# Response functions of many-body condensed matter systems

Marcello Porta  Mathematics Area, SISSA, Via Bonomea 265, 34136 Trieste, Italy
Vieri Mastropietro  Mathematics Department, University of Milan, Via C.
Saldini 50, 20133 Milan, Italy  Alessandro Giuliani  Mathematics and Physics
Department, Roma 3 University, Largo S. L. Murialdo 1, 00146 Roma, Italy

###### Abstract

We discuss rigorous results about the response functions of gapped and gapless
interacting fermionic lattice models, relevant for the study of transport in
condensed matter physics.

###### Contents

  1. 1 Introduction
  2. 2 Interacting lattice models
  3. 3 Dynamics and linear response
  4. 4 Quantum Hall effect
  5. 5 Transport in semimetals
    1. 5.1 Graphene
    2. 5.2 Weyl semimetals
  6. 6 Transport in $1d$ and quasi-$1d$ metallic systems
    1. 6.1 Interacting $1d$ systems
    2. 6.2 Edge modes of interacting quantum Hall systems

##  1 Introduction

A large part of modern condensed matter physics is devoted to the study of the
transport properties of many-body quantum systems. In many physical
applications, the collective behavior of complex systems can only be
understood according to the laws of quantum mechanics. Despite the complexity
of the microscopic structure of the material, in some cases the response
properties of the system appear to be universal, that is they depend only on
fundamental constants of nature. The paradigmatic example is the integer
quantum Hall effect, where the quantization of the transverse conductivity of
a class of ultrathin materials has a topological interpretation. Other
examples include the minimal optical conductivity in graphene and the magneto-
electric effect in Weyl semimetals, whose universal nature can be understood
via the mechanism of anomaly non-renormalization, a key ingredient in the
quantization of gauge theories.

Because of the extremely accurate experimental precision in the measurement of
such transport phenomena, it is important to develop a theoretical
understanding that does not rely on uncontrolled approximations or ad hoc
regularizations. In this view, mathematical physics plays an important role in
the fundamental understanding of the behavior of complex systems from first
principles. As a matter of fact, the study of transport in condensed matter
physics provided a strong motivation for many works in mathematical physics in
recent years, and it stimulated cross-fertilization between various areas of
mathematics and physics. As a concrete example, our precise understanding of
the quantum Hall effect, and nowadays of the field of topological phases of
matter, has been made possible by insights coming from various branches of
mathematics, ranging from the theory of topological invariants to functional
analysis and probability.

In this article we will give an overview of rigorous results about the
transport properties of many-body condensed matter systems. The framework will
be the one of quantum statistical mechanics of interacting lattice models of
fermionic type. We will give an overview of some of the recent progress, which
has been obtained via the combination of several tools in the analysis of
many-body quantum systems, such as Lieb-Robinson bounds, cluster expansion,
renormalization group, adiabatic theory and Ward identities.

The article is organized as follows. In Section 2 we will define the class of
lattice models we will consider, in Fock space. In Section 3 we will introduce
the time-evolution of equilibrium Gibbs states under external perturbations,
we will define linear response, and we will briefly review rigorous results
about the validity of linear response. Then, in the rest of the article we
will discuss rigorous results about the computation of response functions for
many-body systems. In Section 4 we will discuss gapped many-body systems and
the quantization of the Hall conductivity; in Section 5 we will focus on
interacting gapless systems of semimetallic type; while in Section 6 we will
discuss metallic transport, for interacting one-dimensional systems and for
the edge currents of two-dimensional many-body quantum Hall systems.

##  2 Interacting lattice models

We shall describe interacting fermionic particles on a $d$-dimensional
lattice, as models for electrons on crystalline solids in the tight-binding
regime. An interesting question, not discussed in this article, is to justify
the tight-binding approximation from the continuum Schrödinger equation, in
the presence of periodic potentials describing the location of the ions
forming the crystal. We refer the reader to e.g. [39, 44, 73, 102, 104] and
references therein, for the tight-binding reduction in a semiclassical regime,
or to the more recent works [51, 52], for the discussion of graphene-like
structures.

We shall describe such lattice models in the language of statistical
mechanics, in the grand-canonical ensemble. Let $\Gamma_{L}$ be a
$d$-dimensional lattice of side $L$ with periodic boundary condition,
$\Gamma_{L}=\mathbb{Z}^{d}/L\mathbb{Z}^{d}$. We will suppose that each site is
decorated by $M$ colors, that might take into account internal degrees of
freedom such as the spin or the sublattice label. We shall collect together
space labels and color labels in the decorated lattice
$\Lambda_{L}=\Gamma_{L}\times\\{1,\ldots,M\\}$. We shall use the notation
${\bf x}=(x,\sigma)$ for points on $\Lambda_{L}$, with $x\in\Gamma_{L}$ the
space label and $\sigma\in\\{1,\ldots,M\\}$ the color label.

The single-particle Hilbert space for one quantum particle on $\Lambda_{L}$ is
given by $\mathfrak{h}=\ell^{2}(\Lambda_{L})$, which we can view as
$\mathbb{C}^{ML^{d}}$. The dynamics is generated by a self-adjoint Hamiltonian
$H$ on $\mathfrak{h}$, via the Schrödinger equation,

$i\partial_{t}\psi_{t}=H\psi_{t}\;,\qquad\psi_{0}=\psi\in\mathbb{C}^{ML^{d}}\;.$
(2.1)

where we set $\hbar=1$.

Eq. (2.1) describes the dynamics of one particle on $\Lambda_{L}$. Instead, we
will be interested in describing many interacting fermionic particles on
$\Lambda_{L}$. We will work in a grand-canonical setting, in which the number
of particles is not fixed. To this end, we introduce the fermionic Fock space
as $\mathcal{F}=\bigoplus_{n}\mathfrak{h}^{\wedge n}$. Notice that, being $L$
and $M$ finite, the fermionic Fock space $\mathcal{F}$ is finite dimensional.
Vectors in $\mathcal{F}$ are (finite) sequences of complex-valued
antisymmetric functions,
$\psi=(\psi^{(0)},\psi^{(1)},\ldots,\psi^{(n)},\ldots)$, labelled by the
particle number $n$. Let us introduce the fermionic creation and annihilation
operators $a^{*}_{{\bf x}}$ and $a_{{\bf x}}$, for ${\bf x}\in\Lambda_{L}$,
satisfying the canonical anticommutation relations [36, 37]:

$\\{a^{*}_{{\bf x}},a_{{\bf y}}\\}=\delta_{{\bf x},{\bf y}}\;,\qquad\\{a_{{\bf
x}},a_{{\bf y}}\\}=\\{a_{{\bf x}},a_{{\bf y}}\\}=0\;.$ (2.2)

Given the vacuum vector $\Omega=(1,0,\ldots,0)$, the fermionic Fock space can
be generated by iterative application of the creation operators on the vacuum.
Also, operators on $\mathcal{F}$ can be represented as polynomials in the
creation and annihilation operators. A simple example is the number operator
$\mathcal{N}=\sum_{{\bf x}\in\Lambda_{L}}a^{*}_{{\bf x}}a_{{\bf x}}$, which
counts the number of particles in a given sector of the Fock space:
$(\mathcal{N}\psi)^{(n)}=n\psi^{(n)}$.

The dynamics in Fock space is generated by a self-adjoint Hamiltonian
$\mathcal{H}$ on $\mathcal{F}$. Given an operator $\mathcal{O}$, its
Heisenberg evolution generated by $\mathcal{H}$ will be denoted by
$\tau_{t}(\mathcal{O})$:

$\tau_{t}(\mathcal{O})=e^{i\mathcal{H}t}\mathcal{O}e^{-i\mathcal{H}t}\qquad\text{$t\in\mathbb{R}$}\;.$
(2.3)

A typical form of Hamiltonian, that includes many physically relevant cases,
is

$\mathcal{H}=\sum_{X\subseteq\Lambda_{L}}\Phi_{X}\;,$ (2.4)

with $\Phi_{X}=\Phi_{X}^{*}$ given by a polynomial in the fermionic creation
and annihilation operators labelled by ${\bf x}\in X$, vanishing for
$\text{diam}(X)>R$ for some $R>0$. As $L$ varies, Eq. (2.4) actually defines a
sequence of Hamiltonians labelled by $L$; if $R$ is independent of $L$, we
shall say that $\mathcal{H}$ is finite-ranged. An important example of such
Hamiltonians is provided by many-body perturbations of non-interacting models,

$\begin{split}\mathcal{H}&=\mathcal{H}_{0}+\lambda\mathcal{V}\\\
\mathcal{H}_{0}&=\sum_{{\bf x},{\bf y}\in\Lambda_{L}}a^{*}_{{\bf x}}H_{0}({\bf
x};{\bf y})a_{{\bf y}}\\\ \mathcal{V}&=\lambda\sum_{{\bf x},{\bf
y}\in\Lambda_{L}}v({\bf x};{\bf y})a^{*}_{{\bf x}}a^{*}_{{\bf y}}a_{{\bf
y}}a_{{\bf x}}\;,\end{split}$ (2.5)

with $H_{0},v$ finite-ranged. The quadratic operator $\mathcal{H}_{0}$ is the
second quantization of the single-particle Hamiltonian $H_{0}$, while the
quartic operator $\mathcal{V}$ is the second quantization of the many-body
interaction, specified by the two-body potential $v$ and coupling strength
$\lambda$.

The Gibbs state of the system, describing its equilibrium state at inverse
temperature $\beta$ and chemical potential $\mu$, is denoted by
$\langle\cdot\rangle_{\beta,\mu,L}$, and it is defined as:

$\langle\mathcal{O}\rangle_{\beta,\mu,L}=\frac{\operatorname{Tr}_{\mathcal{F}}\mathcal{O}e^{-\beta(\mathcal{H}-\mu\mathcal{N})}}{\operatorname{Tr}_{\mathcal{F}}e^{-\beta(\mathcal{H}-\mu\mathcal{N})}}\;.$
(2.6)

Eq. (2.6) defines the statistical average of the observable associated with
the operator $\mathcal{O}$. Being $\mathcal{F}$ finite dimensional, the
expression (2.6) is obviously well-defined. However, at this level of
generality it is typically extremely hard to extract information from the
Gibbs state. For instance, given two observables $\mathcal{O}_{X}$,
$\mathcal{O}_{Y}$, a typical question is about decay of their connected
correlation function at equilibrium:

$\langle\mathcal{O}_{X};\mathcal{O}_{Y}\rangle_{\beta,\mu,L}=\langle\mathcal{O}_{X}\mathcal{O}_{Y}\rangle_{\beta,\mu,L}-\langle\mathcal{O}_{X}\rangle_{\beta,\mu,L}\langle\mathcal{O}_{X}\rangle_{\beta,\mu,L}\;,$
(2.7)

as $1\ll\text{dist}(X,Y)\ll L$. The answer to this question strongly depends
on the spectral properties of $\mathcal{H}$. Consider the case of weakly
interacting fermionic models, Eq. (2.5). Suppose that $\mu$ belongs to a
spectral gap of $\mathcal{H}_{0}$:

