# Hall response of locally-correlated two-dimensional electrons at low density

Giacomo Morpurgo  Department of Quantum Matter Physics, University of Geneva,
1211 Geneva, Switzerland  Louk Rademaker  Department of Quantum Matter
Physics, University of Geneva, 1211 Geneva, Switzerland  Christophe Berthod
Department of Quantum Matter Physics, University of Geneva, 1211 Geneva,
Switzerland  Thierry Giamarchi  Department of Quantum Matter Physics,
University of Geneva, 1211 Geneva, Switzerland

(October 17, 2023)

###### Abstract

We study the Hall constant in a homogeneous two-dimensional fluid of
correlated electrons immersed in a perpendicular magnetic field, with special
focus on the regime of low carrier density. The model consists of a one-band
tight-binding model and a momentum-independent causal self-energy,
representing interaction-induced correlations effects that are restricted to
be local in space. We write general gauge-invariant equations for the
conductivity tensor at first order in the magnetic field and solve them
numerically —analytically when possible— in a minimal model of anisotropic
square lattice with constant self-energy. Our results show that deviations
from the universal behavior of the Hall constant, as observed in the
semiclassical regime, appear upon entering the quantum regime, where the Fermi
energy and interaction are comparable energy scales.

##  I Introduction

In the wake of graphene’s discovery [1], the study of gated two-dimensional
(2D) semiconductors has developed swiftly, providing a whole family of new
low-density conductors. The achievable carrier concentrations are sufficiently
low —as low as $10^{10}$–$10^{11}$ cm-2— to even realize the elusive
interaction-dominated Wigner crystal [2, 3, 4]. In gated moiré
heterostructures, new experiments provide a wealth of transport data in the
low-density regime, even through density-controlled metal-insulator
transitions [5]. In recent years, 2D magnetic insulators and conductors have
been uncovered as well [6, 7], some of which like CrSBr display puzzling
magnetotransport properties [8]. This situation prompts for further
theoretical research on the magnetotransport in 2D correlated conductors at
low carrier densities.

Doing so is not easy, since one needs to properly describe assemblies of
charge carriers in a regime where Fermi energy, scattering rate, and
temperature are comparable energy scales. This means that some of the usual
approximations such as the Boltzmannian semiclassical theory [9] cannot be
applied. This theory indeed regards carriers as true particles characterized
by well-defined energy and momentum and subject to Fermi statistics. However,
this semiclassical description reaches its limits when the quasiparticle
scattering rate is not small compared to either the quasiparticle energy —i.e.
typically $k_{\mathrm{B}}T$, where $T$ is the temperature— or the effective
Fermi energy $\mu$, which is proportional to the carrier density. In this
case, the spectral function of the electrons is broad and one must use the
quantum theory [10, 11]. However, the general first-principles many-body
approach poses considerable computational challenges and progress has been
difficult.

For “high” densities, the Hall response of correlated electrons was studied in
the Hubbard model using exact numerical methods on small systems [12, 13] or
approximate methods [14, 15, 16], both at weak [12, 14, 13] and strong
magnetic field [15, 16]. Progress could be made for interacting systems in
special situations such as quasi-one dimensional systems, made of weakly
coupled, strongly interacting chains [17, 18, 19] or more recently for ladders
for which the Hall effect can be accessed both numerically and analytically
[20, 21, 22] and probed in cold atomic gases experiments [23].

Extending such methods to the case of vanishing low densities would be highly
non trivial, so we follow here another route: we neglect the nonlocal
correlations between the charge carriers. At the single-particle level, this
means that the electrons are characterized by a dispersion relation
$E_{\bm{k}}$ and a complex self-energy function $\Sigma(\varepsilon)$ that is
local (independent of ${\bm{k}}$), like in the dynamical mean-field theory
[24, *Georges-1996]. The absence of vertex corrections, both at zero [26] and
finite magnetic field [15, 16], simplifies the calculation of the
conductivities significantly. This class of models thus provides an
interesting playground, intermediate between the approximate semiclassical
theory and the intractable quantum theory, where the magnetotransport can be
studied accurately in the thermodynamic limit. In this approach the self-
energy may be regarded as a phenomenological input, or taken from a
microscopic calculation.

In the present paper we use this local self-energy approach to study the Hall
effect in systems with low densities. The plan of the paper is as follows. In
Sec. II we present general gauge-invariant transport equations that capture
the DC Hall response of a 2D single-band tight-binding model of locally-
correlated electrons. Given a dispersion relation and a causal self-energy,
the transport equations can be solved numerically in all regimes of density
and temperature. In Sec. III we solve these equations in a minimal model,
where the correlations are represented by a single scattering rate $\Gamma$.
The simplicity of the model allows us to obtain exact asymptotic results, in
particular in the low-density regime. We compare the resulting Hall constant
$R_{\mathrm{H}}$ with the universal value $R_{\mathrm{H}}^{0}=-1/(n|e|)$.
Quite remarkably we find that in the model with a self-energy having neither
momentum, nor energy dependence, the Hall effect at low density is enhanced to
four times the semiclassical result. Moreover, we find a non-monotonic
evolution of the Hall factor with a pronounced maximum, when the temperature
is raised to values larger than the scattering rate. This effect is most
pronounced at low density, large scattering rate, and for a strongly
anisotropic (quasi-one dimensional) Fermi surface. We discuss further our
findings in Sec. IV. Some of the more technical points can be found in the
Appendices.

##  II System studied and transport equations

###  II.1 Model Hamiltonian and Green’s function

We study a 2D interacting one-band tight-binding model without spin-orbit
coupling immersed in a perpendicular magnetic field $\bm{B}$ as shown in Fig.
1. Its Hamiltonian is

$H=\sum_{\bm{r}_{1}\bm{r}_{2}\sigma}t^{0}_{\bm{r}_{1}\bm{r}_{2}}e^{i\frac{|e|}{\hbar}\int_{\bm{r}_{1}}^{\bm{r}_{2}}d\bm{r}\cdot\bm{A}(\bm{r})}c^{\dagger}_{\bm{r}_{1}\sigma}c^{\phantom{\dagger}}_{\bm{r}_{2}\sigma}+H_{\mathrm{int}},$
(1)

where $t^{0}_{\bm{r}_{1}\bm{r}_{2}}\equiv t^{0}_{\bm{r}_{1}-\bm{r}_{2}}$ is
the translation-invariant zero-field hopping amplitude from site $\bm{r}_{1}$
to site $\bm{r}_{2}$ of the lattice, $c^{\dagger}_{\bm{r}\sigma}$,
$c^{\phantom{\dagger}}_{\bm{r}\sigma}$ are the creation and annihilation
operators at site $\bm{r}$ for spin $\sigma$, and $\bm{A}$ is the vector
potential, which enters via the Peierls substitution. This Hamiltonian leads
to a dispersion relation
$E_{\bm{k}}=\sum_{\bm{\rho}}t^{0}_{\bm{\rho}}e^{-i\bm{k}\cdot\bm{\rho}}$,
where $\bm{k}$ is a wavevector and $\bm{\rho}$ is a vector linking two sites
in the lattice. Although the model can be totally generic, we specialize in
the applications that follow to a two dimensional anisotropic square lattice,
which means that $t^{0}_{\bm{\rho}}$ will become, for nearest neighbors
$t_{x}$ and $t_{y}$, depending on the direction of the coupling. This
anisotropy defines the smallest hopping as one additional energy scale that
can be tuned to be comparable with either Fermi energy, scattering rate, or
temperature. In this way, we scan a class of geometries ranging from the
quasi-1D case to the 2D case (isotropic hoppings).

Moreover, we assume in all this study that the interactions between particles,
described by a Hamiltonian $H_{\mathrm{int}}$, lead only to local correlations
and therefore to a _local self-energy_. The self-energy $\Sigma(\varepsilon)$
thus depends only on the energy $\varepsilon$ and not on the direction and
size of the momentum. As a consequence, the group velocity is not renormalized
by interactions, which in the diagrammatic language can be seen as the absence
of vertex corrections.

Using this self-energy allows us to define the spectral function
$A(\bm{k},\varepsilon)=(-1/\pi)\mathrm{Im}\,G(\bm{k},\varepsilon)$, where the
single particle Green’s function is $G(\bm{k},\varepsilon)=[\varepsilon-
E_{\bm{k}}-\Sigma(\bm{k},\varepsilon)]^{-1}$. For a local self-energy
$\Sigma(\varepsilon)$, the spectral function only depends on $\bm{k}$ through
$E_{\bm{k}}$ and it is convenient to define $A(\bm{k},\varepsilon)\equiv
A(E_{\bm{k}},\varepsilon)$ with

$A(E,\varepsilon)=\frac{-\mathrm{Im}\,\Sigma(\varepsilon)/\pi}{[\varepsilon-E-\mathrm{Re}\,\Sigma(\varepsilon)]^{2}+[\mathrm{Im}\,\Sigma(\varepsilon)]^{2}}.$
(2)

Thus, in our notations, the variable $E$ refers to an energy shell of the
noninteracting dispersion, while the variable $\varepsilon$ refers to the
energy axis, over which the electron spectral weight is redistributed by the
correlations encoded in $\Sigma(\varepsilon)$.

Note that our model is characterized only by the dispersion relation
$E_{\bm{k}}$, which is linked to the geometry of the system, and the self-
energy $\Sigma(\varepsilon)$, which encodes possible interactions and
disorder. As a consequence the conductivity will also be a function of only
$E_{\bm{k}}$ and $\Sigma(\varepsilon)$.

Figure 1:  We consider a two dimensional tight-binding model with direction-
dependent hopping amplitudes $t_{x}$, $t_{y}$ and a magnetic field $B$ along
the $z$ direction. We pass a current $I$ along the $x$ direction and measure
the created electric field (Hall effect) along the direction $y$. Furthermore
we make the assumptions that interactions in the system can only lead to local
correlations and thus a self-energy $\Sigma(\varepsilon)$ depending only on
the energy $\varepsilon$.

###  II.2 Self-consistent chemical potential

In the low-density limit that we study, and especially when the Fermi energy
(i.e., some measure of the particle or hole density) is comparable with
temperature and/or scattering rate, the precise knowledge of the chemical
potential is crucial. We will compute it self-consistently using the relation

$n=\int_{-\infty}^{\infty}d\varepsilon\,f(\varepsilon-\mu)\int_{-\infty}^{\infty}dE\,N_{0}(E)A(E,\varepsilon),$
(3)

where $f(\varepsilon)=(e^{\varepsilon/k_{\mathrm{B}}T}+1)^{-1}$ is the Fermi-
Dirac distribution, $A(E,\varepsilon)$ is the spectral function defined
previously in Eq. (2), and $N_{0}(E)$ is the noninteracting density of states
(DOS), which for a 2D band of spin-1/2 electrons is given by

$N_{0}(E)=2\int\frac{d^{2}k}{(2\pi)^{2}}\,\delta\left(E-E_{\bm{k}}\right).$
(4)

Equation (3) is valid up to first order in the field $B$ (see Appendix A). The
quantity $N(\varepsilon)=\int_{-\infty}^{\infty}dE\,N_{0}(E)A(E,\varepsilon)$
in Eq. (3) is the interacting DOS. $N(\varepsilon)$ is in general nonzero
outside the noninteracting band. As a result, the chemical potential may move
below the noninteracting band at low electron density (above it at low hole
density), even at zero temperature.

