# Topological robustness of quantization of the anomalous Hall conductance
of a two-dimensional disordered Chern insulator

S. G. Novokshonov<EMAIL_ADDRESS>M. N. Mikheev Institute of Metal Physics of
Ural Branch of Russian Academy of Sciences, 620108, 18, S. Kovalevskaya st.,
Ekaterinburg, Russia  Institute of Physics and Technology, Ural Federal
University, 620002, 19, Mira st., Ekaterinburg, Russia

###### Abstract

The robustness of the anomalous Hall conductance, $\sigma_{yx}=\sigma_{H}$,
quantization in the model of a two-dimensional disordered gas of massive Dirac
electrons subjected to an external orthogonal magnetic field is investigated
in the framework of Kubo–Středa formalism. Using the momentum representation
for the averaged one-electron Green functions in a magnetic field, an explicit
analytical expression for the Středa term of $\sigma_{H}$ is obtained. It is
shown that this term is proportional to the topological Chern number, ${\rm
Ch}=\pm 1/2$, if the Fermi level is in the energy gap. In this case, the total
$\sigma_{H}$ takes the half-integer quantized value, $\sigma_{H}=\pm
e^{2}/4\pi\hbar$,that does not depend on either the magnitude of the disorder
or the strength of the external magnetic field. As an example, we calculated
the densities of states and intrinsic anomalous Hall conductance,
$\sigma_{H}^{\rm int}$, of a two-dimensional disordered gas of massive Dirac
electrons subjected to an external magnetic field in the self-consistent Born
approximation. A numerical analysis of the field and energy dependencies of
these expressions is carried out for various values of the parameters of the
model under consideration. In particular, the results of this analysis show
that the Středa term of $\sigma_{H}$ is susceptible to disorder in a
sufficiently wide vicinity of the point of transition to the quantization
regime.

quantum anomalous Hall effect, Chern number, Berry curvature, Chern insulator,
Dirac electrons, Středa term, Moyal product

###### pacs:

72.15.Gd,72.25.-b,72.25.Dc,72.80.Ng

##  I  Introduction

In the last years, a widespread interest was attracted to investigation of the
quantum anomalous Hall effect (quantum AHE or QAHE) in the topological non-
trivial materials weng_etal_2015 ; wang_etal_2015 ; liu_etal_2016 . The
possibility of quantizing Hall conductance without Landau levels was first
demonstrated by Haldane haldane_1988  based on a tight-binding model with a
zero net magnetic flux through the unit cell of the honeycomb lattice.
Subsequently, several authors predicted the quantization of anomalous Hall
conductance in two-dimensional (2D) magnets with spin-orbit interaction
ono_nag_2002 ; ono_nag_2003 ; cul_etal_2003 ; qi_etal_2006 . This effect was
first discovered experimentally by Cui-Zu Chang with co-workers
chang_etal_2013  in thin films of the $({\rm Bi}_{x}{\rm Sb}_{1-x})_{2}{\rm
Te}_{3}$ topological insulator doped by Cr. An overview of recent experimental
and theoretical studies of QAHE can be found in review by Culcer et
al.cul_etal_2020

Assuming $T=0$, the expression for the anomalous Hall conductance
$\sigma_{H}=\sigma_{yx}$ of a clean electron system with broken time reversal
symmetry can be written as ono_nag_2002 ; jung_etal_2002

$\sigma_{H}=\sigma_{0}\sum_{n}\\!\int_{{\cal E}_{n}(\bm{p})<{\cal
E}_{F}}\\!\Omega_{n,z}(\bm{p})\,\frac{d\bm{p}}{2\pi}.$ (1)

Here, $\sigma_{0}=e^{2}/2\pi\hbar$ is the conductance quantum,
$\Omega_{n,z}(\bm{p})=i\langle
n\bm{p}|(\stackrel{{\scriptstyle\leftarrow}}{{\rotatebox[origin={c}]{180.0}{$\varDelta$}_{\\!\\!\bm{p}}}}\times\stackrel{{\scriptstyle\rightarrow}}{{\rotatebox[origin={c}]{180.0}{$\varDelta$}_{\\!\\!\bm{p}}}})_{z}|n\bm{p}\rangle$
is the $z$-component of Berry’s curvature vectorberry_1985  of the electron
states $|n\bm{p}\rangle$ manifold in $n$th energy band. The arrows above the
nabla operators indicate the directions of their actions.

If the Fermi level $\mbox{$\cal E$}_{F}$ is in an energy gap, then the
integration in the each non-zero summand in (1) is performed over the closed
manifold without boundary. As a result, these integrals become equal to
integers Chern numbers $\rm Ch$ (or $\pm 1/2$ for the model of 2D massive
Dirac electrons), and the Hall conductance takes on a quantized value that is
a multiple of $\sigma_{0}$. In this regime, $\sigma_{H}=\sigma_{0}{\rm Ch}$ is
non-zero entirely due to the so-called intrinsic mechanism of AHE proposed by
Karplus and Luttingerkarp_lutt_1954 . This mechanism is caused by non-trivial
topology of the electron states in ideal crystalcul_etal_2003 ; jung_etal_2002
. Therefore, the quantization of $\sigma_{H}$ must be protected against
perturbations introduced by both internal disorder and external fields, for
example, magnetic field. The numerical finite-size scaling analysis of the
anomalous Hall conductance of the Haldane modelhaldane_1988  with Anderson
disorder confirms this conclusionono_nag_2003 . Namely, the $\sigma_{H}$ tends
to the quantized value $e^{2}/2\pi\hbar$ in the thermodynamic limit if the
Fermi level is in the mobility gap.

