###### Abstract

In this thesis, we study chiral topological phases of 2+1 dimensional quantum
matter. Such phases are abstractly characterized by their non-vanishing chiral
central charge $c$, a topological invariant which appears as the coefficient
of a gravitational Chern-Simons (gCS) action in bulk, and of corresponding
gravitational anomalies at boundaries. The chiral central charge is of
particular importance in chiral superfluids and superconductors (CSF/Cs),
where $U\left(1\right)$ particle-number symmetry is broken, and $c$ is, in
some cases, the only topological invariant characterizing the system. However,
as opposed to invariants which can be probed by gauge fields in place of
gravity, the concrete physical implications of $c$ in the context of condensed
matter physics is quite subtle, and has been the subject of ongoing research
and controversy. The first two parts of this thesis are devoted to the
physical interpretation of the gCS action and gravitational anomalies in the
context of CSF/Cs, where they are of particular importance, but have
nevertheless remained poorly understood.

As a first approach, we demonstrate that, at low energy, fermionic excitations
in $p$-wave CSF/Cs experience an emergent relativistic Riemann-Cartan
geometry, described by the superconducting order parameter and background
$U(1)$ gauge field. This description is then used to infer gCS energy-momentum
responses to a space-time dependent order parameter, and relate these to an
order parameter induced gravitational anomaly. The presence of torsion in the
emergent geometry, as well as the multiplicity of low energy spinors in the
case of lattice systems, lead to additional effects which are not of
topological origin, but nevertheless mimic closely the above gCS physics. We
show how these different phenomena can be disentangled.

We then take on a fully non-relativistic analysis of CSFs, obtaining a low
energy effective field theory that consistently captures both their chiral
Goldstone mode and their non-relativistic gCS action. Using the theory we find
that $c$ cannot be extracted from a measurement of the odd viscosity tensor
alone, despite naive expectation based on previous work. Nevertheless, a
related observable, termed ‘improved odd viscosity’, does allow for the bulk
measurement of $c$. Additional results of the same spirit are found in
Galilean invariant CSFs.

Finally, we turn to a seemingly unrelated aspect of chiral topological phases
- their computational complexity. The infamous sign problem leads to an
exponential complexity in Monte Carlo simulations of generic many-body quantum
systems. Nevertheless, many phases of matter are known to admit a sign-
problem-free representative, allowing an efficient classical simulation.
Motivated by long standing open problems in many-body physics, as well as
fundamental questions in quantum complexity, the possibility of intrinsic sign
problems, where a phase of matter admits no sign-problem-free representative,
was recently raised but remains largely unexplored. Here, we establish the
existence of an intrinsic sign problem in a broad class of chiral topological
phases, defined by the requirement that $e^{2\pi ic/24}$ is not the
topological spin of an anyon. Within this class, we exclude the possibility of
‘stoquastic’ Hamiltonians for bosons, and of sign-problem-free determinantal
Monte Carlo algorithms for fermions. We obtain analogous results for phases
that are spontaneously chiral, and present evidence for an extension of our
results that applies to both chiral and non-chiral topological matter.

##

### Acknowledgments

Adding a grain of sand to one of the very many and ever growing summits of the
mountain range known as ‘human knowledge’ is truly a great privilege. The
climb, however, is usually no easy task, and the one that produced this thesis
was no exception. Here, I would like to pause and thank those who supported me
en route.

First, I am grateful to my advisor Ady Stern, who suggested an unorthodox but
tailor made research direction, and collaborated with me on the first and most
challenging part of the work. Ady is a formidable physicist as well as a
generous advisor, and provided a rare combination of support and freedom in my
research. It was also wonderful to work under someone who views laughter as a
way of life.

Subsequent work was performed in mostly long distance but close collaborations
with Sergej Moroz, Carlos Hoyos, Félix Rose, Zohar Ringel and Adam Smith. In
particular, Sergej, Carlos and Zohar served as additional mentors, each with
his own unique style of doing research. I also benefited greatly from
extensive discussions with Paul Wiegmann, Andrey Gromov, Ryan Thorngren,
Weihan Hsiao, Semyon Klevtsov, Barry Bradlyn and Thomas Kvorning.

The members of my PhD advisory committee, Micha Berkooz and David Mross, as
well as my senior group members, Yuval Baum, Ion Cosma Fulga, Jinhong Park,
Raquel Queiroz and Tobias Holder, provided much needed physical and meta-
physical advice along the way. Our administrative staff, Hava Shapira, Merav
Laniado, Einav Yaish, Inna Dombrovsky, Yuval Toledo and Yuri Magidov,
sustained an incredibly efficient and warm work environment. I also thank my
fellow graduate students Eyal Leviatan, Ori Katz, Dan Klein, Avraham Moriel,
Shaked Rozen, Asaf Miron, Yotam Shpira, Adar Sharon, Dan Dviri and, of course,
Yuval Rosenberg, who dragged me to the Weizmann institute when we were kids,
and got me hooked on physics.

Zooming out, I am grateful to my parents Sharona and Gabi, for their continued
support in whatever I choose to do, and to my wife and best friend Adi, for
making my life happy and balanced. Since we became parents, my work would not
have been possible without Adi’s backing, in particular since the spreading of
Coronavirus, which eliminated some of our support systems, as well as the
distinction between work and home. Finally, I thank our boys Adam and Shlomi
for their smiles, laughter, and curiosity - a reminder of why I was drawn to
science in the first place.

### Publications

This thesis is based on the following publications:

  * •

Reference [1]: Omri Golan and Ady Stern. Probing topological superconductors
with emergent gravity. Phys. Rev. B, 98:064503, 2018.

  * •

Reference [2]: Omri Golan, Carlos Hoyos, and Sergej Moroz. Boundary central
charge from bulk odd viscosity: Chiral superfluids. Phys. Rev. B, 100:104512,
2019.

  * •

Reference [3]: Omri Golan, Adam Smith, and Zohar Ringel. Intrinsic sign
problem in fermionic and bosonic chiral topological matter. Phys. Rev.
Research, 2:043032, 2020.

Complementary results are obtained in:

  * •

Reference [4]: Félix Rose, Omri Golan, and Sergej Moroz. Hall viscosity and
conductivity of two-dimensional chiral superconductors. SciPost Phys., 9:6,
2020.

  * •

Reference [5]: Adam Smith, Omri Golan, and Zohar Ringel. Intrinsic sign
problems in topological quantum field theories. Phys. Rev. Research, 2:033515,
2020.

###  1 Introduction and summary

####  1.1 Overview

The study of topological phases of matter began in 1980, when the Hall
conductivity in a two-dimensional electron gas was measured to be an integer
multiple of $e^{2}/h$, to within a relative error of $10^{-7}$ [6],
subsequently reduced below $10^{-10}$ [7]. Following this discovery, it was
theoretically understood that in many-body quantum systems, certain physical
observables must be precisely quantized, under the right circumstances [8, 9].
Around the same time, quantum field theorists extensively studied the
phenomena of anomalies [10, 11, 12], where classical symmetries and
conservation laws are quantum mechanically violated, and discovered the
seemingly exotic anomaly inflow mechanism [13, 14], which physically
interprets anomalies in terms of topological effective actions in higher
space-time dimensions. It was only later understood that topological effective
actions and anomalies actually capture the essential physics of topological
phases of matter, and even classify them [15, 16, 17, 18, 19, 20].

In particular, 2+1D gapped chiral topological phases are characterized by a
gravitational Chern-Simons (gCS) action [21, 22, 23, 24, 25] and corresponding
1+1D gravitational anomalies [10, 12, 26], having the chiral central charge
$c$ as a precisely quantized coefficient, or topological invariant. The chiral
central charge is of particular importance in chiral superfluids and
superconductors [15, 27], where $U\left(1\right)$ particle-number symmetry is
broken spontaneously or explicitly, and $c$ is, in some cases, the only
topological invariant characterizing the system at low energy. However, as
opposed to topological invariants related to gauge fields for internal
symmetries in place of gravity, the concrete physical implications of $c$ (and
even its very definition) in the context of condensed matter physics is quite
subtle, and has been the subject of ongoing research and controversy [28, 15,
29, 30, 31, 32, 17, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55]. The first goal of this thesis is to
physically interpret the chiral central charge in the context of chiral
superfluids and superconductors, where it is of particular importance, but has
nevertheless remained poorly understood. This goal is pursued in Sec.2-3.

A seemingly unrelated aspect of chiral topological phases is the complexity of
simulating them on classical (as opposed to quantum) computers. It is
generally believed that chiral topological matter is ’hard’ to simulate
efficiently with classical resources. Concretely, it is known that chiral
topological phases do not admit local commuting projector Hamiltonians [56,
57, 58, 59, 54], nor do they admit local Hamiltonians with a PEPS state as an
exact ground state [60, 61, 62, 63]. We will be interested in quantum Monte
Carlo (QMC) simulations, arguably the most important tools in computational
many-body quantum physics [64, 65, 66, 67, 68, 69, 70, 71], and in the
infamous sign problem, which is the generic obstruction to an efficient QMC
simulation [72, 73, 74].

The accumulated experience of QMC practitioners suggests that the sign problem
was never solved in chiral topological matter. Since these phases are
abstractly defined by their non-vanishing chiral central charge, one may
suspect that the chiral central charge and related gravitational phenomena
pose an obstruction to sign-problem-free QMC. Such an obstruction is termed
intrinsic sign problem [75, 76], and is of interest beyond the context of
chiral topological matter, as it is widely accepted that long-standing open
problems in many-body quantum physics, such as the nature of high-temperature
superconductivity [77, 78, 79, 80], dense nuclear matter [81, 82, 83], and the
fractional quantum Hall state at filling $5/2$ [84, 85, 86, 87, 88, 55],
remain open because no solution to the sign problem in a relevant model has
thus far been found. Since the aforementioned open problems are all fermionic,
we are particularly motivated to study the possibility of intrinsic sign
problems in fermionic matter. The second goal of this thesis, pursued in
Sec.4, is to establish the existence of an intrinsic sign problem in chiral
topological phases of matter, based on their non-vanishing chiral central
charge, and with an emphasis on fermionic systems.

The following Sec.1.2-1.6 introduce the central concepts described above in
more detail, pose the main questions we address in this thesis, and summarize
the answers we find.

####  1.2 Chiral topological matter

A gapped local many-body quantum Hamiltonian is said to be in a topological
phase of matter if it cannot be deformed to a trivial reference Hamiltonian,
without closing the energy gap or violating locality. If a symmetry is
enforced, only symmetric deformations are considered, and it is additionally
required that the symmetry is not spontaneously broken [89, 90]. For
Hamiltonians defined on a lattice, a natural trivial Hamiltonian is given by
the atomic limit of decoupled lattice sites, where the symmetry acts
independently on each site. In this thesis we consider both lattice and
continuum models.

Topological phases with a unique ground state on the 2-dimensional torus exist
only with a prescribed symmetry group111A subtle point is that the minimal
symmetry group for fermionic systems is fermion parity - the $\mathbb{Z}_{2}$
group generated by $(-1)^{N}$, where $N$ is the fermion number. This should be
contrasted with bosonic systems, which may have no symmetries. and are termed
symmetry protected topological phases (SPTs) [91, 92, 93]. When such phases
are placed on the cylinder, they support anomalous boundary degrees of freedom
which cannot be realized on isolated 1-dimensional spatial manifolds, as well
as corresponding quantized bulk response coefficients. Notable examples are
the integer quantum Hall states, topological insulators, and topological
superconductors [94]. Topological superconductivity and superfluidity will be
discussed in detail in Sec.1.4.

Topological phases with a degenerate ground state subspace on the torus are
termed topologically ordered, or symmetry enriched if a symmetry is enforced
[95, 96]. Beyond the phenomena exhibited by SPTs, these support localized
quasiparticle excitations with anyonic statistics and fractional charge under
the symmetry group. Notable examples are fractional quantum Hall states [97,
98], quantum spin liquids [99], and fractional topological insulators [100,
101].

In this thesis, we consider chiral topological phases, where the boundary
degrees of freedom that appear on the cylinder propagate unidirectionally. At
energies small compared with the bulk gap, the boundary can be described by a
chiral conformal field theory (CFT) [102, 103], while the bulk reduces to a
chiral topological field theory (TFT) [104, 19], see Fig.1(a). Such phases may
be bosonic and fermionic, and may be protected or enriched by an on-site
symmetry, but we will not make use of this symmetry in our analysis \- only
the chirality of the phase will be used.

A notable example for chiral topological phases is given by Chern insulators
[105, 106, 107]: SPTs protected by the $U\left(1\right)$ fermion number
symmetry, which admit free-fermion Hamiltonians. The single particle spectrum
of a Chern insulator on the cylinder is depicted in Fig.1(b). Another notable
example are the topologically ordered Kitaev spin liquids [104, 108], which
can be described by Majorana fermions with a single particle spectrum similar
to Fig.1(b), coupled to a $\mathbb{Z}_{2}$ (fermion-parity) gauge field. Note
that the velocity $v$ of the boundary CFT is a non-universal parameter which
generically changes as the microscopic Hamiltonian is deformed. More
generally, different chiral branches may have different velocities.

Figure 1: Chiral topological phases of matter on the cylinder. (a) The low
energy description of a chiral topological phase is comprised of two, counter
propagating, chiral conformal field theories (CFTs) on the boundary, and a
chiral topological field theory (TFT) in the bulk. (b) Examples: schematic
single-particle spectrum of a Chern insulator and of the Majorana fermions
describing a Kitaev spin liquid. Assuming discrete translational symmetry with
spacing $a$ in the $x$ direction, one can plot the single-particle eigen-
energies $\varepsilon$ on the cylinder as a function of (quasi) momentum
$k_{x}$. This reveals an integer number of chiral dispersion branches whose
eigen-states are supported on one of the two boundary components. In the Chern
insulator (Kitaev spin liquid) these correspond to the Weyl (Majorana-Weyl)
fermion CFT, with $c=\pm 1$ ($c=\pm 1/2$) per branch. The velocity,
$v=\left|\partial\varepsilon/\partial k_{x}\right|$ at the chemical potential
$\mu$, is a non-universal parameter.

The chirality of the boundary CFT and bulk TFT is manifested by their non-
vanishing chiral central charge $c$, which is rational and universal \- it is
a topological invariant with respect to continuous deformations of the
Hamiltonian which preserve locality and the bulk energy gap, and therefore
constant throughout a topological phase [109, 104, 39, 42]. On the boundary
$c$ is defined with respect to an orientation of the cylinder, so the two
boundary components have opposite chiral central charges. Since, as described
below, $c$ is much better understood from the boundary perspective, we
sometimes refer to it as the boundary chiral central charge. A main theme of
this thesis is the study of $c$ from the bulk perspective, and the relation
between the two perspectives implied by the anomaly inflow mechanism.

####  1.3 Geometric physics in chiral topological matter

The non-vanishing of $c$ implies a number of geometric, or ’gravitational’,
physical phenomena [102, 103, 34, 43, 42, 44]. In particular, the boundary
supports a non-vanishing energy current $J_{E}$, which receives a correction

$\displaystyle J_{E}\left(T\right)$ $\displaystyle=J_{E}\left(0\right)+2\pi
T^{2}\frac{c}{24},$ (1.1)

at a temperature $T>0$, and in the thermodynamic limit $L=\infty$, where $L$
is the circumference of the cylinder. Note that we set $K_{\text{B}}=1$ and
$\hbar=1$ throughout. Within CFT, this correction is universal since it is
independent of $v$. Taking the two counter propagating boundary components of
the cylinder into account, and placing these at slightly different
temperatures, leads to a thermal Hall conductance $K_{H}=c\pi T/6$ [110, 15,
111], a prediction that recently led to the first measurements of $c$ [112,
113, 84, 114].

In analogy with Eq.(1.1), the boundary of a chiral topological phase also
supports a non-vanishing ground state (or $T=0$) momentum density
$p\left(L\right)$, which receives a universal correction on a cylinder with
finite circumference $L<\infty$. The details of this finite-size correction
will be described in Sec.4, where it is used to relate the chiral central
charge (as well as the topological spins of anyon excitations) to the
complexity of simulating chiral topological matter on classical computers.

Abstractly, both $T>0$ and $L<\infty$ corrections described above follow
directly from the (chiral) Virasoro anomaly, or Virasoro central extension,
which defines $c$ in 2D CFT. Equivalently, these corrections can be understood
in terms of the ’global’ gravitational anomaly - the complex phase accumulated
by a CFT partition function on the torus under a Dehn twist [102, 103]. This
anomaly is termed ’global’ since the Dehn twist is a large coordinate
transformation, or more accurately, an element of the diffeomorphism group of
the torus, which lies outside of the connected component of the identity. The
Dehn twist is therefore the geometric analog of the large $U\left(1\right)$
gauge transformation used in the celebrated Laughlin argument, and an attempt
has been made to follow this analogy and produce a ’thermal Laughlin argument’
[50].

The chiral central charge also implies a ’local’, or ’perturbative’
gravitational anomaly, which, at least in the context of relativistic QFT in
curved space-time, physically corresponds to the non-conservation of energy-
momentum in the presence of curvature gradients [10, 12, 26]. Through the
anomaly inflow mechanism, or in more physical terms, through bulk+boundary
energy-momentum conservation, this boundary anomaly implies a gravitational
Chern-Simons (gCS) term in the effective action describing the 2+1D bulk of a
chiral topological phase [21, 22, 23, 24, 25]222Whether the gCS term matches
the boundary global gravitational anomaly as well is, to the best of my
knowledge, an open problem.. In turn, the gCS term implies a quantized energy-
momentum-stress response to curvature gradients, in the bulk 333In fact, the
gCS contribution to the energy-momentum-stress tensor is proportional to the
mathematically important Cotton tensor of the metric [115]. .

