# Thermal transport, geometry, and anomalies

Maxim N. Chernodub  Institut Denis Poisson UMR 7013, Université de Tours,
Tours, 37200, France  Pacific Quantum Center, Far Eastern Federal University,
690950 Vladivostok, Russia  Yago Ferreiros  IMDEA Nanociencia, Faraday 9,
28049 Madrid, Spain  Adolfo G. Grushin  Univ. Grenoble Alpes, CNRS, Grenoble
INP, Institut Néel, 38000 Grenoble, France  Karl Landsteiner  Instituto de
Física Teórica UAM/CSIC,
Nicolás Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain  María A. H.
Vozmediano  Instituto de Ciencia de Materiales de Madrid,
CSIC, Cantoblanco, E-28049 Madrid, Spain.

###### Abstract

The relation between thermal transport and gravity was highlighted in the
seminal work by Luttinger in 1964, and has been extensively developed to
understand thermal transport, most notably the thermal Hall effect. Here we
review the novel concepts that relate thermal transport, the geometry of
space-time and quantum field theory anomalies. We give emphasis to the cross-
pollination between emergent ideas in condensed matter, notably Weyl and Dirac
semimetals, and the understanding of gravitational and scale anomalies
stemming from high-energy physics. We finish by relating to recent
experimental advances and presenting a perspective of several open problems.

###### Contents

  1. I Introduction
  2. II Key concepts
  3. III Thermal and electro-thermal transport
    1. III.1 Thermoelectric relations
    2. III.2 Luttinger theory of thermal transport coefficients
      1. III.2.1 Tolman–Ehrenfest pioneering work: Thermodynamic equilibrium in gravitational fields
    3. III.3 Kubo formulas for the thermal and thermoelectric conductivities
    4. III.4 Formal derivation of generalized Luttinger relations
  4. IV Transport and magnetization currents
    1. IV.1 General remarks
    2. IV.2 Luttinger, thermal transport and energy magnetization
  5. V Transport and chiral anomalies
    1. V.1 Anomalies in QFT
    2. V.2 Anomalies in matter
    3. V.3 Anomalies and magnetotransport in Weyl semi-metals
    4. V.4 Anomaly induced magneto-electric and -thermal transport for Weyl semi-metals
    5. V.5 Weak magnetic field
    6. V.6 Thermoelectric transport relations of the anomaly-induced transport coefficients
    7. V.7 Large magnetic field
    8. V.8 Concluding remarks on magneto-transport
  6. VI Axial gauge fields and the chiral anomaly.
    1. VI.1  Axial gauge fields in Weyl semimetals.
    2. VI.2 New anomaly induced transport phenomena in the presence of axial vector fields
    3. VI.3 Curvature and elastic deformations
  7. VII The scale anomaly and thermoelectromagnetic currents
    1. VII.1 Scale anomaly in quantum electrodynamics
    2. VII.2 Scale anomaly and transport effects in the bulk
    3. VII.3 Anomalous thermoelectric current due to the conformal anomaly
    4. VII.4 Scale electromagnetic effects at the boundary
  8. VIII Torsion and chiral transport
    1. VIII.1 Nieh-Yan current
    2. VIII.2 Absence of universal chiral torsional transport
  9. IX Experimental advances
  10. X Conclusions
  11. XI Acknowledgements
  12. A Velocities as Lagrange multipliers
  13. B Matter fields in a curved background
    1. B.1 Curved spaces without torsion
    2. B.2  Spaces with torsion
    3. B.3  The spin current and the energy-momentum tensor of spinor fields.
  14. C Details on conformal anomaly
    1. C.1 Conformal anomaly in curved background
    2. C.2 Scale electric effect
  15. D The formalism of gravitoelectromagnetism
  16. E Black Holes, chiral transport and gravitational anomaly
  17. F Notations

In fact, if the gravitational field didn’t exist,
one could invent one for the purposes of this paper.111From footnote 7 of J.M.
Luttinger ”Theory of thermal transport coefficients” Luttinger (1964).

##  I Introduction

The main conceptual advances in physics have usually been prompted by almost
simultaneous discoveries in its different disciplines. In the past century,
statistical physics, quantum field theory and condensed matter had their main
developments in parallel with the best physicists contributing to them all.
After a long period of atomization and specialization of the different
branches of physics, recent experiments and conceptual developments in
condensed matter physics are fostering a new era of grand unification of low
and high energy physics. The new era started with the experimental realization
of graphene, a two-dimensional crystal made of carbon atoms in 2004 Novoselov
_et al._ (2005); Zhang  _et al._ (2005). As it is well known, the dynamics of
the low energy electronic excitations in graphene is described by the massless
Dirac Hamiltonian in 2+1 dimensions and the interacting system shows
parallelisms with quantum electrodynamics Gonzalez  _et al._ (1994); Kotov
_et al._ (2012). More recently, Dirac and Weyl semimetals, 3D materials
described at low energies by the massless Dirac equation Jia  _et al._ (2016);
Armitage  _et al._ (2018), completed the picture providing new perspectives on
one of the most exciting aspects of QFT: Quantum anomalies Adler (1969); Bell
and Jackiw (1969); Kimura (1969); Bertlmann (1996); Fujikawa and Suzuki (2004)
and anomaly-induced transport phenomena Kharzeev (2014); Landsteiner (2016);
Burkov (2015).

Quantum anomalies were encountered and described in the earlier developments
of QFT associated to the physics of elementary particles. The language and
ideas at the time were based on concepts such as current algebra or the
analytic $S$ matrix, which involved rather sophisticated mathematics and that
have largely fallen into disuse. Despite the successful reviews adapting the
concepts to more modern language Bilal (2008); Harvey (2005), the QFT
literature seems to remain moderately inaccessible to a large fraction of
condensed matter researchers, both experimentalists and theoreticians.
Equivalently, the condensed matter description of topological matter and the
literature involving the anomaly-related transport experiments, use a
phenomenological language which is often difficult to integrate in the QFT
formalism. This is especially so in the case of the mixed axial–gravitational
anomaly and its relation to the thermal transport phenomena observed in Dirac
materials Gooth  _et al._ (2017); Vu  _et al._ (2021).

Quantum anomalies play an important role in high-energy physics Shifman
(1991). The anomalies help to describe experimental data in phenomenology of
fundamental particle interactions and constrain the space of physically self-
consistent models in quantum field theory (QFT). One of the best known
observational consequences of quantum anomalies is highlighted by the axial
anomaly which shortens the lifetime of a neutral pion by opening the
classically forbidden decay channel of this pseudoscalar particle into two
photons, $\pi^{0}\to 2\gamma$ Adler (1969); Bell and Jackiw (1969).

A different type of quantum anomaly appears in interacting field theories that
possess classical invariance under appropriate global rescaling of coordinates
and fields Peskin and Schroeder (1995). This scale symmetry implies the
equivalence of classical processes that develop at different energies since
the classical equations of motion contain no dimensionful parameters. The
symmetry is often broken by the quantum scale anomaly which generates, due to
quantum corrections, a dependence of the theory on the energy scale. The
existence of the conformal anomaly was first identified via its signatures in
correlation functions which signalled a nonzero expectation value of the trace
of the energy-momentum tensor in a purely curved (gravitational) background
Capper and Duff (1974a); Capper  _et al._ (1974). Later it was recognised that
the interactions of matter fields with classical electromagnetic fields also
lead to a conformal anomaly in flat space Deser  _et al._ (1976).
Perturbatively, the quantum scale anomaly gives rise to non-zero beta
functions that describe how the couplings of the model change with the energy
of the relevant process Shifman (1991). Nonperturbatively, the scale anomaly
can generate a mass gap which sets a new energy or length scale in the theory
Weinberg (1976). In the context of particle physics, both perturbative and
non-perturbative features of the scale anomaly are manifest, for example, in
Quantum Chromodynamics Savvidy (1977); Cornwall (1982). As we will see in this
review, the perturbative scale anomaly has also consequences for condensed
matter.

On the purely theoretical side, the presence of quantum anomalies allows to
narrow down the class of feasible gauge models of particle physics which are
consistent with the experimentally established conservation laws. The
anomalous contributions coming from all matter fields should mutually cancel
each other thus making a tight constraint on the particle content of the model
and the coupling of the particles with interaction carriers Bouchiat  _et al._
(1972). For example, the existence of gauge invariance, which is important for
the renormalizability of the theory, requires the cancellation of the
triangular diagrams Adler (1969); Bell and Jackiw (1969). Similar self-
consistency requirements imply the absence of the axial gauge fields in the
standard model of fundamental interactions.

Here is the place where condensed matter realizations of Dirac and Weyl
fermions go beyond the constraints of high energy physics. For example axial
gauge fields can be realized in Weyl metals by applying strain on the crystal
Cortijo  _et al._ (2015, 2016). This gives unique opportunities to
experimentally probe certain types anomalies related to these gauge fields
Pikulin  _et al._ (2016); Grushin  _et al._ (2016); Ilan  _et al._ (2020).
Similarly, in the realm of geometry, it is commonly accepted that torsion as a
dynamical field is not realized in nature. But in condensed matter systems
certain crystal defects enter the electronics of Weyl and Dirac metals as a
torsion field Kleinert (1989); Katanaev and Volovich (1992) whose coupling to
the fermions gives rise to new interesting physical phenomena Ran  _et al._
(2009).

The main purpose of this review is to present the basic notions underlying new
developments in condensed matter in a language equally accessible to both high
energy and condensed matter communities. The more technical presentations will
be complemented with an intuitive description that will permit to skip the
details keeping the physical ideas. The main concepts will be emphasized in
boxes at the end of each section. We also include a general introductory
section to ease the jumps between the different disciplines involved in the
topic. Thorough reviews are already available on the material implementation
of the chiral anomaly Burkov (2015); Landsteiner (2016); Marsh (2017), and on
the condensed matter systems subjected to these anomalies (Dirac and Weyl
semimetals) Jia  _et al._ (2016); Yan and Felser (2017); Armitage  _et al._
(2018). In this review we will focus on the gravitational and other geometric
anomalies that emerged more recently, and on the new transport phenomena that
they induce. Even though some of the topics included are still under debate in
either condensed matter or high energy physics (or both), the basic facts
around these geometrical anomalies are well established. Since these facts are
scattered in a variety of quite specialized contexts, it is worthy to put them
together.

The review is organized as follows. We begin in Sec. II by setting up a few
well established facts about the similarities and differences between the
condensed matter description and the quantum field theory counterpart
encountered in the field of anomaly–induced transport.

In Sec. III we fix the notation and some general notions of thermal transport.
The section includes the definition of the thermal and thermoelectric
responses, Wiedemann-Franz and Mott relations, and the Luttinger theory of
thermal transport coefficients.

In Sec. IV we describe the magnetization (electric and energy) currents that
may arise in the absence of time reversal symmetry and see how they affect the
interpretation of the Kubo formulas for the transport coefficients. We also
show how they are framed within the Luttinger formalism.

Section V is devoted to transport induced by triangle anomalies. We discuss
what is an anomaly in QFT (and what is not) and summarize the basic facts
around the chiral and the mixed chiral–gravitational anomalies. Then we expose
the main anomaly–related transport phenomena in general, and discuss in detail
their relation with magnetotransport in Dirac and Weyl semimetals. We also
show that the anomaly–induced thermoelectric transport coefficients obey the
Wiedemann–Franz and Mott relations.

In section VI we introduce the axial gauge fields arising in condensed matter
and describe the new anomaly induced phenomena associated to them.

In Sec. VII, we analyze the physical consequences of the scale anomaly. We
introduce the concepts and the consequences of the anomaly to transport
phenomena both in the bulk and in the boundary of the physical systems. We
also discuss the thermomagnetic transport effects produced by the scale
anomaly in Dirac and Weyl semimetals.

Sec. VIII is devoted to the potentially new anomaly–induced transport
phenomena that arise when considering the torsion degrees of freedom that
occurs naturally in crystalline materials associated to dislocations of the
lattice. We will describe the basic formalism and give an alternative
perspective to some of the works encountered in the literature. In particular
we review in what sense there are no mixed axial–torsional anomalies in the
strict quantum field theory sense, and that there is no universal chiral
transport as a response to geometric torsion.

In Sec. IX we give an overview of the experimental situation. We review the
recent experimental efforts to measure the mixed axial-gravitational anomalies
in thermo-electric and thermal transport in Weyl semimetals in a magnetic
field.

We include various appendices that give technical support to the material
covered in the review. We have included boxes to highlight some of the main
conclusions of each section. In the last Appendix we review our notation for
different quantities.

Finally, let us comment on certain fascinating related issues left aside from
this review. Since we address phenomena related to quantum field theory
anomalies common to high energy and condensed matter physics, we have reduced
our condensed matter systems to the topological materials whose low energy
effective description matches the Dirac equation in three spatial dimensions
describing, basically, Dirac and Weyl semimetals. This leaves aside other
types of “Dirac matter” as nodal Bian  _et al._ (2016) or multi-Weyl
semimetals Huang  _et al._ (2016), Kane fermions Teppe  _et al._ (2016),
topological insulators and superconductors Ryu  _et al._ (2012). We have also
omitted the very interesting analogs Fransson  _et al._ (2016); Zhang  _et
al._ (2018); He  _et al._ (2018); Nomura  _et al._ (2019); Peri  _et al._
(2019); Jia  _et al._ (2019) and gapped quantum materials Levin  _et al._
(2021).

##  II Key concepts

The anomaly related transport phenomena encountered in modern condensed matter
systems are sourced in different areas of physics which have their own terms
to describe them. This broadness testifies to the richness of the field, but
it also nourishes misunderstandings. Same phenomena are called different
names, and different phenomena can be mistaken as the same. An additional
difficulty lies on the fact that the approximations assumed in the different
areas of physics are often not explicitly established and their regime of
validity can be incompatible with the experimental conditions. Since this
review aims to provide a common basis for the understanding of how anomalies
evolved from high energy to condensed matter, we begin with an attempt to
clarify the main concepts participating in the field.

In this section we will establish the similarities and differences between the
quantum field theory description of chiral fermions as relativistic particles
and the condensed matter “Dirac fermions”.

The mere idea of the name “Dirac fermions” is somehow astonishing in particle
physics; what else can they be? Fermions $\equiv$ Dirac spinors. In condensed
matter the electrons are fermions in the sense that they obey the Pauli
exclusion principle, and the distribution function used in the statistical
averages is the Fermi-Dirac distribution. In the standard model for metals,
the Landau Fermi liquid Landau (1957), the electronic dynamics is described by
the Hamiltonian $H=p^{2}/2m$.

Dirac dynamics became popular in condensed matter with the advent of graphene
since the low-energy description of the electronic degrees of freedom
(quasiparticles) around band crossings (point-like Fermi surface and linear
dispersion relation) was the (2+1) Dirac equation describing massless
fermions. In the graphene case, Dirac physics is not related to the physical
spin of the electrons but to a flavor degree of freedom coming from the
structure of the underlying ionic lattice (pseudospin). So, even when
$H=v_{F}{\vec{\sigma}}\cdot{\vec{p}}$, still these fermions are spinless and
the spin is added by doubling the degrees of freedom. Also graphene lives in
(2+1) dimensions where the chirality operator is not defined, so the fermions
are characterized by parity, but not chirality. The advent of Dirac and Weyl
semimetals in (3+1) dimensions brought the condensed matter fermions closer to
their QFT partners but the concepts remain sometimes mixed and obscure since,
again, the ”Diracness” there often arises associated to a pseudospin.

In what follows we will describe and define some of the most important
concepts involved in this review222A very interesting description of Dirac
matter from a quantum field theory point of view is given by E. Witten in
Witten (2016)..

