# Observational constraints on accelerators of ultra-high energy cosmic rays

Sullivan Maraficoa, Jonathan Biteaua, Antonio Condorellia, Olivier Delignya,*,
Quentin Luceb a Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay,
France
b Karlsruhe Institute of Technology, Institute for Experimental Particle
Physics (ETP), Karlsruhe, Germany *<EMAIL_ADDRESS>
###### Abstract

We explore two generic hypotheses for tracing the sources of ultra-high energy
cosmic rays (UHECRs) in the Universe: star formation rate density or stellar
mass density. For each scenario, we infer a set of constraints for the
emission mechanisms in the accelerators, for their energetics and for the
abundances of elements at escape from their environments. From these
constraints, we generate sky maps above 40 EeV expected from a catalog that
comprises 410,761 galaxies out to 350 Mpc and provides a near-infrared flux-
limited sample to map both stellar mass and star formation rate over the full
sky. Considering a scenario of intermittent sources hosted in every galaxy, we
show that the main features observed in arrival directions of UHECRs can in
turn constrain the burst rate of the sources provided that magnetic-horizon
effects are at play in clusters of galaxies.

  * September 2022

Introduction. A correlation between arrival directions of ultra-high energy
cosmic rays (UHECRs) and the flux patterns of star-forming galaxies within a
few Mpc provides currently the most promising evidence for anisotropy above 40
EeV [1, 2, 3]. Its meaning, however, remains to be uncovered. In particular,
the relatively low-signal strength, on the order of 10–15%, prevents one from
drawing firm statements about an origin of UHECRs exclusively from star-
forming galaxies. The aim of this contribution, based on [4, 5], is to
constrain a benchmark scenario that could accommodate the observed
correlation.

Starburst galaxies are responsible for $\simeq 15\%$ of the total star
formation rate (SFR) for redshifts $z<2$ [6, 7]. If UHECR sources are related
to cataclysmic events associated with the deaths of short-lived, massive
stars, the rate of which is traced by the SFR, the signal strength is then
suggestive of sources being hosted in every star-forming galaxy in the
Universe, with starburst galaxies harboring more events than main-sequence
galaxies. We explore such a benchmark scenario by considering that the density
of sources is fairly traced by the SFR density (SFRD) or, alternatively, by
the stellar mass density (SMD). In both cases, we first infer constraints on
the emission processes, energetics, and abundances of elements in the source
environments by fitting, to the energy spectrum and mass-composition data
reported in [8, 9], a benchmark scenario in which the intensity of each
individual nuclear components is assumed to drop off at the same magnetic
rigidity. This is consistent with the basic expectation that electromagnetic
processes accelerate particles up to a maximum energy proportional to their
electric charge $Z$. Based on these results, we map the UHECR intensity in the
sky above 40 EeV that would result from sources following the SFRD or the SMD
by using the most complete flux-limited catalog of galaxies over the entire
sky [10, 11]. Considering an intermittent nature for the UHECR sources then
enables us to match the arrival directions of UHECRs for some viable range of
rate density.

Figure 1: UHECR energy flux on Earth as modeled by the best-fit parameters for
the benchmark scenario, for five detected mass groups and for all particles.
Left: SMD [4]. Right: SFRD [5].

Source properties required to match observations. The benchmark model follows
here from [4], in which the energy spectrum of the ejected particles as well
as the amount of ejected nuclei $A$ may differ strongly from those injected
into the electromagnetic field due to in-source interactions, see e.g. [12,
13]. This has two important consequences: a) the ejected spectrum of the
charged nuclei in $E^{-\gamma_{A}}$ can be much harder than that injected, due
to the escape mechanism; b) the interactions can produce a copious flux of
secondary neutrons of energy $E_{\rm n}=E/A$. These neutrons can escape freely
from the magnetic confinement zones, with an ejection spectrum
$E^{-\gamma_{\rm p}}$ much softer than that of nuclei, to decay into protons
on their way to the Earth. Consequently, the ejection rate of the sources per
comoving volume unit and per energy unit of elements is modeled by an
exponentially-suppressed power law, the spectral index of which is possibly
different for protons ($\gamma_{\rm p}$) and nuclei ($\gamma_{A}$).

For reproducing simultaneously the energy spectrum and mass-composition data
above $\simeq 0.6$ EeV, the modelled abundance of nuclear elements is found to
be dominated by intermediate-mass ones, ranging from He to Si, accelerated to
$E^{Z}_{\mathrm{max}}\simeq 5Z\,\mathrm{EeV}$ and escaping from the source
environments with a very hard spectral index $\gamma_{A}$, consistent with
previous studies above 5 EeV [14]. Such a hard spectral index is called for to
account for the quasi-monoelemental increase of the mass with energy. In the
SMD scenario, fitting the model to the data indeed requires the spectral index
for protons ($\gamma_{\rm p}\simeq 3.3$) to be much softer than that of nuclei
($\gamma_{A}\simeq-0.45$) [4]. The results are illustrated in the left panel
of Fig. 1. In the SFRD scenario, however, the large increase of star-forming
activity up to the cosmic noon ($z\simeq 2$) results in a large increase of
secondary protons created en route subsequently to nuclei interactions. These
secondary protons could accommodate the observed spectrum (right panel of Fig.
1), on the condition that $\gamma_{A}\simeq-2$. Inferring a set of constraints
on $\gamma_{\rm p}$ and $\gamma_{A}$ would require considering the proton
spectrum to even lower energies.

UHECR accelerators from Galactic and extragalactic transient events? If
protons of extragalactic origin contribute to the sub-ankle component, other
elements are needed to make up the all-particle spectrum shown in Fig. 1.
Interestingly, a steep fall-off of the Fe component above $10^{17}$ eV has
been reported in [9]. This is along the lines of a scenario for Galactic CRs
characterised by a rigidity-dependent maximum acceleration energy to explain
the knee structures. At the same time, the presence of nuclei of the CNO group
(and of He nuclei, although with much larger uncertainties) is observed. We
advocate here that such intermediate-mass elements could be remnants of an old
event producing UHECRs in the Galaxy, in line with the starting assumption
that UHECR sources could be harbored in any galaxy. In fact, little is known
about the confinement of CRs in the Galaxy by the magnetic field at these
energies. Adopting the widely-used value of the confinement volume of the
Galactic disk, $\pi(20\,\mathrm{kpc})^{2}\times 300\,\mathrm{pc}\simeq
10^{67}$ cm3, intermittent sources such as long gamma-ray bursts releasing
$\simeq 10^{51}$ erg every million years in the Milky Way [15] would supply a
luminosity density of a few $10^{-30}$ erg s-1 cm-3. For a confinement time of
the order of $10^{4}$ or $10^{5}$ yr for N nuclei between $10^{17}$ and
$10^{18}$ eV [16, 17], such a luminosity density can accommodate the missing
energetics between the all-particle and proton spectra in Fig. 1 below the
ankle energy. Within this scenario, in which any galaxy could host transient
accelerators, UHECRs from the Galactic event would have escaped so that those
observed above the ankle energy are indeed from extragalactic events – see
e.g. [18, 19] for such a scenario. We explore below the constraints that can
be obtained on the luminosity required for the UHECR sources by finding the
best match between the observed and the expected sky map at the highest
energies.

