\omega=0=q$, but the limit $q\rightarrow 0$ at $\omega\neq 0$ will be
regular. In both cases, the limit $q\rightarrow 0$ at $\omega\neq 0$ of (G.19)
reduces to the known result
$\eta_{\text{o}}\left(\omega,\mathbf{0}\right)=\eta_{\text{o}}^{\left(1\right)}\left(\omega,0\right)\sigma^{xz}$
[173, 174, 175, 160, 176].

####  G.3 $PT$ symmetry

The combination $PT$ of parity and time-reversal is a symmetry in any system
in which $T$ is broken (perhaps spontaneously) due to some kind of angular
momentum, as in QH states, $\ell$-wave SFs, and active chiral fluids [183].
Here we consider the implications of $PT$ symmetry on (G.19).

From the definition (G.14) it is clear that $\tau^{0}$ and $\tau^{z}$ are
$P$-even, while $\tau^{x}$ is $P$-odd. Therefore, $\tau^{xz},\tau^{0x}$ are
$P$-odd while $\tau^{0z}$ is $P$-even (and all three are $T$-even). Since
$\eta_{\text{o}}$ is $T$-odd and $P$-even, and using (G.18), it follows that
$\eta_{\text{o}}^{\left(1\right)}$ and $\eta_{\text{o}}^{\left(2\right)}$ are
$P,T$-odd, while $\eta_{\text{o}}^{\left(3\right)}$ is $T$-odd but $P$-even.
In particular, $\eta_{\text{o}}^{\left(3\right)}$ is $PT$-odd, and must vanish
in $PT$-symmetric systems. The odd viscosity tensor in $SO\left(2\right)$ and
$PT$ symmetric systems is therefore given by

$\displaystyle\eta_{\text{o}}\left(\omega,\mathbf{q}\right)=$
$\displaystyle\eta_{\text{o}}^{\left(1\right)}\left(\omega,q^{2}\right)\sigma^{xz}$
(G.20)
$\displaystyle+\eta_{\text{o}}^{\left(2\right)}\left(\omega,q^{2}\right)\left[\left(q_{x}^{2}-q_{y}^{2}\right)\sigma^{0x}-2q_{x}q_{y}\sigma^{0z}\right].$

This form is confirmed by previous results for QH states [34], and by the
results presented in Sec.3 for $\ell$-wave SFs. The same form is obtained at
$\mathbf{q}=\mathbf{0}$, but in the presence of vector, or pseudo-vector,
anisotropy $\mathbf{b}$, in which case we find

$\displaystyle\eta_{\text{o}}\left(\omega\right)=$
$\displaystyle\eta_{\text{o}}^{\left(1\right)}\left(\omega\right)\sigma^{xz}$
(G.21)
$\displaystyle+\eta_{\text{o}}^{\left(2\right)}\left(\omega\right)\left[\left(b_{x}^{2}-b_{y}^{2}\right)\sigma^{0x}-2b_{x}b_{y}\sigma^{0z}\right],$

which explains the tensor structure found in [263, 264].

####  G.4 Frequency dependence and reality conditions

In closed and clean systems, like the $\ell$-wave SFs studied in Sec.3, the
viscosity can be obtained from an induced action

$\displaystyle S_{\text{ind}}$
$\displaystyle\supset\frac{1}{2}\int\text{d}t\text{d}t^{\prime}\text{d}^{2}\mathbf{x}\text{d}^{2}\mathbf{x}^{\prime}H_{ij}\left(t,\mathbf{x}\right)\eta^{ij,kl}\left(t-t^{\prime},\mathbf{x}-\mathbf{x}^{\prime}\right)\partial_{t^{\prime}}H_{kl}\left(t^{\prime},\mathbf{x}^{\prime}\right)$
(G.22)
$\displaystyle=\frac{1}{2}\int\frac{\text{d}\omega}{2\pi}\frac{\text{d}^{2}\mathbf{q}}{\left(2\pi\right)^{2}}H_{ij}\left(-\omega,-\mathbf{q}\right)i\omega\eta^{ij,kl}\left(\omega,\mathbf{q}\right)H_{kl}\left(\omega,\mathbf{q}\right).$

As a result, $\eta$ satisfies the additional property,

$\displaystyle\eta^{ij,kl}\left(\omega,\mathbf{q}\right)=$
$\displaystyle-\eta^{kl,ij}\left(-\omega,-\mathbf{q}\right),$ (G.23)

which, along with (G.7)-(G.6) and the fact that $\eta$ is even in
$\mathbf{q}$, implies that $\eta_{\text{o}}$ ($\eta_{\text{e}}$) is even (odd)
in $\omega$,

$\displaystyle\eta_{\text{e}}^{ij,kl}\left(\omega,\mathbf{q}\right)=$
$\displaystyle-\eta_{\text{e}}^{ij,kl}\left(-\omega,\mathbf{q}\right),$ (G.24)
$\displaystyle\eta_{\text{o}}^{ij,kl}\left(\omega,\mathbf{q}\right)=$
$\displaystyle+\eta_{\text{o}}^{ij,kl}\left(-\omega,\mathbf{q}\right).$

This result, along with (G.3) and the fact that $\eta$ is even in
$\mathbf{q}$, implies that $\eta_{\text{o}}$ ($\eta_{\text{e}}$) is real
(imaginary),

$\displaystyle\eta_{\text{e}}^{ij,kl}\left(\omega,\mathbf{q}\right)\in$
$\displaystyle i\mathbb{R},$ (G.25)
$\displaystyle\eta_{\text{o}}^{ij,kl}\left(\omega,\mathbf{q}\right)\in$
$\displaystyle\mathbb{R}.$

These general properties are satisfied by the odd viscosity tensor computed in
Sec.3. These are also compatible with the examples worked out in [175], as
well with viscosity-conductivity relations that hold in Galilean invariant
systems (in conjugation with known properties of the conductivity) [248, 175,
163].

We note that some care is required when interpreting (G.24)-(G.25) around
singularities of $\eta$. For example, the first equation in (G.24) naively
implies that $\eta_{\text{e}}\left(0,\mathbf{q}\right)=0$, which in particular
implies that the bulk and shear viscosities
$\eta_{\text{e}}\left(0,\mathbf{0}\right)=\zeta\sigma^{0}\otimes\sigma^{0}+\eta^{\text{s}}\left(\sigma^{x}\otimes\sigma^{z}+\sigma^{z}\otimes\sigma^{x}\right)$
vanish in the closed, clean, case. This however, is not quite correct, due to
a possible singularity of $\eta_{\text{e}}$ at $\omega=0$, as well as the
usual infinitesimal imaginary part of $\omega$ required to obtain the retarded
response. For example, for free fermions, reference [175] finds
$\eta^{\text{s}}\left(\omega,\mathbf{0}\right)\sim\frac{i}{\omega+i\epsilon}=\pi\delta\left(\omega\right)+i\text{PV}\frac{1}{\omega}$
(where PV is the principle value), which has an infinite real part at
$\omega=0$, in analogy with the Drude behavior of the conductivity.

####  G.5 Odd viscosity from Gaussian integration: a technical result

We now restrict attention to $\ell$-wave SFs. The effective Lagrangian,
perturbatively expanded to second order, and in the absence of the
$U\left(1\right)$ background, takes the form

$\displaystyle\mathcal{L}_{\text{eff}}=$
$\displaystyle\frac{1}{2}\theta\mathcal{G}^{-1}\theta+\mathcal{V}\theta+\mathcal{C},$
(G.26)

where the Green’s function $\mathcal{G}$ is independent of $H$, the vertex
$\mathcal{V}$ is linear in $H$, and the contact term $\mathcal{C}$ is
quadratic in $H$. Performing Gaussian integration over $\theta$ yields the
induced Lagrangian

$\displaystyle\text{$\mathcal{L}$}_{\text{ind}}=$
$\displaystyle-\frac{1}{2}\mathcal{V}\mathcal{G}\mathcal{V}+\mathcal{C},$
(G.27)

and comparing with (G.22) one can read off $\eta_{\text{o}}$. In Appendix H we
write explicit expressions for a Galilean invariant
$\mathcal{L}_{\text{eff}}$, which we then expand to obtain explicit
expressions for $\mathcal{G}^{-1},\mathcal{V},\mathcal{C}$. Appendix I then
describes the resulting $\mathcal{L}_{\text{ind}}$. Here we take a
complementary approach and obtain the general form of $\eta_{\text{o}}$ from
(G.27), using the formalism developed above, based only on $SO\left(2\right)$
and $PT$ symmetries.

The motivation for the analysis in this section is the following. The power
counting described in Sec.3.2 is designed such that the $O\left(p^{n}\right)$
Lagrangian $\mathcal{L}_{n}\subset\mathcal{L}_{\text{eff}}$ produces
$O\left(p^{n}\right)$ contributions to $\mathcal{L}_{\text{ind}}$. Therefore,
naively, one expects the $O\left(q^{2}\right)$ odd viscosity to depend on
$\mathcal{L}_{0},\mathcal{L}_{2},$ and $\mathcal{L}_{3}$ (since
$\mathcal{L}_{1}=0$). Using the notation
$\eta_{\text{o}}=\eta_{\mathcal{V}}+\eta_{\mathcal{C}}$ for the parts of
$\eta_{\text{o}}$ due to $-\mathcal{V}\mathcal{G}\mathcal{V}/2$ and
$\mathcal{C}$, respectively, the result of this section is that
$\eta_{\mathcal{V}}$, to $O\left(q^{2}\right)$, is actually independent of
$\mathcal{L}_{3}$.

We now describe the details. For $\eta_{\mathcal{C}}$, we cannot do better
than the general discussion thus far - it is given by (G.19), with
$\eta_{\text{o}}^{\left(3\right)}=0$, and both
$\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)}$ are real
and regular at $\omega=0=q$, since $\mathcal{L}_{\text{eff}}$, and
$\mathcal{C}$ in particular, are obtained by integrating out gapped degrees of
freedom (the Higgs modes and the fermion $\psi$). For $\eta_{\mathcal{V}}$,
however, we can do better. We first write more explicitly

$\displaystyle\theta\mathcal{G}^{-1}\theta$
$\displaystyle=\frac{1}{2}\theta\left(-\omega,-\mathbf{q}\right)\mathcal{G}^{-1}\left(\omega,\mathbf{q}\right)\theta\left(\omega,\mathbf{q}\right),$
(G.28) $\displaystyle\mathcal{V}\theta$
$\displaystyle=\theta\left(-\omega,-\mathbf{q}\right)V^{ij}\left(\omega,\mathbf{q}\right)H_{ij}\left(\omega,\mathbf{q}\right).$

Based on $SO\left(2\right)$ and $PT$ symmetries, the objects
$\mathcal{G}^{-1},V^{ij}$ take the forms

$\displaystyle\mathcal{G}^{-1}\left(\omega,\mathbf{q}\right)=$ $\displaystyle
D\left(\omega^{2},q^{2}\right),$ (G.29) $\displaystyle
V^{ij}\left(\omega,\mathbf{q}\right)=$ $\displaystyle i\omega
a\left(\omega^{2},q^{2}\right)\left(\rho^{0}\right)^{ij}+i\omega
b\left(\omega^{2},q^{2}\right)\left(\rho^{z}\right)^{ij}+s_{\theta}c\left(\omega^{2},q^{2}\right)\left(\rho^{x}\right)^{ij},$

where

$\displaystyle\left(\rho^{0}\right)^{ij}=$ $\displaystyle\delta^{ij},$ (G.30)
$\displaystyle\left(\rho^{x}\right)^{ij}=$ $\displaystyle
q_{\bot}^{(i}q^{j)},$ $\displaystyle\left(\rho^{z}\right)^{ij}=$
$\displaystyle q^{i}q^{j},$

are, in this context, more convenient than the $\tau$s (G.14), and $a,b,c,D$
are general functions of their arguments which are $P,T$-even, real, and
regular at $\omega=0=q$, as follows from the same properties of
$\mathcal{L}_{\text{eff}}$. In particular, we will use the following
expansions

$\displaystyle a\left(0,q^{2}\right)$
$\displaystyle=a_{0}+a_{1}q^{2}+O\left(q^{4}\right),$ (G.31) $\displaystyle
b\left(0,q^{2}\right)$ $\displaystyle=b_{0}+O\left(q^{2}\right),$
$\displaystyle c\left(0,q^{2}\right)$
$\displaystyle=c_{0}+c_{1}q^{2}+O\left(q^{4}\right),$ $\displaystyle
D\left(0,q^{2}\right)$
$\displaystyle=D_{1}q^{2}+D_{2}q^{4}+O\left(q^{6}\right),$

where $D_{0}=0$ because $\theta$ enters $\mathcal{L}_{\text{eff}}$ only
through its derivatives. The odd viscosity $\eta_{\mathcal{V}}$ is then given
by

$\displaystyle\eta_{\mathcal{V}}\left(\omega,\mathbf{q}\right)=$
$\displaystyle-\frac{1}{2i\omega}\frac{V\left(-\omega,-\mathbf{q}\right)\otimes
V\left(\omega,\mathbf{q}\right)-V\left(\omega,\mathbf{q}\right)\otimes
V\left(-\omega,-\mathbf{q}\right)}{D\left(\omega,\mathbf{q}\right)}$ (G.32)
$\displaystyle=$
$\displaystyle\frac{2s_{\theta}c\left(\omega^{2},q^{2}\right)}{D\left(\omega^{2},q^{2}\right)}\left[a\left(\omega^{2},q^{2}\right)\rho^{0x}+b\left(\omega^{2},q^{2}\right)\rho^{zx}\right],$

which is of the form (G.19), with $\eta_{\text{o}}^{\left(3\right)}=0$ and

$\displaystyle\eta_{\mathcal{V}}^{\left(1\right)}\left(\omega,q^{2}\right)=$
$\displaystyle-\frac{s_{\theta}c\left(\omega^{2},q^{2}\right)}{2D\left(\omega^{2},q^{2}\right)}b\left(\omega^{2},q^{2}\right)q^{4},$
(G.33)
$\displaystyle\eta_{\text{$\mathcal{V}$}}^{\left(2\right)}\left(\omega,q^{2}\right)=$
$\displaystyle\frac{s_{\theta}c\left(\omega^{2},q^{2}\right)}{D\left(\omega^{2},q^{2}\right)}\left[a\left(\omega^{2},q^{2}\right)+b\left(\omega^{2},q^{2}\right)q^{2}\right].$