  * •

The Noether theorem and quantum anomalies

According to the first Noether theorem333 See ref. Quigg (2019) for a
historical review., continuum symmetries in a Lagrangian system generate
conserved currents and charges. Well known examples of external symmetries are
space (time) translations giving rise to momentum (energy) conservation or
rotational invariance generating conserved angular momentum in classical and
quantum mechanics. Perhaps more interestingly, internal symmetries as phase
(gauge) rotations of the wave function in quantum mechanics give rise to
particle current and particle number conservation - electric charge
conservation if the particles are charged -. In the process of quantization,
classically conserved currents are promoted from vector fields to composite
local operators - several quantum fields located at the same point - which
often need to be regularized. An anomaly occurs when you cannot find a
regulator that retains all the symmetries of the classical action. The best
example is the case of a massless Dirac system which will be discussed in the
next paragraph. As we will see, the system has two classically conserved
currents eqs. (6), (7) associated to gauge invariance and chiral symmetry. Due
to the quantum (anti) commutation relation
$\\{\Psi_{\alpha}(x),\Psi^{+}_{\beta}(y)\\}=\delta_{\alpha\beta}\delta^{4}(x-y)$,
with two quantum fields located at the same point the currents need to be
regularized. It so happens that no regularization procedure can keep both
currents conserved. In this simplest case of having two $U(1)_{V}$ and
$U(1)_{A}$ currents, we can choose to keep the vector current conserved, in
which case the divergence of the axial current turns out to be Bertlmann
(1996):

$\partial_{\mu}J_{A}^{\mu}=\frac{1}{4\pi^{2}}{\vec{E}}\cdot{\vec{B}},$ (1)

where ${\vec{E}},{\vec{B}}$ are background electromagnetic fields. We absorb
electric charge $e$ into the definition of the electromagnetic fields,
$\vec{E}$ and $\vec{B}$, and restore it when it is convenient.

  * •

Chirality and Weyl fermions

The concept of chirality is clear in particle physics. Fermion fields
transform as a spinor representations of the Lorentz group, they have an
inherent spin 1/2 (or semi–integer in general, but we will restrict ourselves
to spin 1/2), and their dynamics is described by

${\cal L}=\bar{\Psi}(i\gamma^{\mu}\partial_{\mu}-m)\Psi,$ (2)

where $\partial_{\mu}\equiv\partial/\partial x^{\mu}$ is the coordinate
derivative, $\gamma^{\mu}$ are (3+1) dimensional Dirac matrices satisfying

$\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=-2\eta^{\mu\nu}{\cal I}_{4},$ (3)

and ${\cal I}_{N}$ is a $N\times N$ unit matrix in the spinor space. The Dirac
equation

$(i\gamma^{\mu}\partial_{\mu}-m)\Psi=0,$ (4)

can be written as the Schrodinger equation $i\partial_{t}\Psi=H\Psi$ with
$H=\beta m+{\vec{\alpha}}\cdot{\vec{p}}$, where $\beta=\gamma^{0}$ and
$\vec{\alpha}=\beta.\vec{\gamma}$.

In odd space dimensions a $\gamma^{5}$ matrix exists that anti-commutes with
all the $\gamma^{\mu}$ matrices and defines the chirality projection operator
$P_{L,R}=(1\pm\gamma^{5})/2$.

In the Weyl representation of the gamma matrices:

$\gamma^{0}=\left(\begin{array}[]{cc}0&{\cal I}_{2}\\\ {\cal
I}_{2}&0\end{array}\right)\,,\qquad\gamma^{i}=\left(\begin{array}[]{cc}0&\sigma^{i}\\\
-\sigma^{i}&0\end{array}\right)\,,\qquad\gamma_{5}=\left(\begin{array}[]{cc}{\cal
I}_{2}&0\\\ 0&-{\cal I}_{2}\end{array}\right)\,,$ (5)

and when $m=0$, the Dirac Hamiltonian splits into two bi-dimensional blocks
$H_{R,L}=\pm{\vec{\sigma}}\cdot{\vec{p}}$. The eigenfunctions are two
dimensional spinors of well defined chirality: $\Psi_{L,R}=P_{L,R}\Psi$. These
are the Weyl fermions.

One way to characterise the eigenstates is by noting that the positive energy
states of right-handed fermions have their (pseudo)spin aligned with their
momentum, whereas for left-handed fermions the (pseudo)spin of positive energy
states is anti-aligned with the momentum. In high-energy physics positive
energy states are identified as particles whereas negative energy states are
re-interpreted as anti-particles. In condensed matter physics anti-particle
states are referred to as holes. The Nielsen-Ninomiya theorem Nielsen and
Ninomiya (1981, 1983a) ensures that chiral states in a lattice always arise in
pairs of opposite chiralities.

As a consequence of chiral symmetry, the action of a massless 3+1 fermion is
invariant under independent phase rotations of left and right fermions. We can
define two currents that are conserved at the classical level:

$\displaystyle J_{V}^{\mu}(x)$ $\displaystyle=$
$\displaystyle\bar{\Psi}(x)\gamma^{\mu}\Psi(x)=J_{L}+J_{R}\,,$ (6)
$\displaystyle J_{5}^{\mu}(x)$ $\displaystyle=$
$\displaystyle\bar{\Psi}(x)\gamma^{\mu}\gamma^{5}\Psi(x)=J_{L}-J_{R}\,.$ (7)

The vector $J_{V}$ current is associated to gauge invariance (conservation of
the total electric charge), and the axial $J_{5}$ current is associated to the
chiral symmetry which leads, at the classical level, to conservation of the
difference between the numbers of right- and left-handed fermions.

Figure 1: Schematic dispersion relation of graphene. The Fermi level is
located at the band crossings and the energy is bounded defining a compact
Fermi sea.

Massless Dirac fermions appear in condensed matter as the low energy effective
description of electronic systems whose Fermi level sits near a two-band
crossing. The Nielsen-Ninomiya theorem Nielsen and Ninomiya (1981) ensures
that these crossings appear always in pairs of opposite chiralities. Real
crystals typically host more than one pair of chiral fermions, usually close
to high-symmetry points or planes. Restricting to a single pair, Dirac semi-
metals are those three-dimensional crystals where the two chiralities sit at
the same point in energy-momentum space (as massless Dirac particles). To
realize a Weyl semimetal (isolated Weyl nodes) either inversion or time-
reversal (or both) need to be absent, which allows the two chiralities to be
separated in either energy or momentum (or both). A detailed description of
these conditions and materials that realize them can be found in several
recent reviews Jia  _et al._ (2016); Yan and Felser (2017); Armitage  _et al._
(2018).

Fig. 1 shows the band structure of graphene, the material which made “Dirac
fermions” popular in condensed matter. There are two inquivalent momenta, or
”valleys”, in the Brillouin zone where the dispersion is linear, marked as L,
R in the figure. These are refered to as massless Dirac points. As mentioned
in the introduction of this section, being a two dimensional material, the two
valleys of graphene are not Weyl fermions. The dispersion relation shown is
visually similar to the band structure of a (say, constant $k_{z}$) cut of a
three dimensional Weyl semimetal. As seen in Fig. 1, the bands of the two
valleys are connected in the Brillouin zone.

In condensed matter realizations, in addition to the two band crossings
described by the Dirac or Weyl equation (linear dispersion relation), the name
of Dirac matter may be used to refer to other systems described by more exotic
equations involving also Dirac matrices. Most noticeable are multi-Weyl
fermions with different dispersion relations along different symmetry axis,
higher-order polynomial dispersion relations, nodal Weyl semimetals which host
a line of nodal points, and multifold fermions, which have point degeneracies
of multiple bands Has (2021) modelled by higher-spin generalizations of Weyl
fermions. The phenomena described in this review do not necessarily apply to
these systems, defining in turn an interesting research frontier. In what
follows, when we refer to Dirac matter we mean systems described by the QFT
massless Dirac equation with linear dispersion relation.

  * •

The vacuum

The QFT vacuum refers to fields at zero temperature $T$ and chemical potential
$\mu$. Originally, anomalies were formulated in vacuum and the role at finite
temperature $T$ and chemical $\mu$ has only been fully appreciated recently.
In condensed matter realizations, $T=0$ and $\mu=0$ are fined tuned, possibly
unrealistic, limits. Most importantly, the QFT fermionic vacuum is the Dirac
sea, an infinite tower of negative energy occupied levels. In contrast, the
number of occupied electronic states below the Fermi surface (the condesned
matter vacuum), is always finite (see Fig. 1). So in the QFT intuitive
description of the chiral anomaly, left (right) particles are annihilated and
created from the vacuum. Strong enough fields can also create pair of states
from the vacuum as in the Schwinger mechanism Schwinger (1951). In the
condensed matter realization, the two chiralities are connected through the
bottom of the band (see Fig. 1) so the “spectral flow” of QFT Landsteiner
(2016) reduces to a shift of particles in the same band.

  * •

A note on Lorentz invariance

The Dirac equation (4) assumes the invariance of the theory under the group of
Lorentz transformations, which include boosts and spatial rotations as well as
time-reversal and parity transformations. The Lorentz invariance reflects
itself in the form of the kinetic term,
$\gamma^{\mu}\partial_{\mu}\equiv\gamma^{0}\partial_{t}+c{\vec{\gamma}}\cdot{\vec{\nabla}}$,
where $c$ is the speed of light. Since both the massless fermions and photons
propagate with the same velocity $c$, the full interacting theory, which
includes the kinetic terms for photons and particles, is invariant under
Lorentz boosts.

In Dirac and Weyl semimetals, however, the quasiparticle excitations propagate
with the Fermi velocity $v_{F}$, which is much slower than the speed of light.
The quasiparticles are described by the same Dirac equation (4) which is,
however, modified by the substitution $c\to v_{F}$ in the derivative term:
$\gamma^{\mu}\partial_{\mu}\to\gamma^{0}\partial_{t}+v_{F}{\vec{\gamma}}\cdot{\vec{\nabla}}$,
where we assumed for a moment the isotropy and homogeneity of the spacetime as
seen by the low-energy quasiparticles. Due to the inequivalence of the
velocities of matter fields and interaction carriers, $v_{F}\neq c$, the
kinetic term of quasiparticles (2) loses the invariance under the Lorentz
transformations.

The Lorentz symmetry breaking in condensed matter enriches the physics of the
Dirac quasiparticles without spoiling the self-consistency of the appropriate
low-energy models. We refer to Refs. Armitage  _et al._ (2018); Grushin (2019)
for a more detailed discussion on this topic which is touched in our review
only in bypassing.

  * •

The Bloch theorem

Intuitively a (transport) current moves ”something” from a place A in space to
another place B. To make this statement a bit more meaningful we also demand
that the ”something” is stable and does not decay in the timespan in which the
transport process takes place. A current transports therefore (approximately)
conserves charges. Energy and electric charge are exactly conserved, spin or
axial charge might be only approximately conserved. If the physical system
under consideration is in its lowest energy state, e. g. is in thermodynamic
equilibrium, one would not expect such motion to be possible, since this would
be a realization of a form of a perpetuum mobile. This intuition is formalized
in a theorem444One should not confuse this non-current theorem with a better
known Bloch theorem on the quasimomentum description of solutions of
Schrödinger equation in a periodic potential. stated by Felix Bloch in the
1930s which was later summarized by D. Bohm in Ref. Bohm (1949). Originally it
states that it is not possible for a condensed matter system residing in a
thermodynamic limit to support an electric current in equilibrium Bohm (1949).
A more modern formulation states that the net current corresponding to an
exactly conserved $U(1)$ charge has to vanish in thermodynamic equilibrium. A
quick argument goes as follows. The coupling of the current in the Hamiltonian
is

$H_{\mathrm{int}}=\int d^{3}r\vec{A}\cdot\vec{J}\,.$ (8)

For an exactly conserved $U(1)$ charge the field $\vec{A}$ is a gauge field.
This means that a spatially constant and time independent value of the gauge
potential $\vec{A}$ should have no physical consequence. If the current has
however an expectation value $\langle{\vec{J}}\rangle\neq 0$ then an
infinitesimally small variation of the gauge potential $\delta{\vec{A}}$
results in the energy change

$\langle\delta H_{\mathrm{int}}\rangle=\delta{\vec{A}}\cdot\int
d^{3}x\langle\vec{J}\rangle,$ (9)

which can be negative and thus would allow to lower the energy of the system.
This contradicts the assumption that the system is in the lowest energy state.
It follows that for an exactly conserved $U(1)$ current

$\int d^{3}r\,\vec{J}(\vec{r})=0\,.$ (10)

We note that this condition does not rule out circular (persistent)
magnetization currents nor does it rule out superconducting currents. In the
former case the system resides in a finite (topologically equivalent to a
ring) volume far from the thermodynamic limit. The geometry of the system
imposes a quantization constraint on the gauge field ${\vec{A}}$ via the
Aharonov-Bohm phase which does not allow for infinitesimally small variations
of the gauge potential $\delta{\vec{A}}$. Therefore, the Bloch no-current
theorem does not work for persistent currents. In a superconductor, the gauge
symmetry is spontaneously broken and a state with a non-vanishing supercurrent
can only be a metastable state.

On the other hand if the current is not exactly conserved or is affected by an
anomaly the vector $\vec{A}$ is strictly speaking not a gauge field but a
physical observable by itself. Therefore the system with $\vec{A}\neq 0$ is
not equivalent to the ground state in the system $\vec{A}=0$ but rather a
different physical system with a new Hamiltonian. Therefore the Bloch theorem
does not apply in this situation and equilibrium currents are possible for
this type of currents. A recent quantum field theory discussion of the Bloch
theorem can be found in Yamamoto (2015), and a modern formulation for lattice
Hamiltonians is in Watanabe (2019). A generalization of the Bloch theorem to
the energy current of Hamiltonian lattice systems has been given in Kapustin
and Spodyneiko (2019).

• The massless Dirac Hamiltonian in odd space dimensions, (Dirac semimetal)
may split into two Weyl Hamiltonians (Weyl semimetal). The two Weyl
eigenspinors have well defined, and opposite chiralities. • There is no chiral
anomaly in even spacial dimensions (graphene). • The Bloch theorem (no
transport currents in equilibrium in the thermodynamic limit) applies only to
exactly conserved currents associated with the gauge symmetry.

##  III Thermal and electro-thermal transport

Thermal transport refers generically to transport of energy, charge, or any
other property under the effect of a gradient of temperature. In metals most
transport phenomena are dominated by the free electrons that conduct energy,
and charge. In principle phonons can also contribute to the thermal
conductivity. Since we are mainly interested in the thermal transport
properties of fermions (electrons) described by the Dirac equation we will not
consider the phonon contribution. For standard metals describable as Fermi
liquids, the thermal conductivity is proportional to the electric conductivity
(Wiedemann-Franz law). Generically this can be seen as a sign of the existence
of long lived quasi-particles. When these quasi particles obey a Dirac or Weyl
equation they are subject to the effects of anomalies. As we will see in
section V the Wiedemann-Franz law holds for the anomaly induced conductivities
in the presence of a magnetic field (magneto–thermal transport).

We start from the phenomenological equations describing electric and energy
transport as a response to electric fields and gradients of chemical potential
and temperature Luttinger (1964)

$\displaystyle{\vec{J}}_{\textrm{tr}}=\bm{L}^{(1)}\left({\vec{E}}-T{\vec{\nabla}}\frac{\mu}{T}\right)+\bm{L}^{(2)}T{\vec{\nabla}}\frac{1}{T},$
(11)
$\displaystyle{\vec{J}}_{\textrm{tr}}^{\,\mathcal{E}}=\bm{L}^{(3)}\left({\vec{E}}-T{\vec{\nabla}}\frac{\mu}{T}\right)+\bm{L}^{(4)}T{\vec{\nabla}}\frac{1}{T}.$
(12)

The currents $\vec{J}_{tr}$ and $\vec{J}^{\mathcal{E}}_{tr}$ are the particle
current and energy current respectively, with the particle current related to
the electrical current as $\vec{J}^{e}_{tr}=-e\vec{J}_{tr}$ ($-e$ is the
electron charge). The tensors $\bm{L}^{(i)}$ are the transport coefficients,
and $\vec{E}$ is the electric field. The Onsager relations Onsager (1931)
relate the second and third transport coefficients as
$\bm{L}^{(2)}=\bm{L}^{(3)}$, or $L^{(2)}_{ij}(B)=L^{(3)}_{ji}(-B)$ in the
presence of a magnetic field (see Sec. IV). The first terms inside the
parenthesis in Eqs. (11) and (12) represent the Einstein relation Sutherland
(1905); Einstein (1905); von Smoluchowski (1906); Luttinger (1964), which
states that in equilibrium an electric field is compensated by an opposing
gradient of chemical potential and temperature $\vec{E}=T\vec{\nabla}(\mu/T)$.
The energy current ${\vec{J}}_{\textrm{tr}}^{\,\mathcal{E}}$ results from the
combination of heat current $\vec{J}^{Q}$ and energy transported by the
particle current ${\vec{J}}_{\textrm{tr}}$; the heat current can therefore be
written as

${\vec{J}}_{\textrm{tr}}^{\,Q}={\vec{J}}_{\textrm{tr}}^{\,\mathcal{E}}-\mu{\vec{J}}_{\textrm{tr}}.$
(13)

The currents in Eqs. (11) and (12) are transport currents which means they
represent net transport through a cross section of the given material. As
such, they should vanish in equilibrium. It is important to carefully
distinguish between the transport currents $\vec{J}_{\textrm{tr}}$ relevant to
Eqs. (11) and (12), and the total local currents $\vec{J}$ at any given point
in the sample Smrcka and Streda (1977); Cooper  _et al._ (1997); Bradlyn and
Read (2015). While in time-reversal invariant systems these coincide, breaking
time-reversal symmetry can lead to the appearance of magnetization electric
and energy currents. These are defined as divergenceless circulating currents
which average to zero and therefore do not carry any transport Cooper  _et
al._ (1997). We will discuss the presence of magnetization currents in more
detail in Sec. IV.