Steady images of intermittent sources. Transient events from distant sources
can appear as persistent ones at ultra-high energies because of the temporal
spread induced by intervening magnetic fields (see, e.g. [20] for a review).
The source image of a burst of duration $\delta t$ is then visible during a
period of time $\Delta\tau\gg\delta t$ , which can be shown to scale with the
amplitude of the magnetic field $B$, the particle rigidity $R$, the source
distance $D$ and the coherence length of the field $\lambda_{B}$ as [21]

$c\Delta\tau\simeq 4.4\times 10^{3}\left(\frac{B}{\rm
10\,nG}\right)^{2}\;\left(\frac{R}{10\,\rm
EV}\right)^{-2}\;\left(\frac{D}{1\rm\,Mpc}\right)^{2}\;\left(\frac{\lambda_{B}}{10\rm\,kpc}\right)\;{\rm
yr}.$ (1)

The extragalactic magnetic-field amplitude is currently bracketed between
$O(0.1\,$nG) and $O(0.1\,$fG) [22, 23, 24]. Here, we consider an intermediate
value $B_{\rm XG}=0.1\,$pG. The time spread induced by such a field is
sizeable for sources at cosmological distances, but remains negligible for
sources within the GZK horizon. The values for the turbulent component of the
Galactic magnetic field are taken from one of the most up-to-date model [25].
While this field dominates the angular deflections of UHECRs, its spatial
extension is too limited to imprint sizeable time spreads for extragalactic
sources. Finally, the last magnetic field considered here is that, largely
under-constrained, from the Local Sheet – referring to the structure of the
cosmic web that embeds the Local Group of galaxies. We consider in this
exploratory study $B_{\rm LS}$ between 10 and 25 nG, and $\lambda_{\rm B_{\rm
LS}}=10{\,\rm kpc}$, extending over 1 Mpc. Such values are consistent with
those inferred from MHD simulations [26], although these simulations are
highly seed-dependent. With these values, the Local Sheet field is the
dominant one that governs the time spread of UHECRs, amounting to more than
100,000 years for sources beyond 1 Mpc.

The sky of transient UHECR sources. To extract randomly UHECR sources, we use
the catalog of galaxies established in [10] that comprises 410,761 galaxies
out to 350 Mpc, distance at which 50% of the stellar mass ($M_{\star}$) is
below the 2MASS sensitivity limit. This catalog is designed to account for
incompleteness as a function of luminosity distance, limited by the 2MASS
sensitivity threshold, and of angular distance to the Galactic plane, where
source confusion and obscuration mask background galaxies in the so called
Zone of Avoidance. It provides an unprecedented view on $M_{\star}$ and SFR on
scales ranging from the very outskirts of the Milky Way out to the most
distant superclusters identified in Cosmic Flows [27].

Figure 2: Left: Median sky map in the scenario of intermittent sources traced
by the SFRD, using a 12.6-degree gaussian smearing, for $k=2~{}10^{-5}$
$\textup{M}_{\odot}^{-1}$ [5]. Right: Observed arrival directions [2].

The Virgo cluster, at a distance of $\simeq 16~{}$Mpc, is the major cluster of
galaxies that is within the UHECR horizon above $\simeq 40~{}$EeV.
Accelerators located in the galaxies of this cluster may not, however,
contribute to the UHECR intensity observed on Earth due to the longer
confinement time scales encountered in this environment. Magnetic-field
amplitudes on the order of several $\upmu$G are expected at cluster centers
and are indeed observed in the Coma cluster (central value of $4.7\pm
0.7~{}\upmu$G) [28]. A similar amplitude is expected in Virgo based on its
pressure profile inferred from Planck and X-ray observations [29]. Under the
approximation of a coherence length and extent similar to those of Coma, the
$B$-field in the Virgo-cluster environment is then sufficient to confine
cosmic rays longer than their energy-loss time for the particle rigidities of
interest. Consequently, we consider in this exploratory study that UHECRs from
galaxies in the Virgo cluster could never reach Earth. A more detailed and
careful treatment of the magnetic confinement in galaxy clusters is underway
[30]. Note also that propagation effects could prevents UHECRs from far-enough
distant sources located in the Local Sheet to reach us [31].

In the scenario of intermittent sources, the probability to observe $N$
bursting sources from a given host galaxy follows a Poisson distribution, with
$N=ks\Delta\tau$. The rate of bursting sources is governed by the parameter
$k$, expressed in $\textup{M}_{\odot}^{-1}$ units for $s$ tracing the SFRD or
in $\textup{M}_{\odot}^{-1}~{}\mathrm{yr}^{-1}$ units for $s$ tracing the SMD.
Using the best-fit parameters obtained from the combined fit of the energy
spectrum and mass-composition data, the flux from each direction in the sky
can be evaluated by sampling randomly the Poisson process for a given value of
$k$. As a fair illustration of the various different maps that result from the
stochastic process, we show in the left panel of Fig. 2 the median map for
$k=2~{}10^{-5}$ $\textup{M}_{\odot}^{-1}$ in the SFRD scenario above 40 EeV,
using a 12.6-degree gaussian smearing (that corresponds to a 20-degree top hat
smearing). The model displays overdensities around the supergalactic plane, as
expected from the galaxies in the Local Sheet that are inferred to be
responsible for the indication of anisotropies at the $4.2\sigma$ confidence
level at these energies (right panel) [2]. Similar qualitative agreement can
be found in the SMD case for $k\simeq 10^{-15}$
$\textup{M}_{\odot}^{-1}~{}\mathrm{yr}^{-1}$.

Figure 3: Constraints on the density of UHECR source rate as a function of the
total energy emitted per source, with a few putative sources (CC-SN
($I_{\mathrm{b/c}}$): core-collapse supernovae (of type $I_{\mathrm{b/c}}$),
L(H)L-lGRB: low (high) luminosity long gamma ray bursts, TDE: tidal disruption
events.) From [5].

Constraints on UHECR accelerators. We finally compare our results to the rate
density and the total energy emitted in photons of a few source candidate in
Fig. 3. The dark green band in diagonal corresponds to the UHECR luminosity
obtained in this work for both SMD and SFRD scenarios, indistinctly. In the
absence of firmly-prescribed correspondences between UHECR and photon
luminosity, the lighter green band extends to a factor 10 the putative
proportionality. All five selected source classes are observed within the
green region. The yellow band, on the other hand, results from the constraints
obtained on $k$. Objects such as low-luminosity long gamma ray bursts (LL-
lGRB) in the SFRD case and events such as tidal disrupted ones (TDE) in the
SMD case are observed to match the two sets of constraints. Interestingly,
their energetics and source rate lie in the range of the fiducial values
needed to fill the sub-ankle energy flux with a Galactic event.

In deriving the constraints, the role of magnetic fields in galaxy clusters
and in the Local Sheet is key. The understanding of these fields will benefit
from X-ray and radio observations with eROSITA and SKA and will allow for
deciding whether the scenario of UHECR transient sources harbored in any
galaxy is viable or not.

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# Emergent anomalies and generalized Luttinger theorems in metals and
semimetals

Chong Wang  Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L
2Y5, Canada  Alexander Hickey  Department of Physics and Astronomy, University
of Waterloo, Waterloo, Ontario N2L 3G1, Canada  Xuzhe Ying  Department of
Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1,
Canada  A.A. Burkov  Department of Physics and Astronomy, University of
Waterloo, Waterloo, Ontario N2L 3G1, Canada  Perimeter Institute for
Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

###### Abstract

Luttinger’s theorem connects a basic microscopic property of a given metallic
crystalline material, the number of electrons per unit cell, to the volume,
enclosed by its Fermi surface, which defines its low-energy observable
properties. Such statements are valuable since, in general, deducing a low-
energy description from microscopics, which may perhaps be regarded as the
main problem of condensed matter theory, is far from easy. In this paper we
present a unified framework, which allows one to discuss Luttinger theorems
for ordinary metals, as well as closely analogous exact statements for
topological (semi)metals, whose low-energy description contains either
discrete point or continuous line nodes. This framework is based on the ’t
Hooft anomaly of the emergent charge conservation symmetry at each point on
the Fermi surface, a concept recently proposed by Else, Thorngren and Senthil
Else  _et al._ (2021). We find that the Fermi surface codimension $p$ plays a
crucial role for the emergent anomaly. For odd $p$, such as ordinary metals
($p=1$) and magnetic Weyl semimetals ($p=3$), the emergent symmetry has a
generalized chiral anomaly. For even $p$, such as graphene and nodal line
semimetals (both with $p=2$), the emergent symmetry has a generalized parity
anomaly. When restricted to microscopic symmetries, such as $U(1)$ and lattice
symmetries, the emergent anomalies imply (generalized) Luttinger theorems,
relating Fermi surface volume to various topological responses. The
corresponding topological responses are the charge density for $p=1$, Hall
conductivity for $p=3$, and polarization for $p=2$. As a by-product of our
results, we clarify exactly what is anomalous about the surface states of
nodal line semimetals.