The fundamental problem of the influence of an external magnetic field on the
QAHE was investigated by Böttcher et al.boet_etal_2019 ; boet_etal_2020 .
Using the effective field theory, the authors of these works showed that
quantization of the anomalous Hall conductance $\sigma_{H}^{\rm a}$ survives
in an external orthogonal magnetic field and can be distinguished from the
quantization of normal Hall conductance due to its parity $\sigma_{H}^{\rm
a}(B)=\sigma_{H}^{\rm a}(-B)$.

In both these cases, the quantized anomalous Hall conductance is proportional
to the Chern number. But the equation (1) holds for a free system in the clean
limit when the momentum is a well-defined quantum number. However, there is an
alternative expression for the Chern number in terms of one-electron Green’s
functions (GF)thoul_etal_1982 ; q_niu_etal_1985 ; volovik_1988 ; volovik_2003
, which can be generalized to the case of disorder, electron-electron
interaction, and external magnetic fieldishi_mat_1987 ; matsu_1987 ;
zubkov_2016 . The most general expression for the Chern number in terms of
one-electron GF was obtained by Zubkov et al.zubkov_2016 ; zub_wu_2020 ;
zhang_zub_2019  in the Wigner representation, which is convenient for studying
spatially nonuniform systems.

Thus, the understanding of the physical nature of the quantization of the
anomalous Hall effect has reached a fairly high level by now. Nevertheless,
there are a number of problems in this area that need to be addressed. In
particular, the question of the influence of disorder and external magnetic
field on the quantization of the anomalous Hall conductance and its behavior
in vicinity of the plateau has not been sufficiently studied. The present work
is devoted to solving this problem.

The main text of the article is organized as follows. Sec. II describes a
model of massive Dirac electrons for a two-dimensional disordered Chern
insulator, and also presents the Kubo-Středa formulas for the Hall conductance
of non-interacting disordered system. In Sec. III, the calculation of the
self-energies and densities of states (DOS) of electrons necessary for the
subsequent analysis is performed in the self-consistent Born approximation
(SCBA). In Sec. IV, we derive the relation between the Středa term of the Hall
conductance and the topological Chern number, which is valid in the presence
of disorder and an external magnetic field. In Sec. V, using the results
obtained above, we calculate in SCBA the intrinsic anomalous Hall conductance
of a two-dimensional disordered gas of massive Dirac fermions subject to an
orthogonal external magnetic field. Finally, we discuss our results in Sec.
VI.

##  II  Model

Let us consider a two-dimensional (2D, $||\rm OXY$) disordered Chern insulator
subject to an external orthogonal ($||\rm OZ$) magnetic field
$\bm{B}=\rotatebox[origin={c}]{180.0}{$\varDelta$}\times\bm{A}$. The one-
particle Hamiltonian of the minimal model of the system under consideration
has the following formbernevig_2013

$\mbox{$\cal
H$}+U=\bm{d}_{\mbox{\boldmath\scriptsize$\pi$}}\cdot\mbox{\boldmath$\sigma$}+U=v(\mbox{\boldmath$\pi$}\cdot\mbox{\boldmath$\sigma$})+M\sigma_{z}+U(\bm{r}),$
(2)

Here, the mechanical momentum $\mbox{\boldmath$\pi$}=\bm{p}-e\bm{A}/c$ of an
electron is treated as a two-dimensional vector in the plane of the considered
system, $\mbox{\boldmath$\sigma$}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the
3D-vector formed by the Pauli (pseudo-) spin matrices, $v$ is a model
parameter, which has the meaning of some characteristic velocity, and $M$ is
proportional to Dirac mass $|M|/v^{2}$. The inclusion of the term
$M\sigma_{z}$ leads to violation of the time reversal symmetry of the
Hamiltonian $\cal H$ (2) and also opens the gap $\mbox{$\cal E$}_{g}=2|M|$ in
its spectrum. Finally, $U(\bm{r})$ is a white-noise Gaussian random potential
field.

In absence of an external magnetic field, the electron dispersion in the clean
model (2) is the energy spectrum of two-dimensional massive Dirac electrons

$\mbox{$\cal E$}_{s}(\bm{p})=s\mbox{$\cal
E$}(\bm{p})=s\sqrt{M^{2}+2mv^{2}\mbox{$\cal E$}_{\bm{p}}},$ (3)

where $\mbox{$\cal E$}_{\bm{p}}=\bm{p}^{2}/2m$ is the kinetic energy of a free
non-relativistic electron. The quantum number $s=\pm 1$ determines the
helicity of the electron eigenstate with energy (3), that is, the projection
of its (pseudo-) spin onto the direction of the vector $\bm{d}_{\bm{p}}$. In
an orthogonal external magnetic field ($\bm{B}||\rm OZ$), the spectrum (3) is
transformed into two series of Landau levels

$\mbox{$\cal E$}_{s,n}=s\mbox{$\cal
E$}_{n}=s\sqrt{M^{2}+2mv^{2}\hbar\omega_{c}n}\,,\quad~{}\begin{array}[]{l}s=+1,~{}~{}~{}n\geqslant
1,\\\ s=-1,~{}~{}~{}n\geqslant 0,\end{array}$ (4)

where $\omega_{c}=|e|B/mc$ is the cyclotron frequency of a free electron, $n$
is the integer number of the Landau level $\mbox{$\cal E$}_{n}$. The typical
one-particle dispersion (3) and two fans of the Landau levels (4) are shown in
Fig. 1.