Setting $\omega=0$ and expanding in $q$, we find

$\displaystyle\eta_{\text{$\mathcal{V}$}}^{\left(1\right)}\left(0,q^{2}\right)=$
$\displaystyle-\frac{s_{\theta}c_{0}b_{0}}{2D_{1}}q^{2}+O\left(q^{4}\right),$
(G.34)
$\displaystyle\eta_{\text{$\mathcal{V}$}}^{\left(2\right)}\left(0,q^{2}\right)=$
$\displaystyle\frac{s_{\theta}}{D_{1}}\left[a_{0}c_{0}q^{-2}+\left(a_{0}c_{1}+a_{1}c_{0}+b_{0}c_{0}-a_{0}c_{0}\frac{D_{2}}{D_{1}}\right)\right]+O\left(q^{2}\right).$

Having identified the coefficients $a_{0},a_{1},b_{0},c_{0},c_{1},D_{1},D_{2}$
that determine $\eta_{\mathcal{V}}$ to $O\left(q^{2}\right)$, we now determine
the order in the derivative expansion of $\mathcal{L}_{\text{eff}}$ in which
these enter. Explicitly, the above coefficients are defined by

$\displaystyle\mathcal{L}_{\text{eff}}\supset$
$\displaystyle\frac{1}{2}\theta\left(-\omega,-\mathbf{q}\right)\left(D_{1}q^{2}+D_{2}q^{4}\right)\theta\left(\omega,\mathbf{q}\right)$
(G.35)
$\displaystyle+\theta\left(-\omega,-\mathbf{q}\right)\left[i\omega\left(a_{0}+a_{1}q^{2}\right)\delta^{ij}+i\omega
b_{0}q^{i}q^{j}+s_{\theta}\left(c_{0}+c_{1}q^{2}\right)q^{i}q_{\bot}^{j}\right]H_{ij}\left(\omega,\mathbf{q}\right).$

We see that $c_{1}$ enters $\mathcal{L}_{\text{eff}}$ at
$O\left(p^{3}\right)$, while all other coefficients enter at a lower order,
and come from $\mathcal{L}_{0},\mathcal{L}_{2}$. In particular,
$\eta_{\text{$\mathcal{V}$}}^{\left(1\right)}$ in (G.34) is independent of
$\mathcal{L}_{3}$. Even though $c_{1}$ is the coefficient of an
$O\left(p^{3}\right)$ term, it is actually due to $\mathcal{L}_{2}$. Using
(F.2) we identify $c_{0}\theta
q^{2}q^{i}q_{\bot}^{j}H_{ij}=-s_{\theta}c_{1}\partial^{i}\theta\partial^{2}\omega_{i}$,
which must be a part of

$\displaystyle\frac{c_{1}}{2}\left(\partial_{i}\theta-
A_{i}-s_{\theta}\omega_{i}\right)\partial^{2}\left(\partial_{i}\theta-
A_{i}-s_{\theta}\omega_{i}\right).$ (G.36)

This is an $O\left(p^{2}\right)$ term, and in fact comes from
$\mathcal{L}_{2}^{\left(2\right)}\subset\mathcal{L}_{2}$, see (H.7). Thus,
both
$\eta_{\text{$\mathcal{V}$}}^{\left(1\right)},\eta_{\text{$\mathcal{V}$}}^{\left(2\right)}$
in (G.34) are completely independent of $\mathcal{L}_{3}$.

###  Appendix H Effective action and its perturbative expansion

####  H.1 Zeroth order

It is useful to write the zeroth order scalar $X$ as

$\displaystyle X=$
$\displaystyle\left(\partial_{t}\theta-\mathcal{A}_{t}-\frac{s_{\theta}}{2m}B\right)-\frac{1}{2m}G^{ij}\left(\partial_{i}\theta-\mathcal{A}_{i}\right)\left(\partial_{j}\theta-\mathcal{A}_{j}\right),$
(H.1)

where

$\displaystyle\mathcal{A}_{\mu}$
$\displaystyle=A_{\mu}+s_{\theta}\omega_{\mu}.$ (H.2)

We will also use
$\mathcal{B}=B+\frac{s_{\theta}}{2}R,\;\mathcal{E}_{i}=E_{i}+s_{\theta}E_{\omega,i}$
for the magnetic and electric fields obtained from $\mathcal{A}_{\mu}$, where
$E_{\omega,i}=\partial_{t}\omega_{i}-\partial_{i}\omega_{t}$. Expanding
$\mathcal{L}_{0}=P\left(X\right)$ to second order in the fields, one finds (up
to total derivatives)

$\displaystyle\sqrt{G}\mathcal{L}_{0}=$
$\displaystyle\frac{1}{2}\frac{n_{0}}{m}\theta\left[\partial^{2}-c_{s}^{-2}\partial_{t}^{2}\right]\theta$
(H.3)
$\displaystyle+\left[-\frac{n_{0}}{m}\left(\partial_{i}\mathcal{A}^{i}-c_{s}^{-2}\partial_{t}\left(\mathcal{A}_{t}+\frac{s_{\theta}}{2m}B\right)\right)-n_{0}\partial_{t}\sqrt{G}\right]\theta$
$\displaystyle+\left[-n_{0}\sqrt{G}\mathcal{A}_{t}-\frac{1}{2}\frac{n_{0}}{m}\left(\mathcal{A}^{2}-c_{s}^{-2}\left(\mathcal{A}_{t}+\frac{s_{\theta}}{2m}B\right)^{2}\right)+P_{0}\sqrt{G}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\theta\mathcal{G}^{-1}\theta+\mathcal{V}\theta+\mathcal{C},$

where
$\partial^{2}=\partial^{i}\partial_{i},\;\mathcal{A}^{2}=\mathcal{A}_{i}\mathcal{A}^{i}$,
and we defined the inverse Green’s function $\mathcal{G}^{-1}$, vertex
$\mathcal{V}$, and contact terms $\mathcal{C}$, respectively. These are used
in section I below to obtain $S_{\text{ind}}$.

In (H.3), the geometric objects $\sqrt{G}$ and $\omega_{\mu}$ should be
interpreted as expanded to the required order according to (F.1)-(F.2). In
particular, the term $-n_{0}\sqrt{G}\mathcal{A}_{t}$ includes
$-s_{\theta}n_{0}\sqrt{G}\omega_{t}$, which produces the leading contribution
to $\eta_{\text{o}}^{\left(1\right)}$. To see this, we expand

$\displaystyle\sqrt{G}\omega_{t}=$
$\displaystyle-\frac{1}{2}\partial_{t}\left(\varepsilon^{AB}H_{AB}\right)+\frac{1}{2}\partial_{t}\left(\varepsilon^{AB}H_{AB}\right)H_{i}^{\;i}-\frac{1}{2}\varepsilon^{AB}H_{iA}\partial_{t}H_{B}^{\;i}+O\left(H^{3}\right)$
(H.4) $\displaystyle=$
$\displaystyle-\frac{1}{2}\partial_{t}\left(\varepsilon^{AB}H_{AB}\right)-\frac{1}{2}\varepsilon^{AB}H_{Ai}\partial_{t}H_{B}^{\;i}+O\left(H^{3}\right),$

which is identical to the expansion (F.1) of $\omega_{t}$, apart from
$H_{iA}\leftrightarrow H_{Ai}$. Ignoring total derivatives, this reduces to

$\displaystyle\sqrt{G}\mathcal{L}_{0}$ $\displaystyle\supset-
s_{\theta}n_{0}\sqrt{G}\omega_{t}$ (H.5)
$\displaystyle=-\frac{1}{2}s_{\theta}n_{0}\left[\partial_{t}\left(\varepsilon^{AB}H_{AB}\right)H_{i}^{\;i}-\varepsilon^{AB}\delta^{ij}H_{(Ai)}\partial_{t}H_{(Bj)}\right]+O\left(H^{3}\right)$
$\displaystyle=\frac{1}{2}s_{\theta}n_{0}\varepsilon^{AB}H_{Ai}\partial_{t}H_{B}^{\;i}+O\left(H^{3}\right).$

Comparing with (G.13) and (G.22), the second term in the second line
corresponds to $\eta_{\text{o}}^{\left(1\right)}=-s_{\theta}n_{0}/2$. The
first term in the second line depends on the anti-symmetric part of $H$, and
shows that the full expression (H.5) actually corresponds to a torsional Hall
(or odd) viscosity [155, 156] $\zeta_{H}=-s_{\theta}n_{0}$, which can be read
off from the third line. The appearance of the torsional Hall viscosity at the
level of $S_{\text{eff}}$ (but not at the level of $S_{\text{ind}}$, see
section I) can be understood from the mapping of [1] of the $p$-wave SF to a
Majorana spinor in Riemann-Cartan space-time.

####  H.2 Second order

The full expression for $\mathcal{L}_{2}$ is given by
$\mathcal{L}_{2}=\sum_{i=1}^{6}\mathcal{L}_{2}^{\left(i\right)}$ , where [199]

$\displaystyle\mathcal{L}_{2}^{\left(1\right)}=$ $\displaystyle
F_{1}\left(X\right)R,$ (H.6) $\displaystyle\mathcal{L}_{2}^{\left(2\right)}=$
$\displaystyle
F_{2}\left(X\right)\left(mK_{\;i}^{i}-\nabla^{2}\theta\right)^{2},$
$\displaystyle\mathcal{L}_{2}^{\left(3\right)}=$ $\displaystyle
F_{3}\left(X\right)\left\\{-m^{2}\left(G^{ij}\partial_{t}K_{ij}-K^{ij}K_{ij}\right)-m\nabla_{i}E^{i}+\frac{1}{4}F^{ij}F_{ij}\right.$
$\displaystyle\left.+2m\left[\partial_{i}K_{\;j}^{j}-\nabla^{j}\left(K_{ji}+\frac{1}{2m}F_{ji}\right)\right]\nabla^{i}\theta+R_{ij}\nabla_{i}\theta\nabla_{j}\theta\right\\},$
$\displaystyle\mathcal{L}_{2}^{\left(4\right)}=$ $\displaystyle
F_{4}\left(X\right)G^{ij}\partial_{i}X\partial_{j}X,$
$\displaystyle\mathcal{L}_{2}^{\left(5\right)}=$ $\displaystyle
F_{5}\left(X\right)\left[\left(\partial_{t}-\frac{1}{m}\nabla^{i}\theta\partial_{i}\right)X\right]^{2},$
$\displaystyle\mathcal{L}_{2}^{\left(6\right)}=$ $\displaystyle
F_{6}\left(X\right)\left(mK_{\;i}^{i}-\nabla^{2}\theta\right)\left[\left(\partial_{t}-\frac{1}{m}\nabla^{i}\theta\partial_{i}\right)X\right].$

The terms $\mathcal{L}_{2}^{\left(5\right)}$ and
$\mathcal{L}_{2}^{\left(6\right)}$ were not written explicitly in [199]
because, on shell (on the equation of motion for $\theta$), they are
proportional to $\mathcal{L}_{2}^{\left(4\right)}$ up to $O\left(p^{4}\right)$
corrections, and can therefore be eliminated by a redefinition of $F_{4}$.
However, for the purpose of comparing the general $S_{\text{eff}}$ with the
microscopic expression (J.19), it is convenient to work off shell and keep all
terms explicit.

Specializing to 2+1 dimensions and expanding to second order in fields, one
finds