##  I Introduction

The metal-insulator dichotomy is one of the basic facts of solid state
physics. It is also one of the simplest and most fundamental macroscopic
manifestations of quantum mechanics. If electrons are treated as
noninteracting, the existence of metals and insulators follows from the
quantum mechanics of a particle in periodic potential, which leads to the
existence of bands and band gaps in the electron energy spectrum, and from the
Pauli principle. Whether a given crystalline material is a metal or an
insulator is then determined by a single parameter, the number of electrons
per unit cell per spin (filling). When the number is an integer, the material
is an insulator (we will not distinguish between insulators and accidental
compensated semimetals due to band overlap). Otherwise it is a metal, with a
Fermi surface of gapless excitations, which encloses a volume in momentum
space, directly determined by the fractional part of the filling, which we
denote as $\nu$ henceforth.

The relation between the Fermi surface volume and the filling is in fact much
deeper than simple counting of filled single-particle states of noninteracting
electrons might suggest. It holds even when the electron-electron interactions
are taken into account, a statement known as Luttinger’s theorem Luttinger and
Ward (1960). Recent work has made it clear that this statement has a
topological origin Volovik (2003, 2007); Oshikawa (2000); Hastings (2004);
Furuya and Oshikawa (2017); Cheng  _et al._ (2016); Cho  _et al._ (2017);
Metlitski and Thorngren (2018); Jian  _et al._ (2018); Song  _et al._ (2021);
Else  _et al._ (2021), in particular connecting it with the concepts of
quantum anomalies and higher-dimensional symmetry protected topological (SPT)
phases. In this sense $\nu$ may be viewed as a topological, although
unquantized and continuously tunable, invariant, characterizing a metallic
phase, which is sandwiched between two insulators, corresponding to integer
values of $\nu$.

On another front, it has recently been discovered that there exists another
way a metallic phase may arise: as an intermediate phase between a topological
and an ordinary insulator in three dimensions (3D), when a direct transition
is impossible Volovik (2003, 2007); Murakami (2007); Wan  _et al._ (2011);
Burkov and Balents (2011); Burkov  _et al._ (2011); Armitage  _et al._ (2018).
Such materials are known as topological semimetals, or topological metals, and
exist at integer electron fillings per unit cell, which would normally imply
an insulator as the Luttinger invariant $\nu$ is zero. In this case however,
there exist other topological invariants, analogous to $\nu$ Gioia  _et al._
(2021). Just like $\nu$, these invariants are unquantized and continuously
tunable, but are topological in origin and require gapless modes to be
present. In the simplest case of a magnetic Weyl semimetal, the unquantized
invariant is the Hall conductance per atomic plane Burkov and Balents (2011),
which, when not equal to an integer multiple of $e^{2}/h$, requires gapless
Weyl points whenever the Luttinger invariant $\nu$ is zero. In other types of
topological semimetals the invariants are more subtle, taking the form of
fractional electric charges carried by topological defects of crystalline
symmetries in point node semimetals Gioia  _et al._ (2021), or fractional
polarization in the line node case Ramamurthy and Hughes (2017); Rui  _et al._
(2018). In all cases, however, the idea is very similar to the Luttinger
invariant in ordinary metals: a “fractional” value of the invariant
necessarily requires either gapless modes or topological order (i.e. long-
range entanglement). The invariant is continuously tunable between two
“trivial” values, which correspond to insulators. These are either ordinary
insulators at different integer unit cell filling in the case of ordinary
metals, or insulators with different electronic structure topology in the case
of topological semimetals.

This similarity suggests that there should exist a common framework to
describe the topological properties of all metallic phases, including both
ordinary and topological metals. The goal of this paper is to describe such a
framework. We demonstrate that metals may be described by ’t Hooft anomalies
of emergent symmetry groups, which characterize their low-energy excitation
spectrum. The ’t Hooft anomalies connect the corresponding metallic phases to
higher-dimensional SPT insulators. While the SPT insulators themselves are
characterized by quantized topological response terms, the unquantized
anomalies in metals arise due to unquantized charges of gauged crystalline
symmetries (such as translations and rotations), which enter the corresponding
quantized response terms.

The nature of the emergent anomaly and the corresponding Luttinger-like
topological invariants is decided by the codimension of the Fermi surface
$p=d-d_{FS}$, where $d$ is the space dimension and $d_{FS}$ is the Fermi
surface dimension. For odd $p$, the emergent symmetry has a generalized chiral
anomaly — examples include ordinary metals ($p=1$) and magnetic Weyl semimetal
in 3D ($p=3$). The corresponding topological invariants in the (generalized)
Luttinger theorems are the charge density ($p=1$) and the Hall conductivity
($p=3$) respectively. For even $p$, the emergent symmetry has a generalized
parity anomaly. Examples with $p=2$ include nodal (Dirac) semimetals in two
dimensions (2D) and nodal line semimetals in 3D. In both cases, we show that
the corresponding topological invariant in the generalized Luttinger theorem
is the electric polarization — we explain the exact meaning of this statement,
and along the way we also clarify what is “anomalous” about the surface states
of these $p=2$ semimetals. While most of the above examples have been
discussed in the literature Luttinger and Ward (1960); Else  _et al._ (2021);
Burkov and Balents (2011); Ramamurthy and Hughes (2017); Rui  _et al._ (2018);
Wang  _et al._ (2020); Song  _et al._ (2021), our work provides a unified
framework to understand various different systems, and clarifies the
topological aspects of the nodal line semimetals.

The rest of the paper is organized as follows. In Section II we discuss odd-
codimension Fermi surfaces, including ordinary metals and magnetic Weyl
semimetals. In Section III we discuss even-codimension Fermi surfaces,
including graphene-like systems in 2D and nodal line semimetals in 3D. In
particular, we provide a detailed discussion of the nodal line semimetals and
clarify some important issues such as the anomaly of the surface states. We
conclude in Section IV with a discussion of our results.

##  II Odd Fermi surface codimension: generalized chiral anomaly

###  II.1 Ordinary metals: $p=1$

####  II.1.1 One-dimensional metals

Let us start with the simplest system, a one-dimensional (1D) spinless metal.
In this case the low-energy electronic structure, ignoring interactions for
now, consists of two Fermi points at $k_{\pm}=\pm k_{F}$. Each Fermi point
hosts gapless chiral modes, left (L) and right (R) handed, as depicted in Fig.
1. The equilibrium part of the response of this 1D metal has the form

$S=-i\int d\tau dx\,\rho A_{0},$ (1)

where $\rho$ is the equilibrium charge density, $A_{0}$ is the imaginary-time
scalar potential and we use $\hbar=c=e=1$ units throughout. The charge density
is related to the Fermi sea volume as

$\rho=\frac{2k_{F}}{2\pi}=\frac{\nu}{a},$ (2)

where $\nu$ is the filling (number of electrons per unit cell), introduced
before, and $a$ is the lattice constant.