###  II.3 Hall constant

The Hall effect is the appearance of an electric field in the direction
transverse to the current, which defines a transverse resistivity according to
$E_{y}=\rho_{yx}J_{x}$. In the presence of nontrivial topology or broken time-
reversal symmetry, an “anomalous” Hall effect occurs in zero magnetic field
[27]. The models considered in this work are space- and time-inversion
symmetric and have no spin-orbit coupling, and are therefore characterized by
an ordinary Hall response, i.e., a resistivity $\rho_{yx}(B)$ proportional to
$B$. The Hall constant is then defined as $R_{\mathrm{H}}=\lim_{B\to
0}\rho_{yx}(B)/B$. The transverse resistivity is related to the conductivity
via $\rho_{yx}=\sigma_{xy}/(\sigma_{xx}\sigma_{yy}+\sigma_{xy}^{2})$, where
the property $\sigma_{yx}=-\sigma_{xy}$ has been used 111The property
$\sigma_{yx}=-\sigma_{xy}$ follows from space-inversion symmetry, which
changes $J_{x}$ into $-J_{x}$ and $J_{y}$ into $-J_{y}$, and time-reversal
symmetry, which changes $B$ into $-B$.. An ordinary Hall response also implies
that $\sigma_{xx}$ and $\sigma_{yy}$ vanish at first order in $B$. Hence the
conductivity tensor has the structure

$\sigma_{\alpha\beta}(B)=\delta_{\alpha\beta}\sigma^{(0)}_{\alpha}+(1-\delta_{\alpha\beta})\sigma^{(1)}_{\alpha\beta}B+O(B^{2}).$
(5)

Using this, we obtain a representation of the Hall constant in terms of the
expansion coefficients $\sigma^{(0)}_{\alpha}$ and $\sigma^{(1)}_{xy}$:

$R_{\mathrm{H}}=\lim_{B\to
0}\frac{1}{B}\frac{\sigma_{xy}}{\sigma_{xx}\sigma_{yy}+\sigma_{xy}^{2}}=\frac{\sigma^{(1)}_{xy}}{\sigma^{(0)}_{x}\sigma^{(0)}_{y}}.$
(6)

Following Ohm’s law, $\sigma_{\alpha\beta}$ gives the linear response to the
electric field and must therefore be calculated at vanishing electric field.
In the next section, we first write expressions for $\sigma_{\alpha\beta}$
that contain all orders in the magnetic field $B$ using the Kubo formula. We
then perform an expansion in $B$ in order to extract $\sigma^{(0)}_{\alpha}$
and $\sigma^{(1)}_{xy}$.

###  II.4 DC limit of the Kubo conductivity tensor

The macroscopic (i.e., $q=0$ or spatially-averaged) complex AC conductivity at
frequency $\omega$ is given by the Kubo formula [29]

$\sigma_{\alpha\beta}(\omega)=\frac{i}{\omega}\frac{1}{V}\left[C^{R}_{J_{\alpha}J_{\beta}}(\omega)-C^{R}_{J_{\alpha}J_{\beta}}(0)\right].$
(7)

In this expression, $J_{\alpha}$ is the macroscopic current operator in the
direction $\alpha$ (given in Appendix A) and
$C^{R}_{J_{\alpha}J_{\beta}}(\omega)$ is the retarded current-current
correlation function,

$C^{R}_{J_{\alpha}J_{\beta}}(\omega)=\int_{-\infty}^{\infty}dt\,e^{i\omega
t}(-i/\hbar)\theta(t)\langle[J_{\alpha}(t),J_{\beta}(0)]\rangle.$ (8)

This is an extensive quantity that is divided by the volume $V$ —the surface,
in our 2D case— to yield an intensive conductivity. The second term in Eq. (7)
is the diamagnetic response, which is real in normal metals and ensures that
the DC conductivity is finite.

Our target is the DC conductivity, that is purely real and relates to the
imaginary part of the correlation function:

$\sigma_{\alpha\beta}=\lim_{\omega\to
0}\mathrm{Re}\,\sigma_{\alpha\beta}(\omega)=-\lim_{\omega\to
0}\frac{1}{\omega}\mathrm{Im}\,\frac{1}{V}C^{R}_{J_{\alpha}J_{\beta}}(\omega).$
(9)

In order to evaluate $\sigma_{\alpha\beta}$, we express the currents in terms
of the operators $c^{\dagger}_{\bm{r}\sigma}$ and $c_{\bm{r}\sigma}$ and,
thanks to the absence of vertex corrections, recast the correlation function
as an expression involving only the interacting single-particle Green’s
function. The latter is neither gauge- nor translation-invariant in the
presence of a magnetic field. Nevertheless, the correlation function is
brought to a manifestly gauge-invariant form after introducing a modified
gauge- and translation-invariant Green’s function. The details are given in
Appendix A.

The resulting expression of the current-current correlation function involves
a transport function $\Phi_{\alpha\beta}$ that depends on $E_{\bm{k}}$ but not
on $\Sigma(\varepsilon)$, analogous to the DOS in (4), and the spectral
function (2) that depends on $\Sigma(\varepsilon)$ but not on $E_{\bm{k}}$:

$C^{R}_{J_{\alpha}J_{\beta}}(\omega)=V\int_{-\infty}^{\infty}d\varepsilon_{1}d\varepsilon_{2}\,\frac{f(\varepsilon_{1}-\mu)-f(\varepsilon_{2}-\mu)}{\hbar\omega+\varepsilon_{1}-\varepsilon_{2}+i0}\\\
\times\int_{-\infty}^{\infty}dE_{1}dE_{2}\,\Phi_{\alpha\beta}(E_{1},E_{2},B)A(E_{1},\varepsilon_{1})A(E_{2},\varepsilon_{2}).$
(10)

The number of energy integrals is doubled with respect to Eq. (3), reflecting
the occurrence of two current operators. This is also the case for the
transport function, which now involves two momentum sums:

$\Phi_{\alpha\beta}(E_{1},E_{2},B)=-\left(\frac{e}{\hbar}\right)^{2}\frac{1}{V}\sum_{\bm{k}\bm{q}\sigma}\delta\left(E_{1}-E_{\bm{k}-\frac{\bm{q}}{2}}\right)\delta\left(E_{2}-E_{\bm{k}+\frac{\bm{q}}{2}}\right)\\\
\times\sum_{\bm{\rho}_{1}\bm{\rho}_{2}}\rho_{1\alpha}t^{0}_{\bm{\rho}_{1}}e^{i(\bm{k}+\bm{q}/2)\cdot\bm{\rho}_{1}}\rho_{2\beta}t^{0}_{\bm{\rho}_{2}}e^{i(\bm{k}-\bm{q}/2)\cdot\bm{\rho}_{2}}\\\
\times\frac{1}{N}\sum_{\bm{r}}e^{i\bm{q}\cdot\bm{r}}e^{\frac{i|e|}{2\hbar}\bm{B}\cdot\bm{r}\times(\bm{\rho}_{1}-\bm{\rho}_{2})}.$
(11)

Although the transport function contains all orders in $B$, Eq. (10) is not
valid to all orders in $B$, as it misses the Landau-level physics. On a
lattice, this breaks the spectral function into a non-perturbative fractal
Hofstadter spectrum [30]. It is nevertheless possible to show that the
variation of the spectral function is quadratic in $B$ (Appendix A).
Therefore, Eqs. (10) and (11) are valid at first order in $B$.

The DC limit of the conductivity involves the imaginary part of
$C^{R}_{J_{\alpha}J_{\beta}}(\omega)$, which lets two terms appear. The first
combines $-\pi\delta(\hbar\omega+\varepsilon_{1}-\varepsilon_{2})$ from the
imaginary part of the retarded energy denominator in Eq. (10) with the real
part of the transport function. This term denoted
$\sigma_{\alpha\beta}^{(e)}(B)$ is even in B and reads

$\sigma_{\alpha\beta}^{(e)}(B)=\pi\hbar\int_{-\infty}^{\infty}d\varepsilon\,[-f^{\prime}(\varepsilon-\mu)]\\\
\times\int_{-\infty}^{\infty}dE_{1}dE_{2}\,\mathrm{Re}\,\Phi_{\alpha\beta}(E_{1},E_{2},B)A(E_{1},\varepsilon)A(E_{2},\varepsilon),$
(12)

where $f^{\prime}$ denotes the energy derivative of $f$. It is a “Fermi-
surface” or “on-the-energy-shell” contribution, due to the factor
$-f^{\prime}(\varepsilon-\mu)$ that constrains the energy $\varepsilon$ to
remain within a few $k_{\mathrm{B}}T$ around the chemical potential.

The second term involves the principal value of the energy denominator and the
imaginary part of the transport function. Upon expanding
$1/(\hbar\omega+\varepsilon_{1}-\varepsilon_{2})$ for $\omega\to 0$, a
contribution proportional to $1/(\varepsilon_{1}-\varepsilon_{2})$ appears.
This part is identically zero by symmetry, due to the fact that the imaginary
part of the transport function is odd under the exchange of $E_{1}$ and
$E_{2}$. The subsequent term of the expansion is
$-\hbar\omega/(\varepsilon_{1}-\varepsilon_{2})^{2}$, which once inserted in
Eq. (9) yields a well-behaved limit that is odd in $B$ and is denoted
$\sigma_{\alpha\beta}^{(o)}(B)$:

$\sigma_{\alpha\beta}^{(o)}(B)=\hbar\int_{-\infty}^{\infty}d\varepsilon_{1}d\varepsilon_{2}\,\frac{f(\varepsilon_{1}-\mu)-f(\varepsilon_{2}-\mu)}{(\varepsilon_{1}-\varepsilon_{2})^{2}}\\\
\times\int_{-\infty}^{\infty}dE_{1}dE_{2}\,\mathrm{Im}\,\Phi_{\alpha\beta}(E_{1},E_{2},B)A(E_{1},\varepsilon_{1})A(E_{2},\varepsilon_{2}).$
(13)

Unlike $\sigma_{\alpha\beta}^{(e)}(B)$, this appears to be an “off-the-energy-
shell” term with the energies $\varepsilon_{1}$ and $\varepsilon_{2}$ allowed
to take values far away from the chemical potential.

