expected, these conservation laws turn out to be essentially the ones derived
in section (2.4.1), in the relativistic limit.

######  2.4.2.2 Spin

The Lorentz Lie algebra $so\left(1,2\right)$ is comprised of matrices
$\theta\in\mathbb{R}^{3\times 3}$ with entries $\theta_{\;b}^{a}$ such that
$\theta_{ab}=-\theta_{ba}$. These can be spanned as
$\theta=\frac{1}{2}\theta_{ab}L^{ab}$ where the generators $L^{ab}=-L^{ba}$
are defined such that $\eta L^{ab}$ is the antisymmetric matrix with $1$
($-1$) at position $a,b$ ($b,a$) and zero elsewhere. The spinor representation
of $\theta$ is

$\displaystyle\hat{\theta}=\frac{1}{2}\theta_{ab}\Sigma^{ab},\;\Sigma^{ab}=\frac{1}{4}\left[\gamma^{a},\gamma^{b}\right].$
(2.62)

Local Lorentz transformations act on $\chi,e,\omega$ by

$\displaystyle\chi$ $\displaystyle\mapsto$ $\displaystyle
e^{-\hat{\theta}}\chi,\;e_{a}^{\;\mu}\mapsto
e_{b}^{\;\mu}\left(e^{\theta}\right)_{\;a}^{b},$ $\displaystyle\omega_{\mu}$
$\displaystyle\mapsto$ $\displaystyle
e^{-\hat{\theta}}\left(\partial_{\mu}+\omega_{\mu}\right)e^{\hat{\theta}}.$
(2.63)

The subgroup of $SO\left(1,2\right)$ that is physical in the $p$-wave SC is
$SO\left(2\right)$ generated by $L^{12}$. Using the relations (2.37) between
the $p$-wave SC fields and the geometric fields, and choosing
$\theta=\theta_{12}L^{12}=-2\alpha L^{12}$, the transformation law (2.63)
reduces to the $U\left(1\right)$ transformation (2.43),

$\displaystyle\psi\mapsto e^{i\alpha}\psi,\;\Delta\mapsto
e^{2i\alpha}\Delta,\;A_{\mu}\mapsto A_{\mu}+\partial_{\mu}\alpha.$ (2.64)

The factor of 2 in $\theta_{12}=-2\alpha$ shows that $U(1)$ actually maps to
$Spin(2)$, the double cover of $SO(2)$. Moreover, the fact that $\Delta$ has
$U\left(1\right)$ charge 2 while $\psi$ has $U\left(1\right)$ charge 1
corresponds to $e_{a}^{\;\mu}$ being an $SO\left(1,2\right)$ vector while
$\chi$ is an $SO\left(1,2\right)$ spinor. The Lie algebra version of (2.63) is

$\displaystyle\delta\chi=-\frac{1}{2}\theta_{ab}\Sigma^{ab}\chi,\;\delta
e_{\;\mu}^{a}=-\theta_{\;b}^{a}e_{\;\mu}^{b},\;\delta\omega_{\;b\mu}^{a}=D_{\mu}\theta_{\;b}^{a}.$
(2.65)

Invariance of $S_{\text{RC}}$ under this variation implies the conservation
law

$\displaystyle\nabla_{\mu}\mathsf{J}^{ab\mu}-\mathsf{J}^{ab\rho}T_{\mu\rho}^{\mu}=\mathsf{J}^{[ab]},$
(2.66)

valid on the equations of motion for $\chi$ [156, 41]. This conservation law
relates the anti symmetric part of the energy-momentum tensor to the
divergence of spin current. Essentially, the energy-momentum tensor isn’t
symmetric due to the presence of the background field $\omega$ which
transforms under $SO\left(1,2\right)$. From a different point of view, the
vielbein $e$ acts as a source for the fermionic spin current since it is
charged under $SO\left(1,2\right)$. Inserting the expressions (2.37) into the
$\left(a,b\right)=\left(1,2\right)$ component of (2.66) we obtain (2.44),

$\displaystyle\partial_{\mu}J^{\mu}=-i\psi^{\dagger}\Delta^{j}\partial_{j}\psi^{\dagger}+h.c.$
(2.67)

The other components of (2.66) follow from the symmetry under local boosts,
which is only a symmetry of $S_{\text{SF}}$ when $m^{*}\rightarrow\infty$.
These can be used to obtain the formula (2.61) for the energy current of the
$p$-wave SF, in the limit $m^{*}\rightarrow\infty$, in terms of the momentum
and charge densities.

######  2.4.2.3 Energy-momentum

A diffeomorphism is a smooth invertible map between manifolds. We consider
only diffeomorphisms from space-time to itself and denote the group of such
maps by $Diff$. Since the flat background metric $\delta^{ij}$ decouples in
the relativistic limit, it makes sense to consider all diffeomorphisms, and
not restrict to symmetries of $\delta^{ij}$ as we did in section 2.4.1.2.

Locally, diffeomorphisms can be described by coordinate transformations
$x\mapsto x^{\prime}=f\left(x\right)$. The lie algebra is that of vector
fields $\zeta^{\nu}\left(x\right)$, which means diffeomorphisms in the
connected component of the identity $Diff_{0}$ can be written as
$f\left(x\right)=f_{1}\left(x\right)$ where
$f_{\varepsilon}\left(x\right)=\exp_{x}\left(\varepsilon\zeta\right)=x+\varepsilon\zeta\left(x\right)+O\left(\varepsilon^{2}\right)$
is the flow of $\zeta$ [234]. $Diff$ acts on the geometric fields by the
pullback

$\displaystyle\chi\left(x\right)$ $\displaystyle\mapsto$
$\displaystyle\chi\left(f\left(x\right)\right),\;e_{\;\mu}^{a}\left(x\right)\mapsto\partial_{\mu}f^{\nu}e_{\;\nu}^{a}\left(f\left(x\right)\right),$
$\displaystyle\omega_{\mu}\left(x\right)$ $\displaystyle\mapsto$
$\displaystyle\partial_{\mu}f^{\nu}\omega_{\nu}\left(f\left(x\right)\right).$
(2.68)

The action of $Diff$ on the $p$-wave SF fields is similar, and follows from
(2.68) supplemented by the dictionary (2.37). For $f\in Diff_{0}$ generated by
$\zeta$, the Lie algebra version of (2.68) is given by the Lie derivative,

$\displaystyle\delta\chi$ $\displaystyle=$
$\displaystyle\mathcal{L}_{\zeta}\chi=\zeta^{\mu}\partial_{\mu}\chi,$ (2.69)
$\displaystyle\delta e_{\;\mu}^{a}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{\zeta}e_{\;\mu}^{a}=\partial_{\mu}\zeta^{\nu}e_{\;\nu}^{a}+\zeta^{\nu}\partial_{\nu}e_{\;\mu}^{a},$
$\displaystyle\delta\omega_{\mu}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{\zeta}\omega_{\mu}=\partial_{\mu}\zeta^{\nu}\omega_{\nu}+\zeta^{\nu}\partial_{\nu}\omega_{\mu}.$

Since these variations are not Lorentz covariant, they will give rise to a
conservation law which is not Lorentz covariant. This follows from the fact
that the naive $Diff$ action (2.68) does not commute with Lorentz gauge
transformations, as was described for the simpler case of translations and
$U\left(1\right)$ gauge transformations in section 2.4.1.2. Instead, one
should use the Lorentz-covariant $Diff$ action, which is the pull back from
$f\left(x\right)$ to $x$ followed by a Lorentz parallel transport from
$f\left(x\right)$ to $x$ along the integral curve
$\gamma_{x,\zeta}\left(\varepsilon\right)=\exp_{x}\left(\varepsilon\zeta\right)=f_{\varepsilon}\left(x\right)$,

$\displaystyle\chi\left(x\right)$ $\displaystyle\mapsto$ $\displaystyle
P\chi\left(f\left(x\right)\right),$ (2.70) $\displaystyle
e_{\;\mu}^{a}\left(x\right)$ $\displaystyle\mapsto$ $\displaystyle
P_{\;b}^{a}\partial_{\mu}f^{\nu}e_{\;\nu}^{b}\left(f\left(x\right)\right),$
$\displaystyle\omega_{\mu}\left(x\right)$ $\displaystyle\mapsto$
$\displaystyle
P\left[\partial_{\mu}f^{\nu}\omega_{\mu}\left(f\left(x\right)\right)+\partial_{\mu}\right]P^{-1},$

where $P=\frac{1}{2}P_{ab}\Sigma^{ab}$ and
$P=\mathcal{P}\exp\left(-\int_{\gamma_{x,\zeta}}\omega\right)$ is the spin
parallel transport given by the path ordered exponential. At the Lie algebra
level, this modification of (2.68) amounts to an infinitesimal Lorentz gauge
transformation generated by $\theta_{ab}=-\zeta^{\rho}\omega_{ab\rho}$, which
modifies (2.69) to the covariant expressions

$\displaystyle\delta\chi$ $\displaystyle=$
$\displaystyle\zeta^{\mu}\nabla_{\mu}\chi,$ (2.71) $\displaystyle\delta
e_{\;\mu}^{a}$ $\displaystyle=$
$\displaystyle\nabla_{\mu}\zeta^{a}-T_{\mu\nu}^{a}\zeta^{\nu},$
$\displaystyle\delta\omega_{\mu}$ $\displaystyle=$
$\displaystyle\zeta^{\nu}R_{ab\nu\mu}.$

Since the usual $Diff$ and Lorentz actions on the fields are both symmetries
of $S_{\text{RC}}$, so is the Lorenz-covariant $Diff$ action. This leads
directly to the conservation law

$\displaystyle\nabla_{\mu}\mathsf{J}_{\;\nu}^{\mu}-\mathsf{J}_{\;\nu}^{\rho}T_{\mu\rho}^{\mu}=T_{\nu\mu}^{b}\mathsf{J}_{\;b}^{\mu}+R_{bc\nu\mu}\mathsf{J}^{bc\mu},$
(2.72)

valid on the equations of motion for $\chi$ [41, 156]. We find it useful to
rewrite (2.72) in a way which isolates the effect of torsion,

$\displaystyle\tilde{\nabla}_{\mu}\mathsf{J}_{\;\nu}^{\mu}=C_{ab\nu}\mathsf{J}^{[ab]}+R_{ab\nu\mu}\mathsf{J}^{ab\mu},$
(2.73)

where we note that the curvature also depends on the torsion,
$R=\tilde{R}+\tilde{D}C+C\wedge C$. Equation (2.72) can also be massaged to
the non-covariant form

$\displaystyle\partial_{\mu}\left(\left|e\right|\mathsf{J}_{\;\nu}^{\mu}\right)=\left(e_{a}^{\;\rho}D_{\nu}e_{\;\mu}^{a}\right)\left|e\right|\mathsf{J}_{\;\rho}^{\mu}+R_{\nu\mu
ab}\left|e\right|\mathsf{J}^{ab\mu}.$ (2.74)

Using the dictionary (2.37) and the subsequent paragraph, and (2.60), this
reduces to

$\displaystyle\partial_{\mu}t_{\text{cov}\;\nu}^{\mu}=\left(D_{\nu}\Delta^{j}\right)\frac{1}{2}\psi^{\dagger}\partial_{j}\psi^{\dagger}+h.c+F_{\nu\mu}J^{\mu},$
(2.75)

which is just the energy-momentum conservation law (2.52) for the $p$-wave SF
(with $m^{*}\rightarrow\infty$).

Writing the conservation law in the form (2.74) may not seem natural from the
geometric point of view because it uses the partial derivative as opposed to a
covariant derivative. It is however natural from the $p$-wave SC point of
view, where space-time is actually flat and $e$ is viewed as a bosonic field
with no geometric role, which is the order parameter $\Delta$. This point is
important in the context of the gravitational anomaly in the $p$-wave SC, see
Appendix A.2.

Similar statements hold for other mechanisms for emergent/analogue gravity,
see section I.6 of [27] and [235], and were also made in the gravitational
context without reference to emergent phenomena [236].

####  2.5 Bulk response

#####  2.5.1 Currents from effective action

The effective action for the background fields is obtained by integrating over
the spin-less fermion $\psi$,

$\displaystyle
e^{iW_{\text{SF}}\left[\Delta,A\right]}=\int\text{D}\psi^{\dagger}\text{D}\psi
e^{iS_{\text{SF}}\left[\psi,\psi^{\dagger},\Delta,A\right]}.$ (2.76)

The integral is a fermionic coherent state functional integral, over the
Grassmann valued fields $\psi,\psi^{\dagger}$, and the action $S_{\text{SF}}$
is given in (2.31).

As described in section 2.3, in the relativistic limit $W_{\text{SF}}$ is
equal to the effective action obtained by integrating over a Majorana fermion
coupled to RC geometry,

$\displaystyle e^{iW_{\text{SF}}\left[\Delta,A\right]}$ $\displaystyle=$
$\displaystyle e^{iW_{\text{RC}}\left[e,\omega\right]}$ $\displaystyle=$
$\displaystyle\int\text{D}\left(\left|e\right|^{1/2}\chi\right)e^{iS_{\text{RC}}\left[\chi,e,\omega\right]},$

where $e,\omega$ are given in terms of $\Delta,A$ by (2.37).

It follows from the definition (2.58) of the spin current $\mathsf{J}^{ab\mu}$
and the energy-momentum tensor $\mathsf{J}_{\;a}^{\mu}$ as functional
derivatives of $S_{\text{RC}}$ that their ground state expectation values are
given by

$\displaystyle\left\langle\mathsf{J}_{\;a}^{\mu}\right\rangle=\frac{1}{\left|e\right|}\frac{\delta
W_{\text{RC}}}{\delta
e_{\;\mu}^{a}},\;\left\langle\mathsf{J}^{ab\mu}\right\rangle=\frac{1}{\left|e\right|}\frac{\delta
W_{\text{RC}}}{\delta\omega_{ab\mu}}.$ (2.78)

Using the mapping (2.60) between $\mathsf{J}_{\;a}^{\mu},\;\mathsf{J}^{ab\mu}$
and $t_{\text{cov}\;\nu}^{\mu},\;J^{\mu}$ we see that

$\displaystyle\left\langle
J^{\mu}\right\rangle=4\left|e\right|\left\langle\mathsf{J}^{12\mu}\right\rangle=4\frac{\delta
W_{\text{RC}}\left[e,\omega\right]}{\delta\omega_{12\mu}},$ (2.79)
$\displaystyle\left\langle
t_{\text{cov}\;\nu}^{\mu}\right\rangle=-\left|e\right|e_{\;\nu}^{a}\left\langle\mathsf{J}_{\;a}^{\mu}\right\rangle=-e_{\;\nu}^{a}\frac{\delta
W_{\text{RC}}\left[e,\omega\right]}{\delta e_{\;\mu}^{a}}.$

This is the recipe we will use to obtain the expectation values $\left\langle
J^{\mu}\right\rangle,\left\langle t_{\text{cov}\;\nu}^{\mu}\right\rangle$ from
the effective action $W_{\text{RC}}$ for a Majorana spinor in RC space-time.

Note that in (2.79) there are derivatives with respect to all components of
the vielbein, not just the spatial ones which we can physically obtain from
$\Delta$. For this reason, to get all components of $\left\langle
t_{\text{cov}\;\nu}^{\mu}\right\rangle$, we should obtain $W_{\text{RC}}$ for
general $e$, take the functional derivative in (2.79), and only then set $e$
to the configuration obtained from $\Delta$ according to (2.37). From the
$p$-wave SF point of view, this corresponds to the introduction of a
fictitious background field $e_{0}^{\;\mu}$ which enters $S_{\text{SF}}$ by
generalizing $\psi^{\dagger}iD_{t}\psi$ to
$\psi^{\dagger}\frac{i}{2}e_{0}^{\;\mu}\overleftrightarrow{D_{\mu}}\psi$, and
setting $e_{0}^{\;\mu}=\delta_{t}^{\mu}$ at the end of the calculation, as in
[41].

Before we move on, we offer some intuition for the expressions (2.79). The
first equation in (2.79) follows from the definition $J^{\mu}=-\frac{\delta
S_{\text{SF}}}{\delta A_{\mu}}$ of the electric current and the simple
relation $\omega_{12\mu}=-\omega_{21\mu}=-2A_{\mu}$ between the spin
connection and the $U\left(1\right)$ connection. The second equation in (2.79)
is slightly trickier. It implies that the (relativistic part of the) energy-
momentum tensor $t_{\text{cov}\;\nu}^{\mu}$ is given by a functional
derivative with respect to the order parameter $\Delta$, because $\Delta$ is
essentially the vielbein $e$. This may seem strange, and it is certainly not
the case in an $s$-wave SC, where $\frac{\delta
H}{\delta\Delta}\sim\psi_{\uparrow}^{\dagger}\psi_{\downarrow}^{\dagger}$ has
nothing to do with energy-momentum. In a $p$-wave SC, the operator
$\frac{\delta
H}{\delta\Delta^{j}}\sim\psi^{\dagger}\partial_{j}\psi^{\dagger}$ contains a
spatial derivative which hints that it is related to fermionic momentum. More
accurately, we see from (2.4.1.2) that the operator
$\psi^{\dagger}\partial_{j}\psi^{\dagger}$ enters the energy-momentum tensor
in a $p$-wave SC.

#####  2.5.2 Effective action from perturbation theory

######  2.5.2.1 Setup and generalities

We consider the effective action for a $p$-wave SF on the plane
$\mathbb{R}^{2}$, with the corresponding space-time manifold
$M_{3}=\mathbb{R}_{t}\times\mathbb{R}^{2}$, by using perturbation theory
around the $p_{x}\pm ip_{y}$ configuration
$\Delta=\Delta_{0}e^{i\theta}\left(1,\pm i\right)$ with no electromagnetic
fields $\partial_{\mu}\theta-2A_{\mu}=0$. After $U\left(1\right)$ gauge fixing
$\theta=0$191919In doing so we are ignoring the possibility of vortices, see
[198]., we obtain $\Delta=\Delta_{0}\left(1,\pm i\right),\;A=0$. Let us start
with the $p_{x}+ip_{y}$ configuration, which has a positive orientation, in
which case the corresponding (gauge fixed) vielbein and spin connection are
just $e_{a}^{\;\mu}=\delta_{a}^{\mu}$ and $\omega_{ab\mu}=0$. A perturbation
of the $p_{x}+ip_{y}$ configuration corresponds to
$e_{a}^{\;\mu}=\delta_{a}^{\mu}+h_{a}^{\;\mu}$ with a small $h$ and to a small
spin connection $\omega_{ab\mu}$. In other words, a perturbation of the
$p_{x}+ip_{y}$ configuration without electromagnetic fields corresponds to a
perturbation of flat and torsion-less space-time.