$\displaystyle\sqrt{G}\mathcal{L}_{2}^{\left(1\right)}=$ $\displaystyle
F_{1}^{\prime}\left(\mu\right)R\left(\partial_{t}\theta-\mathcal{A}_{t}-\frac{s_{\theta}}{2m}B\right).$
(H.7) $\displaystyle\sqrt{G}\mathcal{L}_{2}^{\left(2\right)}=$ $\displaystyle
F_{2}\left(\mu\right)\left[-m^{2}H_{i}^{\;i}\partial_{t}^{2}H_{j}^{\;j}+2m\partial_{t}H_{k}^{\;k}\partial^{j}\left(\partial_{j}\theta-\mathcal{A}_{j}\right)-\left(\partial_{i}\theta-\mathcal{A}_{i}\right)\partial^{i}\partial^{j}\left(\partial_{i}\theta-\mathcal{A}_{i}\right)\right],$
$\displaystyle\sqrt{G}\mathcal{L}_{2}^{\left(3\right)}=$ $\displaystyle
F_{3}\left(\mu\right)\left(m^{2}H^{(ij)}\partial_{t}^{2}H_{\left(ij\right)}+\frac{1}{2}B^{2}-2m\varepsilon^{ij}\omega_{i}\partial_{t}\left(\partial_{j}\theta-\mathcal{A}_{j}\right)-B\mathcal{B}\right)$
$\displaystyle+F_{3}^{\prime}\left(\mu\right)\left(\partial_{t}\theta-\mathcal{A}_{t}-\frac{s_{\theta}}{2m}B\right)\left(m^{2}\partial_{t}^{2}H_{i}^{\;i}-m\partial_{i}E^{i}\right),$
$\displaystyle\sqrt{G}\mathcal{L}_{2}^{\left(4\right)}=$ $\displaystyle-
F_{4}\left(\mu\right)\left(\partial_{t}\theta-\mathcal{A}_{t}-\frac{s_{\theta}}{2m}B\right)\partial^{2}\left(\partial_{t}\theta-\mathcal{A}_{t}-\frac{s_{\theta}}{2m}B\right),$
$\displaystyle\sqrt{G}\mathcal{L}_{2}^{\left(5\right)}=$ $\displaystyle-
F_{5}\left(\mu\right)\left(\partial_{t}\theta-\mathcal{A}_{t}-\frac{s_{\theta}}{2m}B\right)\partial_{t}^{2}\left(\partial_{t}\theta-\mathcal{A}_{t}-\frac{s_{\theta}}{2m}B\right),$
$\displaystyle\sqrt{G}\mathcal{L}_{2}^{\left(6\right)}=$ $\displaystyle-
F_{6}\left(\mu\right)\left[m\partial_{t}H_{i}^{\;i}+\partial^{j}\left(\partial_{j}\theta-\mathcal{A}_{j}\right)\right]\partial_{t}\left(\partial_{t}\theta-\mathcal{A}_{t}-\frac{s_{\theta}}{2m}B\right),$

from which one can easily extract the second order corrections to
$\mathcal{G}^{-1},\mathcal{V},\mathcal{C}$, of (H.3). Note that
$\mathcal{L}_{2}^{\left(3\right)}$ includes a term
$\propto\varepsilon^{ij}\omega_{i}\partial_{t}\mathcal{A}_{j}=\varepsilon^{ij}\omega_{i}\partial_{t}\left(A_{j}+s_{\theta}\omega_{j}\right)$.
Comparing with (H.8) below, it is clear that distinguishing
$\mathcal{L}_{2}^{\left(3\right)}$ from $\mathcal{L}_{\text{gCS}}$ is non-
trivial. This is in fact the same problem of extracting the central charge
from the Hall viscosity addressed in Sec.3.4, but at the level of
$S_{\text{eff}}$ (where $\theta$ is viewed as a background field) rather than
$S_{\text{ind}}$ (where $\theta$ has been integrated out). Accordingly, the
central charge can be computed by applying Eq.(3.17) to the response functions
obtained from $S_{\text{eff}}$. Additionally, relying on LGS, one can extract
$F_{3}$ as the coefficient of $H^{(ij)}\partial_{t}^{2}H_{\left(ij\right)}$.
Both approaches produce the same central charge (3.11) in the perturbative
computation of Sec.3.3 and Appendix J.5.

####  H.3 Gravitational Chern-Simons term

The gCS Lagrangian is given explicitly by

$\displaystyle\mathcal{L}_{\text{gCS}}$
$\displaystyle=-\frac{c}{48\pi}\left[\left(\omega_{t}+\frac{B}{2m}\right)R-\varepsilon^{ij}\omega_{i}\partial_{t}\omega_{j}\right]$
(H.8)
$\displaystyle=-\frac{c}{48\pi}\left[\omega\text{d}\omega+\frac{1}{2m}BR\right].$

Its expansion to second order in fields, using (F.1)-(F.2), is

$\displaystyle\sqrt{G}\mathcal{L}_{\text{gCS}}$
$\displaystyle=-\frac{c}{48\pi}\left[\varepsilon^{AB}H_{\left(Ai\right)}\partial_{\bot}^{i}\partial_{\bot}^{j}\partial_{t}H_{\left(Bj\right)}-\frac{1}{m}A_{i}\partial_{\bot}^{i}\partial_{\bot}^{j}\partial_{\bot}^{k}H_{(jk)}\right].$
(H.9)

As opposed to $\sqrt{G}\omega_{t}$, the gCS term $\mathcal{L}_{\text{gCS}}$ is
(locally) $SO\left(2\right)_{L}$ gauge invariant, and accordingly depends only
on the metric, or, within the perturbative expansion, on the symmetric part
$H_{(ij)}$. From this expansion one can read off the gCS contributions to the
odd viscosity $\eta_{\text{o}}$ , and to the odd, mixed, static susceptibility
$\chi_{TJ,\text{o}}$, described in Sec.3.4.

####  H.4 Additional terms at third order

To obtain reliable results at $O\left(p^{3}\right)$ we, in principle, need the
full Lagrangian $\mathcal{L}_{3}$, which includes, but is not equal to,
$\mathcal{L}_{\text{gCS}}$. Nevertheless, we argue that
$\mathcal{L}_{3}-\mathcal{L}_{\text{gCS}}$ does not contribute to the quantity
of interest in this section - $\eta_{\text{o}}$ to $O\left(q^{2}\right)$. We
already demonstrated in Appendix G.5 that the vertex part of the odd viscosity
$\eta_{\mathcal{V}}$ is independent of $\mathcal{L}_{3}$, and it remains to
show that the contact term part $\eta_{\mathcal{C}}$ is independent of
$\mathcal{L}_{3}-\mathcal{L}_{\text{gCS}}$. We do not have a general proof,
but we address this issue in two ways:

  1. 1.

Within the microscopic model (3.8), the perturbative computation of Appendix
J.5 provides an explicit expression for $\eta_{\mathcal{C}}$, which is
completely saturated by the effective action presented thus far. Thus
$\eta_{\mathcal{C}}$ is independent of
$\mathcal{L}_{3}-\mathcal{L}_{\text{gCS}}$ in the particular realization
(3.8).

  2. 2.

The term $\mathcal{L}_{3}$ is $P,T$-odd, and therefore vanishes in an $s$-wave
SF. On the other hand, it suffices to consider the $g$s-wave SF where
$s_{\theta}=0$ (but $\ell\neq 0$), since for $s_{\theta}\neq 0$ the spin
connection included in $\nabla_{\mu}\theta$ will only produce
$O\left(p^{4}\right)$ corrections. By contracting Galilean vectors, we were
able to construct four $P,T$-odd terms in
$\mathcal{L}_{3}-\mathcal{L}_{\text{gCS}}$ for the $g$s-wave SF,

$\displaystyle\mathcal{L}_{3}-\mathcal{L}_{\text{gCS}}\supset$
$\displaystyle\ell\left[C_{1}\left(X\right)\tilde{E}_{i}E_{\omega}^{i}+C_{2}\left(X\right)\varepsilon^{ij}\tilde{E}_{i}E_{\omega,j}+C_{3}\left(X\right)\partial_{i}XE_{\omega}^{j}+C_{4}\left(X\right)\varepsilon^{ij}\partial_{i}XE_{\omega,j}\right].$
(H.10)

where $\tilde{E}_{i}$ is the electric field of the improved $U\left(1\right)$
connection
$\tilde{A}_{t}=A_{t}+\frac{1}{2m}\nabla^{i}\theta\nabla_{i}\theta,\;\tilde{A}_{i}=\partial_{i}\theta-
s_{\theta}\omega_{i}$ [163]. Perturbatively expanding these, we do not find
any $O\left(q^{2}\right)$ contributions to $\eta_{\mathcal{C}}$ (or to
$\eta_{\text{$\mathcal{V}$}}$, in accordance with Appendix G.5).

###  Appendix I Induced action

The arguments presented in Sec.3 suffice to establish the quantization of
$\tilde{\eta}_{\text{o}}$ and $\tilde{\chi}_{TJ,\text{o}}$ directly from
$S_{\text{eff}}$ \- an explicit expression for $S_{\text{ind}}$ is not
required. Nevertheless, it is instructive to compute certain contributions in
$S_{\text{ind}}$ to demonstrate these results explicitly, and also to
reproduce simpler properties of $\ell$-wave SFs. Here we will compute the
contribution of
$\mathcal{L}_{0}+\mathcal{L}_{2}^{\left(1\right)}\subset\mathcal{L}_{\text{eff}}$
to the induced Lagrangian $\mathcal{L}_{\text{ind}}$, and, along the way,
demonstrate explicitly the relation between $\text{var}s=0$ QH states and CSFs
alluded to in Sec.3.5.

The starting point is the induced action due to
$\mathcal{L}_{0}=P\left(X\right)$, obtained from (H.3). It is given by

$\displaystyle\mathcal{L}_{\text{ind}}=$
$\displaystyle-\frac{1}{2}\mathcal{V}\mathcal{G}\mathcal{V}+\mathcal{C}$ (I.1)
$\displaystyle=$ $\displaystyle P_{0}\sqrt{G}-n_{0}\mathcal{A}_{t}$
$\displaystyle+\frac{1}{2}\frac{n_{0}}{m}\frac{\mathcal{B}^{2}-c_{s}^{-2}\mathcal{E}^{2}+\frac{s_{\theta}c_{s}^{-2}}{m}\mathcal{E}^{i}\partial_{i}B-\frac{s_{\theta}^{2}c_{s}^{-2}}{4m^{2}}\left(\partial
B\right)^{2}}{\partial^{2}-c_{s}^{-2}\partial_{t}^{2}}$ $\displaystyle-
n_{0}\frac{m\left(\partial_{t}\sqrt{G}\right)^{2}/2+\left(\mathcal{E}^{i}-\frac{s_{\theta}}{2m}\partial_{i}B\right)\partial_{i}\sqrt{G}}{\partial^{2}-c_{s}^{-2}\partial_{t}^{2}}.$

This expression contains, rather compactly, the entire linear response of the
$\ell$-wave SF to $O\left(p\right)$ in the derivative expansion, as well as
certain $O\left(p^{2}\right)$ contributions [163], and should be interpreted
as expanded to second order using (F.1)-(F.2). In using (F.2), one can set
$H_{[AB]}=0$, since $S_{\text{ind}}$ is $SO\left(2\right)_{L}$ invariant and
the anti-symmetric part $H_{[AB]}$ corresponds to the $SO\left(2\right)_{L}$
phase of the vielbein $E_{A}^{\;i}$. Technically, $H_{[AB]}$ always appears in
the combination
$\partial_{\mu}\left(\theta+s_{\theta}\varepsilon^{AB}H_{AB}/2\right)\subset\nabla_{\mu}\theta$,
so that integrating out $\theta$ eliminates $H_{[AB]}$.

Note that, diagrammatically, equation (I.1) corresponds to linear response at
tree-level. Higher orders in $\theta$ will generate diagrams with $\theta$
running in loops, which can be shown to produce $O\left(p^{3}\right)$
corrections above the leading order to any observable [199], and are therefore
irrelevant for the purpose of $q^{2}$ corrections to $\eta_{\text{o}}$.

The $O\left(p^{0}\right)$ part of (I.1) is obtained by setting $s_{\theta}=0$,
as in an $s$-wave SF,

$\displaystyle\mathcal{L}_{\text{ind},0}=$ $\displaystyle
P_{0}\sqrt{G}-n_{0}A_{t}+\frac{1}{2}\frac{n_{0}}{m}\frac{B^{2}-c_{s}^{-2}E^{2}}{\partial^{2}-c_{s}^{-2}\partial_{t}^{2}}$
(I.2) $\displaystyle-
n_{0}\frac{m\left(\partial_{t}\sqrt{G}\right)^{2}/2+E^{i}\partial_{i}\sqrt{G}}{\partial^{2}-c_{s}^{-2}\partial_{t}^{2}}$

The first line contains the ground state pressure and density $P_{0},n_{0}$,
as well as the London diamagnetic function
$\rho_{\text{e}}=\frac{n_{0}}{m}\frac{1}{q^{2}-c_{s}^{-2}\omega^{2}}$ and the
ideal Drude longitudinal conductivity
$\sigma_{\text{e}}=-\frac{n_{0}}{m}\frac{i\omega
c_{s}^{-2}}{q^{2}-c_{s}^{-2}\omega^{2}}$ of the SF [163]. The second line
contains the mixed response and mixed static susceptibility

$\displaystyle\kappa_{\text{e}}^{ij,k}$
$\displaystyle=-n_{0}\delta^{ij}\frac{iq^{k}}{q^{2}-c_{s}^{-2}\omega^{2}},$
(I.3) $\displaystyle\chi_{TJ,\text{e}}^{ij,t}$
$\displaystyle=n_{0}\delta^{ij}\frac{q^{2}}{q^{2}-c_{s}^{-2}\omega^{2}},$

defined in Sec.3.4, as well as the inverse compressibility
$K^{-1}=-n_{0}m\frac{\omega^{2}}{q^{2}-c_{s}^{-2}\omega^{2}}$ (which agrees
with the thermodynamic expression $K^{-1}=n_{0}^{2}\frac{\partial\mu}{\partial
n_{0}}=n_{0}mc_{s}^{2}$ at $q=0$). In particular, the $\ell$-wave SF is indeed
a superfluid \- the even viscosity $\eta_{\text{e}}$ vanishes to zeroth order
in derivatives (see [175] for a subtlety in separating $K^{-1}$ from
$\eta_{\text{e}}$).

The $O\left(p\right)$ part of the (I.1) is $P,T$-odd and vanishes when
$s_{\theta}=0$. It is given by

$\displaystyle\mathcal{L}_{\text{ind},1}=$ $\displaystyle-
s_{\theta}n_{0}\omega_{t}$ (I.4)
$\displaystyle+\frac{1}{2}\frac{s_{\theta}n_{0}}{m^{2}c_{s}^{2}}\frac{E^{i}\partial_{i}B}{\partial^{2}-c_{s}^{-2}\partial_{t}^{2}}$
$\displaystyle-
s_{\theta}n_{0}\frac{\left(E_{\omega}^{i}-\frac{1}{2m}\partial_{i}B\right)\partial_{i}\sqrt{G}}{\partial^{2}-c_{s}^{-2}\partial_{t}^{2}}.$

The first and third lines produce the following odd viscosity [163],

$\displaystyle\eta_{\text{o}}^{\left(1\right)}=$
$\displaystyle-\frac{1}{2}s_{\theta}n_{0},$ (I.5)
$\displaystyle\eta_{\text{o}}^{\left(2\right)}=$
$\displaystyle\frac{1}{2}s_{\theta}n_{0}\frac{1}{q^{2}-c_{s}^{-2}\omega^{2}},$

and setting $\omega=0$ one obtains the leading terms in equation (3.13). By
using the identity (up to a total derivative)

$\displaystyle
E^{i}\partial_{i}B=\frac{1}{2}\varepsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}\partial^{2}A_{\rho},$
(I.6)

the second line of (I.4) can be written as a non-local CS term

$\displaystyle\mathcal{L}_{\text{ind}}\supset$
$\displaystyle\frac{1}{2}\sigma_{\text{o}}\left(\omega,q\right)\varepsilon^{\mu\nu\rho}A_{\mu}ip_{\nu}A_{\rho}$
(I.7)

with the odd (or Hall) conductivity
$\sigma_{\text{o}}\left(\omega,q\right)=\sigma_{\text{o}}^{0}q^{2}/\left(q^{2}-c_{s}^{-2}\omega^{2}\right)$,
$\sigma_{\text{o}}^{0}=s_{\theta}n_{0}/2m^{2}c_{s}^{2}$ [192, 193, 194, 195,
153, 196, 197, 163], with
$\sigma_{\text{o}}\left(0,q\right)=\sigma_{\text{o}}^{0}$ unquantized, and
$\sigma_{\text{o}}\left(\omega,0\right)=0$, in accordance with the boundary
$U\left(1\right)_{N}$-neutrality [1].