To emphasize the topological character of Eq. (1), it is useful to “gauge” the
translation symmetry Thorngren and Else (2018); Nissinen and Volovik (2019);
Song  _et al._ (2021); Manjunath and Barkeshli (2021). Such gauging may be
viewed as a way to get rid of the dependence on a nonuniversal parameter $a$,
which enters the expression for the density, focusing instead on the universal
number $\nu$. What this means formally is that we imagine a strained crystal,
in which a fixed reciprocal lattice vector $2\pi/a$ is replaced by

$\frac{2\pi}{a}\rightarrow\frac{2\pi}{a}\left(\delta_{\mu
x}-\partial_{\mu}u_{x}-x_{\mu}\right).$ (3)

Here $u_{x}$ is the atomic displacement and $x_{\mu}$ is an integer
translation gauge field, which accounts for the fact that $u_{x}$ is only
defined modulo the lattice constant $a$. The translation gauge field has two
components: spatial $x_{x}$ and temporal $x_{\tau}$. Integral of $x$ over the
spatial cycle gives the total number of unit cells in the 1D crystal

$\int_{x}x=L_{x},$ (4)

where the lattice constant $a$ has been included in the integration measure
$dx/a\rightarrow dx$. The integral over the temporal cycle, on the other hand,
defines the twist of the periodic boundary conditions in the imaginary time
direction, i.e. the distance $\Delta L_{x}$, by which the crystal is shifted
at $\tau=\beta=1/T$ ($T$ is the temperature) with respect to $\tau=0$

$\int_{\tau}x=\Delta L_{x}.$ (5)

Discarding the “nontopological” pieces, Eq. (1) may then be written as Song
_et al._ (2021); Gioia  _et al._ (2021)

$S=i\nu\,\int\epsilon_{\mu\lambda}x_{\mu}A_{\lambda}=i\nu\int\,x\wedge A.$ (6)

Note that the nonuniversal lattice constant $a$ disappears from all observable
quantities, obtained from Eq. (6), given the translation gauge field
definitions Eqs. (4) and (5). Henceforth, we set the lattice constant to
unity.

Figure 1: (Color online) Band dispersion of 1D lattice fermions. Left and
right-handed Weyl branches are shown in black.

Now we note that Eq. (6) may alternatively be obtained from the low-energy
theory only, by invoking its emergent ’t Hooft anomaly. Indeed, at low
energies and assuming perfect translational symmetry (i.e. ignoring impurity
scattering), the numbers of L and R electrons are separately conserved,111This
remains true with electron-electron interactions as long as Umklapp scattering
is irrelevant. which corresponds to an emergent symmetry group $U(1)_{L}\times
U(1)_{R}$. This symmetry, however, is anomalous, in the sense that it may not
be realized as a true microscopic symmetry on a 1D lattice. In our context
this simply means that the L and R quantum numbers are only defined at the
corresponding Fermi points and not globally in the full 1D Brillouin zone
(BZ), as obvious from Fig. 1. A 1D system with such an internal symmetry may
only appear on the boundary of a 2D topological crystalline insulator, whose
topological response theory is given by

$S=\frac{i}{4\pi}\int(A_{R}\wedge dA_{R}-A_{L}\wedge dA_{L}).$ (7)

Each of the two terms in Eq. (7) is a Chern-Simons term with a unit
coefficient, giving rise to a pair of chiral modes of opposite chirality on
the boundary — this is just the familiar chiral anomaly in $(1+1)$ dimensions.

If we restrict the emergent $U(1)_{L}\times U(1)_{R}$ symmetry to the physical
$U(1)\times\mathbb{Z}$ symmetry ($\mathbb{Z}$ being the lattice translation
symmetry), we have $A_{R}=A+k_{F}x$ and $A_{L}=A-k_{F}x$ (recall that momentum
is the “charge” under translation). The chiral anomaly becomes

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{i}{4\pi}\int(A+k_{F}x)\wedge d(A+k_{F}x)$ (8)
$\displaystyle-$ $\displaystyle\frac{i}{4\pi}\int(A-k_{F}x)\wedge
d(A-k_{F}x)=i\nu\int\,x\wedge dA,$

which leads to precisely Eq. (6) on the 1D boundary, assuming the translation
gauge field is flat, i.e. $dx=0$. The reason Eq. (6) actually exists in a
stand-alone 1D system is that the true symmetry of the 1D metal is not
$U(1)\times U(1)$, but $U(1)\times\mathbb{Z}$, where $\mathbb{Z}$ corresponds
to discrete translations. This is reflected in the fact that $x$ in Eq. (6) is
not a $U(1)$ gauge field, but a discrete translation gauge field. Moreover,
the translational symmetry is not an internal symmetry.

These considerations reveal the connection between the Luttinger theorem in a
1D metal and the ’t Hooft anomaly of the emergent $U(1)_{L}\times U(1)_{R}$
symmetry of its low-energy theory. In what follows, we generalize these
arguments to ordinary metals in higher dimensions, as well as to topological
semimetals.

####  II.1.2 Two-dimensional metals

Let us now generalize the above ideas to higher-dimensional metals. For
simplicity we consider the 2D case only, since it contains all of the
essential new features that distinguish higher-dimensional metals from the 1D
case. In 2D the Fermi surface is a continuous 1D manifold. Generalizing the
emergent chiral charge conservation to this case, one assumes independent
charge conservation at every point on the Fermi surface, which is in fact the
standard assumption of the Fermi liquid theory. This implies that the emergent
symmetry group is $LU(1)$, the so-called loop group of $U(1)$ Else  _et al._
(2021). Assuming a single Fermi surface sheet for simplicity, the 1D Fermi
surface acts, in effect, as an extra dimension. Let us denote the coordinate,
which parametrizes the Fermi surface as $\theta\in[0,2\pi]$, and let
$k_{x,y}(\theta)$ be the corresponding components of the Fermi momentum. The
total “space-time” dimension of this system, which includes the extra
$\theta$-dimension, is $2+1+1=4$. The 2D Fermi surface may thus be connected
to the ’t Hooft anomaly of a $(4+1)$-dimensional topological insulator,
described by the following Chern-Simons topological field theory Else  _et
al._ (2021); Ma and Wang (2021)

$S=\frac{i}{6(2\pi)^{2}}\int\mathcal{A}\wedge d\mathcal{A}\wedge
d\mathcal{A},$ (9)

where $\mathcal{A}$ is the probe gauge field living in both real and (Fermi
surface) momentum space. Accordingly, the exterior derivative operation $d$
above contains derivatives with respect to the spacetime coordinates
(including the extra space dimension), as well as the Fermi surface coordinate
$\theta$.

Following previous logic, we now restrict to the physical
$U(1)\times\mathbb{Z}^{2}$ symmetry (charge conservation and two lattice
translations), and set $\mathcal{A}=A+k_{x}x+k_{y}y$. The $LU(1)$ anomaly
becomes

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{i}{6(2\pi)^{2}}\int(A+k_{x}x+k_{y}y)$ (10)
$\displaystyle\wedge$ $\displaystyle d(A+k_{x}x+k_{y}y)\wedge
d(A+k_{x}x+k_{y}y),$

where $x,y$ are gauge fields, corresponding to the translational symmetry in
the $x$ and $y$-directions respectively. The nonvanishing part of this, which
gives rise to the theory of a 2D Fermi liquid, is the following mixed anomaly
term

$S=\frac{i}{2(2\pi)^{2}}\int(k_{x}x+k_{y}y)\wedge d(k_{x}x+k_{y}y)\wedge dA.$
(11)

Assuming $dx=dy=0$, as before, this corresponds to the following boundary
theory

$S=\frac{i}{2(2\pi)^{2}}\int(k_{x}\partial_{\theta}k_{y}-k_{y}\partial_{\theta}k_{x})x\wedge
y\wedge A.$ (12)

Taking into account that

$\frac{1}{2}\int_{0}^{2\pi}d\theta(k_{x}\partial_{\theta}k_{y}-k_{y}\partial_{\theta}k_{x})=V_{F},$
(13)

where $V_{F}$ is the 2D Fermi sea volume, we obtain the following
$2+1$-dimensional topological field theory

$S=\frac{iV_{F}}{(2\pi)^{2}}\int x\wedge y\wedge A=i\nu\int x\wedge y\wedge
A,$ (14)

which, similar to Eq. (6), declares that the charge per unit cell is given by
$\nu=V_{F}/(2\pi)^{2}$. This is precisely the field theory expression of the
Luttinger theorem in a 2D Fermi liquid Song  _et al._ (2021).