$\text{dist}(\sigma(\mathcal{H}_{0}),\mu)\geq\delta\;,$ (2.8)

for some $\delta>0$ uniformly in $L$. For $|\lambda|$ small enough, uniformly
in $\beta$ and $L$, exponential spatial decay estimates for the correlation
functions can be proved thanks to the convergence of the fermionic cluster
expansion, based on the Brydges-Battle-Federbush formula for fermionic
truncated expectations; see e.g. [17, 38, 57, 86, 88, 105, 107]. This result
actually extends to prove exponential decay of Euclidean, or imaginary time,
correlations. Let $\gamma_{t}(\mathcal{O})$ be the imaginary-time evolution of
$\mathcal{O}$:

$\gamma_{t}(\mathcal{O})=e^{t(\mathcal{H}-\mu\mathcal{N})}\mathcal{O}e^{-t(\mathcal{H}-\mu\mathcal{N})}\;.$
(2.9)

In terms of the imaginary-time evolution operator, the Gibbs state satisfies
the Kubo-Martin-Schwinger (KMS) identity:

$\langle\mathcal{A}\mathcal{B}\rangle_{\beta,\mu,L}=\langle\gamma_{\beta}(\mathcal{B})\mathcal{A}\rangle_{\beta,\mu,L}\;,$
(2.10)

which in the present finite-dimensional setting trivially follows from the
cyclicity of the trace. If $\mu$ belongs to a spectral gap of
$\mathcal{H}_{0}$ as in Eq. (2.8), the convergence of the cluster expansion
for $|\lambda|$ small enough allows to prove that, for all $t\in[0,\beta)$:

$|\langle\gamma_{t}(\mathcal{O}_{X});\mathcal{O}_{Y}\rangle_{\beta,\mu,L}|\leq
Ce^{-c\text{dist}(X,Y)-ct_{\beta}}\;,$ (2.11)

with $t_{\beta}=\min(t,\beta-t)$. The constant $C$ depends on the observables,
while the constant $c$ depends only on $\delta$ and on $|\lambda|$. This type
of estimate can actually be used to establish the existence of a spectral gap
for the many-body Hamiltonian $\mathcal{H}$, [43] for $|\lambda|$ small. More
generally, in the absence of a spectral gap, the bound (2.11) still holds at
$\beta<\infty$, however with $\beta$-dependent constants, and with a
$\beta$-dependent smallness condition on $\lambda$.

The requirement of $|\lambda|$ small is due to the fact that the proof of
(2.11) is based on a convergent expansion for Gibbs state of $\mathcal{H}$
around the Gibbs state of $\mathcal{H}_{0}$. If one assumes the existence of a
spectral gap of $\mathcal{H}$, the analogue of (2.11) for $t=0$ at zero
temperature can be proved with different, non-perturbative methods, [70, 99].
Concerning decay of real-time correlation functions for many-body systems, at
the moment no rigorous methods are available to prove such bounds for non-
integrable many-body systems.

In the absence of a spectral gap, one does not expect exponential decay of
correlations uniformly in $\beta$. Nevertheless, in some cases of physical
relevance discussed later on, such as $1d$ metallic systems, and $2d$ and $3d$
semimetals, algebraic decay of ground state correlations can be proved for
weak interactions. This requires the combination of fermionic cluster
expansion with rigorous renormalization group methods, in order to resolve the
infrared singularity that plagues naive perturbation theory around the Gibbs
state of $\mathcal{H}_{0}$, and to cure it by a renormalization of physical
quantities, such as the Fermi velocity or the critical exponents.

Rigorous renormalization group methods for gapless condensed matter systems
have been pioneered in [20, 49], see [21, 88, 107] for reviews. These methods
have been used to study the equilibrium state of interacting Fermi systems
with extended Fermi surface at positive temperatures [45, 46, 28],
exponentially small in the interaction strength, and up to zero temperature
for systems with asymmetric Fermi surface [50]. In one dimensional systems,
these methods have been used to construct the interacting ground state,
starting from the works [22, 34], relying on the exact integrability of the
Luttinger model [94]. This property turns out to imply the vanishing of the
beta function for the effective quartic coupling, which is marginal in the
renormalization group sense. The integrability of the Luttinger model holds at
the Hamiltonian level; such key property is broken by the presence of cut-
offs, typically needed in order to set up any renormalization group analysis,
and this makes the implementation of the integrability of the Luttinger model
within the renormalization group analysis particularly delicate. A different
strategy has been introduced in [24, 29, 89], where the role of integrability
is replaced by the combination of Schwinger-Dyson equations and Ward
identities for a regularized version of the Luttinger field theory, which
allows to compute all correlation functions of the field theory and to
estimate the effect of cut-offs.

More recently, rigorous renormalization group methods have been applied to
study semimetallic systems in two and three dimensions, such as graphene and
Weyl semimetals, and to study transport properties at zero temperature. These
extensions will be at the center of the present paper, and will be discussed
in more detail below.

##  3 Dynamics and linear response

We are interested in the dynamical properties of the interacting lattice
models defined in the previous section. To begin, notice that the Gibbs state
$\rho_{\beta,\mu,L}\propto e^{-\beta(\mathcal{H}-\mu\mathcal{N})}$ is
obviously invariant under the time-evolution generated by $\mathcal{H}$. In
order to have a nontrivial time-evolution of the state, we have to introduce a
perturbation in the Hamiltonian, and evolve the initial equilibrium state
according to the new dynamics. A standard procedure is to follow an adiabatic
protocol: namely, we introduce a slow time dependence in the Hamiltonian,

$\mathcal{H}(\eta t)=\mathcal{H}+\varepsilon g(\eta t)\mathcal{P}\;,$ (3.1)

where: $\mathcal{P}$ is a sum of finite-range potentials, similarly to (2.5);
$\varepsilon$ is a small parameter; $g(t)$ is a switch function, namely a
smooth function such that $g(-\infty)=0$ and $g(0)=1$; and $\eta>0$ allows to
tune the rate of variation of the switch function. We will be interested in
the situation in which $\eta$ is small, uniformly in all the other parameters
in the model. A typical choice of switch function in applications is
$g(t)=e^{t}$, which we shall consider from now on.

The evolution of the system is defined by the Schrödinger-von Neumann equation
for the state:

$\begin{split}i\partial_{t}\rho(t)&=[\mathcal{H}(\eta t),\rho(t)]\;,\qquad
t\leq 0\;,\\\ \rho(-\infty)&=\rho_{\beta,\mu,L}\;.\end{split}$ (3.2)

The question addressed here is: what is the variation of the expectation
values of physical observables due to the slow drive, for $t\leq 0$? To this
end, it is useful to represent the variation of the expectation value of the
observable $\mathcal{O}$ due to the perturbation $\mathcal{P}$ as:

$\begin{split}&\frac{1}{\varepsilon}\Big{(}\operatorname{Tr}\mathcal{O}\rho(0)-\operatorname{Tr}\mathcal{O}\rho(-\infty)\Big{)}\\\
&=-i\int_{-\infty}^{0}dt\,e^{\eta
t}\operatorname{Tr}\,\big{[}\mathcal{O},e^{i\mathcal{H}t}\mathcal{P}e^{-i\mathcal{H}t}\big{]}\rho_{\beta,\mu,L}\\\
&\quad+\mathcal{R}^{\beta,L}_{\mathcal{O},\mathcal{P}}(\varepsilon,\eta)\\\
&\equiv\chi^{\beta,L}_{\mathcal{O},\mathcal{P}}(\eta)+\mathcal{R}^{\beta,L}_{\mathcal{O},\mathcal{P}}(\varepsilon,\eta)\;,\end{split}$
(3.3)

where the first equality follows from the Duhamel expansion in $\varepsilon$
for the non-autonomous dynamics (3.2). The quantity
$\chi^{\beta,L}_{\mathcal{O},\mathcal{P}}(\eta)$ is the linear response of the
observable $\mathcal{O}$ due to the perturbation $\mathcal{P}$, while
$\mathcal{R}^{\beta,L}_{\mathcal{O},\mathcal{P}}(\varepsilon,\eta)$ is of
higher order in $\varepsilon$. In many physical applications, one is
interested in computing the linear response coefficient
$\chi^{\beta,L}_{\mathcal{O},\mathcal{P}}(\eta)$ as $\beta,L\to\infty$, and
possibly $\eta\to 0^{+}$. A complementary question is to prove the smallness
in $\varepsilon$ of the error term
$\mathcal{R}^{\beta,L}_{\mathcal{O},\mathcal{P}}(\varepsilon,\eta)$, uniformly
in $\eta$, $\beta$, $L$, which allows to rigorously justify the validity of
linear response.

For gapped non-interacting systems, the validity of linear response at zero
temperature, i.e. for $\beta\to\infty$, is related to the adiabatic theorem of
quantum mechanics [35, 82]; see [109] for a mathematical review of adiabatic
theory. The standard adiabatic theorem however does not allow to study many-
body quantum systems uniformly in their size. Recently, the adiabatic theorem
has been extended to the many-body setting in [13], in a way that also allows
to prove validity of linear reponse, see [14, 74] for reviews and for further
references. The adaptation from quantum spin systems to fermionic models has
been discussed in [97]. In the case of the quantum Hall effect, the linear
response approximation can actually be proved to be exact, see [84] for non-
interacting fermions and [15] for the extension to many-body systems. The
many-body adiabatic theorem of [13] applies to the setting in which the time-
dependent Hamiltonian $\mathcal{H}(\eta t)$ has a non-degenerate ground state
separated by the rest of the spectrum by a positive gap, for all times and
uniformly in $L$. The extension to time-dependent perturbations that might
close the spectral gap has been obtained in [110]. Finally, the extension to
infinite systems, with a uniform gap or with a bulk gap, has been discussed in
[75, 76].

It is an important open problem to extend the current many-body adiabatic
methods, and the validity of linear response, to positive temperature systems.
The case of small systems coupled to infinite reservoirs at nonzero
temperature, under suitable ergodicity assumptions on the reservoirs, has been
discussed in [1, 2, 77]. Concerning the adiabatic theorem and validity of
linear response for extended many-body systems, recent progress for small
temperatures, vanishing as $\eta\to 0^{+}$ but uniformly in $L$, has been
obtained in [66], via the combination of a rigorous analytic continuation
(Wick rotation) to imaginary times of the full Duhamel expansion for the non-
autonomous dynamics (3.2), and cluster expansion methods to prove the
convergence of the resulting series. It is a challenging open problem to
understand the interplay of the many-body adiabatic dynamics and approach to
equilibrium at fixed positive temperature; see [78, 79] for further recent
insights on the topic.