Though the gCS term is relatively well understood in the context of
relativistic QFT, its concrete physical content in the non-relativistic
setting of condensed matter physics is quite subtle, due to the following
reasons:

  1. 1.

Physically, the actual gravitational field of the earth is usually negligible
in condensed matter experiments. It is therefore clear that the adjective
’gravitational’ used above cannot be taken literally, and requires further
interpretation. Namely, one must find a physical probe relevant in condensed
matter experiments, which will somehow mimic the effects of a strong
gravitational field. This scenario is often referred to as ’analog gravity’ or
’emergent gravity’ [27]. Mathematically, this corresponds to a physically
accessible geometric structure on the space-time occupied by the system of
interest, in the spirit of general relativity444We use the words geometry and
gravity interchangeably from here on.. The most straight forward example is
given by strain - a physical deformation of the sample on which the system
resides [116]. An additional set of examples is given by spin-2
inhomogeneities [117] and collective excitations [28, 27, 29, 30, 33, 45].
Finally, Luttinger’s trick relates temperature gradients to an applied
gravitational field [118, 119, 37, 40, 41, 120, 50].

  2. 2.

Fundamentally, the coupling of a system to gravity generally depends on its
global space-time symmetries in the absence of gravity. For example,
relativistic systems will couple differently from Galilean invariant systems.
Even when the spatial symmetries are fixed, the gravitational background may
vary, e.g Riemannian vs. Riemann-Cartan geometry, which are both relativistic.
Moreover, for systems defined on a lattice, there is no definite, or
universal, prescription for a coupling to gravity at all, as opposed to
lattice gauge fields which are very well understood. The coupling of a system
to gravity therefore relies on more refined information than that used to
classify topological phases of matter. In particular, known results in
relativistic QFT do not directly apply to the non-relativistic condensed
matter systems we are interested in.

  3. 3.

Technically, when describing gravity in terms of a metric, the gCS term is
third order in derivatives, so obtaining effective actions that contain it
consistently, i.e account for all possible terms up to the same order, is
nontrivial.

Naturally, the pioneering approaches to the above difficulties were based on
an adaptation of known results in relativistic QFT [28, 15, 17] (see also [31,
121]), an approach that we carefully and critically follow in Sec.2. A much
more advanced treatment developed over the past decade, primarily in the
context of quantum Hall states [29, 30, 33, 34, 35, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55]. In particular, a non-relativistic
gCS term arises in quantum Hall states, and produces corrections to the odd
viscosity (introduced below) at finite wave-vector, and in curved background
[34, 42, 43, 39, 49]. We follow this observation in Sec.3. We note a couple of
additional central results from the literature:

  1. 1.

In quantum Hall states, the chiral central charge contributes to the braiding
statistics and angular momentum of conical defects [122, 123], the latter was
recently observed in an optical realization of integer quantum Hall states
[124, 53].

  2. 2.

The gCS term is not directly related to $K_{H}=c\pi T/6$ through Luttinger’s
trick, simply because it is too high in derivatives of the background metric
[25]. Moreover, careful analysis in quantum Hall states shows that $K_{H}$
receives no bulk contribution at all [40, 41], and is therefore purely a
boundary phenomenon, as explained below Eq.(1.1). Nevertheless, derivatives of
$K_{H}$ can be computed from the bulk Hamiltonian a la Luttinger [119, 32,
41], resulting in a relative topological invariant for gapped lattice systems
[54]. The latter gives a rigorous 2+1D lattice definition for the chiral
central charge.

####  1.4 Chiral superfluids and superconductors

An important class of 2+1D chiral topological phases appears in chiral
superfluids and superconductors (CSFs and CSCs, or CSF/Cs), where the ground
state is a condensate of Cooper pairs of fermions, which are spinning around
their centre of mass with a non-vanishing angular momentum $\ell\in\mathbb{Z}$
[15, 27], see Fig.2. As reviewed below, CSF/Cs appear in a wide range of
physical systems, all of which have been the subject of extensive and
continued research effort, going back to the classic body of work on
superfluid ${}^{3}\text{He}$ [125]. The interest in CSF/Cs comes from two
fronts, a fermionic/topological front, and a bosonic/symmetry-breaking front,
both resulting directly from the $\ell$-wave condensate. A central theme of
this thesis is the intricate interplay between these two facets of CSF/Cs.

Figure 2: Chiral superfluids and superconductors (CSF/Cs) are comprised of
fermions $\psi$, which form Cooper pairs with non-vanishing relative angular
momentum $\ell\in\mathbb{Z}$ (red arrows), in units of $\hbar$. As a result,
CSF/Cs support boundary degrees of freedom (dashed orange) with a chiral
central charge $c\in\left(\ell/2\right)\mathbb{Z}$.

On the fermionic/topological front, the $\ell$-wave condensate leads to an
energy gap for single fermion excitations, which form a chiral SPT phase of
matter: a topological superconductor [126, 127]. Topological superconductors
are of interest since their chiral central charge
$c\in\left(\ell/2\right)\mathbb{Z}$ can be half-integer, indicating the
presence of chiral Majorana spinors on boundaries, each contributing an
additive $\pm 1/2$ to $c$, where the sign depends on their chirality. In turn,
this implies the presence of Majorana bound states, or zero modes, in the
cores of vortices [15]. The observation of Majorana fermions, which are their
own anti-particles, and may not exist in nature as elementary particles, is of
fundamental interest. Moreover, Majorana bound states are closely related to
non-abelian Ising anyons [128], which have been proposed as building blocks
for topological quantum computers [129, 97].

On the bosonic/symmetry-breaking front, the $\ell$-wave condensate implies an
exotic symmetry breaking pattern, which leads to an unusual spectrum of
bosonic excitations, and as a result, an unusual hydrodynamic description. In
more detail, the condensation of $\ell$-wave Cooper pairs corresponds to a
non-vanishing ground-state expectation value for the operator
$\psi^{\dagger}\left(\partial_{x}\pm
i\partial_{y}\right)^{\left|\ell\right|}\psi^{\dagger}$, where
$\psi^{\dagger}$ is a fermion creation operator555Due to Fermi statistics,
$\ell$ must be odd if $\psi$ is spin-less. An even $\ell$ requires spin-full
fermions forming spin-less (singlet) Cooper pairs,
$\psi_{\uparrow}^{\dagger}\left(\partial_{x}\pm
i\partial_{y}\right)^{\left|\ell\right|}\psi_{\downarrow}^{\dagger}$. Spin-
full fermions can also form spin-1 (triplet) Cooper pairs with odd $\ell$, as
is the case in ${}^{3}\text{He-A}$. Since the geometric physics we are
interested in is independent of the spin of the Cooper pair, we restrict
attention to spin-less fermions for odd $\ell$, and write our expressions per
spin component for even $\ell$. and $\pm=\text{sgn}\left(\ell\right)$. An
$\ell$-wave condensate implies the breaking of time reversal symmetry $T$ and
parity (spatial reflection) $P$ down to $PT$, and of the symmetry groups
generated by particle number $N$ and angular momentum $L$ down to a diagonal
subgroup

$\displaystyle U\left(1\right)_{N}\times SO\left(2\right)_{L}\rightarrow
U\left(1\right)_{L-\left(\ell/2\right)N}.$ (1.2)

In CSFs, this symmetry breaking occurs spontaneously, due to a symmetric two-
body attractive interaction between fermions. This phenomenon is generic, at
least from the perspective of perturbative Fermi-surface renormalization group
[130]. Thin films of ${}^{3}\text{He-A}$ are experimentally accessible
$p$-wave CSFs [131, 132, 133, 134], and there are many proposals for the
realization of various $\ell$-wave CSFs in cold atoms [135, 136, 137, 138,
139, 140, 141]. The spontaneous symmetry breaking (1.2) implies a single
Goldstone field, charged under the broken generator $N+\left(\ell/2\right)L$,
as well as massive Higgs fields, which are $U\left(1\right)_{N}$-neutral, and
carry angular momentum $0$ and $\pm 2\ell$ [142, 143, 144, 1, 145]. In
particular, $p$-wave ($\ell=\pm 1$) superfluids support Higgs fields of
angular momentum $0$ and $\pm 2$, which form a spatial metric, including a
non-relativistic analog of the graviton [28, 15, 27]. This observation will
play a central role in Sec.2. The angular momentum $\ell/2$ carried by the
Goldstone field leads to a $P,T$-odd hydrodynamic description, including an
odd (or Hall) viscosity, which is introduced below, and studied in Sec.3.

An intrinsic CSC is obtained if the $U\left(1\right)_{N}$ symmetry is gauged,
by coupling to a dynamical gauge field. This gauge field physically
corresponds to the 3+1D electromagnetic interaction between electrons, which
are themselves confined to a 2+1D lattice of ions. Experimental evidence for
chiral superconductivity was recently reported in Ref.[146]. One may also
consider an emergent 2+1D gauge field, with a Chern-Simons and/or Maxwell
dynamics. In particular, this leads to CSFs of ’composite fermions’ [15, 147,
148], including field theoretic descriptions of the non-abelian candidates for
the fractional quantum Hall state observed at filling $5/2$ [84], a subject of
ongoing debate [85, 86, 87, 149, 88]. The symmetry breaking pattern (1.2) may
also occur explicitly, due the proximity of a conventional $s$-wave
superconductor (SC) to 2+1D spin-orbit coupled metal, in which case we speak
of a proximity induced CSC, an observation of which was reported in Refs.[150,
151]. Note that in this case the Goldstone and Higgs fields can be viewed as
non-dynamical.

Despite the large body of work on boundary Majorana fermions in CSFs and CSCs
(CSF/Cs), the bulk geometric physics corresponding to these through anomaly
inflow, and presumably captured by a gCS action, remains poorly understood,
due to the difficulties mentioned in Sec.1.3. In fact, most existing
statements, though made in truly pioneering and seminal work [28, 15, 17], are
speculative, and are primarily based on an inaccurate adaptation of known
results in relativistic QFT to $p$-wave CSF/Cs. An understanding of the bulk
geometric physics is of particular importance since, in the simplest case of
spin-less fermions with no additional internal symmetry, the only charge
carried by the boundary Majorana fermions is energy-momentum, and the only
boundary anomalies and bulk topological effective actions are therefore
gravitational666For spin-full $p$-wave CSFs, one can exploit
$SU\left(2\right)$ spin rotation symmetry, and does not have to resort to
gravitational probes [152, 15, 153].. In particular, the boundary Majorana
fermions are always $U\left(1\right)_{N}$-neutral, and it follows that no
$U\left(1\right)_{N}$ boundary anomaly or bulk topological effective action
occurs777An exception to this rule occurs in Galilean invariant systems, where
momentum and $U\left(1\right)_{N}$-current are identified, as we will see in
Sec.3.. Motivated by this state of affairs, the goal of Sec.2-3 is to turn the
insightful ideas of Refs.[28, 15, 17] into concrete physical predictions.

As a first approach to the problem, in Sec.2 we follow Refs.[28, 15, 17] and
utilize the low energy relativistic description of $p$-wave CSF/Cs, which
exists because the $p$-wave condensate $\psi^{\dagger}\left(\partial_{x}\pm
i\partial_{y}\right)\psi^{\dagger}$ is first order in derivatives. The main
questions we ask are:

> What type of space-time geometry emerges in the low energy relativistic
> description of $p$-wave superfluids and superconductors? What are the
> physical implications of the emergent relativistic geometry to these non-
> relativistic systems?

Our answer to the first question is that the fermionic excitations in $p$-wave
CSF/Cs correspond at low energy to a massive relativistic Majorana spinor,
which is minimally coupled to an emergent Riemann-Cartan geometry. This
geometry is described by the $p$-wave order parameter
$\Delta^{i}\sim\delta^{ij}\psi^{\dagger}\partial_{j}\psi^{\dagger}$, made up
of the Goldstone and Higgs fields, as well as a $U\left(1\right)_{N}$ gauge
field. As opposed to the Riemannian geometry previously believed to emerge
[28, 15, 17], Riemann-Cartan space-times are characterized by a non-vanishing
torsion tensor, in addition to the curvature tensor [154]. In condensed matter
physics (or elasticity theory), torsion is well known to describe the density
of lattice dislocations [155, 156, 157, 158]888Similarly, curvature
traditionally describes the density of lattice disclinations, as well a the
curving of a two-dimensional material in three dimensional space. It is also
known that temperature gradients correspond to time-like torsion via
Luttinger’s trick [37, 41]. , and our results provide a new mechanism by which
torsion can emerge \- due to the symmetry breaking pattern (1.2) at $\ell=\pm
1$ . The above statements are relevant if one aims at studying relativistic
fermions in nontrivial space-times using table-top experiments [159], or if
one hopes that emergent relativistic geometry in condensed matter can answer
fundamental questions about the seemingly relativistic geometry of our
universe [27]. Here, however, we are interested in answering the second
question posed above.

As expected, a gCS term appears in the low energy effective action of $p$-wave
CSF/Cs, and we find that it produces a precisely quantized bulk energy-
momentum-stress response to the $p$-wave Higgs fields. Accordingly, a
(perturbative) gravitational anomaly that depends on the Higgs fields appears
on the boundary, implying a $c$-dependent transfer of energy-momentum between
bulk and boundary. The emergence of torsion leads to additional interesting
terms in the bulk effective action. In particular, a non-topological
’gravitational pseudo Chern-Simons’ term produces an energy-momentum-stress
response closely mimicking that of the gCS term, and we show how to
disentangle the two responses in order to extract $c$ from bulk measurements.
In lattice models, the low energy description consists of an even number of
relativistic Majorana spinors - a fermion doubling phenomena. Surprisingly, we
find that these spinors experience slightly different emergent geometries. As
a result, additional ’gravitational Chern-Simons difference’ terms are
possible, which are again not of topological origin, but nevertheless imply
responses which must be carefully distinguished from those of the gCS term.
All other terms in the bulk effective action are either higher in derivatives,
or are lower in derivatives but naively diverge within the relativistic
description. The latter ’UV-sensitive’ terms cannot be reliably interpreted
based on the relativistic description, and require a non-relativistic
treatment. In particular, the relativistic, UV-sensitive, and somewhat
controversial ’torsional Hall viscosity’ [155, 156, 157, 158, 160, 41], is
found in Sec.3 to correspond to the non-relativistic and well understood odd
(or Hall) viscosity of CSF/Cs [161, 162, 163, 36, 164], which is introduced
below.

Before continuing, we note that there has been considerable recent interest in
torsional physics in condensed matter, in systems described at low energy by
3+1D Weyl (or Majorana-Weyl) spinors, namely Weyl semi-metals and 3He-A [165,
166, 167, 168, 169, 170], and in Kitaev’s honeycomb model [171, 172].

####  1.5 Odd viscosity

The odd (or Hall) viscosity $\eta_{\text{o}}$ is a non-dissipative, time
reversal odd, stress response to strain-rate [173, 174, 175, 160, 176], which
can appear even in superfluids (SFs) and incompressible (or gapped) fluids,
where the more familiar and intuitive dissipative viscosity vanishes. The
observable signatures of $\eta_{\text{o}}$ are actively studied in a variety
of systems [177, 178, 179, 180, 181, 182, 183, 184, 185, 186], and recently
led to its measurement in a colloidal chiral fluid [187] and in graphene’s
electron fluid under a magnetic field [188].

In isotropic 2+1 dimensional fluids, the odd viscosity tensor at zero wave-
vector ($\mathbf{q}=0$) reduces to a single component . In analogy with the
celebrated quantization of the odd (or Hall) conductivity in the quantum Hall
(QH) effect [9, 8, 189, 106, 190, 191], this component obeys a quantization
condition

$\displaystyle\eta_{\text{o}}^{\left(1\right)}=-\left(\hbar/2\right)s\cdot
n_{0},\;s\in\mathbb{Q},$ (1.3)

in incompressible quantum fluids, such as integer and fractional QH states
[173, 161, 162]. Here $n_{0}$ is the ground state density, and $s$ is a
rational-valued topological invariant999In fact, an
$SO\left(2\right)_{L}$-symmetry-protected topological invariant., labeling the
many-body ground state, which can be interpreted as the average angular
momentum per particle (in units of $\hbar$, which is henceforth set to 1).

Remarkably, Eq.(1.3) also holds in CSFs, though they are compressible.
Computing $\eta_{\text{o}}^{\left(1\right)}$ in an $\ell$-wave CSF, one finds
Eq.(1.3) with the intuitive angular momentum per fermion, $s=\ell/2$ [161,
162, 163, 36, 164]. Thus, a measurement of $\eta_{\text{o}}^{\left(1\right)}$
at $\mathbf{q}=\mathbf{0}$ can be used to obtain the angular momentum $\ell$
of the Cooper pair, but carries no additional information. It is therefore
clear that the symmetry breaking pattern (1.2) which defines $\ell$, rather
than ground-state topology, is the origin of the quantization $s=\ell/2$ in
CSFs 101010Accordingly, the quantization of $s$ is broken in a mixture of CSFs
with different $\ell$s, where $U\left(1\right)_{N}\times SO\left(2\right)_{L}$
is completely broken. In the mixture
$s\equiv-2\eta_{\text{o}}^{\left(1\right)}/n=\sum_{i}n_{i}\left(\ell_{i}/2\right)/\sum_{i}n_{i}$
retains its meaning as an average angular momentum per particle, but is no
longer quantized. This should be contrasted with multicomponent QH states
[42], where all $n_{i}$s are proportional to the same applied magnetic field
through the filling factors $\nu_{i}\in\mathbb{Q}$, and $s$ remains
quantized..