Figure 1:  (color online) Left panel: The gapped ($\mbox{$\cal E$}_{g}=2|M|$)
one-particle dispersion $\mbox{$\cal E$}_{s}(\bm{p})$ (3). Dashed lines
represent the gapless spectrum of a massless Dirac electron, $p_{0}=|M|/v$.
Right panel: The gapped ($\mbox{$\cal
E$}_{g}=|M|+\sqrt{M^{2}+2mv^{2}\hbar\omega_{c}}$) series of the Landau levels
(4). The horizontal line depicts the anomalous Landau level $\mbox{$\cal
E$}_{-1,0}=-M$ ($M>0$) in the lower subband of the electron spectrum.

According to the Kubo–Středa formalism streda , the expression for the Hall
component of the conductivity tensor of the system under consideration can be
written as

$\displaystyle\sigma_{H}=$ $\displaystyle\,\sigma_{H}^{\rm I}+\sigma_{H}^{\rm
II}=-\frac{\hbar e^{2}}{2\pi}\\!\int\\!{\displaystyle\frac{\partial
f}{\partial\mbox{$\cal E$}}}\mathop{\rm
Tr}\nolimits\big{\langle}V_{y}\mbox{$\sf G$}^{R}V_{x}\mbox{$\sf
G$}^{A}\big{\rangle}d\mbox{$\cal E$}-$ $\displaystyle\,-\frac{\hbar
e^{2}}{2\pi}\mathop{\rm Re}\nolimits\\!\int\\!f(\mbox{$\cal E$})\mathop{\rm
Tr}\nolimits\Big{\langle}V_{y}\mbox{$\sf
G$}^{A}V_{x}{\displaystyle\frac{\partial\mbox{$\sf
G$}^{A}}{\partial\mbox{$\cal E$}}}-(x\rightleftarrows
y)\Big{\rangle}d\mbox{$\cal E$}.$ (5)

Here, $f(\mbox{$\cal E$})$ is the Fermi–Dirac function, $\mbox{$\sf
G$}^{R(A)}(\mbox{$\cal E$})=1/(\mbox{$\cal E$}-\mbox{$\cal H$}-U\pm i0)$ is
the exact (i.e., non-averaged) retarded ($R$) or advanced ($A$) GF operator of
the Schrödinger equation with total Hamiltonian $\mbox{$\cal H$}+U$ (2),
$V_{y(x)}=v\sigma_{y(x)}$ are Cartesian components of the velocity operator,
the symbol $\mathop{\rm Tr}\nolimits=\mathop{\rm tr}\nolimits\mathop{\rm
Sp}\nolimits$ denotes the trace over both the spatial ($\mathop{\rm
Sp}\nolimits$) and (pseudo-) spin ($\mathop{\rm tr}\nolimits$) degrees of
freedom, the symbol $(x\rightleftarrows y)$ denotes term that is obtained from
the previous one due to permutation $V_{x}\rightleftarrows V_{y}$, and,
finally, angular brackets $\langle\ldots\rangle$ denote averaging over the
random field $U$ configurations. The normalization volume (area) is assumed to
be unity.

The first term of conductivity (II), $\sigma_{H}^{\rm I}$, results from the
electrons at the Fermi surface, whereas $\sigma_{H}^{\rm II}$ is determined by
the contribution of all occupied states of the Fermi sea. Středa et al.streda
were first to show that, for spinless electrons, this part of the conductivity
is equal to

$\sigma_{H}^{\rm II}=|e|c\left({\displaystyle\frac{\partial n}{\partial
B}}\right)_{\zeta},$ (6)

where $n$ is the electron concentration, $B$ is a magnetic field orthogonal to
the considered system ($||{\rm OZ}$), and $\zeta$ is the chemical potential of
the electron gas. It should be pointed out that Eq. (6) is exact, and with
thermodynamic Maxwell relation $\sigma_{H}^{\rm II}$ can be expressed through
$\big{(}\partial M/\partial\zeta\big{)}_{B}$, where $M$ is the magnetization
of the electron gas. Detailed discussion of $\sigma_{H}^{\rm II}$ and its
physical interpretation can be found in Pruisken’s surveypruisken_1990 . As
shown in Ref. nov_grosh_2006, , the expression (6) is converted to

$\sigma_{H}^{\rm II}=|e|c\left[\left({\displaystyle\frac{\partial n}{\partial
B}}\right)_{\zeta}-\left({\displaystyle\frac{\partial
M_{p}}{\partial\zeta}}\right)_{B}\right]$ (7)

if the electron spin degree of freedom taken into account. Here, $M_{p}$ is
the spin magnetization of the electron gas. It follows that in the general
case, $\sigma_{H}^{\rm II}$ is determined only by the orbital (diamagnetic)
part of the electron gas magnetization.