In the rest of this article, we will focus on rigorous results about the
linear response contribution $\chi^{\beta,L}_{\mathcal{O},\mathcal{P}}(\eta)$,
for various physical systems. One of the main difficulties in studying this
object is the control of the time integral, in the adiabatic limit $\eta\to
0^{+}$. In fact, it is extremely difficult to gain direct insight on the decay
of real-time correlation functions for non-integrable many-body systems. One
way to approach this problem is by complex deformation, by writing the
transport coefficient in terms of Euclidean, or imaginary-time, correlation
functions; at least for the case of weakly interacting fermionic models,
Euclidean correlation functions can be estimated via fermionic cluster
expansion. Let $\eta_{\beta}\in\frac{2\pi}{\beta}\mathbb{Z}$. Then, the
following identity holds true, as a consequence of the KMS property (2.10) and
of Cauchy theorem for analytic function:

$\chi^{\beta,L}_{\mathcal{O},\mathcal{P}}(\eta_{\beta})=-\int_{0}^{\beta}ds\,e^{-i\eta_{\beta}s}\langle\gamma_{s}(\mathcal{P})\;;\mathcal{O}\rangle_{\beta,\mu,L}\;,$
(3.4)

with $\gamma_{s}(\mathcal{P})$ the Euclidean time evolution of $\mathcal{P}$,
recall (2.9). Given $\eta>0$, let $\eta_{\beta}\geq\eta$ be its best
approximation in $\frac{2\pi}{\beta}\mathbb{Z}$ (obviously,
$\eta_{\beta}-\eta\leq 2\pi/\beta$). Then, using only the boundedness of the
fermionic operators and Lieb-Robinson bounds, see [100] for a review, it is
possible to prove that:

$\Big{|}\chi^{\beta,L}_{\mathcal{O},\mathcal{P}}(\eta_{\beta})-\chi^{\beta,L}_{\mathcal{O},\mathcal{P}}(\eta)\Big{|}\leq\frac{C}{\eta^{d+2}\beta}\;,$
(3.5)

where the constant $C$ depends only on the support of the observable
$\mathcal{O}$ (and it is independent of $L$ if $\mathcal{O}$ is local). See
e.g. [7, 92] for a proof of (3.4), (3.5), and [66] for an extension to higher
commutators. Eqs. (3.4), (3.5) allow to apply methods of Euclidean quantum
field theory to the study of real-time quantum dynamics, and in particular to
the evaluation of transport coefficients as $\beta\to\infty$ and $\eta\to
0^{+}$.

##  4 Quantum Hall effect

The quantum Hall effect is the paradigmatic example of topological transport
phenomenon in condensed matter physics. It has been discovered experimentally
in [114], and the first theoretical explanation dates back to the seminal
works [85, 113]. The quantum Hall effect is a measurement of the charge
current propagating in a $2d$ system, exposed to a transverse magnetic field
and to an in-plane electric field. The measurement is performed at
temperatures of the order of the Kelvin, and for a class of materials of
insulating type.

To describe the experimental observation, let us denote by $j_{i}$, $i=1,2$,
the current density propagating in the two orthogonal directions on the plane.
Let us denote by $E=(E_{1},E_{2})$ the electric field. Then, neglecting higher
order terms, the current in the steady state and the electric field are
related by the linear response formula:

$j=\sigma E\;,$ (4.1)

where $\sigma=(\sigma_{ij})_{1\leq i,j\leq 2}$ is the conductivity matrix. The
entries $\sigma_{11}$, $\sigma_{22}$ are called the longitudinal
conductivities, while the entries $\sigma_{12}$, $\sigma_{21}$ are called the
transverse conductivities. From a phenomelogical viewpoint, the insulating
behavior of the system is associated to $\sigma_{11}=\sigma_{22}=0$: this
means that transport is purely non-dissipative. Furthermore, if the system is
invariant under rotations, one also has $\sigma_{12}=-\sigma_{21}$. In units
such that $e^{2}=\hbar=1$, the integer quantum Hall effect (IQHE) is the
observation that the transverse, or Hall, conductivity satisfies:

$\sigma_{12}\in\frac{1}{2\pi}\cdot\mathbb{Z}\;,$ (4.2)

with a nine-digits experimental precision. The Hall conductivity turns out to
be piecewise constant in the density of charge carries, and to display sudden
jumps from one plateau to another, corresponding to a transition from
different integers in (4.2). This behavior is in glaring contrast with the
expected classical behavior of $\sigma_{12}$, which in an idealized system
would be linear in the density of charge carriers. See e.g. [115] for a review
of the experimental discovery of this phenomenon. Nowadays, quantum Hall
systems are recognized as paradigmatic examples of topological phase of
matter, see e.g. [96] for a recent review on the topic.

The IQHE is a special case of the fractional quantum Hall effect (FQHE), which
predicts the quantization of $\sigma_{12}$ in rational multiples of $1/2\pi$.
In constrast to the IQHE, the theoretical understanding of the FQHE starting
from microscopic models is still an outstanding problem in mathematical
physics. See e.g. [33, 55] for reviews of the effective field theoretic
approach to this phenomenon. In what follows we will restrict the attention to
the IQHE for lattice many-body quantum systems.

Let us consider many-body Hamiltonians of the form (2.5) in dimension $d=2$.
We define the lattice current associated with ${\bf x},{\bf y}$ in
$\Lambda_{L}$ as:

$j_{{\bf x},{\bf y}}=ia^{*}_{{\bf x}}H({\bf x};{\bf y})a_{{\bf
y}}+\text{h.c.}\;.$ (4.3)

These operators are related to the time variation of the density operator
$n_{{\bf x}}=a^{*}_{{\bf x}}a_{{\bf x}}$ via the continuity equation:

$\partial_{t}\tau_{t}(n_{{\bf x}})=\sum_{{\bf
y}\in\Lambda_{L}}\tau_{t}(j_{{\bf x},{\bf y}})\;.$ (4.4)

The right-hand side of (4.4) is a lattice divergence. To see this, let $Q_{X}$
be the charge operator associated with $X\subset\Lambda_{L}$,
$Q_{X}=\sum_{{\bf x}\in X}n_{{\bf x}}$. Then, Eq. (4.4) implies that the time-
variation of $Q_{X}$ is due to a flow across the boundary of $X$:

$\partial_{t}\tau_{t}(Q_{X})=\sum_{\begin{subarray}{c}{\bf x}\in X\\\ {\bf
x}\notin X\end{subarray}}\tau_{t}(j_{{\bf x},{\bf y}})\;.$ (4.5)

Next, we introduce the total current operator as:

$\mathcal{J}_{k}=\frac{1}{2}\sum_{{\bf x},{\bf
y}\in\Lambda_{L}}(x_{k}-y_{k})j_{{\bf x},{\bf y}}\;,\qquad k=1,2\;,$ (4.6)

which can be viewed as $i[\mathcal{H},\mathcal{X}_{k}]$ with $\mathcal{X}_{k}$
the second quantization of the $k$-th component of the position operator.
Notice that in general the position operator is not well defined in the
presence of periodic boundary conditions; nevertheless, being the Hamiltonian
short-ranged, we can interpret $i[\mathcal{H},\mathcal{X}_{k}]$ as the right-
hand side of (4.6). The total current operator can be viewed as the value at
$p=(0,0)$ of:

$\hat{j}_{p,k}=\frac{1}{2}\sum_{{\bf x},{\bf y}\in\Lambda_{L}}\frac{e^{ip\cdot
y}-e^{ip\cdot x}}{ip\cdot(y-z)}(y-x)_{k}j_{{\bf x},{\bf z}}\;,$ (4.7)

which satisfies the momentum-space continuity equation:

$\partial_{t}\tau_{t}(\hat{n}_{p})=\sum_{k=1,2}ip_{k}\tau_{t}(\hat{j}_{p,k})\;.$
(4.8)

We are interested in describing the variation of the average current, after
having introduced a weak electric field. The coupling with the electromagnetic
field is introduced via the standard procedure used in lattice gauge theories,
the Peierls’ substitution:

$H({\bf x};{\bf y})\to H({\bf x};{\bf y})e^{i\int_{x\to y}d\ell\cdot
A(t,\ell)}\;,$ (4.9)

where $A$ is a time-dependent vector potential, defined in $\mathbb{R}^{2}$.
The integral in Eq. (4.9) is taken over a straight line connecting $x$ to $y$.
Let us suppose that $A(t,z)$ is constant in space, $A(t,z)\equiv A(t)$, and
$A(t)=e^{\eta t}A$, so that the vector potential generates the space-
homogeneous electric field $E(t)=-\eta e^{\eta t}A$.

Let $\mathcal{H}(\eta t)$ be the time-dependent Hamiltonian, coupled to the
gauge field via (4.9), and let $\rho(t)$ be the solution of the Schrödinger-
von Neumann equation (3.2). Let $\mathcal{J}_{k}(A(t))$ the current operator
in the presence of the gauge field. We are interested in computing the
variation

$\frac{1}{L^{2}}\operatorname{Tr}\mathcal{J}_{i}(A(0))\rho(0)-\frac{1}{L^{2}}\operatorname{Tr}\mathcal{J}_{i}\rho_{\beta,\mu,L}\;,$
(4.10)

at first order in the electric field. By performing one step of Duhamel
expansion in the gauge field, we see that the expression for the conductivity
matrix is, in the linear response approximation:

$\begin{split}&\sigma_{ij}^{\beta,L}(\eta)\\\ &=\quad\frac{i}{\eta
L^{2}}\Big{[}\Delta^{\beta,L}_{ij}+\int_{-\infty}^{0}ds\,e^{\eta
s}\Big{\langle}\big{[}\mathcal{J}_{i},\tau_{s}(\mathcal{J}_{j})\big{]}\Big{\rangle}_{\beta,\mu,L}\Big{]}\end{split}$
(4.11)

where
$\Delta^{\beta,L}_{i,j}=\langle[\mathcal{J}_{i},\mathcal{X}_{j}]\rangle$. The
mathematical problem that we address is about the computation of:

$\sigma_{ij}=\lim_{\eta\to
0^{+}}\lim_{\beta,L\to\infty}\sigma_{ij}^{\beta,L}(\eta)\;,$ (4.12)

for many-body systems with Hamiltonian (2.5). We will further specify that the
system is in an insulating phase. To this end, it is sufficient to assume that
the chemical potential $\mu$ lies in a spectral gap of the Hamiltonian,
uniformly in $L$.

Let us first briefly discuss the case of non-interacting systems. We refer the
reader to [5, 6] for mathematical reviews on the topic. For $T=0$, the state
of the system is completely specified by the Fermi projector
$P_{\mu}=\chi(H\leq\mu)$, with $\chi(\cdot)$ the characteristic function and
$\mu$ in a spectral gap of $H$. Under this assumption, the matrix elements of
the Fermi projector are known to decay exponentially fast:

$|P_{\mu}({\bf x};{\bf y})|\leq Ce^{-c|x-y|}\;.$ (4.13)

The infinite volume conductivity matrix can be written as a suitable trace per
unit volume via Kubo-Streda formula, see e.g. [5]:

$\sigma_{ij}=\lim_{L\to\infty}\frac{i}{L^{2}}\operatorname{Tr}\mathbbm{1}_{L}P_{\mu}[[P_{\mu},X_{i}],[P_{\mu},X_{j}]]\;,$
(4.14)

where the trace is over $\ell^{2}(\mathbb{Z}^{2};\mathbb{C}^{M})$ and
$\mathbbm{1}_{L}=\chi(x\in[-L/2,L/2]^{2})$. In particular, the longitudinal
conductivity $\sigma_{ii}$ is zero. Instead, $\sigma_{12}$ might be non-
vanishing, and it turns out to have a beautiful mathematical interpretation.
At first, suppose that $H$ commutes with translations on $\mathbb{Z}^{2}$.
Then, the Hamiltonian $H$ can be fibered in momentum space, as follows:

$H=\int_{\mathbb{T}^{2}}^{\oplus}dk\,\hat{H}(k)\;,$ (4.15)

where $\hat{H}(k)\in\mathbb{C}^{M\times M}$ is the Bloch Hamiltonian.
Similarly, the Fermi projector $P_{\mu}$ can be written as the direct integral
of the fibered projectors $\hat{P}_{\mu}(k)=\chi(\hat{H}(k)\leq\mu)$. In terms
of these objects, the Hall conductivity can be rewritten as an integral over
the Brillouin zone:

$\sigma_{12}=i\int_{\mathbb{T}^{2}}\frac{dk}{(2\pi)^{2}}\,\operatorname{Tr}\hat{P}_{\mu}(k)\Big{[}\partial_{1}\hat{P}_{\mu}(k)\,,\,\partial_{2}\hat{P}_{\mu}(k)\Big{]}\;,$
(4.16)

where the trace is now over $\mathbb{C}^{M}$. The key observation [9, 113] is
that the argument of the integral is a curvature, and the resulting expression
is equal $\frac{1}{2\pi}\cdot\mathcal{C}_{1}$, with $\mathcal{C}_{1}$ the
first Chern number of the Bloch bundle, which has $\mathbb{T}^{2}$ as base
space and $\text{Ran}\hat{P}_{\mu}(k)$ as fibers. Thus, the Hall conductivity
can only take values in $\frac{1}{2\pi}\cdot\mathbb{Z}$, and a nonzero value
is associated to a nontrivial topology of the Bloch bundle. A further
perspective on the triviality/nontriviality of the Bloch bundle is through the
localization properties of the Wannier functions, constructed via the Bloch
transform of the eigenstates of the Bloch Hamiltonian (also called Bloch
functions). Informally, a vanishing Hall conductivity is equivalent to the
exponential decay of the Wannier functions associated with the energy bands
below the Fermi level, while a nonzero Hall conductivity is equivalent to a
slow algebraic decay, implying the divergence of the squared position
operator. We refer the reader to [98] for a precise statement and for a proof
of this result.