###  II.2 Magnetic Weyl semimetal: $p=3$

Now we demonstrate that identical arguments may be used to describe 3D
topological semimetals. Topological semimetals exist at integer fillings,
which would normally correspond to insulators. This means that the filling
anomaly, which corresponds to the Luttinger theorem in Fermi liquids, does not
exist in this case. However, just as ordinary metals may be viewed as
intermediate phases between insulators, corresponding to different integer
values of the filling $\nu$, topological semimetals may be viewed as
intermediate gapless phases between pairs of topologically-distinct insulators
at the same fixed integer filling. Correspondingly, in each case there exists
a continuously-variable parameter, which tunes between the two insulators and
acts as the coefficient of the corresponding anomaly term, analogously to the
filling $\nu$ in Fermi liquids. While many different kinds of topological
semimetals exist, here we focus on the magnetic Weyl semimetal. In Sec. III.1
we discuss nodal line semimetals in detail. For an in-depth discussion of
other types of topological semimetals from a similar viewpoint, see Ref. Gioia
_et al._ (2021).

A magnetic Weyl semimetal has an odd number of pairs of opposite-chirality
Weyl nodes at the Fermi energy. We consider the simplest case when this number
is one and assume the nodes are separated along the $z$-axis. Let the node
coordinates be $k_{z}^{\pm}=\pm Q\hat{z}$. The separation between the nodes
$2Q$ is a continuously-tunable parameter. $2Q=0$ corresponds to an ordinary
magnetic insulator with zero Hall conductivity, while $2Q=2\pi$ (i.e. a
reciprocal lattice vector) corresponds to a quantum anomalous Hall insulator
with a Hall conductivity

$\sigma_{xy}=\frac{1}{2\pi}\frac{2Q}{2\pi}=\frac{1}{2\pi},$ (15)

which is equivalent to a Hall conductance quantum $1/2\pi$ per $xy$ atomic
plane, i.e. this is a weak topological insulator that may be viewed as a stack
of 2D integer quantum Hall insulators. Intermediate values of $2Q\in(0,2\pi)$
correspond to the Weyl semimetal. In this case the Hall conductance per atomic
plane $\sigma_{xy}=2Q/4\pi^{2}$ takes a noninteger value, which is impossible
in an insulator, unless it has topological order. This is the analog of a
fractional filling in a Fermi liquid metal.

The corresponding emergent anomaly is the familiar chiral anomaly of the
emergent $U(1)_{L}\times U(1)_{R}$ symmetry corresponding the the charge
conservation at each Weyl node. If the $U(1)_{L}\times U(1)_{R}$ symmetry is
exact, the system can only be realized on the boundary of a 4D topological
insulator, described by the following $(4+1)$-dimensional Chern-Simons theory

$S=\frac{i}{6(2\pi)^{2}}\int(A_{R}\wedge dA_{R}\wedge dA_{R}-A_{L}\wedge
dA_{L}\wedge dA_{L}).$ (16)

We now restrict to the physical $U(1)$ and $z$-translation symmetry, and make
the substitution $A_{R}=A+Qz$, $A_{L}=A-Qz$. The anomaly becomes

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{i}{6(2\pi)^{2}}\int(A+Qz)\wedge d(A+Qz)\wedge d(A+Qz)$
(17) $\displaystyle-$ $\displaystyle\frac{i}{6(2\pi)^{2}}\int(A-Qz)\wedge
d(A-Qz)\wedge d(A-Qz)$ $\displaystyle=$ $\displaystyle\frac{iQ}{4\pi^{2}}\int
z\wedge dA\wedge dA.$

This gives a $(3+1)$-dimensional boundary theory

$S=\frac{iQ}{4\pi^{2}}\int z\wedge A\wedge dA,$ (18)

which describes the topological response of the magnetic Weyl semimetal.
Indeed, Eq. (18) encodes precisely the Hall conductance of

$G_{xy}=\frac{2Q}{4\pi^{2}}L_{z}.$ (19)

There exists a close connection between the magnetic Weyl semimetal and the 1D
metal, discussed in Section II.2. This may be seen by inserting a magnetic
flux line carrying a unit flux quantum $\Phi=2\pi$ in the $xy$-plane.
According to Eq. (18), this flux line is described by the action

$S=\frac{i2Q}{2\pi}\int z\wedge A.$ (20)

Comparing this with Eq. (6), we see that the $2\pi$ flux line carries a 1D
metal, with the effective filling $\nu=2Q/2\pi$. This is just another way to
arrive at the familiar result that a Weyl semimetal in an external magnetic
field has the lowest Landau level that crosses the Fermi energy at the
location of the Weyl nodes.

We note that we can derive the emergent anomaly of a magnetic Weyl semimetal
from that of a three-dimensional metal. The latter is described by a
$7$-dimensional [$(3+1)$ from real spacetime, $2$ from Fermi surface momentum,
and $1$ extra “bulk” dimension] Chern-Simons term

$S=\frac{i}{6(2\pi)^{3}}\int\mathcal{A}\wedge d\mathcal{A}\wedge
d\mathcal{A}\wedge d\mathcal{A}.$ (21)

A Weyl fermion can be recovered in the limit of a small spherical Fermi
surface, with a total Berry curvature $\int_{FS}d\mathcal{A}=2\pi$ on the
Fermi surface. This reduces Eq. (21) to a $5$-dimensional Chern-Simons term,
which is exactly the anomaly of a Weyl fermion Eq. (16). This procedure can be
straightforwardly generalized to higher dimensions, from which we can obtain
the emergent anomaly of a general odd-codimension Fermi surface, as we discuss
next.

###  II.3 General dimensions

Using the (momentum space) dimensional reduction scheme described above, the
pattern becomes apparent for general space dimension $d$ and Fermi surface
dimension $d-p$, with $p$ odd. The emergent symmetry corresponds to charge
conservation at each point on the $d-p$-dimensional Fermi surface. The probe
gauge field $\mathcal{A}$ then lives in both real space-time, as well as the
$d-p$-dimensional momentum space. The corresponding ’t Hooft anomaly can be
described by a Chern-Simons term in $(d+1)+(d-p)+1=2d-p+2$ dimension (note the
importance of $p$ being odd):

$S=\frac{i}{N_{d,p}}\int\mathcal{A}(d\mathcal{A})^{d-(p-1)/2},$ (22)

where all the products in the integrand are wedge products, and the
normalization factor

$N_{d,p}=(d-(p-3)/2)!(2\pi)^{d-(p-1)/2}.$ (23)

If we restrict to the physical $U(1)$ and translation symmetries (with gauge
fields represented by $A$ and $x_{1},x_{2}...$), the anomaly term becomes a
response term in the physical $d+1$ space-time dimensions

$S=\frac{iV^{F}_{i_{1}...i_{d+1-p}}}{(\frac{p+1}{2})!(2\pi)^{d-(p+1)/2}}\int
A(dA)^{(p-1)/2}x_{i_{1}}x_{i_{2}}...x_{i_{d+1-p}},$ (24)

where $V^{F}_{i_{1}...i_{d+1-p}}$ is the Fermi surface volume, projected to
the $i_{1}...i_{d+1-p}$ subspace. Physically this represents a nontrivial
Chern-Simons type of response per unit length scale.

##  III Even Fermi surface codimension: generalized parity anomaly

###  III.1 Nodal line semimetal: $p=2$

We now discuss the case of nodal line semimetals. Since this has not received
as much attention in the literature as point node semimetals, we discuss this
case in more detail. Nodal line semimetals with nontrivial topology arise only
when bands are nondegenerate Burkov  _et al._ (2011); Ramamurthy and Hughes
(2017). This requires breaking of either time reversal or inversion symmetry
and nonvanishing spin-orbit coupling. The gaplessness of the nodal line is
then protected by mirror reflection symmetry in the plane, containing the
nodal line. Given this, we adopt a simple two-band model, with the following
cubic-lattice Hamiltonian Wang and Nandkishore (2017); Shapourian  _et al._
(2018)

$\displaystyle{\cal H}({{\bf{k}}})$ $\displaystyle=$
$\displaystyle\left[6-t_{1}-2(\cos k_{x}+\cos k_{y}+\cos
k_{z})\right]\sigma_{x}$ (25) $\displaystyle+$ $\displaystyle 2t_{2}\sin
k_{z}\sigma_{y}+t_{3}(2-\cos k_{x}-\cos k_{y}).$

The nodal line in this model appears in the $xy$-plane and is protected by the
mirror reflection symmetry within this plane, where the mirror reflection
operator is $\sigma_{x}$. It is implicit that the time reversal symmetry is
broken, since the bands are nondegenerate. Exactly this sort of Hamiltonian
describes the low-energy electronic structure in, for example, the magnetic
multilayer model of Ref. Burkov  _et al._ (2011).