To demonstrate explicitly that $c$ cannot be extracted from the odd viscosity
alone, it suffices to add the $O\left(p^{2}\right)$ term
$\mathcal{L}_{2}^{\left(1\right)}=F_{1}\left(X\right)R\subset\mathcal{L}_{2}$.
The situation is particularly simple for the special case
$F_{1}\left(X\right)=-s_{\theta}^{2}P^{\prime}\left(X\right)/4m$. Then

$\displaystyle P\left(X-\frac{s_{\theta}^{2}}{4m}R\right)$
$\displaystyle=P\left(X\right)-\frac{s_{\theta}^{2}}{4m}P^{\prime}\left(X\right)R+O\left(p^{4}\right)$
(I.8)
$\displaystyle=P\left(X\right)+F_{1}\left(X\right)R+O\left(p^{4}\right),$

which shows that $F_{1}\left(X\right)R$ can be absorbed into $P\left(X\right)$
by a modification of $X$. The scalar $X-\frac{s_{\theta}^{2}}{4m}R$ is useful
because, unlike $X$, it depends on $A_{\mu}$ and $\omega_{\mu}$ only through
the combination $\mathcal{A}_{\mu}=A_{\mu}+s_{\theta}\omega_{\mu}$. This is
evident in (H.1), where $B$ rather than $\mathcal{B}=B+\frac{s}{2}R$ appears.
It is then clear that, to $O\left(p^{3}\right)$, adding
$\mathcal{L}_{2}^{\left(1\right)}=F_{1}\left(X\right)R=-\frac{s_{\theta}^{2}}{4m}P^{\prime}\left(X\right)R$
to $\mathcal{L}_{0}=P\left(X\right)$ amounts to changing $B$ to $\mathcal{B}$
in the induced Lagrangian (I.1),

$\displaystyle\mathcal{L}_{\text{ind}}=$ $\displaystyle
P_{0}\sqrt{G}-n_{0}\mathcal{A}_{t}$ (I.9)
$\displaystyle+\frac{1}{2}\frac{n_{0}}{m}\frac{\mathcal{B}^{2}-c_{s}^{-2}\mathcal{E}^{2}+\frac{s_{\theta}c_{s}^{-2}}{m}\mathcal{E}^{i}\partial_{i}\mathcal{B}-\frac{s_{\theta}^{2}c_{s}^{-2}}{4m^{2}}\left(\partial\mathcal{B}\right)^{2}}{\partial^{2}-c_{s}^{-2}\partial_{t}^{2}}$
$\displaystyle-
n_{0}\frac{m\left(\partial_{t}\sqrt{G}\right)^{2}/2+\left(\mathcal{E}^{i}-\frac{s_{\theta}}{2m}\partial_{i}\mathcal{B}\right)\partial_{i}\sqrt{G}}{\partial^{2}-c_{s}^{-2}\partial_{t}^{2}}.$

The only contribution to $\eta_{\text{o}}$, beyond (I.5), comes from the term
proportional to $\mathcal{E}^{i}\partial_{i}\mathcal{B}$. By using the
identity (I.6) for $\mathcal{A}_{\mu}$, this term can be written as the sum of
non-local CS, WZ1, and WZ2 terms, which generalizes (I.7) to

$\displaystyle\mathcal{L}_{\text{ind}}\supset$
$\displaystyle\frac{1}{2}\sigma_{\text{o}}\left(\omega,q\right)\varepsilon^{\mu\nu\rho}\left(A_{\mu}+s_{\theta}\omega_{\mu}\right)ip_{\nu}\left(A_{\rho}+s_{\theta}\omega_{\rho}\right).$
(I.10)

Most importantly, this includes a non-local version of WZ2, which is
indistinguishable from $\mathcal{L}_{\text{gCS}}$ at $\omega=0$, where
$\sigma_{\text{o}}\left(0,q\right)=\sigma_{\text{o}}^{0}$ is a constant.
Noting that
$F_{1}^{\prime}=-s_{\theta}^{2}P^{\prime\prime}/4m=-\left(s_{\theta}/2\right)\sigma_{\text{o}}^{0}$,
and comparing to (H.8), it follows that $c$ and $F_{1}^{\prime}$ will enter
the $\omega=0$ odd viscosity only through the combination $c+48\pi
s_{\theta}F_{1}^{\prime}$. In more detail, the odd viscosity tensor due to
$\mathcal{L}_{0}+\mathcal{L}_{2}^{\left(1\right)}+\mathcal{L}_{\text{gCS}}$,
is given by

$\displaystyle\eta_{H}^{\left(1\right)}\left(\omega,q^{2}\right)=$
$\displaystyle-\frac{1}{2}s_{\theta}n_{0}-\left(\frac{c}{24}\frac{1}{4\pi}+\frac{s_{\theta}}{2}F_{1}^{\prime}\frac{q^{2}}{q^{2}-c_{s}^{-2}\omega^{2}}\right)q^{2}+O\left(q^{4}\right),$
(I.11) $\displaystyle\eta_{H}^{\left(2\right)}\left(\omega,q^{2}\right)=$
$\displaystyle\frac{1}{2}s_{\theta}n_{0}\frac{1}{q^{2}-c_{s}^{-2}\omega^{2}}+\left(\frac{c}{24}\frac{1}{4\pi}+\frac{s_{\theta}}{2}F_{1}^{\prime}\frac{q^{2}}{q^{2}-c_{s}^{-2}\omega^{2}}\right)+O\left(q^{2}\right),$

which, at $\omega=0$, is a special case of equation (12) of the main text.

Equation (I.11) remains valid away from the special point
$F_{1}=-s_{\theta}^{2}P^{\prime}/4m$, even though (I.10) does not. Examining
the perturbatively expanded $\mathcal{L}_{0}$ (H.3) and
$\mathcal{L}_{2}^{\left(1\right)}$ (H.7), we see that a general $F_{1}$
amounts to replacing $B$ in (I.1) with $B+\alpha\frac{s_{\theta}}{2}R$, where
$\alpha=-\frac{4mF_{1}^{\prime}}{s_{\theta}^{2}P^{\prime\prime}}\neq 1$
generically (as well as in the microscopic model (J.24)-(3.11)). The general
induced Lagrangian due to $\mathcal{L}_{0}+\mathcal{L}_{2}^{\left(1\right)}$,
valid to $O\left(p^{3}\right)$, is then given by

$\displaystyle\mathcal{L}_{\text{ind}}=$ $\displaystyle
P_{0}\sqrt{G}-n_{0}\mathcal{A}_{t}$ (I.12)
$\displaystyle+\frac{1}{2}\frac{n_{0}}{m}\frac{\mathcal{B}^{2}-c_{s}^{-2}\mathcal{E}^{2}+\frac{s_{\theta}c_{s}^{-2}}{m}\mathcal{E}^{i}\partial_{i}\left(B+\alpha\frac{s_{\theta}}{2}R\right)-\frac{s_{\theta}^{2}c_{s}^{-2}}{4m^{2}}\left(\partial
B\right)^{2}}{\partial^{2}-c_{s}^{-2}\partial_{t}^{2}}$ $\displaystyle-
n_{0}\frac{m\left(\partial_{t}\sqrt{G}\right)^{2}/2+\left[\mathcal{E}^{i}-\frac{s_{\theta}}{2m}\partial_{i}\left(B+\alpha\frac{s_{\theta}}{2}R\right)\right]\partial_{i}\sqrt{G}}{\partial^{2}-c_{s}^{-2}\partial_{t}^{2}},$

and, along with the $\mathcal{L}_{\text{gCS}}$ , produces the odd viscosity
(I.11). This expression does not depend on $A_{\mu},\omega_{\mu}$ only through
$\mathcal{A}_{\mu}$, but the terms contributing to (I.11) still vanish
$s_{\theta}=0$, which is why the improved odd viscosity due to (I.12)
vanishes. In addition to $\mathcal{L}_{2}^{\left(1\right)}$, the second order
terms $\mathcal{L}_{2}^{\left(2\right)},\mathcal{L}_{2}^{\left(3\right)}$
(H.6) also produce $q^{2}$ corrections to the odd viscosity, but not to the
improved odd viscosity.

Though equation (I.10) describes only a part of $\mathcal{L}_{\text{ind}}$,
and is non-generic, it does reveal the analogy between $\ell$-wave SFs and
$\text{var}s=0$ QH states, described in Sec.3.5, in a very simple setting.
Indeed, comparing (I.10) with Eq.(3.7), we see that $\ell$-wave SFs are
analogous to $\text{var}s=0$ QH states, with $\overline{s}=s_{\theta}=\ell/2$,
but with a non-local, non-quantized, Hall conductivity, in place of the
filling factor $\nu/2\pi$. Additionally, both QH states and $\ell$-wave SFs
have the same gCS term (H.8), with $c$ the boundary chiral central charge.

###  Appendix J Detailed analysis of the microscopic model Eq.(3.8)

####  J.1 Symmetry

The microscopic action $S_{\text{m}}$ of Eq.(3.8) is invariant under
$U\left(1\right)_{N}$ gauge transformations,

$\displaystyle\psi\mapsto e^{-i\alpha}\psi,\;\Delta^{j}\mapsto
e^{-2i\alpha}\Delta^{j},\;A_{\mu}\mapsto A_{\mu}+\partial_{\mu}\alpha,$ (J.1)

which implies the current conservation $\partial_{\mu}(\sqrt{G}J^{\mu})=0$,
where $\sqrt{G}J^{\mu}=-\delta S/\delta A_{\mu}$. It is also clear that
$S_{\text{m}}$ is invariant under time-independent spatial diffeomorphisms,
generated by $\delta x^{i}=\xi^{i}\left(\mathbf{x}\right)$, if $\psi$
transforms as a function, $A_{\mu}$ as a 1-form, $\Delta^{j}$ as a vector, and
$G_{ij}$ as a rank-2 tensor. As described in section 3.1, due to its Galilean
symmetry in flat space, $S_{\text{m}}$ is also invariant under time-dependent
spatial diffeomorphisms $\delta x^{i}=\xi^{i}\left(\mathbf{x},t\right)$,
provided one modifies the transformation rule of $A_{i}$ to Eq.(3.1).

####  J.2 Effective action and fermionic Green’s function

Starting with the microscopic action (3.8), the effective action for the order
parameter $\Delta$ in the $A,G$ background is obtained by integrating out the
(generically) gapped fermion $\psi$,

$\displaystyle
e^{iS_{\text{eff,m}}\left[\Delta;A,G\right]}=\int\text{D}\left(G^{1/4}\psi\right)\text{D}\left(G^{1/4}\psi^{\dagger}\right)e^{iS_{\text{m}}\left[\psi;\Delta,A,G\right]},$
(J.2)

where $G^{1/4}=\left(\text{det}G_{ij}\right)^{1/4}$ is the square root of the
volume element $\sqrt{G}$. The form of the measure is fixed by the fact that
the fundamental fermionic degree of freedom is the fermion-density
$\tilde{\psi}=G^{1/4}\psi$, which satisfies the usual canonical commutation
relation
$\left\\{\tilde{\psi}^{\dagger}\left(\mathbf{x}\right),\tilde{\psi}\left(\mathbf{y}\right)\right\\}=\delta^{\left(2\right)}\left(\mathbf{x}-\mathbf{y}\right)$
as an operator [242, 243, 34, 1]. This is to be contrasted with
$\left\\{\psi^{\dagger}\left(\mathbf{x}\right),\psi\left(\mathbf{y}\right)\right\\}=\delta^{\left(2\right)}\left(\mathbf{x}-\mathbf{y}\right)/\sqrt{G\left(\mathbf{x}\right)}$
which ties the fermion to the background metric. In terms of $\tilde{\psi}$
the action (3.8) takes the form

$\displaystyle S_{\text{m}}$
$\displaystyle=\int\text{d}^{2}x\text{d}t\left[\tilde{\psi}^{\dagger}\frac{i}{2}\overleftrightarrow{\nabla}_{t}\tilde{\psi}-\frac{1}{2m}G^{ij}\nabla_{i}\tilde{\psi}^{\dagger}\nabla_{j}\tilde{\psi}+\left(\frac{1}{2}\Delta^{i}\tilde{\psi}^{\dagger}\nabla_{i}\tilde{\psi}^{\dagger}+h.c\right)-\mathcal{U}\right],$
(J.3)

where $\nabla_{\mu}=\partial_{\mu}+iA_{\mu}-\frac{1}{4}\partial_{\mu}\log G$
is the covariant derivative for densities, and
$\mathcal{U}=\frac{1}{2\lambda}\sqrt{G}G_{ij}\Delta^{i*}\Delta^{j}$. Passing
to the BdG form of the fermionic part of the action, in terms of the Nambu
spinor-density
$\tilde{\Psi}^{\dagger}=\left(\tilde{\psi}^{\dagger},\tilde{\psi}\right)$
(which is a Majorana spinor-density [1]), one finds