###  II.5 Expansion for weak field

The transport function in Eq. (11) contains all the magnetic-field dependence,
but it is cumbersome for numerical evaluation. We therefore consider its
expansion in powers of $B$. The zero-field limit is easy to grasp: the
$\bm{r}$ sum forces $\bm{q}$ to vanish, which in turn forces $E_{1}$ and
$E_{2}$ to be equal; the remaining $\bm{\rho}_{1}$ and $\bm{\rho}_{2}$ sums
yield derivatives of the dispersion $E_{\bm{k}}$. As a result, the transport
function is real and Eq. (12) yields the first coefficient appearing in Eq.
(5):

$\displaystyle\sigma_{\alpha}^{(0)}$
$\displaystyle=\pi\hbar\int_{-\infty}^{\infty}d\varepsilon\,[-f^{\prime}(\varepsilon-\mu)]\int_{-\infty}^{\infty}dE\,\Phi_{\alpha}^{(0)}(E)A^{2}(E,\varepsilon)$
(14) $\displaystyle\Phi_{\alpha}^{(0)}(E)$
$\displaystyle=\left(\frac{e}{\hbar}\right)^{2}2\int\frac{d^{2}k}{(2\pi)^{2}}\left(\frac{\partial
E_{\bm{k}}}{\partial k_{\alpha}}\right)^{2}\delta(E-E_{\bm{k}}).$ (15)

Details are provided in Appendix B. The interpretation of this formula is that
the DC conductivity is controlled by zero-energy electronic transitions close
to the chemical potential. On every shell of energy $E$, the transitions are
weighted by the transport function, which is proportional to the squared
$\alpha$ component of the group velocity averaged on that shell, and by the
square of the spectral function, which sets the amount of electronic spectral
weight available for the initial and final states of the transition.

At first order in $B$, the transport function is imaginary and we only get a
contribution from Eq. (13). Remarkably, we find that the off-shell character
of this term is not fundamental, as the expression can be recast _exactly_ in
an on-shell form (Appendix B):

$\displaystyle\sigma_{xy}^{(1)}$
$\displaystyle=\hbar\int_{-\infty}^{\infty}d\varepsilon\,[-f^{\prime}(\varepsilon-\mu)]\int_{-\infty}^{\infty}dE\,\Phi^{(1)}(E)A^{3}(E,\varepsilon)$
(16) $\displaystyle\Phi^{(1)}(E)$
$\displaystyle=\left(\frac{|e|}{\hbar}\right)^{3}\frac{2\pi^{2}}{3}\int\frac{d^{2}k}{(2\pi)^{2}}\left[2\frac{\partial
E_{\bm{k}}}{\partial k_{x}}\frac{\partial E_{\bm{k}}}{\partial
k_{y}}\frac{\partial^{2}E_{\bm{k}}}{\partial k_{x}\partial k_{y}}\right.$
$\displaystyle\quad\left.-\left(\frac{\partial E_{\bm{k}}}{\partial
k_{x}}\right)^{2}\frac{\partial^{2}E_{\bm{k}}}{\partial
k_{y}^{2}}-\left(\frac{\partial E_{\bm{k}}}{\partial
k_{y}}\right)^{2}\frac{\partial^{2}E_{\bm{k}}}{\partial
k_{x}^{2}}\right]\delta(E-E_{\bm{k}}).$ (17)

The transformation from off-shell form to on-shell form relies explicitly on
the causal character of the spectral function, which in turns requires the
causality of the self-energy. The latter must therefore respect the
Kramers–Kronig relations. We have not been able to bring the full Eq. (13) to
on-shell form, a frustration that has already been expressed in the literature
[11].

Let us make some comments on our results, and in particular the expression for
$\sigma_{xy}^{(1)}$, in connection with previous works. Our derivation
recovers some known results that are scattered across the relevant literature,
but proceeds differently from previous works. In Refs. 10, 11, 31, the limit
of vanishing magnetic field is taken from the outset, before the DC limit,
which allows the authors to relate the Hall response with a three-current
correlation function. Within the family of locally-correlated models, the DC
limit can be taken before expanding in powers of the magnetic field. The
resulting expression for the transverse conductivity is off-shell, as found by
other authors [11, 15, 16]. As we showed, at first order in the magnetic
field, this apparently off-shell contribution can be rigorously recast, after
the expansion in $B$, as a on-shell expression involving the _third power_
[31, 14] of the spectral function in (16), which is reminiscent of the three-
current correlation function that characterizes the transverse response when
the low-field expansion is performed before taking the DC limit [10, 11, 31].
This equivalence relies explicitly on the causality of the self-energy
function.

Second, the average between the $x$ and $y$ directions appearing at the second
line of Eq. (17) reflects the gauge invariance of the formula and is crucial
in our anisotropic setting. Related expressions based on the Boltzmann theory
in the literature lack this symmetrization and should therefore be corrected
for anisotropic systems [9, 32, 33]. In Appendix C, we show how to recover
from Eqs. (14–17) the Boltzmann theory in the isotropic relaxation-time
approximation.

##  III Results for the anisotropic square lattice

Results are presented here for the Hall constant of the anisotropic tight-
binding model depicted in Fig. 1, with a special emphasis on the regime of low
electron density. For the self-energy, we use the simplest causal model,
$\Sigma(\varepsilon)=-i\Gamma$, which represents quasiparticles with an
energy-independent relaxation time $\tau=\hbar/(2\Gamma)$. While elementary,
this model has two advantages. It contains a single energy scale to describe
the scattering, like the semiclassical theory in the isotropic relaxation-time
approximation, such that differences between the quantum and semiclassical
approaches cannot be ascribed to differences in the scattering. Furthermore,
it allows for a better numerical control, as certain integrals can be
performed exactly (see Appendix D).

One peculiarity of this self-energy is that the spectral function —and a
fortiori the interacting DOS— have $\sim 1/\varepsilon^{2}$ tails without low-
or high-energy cutoff. Consequently, the chemical potential is also unbounded
in the limit of vanishing density, even at zero temperature. Microscopic
lattice models generally have self-energies with a finite support, possibly
different from the support of the noninteracting band. The low-density
behavior of $R_{\mathrm{H}}$ may depend on the details of the self-energy in
nontrivial ways, and additional work is needed in order to determine how the
results presented here would change if a cutoff were introduced in the self-
energy.

Figure 2:  (a) Density of states, (b) zeroth-order transport functions, and
(c) first-order transport function for the model shown in Fig. 1 with
$t_{y}/t_{x}=0.1$. Energies are measured in units of $t_{x}$ and
$\mathfrak{a}=a_{x}a_{y}$ is the area of the unit cell. The dashed lines show
the leading behavior close to the band edge $E=-D$, i.e.,
$N_{0}(E)=1/(2\pi\mathfrak{a}\sqrt{|t_{x}t_{y}|})$,
$\Phi^{(0)}_{\alpha}(E)=(e/\hbar)^{2}(a_{\alpha}^{2}/\mathfrak{a})|t_{\alpha}|/\sqrt{|t_{x}t_{y}|}\frac{1}{\pi}(E+D)$,
and
$\Phi^{(1)}(E)=-(|e|/\hbar)^{3}\mathfrak{a}\sqrt{|t_{x}t_{y}|}\frac{4\pi}{3}(E+D)$.

###  III.1 DOS and transport functions

Before discussing the conductivities, we briefly describe the DOS and the
transport functions of the system sketched in Fig. 1. The DOS is displayed in
Fig. 2(a) for $t_{y}/t_{x}=0.1$. It is particle-hole symmetric with a half-
bandwidth $D=2(|t_{x}|+|t_{y}|)$. There are two logarithmic van Hove
singularities at energies $\pm 2(|t_{x}|-|t_{y}|)$, that merge into one in the
2D case and become two square-root singularities at the band edges in the 1D
case. The graph of $\Phi_{\alpha}^{(0)}(E)$ is displayed in Fig. 2(b). Like
the DOS, it is particle-hole symmetric. Where the DOS has van-Hove
singularities, it displays logarithmic singularities of the first derivative.
Figure 2(c) shows $\Phi^{(1)}(E)$, that is odd for the model of Fig. 1, which
is consistent with the fact that the Hall constant must vanish at half filling
in particle-hole symmetric systems. Interestingly, we find that
$\Phi^{(1)}(\mu)$ is equal to $(|e|/\hbar)^{3}(4\Gamma^{2}/3)A_{l}$, where
$A_{l}$ is the “Stokes area” introduced by Ong [33], if the latter is
calculated for an isotropic scattering time $\tau_{\bm{k}}=\hbar/(2\Gamma)$.

Figure 3:  Zero-temperature Hall constant in the anisotropic square lattice
with $t_{y}=0.01t_{x}$ and $\Gamma=0.1t_{x}$. Dotted lines show Eq. (18) and
$n_{c}$ indicates Eq. (20). The inset highlights the behavior at low electron
density. $\mathfrak{a}$ is the area of the unit cell.

###  III.2 Zero-temperature Hall constant

The impact of the correlations may be quantified by comparing the Hall
constant $R_{\mathrm{H}}$ with the universal value
$R_{\mathrm{H}}^{0}=-1/(n|e|)$. Even in the semiclassical regime, band-
structure effects are known to reduce the “Hall factor”
$r=R_{\mathrm{H}}/R_{\mathrm{H}}^{0}$ to values smaller than unity [33]. For
nearly-free electrons, Ref. 10 found that the momentum dependence of the self-
energy renormalizes the Hall constant by a factor $r=(N_{0}/N)^{2}$, with
$N_{0}$ and $N$ the bare and renormalized Fermi-level densities of states,
which is also generally smaller than unity. Another finding was that the
energy dependence of the self-energy yields no renormalization.

For our model, Fig. 3 shows a typical evolution of the Hall constant with
varying carrier density at $T=0$, for the parameters $t_{y}/t_{x}=0.01$ and
$\Gamma/t_{x}=0.1$. $R_{\mathrm{H}}$ is negative (positive) for electron
(hole) conduction and diverges at low electron or hole density. It vanishes at
$n=1$ due to the particle-hole symmetry of the model. At low electron/hole
density, the divergences seen are expected from the semiclassical theory,
which predicts $-1/n$, respectively $1/(2-n)$ behavior. However, a closer
inspection reveals that the divergence is four times faster than the
semiclassical prediction (see inset of Fig. 3). A Hall factor $r$ _higher_
than unity is normally not expected [10, 33]. Here, the value $r=4$ occurs in
the “quantum regime”, where the semiclassical assumption of sharp
quasiparticles is no longer justified.

In order to better understand the low-density asymptotic behaviors in these
two regimes, we consider first the semiclassical case by taking the limit
$\Gamma\to 0^{+}$ in the transport equations at $T=0$ (see Appendix E.1). We
find the Hall constant

$R_{\mathrm{H}}=\frac{3}{2\pi^{2}\hbar}\frac{\Phi^{(1)}(\mu)}{\Phi^{(0)}_{x}(\mu)\Phi^{(0)}_{y}(\mu)}\quad(T=0,\Gamma\to
0).$ (18)

This result simplifies even further in the limit of low density: as $n$
approaches zero, the chemical potential approaches the lower band edge, where
the DOS becomes constant and the transport functions vanish linearly (see Fig.
2). We can therefore substitute in Eq. (18) the limiting expressions given in
the legend of Fig. 2 and use the limiting value of the density
$n=N_{0}(-D)(\mu+D)$, which yields the semiclassical result
$R_{\mathrm{H}}^{0}=-1/(|e|n)$. Away from the band-edge, $R_{\mathrm{H}}$ as
given by Eq. (18) is smaller than $R_{\mathrm{H}}^{0}$, irrespective of the
anisotropy $t_{y}/t_{x}$. This is consistent with the expectation that band-
structure effects reduce the Hall constant in the semiclassical approximation
[33]. The full semiclassical result of Eq. (18) is displayed in Fig. 3 as the
dotted lines, to compare to the full quantum result obtained using Eq.
(16)-(17).