##  III  One-electron Green’s function and densities of states

The terms of the perturbation theory series for the electrical conductivity
(II) are expressed in terms of the averaged retarded (advanced) GFs, which can
be represented in the Dyson form

$G^{R(A)}(\mbox{$\cal E$})=\big{\langle}\mbox{$\sf G$}^{R(A)}(\mbox{$\cal
E$})\big{\rangle}=\frac{1}{\mbox{$\cal E$}-\mbox{$\cal
H$}-\Sigma^{R(A)}(\mbox{$\cal E$})},$ (8)

where $\Sigma^{R(A)}(\mbox{$\cal E$})$ is the electron self-energy operator.
Below, in specific calculations, we restrict ourselves to the SCBA in which
$\Sigma^{R(A)}(\mbox{$\cal E$})=W\mathop{\rm Sp}\nolimits G^{R(A)}(\mbox{$\cal
E$})$, where $W$ is the amplitude of the pair correlator of the Gaussian
random field $U$. It is easy to verify that, in this approximation,
$\Sigma(\mbox{$\cal E$})$ is diagonal in spin space and has the following
matrix structurefootnote_1

$\Sigma=\Sigma_{e}+\Sigma_{m}\sigma_{z},\quad~{}~{}\Sigma_{e}=\frac{W}{2}\mathop{\rm
Tr}\nolimits G,\quad~{}~{}\Sigma_{m}=\frac{W}{2}\mathop{\rm
Tr}\nolimits\sigma_{z}G.$ (9)

Hence, it follows that the averaged GF (8) can be obtained from the GF of the
clean model $G^{0}=1/(\mbox{$\cal E$}-\mbox{$\cal H$})$ using the
substitutionsdug_etal_2005 ; nov_grosh_2006

$\mbox{$\cal E$}\mapsto\tilde{}\mbox{$\cal E$}=\mbox{$\cal
E$}-\Sigma_{e},\qquad M\mapsto\mbox{$\tilde{M}$}=M+\Sigma_{m}.$ (10)

Here, $\Sigma_{e}=\Delta_{e}\mp i\hbar/2\tau_{e}$ describes the perturbation
(shift $\Delta_{e}$ and broadening $\hbar/\tau_{e}$) of the one-electron
energy levels by a random field. The real part of the
$\Sigma_{m}=\Delta_{m}\mp i\hbar/2\tau_{m}$ determines the renormalization of
the Dirac electron mass $M$, while its imaginary part $\propto\hbar/\tau_{m}$
makes a contribution to the overall lifetime of the one-electron states in the
spin-split subbands. Thus, the one-particle GF operator averaged over random
field configurations takes the following formsin_etal_2006 ; sin_etal_2007 ;
nov_2019

$G=\frac{\tilde{}\mbox{$\cal
E$}+\mbox{$\tilde{M}$}\sigma_{z}+v(\mbox{\boldmath$\pi$}\cdot\mbox{\boldmath$\sigma$})}{\tilde{\cal
E}^{2}-\tilde{M}^{2}-2mv^{2}\mbox{$\cal H$}_{0}}=\big{[}\tilde{}\mbox{$\cal
E$}+\mbox{$\tilde{M}$}\sigma_{z}+v(\mbox{\boldmath$\pi$}\cdot\mbox{\boldmath$\sigma$})\big{]}\mbox{$\cal
G$},$ (11)

where

$\mbox{$\cal
H$}_{0}=\frac{1}{2m}(\mbox{\boldmath$\pi$}\cdot\mbox{\boldmath$\sigma$})^{2}=\frac{\mbox{\boldmath$\pi$}^{2}}{2m}+\frac{\hbar\omega_{c}}{2}\sigma_{z}$
(12)

is the Hamiltonian of a free electron with ideal value of Zeeman coupling (g =
2) in an orthogonal magnetic field. In the limit $B\to 0$, $\pi$ and
$\mbox{$\cal H$}_{0}$ in (11), (12) are replaced by $\bm{p}$ and $\mbox{$\cal
E$}_{\bm{p}}$, respectively. It should be emphasized that the second term in
Hamiltonian $\mbox{$\cal H$}_{0}$ (12) appears due to commutation properties
of the operators $\pi$ and $\sigma$ and has no relation to the true Zeeman
energy of an electron in a magnetic field.

Eqs. (9) and (11) form a system of the self-consistent transcendental
equations for the $\Sigma_{e}$ and $\Sigma_{m}$. In the absence of an external
magnetic field, the traces $\mathop{\rm Tr}\nolimits G$ and $\mathop{\rm
Tr}\nolimits\sigma_{z}G$ are proportional to each other; therefore, both of
these self-energies in SCBA are expressed through the same functionnov_2019

$\Sigma_{e}=\gamma_{0}\mbox{$\cal
E$}\Phi,\quad\Sigma_{m}=\gamma_{0}M\Phi,\quad|\gamma_{0}\Phi|<<1$ (13)

that satisfies the equation

$\Phi=\ln\left[(1+\gamma_{0}\Phi)^{2}-(1-\gamma_{0}\Phi)^{2}\frac{{\mbox{$\cal
E$}}^{2}}{M^{2}}\right].$ (14)

Here $\gamma_{0}=W\mbox{$\cal N$}_{F}/2mv^{2}$ is the dimensionless parameter
of disorder, $\mbox{$\cal N$}_{F}=m/2\pi\hbar^{2}$ is the density of states
(DOS) of two-dimensional free non-relativistic spinless electrons in the
absence of a magnetic field. The numerical solution of equation (14) allows us
to calculate the self-energies $\Sigma_{e}$ and $\Sigma_{m}$, as well as the
total DOS $\mbox{$\cal N$}(\mbox{$\cal E$})$ and the difference of partial
DOSs $\mbox{$\cal N$}_{m}(\mbox{$\cal E$})$ with opposite values of the
pseudospin projections onto the OZ-axis (SDOS).