The effective action for a Dirac spinor in a background RC geometry was
recently calculated perturbatively around flat and torsionless space-time,
with a positive orientation, in the context of geometric responses of Chern
insulators [156, 157]. This is equal to $2W_{\text{RC}}$ where $W_{\text{RC}}$
is the effective action for a Majorana spinor in RC geometry.

At this point is seems that we can apply these results in order to obtain the
effective action for the $p$-wave SC, in the relativistic limit. There is
however, an additional ingredient in the perturbative calculation of the
effective action which we did not yet discuss, which is the renormalization
scheme used to handle diverging integrals. We refer to terms in the effective
action that involve diverging integrals as UV sensitive. The values one
obtains for such terms depend on the details of the renormalization scheme, or
in other words, on microscopic details that are not included in the continuum
action.

For us, the continuum description is simply an approximation to the lattice
model, where space is a lattice but time is continuous. This implies a
physical cutoff $\Lambda_{UV}$ for wave-vectors, but not for frequencies. In
particular, such a scheme is not Lorentz invariant, even though the action in
the relativistic limit is. Lorentz symmetry is in any case broken down to
spatial $SO\left(2\right)$ for finite $m^{*}$. For these reasons, UV sensitive
terms in the effective action $W_{\text{RC}}$ for the $p$-wave SC will be
assigned different values than those obtained before, using a fully
relativistic scheme.

The perturbative calculation within the renormalization scheme outlined above
is described in appendix E, where we also demonstrate that it produces
physical quantities that approximate those of the lattice model, and compare
to the fully relativistic schemes used in previous works. In the following we
will focus on the UV insensitive part of the effective action, and in doing so
we will obtain results which are essentially202020See the discussion of
$O\left(\frac{m}{\Lambda_{UV}}\right)$ corrections below. independent of
microscopic details that do not appear in the continuum action. We start by
quoting the fully relativistic results of [156, 157], and then restrict our
attention to the UV insensitive part of the effective action, and describe the
physics of the $p$-wave SC it encodes.

######  2.5.2.2 Effective action for a single Majorana spinor

The results of [156, 157] can be written as

$\displaystyle 2W_{\text{RC}}\left[e,\omega\right]$ $\displaystyle=$
$\displaystyle\frac{\kappa_{H}}{2}\int_{M_{3}}Q_{3}\left(\tilde{\omega}\right)$
$\displaystyle+$
$\displaystyle\frac{\zeta_{H}}{2}\int_{M_{3}}e^{a}De_{a}-\frac{\kappa_{H}}{2}\int_{M_{3}}\tilde{\mathcal{R}}e^{a}De_{a}$
$\displaystyle+$
$\displaystyle\frac{1}{2\kappa_{N}}\int_{M_{3}}\left(\tilde{\mathcal{R}}-2\Lambda+\frac{3}{2}c^{2}\right)\left|e\right|\mbox{d}^{3}x+\cdots$

where

$\displaystyle
Q_{3}\left(\tilde{\omega}\right)=\text{tr}\left(\tilde{\omega}\text{d}\tilde{\omega}+\frac{2}{3}\tilde{\omega}^{3}\right)$
(2.81)

is the Chern-Simons (local) 3-form, $c=C_{abc}\varepsilon^{abc}$ is the
totally antisymmetric piece of the contorsion tensor, and
$\kappa_{H},\zeta_{H},1/\kappa_{N},\Lambda/\kappa_{N}$ are coefficients that
will be discussed further below. The first two lines of (2.5.2.2) are written
in terms of differential forms, and the third line is written in terms of
scalars. By scalars we mean $Diff$ invariant objects. In the differential
forms the wedge product is implicit, as it will be from now on, so
$\tilde{\omega}\wedge\text{d}\tilde{\omega}$ is written as
$\tilde{\omega}\text{d}\tilde{\omega}$ and so on. The integrals over
differential forms can be written as integrals over pseudo-scalars,

$\displaystyle
e^{a}De_{a}=\left(e_{\;\alpha}^{a}D_{\beta}e_{\;\gamma}^{b}\frac{1}{\left|e\right|}\varepsilon^{\alpha\beta\gamma}\right)\left|e\right|\text{d}^{3}x=-oc\left|e\right|\text{d}^{3}x,$
(2.82) $\displaystyle
Q_{3}\left(\tilde{\omega}\right)=\left(\tilde{\omega}_{\;b\alpha}^{a}\partial_{\beta}\tilde{\omega}_{\;a\gamma}^{b}+\frac{2}{3}\tilde{\omega}_{\;b\gamma}^{a}\tilde{\omega}_{\;c\beta}^{b}\tilde{\omega}_{\;a\gamma}^{c}\right)\frac{1}{\left|e\right|}\varepsilon^{\alpha\beta\gamma}\left|e\right|\text{d}^{3}x,$

which are only invariant under the orientation preserving subgroup of $Diff$
which we denote $Diff_{+}$. Here
$o=\text{sgn}\left(\text{det}\left(e\right)\right)$ is the orientation of $e$.
These expressions are odd under orientation reversing diffeomorphisms because
so are $o$ and the pseudo-tensor
$\frac{1}{\left|e\right|}\varepsilon^{\alpha\beta\gamma}$ 212121In this
section $\varepsilon$ always stands for the usual totally anti symmetric
symbol, normalized to 1. Thus
$\varepsilon^{123}=\varepsilon^{xyt}=\varepsilon_{xyt}=1$. Note that
$\varepsilon^{abc}$ is an $SO\left(1,2\right)$ tensor, and an
$O\left(1,2\right)$ pseudo-tensor, while
$\varepsilon^{\mu\nu\rho}=\text{det}\left(e\right)e_{a}^{\;\mu}e_{b}^{\;\nu}e_{c}^{\;\rho}\varepsilon^{abc}$
is a (coordinate) tensor density,
$\frac{1}{\text{det}\left(e\right)}\varepsilon^{\mu\nu\rho}$ is a tensor and
$\frac{1}{\left|e\right|}\varepsilon^{\mu\nu\rho}=\frac{1}{\left|\text{det}\left(e\right)\right|}\varepsilon^{\mu\nu\rho}$
is a pseudo-tensor. .

Equation (2.5.2.2) can be expanded in the perturbations $h_{a}^{\;\mu}$ and
$\omega_{ab\mu}$ to reveal the order in perturbation theory at which the
different terms arise, see appendix E. Additionally, at every order in the
perturbations the effective action can be expanded in powers of derivatives of
the perturbations over the mass $m$. The terms written explicitly above show
up at first and second order in $h,\omega$ and at up to third order in their
derivatives. They also include higher order corrections that make them
$Diff_{+}$ and Lorentz gauge invariant, or invariant up to total derivatives.

All other contributions denoted by $+\cdots$ are at least third order in the
perturbations or fourth order in derivatives. Such a splitting is not unique
[156], but the form (2.5.2.2) has been chosen because it is well suited for
the study of the bulk responses.

Let us now describe the different terms in (2.5.2.2). The first term is the
gravitational Chern-Simons (gCS) term. It has a similar structure to the more
familiar $U\left(1\right)$ CS term $\int A\text{d}A$, and is in fact an
$SO\left(1,2\right)$ CS term, but note that the LC spin connection
$\tilde{\omega}$ is a functional of the vielbein $e$. It is important that the
spin connection in gCS is not $\omega$, since through
$\omega_{\mu}=-2A_{\mu}\Sigma^{12}$ this would imply a quantized Hall
conductivity in a $p$-wave SC, which does not exist [15, 153]. As it is
written in (2.5.2.2), gCS is invariant under $Diff_{+}$, but not under
$SO\left(1,2\right)$ if $M_{3}$ has a boundary. This is the boundary
$SO\left(1,2\right)$ anomaly, which is discussed further in section 2.5.3.
Using the relation
$\tilde{\Gamma}_{\;\beta\mu}^{\alpha}=e_{a}^{\;\alpha}\left(\delta_{b}^{a}\partial_{\mu}+\tilde{\omega}_{\;b\mu}^{a}\right)e_{\;\beta}^{b}$
between $\tilde{\Gamma}$ and $\tilde{\omega}$, one can derive an important
formula,

$\displaystyle
Q_{3}\left(\tilde{\Gamma}\right)-Q_{3}\left(\tilde{\omega}\right)=\text{tr}\left[\frac{1}{3}\left(e\text{d}e^{-1}\right)^{3}+\text{d}\left(\text{d}e^{-1}e\tilde{\Gamma}\right)\right],$

where unusually, $e=\left(e_{\;\mu}^{a}\right)$ is treated in this expression
as a matrix valued function [23, 25]. The variation with respect to $e$ of the
two terms on the right hand side is a total derivative, which means that they
are irrelevant for the purpose of calculating bulk responses. One can
therefore use $Q_{3}\left(\tilde{\Gamma}\right)$, which only depends on the
metric $g_{\mu\nu}$, instead of $Q_{3}\left(\tilde{\omega}\right)$. The form
$\int_{M_{3}}Q_{3}\left(\tilde{\Gamma}\right)$ of gCS is invariant under
$SO\left(1,2\right)$ but not under $Diff_{+}$, as opposed to
$\int_{M_{3}}Q_{3}\left(\tilde{\omega}\right)$. Thus the right hand side of
(LABEL:eq:70) has the effect of shifting the boundary anomaly from
$SO\left(1,2\right)$ to $Diff$.

The second term in (2.5.2.2) has a structure similar to a CS term with $e^{a}$
playing the role of a connection, and indeed some authors refer to it as such
[237]. Nevertheless, it is $SO\left(1,2\right)$ and $Diff_{+}$ invariant, as
can be seen from (2.82). This term was related to the torsional Hall viscosity
in [156], where it was discussed extensively. The third term in (2.5.2.2) is
also $SO\left(1,2\right)$ and $Diff_{+}$ invariant. We refer to this term as
gravitational pseudo Chern-Simons (gpCS), to indicate its similarity to gCS,
and the fact that it is not a Chern-Simons term. The similarity between gCS
and gpCS is demonstrated and put in a broader context in the discussion,
section 2.6. In section 2.5.4 we will see that gCS and gpCS produce similar
contributions to bulk responses. For now, we simply note that both terms are
second order in $h$ and third order in derivatives of $h$.

The third line in (2.5.2.2) contains the Einstein-Hilbert action with a
cosmological constant $\Lambda$ familiar from general relativity, and an
additional torsional contribution $\propto c^{2}$. The coefficient
$1/\kappa_{N}$ of the Einstein-Hilbert term is usually related to a Newton’s
constant $G_{N}=\kappa_{N}/8\pi$. Note that in Riemannian geometry, where
torsion vanishes and $\omega=\tilde{\omega}$, only the gCS term, the Einstein-
Hilbert term, and the cosmological constant survive.

The coefficients $\kappa_{H},\zeta_{H},1/\kappa_{N},\Lambda/\kappa_{N}$ are
given by frequency and wave-vector integrals that arise within the
perturbative calculation, and are described in appendix E. In particular
$\zeta_{H},1/\kappa_{N},\Lambda/\kappa_{N}$ are dimension-full, with mass
dimensions $2,1,3$, and naively diverge. In other words, they are UV
sensitive. On the other hand, $\kappa_{H}$ is dimensionless and UV
insensitive. With no regularization, one finds

$\displaystyle\kappa_{H}=\frac{1}{48\pi}\frac{\text{sgn}\left(m\right)o}{2}.$
(2.84)

Thus, the effective action for a single Majorana spinor can be written as

$\displaystyle
W_{\text{RC}}\left[e,\omega\right]=\frac{1/2}{96\pi}\frac{\text{sgn}\left(m\right)o}{2}W\left[e,\omega\right]+\cdots$
(2.85)

where

$\displaystyle
W\left[e,\omega\right]=\int_{M_{3}}Q_{3}\left(\tilde{\omega}\right)-\int_{M_{3}}\tilde{\mathcal{R}}e^{a}De_{a}$
(2.86)

is the sum of gCS and gpCS, and the dots include UV sensitive terms, or terms
of a higher order in derivatives or perturbations, as described above.

Since the lattice model implies a finite physical cutoff $\Lambda_{UV}$ for
wave-vectors, (2.85) is exact only for $m/\Lambda_{UV}\rightarrow 0$. For non-
zero $m$ there are small $O\left(m/\Lambda_{UV}\right)$ corrections222222All
expressions here are with $\hbar=c_{\text{light}}=1$. Restoring units one
finds
$\frac{m}{\Lambda_{\text{UV}}}\sim\frac{\text{max}\left(t,\mu\right)}{\delta}$
and so $\frac{m}{\Lambda_{\text{UV}}}\ll 1$ in the relativistic regime. to
(2.85). We will keep these corrections implicit for now, and come back to them
in section 2.5.3.

######  2.5.2.3 Summing over Majorana spinors

As discussed in Appendix 2.2, the continuum description of the $p$-wave SC
includes four Majorana spinors labeled by $1\leq n\leq 4$, with masses
$m_{n}$, which are coupled to vielbeins $e_{\left(n\right)}$. Let us repeat
the necessary details. The vielbein $e_{\left(1\right)}$ is associated with
the order parameter $\delta$ of the underlying lattice model, as in (2.37), up
to an unimportant rescaling by the lattice spacing $a$. For this reason we
treat it as a fundamental vielbein and write $e=e_{\left(1\right)}$ in some
expressions. The other vielbeins $\left(e_{\left(n\right)}\right)_{a}^{\;\mu}$
are obtained from $e$ by multiplying one of the columns $\mu=x,y$ or both by
$-1$. This implies that $o=o_{1}=o_{3}=-o_{2}=-o_{4}$, and that the metrics
$g_{\left(n\right)}^{\mu\nu}$ are identical apart from
$g^{xy}=g_{\left(1\right)}^{xy}=g_{\left(3\right)}^{xy}=-g_{\left(2\right)}^{xy}=-g_{\left(4\right)}^{xy}$.
With this in mind, we can sum over the four Majorana spinors and obtain and
effective action for the $p$-wave SC,

$\displaystyle W_{\text{SC}}\left[e,\omega\right]$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{4}W_{\text{RC}}\left[e_{\left(n\right)},\omega\right]$
$\displaystyle=$
$\displaystyle\frac{1/2}{96\pi}\sum_{n=1}^{4}\frac{\text{sgn}\left(m_{n}\right)o_{n}}{2}W\left[e_{\left(n\right)},\omega\right]+\cdots$

Note that the Chern number of the lattice model is given by
$\nu=\sum_{n=1}^{4}\text{sgn}\left(m_{n}\right)o_{n}/2$, but since $W$ also
depends on the different vielbeins $e_{\left(n\right)}$, (2.5.2.3) does not
only depend on $\nu$ in the general case.

Some simplification is possible however. Since
$e_{\left(1\right)}=e_{\left(3\right)}$ and
$e_{\left(2\right)}=e_{\left(4\right)}$ up to a space-time independent
$SO\left(2\right)$ ($U\left(1\right)$) transformation,

$\displaystyle W_{\text{SC}}\left[e,\omega\right]$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{2}\frac{\nu_{l}/2}{96\pi}W\left[e_{\left(l\right)},\omega\right]+\cdots$
$\displaystyle=$
$\displaystyle\sum_{l=1}^{2}\frac{\nu_{l}/2}{96\pi}\int_{M_{3}}Q_{3}\left(\tilde{\omega}{}_{\left(l\right)}\right)+\cdots$

where in the second line, we have only written explicitly gCS terms. Here we
defined

$\displaystyle\nu_{1}$
$\displaystyle=\frac{o_{1}}{2}\left(\text{sgn}\left(m_{1}\right)+\text{sgn}\left(m_{3}\right)\right),$
(2.89) $\displaystyle\nu_{2}$
$\displaystyle=\frac{o_{2}}{2}\left(\text{sgn}\left(m_{2}\right)+\text{sgn}\left(m_{4}\right)\right),$

which are both integers, $\nu_{1},\nu_{2}\in\mathbb{Z}.$ The Chern number of
the lattice model is given by the sum $\nu=\nu_{1}+\nu_{2}$. Thus the lattice
model seems to behave like a bi-layer, with layer index $l=1,2$. In the
topological phases of the model $\nu_{1}=0$, $\nu=\nu_{2}=\pm 1$, and so

$\displaystyle W_{\text{SC}}\left[e,\omega\right]$
$\displaystyle=\frac{\nu/2}{96\pi}W\left[e_{\left(2\right)},\omega\right]+\cdots$
$\displaystyle=\frac{\nu/2}{96\pi}\int_{M_{3}}Q_{3}\left(\tilde{\omega}_{\left(2\right)}\right)+\cdots$

where again, in the second line we have only written explicitly the gCS term.
This result is close to what one may have guessed. In the topological phases
with Chern number $\nu\neq 0$, the effective action contains a single gCS
term, with coefficient $\frac{\nu/2}{96\pi}$. A result of this form has been
anticipated in [28, 15, 31, 17, 121], but there are a few details which are
important to note. First, apart from gCS, $W$ also contains the a gpCS term of
the form $\int_{M_{3}}\tilde{\mathcal{R}}e^{a}De_{a}$, which is possible due
to the emergent torsion. Second, the connection that appears in the CS form
$Q_{3}$ is a LC connection, and not the torsion-full connection $\omega$.
Moreover, this LC connection is not $\tilde{\omega}$, but a modification of it
$\tilde{\omega}{}_{\left(2\right)}$, where the subscript $\left(2\right)$
indicates the effect of the multiple Majorana spinors in the continuum
description of the lattice model. Third, the geometric fields $e,\omega$ are
given by $\Delta,A$.

In the trivial phases $\nu_{1}=-\nu_{2}\in\left\\{-1,0,1\right\\}$, $\nu=0$,
and we find

$\displaystyle W_{\text{SC}}\left[e,\omega\right]$
$\displaystyle=\frac{\nu_{1}/2}{96\pi}\left[W\left[e_{\left(1\right)},\omega\right]-W\left[e_{\left(2\right)},\omega\right]\right]+\cdots$
$\displaystyle=\frac{\nu_{1}/2}{96\pi}\left[\int_{M_{3}}Q_{3}\left(\tilde{\omega}{}_{\left(1\right)}\right)-\int_{M_{3}}Q_{3}\left(\tilde{\omega}{}_{\left(2\right)}\right)\right]+\cdots$

This result is quite surprising. Instead of containing no gCS terms, some
trivial phases contain the difference of two such terms, with slightly
different spin connections. One may wonder if these trivial phases are really
trivial after all. This is part of a larger issue which we now address.