Nevertheless, the gapped fermions in a CSF do carry non-trivial ground-state
topology labeled by the central charge $c\in\left(\ell/2\right)\mathbb{Z}$,
and, based on results in quantum Hall states [34, 43, 42], a $c$-dependent
correction to $\eta_{\text{o}}^{\left(1\right)}$ of Eq.(1.3) is therefore
expected to appear at small non-zero wave-vector,

$\displaystyle\delta\eta_{\text{o}}^{\left(1\right)}\left(\mathbf{q}\right)=-\frac{c}{24}\frac{1}{4\pi}q^{2}.$
(1.4)

This raises the questions:

> In chiral superfluids, can the boundary chiral central charge be extracted
> from a measurement of the bulk odd viscosity? Can it be extracted from any
> bulk measurement?

Providing a definite answer to these questions is the main goal of Sec.3, and
requires a fully non-relativistic treatment of CSFs. The main reason for this
is that the relativistic low energy description misses most of the physics of
the Goldstone field. Analysis of Goldstone physics in CSFs was undertaken in
Refs.[192, 193, 194, 195, 153, 196, 197, 198], most of which revolving around
the non-vanishing, yet non-quantized, Hall (or odd) conductivity in CSFs. More
recently, Refs.[163, 164] considered CSFs in curved (or strained) space,
following the pioneering work [199] on $s$-wave ($\ell=0$) SFs. These works
demonstrated that the Goldstone field, owing to its charge
$L+\left(\ell/2\right)N$, produces the $\mathbf{q}=\mathbf{0}$ odd viscosity
(1.3), and it is therefore natural to expect that a $q^{2}$ correction similar
to (1.4) will also be produced. Nevertheless, Refs.[163, 164] did not consider
the derivative expansion to the high order at which $q^{2}$ corrections to
$\eta_{\text{o}}$ would appear, nor did they detect any bulk signature of $c$
at lower orders. In Sec.3 we obtain a low energy effective field theory that
consistently captures both the chiral Goldstone mode and the gCS term, thus
unifying and extending the seemingly unrelated analysis of Refs.[199, 163,
164] and Sec.2. Using the theory we show that $c$ cannot be extracted for a
measurement of the odd viscosity alone, as suggested by Eq.(1.4).
Nevertheless, a related observable, termed ’improved odd viscosity’, does
allow for the bulk measurement of $c$. Additional results of the same spirit
are found in Galilean invariant CSFs.

####  1.6 Quantum Monte Carlo sign problems in chiral topological matter

Utilizing a random sampling of phase-space according to the Boltzmann
probability distribution, Monte Carlo simulations are arguably the most
powerful tools for numerically evaluating thermal averages in classical many-
body physics [200]. Though the phase-space of an $N$-body system scales
exponentially with $N$, a Monte-Carlo approximation with a fixed desired error
is usually obtained in polynomial time [72, 201]. In Quantum Monte Carlo
(QMC), one attempts to perform Monte-Carlo computations of thermal averages in
quantum many-body systems, by following the heuristic idea that quantum
systems in $d$ dimensions are equivalent to classical systems in $d+1$
dimensions [65, 71].

The difficulty with any such quantum to classical mapping, henceforth referred
to as a method, is the infamous sign problem, where the mapping can produce
complex, rather than non-negative, Boltzmann weights $p$, which do not
correspond to a probability distribution. Faced with a sign problem, one can
try to change the method used and obtain $p\geq 0$, thus curing the sign
problem [73, 74]. Alternatively, one can perform QMC using the weights
$\left|p\right|$, which is often done but generically leads to an exponential
computational complexity in evaluating physical observables, limiting ones
ability to simulate large systems at low temperatures [72].

Conceptually, the sign problem can be understood as an obstruction to mapping
quantum systems to classical systems, and accordingly, from a number of
complexity theoretic perspectives, a generic curing algorithm in polynomial
time is not believed to exist [72, 202, 75, 73, 203, 74]. In many-body
physics, however, one is mostly interested in universal phenomena, i.e phases
of matter and the transitions between them, and therefore representative
Hamiltonians which are free of the sing problem (henceforth ’sign-free’) often
suffice [68]. In fact, QMC simulations continue to produce unparalleled
results, in all branches of many-body quantum physics, precisely because new
sign-free models are constantly being discovered [64, 204, 66, 67, 68, 69, 70,
71]. Designing sign-free models requires design principles (or “de-sign”
principles) [68, 205] \- easily verifiable properties that, if satisfied by a
Hamiltonian and method, lead to a sign-free representation of the
corresponding partition function. An important example is the condition
$\left\langle i\right|H\left|j\right\rangle\leq 0$ where $i\neq j$ label a
local basis, which implies non-negative weights $p$ in a wide range of methods
[68, 203]. Hamiltonians satisfying this condition in a given basis are known
as stoquastic [202], and have proven very useful in both application and
theory of QMC in bosonic (or spin, or ’qudit’) systems [68, 72, 202, 75, 73,
203, 74].

Fermionic Hamiltonians are not expected to be stoquastic in any local basis
[72, 71], and alternative methods, collectively known as determinantal quantum
Monte-Carlo (DQMC), are therefore used [206, 65, 77, 71, 70]. The search for
design principles that apply to DQMC, and applications thereof, has naturally
played the dominant role in tackling the sign problem in fermionic systems,
and has seen a lot of progress in recent years [207, 205, 208, 209, 210, 70,
71]. Nevertheless, long standing open problems in quantum many-body physics
continue to defy solution, and remain inaccessible for QMC. These include the
nature of high temperature superconductivity and the associated repulsive
Hubbard model [77, 78, 79, 80], dense nuclear matter and the associated
lattice QCD at finite baryon density [81, 82, 83], and the enigmatic
fractional quantum Hall state at filling $5/2$ and its associated Coulomb
Hamiltonian [84, 85, 86, 87, 88, 55], all of which are fermionic.

One may wonder if there is a fundamental reason that no design principle
applying to the above open problems has so far been found, despite intense
research efforts. More generally,

> Are there phases of matter that do not admit a sign-free representative? Are
> there physical properties that cannot be exhibited by sign-free models?

We refer to such phases of matter, where the sign problem simply cannot be
cured, as having an intrinsic sign problem [75]. From a practical perspective,
intrinsic sign problems may prove useful in directing research efforts and
computational resources. From a fundamental perspective, intrinsic sign
problems identify certain phases of matter as inherently quantum - their
physical properties cannot be reproduced by a partition function with positive
Boltzmann weights.

To the best of our knowledge, the first intrinsic sign problem was discovered
by Hastings [75], who proved that no stoquastic, commuting projector,
Hamiltonians exist for the ’doubled semion’ phase [211], which is bosonic and
topologically ordered. In Ref.[5], we generalize this result considerably -
excluding the possibility of stoquastic Hamiltonians in a broad class of
bosonic non-chiral topological phases of matter. Additionally, Ref.[76]
demonstrated, based on the algebraic structure of edge excitations, that no
translationally invariant stoquastic Hamiltonians exist for bosonic chiral
topological phases.

In Sec.4, we will establish a new criterion for intrinsic sign problems in
chiral topological matter, and take the first step in analyzing intrinsic sign
problems in fermionic systems. First, based on the well established ’momentum
polarization’ method for characterizing chiral topological matter [212, 213,
38, 214], we obtain a variant of the result of Ref.[76] \- excluding the
possibility of stoquastic Hamiltonians in a broad class of bosonic chiral
topological phases. We then develop a formalism with which we obtain analogous
results for systems comprised of both bosons and fermions - excluding the
possibility of sign-free DQMC simulations.

Phase of matter | Parameterization | $\boldsymbol{c}$ | $\boldsymbol{\left\\{h_{a}\right\\}}$ | Intrinsic sign problem?
---|---|---|---|---
Laughlin (B) [55] | Filling $1/q,\;(q\in 2\mathbb{N})$ | $1$ | $\left\\{a^{2}/2q\right\\}_{a=0}^{q-1}$ | In $98.5\%$ of first $10^{3}$
Laughlin (F) [55] | Filling $1/q,\;(q\in 2\mathbb{N}-1)$ | $1$ | $\left\\{\left(a+1/2\right)^{2}/2q\right\\}_{a=0}^{q-1}$ | In $96.7\%$ of first $10^{3}$
Chern insulator (F) [3] | Chern number $\nu\in\mathbb{Z}$ | $\nu$ | $\left\\{\nu/8\right\\}$ | For $\nu\notin 12\mathbb{Z}$
$\ell$-wave superconductor (F) [2] | Pairing channel $\ell\in 2\mathbb{Z}-1$ | $-\ell/2$ | $\left\\{-\ell/16\right\\}$ | Yes
Kitaev spin liquid (B) [104] | Chern number $\nu\in 2\mathbb{Z}-1$ | $\nu/2$ | $\left\\{0,1/2,\nu/16\right\\}$ | Yes
$SU\left(2\right)_{k}$ Chern-Simons (B) [215] | Level $k\in\mathbb{N}$ | $3k/\left(k+2\right)$ | $\left\\{a\left(a+2\right)/4\left(k+2\right)\right\\}_{a=0}^{k}$ | In $91.6\%$ of first $10^{3}$
$E_{8}$ $K$-matrix (B) [216] | Stack of $n\in\mathbb{N}$ copies | $8n$ | $\left\\{0\right\\}$ | For $n\notin 3\mathbb{N}$
Fibonacci anyon model (B) [215] |  |  $14/5$ (mod $8$) | $\left\\{0,2/5\right\\}$ | Yes
Pfaffian (F) [217] |  | $3/2$ | $\left\\{0,1/2,1/4,3/4,1/8,5/8\right\\}$ | Yes
PH-Pfaffian (F) [217] |  | $1/2$ | $\left\\{0,0,1/2,1/2,1/4,3/4\right\\}$ | Yes
Anti-Pfaffian (F) [217] |  | $-1/2$ | $\left\\{0,1/2,1/4,3/4,3/8,7/8\right\\}$ | Yes

Table 1: Examples of intrinsic sign problems based on the criterion $e^{2\pi
ic/24}\notin\left\\{\theta_{a}\right\\}$, in terms of the chiral central
charge $c$ and the topological spins $\theta_{a}=e^{2\pi ih_{a}}$. The number
of spins $h_{a}$ is equal to the dimension of the ground state subspace on the
torus. We mark bosonic/fermionic phases by (B/F). The quantum Hall Laughlin
phases corresond to $U(1)_{q}$ Chern-Simons theories. The $\ell$-wave
superconductor is chiral, e.g $p+ip$ for $\ell=1$, and comprised of a single
flavour of spinless fermions. The data shown refers only to the SPT phase
formed by gapped fermionic excitations, see Sec.1.4. Data for the spinfull
case is identical to that of the Chern insulator, with $-\ell$ odd (even) in
place of $\nu$, for triplet (singlet) pairing. The modulo 8 ambiguity in the
central charge of the Fibonacci anyon model corresponds to the stacking of a
given realization with copies of the $E_{8}$ $K$-matrix phase. Data for the
three quantum Hall Pfaffian phases is given at the minimal filling 1/2. The
physical filling 5/2 is obtained by stacking with a $\nu=2$ Chern insulator,
and an intrinsic sign problem appears in this case as well.

All of the above mentioned topological phases are gapped, 2+1 dimensional, and
described at low energy by a topological field theory [218, 104, 19]. The
class of such phases in which we find an intrinsic sign problem is defined in
terms of robust data characterizing them: the chiral central charge $c$, a
rational number, as well as the set $\left\\{\theta_{a}\right\\}$ of
topological spins of anyons, a subset of roots of unity. Namely, we find that

> An intrinsic sign problem exists if $e^{2\pi ic/24}$ is not the topological
> spin of some anyon, i.e $e^{2\pi ic/24}\notin\left\\{\theta_{a}\right\\}$.

The above criterion applies to most chiral topological phases, see Table 1 for
examples. In particular, we identify an intrinsic sign problem in $96.7\%$ of
the first one-thousand fermionic Laughlin phases, and in all non-abelian
Kitaev spin liquids. We also find intrinsic sign problems in $91.6\%$ of the
first one-thousand $SU\left(2\right)_{k}$ Chern-Simons theories. Since, for
$k\neq 1,2,4$, these allow for universal quantum computation by manipulation
of anyons [219, 97], our results support the strong belief that quantum
computation cannot be simulated with classical resources, in polynomial time
[220]. This conclusion is strengthened by examining the Fibonacci anyon model,
which is known to be universal for quantum computation [97], and is found to
be intrinsically sign-problematic.

We stress that both $c$ and $\left\\{\theta_{a}\right\\}$ have clear
observable signatures in both the bulk and boundary of chiral topological
matter, some of which have been experimentally observed. The chiral central
charge was extensively discussed in previous section, including its
observation in Refs.[112, 113, 84, 114, 53]. The topological spins determine
the exchange statistics of anyons, predicted to appear in interferometry
experiments [97]. Experimental observation remained elusive [221] until it was
recently reported in the Laughlin 1/3 quantum Hall state [222]. Additionally,
a measurement of anyonic statistics via current correlations [223] was
recently reported in the Laughlin 1/3 state [224].

###  2 Probing topological superconductors with emergent relativistic gravity

In this section, we restrict our attention to spin-less $p$-wave chiral
superfluids and superconductors (CSF/Cs), and analyze the relativistic
geometry, or gravity, that emerges at low energy. We seek answers to the
questions posed in Sec.1.4.

####  2.1 Approach and main results

#####  2.1.1 Model and approach

As a starting point, we consider a simple model for spin-less $p$-wave CSF/Cs
[225]. The action is given by

$\displaystyle S\left[\psi;\Delta\right]=$
$\displaystyle\int\text{d}^{2+1}x\left[\psi^{\dagger}\left(i\partial_{t}+\frac{\delta^{ij}\partial_{i}\partial_{j}}{2m^{*}}-m\right)\psi+\left(\frac{1}{2}\psi^{\dagger}\Delta^{j}\partial_{j}\psi^{\dagger}+h.c\right)\right],$
(2.1)

and describes the coupling of a spin-less fermion $\psi$, with effective mass
$m^{*}$ and chemical potential $-m$, to a $p$-wave order parameter
$\Delta=\left(\Delta^{x},\Delta^{y}\right)\in\mathbb{C}^{2}$. The order
parameter corresponds to the condensate of Cooper pairs described in Sec.1.4.
In a proximity induced CSC, the order parameter can be thought of as a non-
dynamical background field, as it appears in Eq.(2.1). In intrinsic CSCs, and
in CSFs, the order parameter is a quantum field which mediates an attractive
interaction between fermions, and a treatment of the dynamics of $\Delta$ is
deferred to Sec.2.6 and 3. The ground-state, or unperturbed, configuration of
$\Delta$ is the $p_{x}\pm ip_{y}$ configuration
$\Delta=\Delta_{0}e^{i\theta}\left(1,\pm i\right)$, where $\Delta_{0}>0$ and
$\theta$ are constants. The phase $\theta$ corresponds to the Goldstone field
implied by Eq.(1.2), while the orientation (or chirality) $o=\pm$ corresponds
to the breaking of reflection and time reversal symmetries to their product,
$P\times T\rightarrow PT$.

One may view the model (2.1) as ’microscopic’, as will be done in Sec.3, but
here we will think of it as the low energy description of a lattice model,
introduced and analyzed in Sec.2.2. In the ’relativistic regime’ where the
order parameter is much larger than the single particle scales, the lattice
model is essentially a lattice regularization of four, generically massive,
relativistic Majorana spinors, centered at the particle-hole invariant points
$k=-k$ in the Brillouin zone. Around each of these four points, the low energy
description is given by an action of the form (2.1). In the relativistic
regime the effective mass $m^{*}$ is large, and in the limit
$m^{*}\rightarrow\infty$ Eq.(2.1) reduces to a relativistic action, with mass
$m$ and speed of light $c_{\text{light}}=\Delta_{0}/\hbar$, for the Nambu
spinor $\Psi^{\dagger}=\left(\psi^{\dagger},\psi\right)$, which is a Majorana
spinor. The different Majorana spinors, associated with the four particle-hole
invariant points, have different orientations $o_{n}$ and masses $m_{n}$,
where $1\leq n\leq 4$.

The chiral central charge of the lattice model can be deduced from its Chern
number [15, 104, 27, 107]. The $n$th Majorana spinor contributes
$c_{n}=o_{n}\text{sgn}\left(m_{n}\right)/4$, and summing over $n$ one obtains
the central charge of the lattice model
$c=\sum_{n=1}^{4}c_{n}=\sum_{n=1}^{4}o_{n}\text{sgn}\left(m_{n}\right)/4$,
which gives the topological phase diagram purely in terms of the low energy
relativistic data $o_{n},m_{n}$, see Sec.2.2. This formula motivates a study
of the geometric physics associated with $c$, purely within the low energy
relativistic description, which we now pursue.