$\displaystyle S_{\text{m}}=\int\text{d}^{2}x\text{d}t$
$\displaystyle\left\\{\frac{1}{2}\tilde{\Psi}^{\dagger}\gamma^{0}\left[i\gamma^{0}\partial_{t}-A_{t}+\frac{1}{2m}\nabla_{i}G^{ij}\nabla_{j}\right.\right.+\left.\left.\frac{i}{2}\gamma^{\tilde{A}}\left(e_{\tilde{A}}^{\;i}\partial_{i}+\partial_{i}e_{\tilde{A}}^{\;i}\right)\right]\tilde{\Psi}-\mathcal{U}\right\\}$
$\displaystyle=\int\text{d}^{2}x\text{d}t$
$\displaystyle\left\\{\frac{1}{2}\tilde{\Psi}^{\dagger}\gamma^{0}\mathcal{G}^{-1}\tilde{\Psi}-\mathcal{U}\right\\},$

where derivatives act on all fields to the right; $\tilde{A}=1,2$ is an index
for $U\left(1\right)_{N}$, viewed as a copy of $SO\left(2\right)$; the gamma
matrices are
$\gamma^{0}=\sigma^{z},\;\gamma^{1}=-i\sigma^{x},\;\gamma^{2}=i\sigma^{y}$,
satisfying $\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=2\eta^{\mu\nu}$ with
$\eta^{\mu\nu}=\text{diag}\left[1,-1,-1\right]$, and
$\text{tr}\left(\gamma^{0}\gamma^{1}\gamma^{2}\right)=2i$; and

$\displaystyle
e_{\tilde{A}}^{\;i}=\begin{pmatrix}\text{Re}\Delta^{x}&\text{Re}\Delta^{y}\\\
\text{Im}\Delta^{x}&\text{Im}\Delta^{y}\end{pmatrix}$ (J.4)

is the emergent vielbein [28, 1], to be distinguished from the background
vielbein $E_{A}^{\;i}$ (with an $SO\left(2\right)_{L}$ index $A=1,2$) that was
introduced in Sec.3.1 and that will be used momentarily. We also defined the
inverse Green’s function $\mathcal{G}^{-1}$. The effective action (J.2) is
then given by the logarithm of the Pfaffian

$\displaystyle S_{\text{eff,m}}=$
$\displaystyle-i\log\text{Pf}\left(i\gamma^{0}\mathcal{G}^{-1}\right)-\int\text{d}^{2}x\text{d}t\mathcal{U}$
(J.5) $\displaystyle=$
$\displaystyle-\frac{i}{2}\log\text{Det}\left(i\gamma^{0}\mathcal{G}^{-1}\right)-\int\text{d}^{2}x\text{d}t\mathcal{U}.$

####  J.3 Fermionic ground state topology

For a given $\Delta^{j}$, the fermion $\psi$ is gapped, unless the chemical
potential $\mu$ or chirality
$\ell=\text{sgn}\left(\text{Im}\left(\Delta^{x*}\Delta^{y}\right)\right)$ are
tuned to 0, and forms a fermionic topological phase characterized by the bulk
Chern number. Assuming $A_{\mu}=0$ and space-time independent
$\Delta^{i},G^{ij}$, it is given by [27]

$\displaystyle C=$
$\displaystyle\frac{1}{24\pi^{2}}\mbox{tr}\int\mbox{d}^{3}q\varepsilon^{\alpha\beta\gamma}\left(\mathcal{G}\partial_{\alpha}\mathcal{G}^{-1}\right)\left(\mathcal{G}\partial_{\beta}\mathcal{G}^{-1}\right)\left(\mathcal{G}\partial_{\gamma}\mathcal{G}^{-1}\right)\in\mathbb{Z},$
(J.6)

and determines the boundary chiral central charge $c=C/2$ [15, 104, 27, 107].
Here the fermionic Green’s function $\mathcal{G}$ is Fourier transformed to
Euclidian 3-momentum $q=\left(iq_{0},\mathbf{q}\right)$ (see (J.16)). For the
particular model (3.8) one finds

$\displaystyle
c=-\left(\ell/4\right)\left(\text{sgn}\left(\mu\right)+\text{sgn}\left(m\right)\right)\in\left\\{0,\pm
1/2\right\\},$ (J.7)

see [15, 27, 1] for similar expressions. Note that the central charge is well
defined for both $m>0$ and $m<0$, even though the single particle dispersion
is not bounded from below in the latter, and many physical quantities naively
diverge (we will see below that certain physical quantities diverge also with
$m>0$). A negative mass can occur as an effective mass in lattice models, in
which case the lattice spacing provides a natural cutoff (which must be smooth
in momentum space for (J.6) to hold). In any case, a negative mass makes it
possible to obtain both fundamental central charges $c=\pm 1/2$, for fixed
$\ell$, within the model (3.8). All possible $c\in\left(1/2\right)\mathbb{Z}$
can then be obtained by stacking layers of the model (3.8) with the same
$\ell$ but different $m,\mu$. Thus the model (3.8) suffices to generate a
representative for all topological phases of the $p$-wave CSF. For
concreteness, below we will work only with $m>0$, in which case $c$ is given
by Eq.(3.9).

####  J.4 Symmetry breaking and bosonic ground state in the presence of a
background metric

For time independent fields $A,G,\Delta$ the effective action reduces to

$\displaystyle S_{\text{eff,m}}\left[\Delta;G\right]=$
$\displaystyle-\int\text{d}^{2}x\text{d}t\varepsilon_{0}\left[\Delta;G\right],$
(J.8)

where $\varepsilon_{0}$ is the ground-state energy-density as a function of
the fields. In flat space $G_{ij}=\delta_{ij}$, with $A_{t}=-\mu$ and
$A_{i}=0$, and assuming $\Delta$ is constant, it is given by [27, 1]

$\displaystyle\varepsilon_{0}=$
$\displaystyle\frac{1}{2}\int\frac{\text{d}^{2}\mathbf{q}}{\left(2\pi\right)^{2}}\left(\xi_{\mathbf{q}}-\sqrt{\xi_{\mathbf{q}}^{2}+g^{ij}q_{i}q_{j}}\right)+\frac{1}{2\lambda}\delta_{ij}g^{ij},$
(J.9)

where

$\displaystyle\xi_{\mathbf{q}}=\left|\mathbf{q}\right|^{2}/2m-\mu$ (J.10)

is the single particle dispersion, and
$g^{ij}=\Delta^{(i}\Delta^{j)*}=\delta^{\tilde{A}\tilde{B}}e_{\tilde{A}}^{\;i}e_{\tilde{B}}^{\;j}$
is the emergent metric \- a dynamical metric to be distinguished from the
background metric $G^{ij}$. The ground state configuration of $g^{ij}$ is
determined by minimizing $\varepsilon_{0}$, while the overall phase $\theta$
of the order parameter and the chirality $\ell$, of which $g^{ij}$ is
independent, are left undetermined. Thus $g^{ij}$ corresponds to a massive
Higgs field, while $\theta$ is a Goldstone field. The energy-density (J.9) is
UV divergent, and requires regularization. We do this in the simplest manner,
by introducing a momentum cutoff $q^{2}<\Lambda^{2}$. Since the divergence
disappears for $g^{ij}=0$ (assuming $m>0$), this can be thought of as a small,
but non-vanishing, range $1/\Lambda$ for the interaction mediated by $\Delta$.
With a finite $\Lambda$, the energy-density is well defined and has a unique
global minimum at $g^{ij}=\Delta_{0}^{2}\delta^{ij}$, with $\Delta_{0}$
determined by the self-consistent equation

$\displaystyle\frac{1}{4}\int^{\Lambda}\frac{\text{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\frac{\left|\boldsymbol{q}\right|^{2}}{\sqrt{\xi_{\boldsymbol{q}}^{2}+\Delta_{0}^{2}\left|\boldsymbol{q}\right|^{2}}}=\frac{1}{\lambda}.$
(J.11)

For $\mu>0$ the non-interacting system has a Fermi surface, and a solution
exists for all $\lambda>0$, which is the statement of the BCS instability. For
$\mu<0$, the non-interacting system is gapped, and a solution exists if the
interaction is large enough compared with the gap,
$\lambda\Lambda^{-4}\gtrsim\left|\mu\right|$. Consider now the case of a
general constant metric $G_{ij}$, and let us introduce a constant vielbein $E$
such that $G_{ij}=E_{\;i}^{A}\delta_{AB}E_{\;j}^{B}$. The inverse transpose
$E^{-T}=\left(E^{-1}\right)^{T}$ is given in coordinates by $E_{A}^{\;i}$. We
also introduce the internal order parameter
$\Delta^{A}=E_{\;i}^{A}\Delta^{i}$. The action (J.3) then reduces to

$\displaystyle S_{\text{m}}$
$\displaystyle=\int\text{d}^{2}x\text{d}t\left[\tilde{\psi}^{\dagger}i\partial_{t}\tilde{\psi}-\frac{\delta^{AB}}{2m}E_{A}^{\;i}\partial_{i}\tilde{\psi}^{\dagger}E_{B}^{\;j}\partial_{j}\tilde{\psi}+\left(\frac{1}{2}\Delta^{A}E_{A}^{\;i}\tilde{\psi}^{\dagger}\partial_{i}\tilde{\psi}^{\dagger}+h.c\right)-\sqrt{G}\frac{1}{2\lambda}\delta_{AB}\Delta^{A*}\Delta^{B}\right].$
(J.12)

This is identical to the flat space case, with $\partial_{i}$ replaced by
$E_{A}^{\;i}\partial_{i}$. We also need to change the UV cutoff to
$\delta^{AB}E_{A}^{\;i}q_{i}E_{B}^{\;j}q_{j}=G^{ij}q_{i}q_{j}<\Lambda^{2}$.
This in natural since we interpret $\Lambda^{2}$ as a range of the interaction
mediated by $\Delta$, which should be defined in terms of the geodesic
distance rather than the Euclidian distance. It follows that the flat space
result (J.9) is modified to

$\displaystyle\varepsilon_{0}$
$\displaystyle=\frac{1}{2}\int_{\left|E^{-T}\mathbf{q}\right|^{2}<\Lambda^{2}}\frac{\text{d}^{2}\mathbf{q}}{\left(2\pi\right)^{2}}\left(\xi_{E^{-T}\mathbf{q}}-\sqrt{\xi_{E^{-T}\mathbf{q}}^{2}+g^{AB}E_{A}^{\;i}E_{B}^{\;j}q_{i}q_{j}}\right)+\sqrt{G}\frac{1}{2\lambda}\delta_{AB}g^{AB}$
(J.13)
$\displaystyle=\sqrt{G}\left[\frac{1}{2}\int_{q^{2}<\Lambda^{2}}\frac{\text{d}^{2}\mathbf{k}}{\left(2\pi\right)^{2}}\left(\xi_{\mathbf{k}}-\sqrt{\xi_{\mathbf{k}}^{2}+g^{AB}k_{A}k_{B}}\right)+\frac{1}{2\lambda}\delta_{AB}g^{AB}\right],$

where $\mathbf{k}=E^{-T}\mathbf{q}$, or $k_{A}=E_{A}^{\;i}q_{i}$, and
$g^{AB}=\Delta^{(A}\Delta^{B)*}=\delta^{\tilde{A}\tilde{B}}e_{\tilde{A}}^{\;A}e_{\tilde{B}}^{\;B}$
is the internal emergent metric. This is identical to the $G_{ij}=\delta_{ij}$
result (J.9), apart from the volume element $\sqrt{G}$, and the fact that it
is the internal metric $g^{AB}$ that appears, rather than $g^{ij}$. It is then
clear that minimizing (J.13) with respect to $g^{AB}$ gives

$\displaystyle g^{AB}=\Delta_{0}^{2}\delta^{AB}\text{, or
}g^{ij}=\Delta_{0}^{2}G^{ij},$ (J.14)

with the same $\Delta_{0}$ of (J.11), which is $G$ independent. Thus, the
emergent metric is proportional to the background metric in the ground state.
This solution corresponds to emergent vielbeins $e_{\tilde{A}}^{\;A}\in
O\left(2\right)$, or order parameters
$\Delta^{A}=\Delta_{0}e^{2i\theta}\left(1,\pm i\right)$, which is the
$p_{x}\pm ip_{y}$ configuration, and implies the SSB pattern

$\displaystyle\left(\mathbb{Z}_{2,T}\ltimes
U\left(1\right)_{N}\right)\times\left(\mathbb{Z}_{2,P}\ltimes
SO\left(2\right)_{L}\right)\rightarrow\begin{cases}\mathbb{Z}_{2,PT}\ltimes
U\left(1\right)_{L-\frac{\ell}{2}N}&\ell\in 2\mathbb{Z}+1\\\
\mathbb{Z}_{2,PT}\ltimes
U\left(1\right)_{L-\frac{\ell}{2}N}\times\mathbb{Z}_{2,\left(-1\right)^{N}}&\ell\in
2\mathbb{Z}\end{cases},$ (J.15)

described less formally in Eq.(1.2). Note that fermion parity
$\mathbb{Z}_{2,\left(-1\right)^{N}}$ is the $\mathbb{Z}_{2}$ subgroup of
$U\left(1\right)_{L-\frac{\ell}{2}N}$ for odd $\ell$. For $\Delta^{j}$, we
find the ground state configuration Eq.(3.10). As described in Sec.3.3, we
will ignore the massive Higgs fluctuations, and obtain
$S_{\text{eff}}\left[\theta;A,G\right]$ by plugging the ground state
configuration (3.10) into the functional Pfaffian (J.5).