#####  2.5.3 Symmetries of the effective action

By considering the gauge symmetry of the effective action we can reconstruct
the topological phase diagram appearing in Fig.5 from (2.5.2.3). This will
also help us understand which of our results are special to the relativistic
limit, and which should hold throughout the phase diagram. By gauge symmetry
we refer in this section to the $SO\left(2\right)$ subgroup of
$SO\left(1,2\right)$, which corresponds to the physical $U\left(1\right)$
symmetry of the $p$-wave SC. Equation (LABEL:eq:70) shows that we can
equivalently consider $Diff$ symmetry. The physical reason for this
equivalence is that the $p$-wave order parameter is charged under both
symmetries, and therefore maps them to one another.

The effective action was calculated within perturbation theory on the space-
time manifold $M_{3}=\mathbb{R}_{t}\times\mathbb{R}^{2}$, but for this
discussion, we use its locality to assume it remains locally valid on more
general $M_{3}$, which may be closed (compact and without a boundary) or have
a boundary. A closed space-time is most simply obtained by working on
$M_{3}=\mathbb{R}_{t}\times M_{2}$ with $M_{2}$ closed, and with background
fields $\Delta,A$ which are periodic in time, such that $\mathbb{R}_{t}$ can
be compactified to a circle.

As described in appendix B.6, a non singular order parameter endows $M_{3}$
with an orientation and a spin structure, and in particular requires that
$M_{2}$ be orientable [238], which we assume. Thus, for example, we exclude
the possibility of $M_{2}$ being the Mobius strip. Moreover, a non singular
order parameter on a closed $M_{2}$ requires that $M_{2}$ contain
$\left(g-1\right)o$ magnetic monopoles [15], where $g$ is the genus of
$M_{2}$, and we assume that this condition is satisfied. For example, if
$M_{2}$ is the sphere then it must contain a single monopole or anti-monopole
depending on the orientation $o$ [239, 240].

######  2.5.3.1 Quantization of coefficients

The first fact about the gCS term that we will need, is that gauge symmetry of
$\alpha\int_{M_{3}}Q_{3}\left(\tilde{\omega}\right)$ for all closed $M_{3}$
requires that $\alpha$ be quantized such that
$\alpha\in\frac{1}{192\pi}\mathbb{Z}$, see equation (2.27) of [24]. In order
to understand how generic is our result (2.5.2.3), we will check what
quantization condition on $\alpha_{1},\alpha_{2}$ is required for gauge
symmetry of
$\alpha_{1}\int_{M_{3}}Q_{3}\left(\tilde{\omega}_{\left(1\right)}\right)+\alpha_{2}\int_{M_{3}}Q_{3}\left(\tilde{\omega}_{\left(2\right)}\right)$
on all closed $M_{3}$. Following the arguments of [24] we find that
$\alpha_{1}+\alpha_{2}\in\frac{1}{192\pi}\mathbb{Z}$, but
$\alpha_{1},\alpha_{2}\in\mathbb{R}$ are not separately restricted, see
appendix C. It is therefore natural to define $\alpha=\alpha_{1}+\alpha_{2}$
and rewrite

$\displaystyle\alpha_{1}\int_{M_{3}}Q_{3}\left(\tilde{\omega}_{\left(1\right)}\right)+\alpha_{2}\int_{M_{3}}Q_{3}\left(\tilde{\omega}_{\left(2\right)}\right)$
(2.92)
$\displaystyle=\alpha\int_{M_{3}}Q_{3}\left(\tilde{\omega}_{\left(2\right)}\right)+\alpha_{1}\int_{M_{3}}\left[Q_{3}\left(\tilde{\omega}_{\left(1\right)}\right)-Q_{3}\left(\tilde{\omega}_{\left(2\right)}\right)\right],$

where $\alpha\in\frac{1}{192\pi}\mathbb{Z}$ but $\alpha_{1}\in\mathbb{R}$.
Comparing with the result (2.5.2.3), we identify $\alpha=\frac{\nu/2}{96\pi}$,
$\alpha_{1}=\frac{\nu_{1}/2}{96\pi}$, and we conclude that $\nu$ must be
precisely an integer and equal to the Chern number, while $\nu_{1}$ need not
be quantized. We therefore interpret the $O\left(m/\Lambda_{\text{UV}}\right)$
corrections to $\alpha=\frac{\nu/2}{96\pi}$ produced in our computation as
artifacts of our approximations232323Specifically, in obtaining the
relativistic continuum approximation we split the Brillouin zone $BZ$ into
four quadrants and linearized the lattice Hamiltonian (2.16) in every
quadrant. Applying any integral formula for the Chern number to the
approximate Hamiltonian will give a result
$\nu_{\text{apprx}}=\frac{1}{2}\sum_{n=1}^{4}o_{n}\text{sgn}\left(m_{n}\right)+O\left(m/\Lambda_{\text{UV}}\right)$
which is only approximately quantized in the relativistic regime, simply
because the approximate Hamiltonian is discontinuous on $BZ$. Nevertheless,
the known quantization $\nu\in\mathbb{Z}$ and the fact that
$\nu_{\text{apprx}}\approx\nu$ are enough to obtain the exact result
$\nu=\frac{1}{2}\sum_{n=1}^{4}o_{n}\text{sgn}\left(m_{n}\right)$., which must
vanish due to gauge invariance. On the other hand, we interpret the
quantization $\alpha_{1}=\frac{\nu_{1}/2}{96\pi}$ as a special property of the
relativistic limit with both $m^{*}\rightarrow\infty$ and $m\rightarrow 0$,
which should not hold throughout the phase diagram.

So far we have only considered gCS terms. As already explained, the gpCS term
is gauge invariant on any $M_{3}$, and we therefore see no reason for the
quantization of its coefficient. Explicitly,
$-\beta\int_{M_{3}}\tilde{\mathcal{R}}e^{a}De_{a}$ is gauge invariant for all
$\beta\in\mathbb{R}$. Thus we interpret the approximate quantization of the
coefficients of gpCS terms as a special property of the relativistic limit,
which should not hold throughout the phase diagram. We note that even for a
relativistic spinor any $\beta\in\mathbb{R}$ can be obtained, by adding a non
minimal coupling to torsion [156].

In light of the above, it is natural to interpret (2.5.2.3) as a special case
of

$\displaystyle
W_{\text{SC}}\left[e,\omega\right]=\frac{\nu/2}{96\pi}\int_{M_{3}}Q_{3}\left(\tilde{\omega}_{\left(2\right)}\right)$
(2.93)
$\displaystyle+\alpha_{1}\int_{M_{3}}\left[Q_{3}\left(\tilde{\omega}_{\left(1\right)}\right)-Q_{3}\left(\tilde{\omega}_{\left(2\right)}\right)\right]$
$\displaystyle-\beta_{1}\int_{M_{3}}\tilde{\mathcal{R}}_{\left(1\right)}e_{\left(1\right)}^{a}De_{\left(1\right)a}-\beta_{2}\int_{M_{3}}\tilde{\mathcal{R}}e_{\left(2\right)}^{a}De_{\left(2\right)a}+\cdots$

where $\nu\in\mathbb{Z}$ is the Chern number and
$\alpha_{1},\beta_{1},\beta_{2}$ are additional, non quantized, yet
dimensionless, response coefficients. In the relativistic limit
$\alpha_{1},\beta_{1},\beta_{2}$ happen to be quantized, but this is not
generic. Only the first gCS term encodes topological bulk responses,
proportional to the Chern number $\nu$, and below we will see that only this
term is related to an edge anomaly. We can also write (2.93) more
symmetrically,

$\displaystyle W_{\text{SC}}\left[e,\omega\right]$ (2.94)
$\displaystyle=\sum_{l=1}^{2}\left[\alpha_{l}\int_{M_{3}}Q_{3}\left(\tilde{\omega}_{\left(l\right)}\right)-\beta_{l}\int_{M_{3}}\tilde{\mathcal{R}}_{\left(l\right)}e_{\left(l\right)}^{a}De_{\left(l\right)a}\right]+\cdots$

but here we must keep in mind the quantization condition
$\alpha_{1}+\alpha_{2}=\frac{\nu/2}{96\pi}\in\frac{1}{192\pi}\mathbb{Z}$.

This equation should be compared with the result in the relativistic limit
(2.5.2.3), where $\alpha_{l},\beta_{l}$ are all quantized, and
$\alpha_{l}=\beta_{l}$. We note that the quantization of
$\alpha_{l},\beta_{l}$ in the relativistic limit can be understood on
dimensional grounds: in this limit there are simply not enough dimension-full
quantities which can be used to construct dimensionless quantities, beyond
$\text{sgn}\left(m_{n}\right)$ and $o_{n}$. Of course, this does not explain
why $\alpha_{l}=\beta_{l}$ in the relativistic limit.

######  2.5.3.2 Boundary anomalies

We can strengthen the above conclusions by considering space-times $M_{3}$
with a boundary. The second fact about the gCS term that we will need is that
it is not gauge invariant when $M_{3}$ has a boundary, even with a properly
quantized coefficient. In more detail, the $SO\left(2\right)$ variation of gCS
is given by

$\displaystyle\delta_{\theta}\int_{M_{3}}Q_{3}\left(\tilde{\omega}\right)=-\text{tr}\int_{\partial
M_{3}}\mbox{d}\theta\tilde{\omega}.$ (2.95)

Up to normalization, the boundary term above is called the consistent Lorentz
anomaly, which is one of the forms in which the gravitational anomaly
manifests itself [12]242424Generally speaking, consistent anomalies are given
by symmetry variations of functionals. We will also discuss below the more
physical covariant anomalies, which correspond to the actual inflow of some
charge from bulk to boundary. The anomaly $\text{tr}\int_{\partial
M_{3}}\mbox{d}\theta\tilde{\omega}$ is a local functional that can either be
written as the gauge variation of a local bulk functional, as it is written
above, or as the gauge variation of a nonlocal boundary functional
$F\left[\tilde{\omega}\right]$, such that
$\delta_{\theta}F\left[\tilde{\omega}\right]=\int_{\partial
M_{3}}\mbox{d}\theta\tilde{\omega}$, but cannot be written as the gauge
variation of a local boundary functional [241]. The difference of two gCS
terms is also not gauge invariant,

$\displaystyle\delta_{\theta}\left[\int_{M_{3}}Q_{3}\left(\tilde{\omega}_{\left(1\right)}\right)-\int_{M_{3}}Q_{3}\left(\tilde{\omega}_{\left(2\right)}\right)\right]$
(2.96) $\displaystyle=-\text{tr}\int_{\partial
M_{3}}\mbox{d}\theta\left(\tilde{\omega}_{\left(1\right)}-\tilde{\omega}_{\left(2\right)}\right),$

but here there is a local boundary term that can produce the same variation,
given by $\text{tr}\left(\tilde{\omega}_{(1)}\tilde{\omega}_{(2)}\right)$.

The physical interpretation is as follows. Since
$F\left[\tilde{\omega}\right]$ is non local it can be interpreted as the
effective action obtained by integrating over a gapless, or massless, boundary
field coupled to $e$. These are the boundary chiral Majorana fermions of the
$p$-wave SC. The statement that $F$ cannot be local implies that this boundary
field cannot be gapped. In this manner the existence of the gCS term in the
bulk effective action, with a coefficient that is fixed within a topological
phase, implies the existence of gapless degrees of freedom that cannot be
gapped within a topological phase. We study this bulk-boundary correspondence
in more detail in Appendix A. Naively, the difference of two gCS terms implies
the existence of two boundary fermions with opposite chiralities, one of which
is coupled to $e_{\left(1\right)}$ and the other coupled to
$e_{\left(2\right)}$. The boundary term $\int_{\partial
M_{3}}\text{tr}\left(\tilde{\omega}_{\left(1\right)}\tilde{\omega}_{\left(2\right)}\right)$
can only be generated if the two counter propagating fermions are coupled, and
its locality indicates that this coupling can open a gap. Thus the term
$\int_{\partial
M_{3}}\text{tr}\left(\tilde{\omega}_{\left(1\right)}\tilde{\omega}_{\left(2\right)}\right)$
represents the effect of a generic interaction between two counter propagating
chiral Majorana fermions.

Again, as opposed to the gCS term, the gpCS term is gauge invariant on any
$M_{3}$, and is therefore unrelated to edge anomalies. Thus, in the effective
action (2.93), only the first gCS term is related to an edge anomaly.

######  2.5.3.3 Time reversal and reflection symmetry of the effective action

Time reversal $T$ and reflection $R$ are discussed in appendices B.5.2 and
B.5.3. The orientation $o$ of the order parameter is odd under both $T,R$, and
it follows that so are the coefficients $\nu_{l}$. Therefore $\nu_{l}$ are
$T,R$-odd response coefficients. More generally, $\alpha_{l},\beta_{l}$ in
(2.94) are $T,R$-odd response coefficients. As described in section 2.5.2.2,
integrals over differential forms are also odd under the orientation reversing
diffeomorphisms $T,R$, and therefore $W_{\text{SC}}$ is invariant under $T,R$.

#####  2.5.4 Calculation of currents

To derive the currents we start with the expression

$\displaystyle\alpha_{1}\int_{M_{3}}Q_{3}\left(\tilde{\omega}\right)-\beta_{1}\int_{M_{3}}\tilde{\mathcal{R}}e^{a}De_{a}+\cdots$
(2.97)

which is the effective action for the layer $l=1$. We then sum the results
over $l=1,2$, as in (2.94), to get the full low energy response of the lattice
model, keeping in mind that
$\alpha_{1}+\alpha_{2}=\frac{\nu/2}{96\pi}\in\frac{1}{192\pi}\mathbb{Z}$.

######  2.5.4.1 Bulk response from gravitational Chern-Simons terms

For the purpose of calculating the contribution of gCS to the bulk energy-
momentum tensor it is easier to use $Q_{3}\left(\tilde{\Gamma}\right)$ instead
of $Q_{3}\left(\tilde{\omega}\right)$. The result is [22, 115, 25]

$\displaystyle\left\langle\mathsf{J}_{\;a}^{\mu}\right\rangle_{\text{gCS}}=\frac{1}{\left|e\right|}\frac{\delta}{\delta
e_{\;\mu}^{a}}\left[\alpha_{1}\int_{M_{3}}Q_{3}\left(\tilde{\Gamma}\right)\right]=4\alpha_{1}\tilde{C}_{\;a}^{\mu},$
(2.98)

where $\tilde{C}$ is the Cotton tensor, which can be written as

$\displaystyle\tilde{C}^{\mu\nu}=-\frac{1}{\sqrt{g}}\varepsilon^{\rho\sigma(\mu}\tilde{\nabla}_{\rho}\tilde{\mathcal{R}}_{\sigma}^{\nu)}.$
(2.99)

Relevant properties of the Cotton tensor are
$\tilde{\nabla}_{\mu}\tilde{C}^{\mu\nu}=0$, $\tilde{C}_{\;\mu}^{\mu}=0$, and
$C^{[\mu\nu]}=0$. It follows from (2.98) that

$\displaystyle\left\langle
t_{\text{cov}\;\nu}^{\mu}\right\rangle_{\text{gCS}}=-\left|e\right|\left\langle\mathsf{J}_{\;\nu}^{\mu}\right\rangle_{\text{gCS}}=-4\alpha_{1}\left|e\right|\tilde{C}_{\;\nu}^{\mu}.$
(2.100)

For order parameters of the form

$\displaystyle\Delta=e^{i\theta}\left(\left|\Delta^{x}\right|,\pm
i\left|\Delta^{y}\right|\right)$ (2.101)

the metrics for both layers $l=1,2$ are identical. Since $\tilde{C}$ only
depends on the metric it follows that for such order parameters the summation
over $l=1,2$ gives

$\displaystyle\left\langle
t_{\text{cov}\;\nu}^{\mu}\right\rangle_{\text{gCS}}=-\left|e\right|\left\langle\mathsf{J}_{\;\nu}^{\mu}\right\rangle_{\text{gCS}}=-\frac{\nu/2}{96\pi}4\left|e\right|\tilde{C}_{\;\nu}^{\mu}.$
(2.102)

Put differently, the difference of gCS terms in (2.94), with coefficient
$\alpha_{1}$, does not produce a bulk response for such order parameters. This
provides a simple way to separate the topological invariant $\nu$ from the non
quantized $\alpha_{1}$.