In order to access the physics associated with $c$, we perturb the order
parameter out of the $p_{x}\pm ip_{y}$ configuration, and treat
$\Delta=\left(\Delta^{x},\Delta^{y}\right)\in\mathbb{C}^{2}$ as a general
space-time dependent field. This is analogous to applying an electromagnetic
field in order to measure a quantized Hall conductivity in the quantum Hall
effect. Following the observations of Refs.[27, 15], we show in Sec.2.3 that,
in the relativistic limit, the Majorana spinor $\Psi$ experiences such a
general order parameter (along with a general $U\left(1\right)_{N}$ gauge
field) as a non trivial gravitational background, namely Riemann-Cartan
geometry. See also Sec.2.2.2 for the basics of this statement. Some of the
emergent geometry is described by the (inverse) spatial metric

$\displaystyle g^{ij}=-\Delta^{(i}\Delta^{j)*},$ (2.2)

where brackets denote the symmetrization, and the sign is a matter of
convention. The metric $g^{ij}$ corresponds to the Higgs field included in the
order parameter. Parameterizing
$\Delta=e^{i\theta}\left(\left|\Delta^{x}\right|,e^{i\phi}\left|\Delta^{y}\right|\right)$
with the overall phase $\theta$ and relative phase
$\phi\in\left(-\pi,\pi\right]$, the metric is independent of $\theta$ and of
the orientation $o=\text{sgn}\phi=\pm$, which splits order parameters into
$p_{x}+ip_{y}$-like and $p_{x}-ip_{y}$-like. Note that in the $p_{x}\pm
ip_{y}$ configuration the metric is euclidian,
$g^{ij}=-\Delta_{0}^{2}\delta^{ij}$. For our purposes it is important that the
metric be perturbed out of this form, and in particular it is not enough to
take the $p_{x}\pm ip_{y}$ configuration with a space-time dependent Goldstone
field $\theta$.

Figure 3: A comparison of the integer quantum Hall effect (IQHE) and its
energy-momentum analog in $p$-wave CSF/Cs. (a) In the IQHE there is a
perpendicular electric current $\left\langle J\right\rangle$ in response to an
applied electric field $E$, with a quantized Hall conductivity, proportional
to the Chern number $\nu\in\mathbb{Z}$, as encoded in a $U(1)_{N}$ Chern-
Simons term. (b) In $p$-wave CSF/Cs, an energy current $\left\langle
J_{E}\right\rangle$ flows in response to a space dependent order parameter
$\Delta$, as encoded in a gravitational Chern-Simons term. Derivatives of the
curvature $\tilde{\mathcal{R}}$ associated with $\Delta$ play the role of the
electric field in the IQHE, and $\left\langle J_{E}\right\rangle$ is
perpendicular to the curvature gradient $\nabla\tilde{\mathcal{R}}$. The ratio
between the magnitudes of $\left\langle J_{E}\right\rangle$ and
$\nabla\tilde{\mathcal{R}}$ is quantized, and proportional to the chiral
central charge $c\in(1/2)\mathbb{Z}$. As described in the text, the
spontaneous breaking of $U(1)_{N}$ symmetry in $p$-wave CSF/Cs allows for a
gravitational pseudo Chern-Simons term, encoding closely related bulk
responses, which are not topological in nature. (c) The quantized Hall
conductivity implies the existence of a chiral boundary fermion with a
$U\left(1\right)_{N}$ anomaly, which can be described as a Weyl fermion at low
energy. (d) The analogous response in $p$-wave CSF/Cs implies the existence of
a boundary chiral Majorana fermion with a gravitational anomaly, which can be
described as a Majorana-Weyl fermion at low energy.

#####  2.1.2 Topological bulk responses from a gravitational Chern-Simons term

Using the mapping of $p$-wave CSF/Cs to relativistic Majorana fermions in
Riemann-Cartan space-time, we compute and analyze in Sec.2.5 the effective
action obtained by integrating over the bulk fermions in the presence of a
general order parameter $\Delta$, and $U\left(1\right)_{N}$ gauge field. Here
we discuss the main physical implications of this effective action. As already
explained in Sec.1.4, we only describe UV-insensitive physics, which can be
reliably understood within the low energy relativistic description. This
physics is controlled by dimensionless coefficients, including the chiral
central charge $c$ in which we are primarily interested.

As expected, the effective action contains a gCS term [21, 22, 23, 24, 115,
25]

$\displaystyle S_{\text{gCS}}=$
$\displaystyle\alpha\int\text{tr}\left(\tilde{\Gamma}\wedge\text{d}\tilde{\Gamma}+\frac{2}{3}\tilde{\Gamma}\wedge\tilde{\Gamma}\wedge\tilde{\Gamma}\right),$
(2.3)

with coefficient $\alpha=\frac{c}{96\pi}\in\frac{1}{192\pi}\mathbb{Z}$, and
where $\tilde{\Gamma}$ is the Christoffel symbol of the space-time metric
obtained from Eq.(2.2), see Sec.2.5.2.2. Although we obtain this result in the
limit $m^{*}\rightarrow\infty$, we expect it to hold throughout the phase
diagram. This is based on known arguments for the quantization of the
coefficient $\alpha$ due to symmetry, and on the relation with the boundary
gravitational anomaly described below.

The gCS term implies a topological bulk response, where energy-momentum
currents and densities appear due to a space-time dependent order parameter.
To see this in the simplest setting, assume that the order parameter is time
independent, and that the relative phase is $\phi=\pm\frac{\pi}{2}$, as in the
$p_{x}\pm ip_{y}$ configuration, so that
$\Delta=e^{i\theta}\left(\left|\Delta^{x}\right|,\pm
i\left|\Delta^{y}\right|\right)$, $o=\pm$. Then the metric is time
independent, and takes the simple form

$\displaystyle g^{ij}=-\begin{pmatrix}\left|\Delta^{x}\right|^{2}&0\\\
0&\left|\Delta^{y}\right|^{2}\end{pmatrix}.$ (2.4)

On this background, we find the following contributions to the expectation
values of the fermionic energy current $J_{E}^{i}$, and momentum density
$P_{i}$ 111111$P_{x}$ ($P_{y}$) is the density of the $x$ ($y$) component of
momentum.,

$\displaystyle\left\langle J_{E}^{i}\right\rangle_{\text{gCS}}$
$\displaystyle=$
$\displaystyle-\frac{c}{96\pi}\frac{1}{\hbar}\varepsilon^{ij}\partial_{j}\tilde{\mathcal{R}},$
(2.5) $\displaystyle\left\langle P_{i}\right\rangle_{\text{gCS}}$
$\displaystyle=$ $\displaystyle-\frac{c}{96\pi}\hbar
g_{ik}\varepsilon^{kj}\partial_{j}\tilde{\mathcal{R}}.$

Here $\tilde{\mathcal{R}}$ is the curvature, or Ricci scalar, of the metric
$g_{ij}$, which is the inverse of $g^{ij}$, and
$\varepsilon^{xy}=-\varepsilon^{yx}=1$. The curvature for the above metric is
given explicitly by

$\displaystyle\tilde{\mathcal{R}}$
$\displaystyle=-2\left|\Delta^{x}\right|\left|\Delta^{y}\right|\left(\partial_{y}\left(\frac{\left|\Delta^{y}\right|\partial_{y}\left|\Delta^{x}\right|}{\left|\Delta^{x}\right|^{2}}\right)+\partial_{x}\left(\frac{\left|\Delta^{x}\right|\partial_{x}\left|\Delta^{y}\right|}{\left|\Delta^{y}\right|^{2}}\right)\right).$
(2.6)

It is a nonlinear expression in the order parameter, which is second order in
derivatives. Thus the responses (2.5) are third order in derivatives, and
start at linear order but include nonlinear contributions as well. The first
equation in (2.5) is analogous to the response $\left\langle
J^{i}\right\rangle=\frac{\nu}{2\pi}\varepsilon^{ij}E_{j}$ of the IQHE, see
Fig.3(a),(b). The second equation is analogous to the dual response
$\left\langle\rho\right\rangle=\frac{\nu}{2\pi}B$.

#####  2.1.3 Additional bulk responses from a gravitational pseudo Chern-
Simons term

Apart from the gCS term, the effective action obtained by integrating over the
bulk fermions also contains an additional term of interest, which we refer to
as a gravitational pseudo Chern-Simons term (gpCS). To the best of our
knowledge, the gpCS term has not appeared previously in the context of
$p$-wave CSF/Cs. It is possible because $U\left(1\right)_{N}$ symmetry is
spontaneously broken in $p$-wave CSF/Cs. In the geometric point of view, this
translates to the emergent geometry in $p$-wave CSF/Cs being not only curved
but also torsion-full. The gpCS term produces bulk responses which are closely
related to those of gCS, despite it being fully gauge invariant. This gauge
invariance implies that it is not associated with a boundary anomaly, nor does
its coefficient $\beta$ need to be quantized. Hence, gpCS does not encode
topological bulk responses. Remarkably, we find that $\beta$ is quantized and
identical to the coefficient $\alpha=\frac{c}{96\pi}$ of the gCS term in the
limit of $m^{*}\rightarrow\infty$, but we do not expect this value to hold
outside of this limit. Let us now describe the bulk responses from gpCS,
setting $\beta=\frac{c}{96\pi}$. First, we find the following contributions to
the fermionic energy current and momentum density,

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

Up to the sign difference in the first equation, these responses are the same
as those from gCS (2.5). As opposed to gCS, the gpCS term also contributes to
the fermionic charge density $\rho=-\psi^{\dagger}\psi$. For the bulk
responses we have written thus far, every Majorana spinor contributed
$c_{n}=\frac{o_{n}}{4}\text{sgn}\left(m_{n}\right)$, and summing over $n$
produced the central charge $c$. For the density response this is not the
case. Here, the $n$th Majorana spinor contributes

$\displaystyle\left\langle\rho\right\rangle_{\text{gpCS}}=\frac{o_{n}c_{n}}{24\pi}\sqrt{g}\tilde{\mathcal{R}},$
(2.8)

where $\sqrt{g}=\sqrt{\text{det}g_{ij}}$ is the emergent volume element. The
orientation $o_{n}$ in Eq. (2.8) makes the sum over the four Majorana spinors
different from the central charge,
$\sum_{n=1}^{4}o_{n}c_{n}=\sum_{n=1}^{4}\frac{1}{2}\text{sgn}\left(m_{n}\right)\neq
c$. The appearance of $o_{n}$ can be understood by considering the effect of
time reversal, since both the density and curvature are time reversal even.
The response (2.8) also holds when the order parameter is time dependent, in
which case $\tilde{\mathcal{R}}$ will also contain time derivatives. One then
finds a time dependent density, but there is no corresponding current
response, which is due to the breaking of $U(1)_{N}symmetry$.

To gain some insight into the expressions we have written thus far, we write
the operators $P,J_{E}$ more explicitly. For each Majorana spinor (suppressing
the index $n$),

$\displaystyle P_{j}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\psi^{\dagger}\overleftrightarrow{D_{j}}\psi,$ (2.9)
$\displaystyle J_{E}^{j}$ $\displaystyle=$ $\displaystyle
g^{jk}P_{k}+\frac{o}{2}\partial_{k}\left(\frac{1}{\sqrt{g}}\varepsilon^{jk}\rho\right)+O\left(\frac{1}{m^{*}}\right).$

The momentum density is the familiar expression for free fermions, but in the
energy current we have only written explicitly contributions that survive the
limit $m^{*}\rightarrow\infty$. These contributions are only possible due to
the $p$-wave pairing, and are of order $\Delta^{2}$. From the relation (2.9)
between $J_{E}$, $P$ and $\rho$ we can understand that the equality
$\left\langle J_{E}^{j}\right\rangle_{\text{gCS}}=g^{jk}\left\langle
P_{k}\right\rangle_{\text{gCS}}$ expressed in equation (2.5) is a result of
the vanishing contribution of gCS to the density $\rho$. We can also
understand the sign difference between the first and second line of (2.7) as a
result of (2.8). The important point is that a measurement of the charge
density $\rho$ can be used to fix the value of the coefficient $\beta$, which
is generically unquantized, and thus separate the contributions of gpCS to
$P,J_{E}$, from those of gCS. In this manner, one can overcome the obscuring
of gCS by gpCS.

#####  2.1.4 Bulk-boundary correspondence from gravitational anomaly

Among the two terms in the bulk effective action which we described in
Sec.2.1.2-2.1.3, only gCS is related to the boundary gravitational anomaly.
This relation can be explicitly analyzed in the case where
$\Delta=\Delta_{0}e^{i\theta\left(t,x\right)}\left(1+f\left(x,t\right),i\right)$
is a perturbation of the $p_{x}+ip_{y}$ configuration with small $f$, and
there is a domain wall (or boundary) at $y=0$ where the value of $c$ jumps.
For simplicity, assume $c=1/2$ for $y<0$ and $c=0$ for $y>0$. This situation
is illustrated in Fig.3(d). In Appendix A we derive the action for the
boundary, or edge mode,

$\displaystyle
S_{\text{e}}=\frac{i}{2}\int\mbox{d}t\text{d}x\tilde{\xi}\left(\partial_{t}-\left|\Delta^{x}\left(t,x\right)\right|\partial_{x}\right)\tilde{\xi},$
(2.10)

which describes a chiral $D=1+1$ Majorana fermion $\tilde{\xi}$ localized on
the boundary, with a space-time dependent velocity
$\left|\Delta^{x}\left(x,t\right)\right|=\Delta_{0}\left|1+f\left(x,t\right)\right|$.
Classically, the edge fermion $\tilde{\xi}$ conserves energy-momentum in the
following sense,

$\displaystyle\partial_{\beta}t_{\mbox{e}\;\alpha}^{\beta}+\partial_{\alpha}\mathcal{L}_{\mbox{e}}=0.$
(2.11)

Here $t_{\mbox{e}}$ is the canonical energy-momentum tensor for $\tilde{\xi}$,
with indices $\alpha,\beta=t,x$, and $\mathcal{L}_{\mbox{e}}$ is the edge
Lagrangian, $S_{\text{e}}=\int\text{d}t\mathcal{L}_{\mbox{e}}$, see
Sec.2.4.1.2. For $\alpha=t$ ($\alpha=x$), Eq.(2.11) describes the sense in
which the edge fermion conserves energy (momentum) classically. The source
term $\partial_{\alpha}\mathcal{L}_{\mbox{e}}$ follows from the space-time
dependence of $\mathcal{L}_{\text{e}}$ through $\Delta^{x}$. Quantum
mechanically, the action $S_{\text{e}}$ is known to have a gravitational
anomaly, which means that energy-momentum is not conserved at the quantum
level [12]. In the context of emergent gravity, this implies that Eq.(2.11) is
violated for the expectation values,

$\displaystyle\partial_{\beta}\left\langle
t_{\mbox{e}\;\alpha}^{\beta}\right\rangle+\partial_{\alpha}\left\langle\mathcal{L}_{\mbox{e}}\right\rangle=-\frac{c}{96\pi}g_{\alpha\gamma}\varepsilon^{\gamma\beta
y}\partial_{\beta}\tilde{\mathcal{R}}.$ (2.12)

This equation is written with $\hbar=1$ and
$c_{\mbox{light}}=\Delta_{0}/\hbar=1$ for simplicity. Since $\Delta^{x}$
depends on time, $\tilde{\mathcal{R}}$ is not the curvature of the spatial
metric $g_{ij}$, but of a corresponding space-time metric $g_{\mu\nu}$ (2.41),
and is given by
$\tilde{\mathcal{R}}=\ddot{f}-2\dot{f}^{2}+O(f\ddot{f},f\dot{f}^{2})$ in this
case. Note that time dependence in this example is crucial. From gCS we find
for
$\Delta=\Delta_{0}e^{i\theta\left(t,x\right)}\left(1+f\left(x,t\right),i\right)$
the bulk energy-momentum tensor

$\displaystyle\left\langle
t_{\;\alpha}^{y}\right\rangle_{\text{gCS}}=-\frac{c}{96\pi}g_{\alpha\gamma}\varepsilon^{\gamma\beta
y}\partial_{\beta}\tilde{\mathcal{R}},$ (2.13)

which explains the anomaly as the inflow of energy-momentum from the bulk to
the boundary,

$\displaystyle\partial_{\beta}\left\langle
t_{\mbox{e}\;\alpha}^{\beta}\right\rangle+\partial_{\alpha}\left\langle\mathcal{L}_{\mbox{e}}\right\rangle=\left\langle
t_{\;\alpha}^{y}\right\rangle_{\text{gCS}}.$ (2.14)

Since $c$ jumps from 1/2 to 0 at $y=0$ the energy-momentum current (2.13)
stops at the boundary and does not extend to the $y>0$ region. The
gravitationally anomalous boundary mode is then essential for the conservation
of total energy-momentum to hold. As this example shows, bulk-boundary
correspondence follows from bulk+boundary conservation of energy-momentum in
the presence of a space-time dependent order parameter.

####  2.2 Lattice model

In this section we review and slightly generalize a simple lattice model for a
$p$-wave SC [226], which will serve as our microscopic starting point. We
describe its band structure and its symmetry protected topological phases, and
also explain some of the basics of the emergent geometry which can be seen in
this setting.