####  J.5 Perturbative expansion

We now write $E_{A}^{\;i}=\delta_{A}^{i}+H_{A}^{\;i}$ and
$e_{\tilde{A}}^{\;A}=\Delta_{0}\delta_{\tilde{A}}^{A}$ (which corresponds to
$\Delta^{A}=\Delta_{0}\left(1,i\right)^{A}$) and expand (J.2) to second order
in $H,A$. Due to $SO\left(2\right)_{L}$ gauge symmetry, the anti-symmetric
part of $H$ can be interpreted as the Goldstone field,
$\theta=\left(s_{\theta}/2\right)\varepsilon_{AB}H^{AB}$. The $p_{x}-ip_{y}$
configuration $\Delta^{A}=\Delta_{0}\left(1,-i\right)^{A}$ can be incorporated
by changing the sign of one of the gamma matrices $\gamma^{\tilde{A}}$. The
expansion in $H,A$ produces a splitting of the propagator into an unperturbed
propagator and vertices, $\mathcal{G}^{-1}=\mathcal{G}_{0}^{-1}+\mathcal{V}$,
where $\mathcal{V}$ further splits as
$\mathcal{V}=\mathcal{V}_{1}+\mathcal{V}_{2}$, where $\mathcal{V}_{1}$
($\mathcal{V}_{2}$) is first (second) order in the fields. The terms in
$\mathcal{V}_{2}$ are often referred to as contact terms. Using (F.1) we find
the explicit form of $\mathcal{G}_{0}^{-1},\mathcal{V}_{1},\mathcal{V}_{2}$ in
Fourier components,

$\displaystyle\mathcal{G}_{0}^{-1}\left(q\right)=$
$\displaystyle-\gamma^{0}q_{0}-\Delta_{0}\gamma^{j}q_{j}-\xi_{\mathbf{q}},$
(J.16) $\displaystyle\mathcal{V}_{1}\left(q,p\right)=$ $\displaystyle-
A_{t,p}-\Delta_{0}\gamma^{A}\left(H_{A}^{\;i}\right)_{p}q_{i}$
$\displaystyle-\frac{1}{m}\left[q_{i}q_{j}-\frac{1}{4}\left(p_{i}p_{j}-\delta_{ij}p^{2}\right)\right]H_{p}^{ij}+\gamma^{0}\frac{1}{m}A_{p}^{j}q_{j},$
$\displaystyle\mathcal{V}_{2}\left(q,0\right)=$
$\displaystyle-\frac{1}{2m}\left(H_{A}^{\;i}H^{Aj}\right)_{p=0}q_{i}q_{j}-\frac{1}{8m}\left(\partial^{j}H_{A}^{\;A}\partial_{j}H_{B}^{\;B}\right)_{p=0}$
$\displaystyle-\gamma^{0}\frac{2}{m}\left(A_{i}H^{(ij)}\right)_{p=0}q_{j}-\frac{1}{2m}\left(A^{j}A_{j}\right)_{p=0}.$

Here $\left(\cdots\right)_{p}$ denotes the $p$ Fourier component of the field
$\left(\cdots\right)$, and we set $p=0$ in $\mathcal{V}_{2}$ since only this
component will be relevant. The unperturbed Greens’s function is given
explicitly by

$\displaystyle\mathcal{G}_{0}\left(q\right)=-\frac{q_{0}\gamma^{0}+\Delta_{0}q_{i}\gamma^{i}-\xi_{\mathbf{q}}}{q_{0}^{2}-q_{i}q^{i}-\xi{}_{\mathbf{q}}^{2}}.$
(J.17)

The perturbative expansion of $S_{\text{eff}}$ is obtained from (J.5) by using
$\text{log}\left[\text{Det}\left(\cdot\right)\right]=\text{Tr}\left[\log\left(\cdot\right)\right]$,
and expanding the logarithm in $\mathcal{V}$,

$\displaystyle S_{\text{eff,m}}$
$\displaystyle=-i\text{Tr}\left\\{\log\left[i\gamma^{0}\left(\mathcal{G}_{0}^{-1}+\mathcal{V}\right)\right]\right\\}$
$\displaystyle=-\frac{i}{2}\mbox{Tr}\left(\log
i\gamma^{0}\mathcal{G}_{0}^{-1}\right)-\frac{i}{2}\mbox{Tr}\left(\mathcal{G}_{0}\mathcal{V}\right)+\frac{i}{4}\mbox{Tr}\left(\mathcal{G}_{0}\mathcal{V}\right)^{2}+O\left(\mathcal{V}^{3}\right)$
$\displaystyle=-\frac{i}{2}\text{Tr}\left(\mathcal{G}_{0}\mathcal{V}_{1}\right)-\frac{i}{2}\text{Tr}\left(\mathcal{G}_{0}\mathcal{V}_{2}\right)+\frac{i}{4}\text{Tr}\left(\mathcal{G}_{0}\mathcal{V}_{1}\mathcal{G}_{0}\mathcal{V}_{1}\right)+\cdots,$

where in the last line we kept explicit only terms at first and second order
in $H,A$ (the term of zeroth order was described in the previous section).
Writing the functional traces as integrals over Fourier components and traces
over spinor indices, we then find

$\displaystyle S_{\text{eff,m}}=$
$\displaystyle-\frac{i}{2}\text{tr}\int_{q}\mathcal{V}_{1}\left(q,0\right)\mathcal{G}_{0}\left(q\right)-\frac{i}{2}\text{tr}\int_{q}\mathcal{V}_{2}\left(q,0\right)\mathcal{G}_{0}\left(q\right)$
(J.19)
$\displaystyle+\frac{i}{4}\text{tr}\int_{p,q}\mathcal{G}_{0}\left(q-\frac{1}{2}p\right)\mathcal{V}_{1}\left(q,-p\right)\mathcal{G}_{0}\left(q+\frac{1}{2}p\right)\mathcal{V}_{1}\left(q,p\right)+\cdots,$

where $\int_{q}=\int\frac{\text{d}^{2}q\text{d}q_{0}}{\left(2\pi\right)^{3}}$.
We are interested in $S_{\text{eff}}$ to third order in derivatives, which
amounts to expanding the above expression to $O\left(p^{3}\right)$, and
evaluating the resulting traces and integrals. These computations were
performed systematically using Mathematica, and can be found in the
supplemental material of Ref.[2]. The result, focusing on terms relevant for
$\eta_{\text{o}},\tilde{\eta}_{\text{o}}$ to $O\left(q^{2}\right)$, is
compatible with the general effective action of Sec.3.2, as confirmed by
comparing (J.19) to the perturbatively expanded $S_{\text{eff}}$. This
comparison provides explicit expressions for all of the coefficients that
appear in $S_{\text{eff}}$, as we now describe. The ground state pressure
$P\left(\mu\right)$ diverges logarithmically, and is given by

$\displaystyle P$
$\displaystyle=\frac{1}{2}\int^{\Lambda}\frac{\text{d}^{2}q}{\left(2\pi\right)^{2}}\left[\frac{q^{2}}{2m}-\frac{\frac{1}{2}\Delta_{0}^{2}q^{2}+\frac{q^{2}}{2m}\left(\frac{q^{2}}{2m}-\mu\right)}{\sqrt{\Delta_{0}^{2}q^{2}+\left(\frac{q^{2}}{2m}-\mu\right)^{2}}}\right]$
(J.20)
$\displaystyle=-\frac{m^{3}\Delta_{0}^{4}}{4\pi}\left(1-2\frac{\mu}{m\Delta_{0}^{2}}\right)\log\Lambda+O\left(\Lambda^{0}\right).$

Directly computing the ground state density $n_{0}$ and leading odd viscosity
$\eta_{\text{o}}^{\left(1\right)}$ one finds

$\displaystyle n_{0}$
$\displaystyle=\frac{1}{2}\int\frac{\text{d}^{2}q}{\left(2\pi\right)^{2}}\left[1-\frac{\left(\frac{q^{2}}{2m}-\mu\right)}{\sqrt{\Delta_{0}^{2}q^{2}+\left(\frac{q^{2}}{2m}-\mu\right)^{2}}}\right]$
(J.21)
$\displaystyle=\frac{m^{2}\Delta_{0}^{2}}{2\pi}\log\Lambda+O\left(\Lambda^{0}\right),$
$\displaystyle\eta_{\text{o}}^{\left(1\right)}$
$\displaystyle=-\frac{\ell}{16}\int\frac{\text{d}^{2}q}{\left(2\pi\right)^{2}}\frac{\Delta_{0}^{2}q^{2}\left(\frac{q^{2}}{2m}+\mu\right)}{\left[\left(\frac{q^{2}}{2m}-\mu\right)^{2}+q^{2}\Delta_{0}^{2}\right]^{3/2}}$
(J.22) $\displaystyle=-\frac{\ell
m^{2}\text{$\Delta$}_{0}^{2}}{8\pi}\log\Lambda+O\left(\Lambda^{0}\right),$

so the relations $n_{0}=P^{\prime}\left(\mu\right)$, and
$\eta_{\text{o}}^{\left(1\right)}=-\left(\ell/4\right)n_{0}$, described in
Sec.3, are maintained to leading order in the cutoff. As explained in Appendix
J.4, the cutoff $\Lambda$ corresponds to a non-vanishing interaction range,
which softens the contact interaction in the model (3.8). With a space-
independent metric, a smooth cutoff can easily be implemented by replacing

$\displaystyle\Delta^{A}E_{A}^{\;j}\tilde{\psi}_{-\mathbf{q}}^{\dagger}iq_{j}\tilde{\psi}_{\mathbf{q}}^{\dagger}\mapsto\Delta^{A}E_{A}^{\;j}\tilde{\psi}_{-\mathbf{q}}^{\dagger}\left(iq_{j}e^{-q_{k}q_{l}G^{kl}/\Lambda^{2}}\right)\tilde{\psi}_{\mathbf{q}}^{\dagger},$
(J.23)

for example, in the Fourier transformed Eq.(J.12), and should lead to the
exact relations $n_{0}=P^{\prime}\left(\mu\right)$,
$\eta_{\text{o}}^{\left(1\right)}=-\left(\ell/4\right)n_{0}$. However, a
computation of the $q^{2}$ correction to $\eta_{\text{o}}$ requires a space-
dependent metric, where a non-vanishing interaction range involves the
geodesic distance and complicates the vertex $\mathcal{V}$ in (J.16)
considerably. Moreover, all other coefficients in $S_{\text{eff}}$ converge,
and we can therefore work with the simple contact interaction,
$\Lambda=\infty$. For the second derivative $P^{\prime\prime}$, we find

$\displaystyle\frac{P^{\prime\prime}}{m}=$
$\displaystyle\frac{1}{2\pi}\begin{cases}1\\\ \frac{1}{1+2\kappa}\end{cases},$
(J.24)

where $\kappa=\left|\mu\right|/m\Delta_{0}^{2}>0$, and the cases refer to
$\mu>0$ and $\mu<0$. This coefficient determines the odd (or Hall)
conductivity $\sigma_{\text{o}}^{0}=\left(\ell/2\right)P^{\prime\prime}/2m$
and has been computed previously in the literature [192, 193, 194, 195, 153,
196, 197, 163, 198]. Note that $P^{\prime\prime}$ is continuous at $\mu=0$,
while $P^{\prime\prime\prime}$ is not, in accordance with the third order
phase transition found in an exact solution of the model (3.8) in the absence
of background fields [318].

The coefficients $P^{\prime\prime},F_{1}^{\prime},F_{2},F_{3}$ were presented
in Sec.3.3. The remaining coefficients $F_{4},F_{5},F_{6}$, are irrelevant for
the quantities discussed in Sec.3, and are presented here for completeness,

$\displaystyle F_{4}=$
$\displaystyle\frac{1}{24\pi\mu}\begin{cases}\frac{\kappa-2}{2}\\\
\frac{1}{1+2\kappa}\end{cases},\;F_{5}=\frac{1}{24\pi\mu\Delta_{0}^{2}}\begin{cases}1\\\
-\frac{1}{\left(1+2\kappa\right)^{2}}\end{cases},\;F_{6}=-\frac{\kappa}{24\pi\mu}\begin{cases}\frac{1}{2}\\\
\frac{1}{\left(1+2\kappa\right)^{2}}\end{cases}.$ (J.25)

As stated in Sec.3.3, there is a sense in which the relativistic limit
$\kappa\rightarrow 0$, or $m\rightarrow\infty$, reproduces the effective
action of a massive Majorana spinor in Riemann-Cartan space-time (Sec.2.5).
Taking the relativistic limit of the dimensionful coefficients (J.25), one
finds $F_{6}=0$, while $F_{4}=-\Delta_{0}^{2}F_{5}\neq 0$ describe a
relativistic term which is second order in torsion, and was not written
explicitly in Sec.2.5.

###  Appendix K Further details regarding Eq.(4.2)

This appendix involves basic facts in CFT, which can be found in e.g [102,
103].

####  K.1 Definition of $h_{0}$ and ambiguities in its value

A chiral topological phase of matter has a finite-dimensional ground state
subspace on the spatial torus. A basis
$\left\\{\left|a\right\rangle\right\\}_{a=1}^{N}$ for the torus ground state
subspace exists, such that each state $\left|a\right\rangle$ corresponds to a
conformal family in the boundary CFT [278, 212, 213], constructed over a
primary with right/left moving conformal weights
$h_{a}^{\left(l\right)},h_{a}^{\left(r\right)}\geq 0$. The corresponding
chiral and total conformal weights are then given by
$h_{a}=h_{a}^{\left(l\right)}-h_{a}^{\left(r\right)}$ and
$h_{a}^{+}=h_{a}^{\left(l\right)}+h_{a}^{\left(r\right)}$, respectively. The
chiral and total central charges of the CFT are similarly defined in terms of
the left/right moving central charges,
$c=c^{\left(l\right)}-c^{\left(r\right)}$ and
$c^{+}=c^{\left(l\right)}+c^{\left(r\right)}$.