This elegant explanation of the quantization of the Hall conductivity
apparently breaks down in the absence of translation invariance. This is of
course always the case in all physical applications, due to the presence of
unavoidable impurities on the sample. Remarkably, the quantization of
$\sigma_{12}$ survives the presence of disorder. Assuming the existence of a
spectral gap for $H$, this has been proved in [18] via the methods of non-
commutative geometry, and in [10] using functional analytic tools. Later, [5]
extended the result of [10] to the case in which the spectral gap is replaced
by a mobility gap, which is the relevant setting in the presence of Anderson
localization (that is, at strong disorder), an essential ingredient to
establish the existence of plateaux in the plot of $\sigma_{12}$ as a function
of the density of charge carries.

None of the mentioned rigorous results applies to interacting many-body
systems, e.g. $\lambda\neq 0$ in Eq. (2.5). A general field-theoretic approach
to the quantum Hall effect, based on the classification of effective actions
of non-relativistic Fermi gases at low length scales, has been introduced in
[53, 54], see [33, 55] for reviews. This approach describes quantum Hall
systems on domains with nontrivial boundaries, and it allows to understand the
quantization of the Hall conductivity in terms of a cancellation mechanism
between the anomaly of the Chern-Simons field theory describing the scaling
limit of the bulk degrees of freedom, and the anomaly of the chiral Luttinger
liquid describing the scaling limit of the edge degrees of freedom.

A first attempt in the direction of formulating the many-body conductivity
matrix as a topological invariant in the spirit of [113] has been proposed in
[11]. There, following the original intuition of Laughlin [85], it is argued
that the many-body Hall conductivity can be computed by probing the response
of the system to a ‘phase twist’ of the boundary conditions, physically
equivalent to the introduction of a suitable magnetic flux in the system.
Under an extra averaging assumption over a second magnetic flux and the
assumption that the ground state of the system is gapped and non-degenerate,
one can show that the many-body Hall conductivity is quantized [11].

The problem of removing the extra averaging assumption in [11] remained open
for a long time [12]. The proof that the extra averaging assumption can be
relaxed has been given in [71], thus giving the first proof of quantization of
the Hall conductivity for a many-body quantum system up to exponentially small
corrections in the size of the system, under a spectral gap assumption for the
many-body ground state.

Coming back to the formulation of the conductivity matrix as linear response
with respect to a constant electric field, Eq. (4.11), a different route to
quantization is to show that Eq. (4.12) is actually constant in the strength
of the many-body interaction, provided the spectral gap of $\mathcal{H}$ does
not close along the path that connects the Hamiltonian to its noninteracting
counterpart ($\lambda=0$). The persistence of the spectral gap is a
consequence of the convergence of the fermionic cluster expansion, which can
be proved for $|\lambda|$ small enough, and which allows to prove exponential
decay of correlations (2.11). After proving that the conductivity matrix stays
constant along the path that deforms the model into a non-interacting one,
quantization of $\sigma_{12}$ follows from the known results for non-
interacting systems. This approach to quantization has been introduced in
[60], for translation-invariant systems, and it has been extended in [61, 62]
to study the topological phase transition of the Haldane-Hubbard model,
including the construction of an interacting critical line across which
$\sigma_{12}$ has a jump discontinuity. The starting point of the approach is
the rigorous Wick rotation, that allows to represent the conductivity matrix
in terms of Euclidean correlation functions, recall Eqs. (3.4), (3.5). Then,
the constancy of the conductivity matrix along the path of Hamiltonians that
deforms $\mathcal{H}$ into $\mathcal{H}_{0}$ is proved using lattice Ward
identities, following from the conservation of lattice current (4.8), in a way
inspired by the non-renormalization of the so-called topological mass in QED3,
[41].

Another approach to quantization of $\sigma_{12}$ has been proposed in [16],
where the quantization of the Hall conductivity for gapped many-body systems
is proved via a many-body index theorem for the ground state projector.
Assuming a suitable degeneracy condition for the many-body ground state, the
result of [16] also shows fractional quantization of the response coefficient.
Proving the required degeneracy assumption on the many-body projectors
starting from an interacting microscopic model is a difficult open problem in
mathematical physics.

All the above results have been obtained for many-body systems under a
spectral gap assumption. A limitation of this setting is that it does not
allow to prove the emergence of plateaux in the plot of the Hall conductivity
as a function of the density of charge carriers. As for the non-interacting
case, in order to understand the emergence of plateaux, one should include
strong disorder in the microscopic model. It is an important problem to
understand the interplay of interactions and disorder effects in many-body
quantum Hall systems.

##  5 Transport in semimetals

So far, we have focused on two-dimensional gapped many-body systems, for which
transport is non-dissipative, and where the transverse conductivity exhibits
interesting quantization properties. In the absence of a spectral gap, it
turns out that the transport properties of the system strongly depend on the
nature of the low-energy excitations at the Fermi level. In this section we
shall consider translation-invariant models, with Bloch Hamiltonian
$\hat{H}(k)\in\mathbb{C}^{M\times M}$, $k\in\mathbb{T}^{d}$. Given a value of
the chemical potential $\mu$, the Fermi surface is defined as:

$\Big{\\{}k\in\mathbb{T}^{d}\,\,\big{|}\,\,\mu\in\sigma(\hat{H}(k))\Big{\\}}\;.$
(5.1)

An insulator corresponds to the situation in which the Fermi surface is empty.
Instead, a metallic system corresponds to the situation in which the Fermi
surface has co-dimension $1$. The standard example is the one of the lattice
Laplacian $-\Delta_{\mathbb{Z}^{d}}$ and $\mu$ chosen within the spectrum of
the Laplacian. For this class of systems, the zero-temperature conductivity
matrix is typically infinite.

Insulators and metals do not exhaust all possible physical situations.
Semimetals are a class of physical systems for which the Fermi surface is
nonempty but nevertheless the conductivity matrix is finite. Here we shall
review the cases of graphene and of Weyl semimetals.

###  5.1 Graphene

Graphene is a paradigmatic example of semimetal in two dimensions, consisting
of a one-atom thick layer of graphite [56]. Its Fermi surface in the absence
of doping is given by two points, $\\{k_{F}^{+},k_{F}^{-}\\}$. Around the two
Fermi points, the dispersion relation of graphene takes a peculiar shape,
mimicking the dispersion relation of $2+1$ dimensional massless Dirac
fermions, which is responsible for its remarkable transport properties [83].

The simplest tight-binding model for graphene is the Laplacian on the
honeycomb lattice, [116]. The honeycomb lattice can be viewed as the
superposition of two triangular sublattices, generated by the basis vectors
$a_{1}=(1/2)(3,\sqrt{3})$, $a_{2}=(1/2)(3,-\sqrt{3})$:

$\begin{split}\Lambda_{L}^{A}&=\Big{\\{}x_{1}a_{1}+x_{2}a_{2}\,\Big{|}\,0\leq
x_{i}\leq L-1\Big{\\}}\;,\\\
\Lambda_{L}^{B}&=\Lambda_{L}^{A}+(1,0)\;.\end{split}$ (5.2)

with edges connecting sites of $\Lambda^{A}_{L}$ with sites of
$\Lambda^{B}_{L}$ at distance one. Equivalently, the vertex set
$\Lambda_{L}^{A}\cup\Lambda^{B}_{L}$ of the honeycomb lattice can be thought
of as the union over $x\in\Lambda_{L}^{A}$ of the pairs of sites
$(x,x+(1,0))$; that is, as a decorated triangular lattice with two internal
degrees of freedom. The advantage of this representation is that, in contrast
to the honeycomb lattice, the triangular lattice is a Bravais lattice. Thus,
given $x=(x_{1},x_{2})\in\Lambda_{L}^{A}$, the single-particle wave function
$\psi(x)$ is a two-component function, collecting values in the two shifted
sublattices labelled by $x$, $\psi(x)=(\psi_{A}(x),\psi_{B}(x+(1,0)))$.

The Bloch Hamiltonian $\hat{H}(k)$ of the Laplacian on the honeycomb lattice
is:

$\hat{H}(k)=-t\begin{pmatrix}0&\Omega(k)\\\ \Omega(k)^{*}&0\end{pmatrix}\;,$
(5.3)

where $t>0$ is the hopping parameter and the complex-valued function
$\Omega(k)$ is given by:

$\Omega(k)=1+e^{-ik_{1}}+e^{-ik_{2}}\qquad k\in\mathbb{T}^{2}\;.$ (5.4)

The energy bands of the model are $\pm t|\Omega(k)|$, and they touch at the
two inequivalent Fermi points on $\mathbb{T}^{2}$:

$k_{F}^{+}=\Big{(}\frac{2\pi}{3},\frac{4\pi}{3}\Big{)}\;,\qquad
k_{F}^{-}=\Big{(}\frac{4\pi}{3},\frac{2\pi}{3}\Big{)}\;.$ (5.5)

Taking into account the two spin degrees of freedom, the half-filling
condition corresponds to the choice $\mu=0$, for which the Fermi surface is
given by $k_{F}^{+}$ and by $k_{F}^{-}$. This choice of chemical potential is
of course non-generic, but it is the physically relevant one to describe
neutral graphene. In proximity of the Fermi surface, the Bloch Hamiltonian can
be approximated by the expression

$\begin{split}\hat{H}(k^{\prime}+k_{F}^{\omega})&=-\frac{3t}{2}\begin{pmatrix}0&ik^{\prime}_{1}-\omega
k^{\prime}_{2}\\\ -ik^{\prime}_{1}-\omega k^{\prime}_{2}&0\end{pmatrix}\\\
&\quad+O(|k^{\prime}|^{2})\;,\end{split}$ (5.6)

which mimics, at low energy, the Hamiltonian of $2+1$ dimensional massless
Dirac fermions. We shall denote by
$\langle\cdot\rangle_{\beta,L}\equiv\langle\cdot\rangle_{\beta,0,L}$ the
corresponding Gibbs state.