Figure 4:  Hall factor $r=R_{\mathrm{H}}/R_{\mathrm{H}}^{0}$ (left axis) and
chemical potential (right axis) versus electron density at $T=0$ for
$t_{y}/t_{x}=0.1$ and $\Gamma/t_{x}=0.004$.

Instead, in the quantum regime, one must keep $\Gamma$ finite while taking the
limit $n\to 0$. Due to the $\sim 1/\varepsilon^{2}$ tails of the interacting
DOS, this implies sending $\mu$ to $-\infty$. We find that
$\mu\propto-\Gamma/n$ as $n\to 0$ (see Appendix E.1). At the same time, the
normal and Hall conductivities behave as $\sigma_{\alpha}^{(0)}\sim
n^{4}/(4\Gamma^{2})$ and $\sigma_{xy}^{(1)}\sim-n^{7}/(4\Gamma^{4})$, such
that we indeed find

$R_{\mathrm{H}}=-\frac{4}{|e|n}\quad(T=0,n\to 0).$ (19)

It is remarkable that, although the conductivities have completely different
forms in the quantum and semiclassical regimes (in the latter they read
$\sigma_{\alpha}^{(0)}\sim n/\Gamma$ and $\sigma_{xy}^{(1)}\sim-n/\Gamma^{2}$)
the Hall constant is nevertheless independent of the scattering and
proportional to $1/n$ in both cases.

Figure 5:  Hall factor $r=R_{\mathrm{H}}/R_{\mathrm{H}}^{0}$ at $T=0$ and low
electron density for three values of the anisotropy $t_{y}/t_{x}$. Dark blue
corresponds to $r=1$ and turquoise to $r=4$. The semiclassical and quantum
regimes are labelled with “$1/n$” and “$4/n$”, respectively. The solid
(dotted) white lines correspond to $\mu=-D$ ($\mu=-D-\sqrt{2}\Gamma$) and mark
the beginning (approximate maximum) of the crossover between the semiclassical
and quantum regimes.

With the aim to determine the characteristic density $n_{c}$ at which
$R_{\mathrm{H}}$ crosses over from the semiclassical regime to the quantum
regime, we plot in Fig. 4 the density dependence of the Hall factor,
superimposed with the density dependence of the chemical potential. The Hall
factor varies non-monotonously between $r=1$ and $r=4$ as $n$ decreases, going
through a maximum when $\mu$ lies approximately $\sqrt{2}\Gamma$ below the
band edge. Hence the semiclassical to quantum crossover is associated with the
chemical potential approaching and crossing the bottom of the noninteracting
band. From the condition $\mu=-D$, we deduce an approximate expression for
$n_{c}$ (see Appendix E.1):

$n_{c}=\frac{\Gamma}{2\pi^{2}\mathfrak{a}\sqrt{|t_{x}t_{y}|}}\left[1+\ln\left(\frac{2D}{\Gamma}\right)\right].$
(20)

Hence the quantum regime is reached at higher densities for larger scattering
rates and more one-dimensional geometries. These trends are visualized in Fig.
5, where the zero-temperature Hall factor is displayed versus $n$, $\Gamma$,
and $t_{y}/t_{x}$. On going from 2D to quasi-1D, $n_{c}$ increases and the
quantum regime progressively expands.

###  III.3 Temperature dependence

Since at zero temperature the strongest deviation from a semiclassical Hall
effect occurs when the chemical potential lies slightly outside the
noninteracting band, we also expect an enhancement of the Hall factor when
$\mu$ moves in the vicinity of the band edge under the effect of temperature.
As we will see, this enhancement can reach several orders of magnitude in our
model. In models where the scattering rate has a cutoff, a finite temperature
will also drive $\mu$ below the band if the density is sufficiently low, such
that similar effects could also possibly occur.

Starting at near-zero temperature, the conductivities and the Hall constant
vary initially slowly like $T^{2}$. In Fermi liquids, the $T^{2}$ raise of the
resistivity is due to the scattering rate growing like
$(\varepsilon-\mu)^{2}$. Here, since we assume a temperature-independent self-
energy, the $T^{2}$ dependence of the conductivities stems from thermal
excitations rather than from the scattering rate. It occurs both in the
semiclassical and quantum regimes. However, whether the conductivities
increase or decrease like $T^{2}$ depends on the density (see Appendix E.2).
In the quantum regime $n\to 0$, we find that all conductivities _increase_
with a coefficient proportional to $(n/\Gamma)^{2}$. In the semiclassical
regime, the coefficient depends on the slope and curvature of the transport
function and can have different signs in different directions. Figure 6 shows
the typical temperature variation of the conductivities in the quantum regime.
While $\sigma_{x}^{(0)}$ and $\sigma_{y}^{(0)}$ have nearly the same $T^{2}$
coefficient, the coefficient of $\sigma^{(1)}_{xy}$ is larger, consistently
with the asymptotic results given in Appendix E.2.

Figure 6:  Typical temperature dependence of the conductivities in the quantum
regime ($t_{y}=0.1t_{x}$, $\Gamma=0.01t_{x}$, $n=0.001/\mathfrak{a}$). The
dashed lines show Eq. (21). The onset temperatures indicated by arrows
correspond to Eq. (22).

Strikingly, the conductivities shoot up by orders of magnitude at well-defined
onset temperatures that are different for the $x$, $y$, and $xy$ components.
Below the onset, the Fermi tails in Eqs. (14) and (16) do not overlap
significantly with the band, but only with the band _tails_ resulting from the
convolution of the transport function with the second or third power of the
spectral function. Physically, the thermally excited carriers are very few,
occupying the tail of the spectral function. Above the onset, the Fermi tails
overlap with the band and the number of thermally-excited carriers increases
strongly. As the figure shows, this occurs in a regime where
$k_{\mathrm{B}}T>\Gamma$, such that the spectral function is narrower than the
derivative of the Fermi function. Using this fact, and linearizing the
transport functions close to the band edge, as appropriate at low density, we
derive in Appendix E.2 approximations for the conductivities in this regime:

$\displaystyle\sigma_{\alpha}^{(0)}$
$\displaystyle\approx\frac{e^{2}}{h}\frac{a_{\alpha}^{2}}{\mathfrak{a}}\frac{|t_{\alpha}|}{\sqrt{|t_{x}t_{y}|}}\frac{k_{\mathrm{B}}T}{\Gamma}\ln\left(1+e^{\frac{\mu+D}{k_{\mathrm{B}}T}}\right)$
(21a) $\displaystyle\sigma^{(1)}_{xy}$
$\displaystyle\approx-\frac{|e|^{3}}{h^{2}}2\pi\mathfrak{a}\frac{\sqrt{|t_{x}t_{y}|}k_{\mathrm{B}}T}{\Gamma^{2}}\ln\left(1+e^{\frac{\mu+D}{k_{\mathrm{B}}T}}\right).$
(21b)

In order to estimate the onset temperatures, we ignore the weak $T^{2}$
variation below the onset and determine the temperature at which the above
expressions equal the zero-temperature conductivities. The resulting equations
are simplified using the fact that
$\exp\left(\frac{\mu+D}{k_{\mathrm{B}}T}\right)\ll 1$, allowing an explicit
solution:

$\displaystyle k_{\mathrm{B}}T_{\alpha}$
$\displaystyle=\frac{\delta-D}{W\left(\frac{\delta(\delta-D)}{|t_{\alpha}|(|t_{x}t_{y}|)^{1/2}(\pi
n\mathfrak{a})^{3}}\right)}$ (22a) $\displaystyle k_{\mathrm{B}}T_{xy}$
$\displaystyle=\frac{\delta-D}{W\left(\frac{\delta^{2}(\delta-D)}{2(|t_{x}t_{y}|)^{3/2}(\pi
n\mathfrak{a})^{5}}\right)}.$ (22b)

Here $W(\cdot)$ is the Lambert function and we have defined
$\delta=2\Gamma/(\pi n\mathfrak{a})$. These temperatures are displayed in Fig.
6.

Figure 6 shows that $T_{xy}<T_{x},T_{y}$, which can be rationalized as
follows. According to Eq. (21), the conductivities behave as $\sigma_{x,y}\sim
I(T)/\Gamma$ and $\sigma_{xy}\sim I(T)/\Gamma^{2}$ above onset, with the same
temperature dependence $I(T)$ but different powers of $\Gamma$. The $T$
dependence follows from convolving the Fermi factor with the linearized
transport function:
$\int_{-D}^{\infty}d\varepsilon\,[-f^{\prime}(\varepsilon-\mu)](\varepsilon+D)=k_{\mathrm{B}}T\ln\left[1+\exp\left(\frac{\mu+D}{k_{\mathrm{B}}T}\right)\right]$.
The difference stems from the sum rule for $A^{2}(E,\varepsilon)$ being $\sim
1/\Gamma$, while the sum rule for $A^{3}(E,\varepsilon)$ is $\sim
1/\Gamma^{2}$, which reflects the fact that $A^{3}(E,\varepsilon)$ is narrower
than $A^{2}(E,\varepsilon)$. Theses sum rules appear after convolving the
transport functions with $A^{2}$ and $A^{3}$ in Eqs. (14) and (16),
respectively. On the other hand, we have seen that, at $T=0$ and in the
quantum regime, the conductivities behave as $\sigma_{x,y}\sim
n^{4}/\Gamma^{2}$ and $\sigma_{xy}\sim n^{7}/\Gamma^{4}$. The equation for
$T_{x,y}$ is therefore of the form $I(T)/\Gamma=n^{4}/\Gamma^{2}$, while the
equation for $T_{xy}$ is of the form $I(T)/\Gamma^{2}=n^{7}/\Gamma^{4}$. Since
$I(T)$ increases monotonically with increasing $T$, one sees that
$T_{xy}<T_{x},T_{y}$, provided that $n^{3}<\Gamma$. With all numeric and
dimensional factors in place, this condition reads $(\pi
n\mathfrak{a})^{3}<\Gamma/|t_{x}|$. In the regime
$k_{\mathrm{B}}T_{xy}>\Gamma$ where Eq. (22) was obtained, this condition is
met if $n<n_{c}$. We therefore expect that the condition $T_{xy}<T_{x},T_{y}$
is robust throughout the quantum regime.

Figure 7:  (a) Temperature dependence of the Hall factor
$r(T)=R_{\mathrm{H}}(T)/R_{\mathrm{H}}^{0}$ for $t_{y}=0.1t_{x}$,
$\Gamma=0.01t_{x}$ and various densities. (b) Largest value of the
temperature-dependent Hall factor versus density for three values of the
anisotropy $t_{y}/t_{x}$ and three values of the scattering rate $\Gamma$.