When the torus is cut to a cylinder with finite circumference $L$, the ground
state degeneracy is lifted, generically leaving a unique ground state. The
lowest energy eigenstates on the cylinder can also be labeled as
$\left\\{\left|a\right\rangle\right\\}_{a=1}^{N}$. Each $\left|a\right\rangle$
corresponds to a non-universal choice of state in the conformal family labeled
by $h_{a}^{\left(l\right)},h_{a}^{\left(r\right)}$ , which need not be the
primary, as demonstrated explicitly in Appendix L below.

If the boundary is described by an idealized CFT, all $\left|a\right\rangle$s
correspond to primaries, and the corresponding energies are given by
$E_{a}=\left(4\pi v/L\right)\left(h_{a}^{+}-c^{+}/24\right)$, relative to the
ground state energy on the torus, where $v$ is the velocity of the CFT and $L$
is the circumference of the cylinder. These expressions receive exponentially
small corrections of $O\left(Le^{-R/\xi}\right)$ and
$O\left(Re^{-L/\xi}\right)$, where $\xi$ is the bulk correlation length and
$R$ is the length of the cylinder [98]. The cylinder ground state then
corresponds to the CFT ground state, the primary with minimal $h_{a}^{+}$.

More generally, each state $\left|a\right\rangle$ corresponds to either a
primary or a descendent, and has conformal weights
$h_{a}^{\left(l\right)}+n_{a}^{\left(l\right)},h_{a}^{\left(r\right)}+n_{a}^{\left(r\right)}$,
where $n_{a}^{\left(l\right)},n_{a}^{\left(r\right)}\in\mathbb{N}_{0}$. The
corresponding energies $E_{a}$ differ from the idealized $\left(4\pi
v/L\right)\left(h_{a}^{+}-c^{+}/24\right)$, and the choice of conformal family
$a_{0}$ with minimal $E_{a_{0}}$ is non-universal. In terms of
$n_{a}=n_{a}^{\left(l\right)}-n_{a}^{\left(r\right)}$, we then define
$h_{0}:=h_{a_{0}}+n_{a_{0}}$, the chiral conformal weight associated with the
cylinder ground state $\left|a_{0}\right\rangle$. The value of $h_{0}$
therefore carries two ambiguities: a choice of conformal family
$a_{0}\in\left\\{a\right\\}$, and the choice of a state in the conformal
family, $n_{a_{0}}\in\mathbb{N}_{0}$. As described in Sec.4.1.1, the only
universal statement is $\theta_{0}=e^{2\pi
ih_{0}}\in\left\\{\theta_{a}\right\\}$, where $\theta_{a}=e^{2\pi ih_{a}}$ are
the topological spins of bulk anyons.

The result of Ref.[212] for the momentum polarization is given terms of the
low lying cylinder eigenstates $\left|a\right\rangle$,

$\displaystyle\left\langle a\right|T_{R}\left|a\right\rangle=\exp\left[\alpha
N_{x}+\frac{2\pi
i}{N_{x}}\left(h_{a}-\frac{c}{24}\right)+o\left(N_{x}^{-1}\right)\right],$
(K.1)

where the lattice spacing is set to 1, $N_{x}=L$. It follows that the thermal
expectation value $\tilde{Z}/Z=\text{Tr}\left(T_{R}e^{-\beta H}\right)/Z$ is
equal to $\exp\left[\alpha N_{x}+\frac{2\pi
i}{N_{x}}\left(h_{0}-\frac{c}{24}\right)+o\left(N_{x}^{-1}\right)\right]$, if
the temperature $\beta^{-1}$ is much lower than the boundary energy
differences $\sim N_{x}^{-1}$, namely $\beta^{-1}=o\left(N_{x}^{-1}\right)$,
as described in Sec.4.1.

####  K.2 The value of $h_{0}$ in fermionic phases of matter

Fermionic topological phases are microscopically comprised of fermions (and
possibly bosons), and have the fermion parity $\left(-1\right)^{N_{f}}$ as a
global symmetry [319, 19, 320, 321]. It is therefore useful to probe such
phases with a background $\mathbb{Z}_{2}$ gauge field corresponding to
$\left(-1\right)^{N_{f}}$, or a spin structure. For our purposes, this amounts
to considering both periodic and anti-periodic boundary conditions around non-
contractible cycles in space-time.

In Sec.4 we were only interested in locally sign-free QMC representations of
thermal partition functions, and sign-free geometric manipulations that can be
performed to these. We therefore restricted attention to thermal boundary
conditions in the imaginary time direction (see Sec.4.4.1), and to periodic
boundary conditions around the spatial cylinder. These boundary conditions
cannot generically be modified without introducing signs into the QMC weights.

Here we provide a fuller picture by considering the behavior of $h_{0}$ with
both periodic and anti-periodic boundary conditions, in the closed $x$
direction of the spatial cylinder. Since $h_{0}$ is a ground state property,
the time direction is open and does not play a role.

For a fermionic chiral topological phase, the boundary CFT is also fermionic.
The primary conformal weights $\left\\{h_{a}\right\\}$ then depend on the
choice of boundary conditions (in the $x$ direction), and as a result, so will
the set of topological spins $\left\\{\theta_{a}\right\\}$ in which
$\theta_{0}=e^{2\pi ih_{0}}$ is valued. In particular, the vacuum spin
$\theta_{I}=0$ will not be included in $\left\\{\theta_{a}\right\\}$ for
periodic boundary conditions, while for anti-periodic boundary conditions,
both the vacuum $\theta_{I}=1$ and the spin $\theta_{\psi}=-1$ of the
microscopic fermion will appear [102, 213]. Note that $\theta_{\psi}$ does not
correspond to an emergent fermion, as in e.g the toric code [322], and
therefore does not imply an additional ground state on the torus.

As an example, consider the series of Laughlin phases at filling $1/q$ , with
$q\in\mathbb{N}$, all of which have the chiral central charge $c=1$. These
correspond to $U\left(1\right)_{q}$ Chern-Simons theories. First, for $q\in
2\mathbb{N}$ the phase is bosonic, and we consider only periodic boundary
conditions. The primary conformal weights are given by $h_{a}=a^{2}/2q$ [323,
55], with $a\in\mathbb{N}_{0}$. The topological spins $\theta_{a}=e^{2\pi
ih_{a}}$ depend only on $a\mod q$, and the $q$ spins
$\left\\{\theta_{a}\right\\}_{a=0}^{q-1}$ correspond to the $q$ degenerate
ground states that appear on the torus. In particular, the vacuum spin
$\theta_{I}=1$ is obtained for $a=0$.

For $q\in 2\mathbb{N}-1$ the phase is fermionic, and we consider both periodic
and anti-periodic boundary conditions. For periodic boundary conditions the
weights are given by $h_{a}=\left(a+1/2\right)^{2}/2q$ [55]. As in the bosonic
case, $\theta_{a}=e^{2\pi ih_{a}}$ depend only on $a\mod q$, with
$\left\\{\theta_{a}\right\\}_{a=0}^{q-1}$ corresponding to the $q$ degenerate
ground states on the torus. Unlike the bosonic case, the vacuum spin
$\theta_{I}=1$ is not included in $\left\\{\theta_{a}\right\\}_{a=0}^{q-1}$.
For anti-periodic boundary conditions, the weights are given by
$h_{a}=a^{2}/2q$ as in the bosonic case [323]. The set
$\left\\{\theta_{a}\right\\}_{a=0}^{q-1}$ again corresponds to the $q$ torus
ground states, but now $\theta_{\psi}=\theta_{a=q}=-1$ is an additional
topological spin that corresponds to the physical Fermion $\psi$ [215].

The simplest fermionic Laughlin phase is given by $q=1$, and corresponds to an
integer quantum Hall state, or a Chern insulator [105, 106], which is studied
in detail in Appendix L below. The Chern insulator has a unique ground state
on the torus, and accordingly, there is a unique topological spin
$\theta_{\sigma}=e^{2\pi i\left(1/8\right)}$ for periodic boundary conditions
on the cylinder, and two topological spins $\theta_{I}=1,\theta_{\psi}=-1$ for
anti-periodic boundary conditions. Here $\psi$ corresponds the physical
fermions from which the Chern insulator is comprised. The object carrying the
spin $\theta_{\sigma}$ is the complex analog of the celebrated Majorana zero
mode supported on vortices in the bulk of a chiral $p$-wave superconductor
[15, 104].

###  Appendix L Momentum polarization with non CFT boundaries

As reviewed in Sec.4.1, the existing analytic derivation of Eq.(4.2) relies on
the CFT description of the physical boundaries of the cylinder, and of the
line $y=0$ where $T_{R}$ is discontinuous [212]. In this appendix we perform
an analytic and numerical study that shows that, at least for free fermions,
the relevant CFT expressions and the resulting Eq.(4.2), hold even if the
boundary is not described by an idealized CFT. We will however, find a number
of subtleties which have not been demonstrated in the literature, as already
described below Eq.(4.1) and in Appendix K.

####  L.1 CFT finite-size correction in non CFT boundaries

The main ingredient in the analytic derivation of Eq.(4.2) is the expression
(4.1) for the finite size correction to the momentum density in CFT [212]. In
this appendix we show that, at least in the non-interacting case, Equation
(4.1) remains valid, with $\theta_{0}=e^{2\pi
ih_{0}}\in\left\\{\theta_{a}\right\\}$, even when the boundary cannot be
described by a CFT.

We will consider a Chern insulator, such as the prototypical Haldane model
[105]. When the boundary degrees of freedom can be described by a CFT, they
correspond to the Weyl fermion CFT, where $c=\pm 1$ and the primary conformal
weights are $h_{\sigma}=\pm 1/8$ ($h_{I}=0,h_{\psi}=\pm 1/2$) for periodic
(anti-periodic) boundary conditions, as described in Appendix K.2. The sign
corresponds to the two possible chiralities. More generally, on a lattice with
spacing $1$, the boundary supports a complex fermion with an energy dispersion
$\varepsilon_{k}$, where $k=k_{x}$ takes values in the Brillouin zone
$\mathbb{R}/2\pi\mathbb{Z}$ for an infinite circumference $L=\infty$, or its
discretization $\left(2\pi/L\right)\mathbb{Z}_{L}$
($\left(2\pi/L\right)\left(\mathbb{Z}_{L}+1/2\right)$), for $L<\infty$ and
periodic (anti-periodic) boundary conditions. The only requirement on
$\varepsilon_{k}$ is that it be chiral, in the sense that it connects the two
separated bulk energy bands. If the intersections $k_{l}$ and $k_{u}$ with the
lower and upper bulk bands, respectively, satisfy $k_{l}<k_{u}$
($k_{l}>k_{u}$), we say that the boundary is right (left) moving, or has a
positive (negative) chirality, see Fig.12.

Figure 12: Schematic band structure, energy $\varepsilon$ as a function of
momentum $k=k_{x}$ in the periodic $x$ direction, of a Chern insulator on the
cylinder. The figure shows the bulk energy bands (blue) and the chiral
boundary dispersion, with two dispersion branches, on a single boundary
component (orange and red curves). The opposite chirality branches on the
second boundary component are not drawn. The momenta $k_{l}$ and $k_{u}$
correspond to the intersections of the boundary dispersion with the lower and
upper bulk bands, respectively. Since $k_{u}>k_{l}$, both dispersion branches
have a positive chirality. The orange line indicates the idealized linear
dispersion with a single Fermi momentum $k_{F}=0$, which corresponds to the
Weyl fermion CFT. The solid red curve corresponds to a more general chiral
branch, with three Fermi momenta $k_{F,1},k_{F,2},k_{F,3}$, where the
dispersion around $k_{F,2}$ takes a (non-generic) non-linear form. With
periodic boundary conditions around the cylinder, both dispersion branches
produce the same $L^{-2}$ correction to the momentum density in Eq.(L.2), with
a positive chirality $+$, up to a mod 1 ambiguity: $1/12\mapsto
1/12+n,\;n\in\mathbb{N}$.

More generally, the dispersion will contain several dispersion branches
$\left\\{\varepsilon_{j,k}\right\\}_{j=1}^{J}$, but since the momentum density
is additive in $j$ we restrict attention to a single branch. Without loss of
generality, we also fix the chemical potential $\mu=0$, in which case the
Fermi momentum $k_{F}$ satisfies $\varepsilon_{k_{F}}=0$. The value of $k_{F}$
plays an important role is the subsequent analysis.

The simplest dispersion that satisfies the above requirements is the linear
one $\varepsilon_{k}=v\left(k-k_{F}\right)$. For $k_{F}=0$ this corresponds to
the Weyl fermion CFT. The presence of $k_{F}\neq 0$ corresponds to the
addition of a chemical potential $vk_{F}$, which breaks the conformal
symmetry. The generic form is
$\varepsilon_{k}=v\left(k-k_{F}\right)+O\left(k-k_{F}\right)^{2}$. A non-
generic dispersion can take the form
$\varepsilon_{k}=v_{3}\left(k-k_{F}\right)^{3}+O\left(k-k_{F}\right)^{4}$, and
there may be several Fermi momenta if the dispersion is non monotonic, see
Fig.12.

In all cases the many-body ground state momentum is given by summing the
momenta of all filled single Fermion states
$p\left(L\right)=\frac{1}{L}\sum_{\varepsilon{}_{k}<0}k$, where the sum runs
over $k\in\left(2\pi/L\right)\mathbb{Z}_{L}$ such that $\varepsilon_{k}$ is
negative and in the bulk energy gap. In order to obtain $p$ as a continuous
function of $L$, we treat the bulk energy gap as a smooth cutoff
$p\left(L\right)=\frac{1}{L}\sum_{\varepsilon_{k}<0}kC\left(\varepsilon_{k}\right)$,
where the function $C\left(\varepsilon\right)$ goes to $1$ ($0$) fast enough
as $\varepsilon$ goes to $0$ ($\varepsilon_{k_{l}}$ or
$\varepsilon_{k_{u}}$)444444It suffices that
$C^{\prime}\left(\varepsilon\right)$ vanish at
$\varepsilon=0,\varepsilon_{k_{l}},\varepsilon_{k_{u}}$.. The cutoff $C$
represents the smooth delocalization of boundary eigenstates as their energy
nears the bulk energy bands.