The hamiltonian is given in real space by

$\displaystyle
H=-\frac{1}{2}\sum_{\boldsymbol{l}}\left[t\psi_{\boldsymbol{l}}^{\dagger}\psi_{\boldsymbol{l}+x}+t\psi_{\boldsymbol{l}}^{\dagger}\psi_{\boldsymbol{l}+y}+\mu\psi_{\boldsymbol{l}}^{\dagger}\psi_{\boldsymbol{l}}\right.+\left.\delta^{x}\psi_{\boldsymbol{l}}^{\dagger}\psi_{\boldsymbol{l}+x}^{\dagger}+\delta^{y}\psi_{\boldsymbol{l}}^{\dagger}\psi_{\boldsymbol{l}+y}^{\dagger}+h.c\right].$
(2.15)

Here the sum is over all lattice sites $\boldsymbol{l}\in L$ of a 2
dimensional square lattice $L=a\mathbb{Z}\times a\mathbb{Z}$, with a lattice
spacing $a$. $\psi_{\boldsymbol{l}}^{\dagger},\psi_{\boldsymbol{l}}$ are
creation and annihilation operators for spin-less fermions on the lattice,
with the canonical anti commutators
$\left\\{\psi_{\boldsymbol{l}}^{\dagger},\psi_{\boldsymbol{l}^{\prime}}\right\\}=\delta_{\boldsymbol{l}\boldsymbol{l}^{\prime}}$.
$\boldsymbol{l}+x$ denotes the nearest neighboring site to $\boldsymbol{l}$ in
the $x$ direction. The hopping amplitude $t$ is real and $\mu$ is the chemical
potential. Apart from the single particle terms
$t\psi_{\boldsymbol{l}}^{\dagger}\psi_{\boldsymbol{l}+x}+t\psi_{\boldsymbol{l}}^{\dagger}\psi_{\boldsymbol{l}+y}+\mu$,
there is also the pairing term
$\delta^{x}\psi_{\boldsymbol{l}}^{\dagger}\psi_{\boldsymbol{l}+x}^{\dagger}+\delta^{y}\psi_{\boldsymbol{l}}^{\dagger}\psi_{\boldsymbol{l}+y}^{\dagger}$
, with the order parameter
$\delta=\left(\delta^{x},\delta^{y}\right)\in\mathbb{\mathbb{C}}^{2}$. We
think of $\delta$ as resulting from a Hubbard-Stratonovich decoupling of
interactions, in which case we refer to it as intrinsic, or as being induced
by proximity to an $s$-wave SC. In both cases we treat $\delta$ as a bosonic
background field that couples to the fermions.

The generic order parameter is charged under a few symmetries of the single
particle terms. The order parameter has charge 2 under the global
$U\left(1\right)$ group generated by
$Q=-\sum_{\boldsymbol{l}}\psi_{\boldsymbol{l}}^{\dagger}\psi_{\boldsymbol{l}}$,
in the sense that $e^{-i\alpha Q}H\left(e^{2i\alpha}\delta\right)e^{i\alpha
Q}=H\left(\delta\right)$, which physically represents the electromagnetic
charge $-2$ of Cooper pairs121212Since $\delta$ has charge 2, $H$ commutes
with the fermion parity $\left(-1\right)^{Q}$. The Ground state of $H$ will
therefore be labelled by a fermion parity eigenvalue $\pm 1$, in addition to
the topological label which is the Chern number [15, 227]. Fermion parity is a
subtle quantity in the thermodynamic limit, and will not be important in the
following.. The order parameter is also charged under time reversal $T$, which
is an anti unitary transformation satisfying $T^{2}=1$, that acts as the
complex conjugation of coefficients in the Fock basis corresponding to
$\psi_{\boldsymbol{l}},\psi_{\boldsymbol{l}}^{\dagger}$. The equation
$T^{-1}H\left(\delta^{*}\right)T=H\left(\delta\right)$ shows
$\delta\mapsto\delta^{*}$ under time reversal. Finally, $\delta$ is also
charged under the point group symmetry of the lattice, which for the square
lattice is the Dihedral group $D_{4}$. The continuum analog of this is that
the order parameter is charged under spatial rotations and reflections, and
more generally, under space-time transformations (diffeomorphisms), which is
due to the orbital angular momentum 1 of Cooper pairs in a $p$-wave SC. This
observation will be important for our analysis, and will be discussed further
below.

In an intrinsic $p_{x}\pm ip_{y}$ SC, the configuration of $\delta$ which
minimizes the ground state energy is given by
$\delta=\delta_{0}e^{i\theta}\left(1,\pm i\right)$, where $\delta_{0}>0$ is
determined by the minimization, but the sign $o=\pm 1$ and the phase $\theta$
(which dynamically corresponds to a goldstone mode) are left undetermined. See
[27] for a pedagogical discussion of a closely related model within mean field
theory. A choice of $\theta$ and $o$ corresponds to a spontaneous symmetry
breaking of the group $U\left(1\right)\rtimes\left\\{1,T\right\\}$ including
both the $U\left(1\right)$ and time reversal transformations. More accurately,
in the $p_{x}\pm ip_{y}$ SC, the group
$\left(U\left(1\right)\rtimes\left\\{1,T\right\\}\right)\times D_{4}$ is
spontaneously broken down to a certain diagonal subgroup. We discuss the
continuum analog of this and its implications in section 2.4.1.2.

Crucially, we do not restrict $\delta$ to the $p_{x}\pm ip_{y}$ configuration,
and treat it as a general two component complex vector
$\delta=\left(\delta^{x},\delta^{y}\right)\in\mathbb{\mathbb{C}}^{2}$. In the
following we will take $\delta$ to be space time dependent,
$\delta\mapsto\delta_{\boldsymbol{l}}\left(t\right)$, and show that this space
time dependence can be thought of as a perturbation to which there is a
topological response, but for now we assume $\delta$ is constant.

#####  2.2.1 Band structure and phase diagram

Writing the Hamiltonian (2.15) in Fourier space, and in the BdG form in terms
of the Nambu spinor
$\Psi_{\boldsymbol{q}}=\left(\psi_{\boldsymbol{q}},\psi_{-\boldsymbol{q}}^{\dagger}\right)^{T}$
we find

$\displaystyle H$
$\displaystyle=\frac{1}{2}\int_{BZ}\frac{\mbox{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\Psi_{\boldsymbol{q}}^{\dagger}\begin{pmatrix}h_{\boldsymbol{q}}&\delta_{\boldsymbol{q}}\\\
\delta_{\boldsymbol{q}}^{*}&-h_{\boldsymbol{q}}\end{pmatrix}\Psi_{\boldsymbol{q}}+const$
$\displaystyle=\frac{1}{2}\int_{BZ}\frac{\mbox{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}\Psi_{\boldsymbol{q}}^{\dagger}\left(\boldsymbol{d}_{\boldsymbol{q}}\cdot\boldsymbol{\sigma}\right)\Psi_{\boldsymbol{q}}+const,$
(2.16)

with
$h_{\boldsymbol{q}}=-t\cos\left(aq_{x}\right)-t\cos\left(aq_{y}\right)-\mu$
real and symmetric, and
$\delta_{\boldsymbol{q}}=-i\delta^{x}\sin\left(aq_{x}\right)-i\delta^{y}\sin\left(aq_{y}\right)$
complex and anti-symmetric. Here
$\boldsymbol{\sigma}=\left(\sigma^{x},\sigma^{y},\sigma^{z}\right)$ is the
vector of Pauli matrices and $BZ$ is the Brillouin zone
$BZ=\left(\mathbb{R}/\frac{2\pi}{a}\mathbb{Z}\right)^{2}$. By definition, the
Nambu spinor obeys the reality condition
$\Psi_{\boldsymbol{q}}^{\dagger}=\left(\sigma^{x}\Psi_{-\boldsymbol{q}}\right)^{T}$,
and is therefore a Majorana spinor, see appendix B.5.1. Accordingly, the BdG
Hamiltonian is particle-hole (or charge conjugation) symmetric,
$\sigma^{x}H\left(\boldsymbol{q}\right)^{*}\sigma^{x}=-H\left(-\boldsymbol{q}\right)$,
and therefore belongs to symmetry class D of the Altland-Zirnbauer
classification of free fermion Hamiltonians [107]. The constant in (2.16) is
$\frac{1}{2}\text{tr}h=\frac{V}{2}\int\frac{\text{d}^{2}\boldsymbol{q}}{\left(2\pi\right)^{2}}h_{\boldsymbol{q}}$
where $V$ is the infinite volume. This operator ordering correction is
important as it contributes to physical quantities such as the energy density
and charge density, but we will mostly keep it implicit in the following. The
BdG band structure is given by
$E_{\boldsymbol{q},\pm}=\pm\frac{1}{2}E_{\boldsymbol{q}}$ where

$\displaystyle
E_{\boldsymbol{q}}=\left|\boldsymbol{d}_{\boldsymbol{q}}\right|=\sqrt{h_{\boldsymbol{q}}^{2}+\left|\delta_{\boldsymbol{q}}\right|^{2}}.$
(2.17)

For the $p_{x}\pm ip_{y}$ configuration
$\left|\delta_{\boldsymbol{q}}\right|^{2}=\delta_{0}^{2}\left(\sin^{2}aq_{x}+\sin^{2}aq_{y}\right)$,
and therefore $E_{\boldsymbol{q}}$ can only vanish at the particle-hole
invariant points
$a\boldsymbol{K}^{\left(1\right)}=\left(0,0\right),a\boldsymbol{K}^{\left(2\right)}=\left(0,\pi\right),a\boldsymbol{K}^{\left(3\right)}=\left(\pi,\pi\right),a\boldsymbol{K}^{\left(4\right)}=\left(\pi,0\right)$,
which happens when $\mu=-2t,0,2t,0$. Representative band structures are
plotted in Fig.4. For $\delta_{0}\ll t$ the spectrum takes the form of a
gapped single particle Fermi surface with gap $\sim\delta_{0}$, while for
$\delta_{0}\gg t$ one obtains Four regulated relativistic fermions centered at
the points $\boldsymbol{K}^{\left(n\right)},\;1\leq n\leq 4$ with masses
$m_{n}=-2t-\mu,-\mu,2t-\mu,-\mu$, speed of light
$c_{\text{light}}=a\delta_{0}/\hbar$, bandwidth $\sim\delta_{0}$ and momentum
cutoff $\sim a^{-1}$.

(a)  (b)

Figure 4: Generic band structure of the lattice model. (a) When the order
parameter is much smaller than the single particle bandwidth $\delta\ll t$,
the spectrum takes the form of a gapped single particle Fermi surface with gap
$\sim\delta$. This regime describes the onset of superconductivity, and it is
appropriate to refer to $\delta$ as the “gap function”. (b) When the order
parameter is much larger than the single particle scales $\delta\gg t,\mu$,
the spectrum takes the form of four regulated relativistic fermions centered
at the particle-hole invariant points
$\left(0,0\right),\left(0,\pi\right),\left(\pi,0\right),\left(\pi,\pi\right)$,
in units of the inverse lattice spacing $a^{-1}$. We will be working in this
regime.

With generic $\mu,\delta_{0}$ the spectrum is gapped, and the Chern number
$\nu$ labeling the different topological phases is well defined. It can be
calculated by
$\nu=\int_{BZ}\frac{\text{d}^{2}k}{2\pi}\text{tr}\left(\mathcal{F}\right)$
where $\mathcal{F}$ is the Berry curvature on the Brillouin zone $BZ$ [107]. A
more general definition is $\nu=\frac{1}{24\pi^{2}}\int_{\mathbb{R}\times
BZ}\text{tr}\left(G\text{d}G^{-1}\right)^{3}$131313More explicitly,
$\nu=\frac{1}{24\pi^{2}}\mbox{tr}\int_{\mathbb{R}\times
BZ}\mbox{d}^{3}k\varepsilon^{\alpha\beta\gamma}\left(G\partial_{\alpha}G^{-1}\right)\left(G\partial_{\beta}G^{-1}\right)\left(G\partial_{\gamma}G^{-1}\right)$.,
where $G\left(k_{0},k_{x},k_{y}\right)$ is the single particle propagator
[27], which remains valid in the presence of weak interactions, as long as the
gap does not close. For two band Hamiltonians such as (2.16), $\nu$ reduces to
the homotopy type of the map
$\hat{\boldsymbol{d}_{\boldsymbol{q}}}=\boldsymbol{d}_{\boldsymbol{q}}/\left|\boldsymbol{d}_{\boldsymbol{q}}\right|$
from $BZ$ (which is a flat torus) to the sphere,

$\displaystyle\nu=\frac{1}{\left(2\pi\right)^{2}}\int_{BZ}\text{d}^{2}\boldsymbol{q}\hat{\boldsymbol{d}_{k}}\cdot\left(\partial_{q_{y}}\hat{\boldsymbol{d}_{\boldsymbol{q}}}\times\partial_{q_{y}}\hat{\boldsymbol{d}_{\boldsymbol{q}}}\right)\in\mathbb{Z}.$
(2.18)

One obtains $\nu=0$ for $\left|\mu\right|>2t$, $\nu=\pm 1$ for
$\mu\in\left(0,2t\right)$ and $\nu=\mp 1$ for $\mu\in\left(-2t,0\right)$. The
topological phase diagram is plotted in Fig.5(a).

Away from the $p_{x}\pm ip_{y}$ configuration, the topological phase diagram
is essentially unchanged. For $\text{Im}\left(\delta^{x*}\delta^{y}\right)\neq
0$, gap closings happen at the same points $\boldsymbol{K}^{\left(n\right)}$
and the same values of $\mu$ described above. $\nu$ takes the same values,
with the orientation
$o=\text{sgn}\left(\text{Im}\left(\delta^{x*}\delta^{y}\right)\right)$,
described below, generalizing the sign $\pm 1$ that characterizes the
$p_{x}\pm ip_{y}$ configuration. For
$\text{Im}\left(\delta^{x*}\delta^{y}\right)=0$ the spectrum is always
gapless. The topological phase diagram is most easily understood from the
formula $\nu=\frac{1}{2}\sum_{n=1}^{4}o_{n}\text{sgn}\left(m_{n}\right)$ where
$o_{n}=\pm 1$ are orientations associated with the relativistic fermions which
we describe below [228].

It will also be useful consider a slight generalization of the single particle
part of the lattice model, with un-isotropic hopping
$t^{x}\psi_{\boldsymbol{l}}^{\dagger}\psi_{\boldsymbol{l}+x}+t^{y}\psi_{\boldsymbol{l}}^{\dagger}\psi_{\boldsymbol{l}+y}$.
This changes the masses to
$m_{1}=-\left(t_{1}+t_{2}\right)-\mu,m_{2}=t_{1}-t_{2}-\mu,m_{3}=t_{1}+t_{2}-\mu,m_{4}=-\left(t_{1}-t_{2}\right)-\mu$.
In particular, the degeneracy between the masses $m_{2},m_{4}$ breaks, and
additional trivial phases appear around $\mu=0$. See Fig.5(b).

(a)  (b)

Figure 5: The topological phase diagram of the lattice model is simplest to
understand from the formula
$\nu=\frac{1}{2}\sum_{n=1}^{4}o_{n}\text{sgn}\left(m_{n}\right)$ for the Chern
number in terms of the masses and orientations of low energy relativistic
fermions. (a) Topological phase diagram for isotropic hopping $t$. Units on
the vertical axis are arbitrary, the topological phase diagram only depends on
the orientation
$o=\text{sgn}\left(\text{Im}\left(\delta^{x*}\delta^{y}\right)\right)$. (b)
Topological phase diagram for anisotropic hopping $t^{x}\neq t^{y}$,
additional trivial phases exist around $\mu=0$. Here
$t=\frac{t^{x}+t^{y}}{2}$.

#####  2.2.2 Basics of the emergent geometry

A key insight which we will extensively use, originally due to Volovik, is
that the order parameter is in fact a vielbein. In the present space-time
independent situation, this vielbein is just a $2\times 2$ matrix which
generically will be invertible

$\displaystyle
e_{A}^{\;\;j}=\left(\begin{array}[]{cc}\mbox{Re}(\delta^{x})&\mbox{Re}(\delta^{y})\\\
\mbox{Im}(\delta^{x})&\mbox{Im}(\delta^{y})\end{array}\right)\in
GL\left(2\right),$ (2.21)

where $A=1,2,\;j=x,y$. More accurately, $e_{A}^{\;\;j}$ is invertible if
$\text{det}\left(e_{A}^{\;\;i}\right)=\text{Im}\left(\delta^{x*}\delta^{y}\right)\neq
0$. We refer to an order parameter as singular if
$\text{Im}\left(\delta^{x*}\delta^{y}\right)=0$. From the vielbein one can
calculate a metric, which in the present situation is a general symmetric
positive semidefinite matrix

$\displaystyle
g^{ij}=e_{A}^{\;\;i}\delta^{AB}e_{B}^{\;\;j}=\delta^{(i}\delta^{j)*}=\left(\begin{array}[]{cc}\left|\delta^{x}\right|^{2}\mbox{Re}\left(\delta^{x}\delta^{y*}\right)\\\
\mbox{Re}\left(\delta^{x}\delta^{y*}\right)&\left|\delta^{y}\right|^{2}\end{array}\right).$
(2.24)

Every vielbein determines a metric uniquely, but the converse is not true.
Vielbeins $e,\tilde{e}$ that are related by an internal reflection and
rotation $e_{A}^{\;j}=\tilde{e}{}_{B}^{\;\;j}L_{\;A}^{B}$ with $L\in
O\left(2\right)$ give rise to the same metric. By diagonalization, it is also
clear that any metric can be written in terms of a vielbein. Therefore the set
of (constant) metrics can be parameterized by the coset
$GL\left(2\right)/O\left(2\right)$. To see this explicitly we parameterize
$\delta=e^{i\theta}\left(\left|\delta^{x}\right|,e^{i\phi}\left|\delta^{y}\right|\right)$
with the overall phase $\theta$ and relative phase
$\phi\in\left(-\pi,\pi\right]$. Then

$\displaystyle
g^{ij}=\left(\begin{array}[]{cc}\left|\delta^{x}\right|^{2}&\left|\delta^{x}\right|\left|\delta^{y}\right|\cos\phi\\\
\left|\delta^{x}\right|\left|\delta^{y}\right|\cos\phi&\left|\delta^{y}\right|^{2}\end{array}\right)$
(2.27)

is independent of $\theta$ which parametrizes $SO\left(2\right)$ and
$\text{sgn}\phi$ which parametrizes $O\left(2\right)/SO\left(2\right)$. Note
that the group $O\left(2\right)$ of internal rotations and reflections is just
$U\left(1\right)\rtimes\left\\{1,T\right\\}$ acting on $e_{A}^{\;\;j}$. In
more detail, $\delta\mapsto e^{2i\alpha}\delta$ (or $\delta\mapsto\delta^{*}$)
corresponds to $e_{A}^{\;\;i}\mapsto L_{\;A}^{B}e_{B}^{\;\;i}$ with

$\displaystyle L=\begin{pmatrix}\cos 2\alpha&\sin 2\alpha\\\ -\sin
2\alpha&\cos 2\alpha\end{pmatrix}\left(\text{or }L=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\right).$ (2.28)

The internal reflections, corresponding to a reversal of time, flip the
orientation of the vielbein
$o=\text{sgn}\left(\text{det}\left(e_{A}^{\;\;i}\right)\right)=\text{sgn}\left(\text{Im}\left(\delta^{x*}\delta^{y}\right)\right)$,
and therefore every quantity that depends on $o$ is time reversal odd. We will
also refer to $o$ as the orientation of the order parameter. An order
parameter with a positive (negative) orientation can be thought of as
$p_{x}+ip_{y}$-like ($p_{x}-ip_{y}$-like).