At temperatures between $T_{xy}$ and $T_{y}$, the sharp increase of
$\sigma_{xy}$ drives a correspondingly large increase of $R_{\mathrm{H}}$,
since the normal conductivities remain low. One expects a peak in
$R_{\mathrm{H}}(T)$ that reaches its maximum close to $T_{y}$. Evaluating Eq.
(21) at $T=T_{y}$, we obtain the following estimate for the maximum Hall
factor:

$r_{\max}\approx\frac{4\Gamma}{|t_{y}|(\pi n\mathfrak{a})^{3}}.$ (23)

This can be huge in the low-density limit, especially for a quasi-1D
dispersion. Figure 7(a) displays $R_{\mathrm{H}}(T)$ in the anisotropic case
$t_{y}/t_{x}=0.1$ with $\Gamma/t_{x}=0.01$ and at various densities that scan
from deep in the quantum regime up to the crossover into the semiclassical
regime (see Fig. 5). In the quantum regime, $r(T)$ starts at the value $4$,
raises slowly as $T^{2}$ (Appendix E.2), develops a strong maximum, and
finally drops to the high-temperature value
$r(k_{\mathrm{B}}T\gg\Gamma)\approx 1$ (Appendix E.2). Consistently with Eq.
(23), the height of the maximum is very large at small $n$ and drops rapidly
as $n$ increases. Upon approaching the semiclassical regime, the initial value
$r(0)$ grows (Fig. 5) and the maximum disappears. Finally, deep in the
semiclassical regime (not shown), $r(T)$ varies with almost no temperature
dependence from a value close to $1$ at $T=0$ to a value close to $1$ at high
temperature.

Figure 7(b) shows the maximum Hall factor determined numerically versus $n$,
for three values of $t_{y}$ and three values of $\Gamma$. The power law $\sim
n^{-3}$ predicted by Eq. (23) is observed at low $n$, while for larger $n$ the
behavior changes. The increase of $r_{\max}$ with increasing $\Gamma$ and
decreasing $t_{y}$ is also qualitatively consistent with Eq. (23).

##  IV Summary and Conclusion

Measuring the Hall constant remains the number-one experimental approach for
determining the carrier density of conductors, including correlated ones (see
for instance Ref. 34). This is grounded in the universality of this property,
as predicted by the Boltzmann semiclassical theory of transport. The
conditions under which this universality breaks down are not well known. The
semiclassical approach relies on the carriers being true particles with
fermionic statistics, which is likely a good approximation when interactions
are weak and carrier density is sufficiently high. Deviations may be expected
at low density, even before the regime of Wigner crystallization is reached,
when the Fermi energy is comparable with the scattering rate, because the
carriers cannot be treated as sharp quasiparticles. In this case, the
semiclassical theory gives way to the Kubo–Greenwood framework, but a complete
solution is generally out of reach. One exception, in which a closed set of
equations may be written for the Hall constant, is the class of models, where
nonlocal correlations between the electrons are either absent, like in
dynamical mean-field theory, or negligible. We have collected these equations
in Sec. II, for the case of a 2D single-band system in perpendicular magnetic
field [Eqs. (2–6), and (14–17)]. Our goal is to explore the predictions of
these equations in the regime of low density, which is relevant in the present
context of flourishing research activity on gated 2D conductors.

As a first application, we have studied a minimal two-parameters model. The
first parameter, $t_{y}/t_{x}$, controls the electronic dimensionality and
allows one to study 2D to 1D crossovers. The second, $\Gamma/t_{x}$, controls
the scattering. We have studied $r\equiv R_{\mathrm{H}}/R_{\mathrm{H}}^{0}$ as
a function of density and temperature and found deviations from the
semiclassical regime characterized by $r=1$. As a rule of thumb, the
semiclassical behavior is realized at higher dimensionality, weak scattering,
high density, and high temperature, while significant deviations are found at
low dimensionality, strong scattering, low density, and low temperature. There
are two kinds of deviations from universality. At zero temperature, when the
chemical potential moves outside the noninteracting band, the Hall constant
crosses over from $1/n$ to $4/n$. This quantitative change does not completely
break the universality, in the sense that the Hall constant remains
asymptotically independent of band structure and scattering rate. However, the
crossover region is non-universal, depending on both band structure and
scattering rate [see Eq. (20)]. A second kind of deviation arises at finite
temperature, when thermally-excited carriers —that were at lower temperatures
confined to interaction-induced tails of the density of states— begin to
occupy the extreme levels of the noninteracting band. The ensuing boost of
conductivity, that occurs first for the Hall conductivity, produces a peak in
the Hall factor $r$.

This model is peculiar in that the scattering rate has no dependence on the
energy, which allows charge carriers to exist at arbitrary energies below the
free-carrier band. In a way, it may be seen as the simplest generalization of
the Boltzmann theory in the isotropic relaxation time approximation: it
reduces to the latter in the semiclassical regime, and allows one to describe
the crossover into the quantum regime without additional parameters. This
circumstance enables good analytical and numerical control of the equations as
well as exact asymptotic expressions, but may also lead to non-universal
results. Nevertheless, the model has the merit of showing that significant
deviations from the universal Hall constant paradigm can occur at low density
and gives a first hint on where to search the strongest deviations: in
conditions where the chemical potential varies in the vicinity of the free-
carrier band edge. Of course, how the energy dependence of the self-energy
would influence the low-density Hall constant is an intriguing open question
that is left for future work.

The peak in the temperature-dependent Hall factor could be searched in known
2D materials. With the typical values $\mathfrak{a}=(4~{}\text{\AA})^{2}$,
$t_{x}=100$ meV, and $\Gamma=2$ meV, Eq. (20) gives $n_{c}=4\times 10^{12}$
cm-2 in the least favorable isotropic case. Hence the quantum regime is well
within the range of carrier concentrations that can be achieved with the
field-effect technology. Assuming $n=10^{12}$ cm-2, Eq. (22) gives $T_{x}=278$
K and Eq. (23) gives $r_{\max}\approx 6\times 10^{5}$. The exact values given
by the model are $T_{x}=319$ K and $r_{\max}=1.1\times 10^{5}$. At that
temperature, the conductivity is however very small: $\sigma_{x}\sim
10^{-4}~{}\mu$S. Of course, it is not obvious that the predictions of the
model should be trusted when applied to real materials, in which the self-
energy surely has a complex energy dependence. Further studies are therefore
needed, using realistic band structures and self-energies inspired by
microscopic models of interacting electrons.

###### Acknowledgements.

We are thankful to Fan Wu and Alberto Morpurgo for drawing our attention to
the Hall effect at low density. This work was supported by the Swiss National
Science Foundation under Division II (Grant No. 2000020-188687). LR is funded
by the Swiss National Science Foundation through Starting Grant TMSGI2_211296.

##  Appendix A Current-current correlation function

In this appendix, we describe the steps leading to Eqs. (10) and (11). We also
list a few symmetries of the transport function
$\Phi_{\alpha\beta}(E_{1},E_{2},B)$, that are useful to establish the symmetry
properties of the conductivity tensor.

The local current density is given by the functional derivative of the
Hamiltonian, Eq. (1), with respect to the vector potential:
$j_{\alpha}(\bm{r})=-\delta H/\delta A_{\alpha}(\bm{r})|_{\bm{E}=0}$. The
functional derivative is evaluated at $\bm{E}=0$, because the Kubo formula
gives the linear response to the electric field. It is therefore convenient to
split the vector potential as
$\bm{A}=\bm{A}_{\mathrm{el}}+\bm{A}_{\mathrm{mag}}$, where
$\bm{A}_{\mathrm{el}}$ is irrotational and vanishes for $\bm{E}=0$, while
$\bm{\nabla}\times\bm{A}_{\mathrm{mag}}=\bm{B}$. The interaction
$H_{\mathrm{int}}$ being independent of the vector potential, the functional
derivative is straightforward and yields the total current:

$J_{\alpha}=-\frac{i|e|}{\hbar}\sum_{\bm{r}_{1}\bm{r}_{2}\sigma}(\bm{r}_{2}-\bm{r}_{1})_{\alpha}t^{0}_{\bm{r}_{1}\bm{r}_{2}}e^{iA_{\bm{r}_{1}\bm{r}_{2}}}c^{\dagger}_{\bm{r}_{1}\sigma}c_{\bm{r}_{2}\sigma}$
(24)

with
$A_{\bm{r}_{1}\bm{r}_{2}}=\frac{|e|}{\hbar}\int_{\bm{r}_{1}}^{\bm{r}_{2}}d\bm{r}\cdot\bm{A}_{\mathrm{mag}}(\bm{r})$.
The retarded correlation function, Eq. (8), is the analytic continuation of
the corresponding Matsubara function:
$C^{R}_{J_{\alpha}J_{\beta}}(\omega)=C_{J_{\alpha}J_{\beta}}(i\Omega_{n}\to\hbar\omega+i0)$
[29]. Here $\Omega_{n}=2n\pi k_{\mathrm{B}}T$ are even Matsubara frequencies
and
$C_{J_{\alpha}J_{\beta}}(i\Omega_{n})=\int_{0}^{1/k_{\mathrm{B}}T}d\tau\,e^{i\Omega_{n}\tau}C_{J_{\alpha}J_{\beta}}(\tau)$
with the imaginary-time function $C_{J_{\alpha}J_{\beta}}(\tau)=-\langle
T_{\tau}J_{\alpha}(\tau)J_{\beta}(0)\rangle$ given explicitly by

$C_{J_{\alpha}J_{\beta}}(\tau)=\left(\frac{e}{\hbar}\right)^{2}\sum_{\bm{r}_{1}\bm{r}_{2}\sigma}\sum_{\bm{r}_{3}\bm{r}_{4}\sigma^{\prime}}(\bm{r}_{2}-\bm{r}_{1})_{\alpha}t^{0}_{\bm{r}_{1}\bm{r}_{2}}(\bm{r}_{4}-\bm{r}_{3})_{\beta}t^{0}_{\bm{r}_{3}\bm{r}_{4}}\\\
\times e^{iA_{\bm{r}_{1}\bm{r}_{2}}}e^{iA_{\bm{r}_{3}\bm{r}_{4}}}\langle
T_{\tau}c^{\dagger}_{\bm{r}_{1}\sigma}(\tau)c_{\bm{r}_{2}\sigma}(\tau)c^{\dagger}_{\bm{r}_{3}\sigma^{\prime}}(0)c_{\bm{r}_{4}\sigma^{\prime}}(0)\rangle.$
(25)

$T_{\tau}$ indicates imaginary-time ordering and $\langle\,\cdots\rangle$ a
thermal average with respect to $H$ at $\bm{E}=0$ and finite $\bm{B}$. The
assumption of local correlations in a paramagnetic state implies that the
thermal average factorizes as

$\langle
T_{\tau}c^{\dagger}_{\bm{r}_{1}\sigma}(\tau)c_{\bm{r}_{2}\sigma}(\tau)c^{\dagger}_{\bm{r}_{3}\sigma^{\prime}}(0)c_{\bm{r}_{4}\sigma^{\prime}}(0)\rangle\\\
=\langle
T_{\tau}c^{\dagger}_{\bm{r}_{1}\sigma}(\tau)c_{\bm{r}_{4}\sigma^{\prime}}(0)\rangle\langle
T_{\tau}c_{\bm{r}_{2}\sigma}(\tau)c^{\dagger}_{\bm{r}_{3}\sigma^{\prime}}(0)\rangle\\\
=-\delta_{\sigma\sigma^{\prime}}G(\bm{r}_{4},\bm{r}_{1},-\tau)G(\bm{r}_{2},\bm{r}_{3},\tau),$

where $G(\bm{r}_{1},\bm{r}_{2},\tau)=-\langle
T_{\tau}c_{\bm{r}_{1}\sigma}(\tau)c^{\dagger}_{\bm{r}_{2}\sigma}(0)\rangle$ is
the interacting single-particle Green’s function.