The Cotton tensor takes a simpler form if the geometry is a product geometry,
where the metric is of the form
$\text{d}s^{2}=g_{\alpha\beta}\left(x^{\alpha}\right)\text{d}x^{\alpha}\text{d}x^{\beta}+\sigma\text{d}z^{2}$.
Here $\sigma=\pm 1$ depends on whether $z$ is a space-like or time-like
coordinate, and we will use both in the following. The two coordinates
$x^{\alpha}$ are space-like if $z$ is time-like and mixed if $z$ is space-
like. In this case the curvature is determined by the curvature scalar, which
corresponds to the curvature scalar of the two dimensional metric
$g_{\alpha\beta}$. In particular
$\mathcal{R}_{\;\beta}^{\alpha}=\frac{1}{2}\mathcal{R}\delta_{\beta}^{\alpha}$
and the other components of $\mathcal{R}_{\;\nu}^{\mu}$ vanish. Then

$\displaystyle\left\langle\mathsf{J}^{\alpha
z}\right\rangle_{\text{gCS}}=\left\langle\mathsf{J}^{z\alpha}\right\rangle_{\text{gCS}}=\alpha_{1}\frac{1}{\left|e\right|}\varepsilon^{z\alpha\beta}\partial_{\beta}\tilde{\mathcal{R}},$
(2.103)

and the other components vanish. In terms of $t_{\text{cov}\;\nu}^{\mu}$,

$\displaystyle\left\langle
t_{\text{cov}\;z}^{\alpha}\right\rangle_{\text{gCS}}=-\alpha_{1}\sigma\varepsilon^{z\alpha\beta}\partial_{\beta}\tilde{\mathcal{R}},$
(2.104) $\displaystyle\left\langle
t_{\text{cov}\;\alpha}^{z}\right\rangle_{\text{gCS}}=-\alpha_{1}g_{\alpha\beta}\varepsilon^{z\beta\gamma}\partial_{\gamma}\tilde{\mathcal{R}}.$

Taking $z=t$ is natural in the context of the $p$-wave SC, since the emergent
metric (2.41) is always a product metric if $\Delta$ is time independent.
Then, with a general time independent order parameter,

$\displaystyle\left\langle J_{E}^{i}\right\rangle_{\text{gCS}}=\left\langle
t_{\text{cov}\;t}^{i}\right\rangle_{\text{gCS}}=-\alpha_{1}\varepsilon^{ij}\partial_{j}\tilde{\mathcal{R}},$
(2.105) $\displaystyle\left\langle
P_{i}\right\rangle_{\text{gCS}}=\left\langle
t_{\text{cov}\;i}^{t}\right\rangle_{\text{gCS}}=-\alpha_{1}g_{ik}\varepsilon^{kj}\partial_{j}\tilde{\mathcal{R}},$

where $\tilde{\mathcal{R}}$ is the curvature associated with the spatial
metric $g^{ij}=-\Delta^{(i}\Delta^{j)*}$. Again, for order parameters of the
form (2.101) the metrics for both layers $l=1,2$ are identical, and the
summation over $l=1,2$ produces

$\displaystyle\left\langle
J_{E}^{i}\right\rangle_{\text{gCS}}=-\frac{\nu/2}{96\pi}\varepsilon^{ij}\partial_{j}\tilde{\mathcal{R}},$
(2.106) $\displaystyle\left\langle
P_{i}\right\rangle_{\text{gCS}}=-\frac{\nu/2}{96\pi}g_{ik}\varepsilon^{kj}\partial_{j}\tilde{\mathcal{R}}.$

These are the topological bulk responses described in section 2.1.2. It is
also usefull to consider order parameters of the form

$\displaystyle\Delta=\Delta_{0}e^{i\theta}\left(1,e^{i\phi}\right),$ (2.107)

where $\phi$ is space dependent. Here the metrics satisfy
$g^{xy}=g_{\left(1\right)}^{xy}=-g_{\left(2\right)}^{xy}=\Delta_{0}^{2}\cos\phi$,
with the other components constant, and therefore the Ricci scalars satisfy
$\mathcal{R}=\mathcal{R}_{\left(1\right)}=-\mathcal{R}_{\left(2\right)}$. The
summation over $l=1,2$ for such order parameters then gives

$\displaystyle\left\langle
J_{E}^{i}\right\rangle_{\text{gCS}}=-\left(\alpha_{1}-\alpha_{2}\right)\varepsilon^{ij}\partial_{j}\tilde{\mathcal{R}}.$
(2.108)

Unlike the sum $\alpha_{1}+\alpha_{2}=\frac{\nu/2}{96\pi}$, the difference
$\alpha_{1}-\alpha_{2}=2\alpha_{1}-\frac{\nu/2}{96\pi}$ is not quantized. The
response (2.108) is therefore not a topological bulk response. Measuring
$\left\langle J_{E}\right\rangle$ for an order parameter such that
$\mathcal{R}=\mathcal{R}_{\left(1\right)}=\mathcal{R}_{\left(2\right)}$, and
then for an order parameter such that
$\mathcal{R}=\mathcal{R}_{\left(1\right)}=-\mathcal{R}_{\left(2\right)}$,
allows one to fix both $\alpha_{1},\alpha_{2}$, or both $\nu$ and
$\alpha_{1}$.

To demonstrate how closely (2.108) can resemble a topological bulk response,
we go back to the lattice model. In the relativistic limit we found that some
trivial phases, where $\nu=0$, have $\alpha_{1}=\frac{\nu_{1}/2}{192\pi}\neq
0$. It follows that these trivial phases have in the relativistic limit a
quantized response

$\displaystyle\left\langle
J_{E}^{i}\right\rangle_{\text{gCS}}=-2\alpha_{1}\varepsilon^{ij}\partial_{j}\tilde{\mathcal{R}}=-\frac{\nu_{1}}{96\pi}\varepsilon^{ij}\partial_{j}\tilde{\mathcal{R}},$
(2.109)

for order parameters $\Delta=\Delta_{0}e^{i\theta}\left(1,e^{i\phi}\right)$.

Another case of interest is when $z$ is a spatial coordinate. As an example,
we take $z=y$. This decomposition is less natural in the $p$-wave SC, as can
be seen from (2.41). It allows for time dependence, but restricts the
configuration the order parameter can take at any given time. A simple example
for an order parameter that gives rise to a product metric with respect to $y$
is $\Delta=\Delta_{0}e^{i\theta\left(t,x\right)}\left(1+f\left(t,x\right),\pm
i\right)$, which is a perturbation of the $p_{x}\pm ip_{y}$ configuration with
a small real function $f$. Then

$\displaystyle\left\langle
t_{\text{cov}\;\alpha}^{y}\right\rangle_{\text{gCS}}=-\frac{\nu/2}{96\pi}g_{\alpha\beta}\varepsilon^{\beta\gamma
y}\partial_{\gamma}\tilde{\mathcal{R}},$ (2.110)

where we have summed over $l=1,2$. This an interesting contribution to the
$x$-momentum current and energy current in the $y$ direction. If we consider,
as in Fig.3, a boundary or domain wall at $y=0$, between a topological phase
and a trivial phase where $\nu=0$, we see that there is an inflow of energy
and $x$-momentum into the boundary from the topological phase. This shows that
energy and $x$-momentum are accumulated on the boundary, at least locally,
which corresponds to the boundary gravitational anomaly. We complete the
analysis of this situation from the boundary point of view in Appendix A.3.

######  2.5.4.2 Bulk response from the gravitational pseudo Chern-Simons term

The gpCS term $-\beta_{1}\int_{M_{3}}\tilde{\mathcal{R}}e^{a}De_{a}$
contributes to the energy-momentum tensor, and also provides a contribution to
the spin density,

$\displaystyle\left\langle\mathsf{J}^{\mu\nu}\right\rangle_{\text{gpCS}}=$
$\displaystyle\beta_{1}\left\\{\frac{1}{\left|e\right|}\varepsilon^{\mu\nu\rho}\partial_{\rho}\tilde{\mathcal{R}}-\frac{1}{\left|e\right|}\varepsilon^{\mu\rho\sigma}\tilde{\mathcal{R}}T_{\rho\sigma}^{\nu}\right.+\left.2o\left[\left(\tilde{\nabla}^{\mu}\tilde{\nabla}^{\nu}-g^{\mu\nu}\tilde{\nabla}^{2}\right)-\tilde{\mathcal{R}}^{\mu\nu}\right]c\vphantom{\frac{1}{\left|e\right|}}\right\\}.$
(2.111)
$\displaystyle\left\langle\mathsf{J}^{ab\mu}\right\rangle_{\text{gpCS}}=$
$\displaystyle\beta_{1}o\tilde{\mathcal{R}}\varepsilon^{abc}e_{c}^{\;\mu}$

These are calculated in appendix D.

Using (2.79), the above contribution to the spin density corresponds to a
contribution to the charge density,

$\displaystyle\left\langle
J^{t}\right\rangle_{\text{gpCS}}=4\beta_{1}o\left|e\right|\tilde{\mathcal{R}}.$
(2.112)

The most notable feature of this density is that it is not accompanied by a
current, even for time dependent background fields, where
$\partial_{\mu}\left\langle J^{\mu}\right\rangle=\partial_{t}\left\langle
J^{t}\right\rangle\neq 0$. This represents the non conservation of fermionic
charge in a $p$-wave SC (2.44). The appearance of $o$ can be understood from
(2.82). One can also understand the appearance of $o$ based on time reversal
symmetry. Since both $J^{t}$ and $\tilde{\mathcal{R}}$ are time reversal even,
the coefficient of the above response cannot be $\beta_{1}$, which is time
reversal odd.

We now discuss the energy-momentum contributions
$\left\langle\mathsf{J}^{\mu\nu}\right\rangle_{\text{gpCS}}$ in (2.111), with
the purpose of comparing them to the gCS contributions
$\left\langle\mathsf{J}^{\mu\nu}\right\rangle_{\text{gCS}}$. To do this in the
simplest setting, we restrict to a product geometry with respect to the
coordinate $z$ as described in the previous section. We will also assume for
simplicity that torsion vanishes, and generalize to non-zero torsion in
appendix D. For a torsion-less product geometry
$\left\langle\mathsf{J}^{\mu\nu}\right\rangle_{\text{gpCS}}$ reduces to

$\displaystyle-\left\langle\mathsf{J}^{\alpha
z}\right\rangle_{\text{gpCS}}=\left\langle\mathsf{J}^{z\alpha}\right\rangle_{\text{gpCS}}=\beta_{1}\frac{1}{\left|e\right|}\varepsilon^{z\alpha\beta}\partial_{\beta}\tilde{\mathcal{R}}.$
(2.113)

Note that while the gpCS term vanishes in a torsion-less geometry, the
currents it produces, given by its functional derivatives, do not. Comparing
with (2.103), we see that
$\left\langle\mathsf{J}^{z\alpha}\right\rangle_{\text{gpCS}}\propto\left\langle\mathsf{J}^{z\alpha}\right\rangle_{\text{gCS}}$,
while $\left\langle\mathsf{J}^{\alpha
z}\right\rangle_{\text{gpCS}}\propto-\left\langle\mathsf{J}^{\alpha
z}\right\rangle_{\text{gCS}}$, with the proportionality constant
$\alpha_{1}/\beta_{1}$, that goes to 1 in the relativistic limit. This
demonstrates the similarity between the gpCS and gCS terms.

In particular, we find in a time independent situation the following
contributions to the energy current and momentum density,

$\displaystyle\left\langle J_{E}^{i}\right\rangle_{\text{gpCS}}=\left\langle
t_{\text{cov}\;t}^{i}\right\rangle_{\text{gpCS}}=\beta_{1}\varepsilon^{ij}\partial_{j}\tilde{\mathcal{R}},$
(2.114) $\displaystyle\left\langle
P_{i}\right\rangle_{\text{gpCS}}=\left\langle
t_{\text{cov}\;i}^{t}\right\rangle_{\text{gpCS}}=-\beta_{1}g_{ik}\varepsilon^{kj}\partial_{j}\tilde{\mathcal{R}}.$

Comparing with (2.105), we see that $\left\langle
P_{i}\right\rangle_{\text{gpCS}}\propto\left\langle
P_{i}\right\rangle_{\text{gCS}}$, while $\left\langle
J_{E}^{i}\right\rangle_{\text{gpCS}}\propto-\left\langle
J_{E}^{i}\right\rangle_{\text{gCS}}$. This sign difference can be understood
from the density response (2.112), and the relation (2.61) between the
operators $J_{E}$ and $P$, in the relativistic limit. With vanishing torsion
it reduces to

$\displaystyle
J_{E}^{j}-g^{jk}P_{k}=\frac{o}{2}\varepsilon^{jk}\partial_{k}\left(\frac{1}{\left|e\right|}J^{t}\right).$
(2.115)

Thus the gCS contributions (2.105) satisfy $\left\langle
J_{E}^{j}\right\rangle_{\text{gCS}}-g^{jk}\left\langle
P_{k}\right\rangle_{\text{gCS}}=0$ because gCS does not contribute to the
density. On the other hand, the gpCS does contribute to the density, which is
why $\left\langle J_{E}^{j}\right\rangle_{\text{gpCS}}-g^{jk}\left\langle
P_{k}\right\rangle_{\text{gpCS}}\neq 0$. This conclusion holds regardless of
the value of the coefficient $\beta_{1}$ of gpCS. One can therefore fix the
value of $\beta_{1}$ by a measurement of the density, and thus separate the
topological bulk responses (gCS) from the non-topological bulk responses
(gpCS).

More accurately, we have seen that the lattice model behaves as a bi-layer
with layer index $l=1,2$, and there are actually two coefficients
$\beta_{1},\beta_{2}$. As in the previous section, one can extract both
$\beta_{1},\beta_{2}$ by first considering an order parameter (2.101) such
that $\mathcal{R}=\mathcal{R}_{\left(1\right)}=\mathcal{R}_{\left(2\right)}$,
and then considering an order parameter (2.107) such that
$\mathcal{R}=\mathcal{R}_{\left(1\right)}=-\mathcal{R}_{\left(2\right)}$.

Another case of interest is when $z$ is a spatial coordinate, and as in the
previous section we take $z=y$,
$\Delta=\Delta_{0}e^{i\theta\left(t,x\right)}\left(1+f\left(t,x\right),\pm
i\right)$. We then find from (2.113),
$\left\langle\mathsf{J}^{y\alpha}\right\rangle_{\text{gpCS}}=\beta_{1}\frac{1}{\left|e\right|}\varepsilon^{z\alpha\beta}\partial_{\beta}\tilde{\mathcal{R}}$,
or

$\displaystyle\left\langle
t_{\text{cov}\;\alpha}^{y}\right\rangle_{\text{gpCS}}=-\beta_{1}g_{\alpha\beta}\varepsilon^{\beta\gamma
y}\partial_{\gamma}\tilde{\mathcal{R}}.$ (2.116)

In the presence of a boundary (or domain wall) at $y=0$, this describes an
inflow of energy and $x$-momentum from the bulk to the boundary, such that
$\left\langle
t_{\text{cov}\;\alpha}^{y}\right\rangle_{\text{gpCS}}\propto\left\langle
t_{\text{cov}\;\alpha}^{y}\right\rangle_{\text{gCS}}$. After summing over
$l=1,2$ one finds the proportionality constant
$\frac{\alpha_{1}+\alpha_{2}}{\beta_{1}+\beta_{2}}$, that goes to 1 in the
relativistic limit. Nevertheless, we argue that $\left\langle
t_{\text{cov}\;\alpha}^{y}\right\rangle_{\text{gCS}}$ corresponds to a
boundary gravitational anomaly while $\left\langle
t_{\text{cov}\;\alpha}^{y}\right\rangle_{\text{gpCS}}$ does not, in accordance
with section 2.5.3.2. The relation between gCS and the boundary gravitational
anomaly is well known within the gravitational description, and is described
from the $p$-wave SC point of view in Appendix A.3. The fact that
$\left\langle t_{\text{cov}\;\alpha}^{y}\right\rangle_{\text{gpCS}}$ is
unrelated to any boundary anomaly follows from the fact that it is
$SO\left(1,2\right)$ and $Diff$ invariant. Due to this invariance the bulk
gpCS term produces not only the bulk currents (2.111), but also boundary
currents, such that bulk+boundary energy-momentum is conserved. In a product
geometry with $z=y$ we find the boundary currents

$\displaystyle\left\langle\mathsf{j}^{\alpha\beta}\right\rangle_{\text{gpCS}}=$
$\displaystyle-\beta_{1}\frac{1}{\left|e\right|}\varepsilon^{\alpha\beta
y}\tilde{\mathcal{R}},$ (2.117)
$\displaystyle\left\langle\mathsf{j}^{ab\mu}\right\rangle_{\text{gpCS}}=$
$\displaystyle 0,$

which are calculated in appendix D. We see that

$\displaystyle\tilde{\nabla}_{\alpha}\left\langle\mathsf{j}^{\alpha\beta}\right\rangle_{\text{gpCS}}=\left\langle\mathsf{J}^{y\beta}\right\rangle_{\text{gpCS}}.$
(2.118)

This conservation law is the statement of bulk+boundary conservation of
energy-momentum within the gravitational description. It can be understood
from (2.73), by noting that the source terms in (2.73) vanish because
$\left\langle\mathsf{j}^{ab\mu}\right\rangle_{\text{gpCS}}=0$, and because we
assumed torsion vanishes. The additional source term
$\left\langle\mathsf{J}^{y\beta}\right\rangle_{\text{gpCS}}$, absent in
(2.73), represents the inflow from the bulk. In Appendix A.3 we translate
(2.118) to the language of the $p$-wave SC.

####  2.6 Discussion

###### Beyond the relativistic limit

In this section we have shown that there is a topological bulk response of
$p$-wave CSF/Cs to a perturbation of their order parameter, which follows from
a gCS term, and we described a corresponding gravitational anomaly of the edge
states. The coefficient of gCS was found to be $\alpha=\frac{c}{96\pi}$ where
$c$ is the chiral central charge, as anticipated. We provided arguments, based
on symmetry and topology, for the validity of these results beyond the
relativistic limit in which they were computed. The appearance of torsion in
the emergent geometry brought about a surprise: an additional term, closely
related but distinct from gCS, which we referred to as gravitational pseudo
Chern-Simons (gpCS), with a dimensionless coefficient
$\beta=\frac{c}{96\pi}=\alpha$. The gpCS term is completely invariant under
the symmetries we considered, and is therefore unrelated to edge anomalies.
Therefore, the quantization of $\beta$ seems to be a property of the
relativistic limit252525The quantization of $\beta$ can be understood on
dimensional grounds \- in the relativistic limit there are simply not enough
dimension-full quantities which can be used to construct dimensionless
quantities, beyond $\text{sgn}\left(m\right)$ and $o$. Of course, this does
not explain why $\beta=\alpha$ in the relativistic limit. , which will not
hold throughout the phase diagram.

The above results are based on a careful mapping of $p$-wave CSF/Cs, in the
regime where the order parameter is large, to relativistic Majorana spinors in
a curved and torsion-full space-time. Though the relativistic limit captures a
rich geometric physics in CSF/Cs, it is in fact very limited in its ability to
describe the full non-relativistic system, as discussed in Appendix 2.2.4.
This raises the question of whether the results obtained in this section truly
characterize $p$-wave CSF/Cs, especially in the physically important weak
coupling limit where $\Delta$ is small, which is opposite to the relativistic
limit. Moreover, it is natural to ask whether the results obtained here apply
to $\ell$-wave CSF/Cs with $\left|\ell\right|>1$, which do not admit a
relativistic low energy description. These questions will be answered, in
part, in Sec.3, where we perform a general non-relativistic analysis of
continuum models for $\ell$-wave CSF/Cs. In order to address these questions
in the context of lattice models, numerical studies seem to be required.
Recently, such studies where initiated in the context of emergent gravity in
Kitaev’s honeycomb model [171, 172], which is closely related to the $p$-wave
CSF/Cs discussed in this section. For non-topological physical properties, the
results show that the lattice model is well described by relativistic
predictions, but only in the relativistic regime, where the masses of low
energy relativistic fermions are small and the correlation length is large, as
might be expected. An interesting question is whether the topological physics
associated with the gCS term is more robustly captured by the relativistic
predictions made in this section.