$\mbox{$\cal N$}(\mbox{$\cal E$})=\frac{1}{\pi}\mathop{\rm
Im}\nolimits\mathop{\rm Tr}\nolimits G^{A}(\mbox{$\cal
E$}),\quad~{}~{}\mbox{$\cal N$}_{m}(\mbox{$\cal E$})=\frac{1}{\pi}\mathop{\rm
Im}\nolimits\mathop{\rm Tr}\nolimits\sigma_{z}G^{A}(\mbox{$\cal E$}).$ (15)

Some results of this calculation at the different values of the parameter
$\gamma_{0}$ are shown in Fig. 2. As can be seen from this figure, the energy
gap between valence and conductivity bands narrows as $\gamma_{0}$ increases.

Figure 2:  (color online) Left top panel: The energy dependence of the DOS
$\mbox{$\cal N$}_{e}(\mbox{$\cal E$})=\mbox{$\cal N$}_{\uparrow}(\mbox{$\cal
E$})+\mbox{$\cal N$}_{\downarrow}(\mbox{$\cal E$})$ calculated in SCBA for
various values of the disorder parameter $\gamma_{0}=W\mbox{$\cal
N$}_{F}/2mv^{2}$. Three curves are calculated for $B=0$ and
$\gamma_{0}=0.0~{}~{}(1),~{}0.02~{}~{}(2),~{}0.04~{}~{}(3)$. Vertical dashed
lines represent unperturbed edges of the electronic spectrum $\mbox{$\cal
E$}=\pm|M|$.
Left bottom panel: The same for the SDOS $\mbox{$\cal N$}_{m}(\mbox{$\cal
E$})=\mbox{$\cal N$}_{\uparrow}(\mbox{$\cal E$})-\mbox{$\cal
N$}_{\downarrow}(\mbox{$\cal E$})$.
Central panel: The energy dependence of the DOS calculated in SCBA for
$\gamma_{0}=0.02$ and $\hbar\omega_{c}/M=0.50,\;0.33,\;0.25,\;0.17$ (from top
to bottom). The dashed curves depict the energy dependence of DOS for
$\gamma_{0}=0.02$ and $B=0$. For clarity, the curves are shifted relative to
each other along the abscissa axis.
Right panel: The same for the SDOS $\mbox{$\cal N$}_{m}(\mbox{$\cal E$})$.

Now, we consider the calculation of the electron self-energies (9) and DOSs
(15) in the case of $B\neq 0$. The spectrum of Hamiltonian $\mbox{$\cal
H$}_{0}$ (12) consists of two unbounded from above sets of the equidistant
Landau levels $\mbox{$\cal E$}_{n}=\hbar\omega_{c}(n+1/2\pm 1/2)$, where
$n=0,1,2,\ldots$. As a consequence, the traces that determine the electronic
self-energies (9) diverge logarithmically. These divergences are non-physical
and can be eliminated by the appropriate redefinition of the quantities $\cal
E$, $M$, $\Sigma_{e}$, and $\Sigma_{m}$ (see, for example, Ref. pruisken_1990,
). As a result, the traces $\mathop{\rm Tr}\nolimits G$ and $\mathop{\rm
Tr}\nolimits\sigma_{z}G$ can be written asnov_2019

$\left[\begin{array}[]{c}\mathop{\rm Tr}\nolimits G\\\\[2.0pt] \mathop{\rm
Tr}\nolimits\sigma_{z}G\end{array}\right]=\frac{\mbox{$\cal
N$}_{F}}{mv^{2}}\left\\{\left[\begin{array}[]{c}\tilde{\cal E}\\\
\tilde{M}\end{array}\right]\Phi(z)+\left[\begin{array}[]{c}\tilde{M}\\\
\tilde{\mbox{$\cal E$}}\end{array}\right]\frac{1}{2z}\right\\},$ (16)

where

$\Phi(z)=\psi(z)+\frac{1}{2z}+\ln\frac{2mv^{2}\hbar\omega_{c}}{M^{2}},\quad
z=\frac{\mbox{$\tilde{M}$}^{2}-\tilde{\cal E}^{2}}{2mv^{2}\hbar\omega_{c}},$
(17)

and $\psi(z)=d/dz\ln\Gamma(z)$ is the digamma-functionabram_steg_1964 . The
last term in the right-hand side of Eq. (17) eliminates the divergence of the
digamma-function as $B\to 0$ and provides the correct passage of (17) to this
limit.

Some results of the numerical solution of the SCBA-equations (9), (16), (17)
at the different values of the magnetic induction are shown in Fig. 2. As can
be seen from this figure, switching-on of an external magnetic field leads to
violation of the electron-hole symmetry due to the presence of the anomalous
Landau level $\mbox{$\cal E$}_{-1,0}=-M$ (4) at the top of the valence band.
This manifests itself in a shift of the electronic spectrum gap $\mbox{$\cal
E$}_{g}$ towards higher energies compared to its location in the absence of a
magnetic field. Finally, the de Haas–van Alphen type oscillations in the
energy dependence of the DOSs appear as the magnetic field $B$ increases.