To obtain the $L$ dependence of $p\left(L\right)$, we will use the Euler-
Maclaurin formula

$\displaystyle\sum_{n=n_{1}}^{n_{2}}f\left(n\right)$
$\displaystyle=\int_{n_{1}}^{n_{2}}f\left(x\right)dx+\frac{f\left(n_{2}\right)+f\left(n_{1}\right)}{2}$
(L.1) $\displaystyle+$
$\displaystyle\frac{1}{6}\frac{f^{\prime}\left(n_{2}\right)-f^{\prime}\left(n_{1}\right)}{2!}-\frac{1}{30}\frac{f^{\prime\prime\prime}\left(n_{2}\right)-f^{\prime\prime\prime}\left(n_{1}\right)}{4!}+\mathsf{R},$

where the remainder satisfies
$\left|\mathsf{R}\right|\leq\frac{2\zeta\left(5\right)}{\left(2\pi\right)^{5}}\int_{n_{1}}^{n_{2}}\left|f^{\left(5\right)}\left(x\right)\right|dx$.
We begin by considering periodic boundary conditions, where we set
$f\left(n\right)=\left(2\pi n/L^{2}\right)C\left(\varepsilon_{2\pi
n/L}\right)$. Assuming a single, vanishing, Fermi momentum $k_{F}=0$, we set
$\left(n_{1},n_{2}\right)=\left(-\infty,0\right)$ for positive chirality, and
$\left(n_{1},n_{2}\right)=\left(0,\infty\right)$ for negative chirality.
Equation (L.1) then gives

$\displaystyle p\left(L\right)=$ $\displaystyle
p\left(\infty\right)\pm\frac{2\pi}{L^{2}}\frac{1}{12}+O\left(\frac{1}{L^{4}}\right),\;\text{as
}L\rightarrow\infty,$ (L.2)

where
$p\left(\infty\right)=\int_{\varepsilon_{k}<0}kC\left(\varepsilon_{k}\right)dk/2\pi$
and $\pm=\text{sgn}\left(k_{u}-k_{l}\right)$ is the chirality. The $1/L^{2}$
correction in (L.2) comes from $f^{\prime}\left(0\right)=2\pi/L^{2}$ in (L.1).
We see that the leading finite size correction $h_{0}-c/24$ is unchanged from
its CFT value $h_{\sigma}-c/24=\pm 1/12$, even when a CFT description does not
apply.

The case of a single non-zero Fermi momentum $k_{F}\neq 0$ is more
interesting, as it demonstrates that the integer part of $h_{0}$ can change as
a function of $L$ and $k_{F}$. The direct derivation of the end result from
the Euler-Maclaurin formula is surprisingly lengthy, so we omit it and present
a more direct route to the end result. To be concrete, assume a positive
chirality and $k_{F}>0$. The Euler-Maclaurin formula leads to cutoff
independent results, so we can restrict attention to cutoff functions
$C\left(\varepsilon_{k}\right)$ which are identically 1 for $0<k<k_{F}$. Since
these can serve as cutoff functions for the case $k_{F}=0$ as well, we can
deduce the $k_{F}\neq 0$ momentum density $p\left(L,k_{F}\right)$ from the
$k_{F}=0$ momentum density $p\left(L\right)$,

$\displaystyle p\left(L,k_{F}\right)$
$\displaystyle=\frac{1}{L}\sum_{k<k_{F}}kC\left(\varepsilon_{k}\right)$ (L.3)
$\displaystyle=\frac{1}{L}\sum_{k<0}kC\left(\varepsilon_{k}\right)+\frac{1}{L}\sum_{0<k<k_{F}}k$
$\displaystyle=p\left(L\right)+\frac{2\pi}{L^{2}}\sum_{l=1}^{n}l,$

where $n=\left\lfloor k_{F}L/2\pi\right\rfloor$. Using Eq.(L.2), we then have

$\displaystyle p\left(L,k_{F}\right)=$ $\displaystyle
p\left(\infty\right)+\frac{2\pi}{L^{2}}\left[\frac{1}{12}+\sum_{l=1}^{n}l\right]+O\left(\frac{1}{L^{4}}\right),$
(L.4)

where $p\left(\infty\right)$ is the momentum density at $L=\infty$ and
$k_{F}=0$. We see that the value of $h_{0}-c/24$ is only equal to the
idealized CFT result $h_{\sigma}-c/24=1/12$ modulo 1, while the integer part
jumps periodically as a function of $k_{F}$ at fixed number of sites $L$, or
as the number of sites $L$ at fixed $k_{F}$. Treating $k_{F}$ as fixed and
valued in $(-\pi,\pi]$, the period in $L$ is given by
$q=\left|2\pi/k_{F}\right|\geq 2$, which need not be an integer. As described
in Appendix K, the mod 1 ambiguity is attributed to $h_{0}$ rather than $c$,
which corresponds to the topological spin $\theta_{0}=\theta_{\sigma}=e^{2\pi
i\left(1/8\right)}$.

The interpretation of Eq.(L.4) is straight forward. As the number of sites $L$
increases, the single particle momenta $\left(2\pi/L\right)\mathbb{Z}_{L}$
become denser in the Brillouin zone $\mathbb{R}/2\pi\mathbb{Z}$. The $n$th
jump in $h_{0}$ correspond to the motion of a single particle state with
momentum $2\pi n/L$ through $k_{F}$ and into the Fermi sea, adding a momentum
density $2\pi n/L^{2}$ to the ground state.

Figure 13 presents the results of numerical computations of the momentum
polarization Eq.(4.2) in a Chern insulator with $k_{F}\neq 0$. Details of the
model and computations can be found in the supplemental material of Ref.[3].
In particular, Fig.13(a) verifies Eq.(L.4).

Figure 13: Numerical results for the momentum polarization Eq.(4.2), in a
Chern insulator with $k_{F}\neq 0$. Black dots mark numerically obtained
values of $N_{x}\arg\tilde{Z}$ as a function of $N_{x}^{2}$. (a) Periodic
boundary conditions. Purple lines indicate linear fits, with approximately the
same slope and intercepts $2\pi\left(1/12+\sum_{l=1}^{n}l\right)$ with the $y$
axis, with $n=0,1,2,3$, in accordance with Eq.(L.4). This allows for the
extraction of the topological spin $\theta_{\sigma}=e^{2\pi
i\left(1/8\right)}$. To illustrate the possibility of accidental degeneracies,
we choose $k_{F}=3/34$, where a degeneracy occurs for $N_{x}=34$, and the
average value of $N_{x}\arg\tilde{Z}$ between the two ground states is
obtained. A grey dotted line indicates the average of the two neighboring
purple lines. (b) Anti-periodic boundary conditions. Colored lines indicate
linear fits, with approximately the same slope, and intercepts
$2\pi\left[-1/24+\sum_{l=1}^{n}\left(l-1/2\right)\right]$, with $n=1,2,3,4$,
in accordance with Eq.(L.5). The value $n=0$ is not obtained as it occurs only
for small circumferences $N_{x}<6$ where $N_{x}\arg\tilde{Z}$ is not computed.
Orange lines correspond to the fermion spin $\theta_{\psi}=-1$, while blue
lines correspond to the vacuum spin $\theta_{I}=0$. To illustrate the
possibility of accidental degeneracies, we choose $k_{F}=\left(5/2\right)/24$,
where an accidental degeneracy occurs for $N_{x}=24$.

A subtle point, not mentioned above, is that when $k_{F}L/2\pi\in\mathbb{Z}$,
which happens only when $k_{F}/2\pi=a/b$ is rational and $L\in b\mathbb{N}$, a
single particle state with momentum exactly $k_{F}$ exists, leading to an
accidental degeneracy on the cylinder, between two many-body ground states
with momentum densities given by Eq.(L.4) with $n$ and $n+1$. The momentum
polarization (4.2) then gives the average momentum density in the two ground-
states, as visualized by the grey dotted line in Fig.13(a). For such system
sizes the value $\theta_{0}=-\theta_{\sigma}$ may be obtained rather than the
generic $\theta_{0}=\theta_{\sigma}$.

The same analysis can be performed for anti-periodic boundary conditions,
where we sum over single particle momenta
$k\in\frac{2\pi}{L}\left(\mathbb{Z}_{L}+\frac{1}{2}\right)$. Equation (L.4) is
then modified to

$\displaystyle p\left(L,k_{F}\right)=$ $\displaystyle p\left(\infty\right)$
(L.5)
$\displaystyle+\frac{2\pi}{L^{2}}\left[-\frac{1}{24}+\sum_{l=1}^{n}\left(l-\frac{1}{2}\right)\right]+O\left(\frac{1}{L^{4}}\right),$

where $n=\left\lfloor\frac{k_{F}L}{2\pi}-\frac{1}{2}\right\rfloor$. As a
function of $L$, jumps in $h_{0}-c/24$ occur with the same period
$q=\left|2\pi/k_{F}\right|\geq 2$, but are shifted by $q/2$. Moreover,
$h_{0}-c/24$ now attains two values modulo 1, namely $h_{I}-c/24=-1/24$ and
$h_{\psi}-c/24=1/2-1/24$.

Equation (L.5) therefore demonstrates explicitly the statements made in
Appendix K.1. For $k_{F}=0$, the cylinder ground state of the Chern insulator
corresponds to the idealized Weyl fermion CFT. A single value $h_{0}=0$ is
attained, which is the conformal weight $h_{I}$ of the CFT vacuum. A non-
vanishing $k_{F}$ corresponds to the addition of a chemical potential to the
CFT, which changes the energies of the CFT states, favoring a CFT exited state
over the CFT vacuum. The cylinder ground state of the Chern insulator may then
correspond to any CFT state in the conformal family of either the vacuum $I$
or fermion $\psi$, which need not be primary. From the bulk TFT perspective,
we see that $\theta_{0}=e^{2\pi ih_{0}}$ may be equal to either of the
topological spins $\theta_{I}=1,\theta_{\psi}=-1$ as a function of $L$.

As in the case of periodic boundary conditions, accidental degeneracies on the
cylinder occur when $k_{F}L/2\pi\in\mathbb{Z}+1/2$, changing the value of
$h_{0}$ attained from the momentum polarization to its average over the
degenerate states. For such system sizes, the value
$\theta_{0}=\pm\sqrt{\theta_{I}\theta_{\psi}}$ is obtained than the generic
$\theta_{0}\in\left\\{\theta_{I},\theta_{\psi}\right\\}$.

Equation (L.5) is verified numerically in Fig.13(b), which demonstrates that
the value of $\theta_{0}=e^{2\pi ih_{0}}$, obtained from the momentum
polarization (4.2), takes different values in the set
$\left\\{\theta_{a}\right\\}$ of topological spins as a function of system
size $L$, apart from accidental degeneracies.

####  L.2 No finite-size correction at finite temperature

The line $y=0$ where $T_{R}$ jumps can be interpreted as an additional
boundary component at the ’entanglement temperature’ $\beta_{*}^{-1}$.
Reference [212] used the modular transformations of CFT partition functions to
demonstrate that when $\beta_{*}\ll L/v$, this boundary component does not
contribute to the $1/L$ correction to $\log\tilde{Z}$. Here, we note that the
same result holds for free fermions with a general dispersion
$\varepsilon_{k}$. The contribution of the additional boundary component to
$\log\tilde{Z}$ is given by
$\log\tilde{Z}{}_{*}\left(L\right)=Lf_{*}\left(L\right)$, with the free energy
density

$\displaystyle f_{*}\left(L\right)$
$\displaystyle=\frac{1}{L}\sum_{k}\log\left(1+e^{iak}e^{-\beta_{*}\varepsilon_{k}}\right)C\left(\varepsilon_{k}\right).$
(L.6)

Using Eq.(L.1) one finds
$f_{*}\left(L\right)=f_{*}\left(\infty\right)+O\left(L^{-4}\right)$ for both
periodic and anti-periodic boundary conditions, which implies
$\log\tilde{Z}{}_{*}=Lf_{*}\left(\infty\right)+O\left(L^{-3}\right)$, with no
$1/L$ contribution. The complex number
$f_{*}\left(\infty\right)=\int\log\left(1+e^{iak}e^{-\beta_{*}\varepsilon_{k}}\right)C\left(\varepsilon_{k}\right)\text{d}k/2\pi$
contributes to the non-universal $\alpha$ in Eq.(4.2).

###  Appendix M Cutting the torus along an arbitrary vector

For $\mathbf{d}=\left(d_{x},0\right)$ we restrict to
$N_{x}=n_{x}d_{x},\;n\in\mathbb{N}$, viewing $d_{x}$ as an enlarged lattice
spacing, and treating $n$ as a reduced number of sites along the circumference
in place of $N_{x}$. The same logic applies to
$\mathbf{d}=\left(0,d_{y}\right)$. For $\mathbf{d}=\left(d_{x},d_{y}\right)$
with both $d_{x},d_{y}\neq 0$, we restrict to system sizes
$\left(N_{x},N_{y}\right)=n\mathbf{d}$, such that the line
$l=\text{span}_{\mathbb{R}}\mathbf{d}$ is a diagonal of the rectangle
$nN_{x}\times nN_{y}$ and corresponds to a circle on the torus
$\left(\mathbb{R}/N_{x}\mathbb{Z}\right)\times\left(\mathbb{R}/N_{y}\mathbb{Z}\right)$.
Cutting $X$ along this line produces a cylinder $C$ of circumference
$L=n\left|\mathbf{d}\right|$. We then view $\left|\mathbf{d}\right|$ as a
lattice spacing and $n$ as a the number of sites along the circumference. Note
that the distance between the boundary components of the resulting cylinder is
$R=nd_{x}d_{x}/\left|\mathbf{d}\right|$, and the thermodynamic limit is indeed