In the absence of interactions, the Gibbs state of the system, and more
generally the Euclidean correlation functions of the system, are completely
specified by the two-point function,

$\big{\langle}{\bf T}\,\gamma_{t}(a^{+}_{{\bf x}})\gamma_{s}(a^{-}_{{\bf
y}})\big{\rangle}^{0}_{\beta,L}\qquad t,s\in[0,\beta)\;,$ (5.7)

with ${\bf T}$ the fermionic imaginary-time ordering and where ${\bf x},{\bf
y}\in\Lambda_{L}^{A}\times\\{A,B\\}$ label points on the honeycomb lattice,
understood as points on a decorated triangular lattice. In the absence of
interactions the Gibbs state is quasi-free, and hence higher order, time-
ordered Euclidean correlations can be computed from (5.7) by application of
the fermionic Wick’s rule. Let us focus on the ground state properties of the
system, in the infinite volume limit. We denote by
$\langle\cdot\rangle^{0}_{\infty}=\lim_{\beta\to\infty}\lim_{L\to\infty}\langle\cdot\rangle^{0}_{\beta,L}$
the (thermal) ground state average for the non-interacting system. Due to the
absence of a spectral gap at the Fermi level, it turns out that the ground
state correlation functions have a non-integrable space-time decay:

$\Big{|}\big{\langle}{\bf T}\,\gamma_{t}(a^{+}_{{\bf
x}})\gamma_{s}(a^{-}_{{\bf
y}})\big{\rangle}^{0}_{\infty}\Big{|}\sim\frac{1}{\|x-y\|^{2}+|t-s|^{2}}\;.$
(5.8)

This asymptotic behavior can be easily checked from an explicit computation,
starting from the Bloch Hamiltonian (5.3).

Let us now focus on many-body models for graphene. The same qualitative
algebraic decay of correlations can be proved for weakly interacting graphene
[58], with Hamiltonian $\mathcal{H}=\mathcal{H}_{0}+\lambda\mathcal{V}$, where
$\mathcal{H}_{0}$ is the second quantization of the lattice Laplacian,
$\mathcal{V}$ is a short-ranged many-body Hamiltonian, and $|\lambda|$ is
small. The analysis of [58] is based on rigorous renormalization group
methods, and on the convergence of the fermionic cluster expansion. The
universality of the critical exponents of graphene in the presence of weak
short range interactions ultimately relies on the fact that short-range
interactions are irrelevant in the renormalization group sense for this class
of systems. Nevertheless, many-body interactions do affect the large scale
behavior of the system, in the sense of a non-trivial wave function
renormalization and a non-trivial renormalization of the Fermi velocity.

The slow decay of correlations has remarkable consequences on the transport
properties of the system. Let $\langle\cdot\rangle_{\beta,L}$ be the Gibbs
state of weakly interacting graphene. We shall focus on the conductivity
matrix of the system, following [59]. Let
$\sigma^{\beta,L}_{ij}(\eta_{\beta})$ be the conductivity matrix of the system
(4.11) after Wick’s rotation:

$\begin{split}&\sigma^{\beta,L}_{ij}(\eta_{\beta})=\\\
&\frac{1}{\eta_{\beta}L^{2}A}\Big{[}i\Delta^{\beta,L}_{ij}+\int_{-\beta/2}^{\beta/2}ds\,e^{-i\eta_{\beta}s}\langle{\bf
T}\mathcal{J}_{i}\;;\gamma_{s}(\mathcal{J}_{j})\rangle_{\beta,L}\Big{]}\;,\end{split}$
(5.9)

where $A=3\sqrt{3}/2$ is the area of the lattice fundamental cell. Let
$\sigma_{ij}(\eta)=\lim_{\beta\to\infty}\lim_{L\to\infty}\sigma^{\beta,L}_{ij}(\eta_{\beta})$
be the ground-state conductivity matrix. Lattice Ward identities, following
from the conservation of the lattice current (4.8), can be used to express
$\sigma_{ij}(\eta)$ in terms of the Euclidean current-current correlation
function [59]:

$\begin{split}&\sigma_{ij}(\eta)=\\\
&\lim_{\beta,L\to\infty}\frac{1}{\eta_{\beta}L^{2}A}\Big{[}\int_{-\frac{\beta}{2}}^{\frac{\beta}{2}}ds\,\Big{(}e^{-i\eta_{\beta}s}-1\Big{)}\langle{\bf
T}\mathcal{J}_{i}\;;\gamma_{s}(\mathcal{J}_{j})\rangle_{\beta,L}\;.\end{split}$
(5.10)

The expression (5.10) is the starting point of a renormalization group
analysis of the conductivity matrix of graphene. A key role in the analysis is
played by the function:

$K_{ij}(\eta)=\lim_{\beta,L\to\infty}\frac{1}{L^{2}A}\int_{-\frac{\beta}{2}}^{\frac{\beta}{2}}ds\,e^{-i\eta_{\beta}s}\langle{\bf
T}\mathcal{J}_{i}\;;\gamma_{s}(\mathcal{J}_{j})\rangle_{\beta,L}$ (5.11)

in terms of which we can rewrite (5.10) as:

$\sigma_{ij}=\lim_{\eta\to
0^{+}}\frac{1}{\eta}\Big{(}K_{ij}(\eta)-K_{ij}(0)\Big{)}\;.$ (5.12)

The off-diagonal part of the conductivity matrix turns out to be zero, by
time-reversal symmetry [59]. Instead, the diagonal part of the conductivity
matrix, called the longitudinal conductivity, is expected to be non-zero: this
is related to the presence of a non-zero value of the minimal conductivity in
graphene [101]. Remarkably, as measured in [101], the minimal longitudinal
conductivity of graphene appears to be universal:

$\sigma_{ii}=\frac{e^{2}}{h}\frac{\pi}{4}\;,$ (5.13)

or equivalently $\sigma_{ii}=1/8$ in units such that $e^{2}=\hbar=c=1$. Thus,
up to very good precision, the longitudinal conductivity appears to depend
only on fundamental constants.

Concerning theoretical predictions, for non-interacting systems the value
(5.13) can be obtained via an explicit computation [112], starting from the
model (5.3). Remarkably, the expression found in [112] agrees with the
analogous quantity computed for $2+1$ dimensional massless Dirac fermions
[111].

The main challenge is thus to understand the universality of $\sigma_{ii}$ for
many-body models describing interacting graphene. This has been achieved in
[59]; let us sketch the strategy of the proof. Coming back to Eq. (5.12), this
expression shows that if the function that $K_{ij}(\eta)$ is differentiable at
$\eta=0$, then $\sigma_{ij}$ would simply be the derivative at $\eta=0$ of
$K_{ij}(0)$. At the same time, it is not difficult to see that $K_{ii}(\eta)$
is even in $\eta$ [59]. Thus, this symmetry combined with differentiability
would imply a trivial longitudinal conductivity, giving a result in
contradiction with the experimental observation [101].

The key point is that, due to the slow decay of correlations, already visible
in the absence of interactions (5.8), the function $K_{ij}(\eta)$ is
continuous but not differentiable at zero. A careful analysis of the
renormalized expansion for the current-current correlations of weakly
interacting graphene allows to isolate the singular contribution to
$K_{ii}(\eta)$ from its regular part:

$K_{ii}(\eta)=K^{\text{Dirac}}_{ii}(\eta)+K_{ii}^{\text{R}}(\eta)\;,$ (5.14)

where $K^{\text{Dirac}}_{ii}$ is the quantum field theory analogue of (5.11),
associated with the integration over a non-interacting $2+1$ dimensional
massless Dirac field, with renormalized Fermi velocity and wave function, and
with a fixed ultraviolet cutoff. Instead, the remainder term
$K^{\text{R}}(\eta)$ is much less explicit, and it collects irrelevant
contributions generated in the renormalization group analysis, such as those
taking into account the nonlinearity of the energy band away from the Fermi
points. Crucially, the term $K_{ii}^{\text{R}}(\eta)$ turns out to be
differentiable at $\eta=0$. By inspection, $K^{\text{Dirac}}_{ii}(\eta)$ is
even in $\eta$, which ultimately implies that $K_{ii}^{\text{R}}(\eta)$ is
also even, and hence, by its improved regularity, it does not contribute to
$\sigma_{ii}$. We thus find:

$\sigma_{ii}=\lim_{\eta\to
0^{+}}\frac{1}{\eta}\Big{(}K^{\text{Dirac}}_{ij}(\eta)-K^{\text{Dirac}}_{ij}(0)\Big{)}\;.$
(5.15)

The right-hand side of (5.15) is now amenable to an explicit computation. The
remaining interaction dependence, due to the finite renormalizations of the
Dirac propagator, turns out to cancel out exactly, which allows to prove the
validity of (5.13) for weakly interacting graphene [59].

###  5.2 Weyl semimetals

Weyl semimetals are three-dimensional electron systems, whose low energy
properties are well described by $3+1$ dimensional massless Weyl fermions, in
the same way in which massless Dirac fermions emerge in graphene.

The simplest setting one can use to model Weyl semimetals is the one of non-
interacting, three dimensional lattice fermions on a bipartite lattice, with
Bloch Hamiltonian $\hat{H}(k)$ on $\mathbb{T}^{3}$, such that

$\hat{H}(k^{\prime}+k_{F}^{\omega})=v_{1}\sigma_{1}k^{\prime}_{1}+v_{2}\sigma_{2}k^{\prime}_{2}+\omega
v_{3}\sigma_{3}k^{\prime}_{3}+O(|k^{\prime}|^{2})\;,$ (5.16)

where $k_{F}^{\omega}=(0,0,\omega k_{F,3})$ with $\omega=\pm$. The points
$k_{F}^{\pm}$ are called Weyl points. In Eq. (5.16), $\sigma_{i}$, $i=1,2,3$,
are the standard Pauli matrices, and the parameters $v_{i}$ play the role of
emergent velocities for the low-energy excitations. A concrete example of
microscopic lattice model with Bloch Hamiltonian satisfying Eq. (5.16) has
been introduced in [42]. Eq. (5.16) simulates a relativistic Hamiltonian at
energies in proximity of $\mu=0$. The two Weyl points introduce two effective
chiralities for the low energy excitations around $\mu=0$, while the two
sublattices play the role of spin degrees of freedom for the emergent
relativistic field.