##  IV  Topological nature of Středa-like term of conductance

Let us take Středa-like formula (7) as the initial expression for calculating
of the $\sigma_{H}^{\rm II}$ contribution to the intrinsic anomalous Hall
conductance. With this purpose, we have to find the explicit expressions for
the thermodynamic derivatives $\big{(}\partial n/\partial B\big{)}_{\zeta}$
and/or $\big{(}\partial M_{p}/\partial\zeta\big{)}_{B}$ in the presence of an
orthogonal magnetic field $B||{\rm OZ}$ [See Eq. (7)]. For example, we
consider the derivative $\big{(}\partial n/\partial B\big{)}_{\zeta}$. By
definition, the electron concentration is equal to

$n=\int\\!f(\mbox{$\cal E$})\mbox{$\cal N$}(\mbox{$\cal E$})\,d\mbox{$\cal
E$}=\frac{1}{2\pi i}\int_{C}f(z)\mathop{\rm Tr}\nolimits G(z)\,dz.$ (18)

Here, $G(z)$ is the one-electron GF defined on the complex $z$–plane in such a
way that $\lim_{z\to{\cal E}\pm i0}G(z)=G^{R(A)}(\mbox{$\cal E$})$; the
integration contour $C$ encircles counterclockwise the energy intervals
belonging to the spectrum of the eigenvalues of the Hamiltonian (2). Using the
analytical properties of the integrand in Eq. (18), we can deform $C$ as shown
in Fig. 3. This deformed integration contour encircles clockwise the poles of
the Fermi–Dirac function $\zeta_{n}=\zeta+i\pi k_{B}T(2n+1)$ (See Figs. 3.1 or
3.2). The integrals along the parts of $C$ located to the right of the poles
$\zeta_{n}$ vanish as $T\to 0$. In this case, the integration contour in Eq.
(18) looks as shown in Figs. 3.3 or 3.4.

Figure 3:  (color online) Deformation of the integration contour $C$. The
bullets depict the poles of the Fermi–Dirac function $\zeta_{n}=\zeta+i\pi
k_{B}T(2n+1)$. Thick horizontal lines represent two neighboring energy bands
separated by a gap $\mbox{$\cal E$}_{g}$.

Thus, the problem is reduced to finding of an explicit expression for
derivative $\partial\mathop{\rm Tr}\nolimits G/\partial B$. This can be done
in the momentum representation in which Dyson’s equation for the averaged one-
particle GF has the formonoda_etal_2008

$Q_{\bm{p}}\star G_{\bm{p}}=\left(z-\mbox{$\cal
H$}_{\bm{p}}-\frac{\hbar\omega_{c}}{4}s_{z}\otimes
g-\Sigma_{\bm{p}}\right)\star G_{\bm{p}}=1.$ (19)

Here, $G_{\bm{p}}$ is the Fourier transform of a gauge-independent and
translation invariant multiplier of the averaged one-particle GF in coordinate
representation
$G(\bm{r},\bm{r}^{\prime})=\left\langle{\bm{r}|G|\bm{r}^{\prime}}\right\rangle$
(See Appendix A), $\mbox{$\cal H$}_{\bm{p}}$ is the Hamiltonian of a some
clean non-interacting system in which the momentum operator $\pi$ is replaced
by $c$-number momentum $\bm{p}$ (further, we will omit subscript $\bm{p}$).
For clarity, the Zeeman energy $\hbar\omega_{c}s_{z}\otimes g/4$ is written
explicitly in (19), where $s_{z}$ is the Pauli spin $z$-matrix, $g$ is the
matrix of the effective $g$-factors. For example, $g={\rm
diag}\\{g_{e},g_{h}\\}$ in the Bernevig–Hughes–Zhang model for the single
valley Dirac fermions in the HgTe quantum well bern_etal_2006 ; buet_etal_2011
.

Symbol $F_{\bm{p}}\star G_{\bm{p}}$ denotes the Moyal $\star$-product
onoda_etal_2008 ; moyal_1949 ; zach_etal_2005  of the momentum-dependent
functions

$F_{\bm{p}}\star G_{\bm{p}}\stackrel{{\scriptstyle\rm
def}}{{=}}F_{\bm{p}}\exp\left[-i\frac{\hbar^{2}}{2l_{B}^{2}}\epsilon_{\alpha\beta}\stackrel{{\scriptstyle\leftarrow}}{{\partial}}_{\alpha}\stackrel{{\scriptstyle\rightarrow}}{{\partial}}_{\beta}\right]G_{\bm{p}},$
(20)

where $l_{B}=\sqrt{c\hbar/|e|B}$ is the magnetic length,
$\epsilon_{\alpha\beta}$ is the antisymmetric unit tensor of the second rank
[$\alpha\,(\beta)=x,\,y$], and $\partial_{\alpha}=\partial/\partial
p_{\alpha}$. The arrows above the differentiation operators indicate the
directions of their actions. The formulation of the momentum representation
for the description of electron states in a magnetic field and an overview of
the various definitions and basic properties of the Moyal $\star$-product can
be found in Appendix A.