###### Real background geometry and manipulation of the order parameter

In this section we considered $p$-wave CSF/Cs in flat space, and focused on
the emergent geometry described by a general $p$-wave order parameter. It is
also natural to consider the effect of a real background geometry, obtained by
deforming, or straining, the 2-dimensional sample in 3-dimensional space,
possibly in a time dependent manner. Treating this at the level of the lattice
model is beyond the scope of this thesis, but we can take the non-relativistic
action (2.1) as a starting point. On a general deformed sample it generalizes
to

$\displaystyle
S\left[\psi;\Delta,A,G\right]=\int\mbox{d}^{2+1}x\sqrt{G}\left[\psi^{\dagger}\frac{i}{2}\overleftrightarrow{D_{t}}\psi-\frac{1}{2m^{*}}G^{ij}D_{i}\psi^{\dagger}D_{j}\psi+\left(\frac{1}{2}\Delta^{j}\psi^{\dagger}\partial_{j}\psi^{\dagger}+h.c\right)\right],$
(2.119)

which now depends on three background fields: the order parameter
$\Delta^{j}$, the $U\left(1\right)$ connection $A_{\mu}$, which enters
$D_{\mu}=\partial_{\mu}-iA_{\mu}$, and includes a chemical potential, and the
real background metric $G$, coming from the embedding of the 2-dimensional
sample in 3-dimensional space, which corresponds to the strain tensor
$u^{ij}=\left(G^{ij}-\delta^{ij}\right)/2$. This action is written for the
fermion $\psi$, which satisfies
$\left\\{\psi^{\dagger}\left(x\right),\psi\left(y\right)\right\\}=\delta^{\left(2\right)}\left(x-y\right)/\sqrt{G\left(x\right)}$
as an operator. In this problem there are two (inverse) metrics, the real
$G^{ij}$ and emergent $g^{ij}=\Delta^{(i}\Delta^{j)*}$, and it is interesting
to study their interplay. In our analysis we focused on the relativistic
limit, where $m^{*}\rightarrow\infty$. In this limit the metric $G$ completely
decouples from the action, when written in terms of the fundamental fermion
density $\tilde{\psi}=G^{1/4}\psi$ [242, 243, 34] . Thus, results obtained
within the relativistic limit, are essentially unaffected by the background
metric $G$. This conclusion is appropriate as long as the order parameter is
treated as an independent background field, which is always suitable for the
purpose of integrating out the gapped fermion density $\psi$. One then obtains
the bulk currents and densities that we have described, which depend on the
configuration of $\Delta$, and the question that remains is what this
configuration physically is. Two scenarios are of importance.

The first scenario is that of a proximity induced CSC, where the order
parameter is induced by proximity to a conventional $s$-wave SC. In this case
both $G$ and $g$ are background metrics, a scenario similar to a bi-metric
description of anisotropic quantum Hall states [46]. In this case the
magnitude of the order parameter depends on the distance between the sample
and the $s$-wave SC, so if the position of the $s$-wave SC is fixed but the
sample is deformed, a space-time dependent order parameter is obtained. One
can also obtain the same effect by considering a flat sample, where $G$ is
euclidian, and an $s$-wave SC with a curved surface, leading to non-Euclidian
$g$. This provides one rout to a manipulation of the order parameter that will
result in the bulk effects we have described.

The second scenario is that of intrinsic CSCs, and CSFs, where the order
parameter is dynamical. The order parameter splits into a massive Higgs field,
which is the emergent metric $g^{ij}$, and a massless Goldstone field which is
the overall phase $\theta$. The quantum theory of the emergent metric $g^{ij}$
is on its own an interesting problem, which will be discussed in Sec.5.
Nevertheless, as long as the probes $A,G$ are slow compared to the Higgs mass,
$g^{ij}$ can be treated as fixed to its instantaneous ground state
configuration, which in general will depend on the details of the attractive
fermionic interaction. For an interaction that depends only on the geodesic
distance, the ground state configuration is expected to be the curved space
$p_{x}\pm ip_{y}$ configuration, where the pairing term is
$\Delta_{0}e^{i\theta}\psi^{\dagger}\left(E_{1}^{\;j}\partial_{j}\pm
iE_{2}^{\;j}\partial_{j}\right)\psi^{\dagger}$, see Appendix J and Refs.[15,
163, 164, 238, 240]. Here $\Delta_{0}$ is a constant, $\theta$ is the
Goldstone phase, and $E$ is a vielbein for the real metric $G$, such that
$G^{ij}=E_{A}^{\;i}\delta^{AB}E_{B}^{\;j}$, which is a fixed background
field262626The $SO\left(2\right)_{L}$ ambiguity in choosing $E$ can be
incorporated into $\theta$, which has $SO\left(2\right)_{L}$ charge 1. What
this means, in the language of this section, is that the emergent metric is
proportional to the real metric, $g^{ij}=\Delta_{0}^{2}G^{ij}$. It follows
that the responses to the emergent metric $g$ that we have described are, in
this case, responses to the real metric $G$. This suggests a second rout to
the manipulation of the order parameter that will result in the bulk effects
we have described.

Of course, in the intrinsic case one cannot ignore the dynamics of the
Goldstone phase $\theta$, which will be gapless as long as $A$ is treated as a
background field, as appropriate in CSFs. It is then interesting to see how
the results obtained in this section are modified by the dynamics of $\theta$.
The interplay between the topological gCS term and the $\theta$-dependent gpCS
term suggests a non-trivial modification, which will be explored in Sec.3.

###### Towards experimental observation

There are a few basic questions that arise when trying to make contact between
the phenomena described in this section and a possible experimental
observation. Here we take as granted that one has at one’s disposal either a
$p$-wave CSF/C, or a candidate material. The first question is how to
manipulate the Higgs part of the order parameter, which is the emergent
metric, and was discussed above.

The second natural question is how to measure energy currents and momentum
densities. Also relevant, though not accentuated in this section, is a
measurement of the stress tensor, comprised of the spatial components of the
energy-momentum tensor. One possible approach, which provides both a means to
manipulate the order parameter, and a measurement of energy-momentum-stress is
a measurement of the phonon spectrum a la [244, 245, 246]. For the gpCS term,
apart from energy-momentum-stress, there is also the density response (2.8),
which is a simpler quantity for measurement, though not a topological bulk
response. A possible way to avoid the need to measure energy-momentum-stress
may be possible in a Galilean invariant system, where electric current and
momentum density are closely related. The simplest scenario is that of a
$p$-wave CSF on a curved sample (2.119), where one assumes that the emergent
metric follows the real metric, $g^{ij}=\Delta_{0}^{2}G^{ij}$. Here the
electric current is related to the momentum density by

$\displaystyle J^{i}=-\frac{G^{ij}}{m^{*}}P_{j}.$ (2.120)

Our result (2.5) then implies that the expectation value $\left\langle
J^{i}\right\rangle$ has a contribution related to the gCS term,

$\displaystyle\left\langle
J^{i}\right\rangle_{\text{gCS}}=-\frac{G^{ij}}{m^{*}}\left\langle
P_{j}\right\rangle_{\text{gCS}}=\frac{1}{m^{*}}\frac{c}{96\pi}\hbar\varepsilon^{ij}\partial_{j}\tilde{\mathcal{R}}.$

This is not a topological response per se, due to the appearance of $m^{*}$.
But, if $m^{*}$ is known, then the central charge $c$ can be extracted from a
measurement of the electric current, which may be simpler to measure than
energy-momentum-stress. This motivates the study of Galilean invariant CSFs,
which will pursued in Sec.3.

###### Hall viscosity and torsional Hall viscosity

In the effective action (2.5.2.2) for a Majorana spinor in Riemann-Cartan
space-time, there is a term $\left(\zeta_{H}/2\right)\int e_{a}De^{a}$, which
describes an energy-momentum-stress response termed torsional Hall viscosity
[155, 156, 157], due to its similarities with the odd (or Hall) viscosity
$\eta_{\text{o}}$ that occurs in quantum Hall states and CSF/Cs [116, 247, 34,
163, 164]. As opposed to the well understood $\eta_{\text{o}}$, the torsional
Hall viscosity $\zeta_{H}$ remains somewhat controversial [158, 160, 41],
because it only appears if a non-symmetric stress tensor is used, and because
it is UV-sensitive. The latter is also the reason that, in this section, we
avoided from interpreting $\zeta_{H}$ in the context of $p$-wave CSF/Cs.
However, the mapping between $p$-wave CSF/Cs and relativistic Majorana spinors
in Riemann-Cartan space-time developed in this section, along with the
existence of a Hall viscosity on one side of the mapping, and of a torsional
Hall viscosity on the other, strongly suggests that the two types of Hall
viscosity should map to one another, in this context. Moreover, it is known
that the gCS term implies universal corrections to $\eta_{\text{o}}$ at non-
zero wave-vector or in curved background [34, 42, 43], and that the gpCS term
implies curvature corrections to $\zeta_{H}$ [156], which we expect to be non-
universal. These observations provide us with a strong motivation to study the
interplay between torsional Hall viscosity, Hall viscosity, and chiral central
charge in CSFs, which we do in Sec.3.

###  3 Boundary central charge from bulk odd viscosity: chiral superfluids

As discussed in Sec.2.6, our analysis of CSF/Cs within the relativistic limit
raises many questions which require a fully non-relativistic treatment. These
questions revolve around the quantization of coefficients, application of
strain (or a background metric), odd viscosity, dynamics of the Goldstone
mode, and Galilean symmetry, and will be addressed in this section. In
particular, we will answer the questions posed in Sec.1.5, by deriving a low
energy effective field theory that consistently captures both the chiral
Goldstone mode implied the symmetry breaking pattern (1.2), and the gCS term
discussed in Sec.2. This theory unifies and extends the seemingly unrelated
analysis of Refs.[199, 163, 164] and Sec.2.

####  3.1 Building blocks for the effective field theory

In order to probe a CSF, we minimally couple it to two background fields - a
time-dependent spatial metric $G_{ij}$, which we use to apply strain
$u_{ij}=\left(G_{ij}-\delta_{ij}\right)/2$ and strain-rate
$\partial_{t}u_{ij}$, and a $U\left(1\right)_{N}$-gauge field
$A_{\mu}=\left(A_{t},A_{i}\right)$, where we absorb a chemical potential
$A_{t}=-\mu+\cdots$. The microscopic action $S$ is then invariant under
$U\left(1\right)_{N}$ gauge transformations, implying the number conservation
$\partial_{\mu}(\sqrt{G}J^{\mu})=0$, where $\sqrt{G}J^{\mu}=-\delta S/\delta
A_{\mu}$. It is also clear that $S$ is invariant under spatial diffeomorphisms
generated by $\delta x^{i}=\xi^{i}\left(\mathbf{x}\right)$, if $G_{ij}$
transforms as a tensor and $A_{\mu}$ as a 1-form. Less obvious is the fact
that a Galilean invariant fluid is additionally symmetric under $\delta
x^{i}=\xi^{i}\left(t,\mathbf{x}\right)$, provided one adds to the
transformation rule of $A_{i}$ a non-standard piece that dependens on the non-
relativistic mass $m$ [199, 248, 163, 35, 249, 250, 251, 252],

$\displaystyle\delta
A_{i}=-\xi^{k}\partial_{k}A_{i}-A_{k}\partial_{i}\xi^{k}+mG_{ij}\partial_{t}\xi^{j}.$
(3.1)

We refer to $\delta x^{i}=\xi^{i}\left(\mathbf{x},t\right)$ as local Galilean
symmetry (LGS), as it can be viewed as a local version of the Galilean
transformation $\delta x^{i}=v^{i}t$. The LGS implies the momentum
conservation law272727We 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
$\varepsilon^{ij}=\varepsilon^{ijt}$.

$\displaystyle\frac{1}{\sqrt{G}}\partial_{t}\left(\sqrt{G}mJ^{i}\right)+\nabla_{j}T^{ji}=nE_{i}+\varepsilon^{ij}J_{j}B,$
(3.2)

where $\sqrt{G}T^{ij}=2\delta S/\delta G^{ij}$ is the stress tensor and the
right hand side is the Lorentz force. This identifies the momentum density
$P^{i}=mJ^{i}$ \- a familiar Galilean relation.

Since CSFs spontaneously break the rotation symmetry in flat space, in order
to describe them in curved, or strained, space, it is necessary to introduce a
background vielbein. This is a field $E_{\;j}^{A}$ valued in
$GL\left(2\right)$, such that $G_{ij}=E_{\;i}^{A}\delta_{AB}E_{\;j}^{B}$,
where $A,B\in\left\\{1,2\right\\}$. For a given metric $G$ the vielbein $E$ is
not unique - there is an internal
$O\left(2\right)_{P,L}=\mathbb{Z}_{2,P}\ltimes SO\left(2\right)_{L}$
ambiguity, or symmetry, acting by $E_{\;j}^{A}\mapsto O_{\;B}^{A}E_{\;j}^{B}$,
$O\in O\left(2\right)_{P,L}$. The generators $L,P$ correspond to internal
spatial rotations and reflections, and are analogs of angular momentum and
spatial reflection (parity), acting on the tangent space rather than on space
itself. The inverse vielbein $E_{B}^{\;j}$ is defined by
$E_{\;j}^{A}E_{B}^{\;j}=\delta_{B}^{A}$.

The charge $N+\left(\ell/2\right)L$ of the Goldstone field $\theta$ implies
the covariant derivative

$\displaystyle\nabla_{\mu}\theta=\partial_{\mu}\theta-
A_{\mu}-s_{\theta}\omega_{\mu},$ (3.3)

with a geometric spin $s_{\theta}=\ell/2$. Here $\omega_{\mu}$ is the non-
relativistic spin connection, an $SO\left(2\right)_{L}$-gauge field which is
$E_{\;j}^{A}$-compatible, see Appendix F. So far we assumed that the
microscopic fermion $\psi$ does not carry a geometric spin, $s_{\psi}=0$,
which defines the physical system of interest, see Fig.6(a). It will be
useful, however, to generalize to $s_{\psi}\in\left(1/2\right)\mathbb{Z}$. A
non-zero $s_{\psi}$ changes the SSB pattern (1.2), by modifying the geometric
spin of the Goldstone field to $s_{\theta}=s_{\psi}+\ell/2$ and the unbroken
generator to $L-s_{\theta}N$. In the special case $s_{\psi}=-\ell/2$ the
Cooper pair is geometrically spin-less and $L$ is unbroken, as in an $s$-wave
SF, see Fig.6(b). This $s_{\theta}=0$ CSF is, however, distinct from a
conventional $s$-wave SF, because $P$ and $T$ are still broken down to $PT$,
and we therefore refer to it as a geometric $s$-wave (g$s$-wave) CSF, to
distinguish the two. In particular, a central charge $c\neq 0$, which is
$P,T$-odd, is not forbidden by symmetry, and is in fact independent of
$s_{\psi}$. This makes the g$s$-wave CSF particularly useful for our purposes.

We note that $\omega_{\mu}$ transforms as a 1-form under LGS only if $B/2m$ is
added to $\omega_{t}$ [163, 164], which we do implicitly throughout. For
$\psi$, this is equivalent to adding a g-factor $g_{\psi}=2s_{\psi}$ [250].

Figure 6: (a) A CSF is comprised of fermions $\psi$ which carry no geometric
spin, $s_{\psi}=0$, which form Cooper pairs with relative angular momentum
$\ell\in\mathbb{Z}$ (red arrows). The geometric spin $s_{\theta}=\ell/2$ of
the Cooper pair gives rise to the $\mathbf{q}=\mathbf{0}$ odd viscosity (1.3),
with $s=s_{\theta}$. The CSF supports boundary degrees of freedom (dashed
orange) with a chiral central charge $c\in\left(\ell/2\right)\mathbb{Z}$,
which cannot be extracted from the odd viscosity
$\eta_{\text{o}}\left(\mathbf{q}\right)$ alone (3.13). (b) In an auxiliary CSF
the fermion $\tilde{\psi}$ is assigned a geometric spin
$\tilde{s}_{\psi}=-\ell/2$ (blue arrows). The geometric spin of the Cooper
pair therefore vanishes, $\tilde{s}_{\theta}=\ell/2+\tilde{s}_{\psi}=0$, as in
an $s$-wave superfluid, but the central charge is unchanged, $\tilde{c}=c$. As
a result, the small $\mathbf{q}$ behavior of the odd viscosity
$\tilde{\eta}_{\text{o}}\left(\mathbf{q}\right)$ depends only on $c$ (3.14).
The improved odd viscosity of the CSF is defined as the odd viscosity of the
auxiliary CSF, and is given explicitly by (3.17).

####  3.2 Effective field theory

Based on the above characterization of CSFs, the low energy, long wave-length,
behavior of the system can be captured by an effective action
$S_{\text{eff}}\left[\theta;A,G;\ell,c\right]$, formally obtained by
integrating out all massive degrees of freedom - the single fermion
excitations and the Higgs fields. In this Section we describe a general
expression for $S_{\text{eff}}$, compatible with the symmetries, SSB pattern,
and ground state topology of CSFs.

The effective action can be written order by order in a derivative expansion,
with the power counting scheme
$\partial_{\mu}=O\left(p\right),\;A_{\mu},G_{ij}=O\left(1\right),\;\theta=O\left(p^{-1}\right)$
[199, 163]. The spin connection is a functional of $G_{ij}$ that involves a
single derivative (see Appendix F), so $\omega_{\mu}=O\left(p\right)$.
Denoting by $\mathcal{L}_{n}$ the term in the Lagrangian which is
$O\left(p^{n}\right)$ and invariant under all symmetries, we have
$S_{\text{eff}}=\sum_{n=0}^{\infty}\int\text{d}^{2}x\text{d}t\sqrt{G}\mathcal{L}_{n}$.
The desired $q^{2}$ corrections to $\eta_{\text{o}}$ are
$O\left(p^{3}\right)$, which poses the main technical difficulty.

The leading order Lagrangian

$\displaystyle\mathcal{L}_{0}=P\left(X\right),\;X=\nabla_{t}\theta-\frac{1}{2m}G^{ij}\nabla_{i}\theta\nabla_{j}\theta,$
(3.4)

was studied in detail in [163], and contains the earlier results of [192, 193,
194, 195, 153, 196, 197]. Here $X$ is the unique $O\left(1\right)$ scalar,
which reduces to the chemical potential $\mu$ in the ground state(s)
$\partial_{\mu}\theta=0$, and $P$ is an arbitrary function of $X$ that
physically corresponds to the ground state pressure $P_{0}=P\left(\mu\right)$.
The function $P$ also determines the ground state density
$n_{0}=P^{\prime}\left(\mu\right)$, and the leading dispersion of the
Goldstone mode $\omega^{2}=c_{s}^{2}q^{2}$, where
$c_{s}^{2}=\partial_{n_{0}}P_{0}/m=P^{\prime}(\mu)/P^{\prime\prime}(\mu)m$ is
the speed of sound. For $\ell\neq 0$, the spin connection appears in each
$\nabla\theta$, see Eq.(3.3), and so $\mathcal{L}_{0}$ includes
$O\left(p\right)$ contributions, which produce the leading odd viscosity and
conductivity, discussed below. There are no additional terms at
$O\left(p\right)$, so that $\mathcal{L}_{1}=0$ [163].