One of the most peculiar phenomena of QED in $3+1$ dimensions is the presence
of the chiral anomaly. Consider the Lagrangian of QED for a massless field
$\psi$ coupled with a gauge field $A_{\mu}$,

$\mathcal{L}(\psi,A)=\int
dx\,\overline{\psi}_{x}\gamma^{\mu}(i\partial_{\mu}-A_{\mu,x})\psi_{x}\;.$
(5.17)

At the classical level, the invariance of the Lagrangian under $U(1)$ gauge
transformations and under chiral gauge transformations implies the following
conservation laws, by Noether’s theorem:

$\partial_{\mu}j_{\mu}=0\;,\qquad\partial_{\mu}j^{5}_{\mu}=0\;,$ (5.18)

with $j_{\mu}=\overline{\psi}\gamma_{\mu}\psi$ the current and
$j^{5}_{\mu}=\overline{\psi}\gamma_{\mu}\gamma_{5}\psi$ the chiral current. It
is well-known that these conservation laws might be broken in the quantization
of the gauge theory, because of the unavoidable presence of cutoffs; these are
needed to make sense of the theory, and might break the gauge symmetries of
the model. The typical question is whether the conservation laws are restored
after removing the cutoffs. As discovered in [3, 19], this is not the case for
the conservation of the chiral current. One has, in momentum space, and in
units such that $e^{2}=\hbar=c=1$:

$\begin{split}&(p_{1,\mu}+p_{2,\mu})\langle
j^{5}_{\mu,p_{1}+p_{2}}\;;j_{\nu,-p_{1}}\;;j_{\sigma,-p_{2}}\rangle^{QED}\\\
&\qquad=-\frac{1}{2\pi^{2}}\varepsilon_{\alpha\beta\nu\sigma}p_{1,\alpha}p_{2,\beta}\end{split}$
(5.19)

where $\varepsilon$ is the totally antisymmetic tensor, and
$\langle\cdot\rangle^{QED}$ denotes the average taken over the QED’s path
integral, expressed in terms of a (formal) sum over connected Feynman
diagrams, after renormalization. The fact that the right-hand side of (5.19)
is nonzero implies that the classical conservation law of the chiral current
in (5.18) does not survive quantization: it is an example of anomaly in
quantum field theory. The right-hand side of (5.19) takes an extremely simple
form, and it is manifestly not affected by radiative corrections. This
cancellation has been first discussed, order by order in perturbation theory,
in [4], showing that the only contribution to the right-hand side of (5.19)
arises from the so-called ‘triangle graph’. The physical consequence of the
chiral anomaly (5.19) is the decay of the neutral pion into photons.

In the condensed matter setting, the counterpart of the chiral anomaly (5.19)
has been first discussed in [103]. There, it was predicted that, for three
dimensional lattice models displaying the relativistic structure (5.16), a
condensed matter counterpart of the chiral anomaly in QED should take place,
after coupling the system to an external time-dependent gauge field. Calling
$\dot{N}_{+}$ and $\dot{N}_{-}$ the time variations of the charge around the
Fermi points with $\omega=+$, resp. $\omega=-$, the prediction of [103] is
that:

$\dot{N}_{+}-\dot{N}_{-}=\frac{1}{2\pi^{2}}E\cdot B\;.$ (5.20)

where $E$ and $B$ are the electric and magnetic field associated to the
external gauge field. Eq. (5.20) has to be understood as a steady charge flow
in momentum space from one Weyl node to the other, which mimics the creation
and annihilation of particles with opposite chiralities in QED. In the
condensed matter setting, this phenonenon is made possible by the fact that
the two Weyl nodes are connected via the energy bands. This transport
phenomenon has been experimentally observed in the last years, we refer the
reader to [8] for a review on Weyl semimetals and for their transport
properties.

We shall now describe the mathematical analysis of the phenomenon, following
[63]. We shall consider weakly interacting Weyl semimetals, with Hamiltonian
$\mathcal{H}=\mathcal{H}_{0}+\lambda\mathcal{V}$, where $\mathcal{H}_{0}$ is
the second quantization of a single-particle Hamiltonian with the properties
(5.16), and $\mathcal{V}$ is a short-ranged many-body interaction. We denote
by $\langle\cdot\rangle_{\beta,L}$ the Gibbs state of such Hamiltonian, in the
presence of a suitably chosen staggered chemical potential that ensures the
existence of two Weyl nodes for the interacting system as well.

The first non-trivial issue is to identify an operator which allows to capture
the chiral charge transfer. This is nontrivial, because the charge of the
emergent Weyl fermions cannot be defined exactly using local operators.
Nevertheless, one can identify operators whose correlation functions behave in
the desired way, at large scales. For instance, we consider:

$n^{5}_{{\bf x}}=\frac{iZ^{5}}{2}\big{(}a^{*}_{{\bf x}}a_{{\bf
x}+e_{3}}-a^{*}_{{\bf x}+e_{3}}a_{{\bf x}}\big{)}\;,$ (5.21)

with $e_{3}=(0,0,1)$. The parameter $Z^{5}$ is a suitable normalization, that
has to be fixed consistently with the renormalization of the usual charge.
Observe that, setting $N^{5}=\sum_{{\bf x}}n^{5}_{{\bf x}}$,

$N^{5}=Z^{5}\int\frac{dk}{(2\pi)^{2}}\sin k_{3}\hat{n}_{k}\;,$ (5.22)

with $\hat{n}_{k}$ the density operator in momentum space. In the integral
(5.22), thanks to the presence of $\sin k_{3}$, the contribution due to the
momenta close to $k_{F}^{\pm}$ mimics the difference of charges with opposite
chiralities. The parameter $Z^{5}$ is fixed imposing the equality of the
chiral vertex function with the charge vertex function, at low energy:

$\langle\hat{n}^{5}_{p};\hat{a}_{k+p};\hat{a}^{*}_{k}\rangle_{\infty}=\omega\langle\hat{n}_{p}\;;\hat{a}_{k+p}\;;\hat{a}^{*}_{k}\rangle_{\infty}(1+o(1))\;,$
(5.23)

where $o(1)$ is a quantity that vanishes for $k\to k_{F}^{\omega}$ and $p\to
0$.

We couple the Hamiltonian $\mathcal{H}$ to an external gauge potential
$e^{\eta t}A_{\mu,x}$, via the Peierls’ substitution (4.9), with $\eta>0$ and
$t\leq 0$, and we denote by $\mathcal{H}(\eta t)$ the corresponding time-
dependent Hamiltonian. We are interested in the quadratic variation in the
gauge field for the evolution of the expectation value of $N^{5}$. Notice
that, being the observable $N^{5}$ nonlocal, it also couples to the gauge
field via the introduction of a Wilson line, similarly to (4.9).

Let $\rho(t)$ be the solution of the Schrödinger equation (3.2). We consider
the quadratic response,

$\begin{split}&\Big{[}\partial_{t}\operatorname{Tr}N^{5}(A(t))\rho(t)\Big{]}^{(2)}\\\
&=\int\frac{dp}{(2\pi)^{3}}\hat{A}_{\mu,p}\hat{A}_{\nu,-p}\eta\Gamma^{5;\beta,L}_{\mu,\nu}((\eta,p),(\eta,-p))\;,\end{split}$
(5.24)

and the goal is to compute the response function $\Gamma^{5}_{\mu,\nu}$. The
result of [63] is that, for $|\lambda|$ small enough, setting
$\Gamma^{5}=\lim_{\beta\to\infty}\lim_{L\to\infty}\Gamma^{5;\beta,L}$:

$\begin{split}&2\eta\Gamma_{\mu,\nu}^{5}((\eta,p),(\eta,-p))\\\
&=-\frac{1}{2\pi^{2}}p_{1,\alpha}p_{2,\beta}\varepsilon_{\alpha\beta\mu\nu}+o(|p_{i}|^{2})\end{split}$
(5.25)

where $p_{1}=(\eta,p)$ and $p_{2}=(\eta,-p)$ are Euclidean momenta with four
components. Eq. (5.25), combined with (5.24), agrees with the prediction
(5.20), with the same prefactor. As for the universality of graphene’s
conductivity, the method of proof relies on rigorous renormalization group
methods, allowing us to isolate the scaling limit contribution to
$\Gamma_{\mu,\nu}^{5}$, which can be computed explicitly in terms of the
renormalized triangle graph, from a remainder term, which is not explicit but
enjoys better regularity properties. Universality follows from the combination
of lattice Ward identities, due to the conservation of the lattice current,
with emergent Ward identities, valid for the correlation functions of QED. Eq.
(5.25) can be viewed as a rigorous version of the non-renormalization
mechanism of Adler and Bardeen [4] for QED, for a short-ranged lattice model.
Remarkably, the non-renormalization of the lattice counterpart of the chiral
anomaly survives the breaking of rotation symmetry due to the lattice. The
proof relies on lattice conservation laws, and holds at a nonperturbative
level.

##  6 Transport in $1d$ and quasi-$1d$ metallic systems

In the previous section we discussed condensed matter systems whose Fermi
surface is formed by isolated points, around which the energy dispersion
relation displays a typical ‘relativistic’ shape. The Gibbs state in the
presence of weak many-body interactions can be constructed rigorously, via
renormalization group methods. For these models, many-body interactions of
order four and higher turn out to be irrelevant in the renormalization group
sense, and the large scale behavior for the system is effectively described in
terms of a non-interacting relativistic field, whose covariance is
renormalized by the integration of the degrees of freedom on smaller scales.

Here we discuss the transport properties of one-dimensional lattice models,
where the low energy excitations are again described by emergent relativistic
fields, but where the quartic many-body interaction turns out to be marginal
in the renormalization group sense. This means that the large scale behavior
of the system is described by an interacting quantum field theory. The
analysis of the emergent quantum field theory is facilitated by its special
integrability features, which are typical of the one-dimensional world. In the
following, we will focus on the response functions of one-dimensional quantum
systems, and of the edge currents of two-dimensional quantum Hall systems.

###  6.1 Interacting $1d$ systems

Let us consider interacting spinless fermions on
$\Lambda_{L}=[-L/2,L/2]\cap\mathbb{Z}$ with periodic boundary conditions. The
many-body Hamiltonian is $\mathcal{H}=\mathcal{H}_{0}+\lambda\mathcal{V}$
with:

$\begin{split}\mathcal{H}_{0}&=-t\sum_{x\in\Lambda_{L}}(a^{*}_{x}a_{x+1}+a^{*}_{x+1}a_{x})\\\
\mathcal{V}&=\sum_{x,y\in\Lambda_{L}}a^{*}_{x}a_{x}v(x-y)a^{*}_{y}a_{y}\;,\end{split}$
(6.1)

with $v(x-y)$ finite-ranged and $t>0$. The Gibbs state of the model is denoted
by $\langle\cdot\rangle_{\beta,\mu,L}$, where $\mu$ is chosen within the
spectrum of the infinite volume lattice Laplacian, $\mu\in(-2t,2t)$. For the
special choice of nearest-neighbour interactions, the Hamiltonian
$\mathcal{H}$ is also related to the Hamiltonian of the XXZ quantum spin
chain, via the Jordan-Wigner transformation, and it is therefore integrable
via Bethe ansatz. For generic choices of the interaction potential, however,
the model is not integrable. Unless otherwise stated, in the following we will
not assume any special integrability property.

We shall focus on the transport properties of the density and the current
operator, defined as:

$n_{x}=a^{*}_{x}a_{x}\;,\qquad
j_{x}=-it(a^{*}_{x}a_{x+1}-a^{*}_{x+1}a_{x})\;,$ (6.2)

related by the lattice continuity equation

$\partial_{t}\tau_{t}(n_{x})+\text{d}_{x}\tau_{t}(j_{x})=0\;,$ (6.3)

with $\text{d}_{x}$ the discrete derivative, $\text{d}_{x}f(x)=f(x)-f(x-1)$.