Let us now fix the chemical potential and allow for gradients of temperature,
and consider a sample disconnected from current leads such that no net
electric current flows. In this situation, the transport electric current
vanishes and an electric field is generated. From the transport Eqs. (11) and
(12), and using Eq. (13) we have

$\displaystyle{\vec{E}}={\bf S}{\vec{\nabla}}T,$ (14)
$\displaystyle{\vec{J}}^{Q}_{\textrm{tr}}=-{\bf K}{\vec{\nabla}}T,$ (15)

where the thermopower $\bm{S}$ (or Seebeck coefficient) and thermal
conductivity $\bm{K}$ tensors are functions of the transport coefficients:

$\displaystyle\bm{S}=\frac{1}{T}\big{[}\bm{\rho}\bm{L}^{(2)}-\mu\big{]},$ (16)
$\displaystyle\bm{K}=\frac{1}{T}\big{[}\bm{L}^{(4)}-\bm{L}^{(3)}\bm{\rho}\bm{L}^{(2)}\big{]}.$
(17)

where $\rho$ is the resistivity tensor $\bm{\rho}=\bm{\sigma}^{-1}$, and we
have renamed $\bm{L}^{(1)}\equiv\bm{\sigma}$, which is the most established
notation for the electric conductivity.

Eq. (14) gives rise to the Seebeck effect, which is an expression of the
longitudinal components of (11): generation of an electric potential in the
direction of a temperature gradient.

|
---|---
(a) | (b)
Figure 2: A system consisting of a sample (a) connected to and (b)
disconnected from current leads, in an external magnetic field, attached to
two thermal reservoirs at different temperatures such that a thermal gradient
is generated through the sample. (a) A transport current flows in the
direction perpendicular to both the magnetic field and gradient of
temperature, corresponding to the Nernst current, Eq. (18); (b) An electric
current cannot flow, so the electric fields $E_{x}$ and $E_{y}$ are generated
in the plane perpendicular to the magnetic field direction, corresponding to
the Nernst-Ettingshausen (20) and Nernst (19) coefficients respectively.

The Nernst-Ettingshausen effects occur in the presence of a magnetic field.
The Nernst effect refers to the generation of an electric current
perpendicular to a temperature gradient and to the applied magnetic field,
Fig. 2(a):

$\displaystyle\vec{J}\propto\vec{B}\times\vec{\nabla}T.$ (18)

For a sample disconnected from current leads, a compensating electric field
must necessarily appear such that the transport current vanishes. This
situation is schematically described in Fig. 2. If we choose the gradient of
temperature to point, say, along the $\hat{x}$ direction, $\nabla_{x}T$, and
the magnetic field $\vec{B}$ to point along $\hat{z}$, two coefficients are
usually defined: the Nernst coefficient

$\displaystyle{\cal
N}_{yx}\equiv\frac{E_{y}}{B_{z}\partial_{x}T}=\frac{\rho_{yi}L^{(2)}_{ix}}{B_{z}}.$
(19)

and the Ettingshausen-Nernst coefficient:

$\displaystyle{\cal
N}_{xx}\equiv\frac{E_{x}}{B_{z}\partial_{x}T}=\frac{\rho_{xi}L^{(2)}_{ix}-\mu}{B_{z}},$
(20)

see Fig. 2(b).

###  III.1 Thermoelectric relations

In the standard model for metals, the Fermi liquid, the thermo–electric
coefficients in Eqs. (11),(12) obey certain phenomenological relations, the
main ones being the Wiedemann-Franz law and the Mott relation Ziman (1960);
Jonson and Mahan (1980):

$\displaystyle\bm{K}=LT\bm{\sigma}(T=0),$ (21)
$\displaystyle\bm{S}=LT\bm{\rho}(T=0)\frac{d\bm{\sigma}(T=0)}{d\mu}{\biggl{|}}_{\mu=\epsilon_{F}/e}.$
(22)

The Wiedemann-Franz law (21) establishes that the ratio of the thermal to the
electrical conductivity is the temperature times a universal number, the
Lorenz number $L=\pi^{2}k_{B}^{2}/3e^{2}$, where $k_{B}$ is the Boltzmann
constant. The Mott relation (22) relates the thermopower with the derivative
of the electrical conductivity with respect to the chemical potential at the
Fermi level. Note that the electric conductivity in Eqs. (21) and (22) is
evaluated at zero temperature, as both the Mott rule and the Wiedemann-Franz
law are obtained as an expansion in $k_{B}T\ll|\mu|$. The Mott and Wiedemann-
Franz relations hold for transport currents only Smrcka and Streda (1977);
Jonson and Girvin (1984); Cooper  _et al._ (1997); Qin  _et al._ (2011), which
means that in systems that break time-reversal symmetry the local currents do
not generally fulfill them (see Sec. IV). The validity of these laws has been
established for any system which can be described as a Fermi liquid, provided
the quasiparticles do not exchange energy during collisions. The validity has
also been proven to be valid when a semi-classical description (as the
Boltzmann formalism) of the electronic system is justified. Deviations from
these phenomenological relations in conventional matter are normally
attributed to electron–electron interactions inducing departures from Fermi
liquid behavior or to the emergence of a new phase regime. In particular the
relations might not apply when the systems are in the hydrodynamic regime.

Topological materials have anomalous conductivities (particularly the Hall
conductivity) similar to that occurring in ferromagnetic materials, induced by
the Berry curvature of the bands. The question of whether or not these
anomalous transport coefficients obeyed the Wiedemann-Franz and Mott
relations, arose soon after the recognition of the topological properties.
Berry curvature effects were included in the Boltzmann transport formalism in
a way to fulfil the rules Xiao  _et al._ (2006); Ryu  _et al._ (2012), but the
experimental situation is more uncertain and violations of the thermoelectric
relations have been reported in some materials Pu  _et al._ (2008); Kim
(2014); Liang  _et al._ (2014); Li  _et al._ (2015).

Typically Dirac materials in two and three dimensions are expected to follow
the standard relations in the low $T$ regime and deviate from it at larger
temperatures Liang  _et al._ (2017); Gorbar  _et al._ (2017a); Jaoui  _et al._
(2018). Violations of the Wiedemann-Franz law have been described in these
systems associated to the presence of a hydrodynamic regime where departures
from the standard Fermi liquid behavior are also found Lucas  _et al._ (2016);
Gooth  _et al._ (2018). In sec. 4 we will see that the anomaly-related
transport coefficients fulfil these relations although a different claim has
been made recently in Ref. Das and Agarwal (2020a).

###  III.2 Luttinger theory of thermal transport coefficients

Transport coefficients are computed in the framework of linear response
theory: the currents are obtained as a response to an applied field, usually
by the derivation of the corresponding Kubo formulas. The electrical current
results from the response to an applied electric field, which, if the given
system is in equilibrium, is compensated by a gradient of chemical potential:
$\vec{E}=\vec{\nabla}\mu$. This relation, the Einstein relation, is what sets
the link between linear response to a field (electric field) and the response
to a statistical force (the gradient of the chemical potential) (see Eq.
(11)). It permits to compute the response to a statistical force through
linear response theory.

A question therefore naturally arises in the context of thermal transport:
what is the “thermal field” needed to develop a linear response theory of
thermal transport? What is the equilibrium relation for the other statistical
force, the gradient of temperature? In 1964, Luttinger solved these problems
by showing that the gravitational field can play the role of the “thermal
field” Luttinger (1964). By coupling the gravitational field to the electronic
degrees of freedom he developed the linear response theory for thermal
transport. In this section we review and contextualize Luttinger’s theory and
ideas.

####  III.2.1 Tolman–Ehrenfest pioneering work: Thermodynamic equilibrium in
gravitational fields

The earliest link that opened the door to relate gravitational effects with
thermal transport in condensed matter systems was set up by the works of
Tolman and Ehrenfest in 1930 Tolman (1930); Tolman and Ehrenfest (1930) when
trying to adapt the thermodynamic relations to general relativity. The
underlying idea that “heat has weight” let them to conclude that, in a
gravitational field, thermal equilibrium must be accompanied by a temperature
gradient in order to prevent the flow of heat from regions of higher to those
of lower gravitational potential. The basic equation defining equilibrium in a
curved background was set to be

$\displaystyle T\sqrt{g_{tt}({\vec{r}})}={\rm constant}.$ (23)

In a weak gravitational field $\phi$,

$\displaystyle g_{tt}({\vec{r}})=1+2\phi({\vec{r}})/c^{2},$ (24)

where we restore $c$ as the speed of light, we get the relation

$\displaystyle\frac{1}{T}{\vec{\nabla}}T=-\frac{1}{c^{2}}{\vec{\nabla}}\phi\,.$
(25)

This relation was first obtained by Tolman Tolman (1930) for a spherical mass
distribution and it was later generalized by Tolman and Ehrenfest Tolman and
Ehrenfest (1930) to general -static- gravitational fields. As a consequence,
the temperature variation in a gravitational field inducing the acceleration
$g$ was

$\frac{1}{T}\frac{dT}{dr}=-\frac{g}{c^{2}}.$ (26)

The estimate of the temperature variation at the surface of the earth given by
Tolman in Ref. Tolman (1930) was

$\frac{1}{T}\frac{dT}{dr}=-\frac{g}{c^{2}}\sim-10^{-18}{\mathrm{cm}}^{-1}.$
(27)

Despite the small magnitude of the effect, the underlying ideas provided the
basis for the Luttinger theory of thermal transport widely used in condensed
matter Luttinger (1964).

The relation (25) defining thermodynamic equilibrium in gravitational fields
with specific distributions of matter, was later extended to the presence of a
finite chemical potential Klein (1949). This issue have been revised in recent
works Santiago and Visser (2018, 2019); Lima  _et al._ (2019). We will see a
formal derivation in Sec. III.4.

###  III.3 Kubo formulas for the thermal and thermoelectric conductivities

The previous ideas were used by Luttinger to derive a Kubo formula for the
thermal transport coefficient Luttinger (1964). In linear response theory
Giuliani and Vignale (2005), the transport coefficients can be written in
terms of the retarded current-current two-points function. A common example is
the expression of the electrical conductivity Kubo (1957):

$J^{i}=\sigma^{ij}E_{j}\quad,\quad E_{j}=i\omega A_{j};$ (28)
$\sigma^{ij}(\omega)=\frac{1}{i\omega}\langle J^{i}J^{j}\rangle(\omega,k=0).$
(29)

The Kubo formula is based on the possibility of adding a local source
($A_{\mu}$) to the current ($J^{\mu}$) in the action in the standard way:

$S[J]=S_{0}-\int d^{4}xJ^{\mu}A_{\mu},$ (30)

so that

$J^{\mu}=-\frac{\delta S}{\delta A_{\mu}}.$ (31)

The difficulty to apply this approach to thermal transport lies in finding the
appropriate local source coupling to heat (or energy) density. Early attempts
were made by considering macroscopic thermodynamic variables as local
variables by dividing the system in small portions and making some assumptions
about how these variables develop in time Green (1952); Kubo  _et al._ (1957);
Mori (1958).

Luttinger Luttinger (1964) aimed to give a mechanistic sustain to what he
called Green-Kubo-Mori formulas. Proceeding by analogy with the electrical
conductivity, he proposed the introduction of a (fictitious) inhomogeneous
gravitational field $\phi(r)$ which causes energy or heat currents to flow as
a response to gradients of $\phi(r)$. This formalism is clear in special
relativity, where the energy momentum tensor of a generic matter field $\phi$
is the conserved current associated to metric variations due to Lorentz
invariance:

$T^{\mu\nu}(\Phi)=-\frac{2}{\sqrt{-g}}\frac{\delta S[\Psi,g]}{\delta
g_{\mu\nu}},$ (32)

where $\Psi$ represents matter fields. In particular, the energy current is
given by the components $J_{\mathcal{E}}^{i}=T^{0i}$. For small deviations
from flat space the gravitational potential $\phi$ is proportional to the
zero-zero component of the metric: $g^{00}=1+2\phi$, which couples to the
energy density $T^{00}$. Here we follow Luttinger (1964) and absorbed the
velocity $c$ into the gravitational potential $\phi\rightarrow c^{2}\phi$.
This rescaled gravitational potential is simply the field that couples to the
energy density. The perturbed Hamiltonian to be used in the linear response
formalism is Luttinger (1964)

$H_{\mathrm{int}}(t)=\int d^{3}r\;h({\vec{r}},t)\phi(t,\vec{r}),$ (33)

where $h({\bf r},t)$ is the Hamiltonian density555This follows since the
metric variation $\phi$ couples to the energy density $T^{00}$. On the other
hand the operator measuring the energy density is (equivalent to) the
Hamiltonian density. For example for a Dirac field
$T^{00}=i\bar{\psi}\gamma^{0}\partial^{0}\psi$ which is equivalent to the
Hamiltonian density once the equations of motion are imposed on the field
operator $\psi$.. We can write the coefficients for the local current
densities as responses to the external fields $A_{0}$ and $\phi$

$\displaystyle\vec{J}={\bm{\mathcal{L}}}^{(1)}\vec{E}-{\bm{\mathcal{L}}}^{(2)}\vec{\nabla}\phi,$
(34)
$\displaystyle\vec{J}^{\mathcal{E}}={\bm{\mathcal{L}}}^{(3)}\vec{E}-{\bm{\mathcal{L}}}^{(4)}\vec{\nabla}\phi,$
(35)

which can in turn be written in terms of correlations of the currents. Note we
use ${\bm{\mathcal{L}}}^{(i)}$ in Eqs. (34), (35) to denote the coefficients
that determine the local currents obtained from Kubo formulas, and
$\bm{L}^{(i)}$ in Eqs. (11), (12) to denote the coefficients that determine
transport currents. The tensors ${\bm{\mathcal{L}}}^{(i)}$ and $\bm{L}^{(i)}$
may differ if magnetization currents are present, which are discussed in
section IV.2. The thermal and thermoelectric coefficients
${\bm{\mathcal{L}}}^{(2)}$ and ${\bm{\mathcal{L}}}^{(4)}$ can be computed as
Luttinger (1964)

${\mathcal{L}}^{(2)}_{ij}=\lim_{\omega\to 0}\frac{1}{i\omega}\langle
J^{\mathcal{E}}_{i}J_{j}\rangle(\omega,k=0),$ (36)
${\mathcal{L}}^{(4)}_{ij}=\lim_{\omega\to 0}\frac{1}{i\omega}\langle
J^{\mathcal{E}}_{i}J^{\mathcal{E}}_{j}\rangle(\omega,k=0).$ (37)

The use of the gravitational field coupling to the energy density whose
variations induce energy currents is the first main contribution of
Luttinger’s work. Still the issue remains as how to couple the statistical
force (temperature gradient) to the energy current. The passage from Eqs.
(34), (35) to Eqs. (11), (12), (this is, the relation between
${\bm{\mathcal{L}}}^{(i)}$ and $\bm{L}^{(i)}$) was first presented in the
original work by Luttinger Luttinger (1964). In what follows we will show a
different re–formulation of the Luttinger approach more suitable for making
the connection with other aspects of the review.