At $O\left(p^{2}\right)$ one has

$\displaystyle\mathcal{L}_{2}=$ $\displaystyle
F_{1}\left(X\right)R+F_{2}\left(X\right)\left[mK_{\;i}^{i}-\nabla^{2}\theta\right]^{2}+F_{3}\left(X\right)\left[2m\left(\nabla_{i}K_{\;j}^{j}-\nabla^{j}K_{ji}\right)\nabla^{i}\theta\right]+\cdots,$
(3.5)

where $K_{ij}=\partial_{t}G_{ij}/2$ and $R$ are the extrinsic curvature and
Ricci scalar of the spatial slice at time $t$ [229], the $F$s are arbitrary
functions of $X$, and dots indicate additional terms which do not contribute
to $\eta_{\text{o}}$ up to $O\left(p^{2}\right)$, see Appendix H.2 for the
full expression. The Lagrangian $\mathcal{L}_{2}$ was obtained in [199] for
$s$-wave SFs. For $\ell\neq 0$ the spin connection in $\nabla\theta$ produces
$O\left(p^{3}\right)$ contributions to $\mathcal{L}_{2}$, and, in turn, non-
universal $q^{2}$ corrections to $\eta_{\text{o}}$.

The term $\mathcal{L}_{3}$ is the last ingredient required for reliable
results at $O\left(p^{3}\right)$. Most importantly, it includes the non-
relativistic gCS term [42, 44],
$\mathcal{L}_{3}=\text{$\mathcal{L}$}_{\text{gCS}}+\cdots$, where

$\displaystyle\text{$\mathcal{L}$}_{\text{gCS}}$
$\displaystyle=-\frac{c}{48\pi}\omega\text{d}\omega.$ (3.6)

Here
$\omega\text{d}\omega=\varepsilon^{\mu\nu\rho}\omega_{\mu}\partial_{\nu}\omega_{\rho}$,
and $c$ is the chiral central charge of the boundary degrees of freedom, as
required to match the boundary gravitational anomaly [23, 25, 1]. Unlike the
lower order terms, $\mathcal{L}_{\text{gCS}}$ is independent of $\theta$, and
encodes only the response of the fermionic topological phase to the background
metric. The direct confirmation of (3.6) within a non-relativistic microscopic
model has been anticipated for some time [28, 15], and is a main result of
Appendix J. In Appendices G.5 and H.4 we argue that additional terms in
$\mathcal{L}_{3}$ do not produce $q^{2}$ corrections to $\eta_{\text{o}}$.

There are three symmetry-allowed topological terms that can be added to
$S_{\text{eff}}$ [253, 254, 34, 35, 39, 255, 43, 42, 256, 47, 52]. These are
the $U\left(1\right)$ Chern-Simons (CS) and first and second Wen-Zee (WZ1,
WZ2) terms, which can be added to $\mathcal{L}_{1}$, $\mathcal{L}_{2}$,
$\mathcal{L}_{3}$, respectively 282828These are not LGS invariant, and would
need to be modified along the lines of [248, 249, 251]. ,

$\displaystyle\frac{\nu}{4\pi}\left(A\text{d}A-2\overline{s}\omega\text{d}A+\overline{s^{2}}\omega\text{d}\omega\right).$
(3.7)

As our notation suggests, WZ2 and gCS are identical for the purpose of local
bulk responses, of interest here, but the two are globally distinct [42, 44,
52]. Based on symmetry, and ignoring boundary physics, the independent
coefficients $\nu$, $\nu\overline{s}$, and $\nu\overline{s^{2}}$ obey certain
quantization conditions [24], but are otherwise unconstrained. The absence of
a boundary $U\left(1\right)_{N}$-anomaly then fixes $\nu=0$ [1], but leaves
$\nu\overline{s},\nu\overline{s^{2}}$ undetermined [42, 44, 52]. One can argue
that a Chern-Simons term can only appear for the unbroken generator
$L-s_{\theta}N$, so that $\nu=0$ implies
$\nu\overline{s}=\nu\overline{s^{2}}=0$. Moreover, in the following section we
will see that a perturbative computation within a canonical model for
$\ell=\pm 1$ shows that $\nu\overline{s}=\nu\overline{s^{2}}=0$, which applies
to any deformation of the model (which preserves the symmetries, SSB pattern,
and single fermion gap), due to the quantization of
$\nu\overline{s},\nu\overline{s^{2}}$. Accordingly, we set
$\nu\overline{s}=\nu\overline{s^{2}}=0$ in the following.

####  3.3 Benchmarking the effective theory against a microscopic model

In this section we take a complementary approach and compute $S_{\text{eff}}$
perturbatively, starting from a canonical microscopic model for a $p$-wave
CSF. The perturbative computation verifies the general expression in a
particular example, and determines the coefficients of topological terms which
are not completely fixed by symmetry. It also gives one a sense of the
behavior of the coefficients of non-topological terms as a function of
microscopic parameters such as the chemical potential $\mu$ and mass $m$. In
particular, the results of Sec.2 are reproduced in the relativistic limit
$m\rightarrow\infty$.

Here we will outline the computation and describe its results, omitting many
technical details which can be found in Appendix J. The microscopic model is
given by

$\displaystyle S_{\text{m}}=\int\mbox{d}^{2}x$
$\displaystyle\text{d}t\sqrt{G}\left[\frac{i}{2}\psi^{\dagger}\overleftrightarrow{\nabla_{t}}\psi-\frac{1}{2m}G^{ij}\nabla_{i}\psi^{\dagger}\nabla_{j}\psi+\left(\frac{1}{2}\Delta^{j}\psi^{\dagger}\nabla_{j}\psi^{\dagger}+h.c\right)-\frac{1}{2\lambda}G_{ij}\Delta^{i*}\Delta^{j}\right],$
(3.8)

where the covariant derivative of the spin-less (or single component) fermion
$\psi$ is $\nabla_{\mu}=\partial_{\mu}+iA_{\mu}+is_{\psi}\omega_{\mu}$. Note
that we allow for a geometric spin $s_{\psi}$, as discussed in Sec.3.1. Apart
from the standard non-relativistic kinetic term, the action includes the
simplest attractive two-body interaction [225, 238], mediated by the complex
vector $\Delta^{i}$, the order parameter, with coupling constant $\lambda>0$.
A simplified version of this action was already discussed in Sec.2.6.

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
boundary chiral central charge [15, 27, 107]

$\displaystyle c=-\left(\ell/2\right)\Theta\left(\mu\right)\in\left\\{0,\pm
1/2\right\\}.$ (3.9)

An effective action $S_{\text{eff},\text{m}}\left[\Delta;A,G\right]$ for
$\Delta^{j}$ in the background $A_{\mu},G_{ij}$ is then obtained by
integrating over the fermion. The subscript ’m’ indicates that this is
obtained from the particular microscopic model $S_{\text{m}}$. Since Eq.(3.8)
is quadratic in $\psi,\psi^{\dagger}$, obtaining $S_{\text{eff},\text{m}}$ is
formally straight forward, and leads to a functional Pfaffian.

To zeroth order in derivatives
$S_{\text{eff},\text{m}}\left[\Delta;A,G\right]=-V\left[\Delta;G\right]$,
where the potential $V$ is minimized by the $p_{x}\pm ip_{y}$ order parameter.
In flat space this is given by the familiar
$\Delta^{j}\partial_{j}=\Delta_{0}e^{-2i\theta}\left(\partial_{x}\pm
i\partial_{y}\right)$. Here $\Delta_{0}$ is a fixed function of $m,\mu$ and
$\lambda$, determined by the minimization, while the phase $\theta$ and
chirality $\ell=\pm 1$ are undetermined. In order to write down the $p_{x}\pm
ip_{y}$ configuration in curved space it is necessary to use the background
vielbein [163, 238]

$\displaystyle\Delta^{j}$
$\displaystyle=\Delta_{0}e^{-2i\theta}\left(E_{1}^{\;j}\pm
iE_{2}^{\;j}\right).$ (3.10)

Fluctuations of $\Delta$ away from these configurations correspond to massive
Higgs modes, which should in principle be integrated out to obtain a low
energy action $S_{\text{eff},\text{m}}\left[\theta;A,G\right]$ that can be
compared with the general $S_{\text{eff}}$ of the previous section. We will
simply ignore these fluctuations, and obtain
$S_{\text{eff},\text{m}}\left[\theta;A,G\right]$ by plugging Eq.(3.10) into
$S_{\text{eff},\text{m}}\left[\Delta;A,G\right]$. This will suffice as a
derivation of $S_{\text{eff}}$ from a microscopic model. A proper treatment of
the massive Higgs modes will only further renormalize the coefficients we
find, apart from the central charge $c$.

To practically compare the actions $S_{\text{eff}}$ and
$S_{\text{eff},\text{m}}$ we expand them in fields, to second order around
$\theta=0,A_{\nu}=-\mu\delta_{\nu}^{t},G_{ij}=\delta_{ij}$, and in
derivatives, to third order, see appendices H and J.5. Equating these two
double expansions leads to an overdetermined system of equations for the
phenomenological parameters in $S_{\text{eff}}$ in terms of the microscopic
parameters in $S_{\text{m}}$, with a unique solution. In particular, we find
the dimensionless parameters

$\displaystyle\frac{P^{\prime\prime}}{m}=$
$\displaystyle\frac{1}{2\pi}\begin{cases}1\\\
\frac{1}{1+2\kappa}\end{cases},\;\;\;\;\;F_{1}^{\prime}=\frac{1}{96\pi}\begin{cases}1\\\
\frac{3}{1+2\kappa}\end{cases},\;\;\;\;\;mF_{2}=-\frac{1}{128\pi}\begin{cases}1+2\kappa\\\
\frac{1}{1+2\kappa}\end{cases},$ (3.11) $\displaystyle mF_{3}=$
$\displaystyle\frac{1}{48\pi}\begin{cases}1+\kappa\\\
\frac{1}{1+2\kappa}\end{cases},\;\;\;\;\;c=\begin{cases}-\ell/2\\\
0\end{cases},$

where $\kappa=\left|\mu\right|/m\Delta_{0}^{2}>0$, primes denote derivatives
with respect to $\mu$, and the cases refer to $\mu>0$ and $\mu<0$. We note
that for $\mu>0$ there is a single particle Fermi surface, with energy
$\varepsilon_{F}=\mu$ and wave-vector $k_{F}=\sqrt{2m\mu}$, which for small
$\lambda$ will acquire an energy gap
$\varepsilon_{\Delta}=\Delta_{0}k_{F}\ll\varepsilon_{F}$. In this weak-
coupling regime, it is natural to parametrize the coefficients in (3.11) using
the small parameter $\varepsilon_{\Delta}/\varepsilon_{F}=\sqrt{2/\kappa}$.

The coefficient $P^{\prime\prime}$ determines the leading odd (or Hall)
conductivity and has been computed previously in the literature [192, 193,
194, 195, 153, 196, 197, 163, 198], while $F_{1},F_{2}$ and $F_{3}$, to the
best of our knowledge, have not been computed previously, even for an $s$-wave
SF. Crucially, Eq.(3.11) shows that the coefficient $c$ of the bulk gCS term
matches the known boundary central charge (3.9), which is a main result of the
perturbative computation. It follows that there is no WZ2 term in
$S_{\text{eff},\text{m}}$, so $\nu\overline{s^{2}}=0$, in accordance with the
general discussion of the previous section. We additionally confirm that
$\nu=\nu\overline{s}=0$.

A few additional comments regarding Eq.(3.11) are in order:

  1. 1.

The seeming quantization of $P^{\prime\prime}/m$ and $F_{1}^{\prime}$ for
$\mu>0$ is a non-generic result, as was shown explicitly for
$P^{\prime\prime}/m$ [198].

  2. 2.

The free fermion limit $\kappa\rightarrow\infty$, or $\Delta_{0}\rightarrow
0$, of certain coefficients in (3.11) diverges for $\mu>0$ but not for
$\mu<0$. This signals the breakdown of the gradient expansion for a gapless
Fermi surface, but not for gapped free fermions.

  3. 3.

The opposite limit, $\kappa\rightarrow 0$, or $m\rightarrow\infty$, is the
relativistic limit studied in Sec.2, in which the fermionic part of the model
reduces to a 2+1 dimensional Majorana spinor with mass $\mu$ and speed of
light $\Delta_{0}$, coupled to a Riemann-Cartan geometry described by
$\Delta^{i},\;A_{\mu}$. Accordingly, there is a sense in which the limit
$\kappa\rightarrow\infty$ of $S_{\text{eff,m}}$ reproduces the effective
action of a massive Majorana spinor in Riemann-Cartan space-time (see
Sec.2.5). In particular, in the limit $\kappa\rightarrow 0$ the dimensionless
coefficients (3.11) are all quantized, as expected based on dimensional
analysis. Apart from $c$, only the coefficient $F_{1}^{\prime}$ is
discontinuous at $\mu=0$ within this limit, with a quantized discontinuity
$-\left(\ell/4\right)\left[F^{\prime}_{1}\left(0^{+}\right)-F^{\prime}_{1}\left(0^{-}\right)\right]=\left(\ell/2\right)/96\pi$
that matches the coefficient $\beta$ of the gravitational pseudo Chern-Simons
term of Sec.2. As anticipated in Sec.2, the coefficient $c$ remains quantized
away from the relativistic limit, while $\beta$, or $F_{1}^{\prime}$, does
not. Finally, we note that our perturbative computation of the gCS term
generalizes the computations of Refs.[257, 258, 259, 260, 261] and Appendix E
for relativistic fermions, and reduces to these as $\kappa\rightarrow 0$.

####  3.4 Induced action and linear response

By expanding $S_{\text{eff}}$ to second order in the fields
$\theta,A_{t}-\mu,A_{i},u_{ij}$, and performing Gaussian integration over
$\theta$, we obtain an induced action
$S_{\text{ind}}\left[A_{\mu},u_{ij}\right]$ that captures the linear response
of CSFs to the background fields, see Appendix I for explicit expressions.
Taking functional derivatives one obtains the expectation values
$J^{\mu}=G^{-1/2}\delta S_{\text{ind}}/\delta A_{\mu}$, $T^{ij}=G^{-1/2}\delta
S_{\text{ind}}/\delta u_{ij}$ of the current and stress, and from them the
conductivity $\sigma^{ij}=\delta J^{i}/\delta E_{j}$, the viscosity
$\eta^{ij,kl}=\delta T^{ij}/\delta\partial_{t}u_{kl}$, and the mixed response
function $\kappa^{ij,k}=\delta T^{ij}/\delta E_{k}=\delta
J^{k}/\delta\partial_{t}u_{ij}$. We will also need the static susceptibilities
$\chi_{JJ}^{\mu,\nu},\;\chi_{TJ}^{ij,\nu}$, defined by restricting to time
independent $A_{\mu},u_{ij}$, and computing $\delta J^{\mu}/\delta A_{\nu}$
and $\delta J^{\nu}/\delta u_{ij}$, respectively.

Before we compute $\eta_{\text{o}}$, it is useful to restrict its form based
on dimensionality and symmetries: space-time translations and spatial
rotations, as well as $PT$. The analysis is performed in Appendices G.1-G.4,
and results in the expression

$\displaystyle\eta_{\text{o}}\left(\omega,\mathbf{q}\right)=$
$\displaystyle\eta_{\text{o}}^{\left(1\right)}\sigma^{xz}+\eta_{\text{o}}^{\left(2\right)}\left[\left(q_{x}^{2}-q_{y}^{2}\right)\sigma^{0x}-2q_{x}q_{y}\sigma^{0z}\right],$
(3.12)

which is written in a basis of anti-symmetrized tensor products of the
symmetric Pauli matrices, $\sigma^{ab}=2\sigma^{[a}\otimes\sigma^{b]}$ [174].
As components of the strain tensor, the matrices $\sigma^{x},\sigma^{z}$
correspond to shears, while the identity matrix $\sigma^{0}$ corresponds to a
dilatation. The details of the system are encoded in two independent
coefficients
$\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)}\in\mathbb{C}$,
which are themselves arbitrary functions of $\omega,q^{2}$. In the case of
uniform strain ($\mathbf{q}=\mathbf{0}$), the odd viscosity tensor reduces to
a single component,
$\eta_{\text{o}}\left(\omega,\mathbf{0}\right)=\eta_{\text{o}}^{\left(1\right)}\left(\omega\right)\sigma^{[x}\otimes\sigma^{z]}$,
as is well known [173, 174, 175, 160, 176]. The additional component
$\eta_{\text{o}}^{\left(2\right)}$ has not been discussed much in the
literature [34, 262], and also appears in the presence of vector, or pseudo-
vector, anisotropy [263, 264], in which case $\mathbf{q}$ should be replaced
by a background vector $\mathbf{b}$. The expression (3.12) applies at finite
temperature, out of equilibrium, and in the presence of disorder that
preserves the symmetries on average. For clean systems at zero temperature,
$\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)}$ are both
real, even functions of $\omega$. In gapped systems
$\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)}$ will
usually be regular at $\omega=0=q^{2}$, though an exception to this rule was
recently found in Ref.[4].

For the CSF, we find the $\omega=0$ coefficients

$\displaystyle\eta_{\text{o}}^{\left(1\right)}\left(q^{2}\right)=$
$\displaystyle-\frac{1}{2}s_{\theta}n_{0}-\left(\frac{c}{24}\frac{1}{4\pi}+s_{\theta}C^{\left(1\right)}\right)q^{2}+O\left(q^{4}\right),$
$\displaystyle\eta_{\text{o}}^{\left(2\right)}\left(q^{2}\right)=$
$\displaystyle\frac{1}{2}s_{\theta}n_{0}q^{-2}+\left(\frac{c}{24}\frac{1}{4\pi}+s_{\theta}C^{\left(2\right)}\right)+O\left(q^{2}\right),$
(3.13)

where $C^{\left(1\right)},C^{\left(2\right)}\in\mathbb{R}$ are generically
non-zero, and are given by particular linear combinations of the dimensionless
coefficients
$F_{1}^{\prime}\left(\mu\right),mF_{2}\left(\mu\right),mF_{3}\left(\mu\right)$,
defined in (3.5), see Appendix I for more details.