For the $p_{x}\pm ip_{y}$ configuration,
$\delta=e^{i\theta}\delta_{0}\left(1,\pm i\right)$, one obtains a scalar
metric $g^{ij}=\delta_{0}\delta^{ij}$, independent of the phase $\theta$ and
the orientation $o=\pm$. We see that $\theta,o$ correspond precisely to the
$O\left(2\right)=U\left(1\right)\rtimes\left\\{1,T\right\\}$ degrees of
freedom of the vielbein to which the metric is blind to. Thus the metric
$g^{ij}$ corresponds to the Higgs part of the order parameter, by which we
mean the part of the order parameter on which the ground state energy depends,
in the intrinsic case.

The fact that $U\left(1\right)$ transformations map to internal rotations also
appears naturally in the BdG formalism which we will use in the following.
Consider the Nambu spinor $\Psi=\left(\psi,\psi^{\dagger}\right)^{T}$. It
follows from the $U\left(1\right)$ action $\psi\mapsto e^{i\alpha}\psi$ that
$\Psi\mapsto e^{i\alpha\sigma^{z}}\Psi$ where $\sigma^{z}$ is the Pauli
matrix. We see that $U\left(1\right)$ acts on $\Psi$ as a spin rotation.
Moreover, the fact that $\delta$ has charge $2$ while $\psi$ has charge 1
implies $e$ is an $SO\left(2\right)$ vector while $\Psi$ is a spinor.

#####  2.2.3 Non-relativistic continuum limit

Consider the lattice model (2.15), with a general space time dependent order
parameter
$\delta_{\boldsymbol{l}}=\left(\delta_{\boldsymbol{l}}^{x}\left(t\right),\delta_{\boldsymbol{l}}^{y}\left(t\right)\right)$,
and minimally coupled to electromagnetism,

$\displaystyle
H=-\frac{1}{2}\sum_{\boldsymbol{l}}\left[t\psi_{\boldsymbol{l}}^{\dagger}e^{iA_{\boldsymbol{l},\boldsymbol{l}+x}}\psi_{\boldsymbol{l}+x}+\left(\mu_{\boldsymbol{l}}+A_{t,\boldsymbol{l}}\right)\psi_{\boldsymbol{l}}^{\dagger}\psi_{\boldsymbol{l}}\right.+\left.\delta_{\boldsymbol{l}}^{x}\psi_{\boldsymbol{l}}^{\dagger}e^{iA_{\boldsymbol{l},\boldsymbol{l}+x}}\psi_{\boldsymbol{l}+x}^{\dagger}+\left(x\leftrightarrow
y\right)+h.c\right].$ (2.29)

Here $A_{\boldsymbol{l},\boldsymbol{l}^{\prime}},A_{t,\boldsymbol{l}}$ are the
components of a $U\left(1\right)$ gauge field describing background
electromagnetism, on the discrete space and continuous time. We will work in
the relativistic regime $\delta_{0}\gg t,\mu$ where $\delta_{0}$ is a
characteristic scale for $\delta$. To obtain a continuum description, we split
$BZ$ into four quadrants $BZ=\cup_{n=1}^{4}BZ^{\left(n\right)}$ centered
around the four points $\boldsymbol{K}^{\left(n\right)}$, and decompose the
fermion operator $\psi_{\boldsymbol{l}}$ as a sum
$\psi_{\boldsymbol{l}}=\sum_{n=1}^{4}\psi_{\boldsymbol{l}}^{\left(n\right)}e^{i\boldsymbol{K}^{\left(n\right)}\cdot\boldsymbol{l}}$,
where
$\psi_{\boldsymbol{l}}^{\left(n\right)}e^{i\boldsymbol{K}^{\left(n\right)}\cdot\boldsymbol{l}}$
has non zero Fourier modes only in $BZ^{\left(n\right)}$. Thus the fermions
$\psi^{\left(n\right)}$ all have non zero Fourier modes only in
$BZ^{\left(n\right)}-\boldsymbol{K}^{\left(n\right)}=\left[-\frac{\pi}{2a},\frac{\pi}{2a}\right]^{2}$.
This restriction of the quasi momenta provides the fermions
$\psi^{\left(n\right)}$ with a physical cutoff $\sim a^{-1}$, which will be
important when we compare results from the continuum description to the
lattice model. Assuming $\mu,\delta,A$ have small derivatives relative to
$a^{-1}$, the inter fermion terms in $H$ can be neglected and $H$ splits into
a sum $H\approx\sum_{n=1}^{4}H^{\left(n\right)}$, with $H^{\left(n\right)}$ a
Hamiltonian for $\psi^{\left(n\right)}$. We then expand the Hamiltonians
$H^{\left(n\right)}$ in small $\psi^{\left(n\right)}$ derivatives relative to
$a^{-1}$. The resulting Hamiltonian, focusing on the point
$\boldsymbol{K}^{\left(1\right)}=\left(0,0\right)$, is the $p$-wave superfluid
(SF) Hamiltonian

$\displaystyle
H_{\text{SF}}=\int\text{d}^{2}x\left[\psi^{\dagger}\left(-\frac{D^{2}}{2m^{*}}+m-A_{t}\right)\psi\right.-\left.\left(\frac{1}{2}\psi^{\dagger}\Delta^{j}\partial_{j}\psi^{\dagger}+h.c\right)\right],$
(2.30)

where the fermion field has been redefined such that
$\left\\{\psi^{\dagger}\left(x\right),\psi\left(x^{\prime}\right)\right\\}=\delta^{\left(2\right)}\left(x-x^{\prime}\right)$.
Here $D_{\mu}=\partial_{\mu}-iA_{\mu}$ is the $U\left(1\right)$-covariant
derivative, with the connection $A=A_{j}\text{d}x^{j}$ related to
$A_{\boldsymbol{l},\boldsymbol{l}^{\prime}}$ by
$A_{\boldsymbol{l},\boldsymbol{l}^{\prime}}=\int_{\boldsymbol{l}}^{\boldsymbol{l}^{\prime}}A$,
and $D^{2}=\delta^{ij}D_{i}D_{j}$ with $i,j=x,y$. Note the appearance of the
flat background spatial metric $\delta^{ij}$. The effective mass is related to
the hopping amplitude $1/m^{*}=a^{2}t$, and the order parameter is
$\Delta=a\delta$, so it is essentially the lattice order parameter. The
chemical potential for the $p$-wave SF is $-m$. The coupling to $A$ in the
pairing term is lost, since $\psi^{\dagger}\psi^{\dagger}=0$. For this reason
it is a derivative and not a covariant derivative that appears in
$\psi^{\dagger}\Delta^{j}\partial_{j}\psi^{\dagger}$, and one can verify that
this term is gauge invariant. Moreover, due to the anti-commutator
$\left\\{\psi^{\dagger}\left(x\right),\psi^{\dagger}\left(y\right)\right\\}=0$
any operator put between two $\psi^{\dagger}$s is anti-symmetrized, and in
particular
$\psi^{\dagger}\Delta^{j}\partial_{j}\psi^{\dagger}=\psi^{\dagger}\frac{1}{2}\left\\{\Delta^{j},\partial_{j}\right\\}\psi^{\dagger}$
where $\left\\{\Delta^{j},\partial_{j}\right\\}$ is the anti-commutator of
differential operators. This Hamiltonian is essentially the one considered in
[15] for the $p$-wave SF. The corresponding action is the $p$-wave SF action

$\displaystyle
S_{\text{SF}}\left[\psi,\Delta,A\right]=\int\text{d}^{2+1}x\left[\psi^{\dagger}\left(iD_{t}+\frac{D^{2}}{2m^{*}}-m\right)\psi\right.+\left.\left(\frac{1}{2}\psi^{\dagger}\Delta^{j}\partial_{j}\psi^{\dagger}+h.c\right)\right],$
(2.31)

in which $\psi,\psi^{\dagger}$ are no longer fermion operators, but
independent Grassmann valued fields,
$\left\\{\psi\left(x\right),\psi^{\dagger}\left(x^{\prime}\right)\right\\}=0$.
This action comes equipped with a momentum cutoff $\Lambda_{UV}\sim a^{-1}$
inherited from the lattice model.

For the other points
$\boldsymbol{K}^{\left(2\right)},\boldsymbol{K}^{\left(3\right)},\boldsymbol{K}^{\left(4\right)}$
the SF action obtained is slightly different. The chemical potential for the
$n$th fermion is $-m_{n}$.The order parameter for the $n$th fermion is
$\Delta_{\left(n\right)}^{x}=a\delta^{x}e^{iK_{x}^{\left(n\right)}},\;\Delta_{\left(n\right)}^{y}=a\delta^{y}e^{iK_{y}^{\left(n\right)}}$,
and we note that $e^{iK_{j}^{\left(n\right)}}=\pm 1$. The order parameters for
$\boldsymbol{K}^{\left(1\right)}=\left(0,0\right),\;\boldsymbol{K}^{\left(3\right)}=\left(\pi,\pi\right)$
are related by an overall sign, which is a $U\left(1\right)$ transformation,
and so are the order parameters for
$\boldsymbol{K}^{\left(2\right)}=\left(0,\pi\right),\boldsymbol{K}^{\left(4\right)}=\left(\pi,0\right)$.
Thus the order parameters for $n=1,3$ are physically indistinguishable, and so
are order parameters for $n=2,4$. The order parameters for $n=1$ and $n=2$ are
however physically distinct. First, the orientations
$o_{n}=\text{sgn}\left(\text{Im}\left(\Delta_{\left(n\right)}^{x*}\Delta_{\left(n\right)}^{y}\right)\right)$
are different, with $o_{1}=-o_{2}$. Second, the metrics
$g_{\left(n\right)}^{ij}=\Delta^{(i}\Delta^{j)*}$ are generically different,
with the same diagonal components, but
$g_{\left(1\right)}^{xy}=-g_{\left(2\right)}^{xy}$. We note that if the
relative phase between $\delta^{x}$ and $\delta^{y}$ is $\pm\pi/2$, as in the
$p_{x}\pm ip_{y}$ configuration, then all metrics $g_{\left(n\right)}^{ij}$
are diagonal and therefore equal. These differences between the orientations
and metrics of the different lattice fermions will be important later on.

Similarly, the effective mass tensor which in (2.30), for $n=1$, is
$\left(M^{-1}\right)^{ij}=\frac{\delta^{ij}}{m^{*}}$, has different signatures
for different $n$, but this will not be important for our analysis. For now we
continue working with the action (2.31) for the $n=1$ fermion, keeping the
other lattice fermions implicit until section 2.5.2.3.

#####  2.2.4 Relativistic continuum limit

Since we work in the relativistic regime $\delta\gg t,\mu$ we can treat the
term $\psi^{\dagger}\frac{D^{2}}{2m^{*}}\psi$ as a perturbation and compute
quantities to zeroth order in $1/m^{*}$. Then $S_{\text{SF}}$ reduces to what
we refer to as the relativistic limit of the $p$-wave SF action, given in BdG
form by

$\displaystyle
S_{\text{rSF}}\left[\psi,\Delta,A\right]=\frac{1}{2}\int\mbox{d}^{2+1}x\Psi^{\dagger}\begin{pmatrix}i\partial_{t}+A_{t}-m\frac{1}{2}\left\\{\Delta^{j},\partial_{j}\right\\}\\\
-\frac{1}{2}\left\\{\Delta^{*j},\partial_{j}\right\\}i\partial_{t}-A_{t}+m\end{pmatrix}\Psi.$
(2.32)

It is well known that when $\Delta$ takes the $p_{x}\pm ip_{y}$ configuration
$\Delta=\Delta_{0}e^{i\theta}\left(1,\pm i\right)$ and $A=0$ this action is
that of a relativistic Majorana spinor in Minkowski space-time, with mass $m$
and speed of light $c_{\text{light}}=\frac{\Delta_{0}}{\hbar}$. In the
following, we will see that for general $\Delta$ and $A$, (2.32) is the action
of a relativistic Majorana spinor in curved and torsion-full space-time. We
wil sometimes refer to the relativistic limit as $m^{*}\rightarrow\infty$,
though this is somewhat loose, because in the relativistic regime both $m^{*}$
is large and $m$ is small.

Before we go on to analyze the $p$-wave SF in the relativistic limit, it is
worth considering what of the physics of the $p$-wave SF is captured by the
relativistic limit, and what is not. First, the coupling to $A_{x},A_{y}$ is
lost, so the relativistic limit is blind to the magnetic field. Since
superconductors are usually defined by their interaction with the magnetic
field, the relativistic limit is actually insufficient to describe the
properties of the $p$-wave SF as a superconductor. Of course, a treatment of
superconductivity also requires the dynamics of $\Delta$. Likewise, the term
$\frac{1}{2m^{*}}\psi^{\dagger}D^{2}\psi=\frac{1}{2m^{*}}\psi^{\dagger}\delta^{ij}D_{i}D_{j}\psi$
seems to be the only term in $S_{\text{SF}}$ that includes the flat background
metric $\delta^{ij}$, describing the real geometry of space. It appears that
the relativistic limit is insufficient to describe the response of the system
to a change in the real geometry of space141414In fact, some of the response
to the real geometry can be obtained, see our discussion, section 2.6..
Nevertheless, as is well known, the relativistic limit does suffice to
determine the topological phases of the $p$-wave SC as a free (and weakly
interacting) fermion system. Indeed, the Chern number labeling the different
topological phases can be calculated by the formula
$\nu=\frac{1}{2}\sum_{n=1}^{4}o_{n}\text{sgn}\left(m_{n}\right)$, which only
uses data from the relativistic limit. Here the sum is over the four particle-
hole invariant points of the lattice model, with orientations $o_{n}$ and
masses $m_{n}$. This suggests that at least some physical properties
characterizing the different free fermion topological phases can be obtained
from the relativistic limit. Indeed, in the following we will see how a
topological bulk response and a corresponding boundary anomaly can be obtained
within the relativistic limit.

####  2.3 Emergent Riemann-Cartan geometry

We argue that (2.32) is precisely the action which describes a relativistic
massive Majorana spinor in a curved and torsion-full background known as
Riemann-Cartan (RC) geometry, with a particular form of background fields. We
refer the reader to [154] parts I.1 and I.4.4, for a review of RC geometry and
the coupling of fermions to it, and provide only the necessary details here,
focusing on the implications for the $p$-wave SF. For simplicity we work
locally and in coordinates, and we differ the treatment of global aspects to
appendix B.6.

The action describing the dynamics of a Majorana spinor on RC background in
2+1 dimensional space-time can be written as

$\displaystyle
S_{\text{RC}}\left[\chi,e,\omega\right]=\frac{1}{2}\int\mbox{d}^{2+1}x\left|e\right|\overline{\chi}\left[\frac{i}{2}e_{a}^{\;\mu}\left(\gamma^{a}D_{\mu}-\overleftarrow{D_{\mu}}\gamma^{a}\right)-m\right]\chi.$
(2.33)

Here $\chi$ is a Majorana spinor with mass $m$ obeying, as a field operator,
the canonical anti-commutation relation
$\left\\{\chi\left(x\right),\chi\left(y\right)\right\\}=\frac{\delta^{\left(2\right)}\left(x-y\right)}{\left|e\left(x\right)\right|}$,
where we suppressed spinor indices. As a Grassmann field
$\left\\{\chi\left(x\right),\chi\left(y\right)\right\\}=0$. The field
$e_{a}^{\;\mu}$ is an inverse vielbein which is an invertible matrix at each
point in space-time. The indices $a,b,\dots\in\left\\{0,1,2\right\\}$ are
$SO\left(1,2\right)$ (Lorentz) indices which we refer to as internal indices,
while $\mu,\nu,\dots\in\left\\{t,x,y\right\\}$ are coordinate indices.