We will be interested in the variation of the density and of the current
operators, after introducing a slowly varying external potential $e^{\eta
t}\mu(x)$, or a slowly varying electromagnetic field generated by an external
vector potential $e^{\eta t}A_{x}$, compatible with the periodicity of
$\Lambda_{L}$. That is, the quantities of interest will be:

$\operatorname{Tr}n_{x}\rho(t)\;,\qquad\operatorname{Tr}j_{x}\rho(t)\;,$ (6.4)

where $\rho(t)$ is the solution of the Schrödinger-von Neumann equation, with
time-dependent Hamiltonian:

$\begin{split}\mathcal{H}(\eta t)&=\mathcal{H}+\varepsilon e^{\eta
t}\sum_{x\in\Lambda_{L}}\mu(x)n_{x}\quad\text{(for the density)}\\\
\mathcal{H}(\eta t)&=\mathcal{H}_{0}(A_{t})+\lambda\mathcal{V}\quad\text{(for
the current)}\;,\end{split}$ (6.5)

where $\mathcal{H}_{0}(A_{t})$ is the free Hamiltonian coupled with an
external vector potential $e^{\eta t}A_{x}$, via the Peierls substitution
(4.9). We define the susceptibility $\kappa_{\beta,L}(\eta,p)$ and the Drude
weight $D_{\beta,L}(\eta,p)$ as the linear response of the average density and
of the average current:

$\begin{split}&\kappa_{\beta,L}(\eta,p)\\\
&\quad=\frac{i}{L}\int_{-\infty}^{0}dt\,e^{\eta
t}\langle[\hat{\rho}_{-p},\tau_{t}(\hat{\rho}_{p})]\rangle_{\beta,\mu,L}\\\
&D_{\beta,L}(\eta,p)\\\
&\quad=\frac{-i}{L}\Big{(}\int_{-\infty}^{0}dt\,e^{\eta
t}\langle[\hat{j}_{-p},\tau_{t}(\hat{j}_{p})]\rangle_{\beta,\mu,L}+\Delta_{\beta,L}\Big{)}\;,\end{split}$
(6.6)

with $\Delta^{\beta,L}=\langle[\mathcal{J},\mathcal{X}]\rangle_{\beta,\mu,L}$
(compare with Eq. (4.11)). In terms of these quantities, one can write the
linear response of the current and of the density in (6.4) as:

$\begin{split}\operatorname{Tr}\,n_{0}\rho(0)&=\frac{1}{L}\sum_{p}\hat{\mu}(p)\kappa_{\beta,L}(\eta,p)+O(\mu^{2})\\\
\operatorname{Tr}\,j_{0}\rho(0)&=\frac{1}{L}\sum_{p}\hat{A}_{p}D_{\beta,L}(\eta,p)+O(A^{2})\;.\end{split}$
(6.7)

The Drude weight is related to the conductivity of the model, defined as
$\sigma_{\beta,L}(\eta,p)=(1/\eta)D_{\beta,L}(\eta,p)$. Thus, a finite Drude
weight as $\eta\to 0^{+}$ implies a divergent conductivity, i.e. metallic
behavior. Several alternative definitions of the Drude weight exists, see [92]
for a review; we shall refer to the expression in (6.6) as the canonical Drude
weight.

As mentioned above, for nearest-neighbour interactions the Hamiltonian (6.1)
can be mapped to the XXZ model, which is exactly solvable via Bethe ansatz
[118]. As a consequence, in this special case the Drude weight and the
susceptibility can be explicitly computed at zero temperature. Much less is
known at positive temperature. There, the Drude weight of the XXZ chain can be
proved to be bounded below by a nonzero quantity [119] by the Mazur bound
[95]. More generally, it has been proposed that the Drude weight of many-body
one dimensional systems is nonzero or zero depending on whether the model is
integrable or not; we refer the reader to [32] for a review of this topic.

The question addressed here, which is amenable to a rigorous analysis, is
about the computation of the Drude weight and the susceptibility at zero
temperature, for non-integrable many-body quantum systems. Heuristic insight
on the value of such transport coefficients can be gained from the comparison
of the original lattice model (6.1) with its scaling limit, the Luttinger
model [87]. The Hamiltonian of this one-dimensional model is:

$\begin{split}\mathcal{H}_{\text{L}}&=\sum_{\omega=\pm}\int
dx\,\overline{\psi}_{\omega,x}ic_{\omega}\partial_{x}\psi_{\omega,x}\\\
&\quad+\sum_{\omega}\int
dxdy\,\overline{\psi}_{+,x}\psi_{+,x}w(x-y)\overline{\psi}_{-,y}\psi_{-,y}\end{split}$
(6.8)

where $\overline{\psi}$, $\psi$ are conjugate fermionic fields, with chirality
$\omega=\pm$, and velocities $c_{\omega}=\omega c$, and where $w(\cdot)$ is a
smooth and short-ranged potential. This model arises from the linearization of
the dispersion relation $\varepsilon(k)=-2t\cos(k)$ around the two Fermi
points $k_{F}^{\pm}$, which are solutions of the equation
$\mu=-2t\cos(k_{F}^{\omega})$. It is well-known that the Luttinger model can
be solved by bosonization, [94]: it can be mapped into a non-interacting
bosonic quantum field theory, and the main effect of the interaction is to
remove the discontinuity in the occupation number of the fermionic modes in
the ground state. More generally, the many-body interaction gives rise to the
appearence of interaction-dependent anomalous exponents in the scaling of the
correlation function. Notice that with respect to the original Luttinger
model, here we are considering a many-body interaction that is smooth and
slightly nonlocal. This does not affect the infrared properties of the model,
and introduces an ultraviolet regularization at small scales, which is
particularly convenient in the renormalization group analysis.

The Luttinger model plays a key role in the renormalization group analysis of
the original lattice model (6.1). It captures the large-scale behavior of the
correlation functions of the system, and its exact solvability allows to
compute the critical exponents of the lattice model even in the absence of
integrability; see [88] for a review of the results and of the renormalization
group analysis. In particular, the Euclidean two-point function of the model
(6.1) is given by, at zero temperature and in the infinite volume limit [23]:

$\begin{split}\big{\langle}{\bf
T}\,a_{0}\gamma_{t}(a^{*}_{x})\big{\rangle}_{\infty}&=g_{0}(t,x)\frac{1+O(\lambda)}{(t^{2}+v^{2}x^{2})^{\frac{\eta}{2}}}+R(x,t)\\\
g_{0}(t,x)&=\sum_{\omega=\pm}\frac{e^{i\omega k_{F}x}}{-it+\omega
vx}\;,\end{split}$ (6.9)

where: $v\equiv v(\lambda)$ is the interacting Fermi velocity, given at lowest
order in $\lambda$ by the slope of the energy-dispersion relation at the Fermi
level $\mu$; $R(x,t)$ is a faster-decaying error term as $t\to\infty$,
$x\to\infty$; and $\eta=a_{0}\lambda^{2}+O(\lambda^{3})$ with $a_{0}>0$ is the
anomalous exponent of the two-point function, analytic in $\lambda$ for
$|\lambda|$ sufficiently small (the exact solution of the Luttinger model [94]
provides the optimal existence and analyticity interval).

Renormalization group methods have been used to compute transport coefficients
of $1d$ chains, in particular the susceptibility and the Drude weight [30,
31]. The starting point of the papers [30, 31] is the formulation of these
transport coefficients in terms of Euclidean correlation functions, via the
use of a Wick rotation, that can be rigorously justified [92]. Let:

$\begin{split}D(\eta,p)&=\lim_{\beta\to\infty}\lim_{L\to\infty}D_{\beta,L}(\eta,p)\\\
\kappa(\eta,p)&=\lim_{\beta\to\infty}\lim_{L\to\infty}\kappa_{\beta,L}(\eta,p)\;.\end{split}$
(6.10)

Then, as proved in [30, 31, 92, 90]:

$\begin{split}D(\eta,p)&=\frac{vK}{\pi}\frac{\eta^{2}}{\eta^{2}+v^{2}p^{2}}+R_{D}(\eta,p)\\\
\kappa(\eta,p)&=\frac{K}{\pi
v}\frac{v^{2}p^{2}}{\eta^{2}+v^{2}p^{2}}+R_{\kappa}(\eta,p)\;,\end{split}$
(6.11)

where $K$ is the Luttinger parameter, related to the anomalous exponent of the
two-point function by the formula $2\eta=K+K^{-1}-2$, while the error terms
$R_{D},R_{\kappa}$ are continuous in a neighbourhood of $(0,0)$ and vanish in
the limit $(\eta,p)\to(0,0)$. In particular, setting

$D=\lim_{\eta\to 0^{+}}\lim_{p\to 0}D(\eta,p)\;,\quad\kappa=\lim_{p\to
0}\lim_{\eta\to 0^{+}}\kappa(\eta,p)\;,$ (6.12)

we obtain the following remarkable identity between Drude weight,
susceptibility and Fermi velocity:

$\frac{D}{\kappa}=v^{2}\;.$ (6.13)

The relation (6.13) was first predicted to hold by Haldane in [67] on the
basis of non-rigorous bosonization arguments, and it has been rigorously
established in [26, 27, 30, 31]. The identity (6.13) relates three non-
universal quantities for non-integrable $1d$ systems in an exact way, and it
is one of the defining properties of the Luttinger liquid universality class
[67, 68]. Recently, a similar ‘Haldane relation’ has been discovered in the
completely different setting of an interacting dimer model in classical
statistical mechanics [64], which turns out to belong to the Luttinger liquid
universality class. In the context of classical statistical mechanics models,
other remarkable identities for non-universal scaling exponents of non-
solvable models belonging to the Luttinger liquid universality class have been
proved in [25].

Finally, we conclude by mentioning that spinful fermions can also be studied,
at the price of a considerably more involved renormalization group analysis
[26, 27]. The main extra technical difficulty, solved in [26, 27], is that the
presence of the spin introduces further quartic marginal terms in the
effective quantum field theory description of the model. These terms cannot be
easily described in terms of emergent bosonic modes, and their renormalization
group flow must be controlled via direct inspection of the beta function.

###  6.2 Edge modes of interacting quantum Hall systems

One-dimensional physics also arises at the boundary of two-dimensional
topological insulators. The connection between the bulk topological order of
insulating materials and the emergence of stable, metallic quasi-$1d$ modes is
the content of the bulk-edge duality [69], a central concept in the theory of
topological materials.

To begin, let us consider non-interacting quantum Hall systems, in the half-
plane $\mathbb{Z}\times\mathbb{N}$. Let $H$ be the single-particle
Hamiltonian, a self-adjoint operator on
$\ell^{2}(\mathbb{Z}\times\mathbb{N};\mathbb{C}^{M})$. We impose Dirichlet
boundary conditions at $x_{2}=0$. We shall also view the operator $H$ as the
restriction to the half-plane of another Hamiltonian $H_{\text{B}}$, defined
on $\mathbb{Z}^{2}$.

Let us suppose that $H_{\text{B}}$ is gapped, and that the Fermi level $\mu$
lies in a spectral gap. The bulk-edge duality establishes an identity between
the value of the Hall conductivity of $H_{\text{B}}$, also called the bulk
Hall conductivity, and the emergence of edge modes for $H$. For simplicity,
let us further assume that both $H$ and $H_{\text{B}}$ are invariant under
translations along the direction of the edge. Then, the Hamiltonians admit a
partial Bloch decomposition,

$H=\int^{\oplus}_{\mathbb{T}^{1}}\frac{dk}{2\pi}\,\hat{H}(k)\;,\qquad
H_{\text{B}}=\int^{\oplus}_{\mathbb{T}^{1}}\frac{dk}{2\pi}\,\hat{H}_{\text{B}}(k)\;.$
(6.14)

Edge modes might appear for energies in the ‘bulk gap’, corresponding to the
spectral gap of $H_{\text{B}}$, and are associated to solutions of the
Schroedinger equation for $H_{\text{B}}$, localized in the proximity of
$x_{2}=0$:

$\hat{H}_{\text{B}}(k)\varphi_{\omega}(k)=\varepsilon_{\omega}(k)\varphi_{\omega}(k)\;,$
(6.15)

where $\varphi_{\omega}(k)\equiv\varphi_{\omega}(k,x_{2})$ is exponentially
decreasing in the bulk,

$|\varphi(k,x_{2})|\leq Ce^{-cx_{2}}\;.$ (6.16)

The decay rate $c$ can be estimated in terms of the distance of
$\varepsilon_{\omega}(k)$ to the rest of the spectrum of $\hat{H}(k)$.