The leading term in $\eta_{\text{o}}^{\left(1\right)}$ is the familiar (1.3),
which also appears in gapped states, while the non-analytic leading term in
$\eta_{\text{o}}^{\left(2\right)}$ is possible because the superfluid is
gapless, and does not contribute to the viscosity tensor when $q\rightarrow 0$
at $\omega\neq 0$ [163]. Both leading terms obey the same quantization
condition due to SSB, can be used to extract $s_{\theta}$, and are independent
of $c$. The sub-leading corrections to both
$\eta_{\text{o}}^{\left(1\right)},\eta_{\text{o}}^{\left(2\right)}$ contain
the quantized gCS contributions proportional to $c$, but also the non-
universal coefficients $C^{\left(1\right)},C^{\left(2\right)}$. Thus the
central charge cannot be extracted from a measurement of $\eta_{\text{o}}$
alone.

Noting that the non-universal sub-leading corrections to $\eta_{\text{o}}$
originate from the geometric spin $s_{\theta}=\ell/2$ of the Goldstone field,
one is naturally led to consider the g$s$-wave CSF, where $s_{\theta}=0$, and
the odd viscosity is, to leading order in $q$, purely due to the gCS term

$\displaystyle\tilde{\eta}_{\text{o}}^{\left(1\right)}\left(q^{2}\right)=$
$\displaystyle-\frac{c}{24}\frac{1}{4\pi}q^{2}+O\left(q^{4}\right),$ (3.14)
$\displaystyle\tilde{\eta}_{\text{o}}^{\left(2\right)}\left(q^{2}\right)=$
$\displaystyle\frac{c}{24}\frac{1}{4\pi}+O\left(q^{2}\right).$

Here and below we use $O$ and $\tilde{O}$, for the quantity $O$ in the CSF and
in the corresponding $g$s-wave CSF, respectively. Equation (3.14) follows from
(3.13) by setting $s_{\theta}=0$, but can be understood directly from
$S_{\text{eff}}$. Indeed, for the g$s$-wave CSF, $S_{\text{eff}}$ is identical
to that of the conventional $s$-wave SF up to $O\left(p^{2}\right)$, but
contains the additional $\mathcal{L}_{\text{gCS}}$ at $O\left(\omega
q^{2}\right)$, which is the leading $P,T$-odd term, and produces the leading
odd viscosity (3.14).

Due to the LGS (3.1)-(3.2), the viscosity (3.14) implies also

$\displaystyle\tilde{\chi}_{TJ,\text{o}}^{ij,k}=-\frac{i}{m}\frac{c}{48\pi}q_{\perp}^{i\vphantom{j}}q_{\perp}^{j}q_{\bot}^{k\vphantom{j}}+O\left(q^{4}\right),$
(3.15)

where $q_{\perp}^{i}=\varepsilon^{ij}q_{j}$, and the subscript “o” (“e”)
refers to the $P,T$-odd (even) part of an object, which is odd (even) in
$\ell$. In particular, a steady $P,T$-odd current
$\tilde{J}_{\text{o}}^{k}=-\frac{1}{m}\frac{c}{96\pi}\partial_{\perp}^{k}R+O\left(q^{4}\right)$
flows perpendicularly to gradients of curvature, which has the linearized form
$R=-2\partial_{\perp}^{i}\partial_{\perp}^{j}u_{ij}$. We stress that the
fundamental relation is the momentum density
$\tilde{P}_{\text{o}}^{k}=-\frac{c}{96\pi}\partial_{\perp}^{k}R+O\left(q^{4}\right)$,
which follows from the odd viscosity (3.14) along with momentum conservation,
irrespective of Galilean symmetry. This relation was predicted for $p$-wave
CSFs based on the relativistic limit, see Sec.2.1.2 and 2.6. We conclude that,
in the g$s$-wave CSF, $c$ can be extracted from a measurement of
$\tilde{\eta}_{\text{o}}$, and in the Galilean invariant case, also from a
measurement of the current $\tilde{J}$ in response to (time-independent)
strain.

Though the simple results above do not apply to the physical system of
interest, the CSF, there is a relation between the observables of the CSF and
the corresponding $g$s-wave CSF, which we can utilize. At the level of induced
actions it is given by the simple relation

$\displaystyle\tilde{S}_{\text{ind}}\left[A_{\mu},u_{ij}\right]=S_{\text{ind}}\left[A_{\mu}-\left(\ell/2\right)\omega_{\mu},u_{ij}\right],$
(3.16)

where $\omega_{\mu}$ is expressed through $u_{ij}$ as in Appendix F, and by
taking functional derivatives one finds relations between response functions
[250]. In particular,

$\displaystyle\tilde{\eta}_{\text{o}}^{ij,kl}=$
$\displaystyle\eta_{\text{o}}^{ij,kl}-\frac{\ell}{4}n_{0}\left(\sigma^{xz}\right)^{ij,kl}+\frac{i\ell}{4}\left(\kappa_{\text{e}\vphantom{\bot}}^{ij,(k}q_{\perp}^{l)}-\kappa_{\text{e}\vphantom{\bot}}^{kl,(i}q_{\perp}^{j)}\right)+\frac{\ell^{2}}{16}\sigma_{\text{o}}q_{\perp}^{(i}\varepsilon_{\vphantom{\bot}}^{j)(k}q_{\perp}^{l)},$
(3.17)

where the response functions
$\eta_{\text{o}},\sigma_{\text{o}},\kappa_{\text{e}}$ depend on
$\omega,\mathbf{q}$. In a Galilean invariant system one further has

$\displaystyle\tilde{\chi}_{TJ,\text{o}}^{ij,k}=$
$\displaystyle\chi_{TJ,\text{o}}^{ij,k}-\frac{\ell}{4m}\chi_{TJ,\text{e}}^{ij,t}iq_{\bot}^{k}+\frac{\ell}{2}iq_{\bot}^{(i}\chi_{JJ,\text{e}}^{j),k}+\frac{\ell^{2}}{8m}q_{\bot}^{(i}\chi_{JJ,\text{o}}^{j),t}q_{\bot}^{k},$
(3.18)

and we note the relations
$\chi_{TJ,\text{e}}^{ij,t}=\kappa_{\text{e}}^{ij,k}iq_{k},\;\chi_{JJ,\text{o}}^{j,t}=\sigma_{\text{o}}q_{\bot}^{j},\;\chi_{JJ,\text{e}}^{j,k}=\rho_{\text{e}}q_{\bot}^{j}q_{\bot}^{k}$,
between the above susceptibilities, the response functions
$\kappa_{\text{e}},\sigma_{\text{o}}$, and the London diamagnetic response
$\rho_{\text{e}}$. Though the above expressions are a mouthful, they
correspond to the simple subtraction of an angular momentum $\ell/2$ per
fermion, as expressed by Eq.(3.16).

####  3.5 Discussion

Equations (3.14) and (3.17) are the main results of this Section. They rely on
the SSB pattern (1.2), but not on Galilean symmetry. Equation (3.17) expresses
$\tilde{\eta}_{\text{o}}$ as a bulk observable of CSFs, which we refer to as
the improved odd viscosity. According to (3.14), the leading term in the
expansion of $\tilde{\eta}_{\text{o}}\left(0,\mathbf{q}\right)$ around
$\mathbf{q}=\mathbf{0}$ is fixed by $c$. Since this leading term occurs at
second order in $\mathbf{q}$, in order to extract $c$ one needs to measure
$\sigma_{\text{o}},\chi_{\text{e}}$, and $\eta_{\text{o}}$, at zeroth, first,
and second order, respectively. In a Galilean invariant system, (3.14) and
(3.17) imply (3.15) and (3.18) respectively, which, in turn, show that $c$ can
be extracted in an experiment where $U\left(1\right)_{N}$ fields and strain
are applied, and the resulting number current and density are measured. In
particular, a measurement of the stress tensor is not required. Since
$U\left(1\right)_{N}$ fields can be applied in Galilean invariant fluids by
tilting and rotating the sample [265], we believe that a bulk measurement of
the boundary central charge, through (3.15) and (3.18), is within reach of
existing experimental techniques [132, 131, 133, 134].

The problem of obtaining $c$ from a bulk observable has been previously
studied in QH states, described by (3.6)-(3.7) [253, 254, 34, 35, 39, 255, 43,
42, 256, 44, 47, 52]. It was found that $c$ can only be extracted if
$\text{var}s=\overline{s^{2}}-\overline{s}^{2}=0$, as in Laughlin states, or
in a single filled Landau level. Under this condition, the response to strain,
at fixed $A_{\mu}-\overline{s}\omega_{\mu}$, depends purely on $c$ [42, 44] \-
a useful theoretical characterization, which seems challenging experimentally.
However, the improved odd viscosity (3.17), constructed here, applies also to
$\text{var}s=0$ QH states, with $\ell$ replaced by $-2\overline{s}$, and
defines a concrete bulk observable which is precisely quantized, and
determined by $c$.

###  4 Intrinsic sign problem in chiral topological matter

The question of intrinsic Monte Carlo sign problems was motivated in Sec.1.6,
where we also stated and discussed the criterion $e^{2\pi
ic/24}\notin\left\\{\theta_{a}\right\\}$, which we obtain for intrinsic sign
problems in chiral topological phases. Here we precisely state and derive the
results that hold under this criterion.

In Sec.4.1 we discuss the universal finite-size correction to the boundary
momentum density in chiral topological phases, arriving at the ’momentum
polarization’ Eq.(4.2). Relying on the above, Sec.4.2 then obtains Result 1 \-
an intrinsic sign problem in bosonic chiral topological matter. In Sec.4.3 we
perform a similar analysis for the case where chirality (or time reversal
symmetry breaking), appears spontaneously rather than explicitly, arriving at
Result 2. We then turn to fermionic systems. In Sec.4.4 we develop a formalism
which unifies and generalizes the currently used DQMC algorithms, and the
corresponding design principles. In Sec.4.5, we obtain within this formalism
Result 1F and Result 2F, the fermionic analogs of Results 1 and 2. Section 4.6
describes a conjectured extension of our results that applies beyond chiral
phases, and unifies them with the intrinsic sign problems found in our
parallel work [5]. In Sec.5 we discuss our results and provide an outlook for
future work.

####  4.1 Signs from geometric manipulations

#####  4.1.1 Momentum finite-size correction

In analogy with the $T>0$ correction to the energy current in Eq.(1.1), the
boundary of a chiral topological phase, described by a chiral CFT, also
supports a non-vanishing ground state (or $T=0$) momentum density $p$, which
receives a universal correction on a cylinder with finite circumference
$L<\infty$,

$\displaystyle
p\left(L\right)=p\left(\infty\right)+\frac{2\pi}{L^{2}}\left(h_{0}-\frac{c}{24}\right).$
(4.1)

Equation (4.1) is the main property of chiral topological matter that we use
below, so we discuss it in detail. The appearance of the chiral central charge
is a manifestation of the global gravitational anomaly, as explained in
Sec.1.3. The rational number $h_{0}$ is a chiral conformal weight from the
boundary CFT. Like the chiral central charge, the two boundary components of
the cylinder have opposite $h_{0}$s, see Fig.1. From the bulk TFT perspective,
$h_{0}$ corresponds to the topological spin of an anyon quasi-particle,
defined by the phase $\theta_{0}=e^{2\pi ih_{0}}$ accumulated as the anyon
undergoes a $2\pi$ rotation [104]. The set
$\left\\{\theta_{a}\right\\}_{a=1}^{N}$ of topological spins of anyons is
associated with the $N$-dimensional ground state subspace on the torus, and
the unique $\theta_{0}=e^{2\pi ih_{0}}$ defined by (4.1) corresponds to the
generically unique ground state on the cylinder, with a finite-size energy
separation $\sim 1/L$ from the low lying excited states, see Appendix K.

As the equation $\theta_{0}=e^{2\pi ih_{0}}$ suggests, only $h_{0}\mod 1$ is
universal for a topological phase. The integer part of $h_{0}$ can change as
the Hamiltonian is deformed on the cylinder, while maintaining the bulk gap,
and even as a function of $L$ for a fixed Hamiltonian. Additionally, the
choice of $\theta_{0}$ from the set $\left\\{\theta_{a}\right\\}$ is non-
universal, and can change due to bulk gap preserving deformations, or as a
function of $L$. Both types of discontinuous jumps in $h_{0}$ may be
accompanied by an accidental degeneracy of the ground state on the cylinder.
Therefore, the universal and $L$-independent statement regarding $h_{0}$ is
that, apart from accidental degeneracies, $e^{2\pi
ih_{0}}=\theta_{0}\in\left\\{\theta_{a}\right\\}$ \- a fact that will be
important in our analysis.

The non-trivial behavior of $h_{0}$ described above appears when the boundary
corresponds to a non-conformal deformation of a CFT, by e.g a chemical
potential. As demonstrated analytically and numerically in Appendix L, such
behavior appears already in the simple context of Chern insulators with non-
zero Fermi momenta, as would be the case in Fig.1(b) if the chemical potential
$\mu$ is either raised or lowered.

#####  4.1.2 Momentum polarization

In this section we describe a procedure for the extraction of $h_{0}-c/24$ in
Eq.(4.1), given a lattice Hamiltonian on the cylinder. Since the two boundary
components carry opposite momentum densities, the ground state on the cylinder
does not carry a total momentum, only a ’momentum polarization’. It is
therefore clear that some sort of one-sided translation will be required.

Figure 7: Momentum polarization. (a) Hamiltonian, or spatial, point of view.
The operator $T_{R}$ translates the right half of the cylinder by one unit
cell, a distance $a$, in the $x$ direction. It acts as the identity on the
left boundary component, and as a translation on the right boundary component.
The object $\tilde{Z}/Z$ is the thermal expectation value of $T_{R}$. (b)
Field theory, or space-time, point of view. The object $\tilde{Z}$ is the
partition function on a space-time carrying a screw dislocation. The space-
time region occupied by the boundary components of the spatial cylinder is
colored in orange. The screw dislocation can be described as an additional
boundary component, on which $T_{R}$ acts as a translation, with a high
effective temperature $1/\beta_{*}$.

Following Ref.[212], we define $\tilde{Z}:=\text{Tr}\left(T_{R}e^{-\beta
H}\right)$, which is related to the usual partition function
$Z=\text{Tr}\left(e^{-\beta H}\right)$ ($\beta=1/T$), by the insertion of the
operator $T_{R}$, which translates the right half of the cylinder by one unit
cell in the periodic $x$ direction, see Fig.7(a). The object $\tilde{Z}$
satisfies

$\displaystyle\tilde{Z}$ $\displaystyle=Z\exp\left[\alpha N_{x}+\frac{2\pi
i}{N_{x}}\left(h_{0}-\frac{c}{24}\right)+o\left(N_{x}^{-1}\right)\right],$
(4.2)

where $N_{x}$ is the number of sites in the $x$ direction,
$\alpha\in\mathbb{C}$ is non-universal and has a negative real part, and
$o\left(N_{x}^{-1}\right)$ indicates corrections that decay faster than
$N_{x}^{-1}$ as $N_{x}\rightarrow\infty$. The above expression is valid at
temperatures low compared to the finite-size energy differences on the
boundary, $\beta^{-1}=o\left(N_{x}^{-1}\right)$, see Appendix K.

Equation (4.2) follows analytically from the low energy description of chiral
topological matter in terms of chiral TFT and CFT [212], and was numerically
scrutinized in a large number of examples in Refs.[212, 213, 38, 61, 214], as
well as in Appendix L. Nevertheless, we are not aware of a rigorous proof of
Eq.(4.2) for gapped lattice Hamiltonians. Therefore, in stating our results we
will use the assumption ’the Hamiltonian $H$ is in a chiral topological phase
of matter’, the content of which is that $H$ admits a low energy description
in terms of a chiral TFT with chiral central charge $c$ and topological spins
$\left\\{\theta_{a}\right\\}$, and in particular, Eq.(4.2) holds for any bulk-
gap preserving deformation of $H$, with $e^{2\pi
ih_{0}}\in\left\\{\theta_{a}\right\\}$ (apart from accidental degeneracies on
the cylinder, as explained below Eq.(4.1)). In the remainder of this section
we further discuss the content of Eq.(4.2) and its expected range of validity,
in light of the Hamiltonian and space-time interpretations of $\tilde{Z}$.

From a Hamiltonian perspective, $\tilde{Z}/Z$ is the thermal expectation value
of $T_{R}$, evaluated at a temperature $\beta^{-1}$ low enough to isolate the
ground state. The exponential decay expressed in Eq.(4.2) appears because
$T_{R}$ is not a symmetry of $H$, and $-\text{Re}\left(\alpha\right)$ can be
understood as the energy density of the line defect where $T_{R}$ is
discontinuous, see Fig.7(a). In fact, we expect Eq.(4.2) to hold irrespective
of whether the uniform translation is a symmetry of $H$, or of the underlying
’lattice’ on which $H$ is defined, which may be any polygonalization of the
cylinder (see Ref.[266] for a similar scenario). The only expected requirement
is that the low energy description of $H$ is homogeneous. Furthermore, if
Eq.(4.2) only holds after a disorder averaging of $\tilde{Z}/Z$, our results
and derivations in the following sections remain unchanged.

There is also a simple space-time interpretation of $\tilde{Z}$, which will be
useful in the context of DQMC. The usual partition function
$Z=\text{Tr}\left(e^{-\beta H}\right)$ has a functional integral
representation in terms of bosonic fields $\phi$ (fermionic fields $\psi$)
defined on space, the cylinder $C$ in our case, and the imaginary time circle
$S_{\beta}^{1}=\mathbb{R}/\beta\mathbb{Z}$, with periodic (anti-periodic)
boundary conditions [267]. In $\tilde{Z}=\text{Tr}\left(T_{R}e^{-\beta
H}\right)$, the insertion of $T_{R}$ produces a twisting of the boundary
conditions of $\phi,\psi$ in the time direction, such that $\tilde{Z}$ is the
partition function on a space-time carrying a screw dislocation, see Fig.7(b).