We will also use $A,B,\dots\in\left\\{1,2\right\\}$ for spatial internal
indices and $i,j,\dots\in\left\\{x,y\right\\}$ for spatial coordinate indices

The vielbein $e_{\;\mu}^{a}$, is the inverse of $e_{a}^{\;\mu}$, such that
$e_{\;\mu}^{a}e_{a}^{\;\nu}=\delta_{\mu}^{\nu},\;e_{\;\mu}^{a}e_{b}^{\;\mu}=\delta_{b}^{a}$.
It is often useful to view the vielbein as a set of linearly independent
(local) one-forms $e^{a}=e_{\;\mu}^{a}\text{d}x^{\mu}$. The metric
corresponding to the vielbein is
$g_{\mu\nu}=e_{\;\mu}^{a}\eta_{ab}e_{\;\nu}^{b}$ and the inverse metric is
$g^{\mu\nu}=e_{a}^{\;\mu}\eta^{ab}e_{b}^{\;\nu}$, where
$\eta_{ab}=\eta^{ab}=\text{diag}\left[1,-1,-1\right]$ is the flat Minkowski
metric. Internal indices are raised and lowered using $\eta$, while coordinate
indices are raised and lowered using $g$ and its inverse. Using $e$ one can
replace internal indices with coordinate indices and vice versa, e.g
$v^{a}=e_{\;\mu}^{a}v^{\mu}$. The volume element is defined by
$\left|e\right|=\left|\text{det}e_{\;\mu}^{a}\right|=\sqrt{g}$.
$\left\\{\gamma^{a}\right\\}_{a=0}^{2}$ are gamma matrices obeying
$\left\\{\gamma^{a},\gamma^{b}\right\\}=2\eta^{ab}$, and we will work with
$\gamma^{0}=\sigma^{z},\;\gamma^{1}=-i\sigma^{x},\;\gamma^{2}=i\sigma^{y}$151515The
gamma matrices form a basis for the Clifford algebra associated with $\eta$.
The above choice of basis is a matter of convention. . The covariant
derivative $D_{\mu}=\partial_{\mu}+\omega_{\mu}$161616We use the notation $D$
for spin, Lorentz, and $U\left(1\right)$ covariant derivatives in any
representation, and the exact meaning should be clear from the field $D$ acts
on.  contains the spin connection
$\omega_{\mu}=\frac{1}{2}\omega_{ab\mu}\Sigma^{ab}$, where
$\Sigma^{ab}=\frac{1}{4}\left[\gamma^{a},\gamma^{b}\right]$ generate the spin
group $Spin\left(1,2\right)$ which is the double cover of the Lorentz group
$SO\left(1,2\right)$. Note that $\omega_{ab\mu}=-\omega_{ba\mu}$ and therefore
$\omega_{\;b\mu}^{a}$ is an $SO\left(1,2\right)$ connection. It follows that
$\omega$ is metric compatible, $D_{\mu}\eta_{ab}=0$. It is often useful to
work (locally) with a connection one-form
$\omega=\omega_{\mu}\text{d}x^{\mu}$. $\overline{\chi}$ is the Dirac conjugate
defined as in Minkowski space-time $\overline{\chi}=\chi^{\dagger}\gamma^{0}$.
The derivative $\overleftarrow{D_{\mu}}$ acts only on $\overline{\chi}$ and is
explicitly given by
$\chi\overleftarrow{D_{\mu}}=\partial_{\mu}\overline{\chi}-\overline{\chi}\omega_{\mu}$.

Our statement is that $S_{\text{RC}}\left[\chi,e,\omega\right]$ evaluated on
the fields

$\displaystyle\chi=\left|e\right|^{-1/2}\Psi,\hskip
7.02625pte_{a}^{\;\mu}=\left(\begin{array}[]{ccc}1&0&0\\\
0&\mbox{Re}(\Delta^{x})&\mbox{Re}(\Delta^{y})\\\
0&\mbox{Im}(\Delta^{x})&\text{Im}(\Delta^{y})\end{array}\right),\hskip
7.02625pt\omega_{\mu}=-2A_{\mu}\Sigma^{12},$ (2.37)

reduces precisely to $S_{\text{rSF}}\left[\psi,\Delta,A\right]$ of equation
(2.32), where one must keep in mind that $S_{\text{RC}}$ is written in
relativistic units where $\hbar=1$ and $c_{\text{light}}=\Delta_{0}/\hbar=1$,
which we will use in the following. Moreover, the functional integral over
$\chi$ is equal to the functional integral over $\Psi$. This refines the
original statement by Volovik and subsequent work by Read and Green [15]. We
defer the proof to appendices B.1 and B.3, where we also address certain
subtleties that arise. Here we describe the particular RC geometry that
follows from (2.37), and attempt to provide some intuition for this geometric
description of the $p$-wave SF. Starting with the vielbein, note that the only
nontrivial part of $e_{a}^{\;\mu}$ is the spatial part $e_{A}^{\;j}$, which is
just the order parameter $\Delta$, as in (2.21). The inverse metric we obtain
from our vielbein is

$\displaystyle
g^{\mu\nu}=e_{a}^{\;\mu}\eta^{ab}e_{b}^{\;\nu}=\left(\begin{array}[]{ccc}1&0&0\\\
0&-\left|\Delta^{x}\right|^{2}&-\mbox{Re}\left(\Delta^{x}\Delta^{*y}\right)\\\
0&-\mbox{Re}\left(\Delta^{x}\Delta^{*y}\right)&-\left|\Delta^{y}\right|^{2}\end{array}\right),$
(2.41)

where the spatial part $g^{ij}=-\Delta^{(i}\Delta^{j)*}$ is the Higgs part of
the order parameter, as in (2.24). For the $p_{x}\pm ip_{y}$ configuration the
metric reduces to the Minkowski metric. If $\Delta$ is time independent
$g^{\mu\nu}$ describes a Riemannian geometry which is trivial in the time
direction, but we allow for a time dependent $\Delta$. A metric of the form
(2.41) is said to be in gaussian normal coordinates with respect to space
[229].

The $U\left(1\right)$ connection $A_{\mu}$ maps to a $Spin\left(2\right)$
connection $\omega_{\mu}=-2A_{\mu}\Sigma^{12}=-iA_{\mu}\sigma^{z}$ which
corresponds to spatial spin rotations. This is a special case of the general
$Spin\left(1,2\right)$ connection which appears in RC geometry. The fact that
$U\left(1\right)$ transformations map to spin rotations when acting on the
Nambu spinor $\Psi$ is a general feature of the BdG formalism as was already
discussed in section 2.2.2. From the spin connection $\omega$ it is natural to
construct a curvature, which is a matrix valued two-form defined by
$R_{\;b}^{a}=\text{d}\omega_{\;b}^{a}+\omega_{\;c}^{a}\wedge\omega_{\;b}^{c}$.
In local coordinates $x^{\mu}$ it can be written as
$R_{\;b}^{a}=\frac{1}{2}R_{\;b\mu\nu}^{a}\text{d}x^{\mu}\wedge\text{d}x^{\nu}$,
where the components are given explicitly by
$R_{\;b\mu\nu}^{a}=\partial_{\mu}\omega_{\;b\nu}^{a}-\partial_{\nu}\omega_{\;b\mu}^{a}+\omega_{\;c\mu}^{a}\omega_{\;b\nu}^{c}-\omega_{\;c\nu}^{a}\omega_{\;b\mu}^{c}$.
It follows from (2.37) that in our case the only non zero components are

$\displaystyle R_{12}=-R_{21}=-2F,$ (2.42)

where the two form $F=\text{d}A$ is the $U\left(1\right)$ field strength, or
curvature, comprised of the electric and magnetic fields.

#####  2.3.1 Torsion and additional geometric quantities

Since we treat $A$ and $\Delta$ as independent background fields, so are the
spin connection $\omega$ and vielbein $e$. This situation is referred to as
the first order vielbein formalism for gravity [154]. Apart from the metric
$g$ and the curvature $R$ which we already described, there are a few more
geometric quantities which can be constructed from $e,\omega$, and that will
be used in the following. These additional quantities revolve around the
notion of torsion.

The torsion tensor $T$ is an important geometrical quantity, but a pragmatic
way to view it is as a useful parameterization for the set of all spin
connections $\omega$, for a fixed vielbein $e$. Thus one can work with the
variables $e,T$ instead of $e,\omega$. We will see later on that the bulk
responses in the $p$-wave SC are easier to describe using $e,T$. This is
analogous to, and as we will see, generalizes, the situation in $s$-wave SC,
where the independent degrees of freedom are $A$ and
$\Delta=\left|\Delta\right|e^{i\theta}$, but it is natural to change variables
and work with $\Delta$ and $D_{\mu}\theta=\partial_{\mu}\theta-2A_{\mu}$
instead. We now provide the details.

The torsion tensor, or two-form, is defined in terms of $e,\omega$ as
$T^{a}=De^{a}$, or in coordinates $T_{\mu\nu}^{a}=2D_{[\mu}e_{\nu]}^{a}$.
Since our temporal vielbein $e^{0}=\text{d}t$ is trivial and the connection
$\omega$ is only an $SO\left(2\right)$ connection, $T^{0}=0$ for all $A$ and
$\Delta$. All other components of the torsion are in general non trivial, and
are given by
$T_{ij}^{A}=D_{i}e_{\;j}^{A}-D_{j}e_{\;i}^{A},\;T_{ti}^{A}=-T_{it}^{A}=D_{t}e_{\;i}^{A}$.
This describes the simple change of variables from $\omega$ to $T$.

Going from $T$ back to $\omega$ is slightly more complicated, and is done as
follows. One starts by finding the $\omega$ that corresponds to $T=0$. The
solution is the unique torsion free spin connection
$\tilde{\omega}=\tilde{\omega}\left(e\right)$ which we refer to as the Levi
Civita (LC) spin connection171717The unique torsion free spin connection
$\tilde{\omega}$ is also referred to as the Cartan connection is the
literature.. This connection is given explicitly by
$\tilde{\omega}_{abc}=\frac{1}{2}\left(\xi_{abc}+\xi_{bca}-\xi_{cab}\right)$
where
$\xi_{\;bc}^{a}=2e_{b}^{\;\mu}e_{c}^{\;\nu}\partial_{[\mu}e_{\;\nu]}^{a}$.
Now, for a general $\omega$ the difference
$C_{\;b\mu}^{a}=\omega_{\;b\mu}^{a}-\tilde{\omega}_{\;b\mu}^{a}$ is referred
to as the contorsion tensor, or one-form. It carries the same information as
$T$ and the two are related by $T^{a}=C_{\;b}^{a}\wedge e^{b}$
($T_{\mu\nu}^{a}=2C_{\;b[\mu}^{a}e_{\;\nu]}^{b}$) and
$C_{\mu\alpha\nu}=\frac{1}{2}\left(T_{\alpha\mu\nu}+T_{\mu\nu\alpha}-T_{\nu\alpha\mu}\right)$.
One can then reconstruct $\omega$ from $e,T$ as
$\omega=\tilde{\omega}\left(e\right)+C\left(e,T\right)$. Note that
$\omega,\tilde{\omega}$ are both connections, but $C,T$ are tensors.

For the $p_{x}\pm ip_{y}$ configuration
$\Delta=\Delta_{0}e^{i\theta}\left(1,\pm i\right)$ one finds
$\tilde{\omega}_{12\mu}=-\tilde{\omega}_{21\mu}=-\partial_{\mu}\theta$ (with
all other components vanishing), and it follows that
$C_{12\mu}=D_{\mu}\theta$. These are familiar quantities in the theory of
superconductivity, and one can view $\tilde{\omega}$ and $C$ as
generalizations of these. General formulas are given in appendix B.4.

Using $\tilde{\omega}$ one can define a covariant derivative $\tilde{D}$ and
curvature $\tilde{R}$ just as $D$ and $R$ are constructed from $\omega$. The
quantity $\tilde{R}_{\;\nu\rho\sigma}^{\mu}$ is the usual Riemann tensor of
Riemannian geometry and general relativity. Note that
$\tilde{R}_{\;\nu\rho\sigma}^{\mu}$ depends solely on $g$ which is the Higgs
part of the order parameter $\Delta$. Since $g$ is flat in the $p_{x}\pm
ip_{y}$ configuration, we conclude that a non vanishing Riemann tensor
requires a deviation of the Higgs part of $\Delta$ from the $p_{x}\pm ip_{y}$
configuration. As in Riemannian geometry we can define the Ricci tensor
$\tilde{\mathcal{R}}_{\nu\sigma}=\tilde{R}_{\;\nu\mu\sigma}^{\mu}$ and Ricci
scalar $\tilde{\mathcal{R}}=\tilde{\mathcal{R}}_{\;\nu}^{\nu}$. Examples for
the calculation of $\tilde{\mathcal{R}}$ in terms of $\Delta$ were given in
section 2.1.2.

Another important quantity which can be constructed from $e,\omega$ is the
affine connection
$\Gamma_{\;\beta\mu}^{\alpha}=e_{a}^{\;\alpha}\left(\partial_{\mu}e_{\;\beta}^{a}+\omega_{\;b\mu}^{a}e_{\;\beta}^{b}\right)=e_{a}^{\;\alpha}D_{\mu}e_{\;\beta}^{a}$,
or affine connection (local) one-form
$\Gamma_{\;\beta}^{\alpha}=\Gamma_{\;\beta\mu}^{\alpha}\text{d}x^{\mu}$. It is
not difficult to see that $T$ is the anti symmetric part of $\Gamma$,
$T_{\;\mu\nu}^{\rho}=\Gamma_{\mu\nu}^{\rho}-\Gamma_{\nu\mu}^{\rho}$, and it
follows that the LC affine connection
$\tilde{\Gamma}_{\;\beta\mu}^{\alpha}=e_{a}^{\;\alpha}\tilde{D}_{\mu}e_{\;\beta}^{a}$,
for which $T=0$, is symmetric in its the two lower indices. This is the usual
metric compatible and torsion free connection of Riemannian geometry, given by
the Christoffel symbol
$\tilde{\Gamma}_{\alpha\beta\mu}=\frac{1}{2}\left(\partial_{\mu}g_{\beta\alpha}+\partial_{\beta}g_{\alpha\mu}-\partial_{\alpha}g_{\mu\beta}\right)$.
$\Gamma$ appears in covariant derivatives of tensors with coordinate indices,
for example
$\nabla_{\mu}v^{\alpha}=\partial_{\mu}v^{\alpha}+\Gamma_{\;\beta\mu}^{\alpha}v^{\beta}$,
$\nabla_{\mu}v_{\alpha}=\partial_{\mu}v_{\alpha}-v_{\beta}\Gamma_{\;\alpha\mu}^{\beta}$,
and so on. We also denote by $\nabla$ the total covariant derivative of
tensors with both coordinate and internal indices, which includes both
$\omega$ and $\Gamma$. Thus, for example,
$\nabla_{\mu}v_{\;\nu}^{a}=\partial_{\mu}v_{\;\nu}^{a}+\omega_{\;b\mu}^{a}v_{\;\nu}^{b}-v_{\;\nu}^{a}\Gamma_{\;\mu\alpha}^{\nu}=D_{\mu}v_{\;\nu}^{a}-v_{\;\nu}^{a}\Gamma_{\;\mu\alpha}^{\nu}$.
The most important occurrence of $\nabla$ is in the identity
$\nabla_{\nu}e_{\;\mu}^{a}=0$, which follows from the definition of $\Gamma$
in this formalism, and is sometimes called the first vielbein postulate. It
means that the covariant derivative $\nabla$ commutes with index manipulation
preformed using $e,\eta$ and $g$. To obtain more intuition for what $\Gamma$
is from the $p$-wave SC point of view, we can write it as
$\Gamma_{\;a\mu}^{\alpha}=-D_{\mu}e_{a}^{\;\alpha}$. Then it is clear that the
non vanishing components of $\Gamma_{\;a\mu}^{\alpha}$ are given by
$\Gamma_{\;1\mu}^{j}+i\Gamma_{\;2\mu}^{j}=-D_{\mu}\Delta^{j}$

####  2.4 Symmetries, currents, and conservation laws

In order to map fermionic observables in the $p$-wave SF to those of a
Majorana fermion in RC space-time, it is usefull is to map the symmetries and
the corresponding conservation laws between the two. We start with
$S_{\text{SF}}$, and then review the analysis of $S_{\text{RC}}$ and show how
it maps to that of $S_{\text{SF}}$, in the relativistic limit. The bottom line
is that there is a sense in which electric charge and energy-momentum are
conserved in a $p$-wave SC, and this maps to the sense in which spin and
energy-momentum are conserved for a Majorana spinor in RC space-time.

#####  2.4.1 Symmetries, currents, and conservation laws of the $p$-wave
superfluid action

######  2.4.1.1 Electric charge

$U\left(1\right)$ gauge transformations act on $\psi,\Delta,A$ by

$\displaystyle\psi\mapsto e^{i\alpha}\psi,\;\Delta\mapsto
e^{2i\alpha}\Delta,\;A_{\mu}\mapsto A_{\mu}+\partial_{\mu}\alpha.$ (2.43)

This symmetry of $S_{\text{SF}}\left[\psi,\Delta,A\right]$ implies a
conservation law for electric charge,

$\displaystyle\partial_{\mu}J^{\mu}=-i\psi^{\dagger}\Delta^{j}\partial_{j}\psi^{\dagger}+h.c,$
(2.44)

where $J^{\mu}=-\frac{\delta S}{\delta A_{\mu}}$ is the fermion electric
current. Since $A_{\mu}$ does not enter the pairing term, $J^{\mu}$ is the
same as in the normal state where $\Delta=0$,

$\displaystyle
J^{t}=-\psi^{\dagger}\psi,\;J^{j}=-\frac{\delta^{jk}}{m^{*}}\frac{i}{2}\psi^{\dagger}\overleftrightarrow{D_{k}}\psi.$
(2.45)

Here
$\psi^{\dagger}\overleftrightarrow{D_{k}}\psi=\psi^{\dagger}D_{k}\psi-\left(D_{k}\psi^{\dagger}\right)\psi$.
The conservation law (2.44) shows that the fermionic charge alone is not
conserved due to the exchange of charge between the fermions $\psi$ and Cooper
pairs $\Delta$. If one adds a ($U\left(1\right)$ gauge invariant) term
$S^{\prime}\left[\Delta,A\right]$ to the action and considers $\Delta$ as a
dynamical field, then it is possible to use the equation of motion
$\frac{\delta\left(S^{\prime}+S\right)}{\delta\Delta}=0$ for $\Delta$ and the
definition $J_{\Delta}^{\mu}=-\frac{\delta S^{\prime}}{\delta A_{\mu}}$ of the
Cooper pair current in order to rewrite (2.44) as
$\partial_{\mu}\left(J^{\mu}+J_{\Delta}^{\mu}\right)=0$. This expresses the
conservation of total charge in the $p$-wave SC.