In the absence of the last term in Eq. (25), proportional to the unit matrix,
the Hamiltonian has an exact particle-hole symmetry, which requires all points
on the nodal line to be at the same energy. However, once the last term is
included, this is no longer the case and the electronic structure consists of
electron and hole pockets of zero total Luttinger volume. We first derive the
topological response of the nodal line assuming the particle-hole symmetry and
then argue that relaxing this assumption does not change the response.

Following our logic in the previous examples, we consider an emergent symmetry
$LU(1)$, that corresponds to charge conservation at each point on the nodal
line. The corresponding probe gauge field $\mathcal{A}$ lives in
$(3+1)$-dimensional real space-time as well as $1$-dimensional momentum space.
If there is an emergent anomaly, associated with this symmetry, it is a
topological term in one higher dimension. The most obvious such term in
$(3+1)+1+1=6$ dimensions is the $\theta$-term with $\theta=\pi$:

$S=\pm\frac{i\pi}{6(2\pi)^{3}}\int d\mathcal{A}\wedge d\mathcal{A}\wedge
d\mathcal{A},$ (26)

where the arbitrariness of the sign reflects the $2\pi$ periodicity of the
coefficient. The quantization of the $\theta$ angle comes from the mirror
reflection symmetry, which inverts only one coordinate (the real space $z$)
and sends $\theta\to-\theta$. This is consistent with the fact that the nodal
ring can be trivially gapped if the reflection symmetry is broken.

The above generalized parity anomaly can in fact be derived from the
generalized chiral anomaly of ordinary metals, using a dimensional-reduction
scheme similar to that in Sec. II.2. To see this, consider a slightly
electron-doped nodal line, such that the Fermi energy $\epsilon_{F}>0$ (still
assuming particle-hole symmetry as in the undoped state). In this case we have
a toric Fermi surface, which encloses the nodal line along its length. As
discussed originally in Ref. Else  _et al._ (2021) and reviewed in Sec.
II.1.2, the Fermi surface carries an emergent anomaly, encoded by the
$(6+1)$-dimensional Chern-Simons term

$S=\frac{i}{6(2\pi)^{3}}\int\mathcal{A}\wedge d\mathcal{A}\wedge
d\mathcal{A}\wedge d\mathcal{A}.$ (27)

A characteristic feature of the nodal line is the $\pi$ Berry phase,
accumulated along a path in momentum space, enclosing the nodal line Burkov
_et al._ (2011). Let the toric Fermi surface be parametrized by two angular
variables $\theta,\phi\in[0,2\pi]$, where $\theta$ refers to the azimuthal
direction, which traverses the Fermi surface without enclosing the nodal line,
while $\phi$ is the polar direction, which wraps around the nodal line. Then
we have

$\int_{0}^{2\pi}d\phi\mathcal{A}_{\phi}=\pm\pi.$ (28)

Plugging this into Eq. (27) and taking the limit $\epsilon_{F}\rightarrow 0$,
we again arrive at Eq. (26).

Following the same logic as in the previous cases, we can now restrict to the
microscopic symmetries [$U(1)$, lattice translations and the mirror reflection
$z\to-z$]. Introducing a parameter $\theta\in[0,2\pi]$, such that
$k_{x,y}(\theta)$ are the coordinates of the points on the nodal line, we have
$\mathcal{A}=A+k_{x}x+k_{y}y$. The $6$-dimensional $\theta$-term now becomes

$\displaystyle S$ $\displaystyle=$
$\displaystyle\pm\frac{i\pi}{6(2\pi)^{3}}\int d(A+k_{x}x+k_{y}y)\wedge
d(A+k_{x}x+k_{y}y)$ (29) $\displaystyle\wedge$ $\displaystyle
d(A+k_{x}x+k_{y}y).$

This gives rise to a mixed anomaly term, which is similar to Eq. (11), except
in one extra dimension

$S=\pm\frac{i\pi}{2(2\pi)^{3}}\int d(k_{x}x+k_{y}y)\wedge
d(k_{x}x+k_{y}y)\wedge dA,$ (30)

corresponding to a boundary theory

$\displaystyle S$ $\displaystyle=$
$\displaystyle\pm\frac{i\pi}{2(2\pi)^{3}}\int(k_{x}\partial_{\theta}k_{y}-k_{y}\partial_{\theta}k_{x})x\wedge
y\wedge dA$ (31) $\displaystyle=$ $\displaystyle\pm\frac{iV_{F}}{8\pi^{2}}\int
x\wedge y\wedge dA,$

where $V_{F}$ is the area in momentum space, enclosed by the nodal line.

The physical meaning of Eq. (31) is that it describes spontaneous (i.e.
existing in the absence of an external electric field) electric polarization
Ramamurthy and Hughes (2017); Song  _et al._ (2021), given by

$P=\pm\frac{V_{F}}{8\pi^{2}}.$ (32)

The two signs in front reflect the fact that, in the presence of the mirror
symmetry, there must exist two degenerate values of polarization, related to
each other by the mirror reflection (unless $P=0$). If the nodal line is
expanded to the edges of the BZ, in which case a gap opens, we have
$V_{F}=(2\pi)^{2}$ and $P=\pm 1/2$. This half-quantized polarization is the
only nontrivial value of polarization that an inversion (or mirror) symmetric
insulator can have. A nontrivial “fractional” value of polarization in Eq.
(32) means that the polarization is not uniquely defined, as manifested by the
arbitrary sign in Eq. (32), which in turn requires a gapless spectrum, i.e.
the nodal line.

The precise meaning of polarization Eq. (32) for a gapless system requires a
more careful explanation. Unlike the Chern-Simons type of responses (such as
the Hall conductivity or the charge density), which can be easily defined even
for gapless systems, the $\theta$-term type of responses are sometimes ill-
defined in this case. One well-known example is the 1D metal, which does not
have a well-defined polarization (this is a direct consequence of the $(1+1)$d
chiral anomaly). Similarly, the magnetoelectric polarizability (the $dAdA$
$\theta$-angle) of a 3D Dirac semimetal is not defined. So what exactly do we
mean by Eq. (32) for the nodal line semimetal? One possible answer is that if
we weakly break the mirror reflection symmetry and completely gap out the
nodal line, the resulting insulator has a spontaneous polarization given by
Eq. (32), with the overall sign determined by the sign of the mirror symmetry
breaking. However, one may still ask if we can associate the polarization
directly to some observable in the semimetal phase, instead of having to rely
on neighboring phases to define it (such as mirror-breaking insulators).

The definition of polarization through the Luttinger theorem violation on the
boundary Song  _et al._ (2021) turns out to be useful here. If the surface
state is a 2D Fermi liquid at low energy, the bulk polarization is given by

$P=\rho-\frac{V_{F}}{(2\pi)^{2}}\hskip 5.0pt{\rm{mod}}\hskip 2.0pt1,$ (33)

where $P$ is the polarization density, perpendicular to the surface, $\rho$ is
the surface charge density (per unit cell) and $V_{F}$ is the Luttinger volume
of the surface Fermi liquid. One can think of the first term as the classical
contribution (which interprets polarization as a dipole moment) and the second
term as the quantum correction. 222If the surface is not a Fermi liquid, we
simply replace the second term by the appropriate “anomaly indicator” Song
_et al._ (2021), which is essentially the charge density required to realize
the state in a purely two-dimensional system.

Figure 2: Electronic structure of a finite size nodal line system, described
by Eq. (25) with $t_{1}=0.75,t_{2}=1,t_{3}=0$. The perfectly flat surface
state acquires a dispersion if $t_{3}\neq 0$.