###  III.4 Formal derivation of generalized Luttinger relations

Here we present a formal approach to generalized Luttinger relations. It is
based on introducing external fields to mimic variations in thermodynamic
parameters, the temperature $T$, chemical potential $\mu$ and velocities
$u_{a}$ (see Appendix A). This allows to derive Kubo formulas for electric and
thermal conductivities from a unifying principle or, more generally, for
transport coefficients appearing in an effective hydrodynamic description of a
physical system.

We start from the very basic assumption that we have an action given as a
functional of the metric and a gauge field

$S=S[g,A]\,.$ (38)

The precise form of this function will depend on the underlying physical
theory but we are only interested in the way a system described by $S$ reacts
to changes in the external fields, the metric and the gauge field. We can
define the energy-momentum tensor through variations of the metric (see Eq.
(32)) and the electric current through variations of the gauge field:

$\displaystyle J^{\mu}$ $\displaystyle=-\frac{1}{\sqrt{-g}}\frac{\delta
S}{\delta A_{\mu}(x)}\,.$ (39)

If the action is invariant under diffeomorphisms and gauge transformations the
current and energy-momentum tensor (32) obey conservation laws. The conserved
momentum and charge can be obtained by integrating over the spatial volume

$\displaystyle P_{\mu}$ $\displaystyle=\int d\Sigma\,T^{t}_{\mu}\,,$ (40)
$\displaystyle Q$ $\displaystyle=\int d\Sigma\,J^{t}\,.$ (41)

The time component of the four-vector $P_{\mu}$ is the total energy (the
Hamiltonian) and the spatial components are the (spatial) momenta. Now we can
form the general statistical operator (see Appendix A)

$\rho=\exp\left[-\frac{1}{T}\left(u^{\nu}P_{\nu}-\mu Q\right)\right]\,,$ (42)

where $u^{\nu}$ is a four velocity, $T$ the temperature and $\mu$ the chemical
potential. We will assume an ensemble at rest with $u^{\nu}=(1,0,0,0)$. The
four-velocity is constrained to be a time-like unit vector $u^{2}=1$. This
constraint is solved by writing

$u^{\mu}=\left(\frac{1}{\sqrt{1-\vec{\beta}^{2}}},\frac{\vec{\beta}}{\sqrt{1-\vec{\beta}^{2}}}\right)\,,$
(43)

in terms of a purely spatial velocity $\vec{\beta}=\vec{v}/c$. Here $c$ can be
interpreted as the speed of light in the context of high-energy physics or as
the Fermi-velocity in a Dirac material.

We are interested on the effect that a variation of the gauge field and metric
has on the thermal ensemble. For simplicity and because it is also the most
relevant case in condensed matter applications we consider the case of a small
metric perturbation on top of a flat background geometry
$g^{\mu\nu}=\mathrm{diag(1+\delta g^{tt},-1,-1,-1)}$. By the definitions (32)
and (39), we can find the change of the Hamiltonian under small and constant
variations of the temporal component $A_{t}$ of the gauge field and the
$g^{tt}$ component of the metric. Since in terms of the kinetic ($T$) and the
potential ($V$) the action and the Hamiltonian can be written as $S=T-V$ and
$H=T+V$, respectively, it follows that $\delta S=-\delta H$ if the variation
$\delta$ only concerns $V$, as it does here. From this it follows that

$\delta H=-\frac{1}{2}\delta g^{tt}H+\delta A_{t}Q\,,$ (44)

So far we have changed the microscopic Hamiltonian by switching on small
increments in gravitational and electrostatic potential. Now we want to ask if
we can mimic the effects of these increments by introducing small variations
in the thermodynamic variables $T$ and $\mu$. In other words we now keep the
microscopic Hamiltonian fixed but change that state thermodynamic state
characterised by increments $T+\delta T$ and $\mu+\delta\mu$. We demand
therefore that the statistical operators of the new system with $\delta
g^{tt}$ and $\delta A_{t}$ are equivalent to the statistical operator of the
old system in the new state $T+\delta T$ and $\mu+\delta\mu$

$\exp\left[-\frac{1}{T}(H+\delta H-\mu
Q)\right]=\exp\left\\{-\frac{1}{T}\left[\left(1-\frac{1}{2}\delta
g^{tt}\right)H-(\mu-\delta
A_{t})Q\right]\right\\}=\exp\left[-\frac{1}{T+\delta
T}(H-(\mu+\delta\mu)Q)\right].$ (45)

To first order in $\delta T$ and $\delta\mu$ and introducing the Newtonian
potential $\phi=-\delta g^{tt}/2$ this leads to the conditions

$\displaystyle\phi$ $\displaystyle=-\frac{\delta
T}{T}=T\delta\left(\frac{1}{T}\right)\,,$ (46) $\displaystyle\delta A_{t}$
$\displaystyle=-\delta\mu+\mu\frac{\delta
T}{T}=-T\delta\left(\frac{\mu}{T}\right)\,.$ (47)

Finally, we can also study the consequences of the variation $\delta g^{xt}$.
It leads to

$\delta H=-\delta g^{xt}P_{x}\,,$ (48)

which leads to a shift in the fluid velocity

$\delta u^{x}=-\delta g^{xt}\,.$ (49)

In this way variations in the external fields can be translated into
variations in the thermodynamic variables. While these considerations apply to
equilibrium and for spatially uniform variations they also apply in the
presence of spatial gradients if we assume that the system is in local thermal
equilibrium. The assumption of local thermal equilibrium implies that
gradients are small on length scales of the order of the mean free path and
that thermodynamic relations hold locally. More precisely a system in local
thermal equlibrium can be described by local and time dependent temperature
$T(t,\vec{x})$, chemical potential $\mu(t,\vec{x})$ and fluid velocity
$u^{\mu}(t,\vec{x})$. Under these conditions it is legitimate to substitute
$\delta\rightarrow\vec{\nabla}$ in (46), (47) and (49). These relations imply
that the condition for remaining in equilibrium are

$\displaystyle-\vec{\nabla}A_{t}-T\vec{\nabla}\left(\frac{\mu}{T}\right)=$
$\displaystyle 0\,,$ (50)
$\displaystyle\vec{\nabla}\phi-T\vec{\nabla}\left(\frac{1}{T}\right)=$
$\displaystyle 0$ (51)

These equations are however neither gauge invariant nor relativistically
covariant. Whereas it is clear that gauge invariance implies the substitution
$-\vec{\nabla}A_{t}\rightarrow\vec{E}$, it is not immediately clear how a
covariant term for the gravitational potential can be written. The problem is
that there is no covariant tensor that can be constructed out of the metric
and its first derivatives. Indeed the curvature tensor $R_{\alpha\beta\mu\nu}$
is second order in derivatives on the metric. It is possible however to write
down a term of the correct form with the help of the velocity field $u^{\mu}$.
An appropriate phenomenological constitutive relation that is first order in
derivatives, gauge invariant and covariant is

$J^{\mu}=\rho
u^{\mu}+P_{\perp}^{\mu\nu}\left[\sigma\left(E_{\nu}-T\nabla_{\nu}(\frac{\mu}{T})\right)+\kappa
T\nabla_{\nu}(\frac{1}{T})+\xi u^{\lambda}\nabla_{\lambda}u_{\nu}\ \right]\,.$
(52)

Here $\rho$ is the charge density (or energy density if we consider the energy
current) and $P_{\perp}^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}$ is the projector
transverse to $u^{\mu}$. This ensures the $u_{\mu}J^{\mu}=\rho$ without any
first order term in derivatives. The covariant electric field is
$E_{\mu}=F_{\mu\nu}u^{\nu}$. The last term depends on the frame vector
$u^{\mu}$ only. Expanding around a flat metric
$g_{\mu\nu}=\eta_{\mu\nu}+\delta g_{\mu\nu}$ and in the rest frame
$u^{\mu}=(1,0,0,0)$ this last term becomes

$P^{\mu\nu}_{\perp}u^{\lambda}\nabla_{\lambda}u_{\nu}=-\delta^{\mu
i}\nabla_{t}u_{i}=-\delta^{\mu
i}(\delta\dot{g}_{it}-\frac{1}{2}\partial_{i}\delta g_{tt})=-\delta^{\mu
i}E_{i}^{g}$ (53)

where $E_{i}^{g}$ is the gravito-electric field (see Appendix D). It is
therefore the covariant term that encodes the action of the gravitational
field in analogous way of the electric field. The condition that in
equilibirium the current should vanish implies then $\xi=-\kappa$. The
physical interpretation of $u^{\lambda}\nabla_{\lambda}u_{\nu}$ is a covariant
version of the acceleration.

We note that $h_{ti}$ can be viewed as a gravito-magnetic vector potential
$A_{g,i}$. A vector field as source for the energy current will further be
discussed in section IV.2.

Luttinger’s transport equations

We can, at this point, complement Eqs. (11), (12) by including Luttinger’s
gravitational potential. From the relation (51) we can write

$\displaystyle\vec{J}_{\textrm{tr}}=\bm{L}^{(1)}\left(\vec{E}-T\vec{\nabla}\frac{\mu}{T}\right)+\bm{L}^{(2)}\left(-\vec{\nabla}\phi+T\vec{\nabla}\frac{1}{T}\right),$
(54)
$\displaystyle\vec{J}_{\textrm{tr}}^{\mathcal{E}}=\bm{L}^{(3)}\left(\vec{E}-T\vec{\nabla}\frac{\mu}{T}\right)+\bm{L}^{(4)}\left(-\vec{\nabla}\phi+T\vec{\nabla}\frac{1}{T}\right).$
(55)

From Eqs. (50) and (51) we see that, in equilibrium,
$\vec{J}_{tr}=\vec{J}^{\epsilon}_{tr}=0$. This is correct, since
$\vec{J}_{tr}$, $\vec{J}^{\epsilon}_{tr}$ are transport currents and should
vanish in equilibrium. It is clear now what is the relation between the
coefficients computed by the Kubo-formulas (${\bm{\mathcal{L}}}^{(i)}$), Eqs.
(34) and (35), and the transport coefficients:

${\bm{\mathcal{L}}}^{(i)}=\bm{L}^{(i)}\,,\qquad\mbox{(no magnetisation
current)}\,,$ (56)

There is still a small, or perhaps not that small, caveat here. The relation
${\bm{\mathcal{L}}}^{(i)}=\bm{L}^{(i)}$ only generally holds in time-reversal
invariant systems, where magnetization currents are absent. The Kubo-formula
coefficients capture the total local current at any given point in the sample.
This includes contributions from magnetization currents. But because
magnetization currents do not contribute to transport,
${\bm{\mathcal{L}}}^{(i)}=\bm{L}^{(i)}$ no longer holds. We will discuss this
further in section IV.

• An inhomogeneous gravitational field can be used to generate energy (heat)
currents to derive Kubo formulas of thermal transport coefficients. • In a
system residing in a local thermodynamic equilibrium, small gradients in the
background gravitational and gauge potentials generate gradients in
temperature, chemical potential and flow velocity:
$\displaystyle\vec{\nabla}\phi=-\frac{\vec{\nabla}T}{T}\quad,\quad\vec{\nabla}A_{t}=-T\vec{\nabla}\left(\frac{\mu}{T}\right)\quad,\quad\vec{\nabla}g^{it}=-\vec{\nabla}u^{i}\,.$

##  IV Transport and magnetization currents

###  IV.1 General remarks

In systems where time-reversal symmetry is broken, the definition of transport
currents, measured in transport experiments becomes more subtle due to the
presence of magnetization currents. Magnetization currents are circulating
currents that arise when time-reversal symmetry is broken. These appear for
instance in the presence of an external magnetic field, and do not contribute
to the net transport through the sample. Therefore, in general, the local
particle and energy currents $\vec{J}$, $\vec{J}^{\epsilon}$, as given by the
Kubo formulas, are written as the sum of transport plus magnetization currents

$\vec{J}=\vec{J}_{\mathrm{tr}}+\vec{J}_{\mathrm{mag}},\quad\vec{J}^{\epsilon}=\vec{J}^{\epsilon}_{\mathrm{tr}}+\vec{J}^{\epsilon}_{\mathrm{mag}}.$
(57)

The magnetization currents can be defined as the rotational of a (energy)
magnetization density

$J_{i,\mathrm{mag}}=\epsilon_{ijk}\partial_{j}M_{k},\quad
J^{\epsilon}_{i,\mathrm{mag}}=\epsilon_{ijk}\partial_{j}M^{\epsilon}_{k},$
(58)

The magnetization vanishes outside the sample, and therefore
$J_{i,\mathrm{mag}}$, $J^{\epsilon}_{i,\mathrm{mag}}$ average to zero over a
cross-section of the sample. At vanishing external fields $A_{\mu}=\phi=0$, we
can define the unperturbed magnetization densities
$M_{0}^{i}(\mu,T)=M^{i}(A_{\mu}=\phi=0)$,
$M_{0}^{\epsilon,i}(\mu,T)=M^{\epsilon,i}(A_{\mu}=\phi=0)$. In this situation,
the magnetization is constant in the bulk, but because it vanishes outside the
material, it will have a pronounced variation at the boundary giving rise to
boundary magnetization currents. For non-vanishing external fields,
$A_{t},\phi\neq 0$, the total magnetization density can be written in terms of
the unperturbed magnetization Cooper  _et al._ (1997):

$\vec{M}=(1+\phi)\vec{M}_{0}(\mu,T),\quad\vec{M}^{\epsilon}=(1+2\phi)\vec{M}_{0}^{\epsilon}(\mu,T)+A_{t}\vec{M}_{0}(\mu,T),$
(59)

and the bulk magnetization currents are then given by Cooper  _et al._ (1997)

$J_{i,\mathrm{mag}}=-\epsilon_{ijk}\left(M_{0,j}\partial_{k}\phi+\frac{\partial
M_{0,j}}{\partial\mu}\partial_{k}\mu+\frac{\partial M_{0,j}}{\partial
T}\partial_{k}T\right),$ (60)
$J^{\epsilon}_{i,\mathrm{mag}}=-\epsilon_{ijk}\left(-M_{0,j}E_{k}+2M^{\epsilon}_{0,j}\partial_{k}\phi+\frac{\partial
M_{0,j}^{\epsilon}}{\partial\mu}\partial_{k}\mu+\frac{\partial
M_{0,j}^{\epsilon}}{\partial T}\partial_{k}T\right).$ (61)

To obtain the transport coefficients $\bm{L}^{(i)}$ defined in Eqs. (11) and
(12), one should subtract the magnetization currents (for
$\vec{\nabla}\mu=\vec{\nabla}T=0$) from the local currents (34), (35). The
transport coefficients are then given in terms of the coefficients obtained
from the Kubo-formulas, ${\bm{\mathcal{L}}}^{(i)}$, and the unperturbed
magnetization densities

$\displaystyle\bm{L}^{(1)}$ $\displaystyle=$
$\displaystyle{\bm{\mathcal{L}}}^{(1)}=\bm{\sigma},\hskip
42.67912ptL^{(2)}_{ij}={{\mathcal{L}}}^{(2)}_{ij}+\epsilon_{ijk}M_{0,k},$ (62)
$\displaystyle L^{(3)}_{ij}$ $\displaystyle=$
$\displaystyle{{\mathcal{L}}}^{(3)}_{ij}+\epsilon_{ijk}M_{0,k},\qquad
L_{ij}^{(4)}={{\mathcal{L}}}_{ij}^{(4)}+2\epsilon_{ijk}M^{\epsilon}_{0,k},$
(63)

The transport coefficients are related under time-reversal by Onsager
relations Onsager (1931) as: $\sigma_{ij}(B)=\sigma_{ji}(-B)$,
$L^{(4)}_{ij}(B)=L^{(4)}_{ji}(-B)$, $L^{(2)}_{ij}(B)=L^{(3)}_{ji}(-B)$, where
$B$ denotes the magnetic field or any other parameter that breaks the time-
reversal symmetry. It is important to state that it is the transport
coefficients $\bm{L}^{(i)}$, and not the ones in the local currents,
${\bm{\mathcal{L}}}^{(i)}$, which fulfill the Mott rule and the Wiedemann-
Franz law Smrcka and Streda (1977); Jonson and Girvin (1984); Cooper  _et al._
(1997); Qin  _et al._ (2011).