In the momentum representation, the operations $\mathop{\rm Tr}\nolimits$ and
$\partial/\partial B$ commute with each other. So, we just need to find an
explicit expression for $\partial G/\partial B$ to calculate $\big{(}\partial
n/\partial B)_{\zeta}$. Direct differentiation of Dyson equation (19) with
respect to the magnetic field induction gives the following result

$\displaystyle{\displaystyle\frac{\partial G}{\partial B}}=$
$\displaystyle-G\star{\displaystyle\frac{\partial Q}{\partial B}}\star
G+i\frac{\hbar|e|}{2c}\epsilon_{\alpha\beta}G\star{\displaystyle\frac{\partial
Q}{\partial p_{\alpha}}}\star{\displaystyle\frac{\partial G}{\partial
p_{\beta}}}$ $\displaystyle=$
$\displaystyle-G\star{\displaystyle\frac{\partial Q}{\partial B}}\star
G-i\frac{\hbar|e|}{2c}\epsilon_{\alpha\beta}G\star{\displaystyle\frac{\partial
Q}{\partial p_{\alpha}}}\star G\star{\displaystyle\frac{\partial Q}{\partial
p_{\beta}}}\star G.$ (21)

Here, the last equality is obtained using the first of the identities

${\displaystyle\frac{\partial
G}{\partial\bm{p}}}=-G\star{\displaystyle\frac{\partial
Q}{\partial\bm{p}}}\star G,\qquad{\displaystyle\frac{\partial G}{\partial
z}}=-G\star{\displaystyle\frac{\partial Q}{\partial z}}\star G,$ (22)

which are also derived from Dyson equation (19).

Let us take a trace of Eq. (IV). Then, using the second identity from (22) and
the trace invariance under cyclic permutations, we obtain

$\displaystyle\mathop{\rm Tr}\nolimits{\displaystyle\frac{\partial G}{\partial
B}}=$ $\displaystyle\mathop{\rm Tr}\nolimits{\displaystyle\frac{\partial
Q}{\partial B}}\star{\displaystyle\frac{\partial G}{\partial z}}-\mathop{\rm
Tr}\nolimits G\star{\displaystyle\frac{\partial Q}{\partial B}}\star
G\star{\displaystyle\frac{\partial\Sigma}{\partial z}}-$ $\displaystyle\hskip
14.22636pt-i\frac{\hbar|e|}{2c}\epsilon_{\alpha\beta}\mathop{\rm Tr}\nolimits
G\star{\displaystyle\frac{\partial Q}{\partial p_{\alpha}}}\star
G\star{\displaystyle\frac{\partial Q}{\partial p_{\beta}}}\star G.$ (23)

Now we substitute the explicit expression

${\displaystyle\frac{\partial Q}{\partial
B}}=-\frac{\hbar|e|}{4mc}s_{z}\otimes
g-{\displaystyle\frac{\partial\Sigma}{\partial B}}$ (24)

in the first term of the right-hand side of Eq. (IV) and transfer the Zeeman
term to the left-hand side of this equality. The electron self energy
satisfies relations like Ward identity

${\displaystyle\frac{\partial\Sigma_{n}}{\partial
z}}=\sum_{n^{\prime}}U_{n,n^{\prime}}{\displaystyle\frac{\partial
G_{n^{\prime}}}{\partial
z}},\quad~{}~{}{\displaystyle\frac{\partial\Sigma_{n}}{\partial
B}}=\sum_{n^{\prime}}U_{n,n^{\prime}}{\displaystyle\frac{\partial
G_{n^{\prime}}}{\partial B}},$ (25)

where $U_{n,n^{\prime}}$ is the two-particle irreducible interaction vertex.
Hence, it follows that

$\mathop{\rm Tr}\nolimits{\displaystyle\frac{\partial\Sigma}{\partial
B}}\star{\displaystyle\frac{\partial G}{\partial z}}=\mathop{\rm
Tr}\nolimits{\displaystyle\frac{\partial G}{\partial
B}}\star{\displaystyle\frac{\partial\Sigma}{\partial z}}.$ (26)

This is more strong version of the relation previously proposed by
Pruiskenpruisken_1990 . With this in mind Eq. (IV) can be rewritten as

$\displaystyle\mathop{\rm Tr}\nolimits{\displaystyle\frac{\partial G}{\partial
B}}+$ $\displaystyle\frac{\hbar|e|}{4mc}\mathop{\rm Tr}\nolimits s_{z}\otimes
g{\displaystyle\frac{\partial G}{\partial
z}}=-i\frac{\hbar|e|}{2c}\epsilon_{\alpha\beta}\times$ $\displaystyle\hskip
28.45274pt\times\mathop{\rm Tr}\nolimits{\displaystyle\frac{\partial
Q}{\partial p_{\alpha}}}\star G\star{\displaystyle\frac{\partial Q}{\partial
p_{\beta}}}\star G\star{\displaystyle\frac{\partial Q}{\partial z}}\star G.$
(27)

After integration with respect to $z$ of this equation with Fermi–Dirac weight
function $f(z)$, the terms of its left-hand side determine derivatives of the
electron concentration and spin magnetization included in Eq. (7). Therefore,
after some cosmetic transformations, the expression for the Středa-like term
(7) of the Hall conductance can be written in the following form