The band structure for an $xy$-surface of the nodal line semimetal is
schematically illustrated in Fig. 2. The most important feature is a branch of
fermion modes that exist only inside the region, enclosed by the projection of
the nodal line onto the surface BZ. Other features, such as the flatness of
these modes, are less robust and do not play a role in the following
discussion. Now let us put the surface chemical potential somewhere and
calculate the polarization from Eq. (33). The answer, however, depends on the
surface chemical potential. We denote the polarization, calculated when the
chemical potential is above (below) zero, which is taken as the energy of the
nodal line, as $P_{+}$ ($P_{-}$). We find

$P_{+}-P_{-}=\frac{V_{F}}{4\pi^{2}}.$ (34)

This means that the bulk state cannot have a uniquely defined polarization
(unless $V_{F}=4\pi^{2}$, which is trivial), and therefore cannot be a short-
range entangled insulator. Since the reflection symmetry inverts polarization,
we must have

$P_{+}=-P_{-}=\frac{V_{F}}{8\pi^{2}},$ (35)

which is exactly Eq. (32).

The above discussion also clarifies the anomalous nature of the surface states
of the nodal line semimetal: while a surface Fermi liquid per se appears to be
quite non-anomalous (just like any other 2D metal), the Luttinger theorem is
violated. Such Luttinger theorem violation also happens on the surface of a
ferroelectric insulator [Eq. (33)], but the violation on the surface of a
nodal line semimetal is stronger in the sense that it cannot be removed by
redefining the surface charge density by a constant shift $P$.

We now consider breaking the particle-hole symmetry, so that the energy is no
longer constant on the nodal line. We ask how the above discussion of
polarization could change. Again, consider moving the surface chemical
potential gradually from well below the nodal line to well above. The
difference now is that there are intermediate regions, where the chemical
potential is above only parts of the nodal line. The resulting Fermi liquid
has open Fermi surfaces (arcs) and $V_{F}$ cannot be defined. As a result, the
polarization $P$ cannot be defined in the intermediate region. However, when
$P$ is well defined (when the chemical potential is far away from the nodal
ring), it still takes the values $P_{\pm}=\pm V_{F}/8\pi^{2}$.

Figure 3: (Color online) Stacked domain wall construction. Each domain wall,
separating regions with different sign of the mirror symmetry breaking “mass”
$m$, hosts a spinless particle or hole-like Fermi liquid state. The $\pm$
signs are the corresponding eigenvalues of $\sigma_{x}$.

####  III.1.1 Alternative perspective: Coupled-layer derivation of the
topological response

We can also derive the nodal line topological response Eq. (31) by an entirely
different, more microscopic method. This method allows us to demonstrate more
explicitly that particle-hole symmetry is not essential and the response is
unchanged when it is broken. We start with the nodal line Hamiltonian Eq. (25)
(but still assuming particle-hole symmetry for now) and add a mirror symmetry
breaking “mass term”

$\displaystyle{\cal H}({{\bf{k}}})=\left[6-t_{1}-2(\cos k_{x}+\cos k_{y}+\cos
k_{z})\right]\sigma_{x}$ (36) $\displaystyle+$ $\displaystyle 2t_{2}\sin
k_{z}\sigma_{y}+m\sigma_{z}.$

Let us now consider a domain wall, at which the mass $m$ changes sign from
negative to positive as a function of $z$. As in other, more familiar cases,
topologically nontrivial nature of the gapless nodal line state manifests in a
gapless bound state on such a domain wall. To find this state, we replace
$k_{z}\rightarrow-i\partial/\partial_{z}$, keeping only up to linear order
terms in $k_{z}$. Inclusion of the higher order terms does not affect the
gaplessness of the bound state Sekine and Nagaosa (2020). We thus consider the
following real-space Hamiltonian

$\displaystyle{\cal H}(k_{x},k_{y},z)=(4-t_{1}-2\cos k_{x}-2\cos
k_{y})\sigma_{x}$ (37) $\displaystyle-$ $\displaystyle
2it_{2}\sigma_{y}\frac{\partial}{\partial z}+m(z)\sigma_{z},$

where $m(z\rightarrow-\infty)<0,\,m(z\rightarrow\infty)>0$. A straightforward
and standard calculation gives the wavefunction of the bound state as

$\Psi_{-}(z)=e^{-\frac{1}{2t_{2}}\int_{0}^{z}dz^{\prime}m(z^{\prime})}|\sigma_{x}=-1\rangle,$
(38)

and the corresponding energy eigenvalue

$\epsilon_{-}(k_{x},k_{y})=-(4-t_{1}-2\cos k_{x}-2\cos k_{y}).$ (39)

A domain wall of opposite topological charge, where $m$ changes sign from
positive to negative, binds a state with

$\Psi_{+}(z)=e^{-\frac{1}{2t_{2}}\int_{0}^{z}dz^{\prime}m(z^{\prime})}|\sigma_{x}=1\rangle,$
(40)

and

$\epsilon_{+}(k_{x},k_{y})=4-t_{1}-2\cos k_{x}-2\cos k_{y}.$ (41)

These bound states correspond to spinless 2D electron or hole-like Fermi
liquids. Their topological responses are thus given by Eq. (14), where the
coefficient is the corresponding Luttinger volume, including opposite signs in
the electron and hole cases. Note that these domain wall states by themselves
are not anomalous, in the sense that a purely 2D lattice realization clearly
exists.

Now we note that the gapless nodal line state may be reconstructed by stacking
alternating domain walls with opposite topological charges, while preserving
mirror symmetry, as shown in Fig. 3. To preserve the mirror symmetry, the
inter-domain wall tunneling Hamiltonian must have the form

${\cal H}_{t}(k_{z})=2t_{2}\sigma_{y}\sin k_{z},$ (42)

i.e. exactly the form of the second term in Eq. (36), which corresponds to
inter-unit cell tunneling with a direction-dependent sign of the tunneling
matrix element.

To obtain the topological response of the nodal line we need to sum the
contributions from the domain wall bound states

$S=\pm i\frac{V_{F}}{(2\pi)^{2}}\int\sum_{n}(-1)^{n}A_{0}(z_{n}),$ (43)

where $V_{F}$ is the magnitude of the Luttinger volume of each 2D domain-wall-
bound Fermi liquid, $z_{n}$ is the coordinate of the $n^{\text{th}}$ domain
wall and the arbitrariness of the overall sign reflects the arbitrariness of
assigning the starting point of the summation. Taking the continuum limit and
switching to the translation gauge field notation, following the logic of
Section II.1.1, we arrive at the topological response theory of Eq. (31).

Now let us add the particle-hole symmetry breaking term $t_{3}(2-\cos
k_{x}-\cos k_{y})-\epsilon_{F}$ to the nodal line Hamiltonian. A finite Fermi
energy $\epsilon_{F}$ is necessary to keep charge neutrality once the
particle-hole symmetry is broken. It is clear that the domain wall bound
states remain the same, except for a shift in their energies

$\displaystyle\epsilon_{\pm}(k_{x},k_{y})$ $\displaystyle=$ $\displaystyle
t_{3}(2-\cos k_{x}-\cos k_{y})-\epsilon_{F}$ (44) $\displaystyle\pm$
$\displaystyle(4-t_{1}-2\cos k_{x}-2\cos k_{y}),$

i.e. the Luttinger volumes of the electron and hole-like domain wall states
get changed by equal and opposite amounts, so that overall charge neutrality
is preserved. It follows that the topological response term Eq. (43) remains
unchanged.