Transport currents are then obtained as a response to gradients of the
statistical fields ($\mu$, $T$) by combining Eqs. (62), (63) and (54), (55),
and setting $\vec{E}=\vec{\nabla}\phi=0$. To obtain the local currents as a
response to $\vec{\nabla}\mu$, $\vec{\nabla}T$, the magnetization currents
(60), (61) have to be added to the transport currents. This will give the
total current locally at any point in the sample.

###  IV.2 Luttinger, thermal transport and energy magnetization

In section III.4 we introduced the Luttinger relation Eq. (25) as a gradient
of a gravitational potential, and showed that we can include a thermal vector
potential, in analogy with an electromagnetic field. The inclusion of a
thermal vector potential turns out to be related to several issues and
necessary extensions that concern the original work by Luttinger to capture a
broader set of phenomena. First, unphysical divergences at $T\to 0$ have been
encountered using the Luttinger derivation of the Kubo formula for thermal
transport currents Qin  _et al._ (2011); Tatara (2015). Second, Luttinger did
not incorporate magnetization currents, which were added later Cooper  _et
al._ (1997); Smrcka and Streda (1977); Qin  _et al._ (2011), resulting in (60)
and (61). Lastly, the difference between diamagnetic and paramagnetic
currents, explained below, is not as evident in the Luttinger formalism as
they are in electromagnetism Tatara (2015).

The first two issues were addressed Cooper  _et al._ (1997); Smrcka and Streda
(1977); Qin  _et al._ (2011) by noticing that the subtraction of the
equilibrium magnetization currents from the result of linear response theory
leads to well defined transport currents, see Eqs. (62), (63), even as
temperature approached zero Qin  _et al._ (2011). Tatara observed Tatara
(2015) that a proper treatment of the diamagnetic heat current term will lead
to this conclusion as well. In electromagnetism, the paramagnetic terms arise
from terms in the Hamiltonian proportional to the vector potential $\vec{A}$,
while diamagnetic terms stem from terms proportional to the square of
$\vec{A}$. While the Dirac Hamiltonian only gives rise to paramagnetic terms
because it is linear in momentum, the quadratic Schrödinger Hamiltonian will
generically contain both paramagnetic and diamagnetic terms. In
electromagnetism both terms are important as the paramagnetic current contains
an equilibrium contribution determined by the electrons below the Fermi level
which exactly cancels the diamagnetic current Tatara (2015). By formulating
the Luttinger relation in terms of a vector potential, $\vec{A}_{g}$, Tatara
was able to isolate the diamagnetic and paramagnetic contribution similarly to
electromagnetism. This formulation reinterprets the results in Cooper  _et
al._ (1997); Smrcka and Streda (1977); Qin  _et al._ (2011), sheds light into
the issues and extensions discussed above related to the Luttinger approach,
and emphasizes the parallelism between electromagnetism and
gravitoelectromagnetism.

The emergence of a vector thermal potential $\vec{A}_{g}$ can be understood as
arising from conservation laws Tatara (2015). In the Luttinger formalism the
coupling to the gravitational field is imposed through the Hamiltonian density
$h(\mathbf{r})$ given by $H_{\mathrm{int}}$ as written in (33)

$H_{\mathrm{int}}=\int d^{3}{r}\;\phi(\vec{r})h(\vec{r}).$ (64)

As discussed in Sec. III.4 the responses caused by a thermal gradient can be
calculated in powers of $\nabla\phi$ and identifying
$\vec{\nabla}\phi=-\frac{\vec{\nabla}T}{T}$ Luttinger (1964). Using the
conservation law ${\partial}_{t}h(\vec{r})+\vec{\nabla}\vec{J}_{\epsilon}=0$,
where $\vec{J}_{\epsilon}$ is an energy current, and restricting to steady-
state properties, it is possible to rewrite this equation as

$H_{A_{g}}=-\int
d^{3}{r}\;\;\vec{J}_{\epsilon}(\vec{r},t)\cdot\vec{A}_{g}(\vec{r},t).$ (65)

By identifying $\partial_{t}\vec{A}_{g}=-\frac{\vec{\nabla}T}{T}$, Tatara
showed that the correct diamagnetic contribution is naturally captured, since
$H_{A_{g}}$ depends explicitly on a vector potential. It is important to note
that both formulations, either $H_{\mathrm{int}}$ or $H_{A_{g}}$ are
equivalent. Both can incorporate magnetization currents and can be regular at
$T\to 0$. The formalism in terms of $H_{A_{g}}$ allows to distinguish the
diamagnetic contribution explicitly, and thus establishes a strong link
between gravitoelectric fields and electromagnetic fields. Taking both
$H_{\mathrm{int}}$ and $H_{A_{g}}$ together, a gauge degree of freedom
emerges. So long as we are interested in steady-state properties, we are free
to choose $\partial_{t}\vec{A}_{g}$ and $\vec{\nabla}\phi$ independently, so
long as we identify
$\vec{\nabla}\phi+\partial_{t}\vec{A}_{g}=-\frac{\vec{\nabla}T}{T}$ Tatara
(2015).

The emergence of a vector thermal potential $\vec{A}_{g}$ has been described
by resorting to the Newton-Cartan formalism Shitade (2014); Gromov and Abanov
(2015); Bradlyn and Read (2015); Geracie  _et al._ (2015). The conservation
equations for momentum and energy densities follow from global space and time
translations. To promote these to local symmetries one can define a local
frame in terms of the vierbeins $e^{\mu}_{a}$. The resulting theory is similar
to general relativity complemented by torsion, known as the Einstein-Cartan
formalism described in sec. VIII and in Appendix B. If Lorentz symmetry is
absent, as natural in condensed matter systems, it was suggested that the
Newton-Cartan formalism is appropriate to describe the geometrical properties
of the theory. Refs. Shitade (2014); Gromov and Abanov (2015); Bradlyn and
Read (2015); Geracie  _et al._ (2015) showed that by taking variations of the
action with respect to $e^{\mu}_{a}$ and the metric, one can obtain the
correct thermal transport and magnetization currents. In this formalism both
$\vec{A}_{g}$ and $\phi$ appear naturally, as a consequence of local
coordinate transformations.

It is important to point out that the results of Refs. Shitade (2014); Gromov
and Abanov (2015); Bradlyn and Read (2015); Geracie  _et al._ (2015) can be
interpreted in terms of derivatives of the velocity field, $u_{\mu}$, in
general curved, but torsion-free, backgrounds. This is suggested by the
results of a recent work Ferreiros and Landsteiner (2020). At finite
temperature and chemical potential, gravito-electromagnetic fields can appear,
even in torsion-free backgrounds, as derivatives of the velocity field, see
Eqs. (52) and (53) and the associated discussion. This can not happen in
vacuum, where the only covariant tensor involving first order derivatives of
the vielbein is the torsion tensor. Therefore, at finite chemical potential
and temperature, responses to gravito-electromagnetic fields can naturally
appear in the absence of torsion.

To emphasize this point explicitly let us assume that we have a magnetized
material. The magnetization is a term in the free energy of the form

$F_{B}=\int d^{3}x\,\vec{B}\cdot\vec{M}\,.$ (66)

Instead of the free energy we switch now to the action formalism. This is
always preferable if we want to make the symmetries manifest. The action
should be a covariant object and is integrated over space and time. The
magnetization part of the action is then the covariant form of (66)

$S_{B}=\frac{1}{2}\int
d^{4}x\,\epsilon_{\mu\nu\rho\lambda}F^{\mu\nu}u^{\rho}M^{\lambda}$ (67)

Now we can compute the current by differentiating with respect to the gauge
field

$J_{\mu,\mathrm{mag}}=\epsilon_{\mu\nu\rho\lambda}\partial^{\nu}\left(u^{\rho}M^{\lambda}\right)\,.$
(68)

In the rest frame $u^{\mu}=(1,0,0,0)$ this gives the well-known result

$\vec{J}_{\mathrm{mag}}=\vec{\nabla}\times\vec{M}\,.$ (69)

Now we want to generalize this to the energy current. The gauge field for
energy (and momentum) currents is the metric. We face however again the
problem that there is no covariant tensor built out of the metric and its
first derivatives. Terms with a Riemann or Ricci curvature tensor would lead
to higher order terms in derivatives. The way out of this problem is to
consider the velocity field as defined in Appendix A. For example, let us
consider a system under rotation with a covariant angular velocity
$\Omega_{\mu}=\frac{1}{2}\epsilon_{\mu\nu\rho\lambda}u^{\nu}\nabla^{\rho}u^{\lambda}$.
Then we can write down the action

$S_{\mathrm{rot}}=-\int
d^{4}x\,\Omega_{\mu}M^{\mu}_{\epsilon}=-\frac{1}{2}\int
d^{4}x\,\epsilon_{\mu\nu\rho\lambda}M^{\lambda}u^{\mu}\nabla^{\nu}u^{\rho}\,,$
(70)

where now we take a general covariant derivative

$D^{\nu}u^{\rho}=g^{\nu\sigma}(\partial_{\sigma}u^{\rho}+\Gamma^{\rho}_{\sigma\kappa}u^{\kappa})\,.$
(71)

In the restframe $u^{\mu}=(1,0,0,0)$ the action is therefore

$S_{\mathrm{rot}}=\frac{1}{2}\int
d^{4}x\,\epsilon_{0ijk}\delta^{il}\Gamma^{j}_{l0}M^{k}\,,$ (72)

and to first order in the metric perturbation

$\Gamma^{j}_{l0}=-\frac{1}{2}\delta^{jm}\left(\partial_{l}\delta
g_{m0}+\partial_{0}\delta g_{ml}-\partial_{m}\delta g_{l0}\right)\,.$ (73)

We insert this into the expression for the rotational action (70) noting that
the term with symmetric spatial indices on the metric perturbation vanish due
to contraction with the anti-symmetric epsilon tensor. We can also introduce
the gravito-magnetic field $A^{g}_{i}=\delta g_{it}$. Since now we have
specified a frame it also makes sense to write down the gravito-magnetic
contribution to the free energy defined as $\int
dtF_{\mathrm{g}}=-S_{\mathrm{g}}$

$F_{g}=\frac{1}{2}\int d^{3}x\,\epsilon_{ijk}\partial_{i}A^{g}_{j}M^{k}=\int
d^{3}x\,\vec{B}_{g}\cdot\vec{M}_{\epsilon}.$ (74)

This leads to the gravito-magnetization current

$\vec{J}_{\epsilon}=\vec{\nabla}\times\vec{M}_{\epsilon}\,.$ (75)

We note that the gravito-magnetic potential $\vec{A}_{g}$ is the thermal
vector potential coupling to the energy current, see (65) Tatara (2015).
Finally we can also give a physical interpretation of the gravito-
magnetization $\vec{M}_{\epsilon}$, which is nothing but the energy
magnetization defined in Eq. (58). Since it can be traced back to the coupling
to angular velocity $\vec{\Omega}$, $\vec{M}_{\epsilon}$ is just the angular
momentum $\vec{M}_{\epsilon}=\vec{L}$. For the non-relativistic case, the
magnetization currents will be determined by a mix of spin and orbital
contributions Gromov and Abanov (2015).

• In the presence of magnetic fields (or other time-reversal breaking effects)
magnetization currents appear, which do not contribute to net electrical and
thermal transport. • In this situation, the Wiedemann–Franz and Mott relations
are not fulfilled by the coefficients obtained through the Kubo formulas.
These relations still hold for the transport coefficients defined in Eqs. (62)
and (63). • The most general theory to derive the thermal transport and
magnetization currents contains both a gravitational potential $\phi$ and a
thermal vector potential $\vec{A}_{g}$, generalizing Luttinger’s derivation. •
The thermal vector potential $\vec{A}_{g}$ allows to understand the presence
of magnetization currents in terms of a gravito-magnetic field.

##  V Transport and chiral anomalies

###  V.1 Anomalies in QFT

Figure 3: Anomalies can be easily detected in triangle diagrams arising in
perturbation theory. The corresponding Feynman diagrams are divergent and need
to be regularized. It is the regularization that breaks the classical chiral
symmetry. Depending on the details of the regulator one obtains different
forms of anomalies. On the right panel we show the triangle diagram for three
general symmetries. One might employ a regularization scheme that treats all
three vertices in a symmetric way. On the other hand in figure (b) we show the
triangle leading to the axial anomaly. The $V$-vertices are singled out in
that no anomaly should be associated to the $V$ vertices since they correspond
to the coupling to the electromagnetic gauge field. In that case the anomaly
has to be concentrated in the $A$ vertex. An alternative is to single one one
particular vertex and demand that it couples covariantly to the gauge fields
sitting at the other two vertices. This scheme defines the covariant anomaly.

In what follows we will describe the most general form of triangle anomalies
of Dirac fields with non-Abelian symmetries and the new anomaly-induced
transport phenomena associated to them. Other interesting aspects of the
quantum anomalies not discussed here, including the case of (1+1) dimensional
system and the topological implications of anomalies, are discussed
pedagogically in Ref. Fradkin (2020)

In perturbation theory anomalies appear at one loop in the three-point
function correlation functions of classically conserved currents and the
energy momentum tensor (triangle diagrams, see figure 3) Adler (1969); Bell
and Jackiw (1969); Kimura (1969). Alternatively they appear in the regularized
definition of the path integral measure Fujikawa (1979). In four dimensions
the most general form of triangle anomalies is

$\displaystyle D_{\mu}J^{\mu}_{a}$
$\displaystyle=\frac{d_{abc}}{32\pi^{2}}\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}^{b}F_{\mu\nu}^{c}+\frac{b_{a}}{768\pi^{2}}\epsilon^{\mu\nu\rho\lambda}R^{\alpha}\,_{\beta\rho\lambda}R^{\beta}\,_{\alpha\rho\lambda}\,,$
(76) $\displaystyle D_{\mu}T^{\mu\alpha}$
$\displaystyle=F^{\alpha\beta}_{a}J^{a}_{\beta}+\frac{b_{a}}{384\pi^{2}}\epsilon^{\mu\nu\rho\lambda}D_{\beta}\left(F_{\mu\nu}^{a}R^{\alpha\beta}\,_{\rho\lambda}\right).$
(77)

The indices denoted by latin letters enumerate different currents
$J^{\mu}_{a}$ which couple to gauge fields $A_{\mu}^{a}$. These currents can
be the currents for left- and right-handed fermions, or the axial or vector
like combination thereof. In general they are the Noether currents
corresponding to a symmetry. The $F^{a}_{\mu\nu}$ are the field strengths of
the gauge fields,
$F^{a}_{\mu\nu}=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+f^{abc}A^{b}_{\mu}A^{c}_{\nu}$,
where $f^{abc}$ the structure constants. The symmetries are generated by
Hermitian matrices $T^{a}$ with commutation relations
$[T^{a},T^{b}]=if^{abc}T^{c}$. For Abelian symmetries the structure constants
vanish. The Riemann tensor $R^{\beta}\,_{\alpha\rho\lambda}$ accounts for
curvature in space-time and is defined in eq. (231).

The currents and the energy-momentum tensor are composite operators in terms
of the fundamental fermion fields. In quantum field theory a composite
operator is one that is built out of local products of the fundamental quantum
fields. For free fermions and in flat space they take the simple forms666We
emphasize that these are the expression in flat space. In general curved space
the energy momentum tensor depends in a non-trivial way on the metric (rather
the vielbein) and this has to be taken into account in calculating the
anomaly. For details see the discussion in Kimura (1969); Alvarez-Gaume and
Witten (1984).