The Green’s function as defined above isn’t gauge invariant and it breaks
translational symmetry in the presence of a finite (spatially uniform)
magnetic field. Indeed, a change of the electromagnetic gauge modifies the
Peierls phase according to $A_{\bm{r}_{1}\bm{r}_{2}}\to
A_{\bm{r}_{1}\bm{r}_{2}}+\chi(\bm{r}_{2})-\chi(\bm{r}_{1})$, where
$\chi(\bm{r})$ is the gauge. The gauge invariance of the Hamiltonian requires
that the field operators change as $c_{\bm{r}\sigma}\to
e^{-i\chi(\bm{r})}c_{\bm{r}\sigma}$. Hence the Green’s function changes as
$G(\bm{r}_{1},\bm{r}_{2},\tau)\to
G(\bm{r}_{1},\bm{r}_{2},\tau)e^{i[\chi(\bm{r}_{2})-\chi(\bm{r}_{1})]}$. Let us
therefore introduce a modified Green’s function [35, 16]

$\bar{G}(\bm{r}_{1}-\bm{r}_{2},\tau)=e^{-iA_{\bm{r}_{1}\bm{r}_{2}}}G(\bm{r}_{1},\bm{r}_{2},\tau),$
(26)

that is manifestly gauge invariant 222Using the equation-of-motion method for
expressing the real-space Green’s function, one can verify that $\bar{G}$ is
also translation invariant.. Thanks to its translation invariance in space and
time, $\bar{G}$ admits the Fourier representation
$\bar{G}(\bm{r},\tau)=\frac{k_{\mathrm{B}}T}{N}\sum_{\bm{k}}\sum_{i\omega_{n}}\bar{G}(\bm{k},i\omega_{n})e^{i(\bm{k}\cdot\bm{r}-\omega_{n}\tau)}$,
where $N$ is the number of $\bm{k}$ points and $\omega_{n}=(2n+1)\pi
k_{\mathrm{B}}T$ are odd Matsubara frequencies. Introducing this in Eq. (25),
we get

$C_{J_{\alpha}J_{\beta}}(i\Omega_{n})=-\left(\frac{e}{\hbar}\right)^{2}\frac{1}{N^{2}}\sum_{\bm{k}\bm{q}\sigma}k_{\mathrm{B}}T\sum_{i\omega_{n}}\\\
\times\bar{G}(\bm{k},i\omega_{n})\bar{G}(\bm{k}+\bm{q},i\omega_{n}+i\Omega_{n})\sum_{\bm{r}_{1}\bm{r}_{2}\bm{r}_{3}\bm{r}_{4}}e^{i\bm{k}\cdot(\bm{r}_{4}-\bm{r}_{1})}e^{i(\bm{k}+\bm{q})\cdot(\bm{r}_{2}-\bm{r}_{3})}\\\
\times(\bm{r}_{2}-\bm{r}_{1})_{\alpha}t^{0}_{\bm{r}_{1}\bm{r}_{2}}(\bm{r}_{4}-\bm{r}_{3})_{\beta}t^{0}_{\bm{r}_{3}\bm{r}_{4}}e^{i(A_{\bm{r}_{1}\bm{r}_{2}}+A_{\bm{r}_{3}\bm{r}_{4}}+A_{\bm{r}_{4}\bm{r}_{1}}+A_{\bm{r}_{2}\bm{r}_{3}})}.$

This correlation function is gauge invariant, because the quantity
$A_{\bm{r}_{1}\bm{r}_{2}}+A_{\bm{r}_{2}\bm{r}_{3}}+A_{\bm{r}_{3}\bm{r}_{4}}+A_{\bm{r}_{4}\bm{r}_{1}}$
is proportional to the magnetic flux threading the oriented tetragon defined
by the points $(\bm{r}_{1},\bm{r}_{2},\bm{r}_{3},\bm{r}_{4})$, and it is
therefore independent of the gauge:

$A_{\bm{r}_{1}\bm{r}_{2}}+A_{\bm{r}_{2}\bm{r}_{3}}+A_{\bm{r}_{3}\bm{r}_{4}}+A_{\bm{r}_{4}\bm{r}_{1}}=\frac{|e|}{2\hbar}\bm{B}\cdot(\bm{r}_{1}-\bm{r}_{3})\times(\bm{r}_{2}-\bm{r}_{4}).$
(27)

This is readily verified, for instance in the symmetric gauge
$\bm{A}_{\mathrm{mag}}(\bm{r})=\frac{B}{2}(-y,x,0)$, where
$A_{\bm{r}_{1}\bm{r}_{2}}=\frac{|e|B}{2\hbar}(r_{1x}r_{2y}-r_{1y}r_{2x})=\frac{|e|}{2\hbar}\bm{B}\cdot(\bm{r}_{1}\times\bm{r}_{2})$.

The Green’s function admits an integral representation in terms of its
spectral function:

$\bar{G}(\bm{k},i\omega_{n})=\int_{-\infty}^{\infty}d\varepsilon\,\frac{\bar{A}(\bm{k},\varepsilon)}{i\omega_{n}+\mu-\varepsilon}.$
(28)

$\bar{A}(\bm{k},\varepsilon)$ contains the fractal Hofstadter physics and
cannot be expanded in powers of $B$. However, because the Hofstadter spectrum
is even in $B$, one can conclude that $\bar{A}(\bm{k},\varepsilon)$ formally
deviates from its value at $B=0$ by corrections of order $B^{2}$. Up to first
order in $B$, it is therefore sufficient to replace
$\bar{A}(\bm{k},\varepsilon)$ by its zero-field value, which is
$A(E_{\bm{k}},\varepsilon)$ given in Eq. (2). After inserting the spectral
representation of $\bar{G}$, standard techniques allow one to perform the sum
over odd Matsubara frequencies [29]. A further shift of $\bm{k}$ by
$-\bm{q}/2$ in order to achieve a more symmetric form and introduction of the
relative vectors $\bm{\rho}_{1}=\bm{r}_{2}-\bm{r}_{1}$,
$\bm{\rho}_{2}=\bm{r}_{4}-\bm{r}_{3}$, and $\bm{r}=\bm{r}_{1}-\bm{r}_{3}$
directly leads to Eqs. (10) and (11).

The transport function defined in Eq. (11) obeys the following symmetries:

$\displaystyle\Phi_{\alpha\beta}(E_{1},E_{2},B)$
$\displaystyle=\Phi^{*}_{\alpha\beta}(E_{1},E_{2},-B)$ (29a)
$\displaystyle\Phi_{\alpha\beta}(E_{1},E_{2},B)$
$\displaystyle=\Phi^{*}_{\alpha\beta}(E_{2},E_{1},B)$ (29b)
$\displaystyle\Phi_{\alpha\beta}(E_{1},E_{2},B)$
$\displaystyle=\Phi_{\beta\alpha}(E_{2},E_{1},B).$ (29c)

The first may be checked by changing $(\bm{r},\bm{\rho}_{1},\bm{\rho}_{2})$
into $(-\bm{r},-\bm{\rho}_{1},-\bm{\rho}_{2})$. The second by changing
$(\bm{q},\bm{\rho}_{1},\bm{\rho}_{2},\bm{r})$ into
$(-\bm{q},-\bm{\rho}_{1},-\bm{\rho}_{2},\bm{r}+\bm{\rho}_{1}-\bm{\rho}_{2})$.
The third by changing $(\bm{q},\bm{\rho}_{1},\bm{\rho}_{2},\bm{r})$ into
$(-\bm{q},\bm{\rho}_{2},\bm{\rho}_{1},-\bm{r})$. Using these symmetries, one
verifies that
$\sigma_{\alpha\beta}(\omega,B)=\sigma^{*}_{\beta\alpha}(-\omega,-B)$. In the
DC limit, this gives the Onsager-Casimir relation
$\sigma_{\alpha\beta}(B)=\sigma_{\beta\alpha}(-B)$ [37, 38]. Equation (29a)
implies that $\mathrm{Re}\,\Phi_{\alpha\beta}(E_{1},E_{2},B)$ and
$\mathrm{Im}\,\Phi_{\alpha\beta}(E_{1},E_{2},B)$ are even and odd functions of
$B$, respectively, which fixes the parity of Eqs. (12) and (13) in the field.
Equation (29c) furthermore implies that
$\sigma_{\beta\alpha}^{(e)}(B)=\sigma_{\alpha\beta}^{(e)}(B)$ and
$\sigma_{\beta\alpha}^{(o)}(B)=-\sigma_{\alpha\beta}^{(o)}(B)$. The latter
shows that the part odd in the field is anti-diagonal,
$\sigma_{\alpha\alpha}^{(o)}(B)=0$. Making use of $\sigma_{xy}=-\sigma_{yx}$,
the former shows that the even part is diagonal,
$\sigma_{\alpha\beta}^{(e)}(B)\propto\delta_{\alpha\beta}$.

##  Appendix B Weak-field expansion of the DC conductivity

The magnetic field dependence of the DC conductivity defined by Eqs. (9), (10)
and (11) stems from the quantity

$\Delta(\bm{q},\bm{\rho},B)=\frac{1}{N}\sum_{\bm{r}}e^{i\bm{q}\cdot\bm{r}}e^{-\frac{i|e|}{2\hbar}B\hat{\bm{z}}\cdot\bm{\rho}\times\bm{r}}.$
(30)

Its zero-field limit gives a Kronecker symbol at $\bm{q}=\bm{0}$:

$\Delta(\bm{q},\bm{\rho},0)=\frac{1}{N}\sum_{\bm{r}}e^{i\bm{q}\cdot\bm{r}}=\delta_{\bm{q}\bm{0}}.$
(31)

The transport function at this order is, therefore,

$\Phi_{\alpha\beta}(E_{1},E_{2},0)=-\left(\frac{e}{\hbar}\right)^{2}\frac{1}{V}\sum_{\bm{k}\sigma}\delta\left(E_{1}-E_{\bm{k}}\right)\delta\left(E_{2}-E_{\bm{k}}\right)\\\
\times\sum_{\bm{\rho}_{1}\bm{\rho}_{2}}\rho_{1\alpha}t^{0}_{\bm{\rho}_{1}}e^{i\bm{k}\cdot\bm{\rho}_{1}}\rho_{2\beta}t^{0}_{\bm{\rho}_{2}}e^{i\bm{k}\cdot\bm{\rho}_{2}}.$

Introducing the group velocity $v_{\bm{k}\alpha}=(1/\hbar)\partial
E_{\bm{k}}/\partial k_{\alpha}$, we can substitute
$\sum_{\bm{\rho}}\rho_{\alpha}t^{0}_{\bm{\rho}}e^{i\bm{k}\cdot\bm{\rho}}$ by
$-i\hbar v_{\bm{k}\alpha}$. Since we only consider systems with the inversion
symmetry $E_{\bm{k}}=E_{-\bm{k}}$, the components of the group velocity are
odd functions of $\bm{k}$. Hence $\Phi_{\alpha\beta}$ vanishes by symmetry if
$\alpha\neq\beta$ and we have

$\Phi_{\alpha\beta}(E_{1},E_{2},0)=\delta_{\alpha\beta}\delta(E_{1}-E_{2})e^{2}\frac{1}{V}\sum_{\bm{k}\sigma}v_{\bm{k}\alpha}^{2}\delta\left(E_{1}-E_{\bm{k}}\right),$

which corresponds to Eq. (15).