######  2.4.1.2 Energy-momentum

Energy and momentum are at the heart of our analysis, and obtaining the
correct expressions for these quantities, as well as interpreting correctly
the conservation laws they satisfy, will be crucial.

In flat space, one usually starts with the canonical energy-momentum tensor.
For a Lagrangian $\mathcal{L}\left(\phi,\partial\phi,x\right)$, where $\phi$
is any fermionic of bosonic field, it is given by

$\displaystyle
t_{\;\nu}^{\mu}=\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi}\partial_{\nu}\phi-\delta_{\nu}^{\mu}\mathcal{L},$
(2.46)

and satisfies, on the equation of motion for $\phi$,

$\displaystyle\partial_{\mu}t_{\;\nu}^{\mu}=-\partial_{\nu}\mathcal{L},$
(2.47)

which can be obtained from Noether’s first theorem for space-time
translations. Thus $t_{\;\nu}^{\mu}$ is conserved if and only if the
Lagrangian is independent of the coordinate $x^{\nu}$. This motivates the
identification of $t_{\;t}^{\mu}$ as the energy current, and of
$t_{\;j}^{\mu}$ as the current of the $j$th component of momentum
($j$-momentum). $t_{\;t}^{t}$ is just the Hamiltonian density, or energy
density, and $t_{\;j}^{t}$ is the $j$-momentum density.

It is well known however, that the canonical energy-momentum tensor may fail
to be gauge invariant, symmetric in its indices, or traceless, in situations
where these properties are physically required, and it is also sensitive to
the addition of total derivatives to the Lagrangian. To obtain the physical
energy-momentum tensor one can either “improve“ $t_{\;\nu}^{\mu}$ or appeal to
a geometric (gravitational) definition which directly provides the physical
energy-momentum tensor [154, 230].

We will comment on the coupling of the $p$-wave SF to a real background
geometry our discussion, section 2.6, but here we fix the background geometry
to be flat, and instead continue by introducing the
$U\left(1\right)$-covariant canonical energy-momentum tensor. It can be shown
to coincide with the physical energy-momentum tensor obtained by coupling the
$p$-wave SF to a real background geometry. Since we work with a fixed flat
background geometry in this section, we will only consider space-time
transformations which are symmetries of this background, and it will suffice
to consider space-time translations and spatial rotations.

The $U\left(1\right)$-covariant canonical energy-momentum tensor is relevant
in the following situation. Assume that the $x$ dependence in $\mathcal{L}$ is
only through a $U\left(1\right)$ gauge field to which $\phi$ is minimally
coupled,
$\mathcal{L}\left(\phi,\partial\phi,x\right)=\mathcal{L}\left(\phi,D\phi\right)$.
Then, $t_{\;\nu}^{\mu}$ is not gauge invariant, and therefore physically
ambiguous. This is reflected in the conservation law (2.47) which takes the
non covariant form

$\displaystyle\partial_{\mu}t_{\;\nu}^{\mu}=J^{\mu}\partial_{\nu}A_{\mu},$
(2.48)

where $J^{\mu}=-\frac{\partial\mathcal{L}}{\partial A_{\mu}}$ is the
$U\left(1\right)$ current. This lack of gauge invariance is to be expected, as
this conservation law follows from translational symmetry, and translations do
not commute with gauge transformations. Instead, one should use
$U\left(1\right)$-covariant space-time translations, which are translations
from $x$ to $x+a$ followed by a $U\left(1\right)$ parallel transport from
$x+a$ back to $x$, $\phi\left(x\right)\mapsto
e^{iq\int_{x-a}^{x}A}\phi\left(x-a\right)$ where $\phi\mapsto
e^{iq\alpha}\phi$ under $U\left(1\right)$ and the integral is over the
straight line from $x-a$ to $a$. This is still a symmetry because the
additional $e^{iq\int_{x-a}^{x}A}$ is just a gauge transformation. The
conservation law that follows from this modified action of translations is

$\displaystyle\partial_{\mu}t_{\text{cov}\;\nu}^{\mu}=F_{\nu\mu}J^{\mu},$
(2.49)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field strength, and

$\displaystyle t_{\text{cov}\;\nu}^{\mu}=\frac{\partial\mathcal{L}}{\partial
D_{\mu}\phi}D_{\nu}\phi-\delta_{\nu}^{\mu}\mathcal{L}=t_{\;\nu}^{\mu}-J^{\mu}A_{\nu}$
(2.50)

is the $U\left(1\right)$-covariant version of $t_{\;\nu}^{\mu}$, which we
refer to as the $U\left(1\right)$-covariant canonical energy-momentum tensor.
The right hand side of (2.49) is just the usual Lorentz force, which acts as a
source of $U\left(1\right)$-covariant energy-momentum. We stress that the
covariant and non covariant conservation laws are equivalent, as can be
verified by using the fact that $\partial_{\mu}J^{\mu}=0$ in this case. Both
hold in any gauge, but in (2.49) all quantities are gauge invariant.

For the $p$-wave SF one obtains the $U\left(1\right)$-covariant energy-
momentum tensor

$\displaystyle t_{\text{cov}\;t}^{t}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\psi^{\dagger}\overleftrightarrow{D_{t}}\psi-\mathcal{L}$
$\displaystyle=$
$\displaystyle\frac{\delta^{ij}D_{i}\psi^{\dagger}D_{j}\psi}{2m^{*}}+m\psi^{\dagger}\psi-\left(\frac{1}{2}\psi^{\dagger}\Delta^{j}\partial_{j}\psi^{\dagger}+h.c\right),$
$\displaystyle t_{\text{cov}\;j}^{t}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\psi^{\dagger}\overleftrightarrow{D_{j}}\psi,$
$\displaystyle t_{\text{cov}\;t}^{i}$ $\displaystyle=$
$\displaystyle-\frac{\delta^{ik}\left(D_{k}\psi\right)^{\dagger}D_{t}\psi}{2m^{*}}+\frac{1}{2}\psi^{\dagger}\Delta^{i}\partial_{t}\psi^{\dagger}+h.c,$
$\displaystyle t_{\text{cov}\;j}^{i}$ $\displaystyle=$
$\displaystyle-\frac{\delta^{ik}\left(D_{k}\psi\right)^{\dagger}D_{j}\psi}{2m^{*}}+\frac{1}{2}\psi^{\dagger}\Delta^{i}\partial_{j}\psi^{\dagger}+h.c-\delta_{j}^{i}\mathcal{L}.$

The $U\left(1\right)$-covariant conservation law is slightly more complicated
than (2.49) due the additional background field $\Delta$,

$\displaystyle\partial_{\mu}t_{\text{cov}\;\nu}^{\mu}=\frac{1}{2}\psi^{\dagger}\partial_{j}\psi^{\dagger}D_{\nu}\Delta^{j}+h.c+F_{\nu\mu}J^{\mu},$
(2.52)

where we have used the $U\left(1\right)$ conservation law (2.44), and
$D_{\mu}\Delta^{j}=\left(\partial_{\mu}-2iA_{\mu}\right)\Delta^{j}$. This
conservation law shows that ($U\left(1\right)$-covariant) fermionic energy-
momentum is not conserved due to the exchange of energy-momentum with the
background fields $A,\Delta$. Apart from the Lorentz force there is an
additional source term due to the space-time dependence of $\Delta$.

As in the case of the electric charge, if one considers $\Delta$ as a
dynamical field and uses its equation of motion, (2.52) can be written
as181818$t_{\Delta\;\text{cov}\;\nu}^{\mu}$ is the $U\left(1\right)$-covariant
energy-momentum tensor of Cooper pairs. It is defined by (2.50) with
$\phi=\Delta$ and
$\mathcal{L}=\mathcal{L}^{\prime}\left(\Delta,\Delta^{*},D\Delta,D\Delta^{*}\right)$
being the (gauge invariant) term added to the $p$-wave SF Lagrangian. Here it
is important that the coupling of $\Delta$ to $\psi$ in (2.31) can be written
without derivatives of $\Delta$.

$\displaystyle\partial_{\mu}\left(t_{\text{cov}\;\nu}^{\mu}+t_{\Delta\;\text{cov}\;\nu}^{\mu}\right)=F_{\nu\mu}\left(J^{\mu}+J_{\Delta}^{\mu}\right),$
(2.53)

which is of the general form (2.49).

Note that the spatial part $t_{\text{cov}\;j}^{i}$ is not symmetric,

$\displaystyle
t_{\text{cov}\;y}^{x}-t_{\text{cov}\;x}^{y}=\frac{1}{2}\psi^{\dagger}\left(\Delta^{x}\partial_{y}-\Delta^{y}\partial_{x}\right)\psi^{\dagger}+h.c,$
(2.54)

which physically represents an exchange of angular momentum between $\Delta$
and $\psi$, possible because of the intrinsic angular momentum of Cooper pairs
in a $p$-wave SC. Explicitly, the ($U\left(1\right)$-covariant) angular
momentum current is given by
$J_{\varphi}^{\mu}=t_{\text{cov}\;\varphi}^{\mu}=t_{\text{cov}\;\nu}^{\mu}\zeta^{\nu}$
where $\zeta=x\partial_{y}-y\partial_{x}=\partial_{\varphi}$ is the generator
of spatial rotations around $x=y=0$, and $\varphi$ is the polar angle. From
(2.52) and (2.54) we find its conservation law

$\displaystyle\partial_{\mu}J_{\varphi}^{\mu}=\left(\frac{1}{2}\psi^{\dagger}\partial_{j}\psi^{\dagger}D_{\varphi}\Delta^{j}+h.c+F_{\varphi\mu}J^{\mu}\right)$
(2.55)
$\displaystyle+\frac{1}{2}\psi^{\dagger}\left(\Delta^{x}\partial_{y}-\Delta^{y}\partial_{x}\right)\psi^{\dagger}+h.c,$

which shows that even when the Lorentz force in the $\varphi$ direction
vanishes and $\Delta$ is ($U\left(1\right)$-covariantly) constant in the
$\varphi$ direction, $\Delta$ still acts a source for fermionic angular
momentum, due to the last term.

Even though fermionic angular momentum is never strictly conserved in a
$p$-wave SF, it is well known that a certain combination of fermionic charge
and fermionic angular momentum can be strictly conserved [36, 231, 232, 233].
Indeed, using (2.55) and (2.44),

$\displaystyle\partial_{\mu}\left(J_{\varphi}^{\mu}\mp\frac{1}{2}J^{\mu}\right)=\left(\frac{1}{2}\psi^{\dagger}\partial_{j}\psi^{\dagger}D_{\varphi}\Delta^{j}+h.c+F_{\varphi\mu}J^{\mu}\right)$
$\displaystyle\pm\frac{i}{2}\psi^{\dagger}\left(\Delta^{x}\pm
i\Delta^{y}\right)\left(\partial_{x}\mp
i\partial_{y}\right)\psi^{\dagger}+h.c.$ (2.56)

We see that when $F_{\varphi\mu}=0$, $D_{\varphi}\Delta=0$ and $\Delta^{y}=\pm
i\Delta^{x}$, the above current is strictly conserved

$\displaystyle\partial_{\mu}\left(J_{\varphi}^{\mu}\mp\frac{1}{2}J^{\mu}\right)=0,$
(2.57)

which occurs in the generalized $p_{x}\pm ip_{y}$ configuration
$\Delta=e^{i\theta\left(r,t\right)}\Delta_{0}\left(r,t\right)\left(1,\pm
i\right)$, written in the gauge $A_{\varphi}=0$, and where
$r=\sqrt{x^{2}+y^{2}}$. This conservation law follows from the symmetry of the
generalized $p_{x}\pm ip_{y}$ configuration under the combination of a spatial
rotation by an angle $\alpha$ and a $U\left(1\right)$ transformation by a
phase $\mp\alpha/2$.

#####  2.4.2 Symmetries, currents, and conservation laws in the geometric
description

The symmetries and conservation laws for Dirac fermions have been described
recently in [156]. Here we review the essential details (for Majorana
fermions) and focus on the mapping to the symmetries and conservation laws of
the $p$-wave SF action (2.32), which were described in section 2.4.1.

######  2.4.2.1 Currents in the geometric description

The natural currents in the geometric description are defined by the
functional derivatives of the action $S_{\text{RC}}$ with respect to the
background fields $e,\omega$,

$\displaystyle\mathsf{J}_{\;a}^{\mu}=\frac{1}{\left|e\right|}\frac{\delta
S_{\text{RC}}}{\delta
e_{\;\mu}^{a}},\;\mathsf{J}^{ab\mu}=\frac{1}{\left|e\right|}\frac{\delta
S_{\text{RC}}}{\delta\omega_{ab\mu}}.$ (2.58)

$\mathsf{J}_{\;a}^{\mu}$ is the energy momentum (energy-momentum) tensor and
$\mathsf{J}^{ab\mu}$ is the spin current. Note that we use $\mathsf{J}$ as
opposed to $J$ to distinguish the geometric currents from the $p$-wave SF
currents described in the previous section, though the two are related as
shown below.

Calculating the geometric currents for the action (2.33) one obtains

$\displaystyle 2\mathsf{J}_{\;a}^{\mu}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{\text{RC}}e_{a}^{\;\mu}-\frac{i}{2}\overline{\chi}\left(\gamma^{\mu}D_{a}-\overleftarrow{D_{a}}\gamma^{\mu}\right)\chi,$
(2.59) $\displaystyle 2\mathsf{J}^{ab\mu}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\overline{\chi}\chi e_{c}^{\;\mu}\varepsilon^{abc},$

where
$\mathcal{L}_{\text{RC}}=\overline{\chi}\left[\frac{i}{2}e_{a}^{\;\mu}\left(\gamma^{a}D_{\mu}-\overleftarrow{D_{\mu}}\gamma^{a}\right)-m\right]\chi$
is (twice) the Lagrangian, which vanishes on the $\chi$ equation of motion. We
see that $\mathsf{J}_{\;a}^{\mu}$ is essentially the
$SO\left(1,2\right)$-covariant version of the canonical energy-momentum tensor
of the spinor $\chi$. We also see that the spin current $\mathsf{J}^{ab\mu}$
has a particularly simple form in $D=2+1$, it is just the spin density
$\frac{1}{2}\overline{\chi}\chi$ times a tensor
$-\frac{1}{2}e_{c}^{\;\mu}\varepsilon^{abc}$ that only depends on the
background field $e$. Using the expressions (2.37) for the geometric fields we
find that $\mathsf{J}_{\;a}^{\mu},\;\mathsf{J}^{ab\mu}$ are related simply to
the electric current and the ($U\left(1\right)$-covariant) canonical energy-
momentum tensor described in section 2.4.1, in the limit
$m^{*}\rightarrow\infty$,

$\displaystyle
J^{\mu}=4\left|e\right|\mathsf{J}^{12\mu}=-\psi^{\dagger}\psi\delta_{t}^{\mu},$
(2.60) $\displaystyle
t_{\text{cov}\;\nu}^{\mu}=-\left|e\right|\mathsf{J}_{\;\nu}^{\mu}=\begin{cases}\frac{i}{2}\psi^{\dagger}\overleftrightarrow{D_{\nu}}\psi&\mu=t\\\
\frac{1}{2}\psi^{\dagger}\Delta^{j}\partial_{\nu}\psi^{\dagger}+h.c&\mu=j\end{cases}.$

Here we have simplified $t_{\text{cov}}$ using the equation of motion for
$\psi$, and one can also use the equation of motion to remove time derivatives
and obtain Schrodinger picture operators. For example,
$t_{\text{cov}\;t}^{t}=\frac{i}{2}\psi^{\dagger}\overleftrightarrow{D_{t}}\psi=m\psi^{\dagger}\psi-\left(\frac{1}{2}\psi^{\dagger}\Delta^{j}\partial_{j}\psi^{\dagger}+h.c\right)$
is just the ($U\left(1\right)$-covariant) Hamiltonian density in the
relativistic limit. The expression for the energy current
$t_{\text{cov}\;t}^{i}$ is more complicated, and it is convenient to write it
using some of the geometric quantities introduced above

$\displaystyle t_{\text{cov
}t}^{j}=g^{jk}\frac{i}{2}\psi^{\dagger}\overleftrightarrow{D_{k}}\psi-$
$\displaystyle\frac{o}{2}\partial_{k}\left(\frac{1}{\left|e\right|}\varepsilon^{jk}\psi^{\dagger}\psi\right)$
(2.61) $\displaystyle-$
$\displaystyle\left(\psi^{\dagger}\psi\right)g^{jk}C_{12k}.$

This is an expression for the energy current in terms of the momentum and
charge densities, and it will be obtained below as a consequence of Lorentz
symmetry in the relativistic limit. We now describe the symmetries of the
action (2.33) and the conservation laws they imply for these currents. As