$\displaystyle J^{\mu}_{a}(x)$
$\displaystyle=\bar{\Psi}_{f}(x)(T_{a})^{f}_{g}\gamma^{\mu}\Psi^{g}(x)\,,$
(78) $\displaystyle T^{\mu\nu}(x)$
$\displaystyle=\frac{i}{2}[\bar{\Psi}_{f}(x)\gamma^{(\mu}\partial^{\nu)}\Psi^{f}(x)-\partial^{(\nu}\bar{\Psi}_{f}(x)\gamma^{\mu)}\Psi^{f}(x)]\,.$
(79)

For the Abelian vector and axial currents Eqs. (6), (7) of a Dirac fermion,
$T_{a}$ is taken from $\\{\mathbf{1},\gamma_{5}\\}$. The round brackets on the
indices denote symmetrization according to
$A^{(\mu}B^{\nu)}=\frac{1}{2}(A^{\mu}B^{\nu}+A^{\nu}B^{\mu})$. Famously,
quantum field theory suffers from ultraviolet divergences which must be dealt
with by a regularization and renormalization scheme. In particular a composite
operator is only defined up to a specific regularization scheme. The anomaly
has to be understood as the fact that no regularization scheme exists which
allows to realize all classical conservation laws in the quantum theory.
However, different regularization schemes might give rise to different quantum
definitions of the current operator. These ambiguities can be fixed by
demanding certain physically plausible conditions on the currents and
operators. One possible definition of regularization is to demand that the
currents themselves transform as well defined tensors (”covariantly”) under
the symmetry transformations even if the symmetry is anomalous. It turns out
that this is a unique way of regularizing the current operators and it gives
rise to what is called the covariant anomaly Bardeen and Zumino (1984). This
regularization scheme is particularly well suited for answering the question
if a theory does have anomalies. The anomalies in eqs. (76) and (77) are
precisely these covariant anomalies. It turns out that their existence is
completely determined by two simple group invariants. These are the anomaly
coefficients

$\displaystyle d_{abc}$
$\displaystyle=\frac{1}{2}\sum_{r}\mathrm{tr}(T_{a}.\\{T_{b},T_{c}\\})_{r}-\frac{1}{2}\sum_{l}\mathrm{tr}(T_{a}.\\{T_{b},T_{c}\\})_{l}\,,$
(80) $\displaystyle b_{a}$
$\displaystyle=\sum_{r}\mathrm{tr}(T_{a})_{r}-\sum_{l}\mathrm{tr}(T_{a})_{l}\,,$
(81)

where the sums are over the representations under which the right-handed and
left-handed chiral fermions transform and curly brackets denote the anti-
commutator. For a $U(1)$ symmetry, the generator $T_{a}$ in a specific
representation is just as the charge $q_{a}$. The Adler-Bardeen non-
renormalization theorem Adler and Bardeen (1969); Adler (2005) guarantees that
there are no further quantum corrections to these numbers. The existence of
anomalies can be unambiguously determined by the calculations of the anomaly
coefficients $d_{abc}$ and $b_{a}$.

As an example, let us consider the single massless Dirac fermion. At the
classical level a Dirac fermion is the direct sum of a right-handed and a
left-handed fermion. We denote the charge of the left- and right-handed
fermions by $(q_{r},q_{l})$. In a vector-axial basis we have two symmetries
with charges $U(1)_{V}$: $(1,1)$ and $U(1)_{5}$: $(1,-1)$ . Accordingly the
non-vanishing anomaly coefficients are $d_{5VV}=d_{V5V}=d_{VV5}=2$,
$d_{555}=2$ and $b_{5}=2$. We note here that in this case all these anomaly
coefficients take the same value.

Now we run however into something that appears troublesome. The anomaly
coefficient $d_{abc}$ is completely symmetric, and this means that for a Dirac
fermion the covariant vector current is anomalous as well

$\partial_{\mu}J^{\mu}_{V}=\frac{1}{8\pi^{2}}\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}^{V}F_{\rho\lambda}^{5}\,.$
(82)

This equation depends on the field strength formed by the axial gauge fields
$A^{\mu}_{5}$ which couples to the axial current. One might argue that in
nature there are no axial gauge fields so there is no such anomaly. This
argument is however problematic. First axial gauge fields can effectively
appear as happens in the low energy description of Weyl semi-metals (see Sec.
VI) and secondly in a quantum theory charge conservation should be an operator
equation that holds in generic correlation functions.

Since the problematic (82) is derived from (76),(77), one might ask how the
anomalous conservation equation for the currents (76),(77) can be interpreted.
There are at least four related but slightly different possibilities

  1. 1.

First, we can interpret the anomaly equations as the non-conservation of the
current corresponding to the classical symmetry $A$ when appropriate classical
background gauge fields are switched on. An example is the non-conservation of
axial charge in a homogeneous background of parallel electric and magnetic
fields. In this case the anomaly equation becomes

$\dot{\rho}_{5}=\frac{1}{2\pi^{2}}\vec{E}\cdot\vec{B}\,.$ (83)

Axial charge is created ”out of the vacuum” at a constant rate determined by
the electric and magnetic fields. Here the electromagnetic fields are thought
of as external to the quantum dynamics of the Dirac fermions. In what concerns
the quantum dynamics they are just classical c-numbers and not operators. This
is the situation that arises in measurements of magneto conductivities of Weyl
or Dirac semi-metals.

  2. 2.

We can also interpret the anomaly equations as a statement of how the operator
“$\partial_{\mu}J^{\mu}(x)$”, the divergence of the current or the energy
momentum tensor, behaves when inserted in correlation function with other
currents. Here we interpret the left-hand side of an anomaly equation as the
expectation value of the divergence of the anomalous current and the right-
hand side as a (local) functional of the classical sources that couple to the
currents and the energy-momentum tensor. The gauge fields and the metric on
the right-hand side of the anomaly equation are not physical fields but simply
act as sources for insertions of the current into correlation functions
according to Eqs. (32) and (39).

As an example we consider the axial anomaly and write it as

$\langle\partial_{\mu}J_{5}^{\mu}(x)\rangle=\frac{1}{16\pi^{2}}\epsilon^{\mu\nu\rho\lambda}F^{V}_{\mu\nu}(x)F^{V}_{\rho\lambda}(x)\,.$
(84)

Functionally differentiating twice with respect the vector gauge field, i.e.
applying $\frac{\delta^{2}}{\delta V_{\rho}(y)\delta V_{\lambda}(z)}$, gives
the anomalous Ward identify

$\langle\partial_{\mu}J_{5}^{\mu}(x)J^{\nu}(y)J^{\lambda}(z)\rangle=\frac{1}{2\pi^{2}}\epsilon^{\mu\nu\rho\lambda}\partial_{\mu}^{x}\delta(x-z)\partial_{\rho}^{x}\delta(x-z)\,.$
(85)

This makes it clear that the anomaly appears even in the absence of physical
external electric and magnetic fields as a contact term in a correlation
function.

The same logic holds for the gravitational contribution to the axial anomaly.
Even in flat space the gravitational contribution to the axial anomaly does
not vanish. It is manifest in correlation functions of the divergence of the
axial current obtained by differentiating eq. (76) twice with respect to the
metric 777This comprises a triangle diagram with two energy-momentum tensor
insertions and a two point function with an operator that arises due to the
non-linearity of the gravitational coupling. The latter is in complete analogy
to the diamagnetic term that arises in Kubo formulas for the electric
conductivity in models of charged bosonic matter. Kimura (1969); Alvarez-Gaume
and Witten (1984). This type of interpretation is also often called ’t Hooft
anomaly. The anomaly is not absent even if there are no gauge fields present
or if one is in flat space. The external gauge fields only make the anomaly
visible in the expectation value of the divergence of a current. Anomalies are
properties of the quantum theory. The non-renormalization theorem also implies
that the results are independent of the coupling to other matter fields. In
particular anomalies are present in the free theory and can be detected by
computing correlations functions of composite operators (the currents) in the
free theory.

  3. 3.

The anomaly can also be interpreted as an identity between two a priori
different operators in a quantum field theory. This interpretation is
appropriate whenever the the quantum dynamics of the gauge fields themselves
is important. Whereas in the first two cases above the vacuum expectation
value of the divergence of the current still vanishes if the right hand side
is set to zero, this is not possible anymore if the gauge fields are
dynamical. The right hand side in Eqs. (76),(77) contains now fluctuating
quantum fields and therefore under no circumstance the corresponding current
is really conserved. It is always subject to the quantum fluctuations of the
gauge fields888This lies at the heart of the resolution of the so-called
$U(1)_{A}$ (A for axial here) puzzle in QCD ’t Hooft (1986). Because of the
quantum fluctuating gauge field there are quantum corrections to the
divergence of the current due to higher-order loop diagrams, even in Abelian
gauge theories Adler (2005).

  4. 4.

In another situation the anomaly has to be interpreted as an inconsistency in
the quantum theory. This happens when a gauge current is anomalous. What is
meant here is that the current coupling to dynamical gauge fields suffers from
an anomaly. This is a problem indeed. Even without quantizing the gauge fields
the inconsistency can be seen from the equations of motion of a $U(1)$ gauge
field

$\partial_{\mu}F^{\mu\nu}=J^{\nu}\,.$ (86)

Taking the divergence of both sides leads to an inconsistency unless the
operator $\partial_{\mu}J^{\mu}$ is identically zero. This inconsistency
carries over to the quantum theory and to non-Abelian gauge fields. In the
quantum theory a consequence of this type of anomaly is that it would allow
transverse physical photon states to scatter into unphysical longitudinal
polarizations of zero norm in the Hilbert space. Thus unitarity is lost999In
principle, at least for $U(1)$ fields, a possible cure is to make the gauge
field massive by a Stueckelberg mechanism Preskill (1991) since in this case
the transverse polarization becomes a physical state. Therefore anomalies in
currents which couple to dynamical gauge fields have to vanish identically.
Sometimes this can be achieved by using the ambiguities in the regularization
prescription as we will exemplify below in the case of the vector like current
coupling to the dynamical electro-magnetic gauge field.

These different interpretations of anomalies are often referred with specific
nomenclature. For example the so-called “’t Hooft anomalies” correspond to our
second point of view whereas ”ABJ anomalies” (after Adler-Bell-Jackiw) to our
third point and ”gauge anomalies” to our fourth point.

Having established that anomalies can not be eliminated by going to flat space
and switching off all electric and magnetic fields we still have to deal the
worrisome Eq. (82). It is an anomaly in a gauge current and therefore should
actually vanish. So far we have insisted that the currents transform
covariantly under the symmetries even if the symmetry is anomalous. This is
indeed too much to ask for. It is possible to define a vector current that is
exactly conserved, even taking the axial background gauge fields into account,
by changing the regularization scheme. This is the electromagnetic consistent
conserved vector current. The name ”consistent” has nothing to do with the
fact that this current can be consistently coupled to dynamical
electromagnetic gauge fields but rather has a technical meaning in that the
corresponding anomaly solves the so-called Wess-Zumino consistency conditions
Wess and Zumino (1971). Without going further into details we note that (82)
is a so called mixed anomaly that involves two symmetries: the axial and the
vector-like ones. In such a situation of mixed anomalies it is possible to
shift the consistent anomalies between the different symmetries. In particular
one can completely eliminate the anomaly (82) by choosing a regularization
scheme that explicitly preserves the vector-like symmetry in question (e.g.
Pauli-Villars) or by adding appropriate counterterms (Bardeen counterterms) to
the quantum effective action Bardeen (1969). It is however not possible to
eliminate the anomaly totally. Therefore it is sufficient to calculate the
symmetric anomaly coefficients $d_{abc}$ and $b_{a}$ with a covariant
regularization scheme to establish the presence of anomalies. In the case of
the anomaly in the vector current eq. (82) the relation of the conserved
consistent current $\mathcal{J}^{\mu}$ to the covariant current $J^{\mu}$ is

$\displaystyle\mathcal{J}^{\mu}$
$\displaystyle=J^{\mu}-\frac{1}{4\pi^{2}}\epsilon^{\mu\nu\rho\lambda}A^{5}_{\mu}F_{\rho\lambda}\,,$
(87) $\displaystyle\partial_{\mu}\mathcal{J}^{\mu}$ $\displaystyle=0\,.$ (88)

The additional Chern-Simons current arises from the regularization as a local
finite counterterm to the current (Bardeen-Zumino Polynomial) Bardeen and
Zumino (1984). The Chern-Simons current precisely cancels the anomaly in the
covariant current. Note that this also means that now the axial gauge field is
not really a gauge field anymore. It is a physical field and observable as a
current in electric and magnetic fields. This conserved consistent current can
now be coupled to a Maxwell field and has a good physical interpretation as
electric current. On the other hand the anomaly in the axial current can not
be cancelled in the same way since it contains the term

$\partial_{\mu}J^{\mu}_{5}=\dot{\rho}_{5}+\vec{\nabla}\cdot\vec{J}_{5}=\frac{1}{16\pi^{2}}\epsilon^{\mu\nu\rho\lambda}F^{V}_{\mu\nu}F^{V}_{\rho\lambda}=\frac{1}{2\pi^{2}}\vec{E}\cdot\vec{B}\,.$
(89)

The corresponding Chern-Simons current that would cancel this anomaly depends
explicitly on the (true) electromagnetic gauge field $V_{\mu}$ and thus has to
be considered unphysical. The axial anomaly is there because of the non-zero
$d_{VVA}$ coefficient. We note that also the $d_{AAA}$ coefficient is non-
vanishing. This essentially means that the axial symmetry is hopelessly lost
in the quantum theory. Therefore the best possible result is that the vector-
like symmetry is preserved in the quantum theory. The corresponding consistent
current is conserved and can be interpreted as electric current.

In the case of the gravitational anomaly the covariant form indeed makes the
energy-momentum tensor a non-conserved current. But again one can add
counterterms to the quantum effective action which cancel the anomaly in the
energy-momentum tensorBardeen (1969); Alvarez-Gaume and Witten (1984). In
theory one could also keep the anomaly in the energy-momentum tensor and
cancel the anomaly in the axial current, but it is not possible to keep the
conservation laws for both of them simultaneously.

• Anomalies arise in quantum field theory because there is no possible
regularization which preserves all the symmetries present in the classical
theory. • Chiral anomalies can be detected by computing the one-loop exact
anomaly coefficients $d_{abc}$ and $b_{a}$ Eqs. (80) and (81). • Anomalies are
intrinsic properties of the operators. They are manifest in correlation
functions even in the absence of external gauge fields and in flat space-time.
• Anomalies in gauge currents lead to mathematical inconsistencies. •
Anomalies suffer from ambiguities which need to be fixed by specific
regularization conditions. These ambiguities give rise to different forms of
anomalies (covariant or consistent anomalies) but also allow to preserve a
subset of anomalous symmetries.

###  V.2 Anomalies in matter

Now we discuss anomaly induced transport, in particular the so-called chiral
magnetic and chiral vortical effects Vilenkin (1979, 1980); Alekseev  _et al._
(1998); Giovannini and Shaposhnikov (1998); Newman (2006); Fukushima  _et al._
(2008); Son and Surowka (2009); Erdmenger  _et al._ (2009); Banerjee  _et al._
(2011). The chiral magnetic effect (CME) denotes the generation of a current
in parallel to an applied magnetic field $\vec{B}$ and the chiral vortical
effect (CVE) is the generation of a current along a vortex $\vec{\Omega}$ (for
a review see Landsteiner (2016); Kharzeev (2014),).

$\displaystyle\vec{J}_{a}$
$\displaystyle=\sigma_{ab}\vec{B}_{b}+\sigma^{\Omega}_{a}\vec{\Omega}\,,$ (90)
$\displaystyle\vec{J}_{\epsilon}$
$\displaystyle=\sigma_{a}^{\epsilon}\vec{B}_{a}+\sigma^{\epsilon,\Omega}\vec{\Omega}\,.$
(91)

The gauge fields are not necessarily physical fields but can simply be sources
for the currents as outlined before. The vorticity is
$\vec{\Omega}=\frac{1}{2}\vec{\nabla}\times\vec{v}$. Explicit calculations in
free fermion theories Landsteiner  _et al._ (2011a) and in holographic field
theories Landsteiner  _et al._ (2011b) give the results101010We will
specialize these equations to the cases that affect the condensed matter
systems in the next section.