At first order in $B$, we have

$\left.\frac{d}{dB}\Delta(\bm{q},\bm{\rho},B)\right|_{B=0}=-\frac{|e|}{2\hbar}\hat{\bm{z}}\cdot\bm{\rho}\times\bm{\nabla}_{\bm{q}}\frac{1}{N}\sum_{\bm{r}}e^{i\bm{q}\cdot\bm{r}}.$
(32)

This formally involves the gradient of the Kronecker symbol
$\delta_{\bm{q}\bm{0}}$, which is only well-defined after taking the continuum
limit for the momenta. Implying this limit, we evaluate the $\bm{q}$ sum in
Eq. (11) by parts as
$\sum_{\bm{q}}F(\bm{q})\bm{\nabla}_{\bm{q}}\delta_{\bm{q}\bm{0}}=-\sum_{\bm{q}}\delta_{\bm{q}\bm{0}}\bm{\nabla}_{\bm{q}}F(\bm{q})=-\bm{\nabla}_{\bm{q}}F(\bm{q})|_{\bm{q}=\bm{0}}$.
Upon calculating $d\Phi_{\alpha\beta}(E_{1},E_{2},B)/dB$ at $B=0$, the
$\bm{\nabla}_{\bm{q}}$ yields one term from acting on the phase factors and
two terms from acting on the Dirac delta functions. The former vanishes, as it
is proportional to
$(\bm{\rho}_{1}-\bm{\rho}_{2})\times(\bm{\rho}_{1}-\bm{\rho}_{2})$. We get

$\left.\frac{d}{dB}\Phi_{\alpha\beta}(E_{1},E_{2},B)\right|_{B=0}=\frac{i|e|^{3}\hbar}{4}\frac{1}{V}\sum_{\bm{k}\sigma}\\\
\times\left[\delta^{\prime}\left(E_{1}-E_{\bm{k}}\right)\delta\left(E_{2}-E_{\bm{k}}\right)-\delta\left(E_{1}-E_{\bm{k}}\right)\delta^{\prime}\left(E_{2}-E_{\bm{k}}\right)\right]\\\
\times\left(\frac{v_{\bm{k}x}v_{\bm{k}\beta}}{m_{\bm{k}y\alpha}}-\frac{v_{\bm{k}y}v_{\bm{k}\beta}}{m_{\bm{k}x\alpha}}-\frac{v_{\bm{k}\alpha}v_{\bm{k}x}}{m_{\bm{k}y\beta}}+\frac{v_{\bm{k}\alpha}v_{\bm{k}y}}{m_{\bm{k}x\beta}}\right),$
(33)

where $\delta^{\prime}$ stands for the derivative of the Dirac delta and the
mass tensor
$1/m_{\bm{k}\alpha\beta}=(1/\hbar^{2})\partial^{2}E_{\bm{k}}/(\partial
k_{\alpha}\partial k_{\beta})$ has been introduced. This result is purely
imaginary and thus only contributes to $\sigma_{\alpha\beta}^{(o)}(B)$, as
expected. It is also odd under the exchange of $\alpha$ and $\beta$ and
vanishes as it should if $\alpha=\beta$.

When Eq. (33) is inserted in Eq. (13), the resulting $E_{1}$ or $E_{2}$
integral involving $\delta^{\prime}$ can be evaluated by parts using, for
instance,
$\int_{-\infty}^{\infty}dE_{1}\,\delta^{\prime}(E_{1}-E_{\bm{k}})A(E_{1},\varepsilon)=-\int_{-\infty}^{\infty}dE_{1}\,\delta(E_{1}-E_{\bm{k}})A^{\prime}(E_{1},\varepsilon),$
with $A^{\prime}(E,\varepsilon)$ denoting the derivative of the spectral
function with respect to the first argument. Upon performing this
substitution, we encounter the integral

$\displaystyle I$
$\displaystyle=\int_{-\infty}^{\infty}d\varepsilon_{1}d\varepsilon_{2}\,\frac{f(\varepsilon_{1}-\mu)-f(\varepsilon_{2}-\mu)}{(\varepsilon_{1}-\varepsilon_{2})^{2}}$
$\displaystyle\quad\times\left[A(E,\varepsilon_{1})A^{\prime}(E,\varepsilon_{2})-A^{\prime}(E,\varepsilon_{1})A(E,\varepsilon_{2})\right].$

Remarkably, if the spectral function is causal this integral can be recast as

$I=\frac{4\pi^{2}}{3}\int_{-\infty}^{\infty}d\varepsilon\,[-f^{\prime}(\varepsilon-\mu)]A^{3}(E,\varepsilon).$
(34)

This leads to Eqs. (16) and (17). In order to prove Eq. (34), we rewrite

$I=2\int_{-\infty}^{\infty}d\varepsilon_{1}\,f(\varepsilon_{1}-\mu)A(E,\varepsilon_{1})\int_{\infty}^{\infty}d\varepsilon_{2}\,\frac{A^{\prime}(E,\varepsilon_{2})}{(\varepsilon_{1}-\varepsilon_{2})^{2}}\\\
-2\int_{-\infty}^{\infty}d\varepsilon_{1}\,f(\varepsilon_{1}-\mu)A^{\prime}(E,\varepsilon_{1})\int_{\infty}^{\infty}d\varepsilon_{2}\,\frac{A(E,\varepsilon_{2})}{(\varepsilon_{1}-\varepsilon_{2})^{2}},$

where a principal value is understood in the $\varepsilon_{2}$ integrals. We
now use the causality of the self-energy, which implies (and requires)

$\displaystyle\mathrm{Re}\,G(E,\varepsilon_{1})$
$\displaystyle=\int_{\infty}^{\infty}d\varepsilon_{2}\,\frac{A(E,\varepsilon_{2})}{\varepsilon_{1}-\varepsilon_{2}}$
$\displaystyle\frac{d}{d\varepsilon_{1}}\mathrm{Re}\,G(E,\varepsilon_{1})$
$\displaystyle=-\int_{\infty}^{\infty}d\varepsilon_{2}\,\frac{A(E,\varepsilon_{2})}{(\varepsilon_{1}-\varepsilon_{2})^{2}},$

with $G(E,\varepsilon)$ the retarded Green’s function. It follows that

$I=-2\int_{-\infty}^{\infty}d\varepsilon\,f(\varepsilon-\mu)\\\
\times\underbrace{\left[A(E,\varepsilon)\frac{d}{d\varepsilon}\mathrm{Re}\,G^{\prime}(E,\varepsilon)-A^{\prime}(E,\varepsilon)\frac{d}{d\varepsilon}\mathrm{Re}\,G(E,\varepsilon)\right]}_{\displaystyle=-\frac{2\pi^{2}}{3}\frac{d}{d\varepsilon}A^{3}(E,\varepsilon).}$

We have used Eq. (2) and the explicit form of the Green’s function,
$G(E,\varepsilon)=1/[\varepsilon-E-\Sigma(\varepsilon)]$. An integration by
parts then gives Eq. (34).

##  Appendix C Relation with the semiclassical theory

The Boltzmann expressions in the isotropic relaxation-time approximation may
be derived from Eqs. (14–17) under the assumptions that (i) the spectral
function is narrower than the derivative of the Fermi distribution, such that
$-f^{\prime}(\varepsilon-\mu)$ may be replaced by $-f^{\prime}(E-\mu)$ and
(ii) the self-energy takes the form $\Sigma(\varepsilon)=-i\hbar/(2\tau)$
where $\tau$ is the relaxation time. Using the identities
$\int_{-\infty}^{\infty}d\varepsilon\,A^{2}(E,\varepsilon)=1/(2\pi\Gamma)$ and
$\int_{-\infty}^{\infty}d\varepsilon\,A^{3}(E,\varepsilon)=(3/2)/(2\pi\Gamma)^{2}$,
one thus arrives at the approximations

$\displaystyle\sigma_{\alpha}^{(0)}$ $\displaystyle\approx
e^{2}\sum_{\bm{k}\sigma}[-f^{\prime}(E_{\bm{k}}-\mu)]\tau
v_{\bm{k}\alpha}^{2}$ $\displaystyle\sigma_{xy}^{(1)}$
$\displaystyle\approx-|e|^{3}\sum_{\bm{k}\sigma}[-f^{\prime}(E_{\bm{k}}-\mu)]\tau^{2}$
$\displaystyle\quad\times\left[\frac{1}{2}\left(\frac{v_{\bm{k}x}^{2}}{m_{\bm{k}yy}}+\frac{v_{\bm{k}y}^{2}}{m_{\bm{k}xx}}\right)-\frac{v_{\bm{k}x}v_{\bm{k}y}}{m_{\bm{k}xy}}\right],$

which, apart from the $(x,y)$ symmetrization at the last line, coincide with
the semiclassical formula given in the literature [9, 32, 33].

##  Appendix D Exact integrals for the case $\Sigma(\varepsilon)=-i\Gamma$

The self-energy $\Sigma(\varepsilon)=-i\Gamma$ obeys causality, because the
Kramers–Kronig transform of a constant is zero. Consistently, the spectral
function $A(E,\varepsilon)=\frac{\Gamma/\pi}{(\varepsilon-E)^{2}+\Gamma^{2}}$
is properly normalized to
$\int_{-\infty}^{\infty}d\varepsilon\,A(E,\varepsilon)=1$. The integrals that
involve the Fermi distribution and the spectral function have a closed form in
terms of polygamma functions $\psi_{n}(z)$, where
$z=\frac{1}{2}-i\frac{E-\mu+i\Gamma}{2\pi k_{\mathrm{B}}T}$. The $\varepsilon$
integral in Eq. (3) reads

$\displaystyle\int_{-\infty}^{\infty}d\varepsilon\,f(\varepsilon-\mu)A(E,\varepsilon)$
$\displaystyle=\frac{1}{2}+\frac{1}{\pi}\mathrm{Im}\,\psi_{0}(z)$
$\displaystyle=\frac{1}{2}-\frac{1}{\pi}\tan^{-1}\left(\frac{E-\mu}{\Gamma}\right)\quad(T=0).$

The integrals appearing in Eqs. (14) and (16) are trivial at $T=0$ and
otherwise read

$\displaystyle\int_{-\infty}^{\infty}d\varepsilon\,[-f^{\prime}(\varepsilon-\mu)]A^{2}(E,\varepsilon)$
$\displaystyle\qquad=\frac{\zeta}{2(\pi\Gamma)^{2}}\mathrm{Re}\left[\psi_{1}(z)-\zeta\psi_{2}(z)\right]$
$\displaystyle\int_{-\infty}^{\infty}d\varepsilon\,[-f^{\prime}(\varepsilon-\mu)]A^{3}(E,\varepsilon)$
$\displaystyle\qquad=\frac{3\zeta}{8(\pi\Gamma)^{3}}\mathrm{Re}\left[\psi_{1}(z)-\zeta\psi_{2}(z)+\frac{\zeta^{2}}{3}\psi_{3}(z)\right],$

where $\zeta=\frac{\Gamma}{2\pi k_{\mathrm{B}}T}$.