####  III.1.2 Gapped topological orders

The above picture of the nodal line as a stack of 2D Fermi liquid domain wall
states offers a useful prospective on the question of gapping the nodal line
while preserving its anomalous response Eq. (31). Following the “vortex
condensation” approach, originally applied to surface states of 3D topological
insulators Wang  _et al._ (2013); Metlitski  _et al._ (2015) and later
generalized to 3D topological semimetals in Refs. Wang  _et al._ (2020);
Thakurathi and Burkov (2020); Sehayek  _et al._ (2020) (also see Ref. Tam  _et
al._ (2021) for related work), we may start with a gapped superconducting
nodal line state and then attempt to construct an insulator with the same
topological response by condensing vortices in the superconductor. A simple
mean-field picture of this state may be obtained by applying this procedure to
each 2D domain wall state independently. Since each domain wall hosts a
spinless 2D Fermi liquid, the simplest fully gapped superconducting state is
$p$-wave, described by the following Hamiltonian

$H=\sum_{{{\bf{k}}}}\left[\epsilon_{\pm}({{\bf{k}}})c^{{\dagger}}_{{{\bf{k}}}}c^{\vphantom{{\dagger}}}_{{{\bf{k}}}}+\frac{\Delta}{2}(\sin
k_{x}+i\sin
k_{y})c^{{\dagger}}_{{{\bf{k}}}}c^{{\dagger}}_{-{{\bf{k}}}}+\text{h.c.}\right],$
(45)

where $\epsilon_{\pm}({{\bf{k}}})$ are given by Eqs. (39), (41) (we ignore the
particle-hole symmetry breaking here for simplicity) and ${{\bf{k}}}$ is the
2D crystal momentum. Introducing Nambu spinor notation
$\psi_{{{\bf{k}}}}=(c^{\vphantom{{\dagger}}}_{{\bf{k}}},c^{{\dagger}}_{-{{\bf{k}}}})$,
this may be represented as a massive 2D Dirac Hamiltonian

$H=\frac{1}{2}\sum_{{\bf{k}}}\psi^{{\dagger}}_{{\bf{k}}}\left[\epsilon_{\pm}({{\bf{k}}})\tau_{z}+\Delta(\tau_{x}\sin
k_{x}-\tau_{y}\sin k_{y})\right]\psi^{\vphantom{{\dagger}}}_{{\bf{k}}},$ (46)

where $\tau_{a}$ are Pauli matrices in the particle-hole space. This
representation makes it clear that the superconducting domain wall state is a
Read-Green topological superconductor Read and Green (2000), which hosts
chiral Majorana modes at the edges, with chirality determined by the type
(electron or hole-like) of the domain wall state. Consequently, an elementary
flux $hc/2e=\pi$ vortex hosts a zero-energy localized Majorana bound state and
can not be condensed. A double, or flux $2\pi$, vortex does not host any zero-
energy states, but may still not be condensed without breaking symmetries.
This is a consequence of the fact that such vortices experience the nontrivial
band filling $\pm\nu$ of the 2D domain wall Fermi liquid (the signs
distinguish between the electron and hole-like states) as magnetic flux,
leading to a projective representation of the group of translations for such
vortices Balents  _et al._ (2005)

$T_{x}T_{y}=e^{\pm 2\pi i\nu}T_{y}T_{x}.$ (47)

This effect is most obvious when the system has a boundary, in which case the
sign in Eq. (47) depends on exactly how the system is terminated. This, in
turn, means that $2\pi$ flux vortices carry nontrivial momentum and their
condensation leads to breaking of the crystal translational symmetry. When the
filling $\nu$ is rational, i.e. $\nu=p/q$ with $p$ and $q$ mutually prime
integers, flux $2\pi q$ vortices may be condensed without breaking any
symmetries. This leads to an insulator with $\mathbb{Z}_{q}$ topological
order. Stacking such fractionalized insulators leads to a gapped insulator,
which reproduces the response of the gapless nodal line state.

###  III.2 General dimensions

We can now easily generalize to arbitrary space dimension $d$ with an even
codimension $p$ for the Fermi surface. The generalized parity anomaly lives in
$(d+1)+(d-p)+1=2d-p+2$ dimensions and is described by a $\theta$-term at
$\theta=\pi$:

$S=\frac{i\pi}{N_{d,p}}\int(d\mathcal{A})^{d+1-p/2},$ (48)

where all the products in the integrand are wedge product, and the
normalization factor

$N_{d,p}=(d+1-p/2)!(2\pi)^{d+1-p/2}.$ (49)

Unlike the cases with odd $p$, the parity anomaly requires extra discrete
symmetries, such as reflection, to quantize the $\theta$-angle at $\pi$.

When restricted to the microscopic $U(1)$ and translation symmetries, Eq. (48)
becomes

$S=\frac{iV^{F}_{i_{1}...i_{d+1-p}}}{(\frac{p}{2})!(2\pi)^{d-p/2}}\int(dA)^{p/2}x_{i_{1}}x_{i_{2}}...x_{i_{d+1-p}},$
(50)

where $V^{F}_{i_{1}...i_{d+1-p}}$ is the Fermi surface volume projected to the
$i_{1}...i_{d+1-p}$ subspace. Physically this represents a nontrivial
$\theta$-term type of response per unit length scale.

Another familiar case is when $d=p=2$, which is nothing but graphene-like
Dirac semimetals in 2D, with the emergent anomaly being the standard parity
anomaly for each Dirac cone. For example, consider a spinless system with two
Dirac cones, separated by momentum $Q\hat{x}$, protected by reflection
symmetry $y\to-y$. It is straightforward to repeat the analysis for the nodal
line semimetal and conclude that the polarization in the $\hat{y}$ direction
is $P_{y}=\pm Q/2$. In fact the situation is even simpler than the nodal line
case, since polarization can be defined directly in the bulk without having to
go to the boundary, as discussed in Ref. Song  _et al._ (2021).

##  IV Discussion and conclusions

The main goal of this paper was to introduce a common framework, which may be
used to describe topological properties of both ordinary metals and
topological semimetals. We have demonstrated that both may be described in
terms of unquantized (i.e. having continuously-variable coefficients)
anomalies, which in turn may be connected to quantized topological response
terms of higher-dimensional SPT insulators, associated with the emergent
symmetries of the (semi)metal. The unquantized coefficients arise from tunable
charges of crystal symmetry gauge fields (e.g. crystal momentum in the case of
translational symmetry).

Such a common framework is useful, in part, because it emphasizes similarities
between ordinary and topological metals. In both cases the necessary existence
of gapless modes may be connected to topology, a viewpoint advocated early on
by Volovik Volovik (2003, 2007). It also compactly encodes the physical
properties of metals, which are directly connected to topology. This may be
viewed as a way to generalize Luttinger’s theorem to (almost) all bulk metals.

One concrete result along these lines is an improved understanding of nodal
line semimetals in three dimensions. In particular, we have shown that the
area, enclosed by the nodal line, determines the electric polarization of the
semimetal. By carefully defining the concept of polarization for such
semimetals, we have also clarified the anomalous nature of the “drumhead”
surface states of nodal line semimetals.

One specific type of a metal this scheme omits is the so-called type-II Dirac
semimetal Young  _et al._ (2012); Steinberg  _et al._ (2014); Yang and Nagaosa
(2014); Parameswaran  _et al._ (2013). The electronic structure of these
materials contains Dirac points at time reversal invariant momenta (TRIM) at
the edge of the BZ, terminating an axis of nonsymmorphic rotation Young  _et
al._ (2012); Steinberg  _et al._ (2014). A type-II Dirac semimetal fails to be
an insulator due to the nonsymmorphic nature of its point group. The only
continuously-tunable property this system has is the filling, since the Dirac
points are pinned to TRIM. However, the standard filling anomaly of a metal of
course vanishes in this case, since the filling is an integer. It is clear
that, since a nonsymmorphic nature of the space group plays a crucial role,
the corresponding anomaly term must involve the corresponding nonsymmorphic
symmetry gauge field. We leave an in-depth exploration of this to future work,
noting that some closely related ideas have been discussed in Refs. Watanabe
_et al._ (2015, 2016); Parameswaran (2019).

###### Acknowledgements.

CW, AH and XY acknowledge support from the Natural Sciences and Engineering
Research Council (NSERC) of Canada. AAB was supported by Center for
Advancement of Topological Semimetals, an Energy Frontier Research Center
funded by the U.S. Department of Energy Office of Science, Office of Basic
Energy Sciences, through the Ames Laboratory under contract DE-AC02-07CH11358.
Research at Perimeter Institute is supported in part by the Government of
Canada through the Department of Innovation, Science and Economic Development
and by the Province of Ontario through the Ministry of Economic Development,
Job Creation and Trade.

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