$\displaystyle\sigma_{ab}$ $\displaystyle=\frac{d_{abc}\mu_{C}}{4\pi^{2}}\,,$
(92) $\displaystyle\sigma_{a}^{\Omega}$
$\displaystyle=\frac{d_{abc}\mu_{b}\mu_{c}}{4\pi^{2}}+\frac{b_{a}}{12}T^{2}\,,$
(93) $\displaystyle\sigma_{a}^{\epsilon}$
$\displaystyle=\frac{d_{abc}\mu_{b}\mu_{c}}{8\pi^{2}}+\frac{b_{a}}{24}T^{2}\,,$
(94) $\displaystyle\sigma^{\epsilon,\Omega}$
$\displaystyle=\frac{d_{abc}\mu_{a}\mu_{b}\mu_{c}}{12\pi^{2}}+\frac{b_{a}\mu_{a}}{12}T^{2}$
(95)

The remarkable fact is that these transport coefficients are completely
determined by the anomaly coefficients (80) and (81). It means that these
currents are non-zero when the theory has the anomalies (82). Since anomalies
are subject to non-renormalization theorems one expects non-renormalization
theorems to hold for these transport coefficients as well. Perturbative proofs
of non-renormalization have been presented in Golkar and Son (2015); Hou  _et
al._ (2012).

It is important to realize that these non-renormalization theorems hold only
when the gauge fields on the right hand side of the anomaly equation are non-
dynamical classical fields (’t Hooft anomalies). On the other hand if the
gauge fields are taken as quantum fields (ABJ anomalies) as well they give
rise to non-trivial loop corrections. This was first pointed out in Hou  _et
al._ (2012) for the $T^{2}$ terms and later shown to hold for all chiral
conductivities (92)-(95) in Jensen  _et al._ (2013a). In hindsight this is not
surprising since also the divergence of an anomalous current is renormalized
in this situation Adler (2005). The non-renormalization properties of the
chiral conductivities are therefore the same as the ones of the anomalous
divergence of the current itself.

The dependence on the chemical potentials in Eqs. (92) - (95) is also fixed by
demanding the existence of an entropy current with positive divergence
(generalized second law) Son and Surowka (2009); Neiman and Oz (2011) in the
presence of chiral anomalies. The dependence on the temperature on the other
hand can not be fixed by this criterion. Rather the temperature enters as an
integration constant Neiman and Oz (2011). More advanced consideration are
necessary to establish a general relation between the gravitational
contribution to the anomalies and the temperature dependence in (93)-(95).
This has first been established in Jensen  _et al._ (2013b). The approach of
Jensen  _et al._ (2013b) was to study Euclidean quantum field theory on cone-
like geometries. A related setup this time however in a Lorentzian signature
space-time containing a black hole was developed in Stone and Kim (2018). This
involves integration of the anomaly equations (76), (77) in a black-hole space
time with the boundary condition such that the currents and the energy current
vanish on the horizon. We will briefly review this argument in the Appendix E.
Finally the relation of the temperature dependence of the anomalous transport
coefficients to a global version of gravitational anomaly has been established
in Golkar and Sethi (2016) and further studied in Chowdhury and David (2016);
Glorioso  _et al._ (2019).

It might come as a surprise that the gravitational contribution to the chiral
anomaly can enter the chiral conductivities. One should keep in mind that the
theory is anomalous even in flat space, in the absence of gauge fields and
without interactions. It still is somewhat surprising since the gravitational
anomaly is a term of fourth order in derivatives and naively can not influence
terms in transport that are only first order. It is certainly true that in
vacuum one can not form candidate covariant terms of lower degree in
derivatives. In vacuum the only available tensor is the metric and there
exists no covariant tensor that can be formed with only one derivative on the
metric. The chiral transport expression apply however not in vacuum but in a
gapless theory of chiral fermions at finite temperature and chemical
potential. This means that in the low energy theory one has more effective
fields, not only the metric or the gauge fields but also the effective low
energy fields temperature $T(x)$, chemical potential $\mu(x)$ and the velocity
field $u^{\mu}(x)$. This allows now to form new covariant objects, such as the
vorticity $\Omega^{\mu}$ with low degrees of derivatives. We have emphasized
in section III that temperature and velocity variations have the same effect
in local thermal equilibrium as variations in the gravitational background.
This gives an intuitive reason why gravitational effects can be captured by
temperature and vorticity. In particular if a gravito-magnetic field (see
Appendix D) is present as a small perturbation of otherwise flat space-time
$\delta g_{ti}=A^{g}_{i}$, even a fluid at rest $u^{\mu}=(1,0,0,0)$ can have
vorticity $2\vec{\Omega}=\vec{B_{g}}=\vec{\nabla}\times\vec{A}^{g}$. The
vorticity is induced via frame dragging by the gravito-magnetic potential
$\vec{A}_{g}$. In this sense the chiral vortical effect can also be seen as a
chiral gravito-magnetic effect Landsteiner  _et al._ (2011c).

As already mentioned the anomaly induced conductivities (92) - (95) are also
constrained by a generalized second law of thermodynamics. It is possible to
define an entropy current such that the anomalous currents do not contribute
to entropy production. This anomalous contribution to the entropy current is
given by Son and Surowka (2009); Neiman and Oz (2011); Stephanov and Yee
(2016)

$\vec{J}_{S,\mathrm{anom}}=\frac{T}{12}b_{a}\vec{B}_{a}+\frac{\mu_{a}b_{a}}{6}\vec{\Omega}$
(96)

Later we will show that using this expression one can derive the Wiedemann-
Franz law and the Mott formula for anomalous transport in Weyl semimetals by
defining an appropriate heat current based on this entropy current as
$J^{\mu}_{Q}=TJ^{\mu}_{S}$. For clarity let us emphasize that we have not
considered dissipative transport here. Including also dissipative transport
coefficients the total entropy current fulfills $\partial_{\mu}J_{S}^{\mu}\geq
0$.

We note that the expression for the entropy current (96) depends also on the
frame choice. The one quoted here corresponds to the so-called ”no-drag” frame
Stephanov and Yee (2016). This is the frame in which the fluid is at rest for
$u^{\mu}=U^{\mu}$ and $U^{\mu}$ denotes the frame field of a impurities on on
which the fluid looses momentum. In a condensed matter context there are
always impurities which imply the existence of a preferred frame. For
applications of anomalous transport to condensed matter physics the no-drag
frame is therefore the physically preferred one.

A violation of the relation between triangle anomalies and the chiral vortical
effect happens for relativistic spin $3/2$ fermions (gravitinos) Loganayagam
and Surowka (2012). In that case it was shown in Golkar and Sethi (2016) that
a global version of the gravitational anomaly implies the chiral vortical
effect (see also Chowdhury and David (2016); Glorioso  _et al._ (2019)).
Finally we point out that recently the relation between the gravitational
anomaly in Maxwell theory and a photonic analogue of the chiral vortical
effect has been shown to hold if both are calculated with the same regulator
Prokhorov  _et al._ (2020).

###  V.3 Anomalies and magnetotransport in Weyl semi-metals

Figure 4: A schematic view of a Weyl semimetal’s band structure.

In Figure 4 we show a schematic view of the band structure of a Weyl semi-
metal. In this simple description, the band structure has two band crossing
points near the Fermi level. By the Nielsen-Ninomiya theorem the two nodal
points have opposite chirality. To lowest order in a derivative expansion near
the Fermi points, the electronics can be effectively described by two Weyl
fermions of opposite chirality. A useful equivalent view is to combine both
Weyl fermions into one Dirac fermion. Then the Dirac like wave equation for
this four-component spinor is Zyuzin  _et al._ (2012); Grushin (2012); Zyuzin
and Burkov (2012); Goswami and Tewari (2013)

$(i\gamma^{\mu}\partial_{\mu}+\gamma^{\mu}\lambda_{\mu}\gamma_{5})\psi=0\,.$
(97)

The four vector $\lambda_{\mu}=(\lambda_{0},\vec{\lambda})$ describes the
separation of the Weyl nodes of opposite chirality in momentum space. A useful
view is to consider $\lambda_{\mu}$ to be the background value of an axial
gauge field $A_{\mu}^{5}=\lambda_{\mu}$. As we have already emphasized the
description in terms of chiral fermions in a crystal is not exact. Due to the
compactness of the Brillouin zone the bands hosting left- and right-handed
chiral fermions are connected such that a hard scattering process might
scatter a fermionic quasiparticle from one Weyl cone to another one of
opposite chirality. This inter-valley scattering will have as a consequence
that in equilibrium the Fermi energies of left- and right-handed Weyl cones
are equal $\epsilon_{F}^{R}=\epsilon_{F}^{L}=\epsilon_{F}$. In order to apply
the quantum field theoretical transport formulas we need to measured energies
with respect to the normal ordered vacuum. This is defined with respect to the
band crossing points at which the density of states within a Weyl cone
vanishes. Let us denote these points with $(E_{R,L},\vec{K}_{R,L})$. We can
define now the four vector $\lambda_{\mu}$ by
$(\lambda_{0},\vec{\lambda})=\frac{1}{2}(E_{R}-E_{L},\vec{K}_{R}-\vec{K}_{L})$.
In particular the chemical potentials have to be measured relative to the
locations of the Weyl points such that $\mu_{R,L}=\epsilon_{F}-E_{R,L}$. The
total chemical potential is $\mu=\frac{1}{2}(\mu_{R}+\mu_{L})$. For the axial
chemical potential we have $\mu_{5}=\frac{1}{2}(\mu_{R}-\mu_{L})=\lambda_{0}$
in equilibrium.

We now review how the anomalous transport formulas lead to a theory of
magneto-transport in Weyl semi-metals. First we want to get the so called
Bloch theorem out of our way. Bloch argued that a system can not sustain an
electric current in thermal equilibrium. As we briefly reviewed in the
introduction, the Bloch theorem applies only to currents which are exactly
conserved gauge currents. Thus we need not be concerned about currents that
are not gauge currents, e.g. the axial current $J_{5}^{\mu}$ can have an
equilibrium expectation value without being in conflict with Bloch’s argument.
This applies to the so-called Chiral Separation Effect (CSE)

$\vec{J}_{5}=\frac{\mu}{2\pi^{2}}\vec{B}\,.$ (98)

It is a special case of the general anomaly transport formulas (90) with
$d_{5VV}=2$. More worrisome is the expression

$\vec{J}=\frac{\mu_{5}}{2\pi^{2}}\vec{B}\,,$ (99)

the Chiral Magnetic Effect proper. In this form it is determined by the same
totally symmetric anomaly coefficient $d_{5VV}$. Again it simply is a special
case of eq. (90). If we were to interpret the current in eq. (99) as electric
current it would contradict the Bloch theorem. Since the Bloch theorem is a
consequence of gauge invariance it would also indicate a subtle violation of
gauge symmetry. We need to remember however, that there is an ambiguity in the
definition of the current and we need to make sure that we actually use the
properly defined quantum operator. The electric current should be conserved
under all circumstances. Therefore it is correct to identify the conserved
consistent current (87) as electric current with the identification
$A^{5}_{\mu}=\lambda_{\mu}$! This conserved consistent current has an
additional contribution stemming from the Chern-Simons current. The Bloch
theorem can be expected to hold only for the conserved consistent current
$\mathcal{J}^{\mu}$. If we add this contribution we find

$\vec{\mathcal{J}}=\frac{\mu_{5}-\lambda_{0}}{2\pi^{2}}\vec{B}\,.$ (100)

As we explained before in equilibrium we have the relation
$\lambda_{0}=\mu_{5}$ and thus the Bloch theorem is obeyed. The chiral
magnetic current vanishes identically in strict equilibrium. This result has
first been obtained in a holographic calculation Rebhan  _et al._ (2010);
Gynther  _et al._ (2011). In Gynther  _et al._ (2011) it was also shown that
it is a consequence of any gauge invariant regularization in quantum field
theory. Later it was shown to hold in a tight-binding approach in Vazifeh and
Franz (2013) which from the point of view of quantum field theory can be
viewed as a lattice regularization. The presence of the Chern-Simons term in a
tight binding approach has been explicitly calculated in Gorbar  _et al._
(2017b). The identification of consistent and covariant currents in lattice
models becomes more involved in the presence of axial gauge fields Behrends
_et al._ (2019). These issues arise if one seeks a quantum field theoretical
description. However, this is not strictly necessary as momentum-space in a
crystal is compact. So, instead of doing quantum field theory with a normal
ordered vacuum, one might just do many body quantum mechanics an measure
energies with respect to the bottom of the band. This point of view has been
emphasized in Refs. Basar  _et al._ (2014); Behrends  _et al._ (2019). In any
case it is re-assuring that both points of view give rise to the same results.

Since we can not expect to get any measurable effect by simply applying a
magnetic field to a Weyl semi-metal we need to do something more to drive the
system slightly out of equilibrium. To activate the CME we must break the
relation $\mu_{5}=\lambda_{0}$ in some way. Here it is important that the
chemical potential is a state variable whereas the axial gauge field is an
external parameter. The quantum dynamics of the fermions alone can not change
it the external field $A^{5}_{0}=\lambda_{0}$ but only the state variable
$\mu_{5}$. If we do work on the fermions we can change their state, e.g.
applying an electric field leads to an increase of temperature due to Joule
heating. In an analogous way the anomalous transport phenomena have an impact
on the state of the fermionic system and can drive them out of equilibrium
leading to measurable electric and heat currents.

We may imagine injecting electrons at one end of a Weyl semi-metal and
extracting fermions on the other side. This will effectively lead to a
gradient in the chemical potential $\vec{\nabla}\mu$. Alternatively we can
apply an electric field $\vec{E}$. Now we use the anomalous conservation of
the axial current (89) to obtain the increase in axial charge

$\dot{\rho}_{5}=\frac{1}{2\pi^{2}}\vec{E}\cdot\vec{B}-\vec{\nabla}\cdot\vec{J}_{5}=\frac{1}{2\pi^{2}}(\vec{E}-\nabla\mu)\cdot\vec{B}\,.$
(101)

We also have taken into account the contribution of the CSE current eq. (98)
proportional to the chemical potential. Now we relate the axial charge density
to the axial chemical potential via an axial susceptibility
$\chi_{5}=\partial\rho_{5}/\partial\mu_{5}$

$\dot{\rho}_{5}=\chi_{5}\cdot\mu_{5}\,.$ (102)

Fourier transforming in time leads to an increase in the axial chemical
potential away from its equilibrium value

$\delta\mu_{5}=i\frac{(\vec{E}-\vec{\nabla}\mu)\cdot\vec{B}}{2\pi^{2}\chi_{5}\omega}\,.$
(103)

Since we are interested in a steady state situation we have to take the limit
$\omega\rightarrow 0$ in which the axial charge diverges. In a crystal axial
charge is however destroyed due to large momentum scatterings from one Weyl
cone into the other. This takes place at a time scale $\tau_{5}$ which is
called the inter-valley scattering time Son and Spivak (2013). This is also
the time scale at which the growth of the axial chemical potential will be cut
off. Formally this can be represented by the substitution.

$\omega\rightarrow\omega+i/\tau_{5}\,,$ (104)

with $\tau_{5}$ the inter-valley scattering time. In this way one predicts a
finite value of the axial chemical potential which eventually can be plugged
into the formula for the CME to obtain

$\vec{J}=\tau_{5}\frac{(\vec{E}-\vec{\nabla}\mu)\cdot\vec{B}}{4\pi^{4}\chi_{5}}\vec{B}\,.$
(105)

This reasoning predicts a large enhancement of the electric conductivity in a
magnetic field if the inter valley scatting time is a long time scale Nielsen
and Ninomiya (1983b); Li  _et al._ (2016). A microscopic calculation in a
kinetic theory approach was presented in Son and Spivak (2013). Here we have
emphasized that it is based on universal results, anomalous conservation laws
and the corresponding anomaly induced transport phenomena. We also made it
clear that even when there are no external electric fields the effect can be
triggered by driving the system in a state with a gradient in the chemical
potential. This is just a reflection of the well known fact that in linear
response the effect of an electric field is the same as the effect of a
gradient in the chemical potential.

###  V.4 Anomaly induced magneto-electric and -thermal transport for Weyl
semi-metals

|
---|---
(a) | (b)
Figure 5: Schematics of the enhanced transport in an magnetic field due to
anomalies. In panel (a) in addition to the magnetic field a parallel electric
field is applied. This leads to an enhanced current due to the chiral magnetic
effect. In panel (b) a temperature gradient parallel to the magnetic field
activates the temperature component of the chiral magnetic effect in the
energy current (see Eq. (94)) proportional to the gravitational anomaly
coefficient.