The edge modes become relevant for the zero-temperature transport properties
of the system if the Fermi level intersects the edge modes dispersion
relations $\varepsilon_{\omega}(k)$ at some value of the quasi-momentum $k$.
We define the edge modes Fermi momenta as the solutions of:

$\mu=\varepsilon_{\omega}(k_{F}^{\omega})\;.$ (6.17)

Around the Fermi level, the energy-momentum dispersion relation of the edge
modes can be linearized,
$\varepsilon_{\omega}(k^{\prime}+k_{F}^{\omega})-\mu=v_{\omega}k^{\prime}+o(k^{\prime})$,
which suggests that the low energy excitations are effectively described by
massless relativistic particles with velocities $v_{\omega}$, exponentially
localized in proximity of the $x_{2}=0$ edge, recall (6.16).

In this setting, the bulk-edge duality relates the value of the bulk Hall
conductivity and the sum of the chiralities of the edge modes, defined as
$\chi_{\omega}=v_{\omega}/|v_{\omega}|$, encoding the directions of
propagation of the edge currents. This duality is encoded by the following
amazing identity:

$\sigma_{12}=\frac{1}{2\pi}\sum_{\omega}\chi_{\omega}\;.$ (6.18)

As we shall see below, the right-hand side of (6.18) can also be understood as
an edge response function, which describes the variation of the edge current
after introducing a perturbation localized at the boundary.

The first proof of (6.18) has been given in [72]. It has then been extended to
disordered systems whose Fermi energy lies in a spectral gap [47, 108], or in
a mobility gap [48]. More recently, the bulk-edge duality has been extended to
other classes of topological insulators, such as time-reversal invariant
systems [65]; see [106] for extensions to other topological phases. Field
theoretic methods for the classification of topological phases have been
introduced in [54], which allow to understand the bulk-edge duality in terms
of anomaly cancellations between the Chern-Simons effective field theory for
the bulk degrees of freedom, and the Luttinger liquid field theory arising on
the boundary.

Let us now focus on rigorous results for many-body quantum systems. We shall
consider weakly interacting fermions on the cylinder, namely on
$\Gamma_{L}=[0,L]^{2}\cap\mathbb{Z}^{2}$ with periodic boundary conditions in
the horizontal direction and Dirichlet boundary conditions at $x_{2}=0,L$. We
shall allow for internal degrees of freedom, and we shall denote by
$\Lambda_{L}$ the decorated lattice; recall the discussion in Section 2. The
points in $\Lambda_{L}$ are denoted by ${\bf x}=(x,\sigma)$, with
$x\in\Gamma_{L}$ the space coordinate and $\sigma\in\\{1,\ldots,M\\}$ the
color label. Let $H$ be a single-particle Hamiltonian, displaying edge modes,
in the sense of exponentially localized solutions of (6.15), at $x_{2}=0$ and
$x_{2}=L$. As usual, we shall denote by $\mathcal{H}_{0}$ the second
quantization of the Hamiltonian, and we shall consider many-body Hamiltonians
of the form $\mathcal{H}=\mathcal{H}_{0}+\lambda\mathcal{V}$, with
$\mathcal{V}$ a short-ranged many-body interaction. We also suppose that the
model is invariant under translations in the horizontal direction, which will
allow us to perform a partial Bloch reduction.

We are interested in the edge transport properties of this model. Let
$j_{i,{\bf x}}$ be the current density, such that $\mathcal{J}_{i}=\sum_{{\bf
x}\in\Lambda_{L}}j_{i,{\bf x}}$, and satisfying the lattice continuity
equation:

$\partial_{t}\tau_{t}(n_{{\bf
x}})+\sum_{i=1}^{2}\text{d}_{i}\tau_{t}(j_{i,{\bf x}})=0\;,$ (6.19)

with $\text{d}_{i}f(x)=f(x)-f(x-e_{i})$ the discrete derivative in the $i$-th
direction. We define the edge current $\mathcal{J}^{\ell}_{x_{1}}$ as:

$\mathcal{J}^{\ell}_{x_{1}}:=\sum_{\sigma}\sum_{x_{2}\leq\ell}j_{1,(x,\sigma)}\;.$
(6.20)

This operator is associated with the charge current flowing in a strip of
width $\ell\ll L$, adjacent to the $x_{2}=0$ boundary. In order to probe the
response of such current, we introduce the time-dependent Hamiltonian:

$\mathcal{H}(\eta t)=\mathcal{H}+e^{\eta t}\sum_{{\bf
y}\in\Lambda_{L}}\mu(y)a^{*}_{{\bf y}}a_{{\bf y}}$ (6.21)

where $\mu(y)\equiv\mu(y_{1})$ for $y_{2}\leq\ell^{\prime}$ and $\mu(y)=0$
otherwise. We will be interested in the variation of the average edge current,
at first order in the external perturbation, in the range of parameters
$1\ll\ell\ll\ell^{\prime}\ll L$.

We define the edge response function as:

$\begin{split}G^{\ell,\ell^{\prime}}_{\beta,L}(\eta,p)&:=-\frac{i}{L}\int_{-\infty}^{0}dt\,e^{\eta
t}\Big{\langle}\Big{[}\tau_{t}\Big{(}\hat{n}^{\ell^{\prime}}_{-p}\Big{)},\hat{\mathcal{J}}^{\ell}_{p_{1}}\Big{]}\Big{\rangle}_{\beta,L}\\\
G^{\ell,\ell^{\prime}}(\eta,p)&:=\lim_{\beta\to\infty}\lim_{L\to\infty}G^{\ell,\ell^{\prime}}(\eta,p)\;,\end{split}$
(6.22)

where
$\hat{n}^{\ell^{\prime}}_{p}=\sum_{\sigma}\sum_{x_{2}\leq\ell^{\prime}}\hat{n}_{p_{1},x_{2},\sigma}$
(the Fourier transform is over the variable $x_{1}$). In terms of this
function, the linear response of the edge current can be written as:

$\operatorname{Tr}\,\mathcal{J}^{\ell}_{0}\rho(0)=\frac{1}{L}\sum_{p}\hat{\mu}(p)G^{\ell,\ell^{\prime}}_{\beta,L}(\eta,p)+O(\mu^{2})\;.$
(6.23)

We will be interested in the edge response function in the limits $\eta\to
0^{+}$, $p\to 0$ and then $\ell,\ell^{\prime}\to\infty$ with
$1\ll\ell\ll\ell^{\prime}\ll L$, which is the relevant setting for studing
slowly varying edge perturbations. Notice that the order of this limits
matters: if one first takes $p\to 0$ and then $\eta\to 0^{+}$ one finds a
trivial result: $\lim_{\ell^{\prime}\to\infty}\lim_{p\to
0}G^{\ell,\ell^{\prime}}(\eta,p)=0$.

From a quantum field theory viewpoint, the large scale properties of the edge
modes are expected to be described by a generalization of the Luttinger model,
called the multi-channel Luttinger model, describing an arbitrary number of
chiral relativistic fermions, interacting via a density-density type
interaction. The Hamiltonian of the model is:

$\begin{split}&\mathcal{H}_{mL}=\sum_{\omega=1}^{M}\int
dx\,\overline{\psi}_{\omega,x}ic_{\omega}\partial_{x}\psi_{\omega,x}\\\
&\;+\sum_{\omega,\omega^{\prime}}\lambda_{\omega,\omega^{\prime}}\int
dxdy\,\overline{\psi}_{\omega,x}\psi_{\omega,x}w(x-y)\overline{\psi}_{\omega^{\prime},y}\psi_{\omega^{\prime},y}\;,\end{split}$
(6.24)

where the sum runs over the labels of $M$ chiral fermions, representing the
edge modes at the Fermi level, with velocities $c_{\omega}$, and the couplings
$\lambda_{\omega,\omega^{\prime}}$ describe the edge mode scattering. As for
the Luttinger model (6.8), this quantum field theory can be studied via
bosonization. It is the starting point of the chiral Luttinger liquid theory
of edge modes [117], a non-rigorous field theory approach to edge transport in
the quantum Hall effect.

Recently, the validity of this effective quantum field theory for the large-
scale properties of edge currents has been rigorously proved; see [7] for the
case of one edge mode, [91] for two counterpropagating edge modes with
opposite spins, and [93] for the generic case of an arbitrary number of edge
states. A key ingredient of the analysis is the vanishing of the beta function
for the multi-channel Luttinger model [93], which allows to control the flow
of the quartic marginal couplings, and to prove that the correlation functions
decay with interaction-dependent anomalous exponents.

The renormalization group analysis, combined with Ward identities for the
lattice model and for the effective quantum field theory, actually allows to
compute the limiting edge response function. We obtain [93], for $|\lambda|$
small enough, and for $1\ll\ell\ll\ell^{\prime}$:

$\lim_{p\to 0}\lim_{\eta\to
0^{+}}G^{\ell,\ell^{\prime}}(\eta,p)=\sum_{\omega}\frac{\chi_{\omega}}{2\pi}+O(e^{-c\ell})\;,$
(6.25)

with $\\{\chi_{\omega}\\}$ the chiralities of the non-interacting edge modes.
In particular, taking the limits $\ell^{\prime}\to\infty$ followed by
$\ell\to\infty$, we see that the edge response function is quantized, and it
is equal to the sum of the chiralities of the edge modes. Thus, combined with
the universality of the Hall conductivity [60], Eq. (6.25) allows to lift the
bulk-edge duality to the realm of weakly interacting quantum Hall systems.
Furthermore, the method also allows to compute the edge Drude weight and the
edge susceptibility. In the case of one and two edge modes, this has been done
in [7, 91]. In particular, these results allow to check the validity of the
Haldane relation (6.13) for the edge response functions in these cases.

Other ways of probing edge currents are possible, involving different material
geometries. For instance, the two-terminal conductance is obtained by
connecting a Hall bar to two leads at different chemical potential, and by
measuring the current propagating in the bar at first order in the difference
of the chemical potential of the source and of the drain. For clean and non-
interacting samples, it is actually expected that the two-terminal conductance
is given by the sum of the absolute values of the chiralities. This quantity
is not expected to be protected against many-body interactions between
counterpropagating edge modes [80, 81]. A possible mechanism to restore
quantization in a way compatible with the bulk-edge duality relies on the
introduction of disorder [80, 81], a relevant perturbation in the
renormalization group sense. See [40] and references therein for recent
experimental studies of the two-terminal conductance in the context of the
fractional quantum Hall effect. The rigorous understanding of this phenomenon
is an interesting open problem in mathematical physics.

Acknowledgements. A. G. and M. P. gratefully acknowledge financial support of
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (ERC CoG UniCoSM, grant agreement n. 724939
for A.G.; and ERC StG MaMBoQ, grant agreement n.80290 for M. P.). A.G. and
V.M. gratefully acknowledge financial support of MIUR, PRIN 2017 project
MaQuMA, PRIN201719VMAST01. Our work has been carried out under the auspices of
the GNFM of INdAM.

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