The above interpretation of $\tilde{Z}$, supplemented by Eq.(4.1), allows for
an intuitive explanation of Eq.(4.2), which loosely follows its analytic
derivation [212]. As seen in Fig.7(b), the line where $T_{R}$ is discontinuous
can be interpreted as an additional boundary component at a high effective
temperature, $\beta_{*}\ll L/v$. Since the effective temperature is much
larger than the finite size energy-differences $2\pi v/L$ on the boundary CFT,
the screw dislocation contributes no finite size corrections to $\tilde{Z}$.
This leaves only the contribution of the boundary component on the right side
of the cylinder, where $T_{R}$ produces the phase $e^{iaLp\left(L\right)}$,
assuming $\beta_{*}\ll L/v\ll\beta$. Equation (4.1) then leads to the
universal finite size correction $\left(2\pi
i/N_{x}\right)\left(h_{0}-c/24\right)$.

####  4.2 Excluding stoquastic Hamiltonians for chiral topological matter

In this section we consider bosonic (or ’qudit’, or spin) systems, and a
single design principle - existence of a local basis in which the many-body
Hamiltonian is stoquastic. A sketch of the derivation of Result 1 is that the
momentum polarization $\tilde{Z}$ is positive for Hamiltonians $H^{\prime}$
which are stoquastic in an on-site and homogenous basis, and this implies that
$\theta_{0}=e^{2\pi ic/24}$ for any Hamiltonian $H$ obtained from $H^{\prime}$
by conjugation with a local unitary.

#####  4.2.1 Setup

The many body Hilbert space is given by $\mathcal{H}=\otimes_{\mathbf{x}\in
X}\mathcal{H}_{\mathbf{x}}$, where the tensor product runs over the sites
$\mathbf{x}=\left(x,y\right)$ of a 2-dimensional lattice $X$, and
$\mathcal{H}_{\mathbf{x}}$ are on-site ’qudit’ Hilbert spaces of finite
dimension $\mathsf{d}\in\mathbb{N}$. With finite-size QMC simulations in mind,
we consider a square lattice with spacing 1, $N_{x}\times N_{y}$ sites, and
periodic boundary conditions, so that
$X=\mathbb{Z}_{N_{x}}\times\mathbb{Z}_{N_{y}}$ is a discretization of the flat
torus
$\left(\mathbb{R}/N_{x}\mathbb{Z}\right)\times\left(\mathbb{R}/N_{y}\mathbb{Z}\right)$.
Generalization to other 2-dimensional lattices is straight forward. On this
Hilbert space a gapped $r$-local Hamiltonian
$H=\sum_{\mathbf{x}}H_{\mathbf{x}}$ is assumed to be given. Here the terms
$H_{\mathbf{x}}$ are supported within a range $r$ of $\mathbf{x}$ \- they are
defined on $\otimes_{\left|\mathbf{y}-\mathbf{x}\right|\leq
r}\mathcal{H}_{\mathbf{y}}$ and act as the identity on all other qudits.

Fix an tensor product basis $\left|s\right\rangle=\otimes_{\mathbf{x}\in
X}\left|s_{\mathbf{x}}\right\rangle$, labeled by strings
$s=\left(s_{\mathbf{x}}\right)_{\mathbf{x}\in X}$, where
$s_{\mathbf{x}}\in\left\\{1,\cdots,\mathsf{d}\right\\}$ labels a basis
$\left|s_{\mathbf{x}}\right\rangle$ for $\mathcal{H}_{\mathbf{x}}$. For any
vector $\mathbf{d}\in X$, the corresponding translation operator
$T^{\mathbf{d}}$ is defined in this basis,
$T^{\mathbf{d}}\left|s\right\rangle=\left|t^{\mathbf{d}}s\right\rangle$, with
$\left(t^{\mathbf{d}}s\right)_{\mathbf{x}}=s_{\mathbf{x}+\mathbf{d}}$. These
statements assert that $\left|s\right\rangle$ is both an on-site and a
homogeneous basis, or on-site homogenous for short. Note that $T^{\mathbf{d}}$
acts as a permutation matrix on the $\left|s\right\rangle$s, and in
particular, has non-negative matrix elements in this basis.

In accordance with Sec.4.1.2, we assume that the low energy description of $H$
is invariant under $T^{\mathbf{d}}$, as defined above. In doing so, we exclude
the possibility of generic background gauge fields for any on-site symmetry
that $H$ may posses, which is beyond the scope of this thesis. Nevertheless,
commonly used background gauge fields, such as those corresponding to uniform
magnetic fields with rational flux per plaquette, can easily be incorporated
into our analysis, by restricting to translation vectors $\mathbf{d}$ in a
sub-lattice of $X$. A restriction to sub-lattice translations can also be used
to guarantee that $T^{\mathbf{d}}$ acts purely as a translation in the low
energy TQFT description. In particular, a lattice translation may permute the
anyon types $a$ 292929We thank Michael Levin for pointing out this
phenomenon.. Since the number of anyons is finite, restricting to large enough
translations will eliminate this effect. An example is given by Wen’s
plaquette model, where different anyons are localized on the even/odd sites of
a bipartite lattice [268], and a restriction to translations that maps the
even (odd) sites to themselves will be made.

Finally, we assume that $H$ is locally stoquastic: it is term-wise stoquastic
in a local basis. This means that a local unitary operator $U$ exists, such
that the conjugated Hamiltonian $H^{\prime}=UHU^{\dagger}$ is a sum of local
terms $H_{\mathbf{x}}^{\prime}=UH_{\mathbf{x}}U^{\dagger}$, which have non-
positive matrix elements in the on-site homogeneous basis, $\left\langle
s\right|H_{\mathbf{x}}^{\prime}\left|\tilde{s}\right\rangle\leq 0$ for all
basis states $\left|s\right\rangle,\left|\tilde{s}\right\rangle$. Note that we
include the diagonal matrix elements in the definition, without loss of
generality.

The term local unitary used above refers to a depth-$D$ quantum circuit, a
product $U=U_{D}\cdots U_{1}$ where each $U_{i}$ is itself a product of
unitary operators with non-overlapping supports of diameter $w$. It follows
that $H^{\prime}$ has a range $r^{\prime}=r+2r_{U}$, where $r_{U}=Dw$ is the
range of $U$. Equivalently, we may take $U$ to be a finite-time evolution with
respect to an $\tilde{r}$-local, smoothly time-dependent, Hamiltonian
$\tilde{H}\left(t\right)$, given by the time-ordered exponential
$U=\text{TO}e^{-i\int_{0}^{1}\tilde{H}\left(t\right)dt}$. The two types of
locality requirements are equivalent, as finite-time evolutions can be
efficiently approximated by finite-depth circuits, while finite-depth circuits
can be written as finite-time evolutions over time $D$ with piecewise constant
$w$-local Hamiltonians [269, 90].

#####  4.2.2 Constraining $c$ and $\left\\{\theta_{a}\right\\}$

In order to discuss the momentum polarization, we need to map the stoquastic
Hamiltonian $H^{\prime}$ from the torus $X$ to a cylinder $C$. This is done by
choosing a translation vector $\mathbf{d}\in X$, and then cutting the torus
$X$ along a line $l$ parallel to $\mathbf{d}$. To simplify the presentation we
restrict attention to the case $\mathbf{d}=\left(1,0\right)$, where
$T^{\mathbf{d}}=T$ (and in the following $T_{R}^{\mathbf{d}}=T_{R}$). All
other cases amount to a lattice-spacing redefinition, see Appendix M. The
cylinder $C=\mathbb{Z}_{N_{x}}\times\left\\{1,\dots,N_{y}\right\\}$ is then
obtained from the torus $X=\mathbb{Z}_{N_{x}}\times\mathbb{Z}_{N_{y}}$ by
cutting along the line
$l=\left\\{\left(i,1/2\right):\;i\in\mathbb{Z}_{N_{x}}\right\\}$. A stoquastic
Hamiltonian on the cylinder can be obtained from that on the torus by removing
all local terms $H^{\prime}_{\mathbf{x}}$ whose support overlaps $l$, see
Fig.8. Note that this procedure may render $H^{\prime}$ acting as $0$ on
certain qudits $\mathcal{H}_{\mathbf{x}}$ with $\mathbf{x}$ within a range
$r^{\prime}$ of $l$, but this does not bother us. Since all terms
$H_{\mathbf{x}}^{\prime}$ are individually stoquastic, this procedure leaves
$H^{\prime}$, now defined on the cylinder, stoquastic. One can similarly map
$H$ and $U$ to the cylinder $C$ such that the relation
$H^{\prime}=UHU^{\dagger}$ remains valid on $C$.

Figure 8: Cutting the torus to a cylinder along the line $l$. Orange areas
mark the supports of Hamiltonian terms $H_{\mathbf{x}}^{\prime}$ which are
removed from $H^{\prime}$, while blue areas mark the supports of terms which
are kept.

Let us now make contact with the momentum polarization Eq.(4.2). Having mapped
$H^{\prime}$ to the cylinder, we consider the ’partition function’

$\displaystyle\tilde{Z}^{\prime}$ $\displaystyle:=\text{Tr}\left(e^{-\beta
H^{\prime}}T_{R}\right),$ (4.3)

where $T_{R}$ is defined by
$T_{R}\left|s\right\rangle=\left|T_{R}s\right\rangle$,
$\left(T_{R}s\right)_{x,y}=s_{x+\Theta\left(y\right),y}$ , and $\Theta$ is a
heavy side function supported on the right half of the cylinder. Though
$\tilde{Z}^{\prime}$ is generally different from
$\tilde{Z}=\text{Tr}\left(e^{-\beta H}T_{R}\right)$ appearing in Eq.(4.2), it
satisfies two useful properties:

  1. 1.

$\tilde{Z}^{\prime}>0$. Both $-H^{\prime}$ and $T_{R}$ have non-negative
entries in the on-site basis $\left|s\right\rangle$, and therefore so does
$e^{-\beta H^{\prime}}T_{R}$.

  2. 2.

$H^{\prime}=UHU^{\dagger}$ is in the same phase of matter as $H$, so
$c^{\prime}=c$ and
$\left\\{\theta_{a}^{\prime}\right\\}=\left\\{\theta_{a}\right\\}$. Moreover,
$h_{0}^{\prime}=h_{0}$ for all $N_{x}$. Treating $U$ as a finite time
evolution, we have
$H\left(\lambda\right)=U\left(\lambda\right)HU\left(\lambda\right)^{\dagger}$,
where
$U\left(\lambda\right):=\text{TO}e^{-i\int_{0}^{\lambda}\tilde{H}\left(t\right)dt}$,
as a deformation from $H$ to $H^{\prime}$ which maintains locality and
preserves the bulk-gap. Moreover, the full spectrum on the cylinder is
$\lambda$-independent, and therefore so is $h_{0}$.

Combining Eq.(4.2), for $H^{\prime}$ instead of $H$, with the two above
properties leads to

$\displaystyle 1=$
$\displaystyle\tilde{Z}^{\prime}/\left|\tilde{Z}^{\prime}\right|$ (4.4)
$\displaystyle=$ $\displaystyle\exp 2\pi
i\left[\epsilon^{\prime}N_{x}+\frac{1}{N_{x}}\left(h_{0}-\frac{c}{24}\right)+o\left(N_{x}^{-1}\right)\right],$

where $\epsilon^{\prime}:=\text{Im}\left(\alpha^{\prime}\right)$ is generally
different from $\epsilon=\text{Im}\left(\alpha\right)$ of Eq.(4.2) since
$H^{\prime}\neq H$. The non-universal integer part of $h_{0}$ can then be
eliminated by raising Eq.(4.4) to the $N_{x}$th power,

$\displaystyle 1$ $\displaystyle=e^{2\pi
i\epsilon^{\prime}N_{x}^{2}}\theta_{0}\left(N_{x}\right)e^{-2\pi
ic/24}+o\left(1\right),$ (4.5)

where we used $\theta_{0}=e^{2\pi ih_{0}}$, and $o\left(1\right)\rightarrow 0$
as $N_{x}\rightarrow\infty$. We also indicated explicitly the possible
$N_{x}$-dependence of $\theta_{0}$, as described in Sec.4.1.1. We proceed
under the assumption that no accidental degeneracies occur on the cylinder, so
that $\theta_{0}\left(N_{x}\right)\in\left\\{\theta_{a}\right\\}$ for all
$N_{x}$, deferring the degenerate case to Appendix N. Now, for rational
$\epsilon^{\prime}=n/m$, the series $e^{2\pi i\epsilon^{\prime}N_{x}^{2}}$
($N_{x}\in\mathbb{N}$) covers periodically a subset $S$ of the $m$th roots of
unity, including $1\in S$. On the other hand, for irrational
$\epsilon^{\prime}$ the series $e^{2\pi i\epsilon^{\prime}N_{x}^{2}}$ is dense
in the unit circle. Combined with the fact that $\theta_{0}\left(N_{x}\right)$
is valued in the finite set $\left\\{\theta_{a}\right\\}$, while $c$ is
$N_{x}$-independent, Equation (4.5) implies that $\epsilon^{\prime}$ must be
rational, and that the values attained by
$\theta_{0}\left(N_{x}\right)e^{-2\pi ic/24}$ cover the set $S$ periodically,
for large enough $N_{x}$. It follows that $1\in
S\subset\left\\{\theta_{a}e^{-2\pi ic/24}\right\\}$. We therefore have

Result 1

If a local bosonic Hamiltonian $H$ is both locally stoquastic and in a chiral
topological phase of matter, then one of the corresponding topological spins
satisfies $\theta_{a}=e^{2\pi ic/24}$. Equivalently, a bosonic chiral
topological phase of matter where $e^{2\pi ic/24}$ is not the topological spin
of some anyon, i.e $e^{2\pi ic/24}\notin\left\\{\theta_{a}\right\\}$, admits
no local Hamiltonians which are locally stoquastic.

The above result can be stated in terms of the topological
$\mathbf{T}$-matrix, which is the representation of a Dehn twist on the torus
ground state subspace, and has the spectrum
$\text{Spec}\left(\mathbf{T}\right)=\left\\{\theta_{a}e^{-2\pi
ic/24}\right\\}_{a}$ [218, 104, 213, 212, 98, 270].

Result 1’

If a local bosonic Hamiltonian $H$ is is both locally stoquastic and in a
chiral topological phase of matter, then the corresponding $\mathbf{T}$-matrix
satisfies $1\in\text{Spec}\left(\mathbf{T}\right)$. Equivalently, a bosonic
chiral topological phase of matter where
$1\notin\text{Spec}\left(\mathbf{T}\right)$, admits no local Hamiltonians
which are locally stoquastic.

The above result is our main statement for bosonic phases of matter. The logic
used in its derivation is extended in Sec.4.3-4.5, where we generalize Result
1 to systems which are fermionic, spontaneously-chiral, or both.

####  4.3 Spontaneous chirality

The invariants $h_{0}$ and $c$ change sign under both time reversal
$\mathcal{T}$ and parity (spatial reflection) $\mathcal{P}$, and therefore
require a breaking of $\mathcal{T}$ and $\mathcal{P}$ down to $\mathcal{PT}$
to be non-vanishing. The momentum polarization Eq.(4.2) is valid if this
symmetry breaking is explicit, i.e $H$ does not commute with $\mathcal{P}$ and
$\mathcal{T}$ separately. Here we consider the case where $H$ is
$\mathcal{P},\mathcal{T}$-symmetric, but these are broken down to
$\mathcal{PT}$ spontaneously, as in e.g intrinsic topological superfluids and
superconductors [27, 2, 271]. We first generalize Eq.(4.2) to this setting,
and then use this generalization to obtain a spontaneously-chiral analog of
Result 1.

Note that the physical time-reversal $\mathcal{T}$ is an on-site anti-unitary
operator acting identically on all qudits, which implies
$\left[\mathcal{T},T_{R}\right]=0$, while $\mathcal{P}$ is a unitary operator
that maps the qudit at $\mathbf{x}$ to that at $P\mathbf{x}$, where $P$ is the
nontrivial element in $O\left(2\right)/SO\left(2\right)$, e.g
$\left(x,y\right)\mapsto\left(-x,y\right)$.

#####  4.3.1 Momentum polarization for spontaneously-chiral Hamiltonians

For simplicity, we begin by assuming that $H$ is ’classically symmetry
breaking’ - it has two exact ground states on the cylinder, already at finite
system sizes. We therefore have two ground states $\left|\pm\right\rangle$,
such that $\left|-\right\rangle$ is obtained from $\left|+\right\rangle$ by
acting with either $\mathcal{T}$ or $\mathcal{P}$. In particular,
$\left|\pm\right\rangle$ have opposite values of $h_{0}$ and $c$. The
$\beta\rightarrow\infty$ density matrix is then $e^{-\beta
H}/Z=\left(\rho_{+}+\rho_{-}\right)/2$, where
$\rho_{\pm}=\left|\pm\right\rangle\left\langle\pm\right|$, and this modifies
the right hand side of Eq.(4.2) to its real part,

$\displaystyle\tilde{Z}:=$ $\displaystyle\text{Tr}\left(T_{R}e^{-\beta
H}\right)$ (4.6) $\displaystyle=$ $\displaystyle Ze^{-\delta N_{x}}\cos
2\pi\left[\epsilon
N_{x}+\frac{2\pi}{N_{x}}\left(h_{0}-\frac{c}{24}\right)+o\left(N_{x}^{-1}\right)\right],$

where $-\delta\pm 2\pi i\epsilon$ are the values of the non-universal $\alpha$
obtained from Eq.(4.2), by replacing $e^{-\beta H}$ by $\rho_{\pm}$ . Indeed,
it follows from $\left[\mathcal{T},T_{R}\right]=0$ that if two density
matrices are related by $\rho_{-}=\mathcal{T}\rho_{+}\mathcal{T}^{-1}$, then
$\tilde{Z}_{\pm}:=Z_{\pm}\text{Tr}\left(T_{R}\rho_{+}\right)$ are complex
conjugates, $\tilde{Z}_{-}=\tilde{Z}_{+}^{*}$.

Now, for a generic symmetry breaking Hamiltonian $H$, exact ground state
degeneracy happens only in the infinite volume limit [272]. At finite size,
the two lowest lying eigenvalues of $H$ would be separated by an exponentially
small energy difference $\Delta E=O\left(e^{-fN_{x}^{\lambda}}\right)$, with
some $f>0,\lambda>0$. The two corresponding eigenstates would be
$\mathcal{T},\mathcal{P}$-even/odd, of the form
$W\left[\left|+\right\rangle\pm\left|-\right\rangle\right]$, where $W$ is a
$\mathcal{T},\mathcal{P}$-invariant local unitary [90]. One can think of these
statements as resulting from the existence of a bulk-gap preserving and
$\mathcal{T},\mathcal{P}$-symmetric deformation of $H$ to a ’classically