Let us now discuss anomaly induced magneto-electric and -thermal transport for
Weyl semi-metals (see Figure 5). This can and has been studied in a
microscopic approach based on (chiral) kinetic theory and the corresponding
Boltzmann equation Spivak and Andreev (2016); Lundgren  _et al._ (2014). The
relation of the thermal conductivities to the anomalies is however not evident
in such an approach. In contrast, the effective transport theory based on the
anomalous transport currents Eqs. (92)-(95), can track the anomaly
contribution. This has been shown first in a hydrodynamic model in Lucas  _et
al._ (2016). It is however unclear if a hydrodynamic regime with independent
flow for different chiralities can be achieved in a Weyl semi-metal.
Fortunately, an effective theory of coupled charge and energy transport in a
magnetic field based on anomaly induced transport can be developed without
assuming a hydrodynamic regime, as we will review now.

First, we elaborate on the role played by the different relaxation processes.
In general we can distinguish between intravalley, intervalley and electron-
electron or electron-phonon scattering. A key assumption is that there exists
a hierarchy of relaxation times $\tau<\tau_{5}<\tau_{ee}\approx\tau_{e\phi}$.
The intravalley scattering time $\tau$ is the shortest. It is the timescale in
which electronic particles are scattered but the change is their momentum is
small such that they stay within the same Weyl cone in momentum space. This
scattering distributes the electronic quasiparticles randomly in the phase
space corresponding to a single Weyl cone. Second, we assume an inter-valley
scattering time scale $\tau_{5}$. It is supposed to be significantly longer
than $\tau$. Finally we also assume that electron-electron and electron-phonon
interactions are even rarer than inter-valley scattering events. Note that
these assumptions exclude hydrodynamic behavior of the electrons111111For
hydrodynamic electron flow the electron-electron interactions must be dominant
and thus $\tau_{ee}$ must be the shortest time scale.. Formally we will work
with an idealized situation in which $\tau=0$ and
$\tau_{ee},\tau_{e\phi}=\infty$. That means that momentum is not a relevant
low energy degree of freedom whereas the energies and charges of each
individual Weyl cone are approximately conserved, on a time scale given by the
intervalley scattering time $\tau_{5}$.

To set up an effective transport model we concentrate on the charges conserved
by the fast elastic intravalley scattering. These are the energy and charge
densities. A transport theory should include temperatures and chemical
potentials for each Weyl cone. It corresponds to the quantum numbers that are
conserved up to a timescale of order $\tau_{5}$. In the absence of external
driving forces the temperatures and chemical potentials will equilibrate at
times longer than $\tau_{5}$.

The currents induced by the anomalies are dissipationless and therefore we
expect them to be insensitive to momentum relaxing elastic scattering. Indeed
model calculations in holography Copetti  _et al._ (2017) and in a
hydrodynamic setup Stephanov and Yee (2016) have shown anomalous transport
formulas to be robust against momentum relaxation. In a quasi-particle picture
the robustness of anomalous transport can be argued on the grounds that the
anomalous transport stems from the lowest Landau level. Then we can view the
relevant degrees of freedom as $1+1$ dimensional chiral fermions which
naturally suppresses backscattering.

We base our treatment on anomalous transport of right- and left-handed
fermions in a magnetic field. The corresponding symmetry is the vector-
symmetry $U(1)_{V}=U(1)_{L}\otimes U(1)_{R}$. The magnetic field acts on both
chiralities in the same way.

The anomaly coefficients determining the chiral currents in a magnetic field
are therefore $d_{LLV}$, $d_{RRV}$, $b_{L}$ and $b_{R}$ defined in eqs.
(76),(77). Due to the Nielsen-Ninomiya theorem the number of left-handed Weyl
cones has to be equal to the number of right-handed Weyl cones
($n_{R}=n_{L}$). Under the left-right symmetry $U(1)_{L}\otimes U(1)_{R}$ the
left-handed fermions have charges $(1,0)$ whereas the right-handed fermions
have charge $(0,1)$. The anomaly coefficients are then

$d_{RRV}=-d_{LLV}=b_{R}=-b_{L}=1\,.$ (106)

for a single Weyl cone depending on its chirality. The Nielsen-Ninomiya
theorem guarantees that the vector symmetry is anomaly free, i.e.
$d_{VVV}=b_{V}=0$. Due to this simple structure it is sufficient to consider
only one Weyl cone. The total current can be obtained by summing over the
currents from each Weyl cone.

The anomaly induced energy and charge currents in a magnetic field in a single
Weyl cone are

$\displaystyle\vec{J}^{\epsilon}_{L,R}$
$\displaystyle=\pm\left(\frac{\mu^{2}}{8\pi^{2}}+\frac{T^{2}}{24}\right)\vec{B}\,,$
(107) $\displaystyle\vec{J}_{R,L}$
$\displaystyle=\pm\frac{\mu}{4\pi^{2}}\vec{B}\,.$ (108)

The sign is determined by the anomaly coefficients (106). These expression are
obtained from (90) and (91) by going to the restframe $u^{\mu}=(1,0,0,0)$ and
identifying the energy current as $T^{0i}=J_{\epsilon}^{i}$. Now we want to
study the effect of applying an external electric field (or a gradient in the
chemical potential) and a thermal gradient to the system. Because the anomaly
acts on each Weyl cone we find from anomalous current conservation

$\displaystyle\delta\dot{\epsilon}$
$\displaystyle=-\vec{\nabla}\cdot\vec{J}_{\epsilon}+\vec{E}\cdot\vec{J}\,,$
(109) $\displaystyle\delta\dot{\rho}$
$\displaystyle=-\vec{\nabla}\cdot\vec{J}\pm\frac{1}{4\pi^{2}}\vec{E}\cdot\vec{B}\,.$
(110)

The first line is energy conservation taking into account the Joule-heating
term and the second line is the anomalous conservation law of the number
density of Weyl fermions in a single cone. Using Eqs. (107) and (108) allows
to compute the growth rates of energy and number densities in a single Weyl
cone as

$\displaystyle-i(\omega+\frac{i}{\tau_{5}})\delta\epsilon$
$\displaystyle=\mp\left(\frac{\mu}{4\pi^{2}}(\vec{\nabla}\mu-\vec{E})\cdot\vec{B}+\frac{T}{12}\vec{\nabla}T\cdot\vec{B}\right)\,,$
(111) $\displaystyle-i(\omega+\frac{i}{\tau_{5}})\delta\rho$
$\displaystyle=\mp\frac{1}{4\pi^{2}}(\vec{\nabla}\mu-\vec{E})\cdot\vec{B}\,.$
(112)

We also have already performed a Fourier transform in time. As before we cut
off the infinite growth at zero frequency by the intervalley relaxation time.
This allows us to simply set $\omega=0$ and compute $\delta\epsilon$ and
$\delta\rho$. Now we express the energy and number densities by the chemical
potentials using the susceptibility matrix $\chi$

$\left(\begin{array}[]{c}\delta\epsilon\\\
\delta\rho\end{array}\right)=\chi\cdot\left(\begin{array}[]{c}\delta T\\\
\delta\mu\end{array}\right),$ (113)

where

$\chi=\left(\begin{array}[]{cc}\frac{\partial\epsilon}{\partial
T}&\frac{\partial\epsilon}{\partial\mu}\\\ \frac{\partial\rho}{\partial
T}&\frac{\partial\rho}{\partial\mu}\end{array}\right).$ (114)

We can plug these increments of the temperatures and chemical potentials into
the anomaly induced currents

$\displaystyle\delta\vec{J}^{\epsilon}$
$\displaystyle=\pm\left(\frac{\mu\delta\mu}{4\pi^{2}}+\frac{T}{12}\delta
T\right)\vec{B}\,,$ (115) $\displaystyle\delta\vec{J}$
$\displaystyle=\pm\frac{1}{4\pi^{2}}\delta\mu\vec{B}\,.$ (116)

Finally we can put everything together and compute the induced energy and
charge current for each single Weyl cone. In matrix notation it is
conveniently expressed as

$\displaystyle\left(\begin{array}[]{c}\delta J^{\epsilon}_{i}\\\ \delta
J_{i}\end{array}\right)$
$\displaystyle=\Sigma_{ij}\cdot\left(\begin{array}[]{c}-\partial_{j}T\\\
E_{j}-\partial_{j}\mu\end{array}\right)\,,$ (121) $\displaystyle\Sigma_{ij}$
$\displaystyle=\tau_{5}\left(\begin{array}[]{cc}\frac{T}{12}&\frac{\mu}{4\pi^{2}}\\\
0&\frac{1}{4\pi^{2}}\end{array}\right)\cdot\chi^{-1}\cdot\left(\begin{array}[]{cc}\frac{T}{12}&\frac{\mu}{4\pi^{2}}\\\
0&\frac{1}{4\pi^{2}}\end{array}\right)B_{i}B_{j}\,.$ (126)

The final result does not depend on the chirality. Both chiralities contribute
in the same way.

###  V.5 Weak magnetic field

Let us apply this result now to a gas of free chiral fermions. We also assume
here that the magnetic field is small compared to chemical potential $\mu$ and
and temperature $B$, i.e. $B/T^{2}\ll 1$ and $B/\mu^{2}\ll 1$. The free energy
density is Le Bellac (2000)

$F=-\frac{1}{24\pi^{2}}\left(\mu^{4}+2\pi^{2}T^{2}\mu^{2}+7/15\pi^{4}T^{4}\right),$
(127)

from which the energy $\epsilon=F-\frac{\partial F}{\partial T}+\mu\rho$ and
the number density $\rho=-\frac{\partial F}{\partial\mu}$ and the
susceptibility matrix $\chi$ can be calculated. The conductivity matrix as
defined in (121) is

$\Sigma_{ij}=\left(\begin{array}[]{cc}\kappa_{ij}&\beta_{ij}\\\
\alpha_{ij}&\sigma_{ij}\end{array}\right),$ (128)

where the different entries are the anomaly induced contributions to the
different conductivities

$\displaystyle\mathrm{thermal~{}conductivity}:\kappa_{ij}$
$\displaystyle=\tau_{5}B_{i}B_{j}\frac{15\mu^{2}T-5\pi^{2}T^{3}}{24\Delta}$
(129) $\displaystyle\mathrm{electro-thermal~{}conductivity}:\beta_{ij}$
$\displaystyle=\tau_{5}B_{i}B_{j}\frac{11\pi^{2}T^{2}\mu+15\mu^{3}}{8\pi^{2}\Delta}$
(130) $\displaystyle\mathrm{thermo-electric~{}conductivity}:\alpha_{ij}$
$\displaystyle=-\tau_{5}B_{i}B_{j}\frac{5\mu T}{4\Delta}$ (131)
$\displaystyle\mathrm{electric~{}conductivity}:\sigma_{ij}$
$\displaystyle=\tau_{5}B_{i}B_{j}\frac{21\pi^{2}T^{2}+15\mu^{2}}{8\pi^{2}\Delta}\,,$
(132)

with $\Delta=7\pi^{4}T^{4}+6\pi^{2}T^{2}\mu^{2}+15\mu^{4}$.

###  V.6 Thermoelectric transport relations of the anomaly-induced transport
coefficients

We will now study how the anomaly induced conductivities collected in the
matrix $\Sigma_{ij}$ realize some standard electronic transport properties
such as the Onsager reciprocity relation, the Wiedemann-Franz law and the Mott
relation. Let us start with the Onsager reciprocity relations.

  * •

Onsager relations. They are a consequence of time-reversal symmetry and should
hold on general grounds. We note that the magnetic field breaks time-reversal
symmetry which means that Onsager’s relations hold if one replaces
$\vec{B}\rightarrow-\vec{B}$. The anomaly induced conductivity matrix
$\Sigma_{ij}$ is however quadratic in the magnetic field and therefore
insensitive to the sign of the magnetic field. The conductivity matrix
$\Sigma_{ij}$ is the response due to gradients in temperature, chemical
potential and electric field. Onsager relations are manifest however if the
conductivities are written as responses due to gradients in the thermodynamic
driving forces (Langrange multipliers) $\beta=1/T$ and $\gamma=-\mu/T$. The
Onsager reciprocity relations state then that the conductivity matrix should
be symmetric and positive. By transforming to the $\beta$, $\gamma$ basis

$\displaystyle\left(\begin{array}[]{c}J^{\epsilon}_{i}\\\
J_{i}\end{array}\right)$
$\displaystyle=L_{ij}\cdot\left(\begin{array}[]{c}\partial_{j}(\frac{1}{T})\\\
\frac{E_{j}}{T}-\partial_{j}(\frac{\mu}{T})\end{array}\right),$ (137)

and we find

$\displaystyle L_{ij}$
$\displaystyle=\Sigma_{ij}\cdot\left(\begin{array}[]{cc}T^{2}&0\\\ \mu
T&T\end{array}\right)=\frac{\tau_{5}B_{i}B_{j}}{\Delta}\left(\begin{array}[]{cc}\frac{1}{24\pi^{2}}(5\pi^{4}T^{5}+18\pi^{2}T^{3}\mu^{2}+45T\mu^{4})&\frac{T\mu}{8\pi^{2}}(11\pi^{2}T^{2}+15\mu^{2})\\\
\frac{T\mu}{8\pi^{2}}(11\pi^{2}T^{2}+15\mu^{2})&\frac{3T}{8\pi^{2}}(7\pi^{2}T^{2}+5\mu^{2})\end{array}\right),$
(142)

which is indeed symmetric and positive definite.

  * •

Wiedemann-Franz law and the Lorentz-ratio from anomalies. One of the basic
paradigms of quasiparticle transport is the Wiedemann-Franz law. It states
that at low temperature the ratio of the thermal conductivity for the heat
current and the electric conductivity is given by a constant, the Lorenz
number $L=\pi^{2}k_{B}^{2}/(3e^{2})$ times the temperature $T$. Here we sill
show that also the anomaly induced magnetoconductivities of a chiral fermion
obey the Wiedemann-Franz law. Let us define the heat current by

$\vec{J}_{Q}=\vec{J}^{\epsilon}-\mu\vec{J}$ (143)

where we use the currents from eq. (121). The thermal conductivity in the heat
current is defined by

$J_{Q,i}=-\hat{\kappa}_{ij}\partial_{j}T$ (144)

Alternatively we could use the definition of the anomaly induced entropy
current (96). Note that the net entropy current in equilibrium vanishes since
the contributions form left-handed and right-handed Weyl fermions cancel. But
upon applying the temperature gradient and electric field they are shifted out
of their equilibrium values as in (113). In particular the increment in
temperature is

$\delta
T=\mp\tau_{5}\frac{5\pi^{2}T^{3}+15\mu^{2}T}{2\Delta}\vec{B}.\vec{\nabla}T\,.$
(145)

Using this expression for the entropy current

$J_{S,i}=\pm\frac{\delta
T}{12}B_{i}=-\tau_{5}\frac{5\pi^{2}T^{3}+15\mu^{2}T}{24\Delta}B_{i}B_{j}\partial_{j}T\,.$
(146)

From this expression for the anomaly induced entropy current we can give
another definition of heat current as

$\vec{J}_{Q}=T\vec{J_{S}}\,.$ (147)

While the two definitions of heat current look very different the do indeed
give the same result eq. (146)!

Now we can build the ratio of the diagonal parts of the thermal and electric
conductivities (we also restore the Boltzmann constant $k_{B}$ and the unit of
electric charge $e$)

$\frac{\hat{\kappa}_{ii}}{\sigma_{ii}}=\frac{5k_{B}^{2}\pi^{2}(\pi^{2}k_{B}^{2}T^{3}+3\mu^{2}T)}{e^{2}9(7\pi^{2}k_{B}^{2}T^{2}+5\mu^{2})}=\frac{\pi^{2}k_{B}^{2}}{3e^{2}}T+O(T^{2}/\mu^{2})\,.$
(148)

To leading order in $\mu/T$ this expression gives the usual Lorenz ratio. The
anomaly induced magnetic conductivities fulfil the Wiedemann-Franz law in the
regime of small temperature and large chemical potential.

  * •

Seebeck coefficient and Mott formula. The Seebeck coefficient describes how
much voltage is built up when applying a temperature gradient at vanishing
electric current. From