##  Appendix E Exact asymptotic results and approximations

###  E.1 $T=0$

At zero temperature, the transport equations are

$\displaystyle n$
$\displaystyle=\int_{-\infty}^{\infty}dE\,N_{0}(E)\left[\frac{1}{2}-\frac{1}{\pi}\tan^{-1}\left(\frac{E-\mu}{\Gamma}\right)\right]$
$\displaystyle\sigma_{\alpha}^{(0)}$
$\displaystyle=\pi\hbar\int_{-\infty}^{\infty}dE\,\Phi_{\alpha}^{(0)}(E)A^{2}(E,\mu)$
$\displaystyle\sigma_{xy}^{(1)}$
$\displaystyle=\hbar\int_{-\infty}^{\infty}dE\,\Phi^{(1)}(E)A^{3}(E,\mu).$

In the semiclassical regime, where $\Gamma$ is the smallest energy scale, the
transport functions may be expanded around $E=\mu$. The leading terms give

$\displaystyle\sigma_{\alpha}^{(0)}$
$\displaystyle=\pi\hbar\Phi_{\alpha}^{(0)}(\mu)\int_{-\infty}^{\infty}dE\,A^{2}(E,\mu)=\frac{\hbar}{2\Gamma}\Phi_{\alpha}^{(0)}(\mu)$
$\displaystyle\sigma_{xy}^{(1)}$
$\displaystyle=\hbar\Phi^{(1)}(\mu)\int_{-\infty}^{\infty}dE\,A^{3}(E,\mu)=\frac{3\hbar}{8\pi^{2}\Gamma^{2}}\Phi^{(1)}(\mu).$

This leads to Eq. (18) of the main text. In order to estimate the crossover
density in Fig. 5, we fix $\mu=-D$ in the equation giving the density. Since
the term in square brackets drops rapidly for energies away from the band
edge, we can use the approximation

$\displaystyle n_{c}$ $\displaystyle\approx
N_{0}(-D)\int_{-D}^{D}dE\,\left[\frac{1}{2}-\frac{1}{\pi}\tan^{-1}\left(\frac{E+D}{\Gamma}\right)\right]$
$\displaystyle=\frac{\Gamma}{2\pi^{2}\mathfrak{a}\sqrt{|t_{x}t_{y}|}}\left[1+\ln\left(\frac{2D}{\Gamma}\right)+\mathscr{O}\left(\frac{\Gamma^{2}}{D^{2}}\right)\right].$

In the quantum regime, where $n$ approaches zero at finite $\Gamma$, we can
expand the equations for $\mu\to-\infty$:

$\displaystyle n$
$\displaystyle=\int_{-\infty}^{\infty}dE\,N_{0}(E)\left[-\frac{\Gamma}{\pi\mu}-\frac{\Gamma
E}{\pi\mu^{2}}+\mathscr{O}\left(\frac{1}{\mu^{3}}\right)\right]$
$\displaystyle\sigma_{\alpha}^{(0)}$
$\displaystyle=\pi\hbar\int_{-\infty}^{\infty}dE\,\Phi_{\alpha}^{(0)}(E)\left[\frac{\Gamma^{2}}{\pi^{2}\mu^{4}}+\frac{4\Gamma^{2}E}{\pi^{2}\mu^{5}}+\mathscr{O}\left(\frac{1}{\mu^{6}}\right)\right]$
$\displaystyle\sigma_{xy}^{(1)}$
$\displaystyle=\hbar\int_{-\infty}^{\infty}dE\,\Phi^{(1)}(E)\left[\frac{\Gamma^{3}}{\pi^{3}\mu^{6}}+\frac{6\Gamma^{3}E}{\pi^{3}\mu^{7}}+\mathscr{O}\left(\frac{1}{\mu^{8}},\frac{E^{2}}{\mu^{8}}\right)\right].$

Considering the symmetry of the DOS and transport functions, the relevant
moments are

$\displaystyle\int_{-\infty}^{\infty}dE\,N_{0}(E)$
$\displaystyle=\frac{2}{\mathfrak{a}}$
$\displaystyle\int_{-\infty}^{\infty}dE\,\Phi_{\alpha}^{(0)}(E)$
$\displaystyle=\left(\frac{e}{\hbar}\right)^{2}\frac{4a_{\alpha}^{2}t_{\alpha}^{2}}{\mathfrak{a}}$
$\displaystyle\int_{-\infty}^{\infty}dE\,E\,\Phi^{(1)}(E)$
$\displaystyle=\left(\frac{|e|}{\hbar}\right)^{3}\frac{16\pi^{2}}{3}\mathfrak{a}t_{x}^{2}t_{y}^{2},$

where $a_{x}$ and $a_{y}$ are the lattice parameters and
$\mathfrak{a}=a_{x}a_{y}$. Hence we arrive at

$\displaystyle n$
$\displaystyle=-\frac{1}{\mathfrak{a}}\frac{2\Gamma}{\pi\mu}+\mathscr{O}(1/\mu^{3})$
$\displaystyle\sigma_{\alpha}^{(0)}$
$\displaystyle=\frac{e^{2}a_{\alpha}^{2}t_{\alpha}^{2}}{\hbar\mathfrak{a}}\frac{4\Gamma^{2}}{\pi\mu^{4}}+\mathscr{O}(1/\mu^{6})$
$\displaystyle\sigma_{xy}^{(1)}$
$\displaystyle=\frac{|e|^{3}\mathfrak{a}t_{x}^{2}t_{y}^{2}}{\hbar^{2}}\frac{32\Gamma^{3}}{\pi\mu^{7}}+\mathscr{O}(1/\mu^{9}),$

which leads to Eq. (19).

###  E.2 $T>0$

At low temperature, we rely on a Sommerfeld expansion. For the density in Eq.
(3), we integrate by parts to let the quantity $-f^{\prime}(\varepsilon-\mu)$
appear. We change variable from $\varepsilon$ to
$u=(\varepsilon-\mu)/(k_{\mathrm{B}}T)$, expand the integrand in powers of
$T$, and perform the integrals using
$\int_{-\infty}^{\infty}du\,\frac{e^{u}}{(e^{u}+1)^{2}}\\{1,u,u^{2}\\}=\\{1,0,\pi^{2}/3\\}$.
The condition $n(T)=n(0)$ allows us to deduce $\mu^{\prime}(0)=0$, where
$\mu^{\prime}(0)$ is the first derivative of the chemical potential with
respect to $k_{\mathrm{B}}T$. A closed expression for the second derivative
can also be deduced in the low-density regime, after sending
$\mu_{0}\equiv\mu(T=0)$ to $-\infty$ and using the relation
$\mu_{0}=-2\Gamma/(\pi n\mathfrak{a})$ derived above. This expression is
$\mu^{\prime\prime}(0)=-\pi^{3}n\mathfrak{a}/(3\Gamma)$. Proceeding similarly
with Eqs. (14) and (16), we arrive at the relations

$\displaystyle\frac{\sigma_{\alpha}^{(0)}(T)}{\sigma_{\alpha}^{(0)}(0)}$
$\displaystyle=1+\frac{\pi^{4}}{2\Gamma^{2}}(n\mathfrak{a})^{2}(k_{\mathrm{B}}T)^{2}$
$\displaystyle\frac{\sigma^{(1)}_{xy}(T)}{\sigma^{(1)}_{xy}(0)}$
$\displaystyle=1+\frac{7\pi^{4}}{4\Gamma^{2}}(n\mathfrak{a})^{2}(k_{\mathrm{B}}T)^{2}$
$\displaystyle\frac{r(T)}{r(0)}$
$\displaystyle=1+\frac{3\pi^{4}}{4\Gamma^{2}}(n\mathfrak{a})^{2}(k_{\mathrm{B}}T)^{2},$

that are valid for $T\to 0$ and $n\to 0$ at finite $\Gamma$. At higher
density, when $\mu_{0}$ lies inside the band, instead of the expansion around
$\mu_{0}=-\infty$ we expand the transport functions around $E=\mu_{0}$. Thus
we find
$\mu^{\prime\prime}(0)=-(\pi^{2}/3)N_{0}^{\prime}(\mu_{0})/N_{0}(\mu_{0})$
and, for the conductivities,

$\displaystyle\frac{\sigma_{\alpha}^{(0)}(T)}{\sigma_{\alpha}^{(0)}(0)}$
$\displaystyle=1-\frac{\pi^{2}}{6}\left[\frac{N_{0}^{\prime}(\mu_{0})}{N_{0}(\mu_{0})}\frac{{\Phi_{\alpha}^{(0)}}^{\prime}(\mu_{0})}{\Phi_{\alpha}^{(0)}(\mu_{0})}-\frac{{\Phi_{\alpha}^{(0)}}^{\prime\prime}(\mu_{0})}{\Phi_{\alpha}^{(0)}(\mu_{0})}\right](k_{\mathrm{B}}T)^{2}$
$\displaystyle\frac{\sigma^{(1)}_{xy}(T)}{\sigma^{(1)}_{xy}(0)}$
$\displaystyle=1-\frac{\pi^{2}}{6}\left[\frac{N_{0}^{\prime}(\mu_{0})}{N_{0}(\mu_{0})}\frac{{\Phi^{(1)}}^{\prime}(\mu_{0})}{\Phi^{(1)}(\mu_{0})}-\frac{{\Phi^{(1)}}^{\prime\prime}(\mu_{0})}{\Phi^{(1)}(\mu_{0})}\right](k_{\mathrm{B}}T)^{2}.$

We estimate the onset temperatures $T_{x}$, $T_{y}$, and $T_{xy}$ in Fig. 6 as
follows. In the regime $\Gamma<k_{\mathrm{B}}T\ll D$, the spectral function is
narrower than the derivative of the Fermi function. We therefore replace in
Eqs. (14) and (16) $f^{\prime}(\varepsilon-\mu)$ by $f^{\prime}(E-\mu)$, such
that the $\varepsilon$ integrals become trivial. With the low-density regime
in mind, we further note that for $k_{\mathrm{B}}T\ll D$, the function
$f^{\prime}(E-\mu)$ is sharp compared with the transport function and peaked
near the band bottom. We therefore replace the transport functions by their
linearizations close to the band bottom, that are indicated in the legend of
Fig. 2. We thus arrive at Eq. (21).

We close this appendix by showing that, in the high-temperature regime
$\Gamma\ll k_{\mathrm{B}}T<D$, the Hall constant recovers approximately the
classical value Eq. (18), irrespective of the density. In that regime, we can
replace $f^{\prime}(\varepsilon-\mu)$ by $f^{\prime}(E-\mu)$, because the
derivative of the Fermi function is narrow compared with the spectral
function, and we can replace the transport function by its value at $E=\mu$,
because the derivative of the Fermi function is still much sharper that the
transport function. This directly leads to the result

$R_{\mathrm{H}}\approx\frac{3}{2\pi^{2}\hbar}\frac{\Phi^{(1)}(\mu)}{\Phi^{(0)}_{x}(\mu)\Phi^{(0)}_{y}(\mu)}\quad(\Gamma\ll
k_{\mathrm{B}}T<D),$

which is the same as the zero-temperature Hall constant for $\Gamma\to 0$.

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