$\displaystyle\sigma_{H}^{\rm II}=$ $\displaystyle-\frac{\hbar
e^{2}}{12\pi}\\!\int_{C}\\!f(z)\times$
$\displaystyle\times\epsilon_{\alpha\beta\gamma}\mathop{\rm
Tr}\nolimits{\displaystyle\frac{\partial Q}{\partial p_{\alpha}}}\star
G\star{\displaystyle\frac{\partial Q}{\partial p_{\beta}}}\star
G\star{\displaystyle\frac{\partial Q}{\partial p_{\gamma}}}\star G\,dz.$ (28)

Here, integration is carried out along the contour $C$ shown in Fig. 3.1 or
Fig. 3.2; $\epsilon_{\alpha\beta\gamma}$ is the unit antisymmetric
pseudotensor of the third rank, indices $\alpha,\,\beta,\,\gamma$ run values
0, 1, 2 with $p_{0}=z$, $p_{1}=p_{x}$, and $p_{2}=p_{y}$.

Thus, we obtain the expression for the Středa-like part of the Hall
conductance in which the averaging over random field configurations is
performed exactly. Eq. (IV) is true regardless of the disorder type and the
approximation for the averaged one-particle GF. This is possible owing to
special analytical structure of starting expression for $\sigma_{H}^{\rm II}$
(II) that does not contain the products $\mbox{$\sf G$}^{R}\mbox{$\sf
G$}^{A}$. It should be emphasized that both expressions for $\sigma_{H}^{\rm
II}$ (II) and (IV) have the identical structure. The only difference is that
Eqs. (II) and (IV) include exact and averaged GFs, respectively. In other
words, the averaging procedure of $\sigma_{H}^{\rm II}$ is reduced to the
simple substitute the exact GFs $\sf G$ for the averaged ones
$G=\left\langle{\mbox{$\sf G$}}\right\rangle$. If desired, Eq. (IV) can be
rewritten in the usual operator representation. To do this, it is enough to
replace the $\star$-products in this equation by the usual multiplication and
treat $G$ and $Q$ as functions depending on the operators of the corresponding
dynamic variables.

Eq. (IV) indicates a close connection between the intrinsic anomalous Hall
conductance and topological properties of the one-electron states. Indeed, in
the case $T=0$, the Fermi–Dirac function $f(z)$ coincides with the Heaviside
unit step $\Theta(\mbox{$\cal E$}_{F}-\mathop{\rm Re}\nolimits z)$, and the
expression for $\sigma_{H}^{\rm II}$ (IV) takes the form

$\displaystyle\sigma_{H}^{\rm II}$
$\displaystyle\,=\sigma_{0}\\!\int_{C}\\!\left\\{-\epsilon_{\alpha\beta\gamma}\frac{1}{24\pi^{2}}\\!\iint\times\right.$
$\displaystyle\,\left.\times\mathop{\rm
tr}\nolimits\left[G\star{\displaystyle\frac{\partial Q}{\partial
p_{\alpha}}}\star G\star{\displaystyle\frac{\partial Q}{\partial
p_{\beta}}}\star G\star{\displaystyle\frac{\partial Q}{\partial
p_{\gamma}}}\right]dp_{1}\,dp_{2}\right\\}dp_{0},$ (29)

where the integration contour $C$ is shown in Figs. 3.3 or 3.4 and trace over
the spatial degrees of freedom is presented as the integral with respect to
momenta $p_{1,2}$ (54).

Now imagine that the Fermi level lies inside the energy gap of the system
under consideration. In this case, the integration contour $C$ in Eq. (IV) is
the straight line parallel to the imaginary axis $\mathop{\rm Re}\nolimits
z=\mbox{$\cal E$}_{F}-0$, $-\infty<\mathop{\rm Im}\nolimits z<+\infty$ (See
Fig. 3.4) and the Středa term of the Hall conductance $\sigma_{H}^{\rm II}$
takes the disorder-independent quantized value proportional to the Chern
number, i.e., $\sigma_{H}^{\rm II}=\sigma_{0}{\rm Ch}$, where

${\rm
Ch}=-\frac{\epsilon_{\alpha\beta\gamma}}{24\pi^{2}}\\!\iiint\\!\mathop{\rm
tr}\nolimits\\!\left[G\star{\displaystyle\frac{\partial Q}{\partial
p_{\alpha}}}\star G\star{\displaystyle\frac{\partial Q}{\partial
p_{\beta}}}\star G\star{\displaystyle\frac{\partial Q}{\partial
p_{\gamma}}}\right]\\!d^{3}p.$ (30)

Integration is performed here over the entire $(2+1)$-dimensional space of
points $(\mathop{\rm Im}\nolimits z,p_{x},p_{y})$. Eq. (30) is an analogue of
the expression for the Chern number obtained by Zubkov et al.zubkov_2016 ;
zub_wu_2020 ; zhang_zub_2019  in the Wigner representation, which is necessary
for describing spatially nonuniform systems. Writing the Chern number in the
momentum representation (30) is more appropriate in the practically important
spatially uniform case.

We represent the result of integration along the straight line $\mathop{\rm
Re}\nolimits z=\mbox{$\cal E$}_{F}-0$ in Eq. (30) in the form of increment of
the corresponding antiderivative, i.e.

${\rm Ch}=\Delta F(\mbox{$\cal E$}_{F}\pm i\infty)=F(\mbox{$\cal
E$}_{F}+i\infty)-F(\mbox{$\cal E$}_{F}-i\infty),$ (31)

where [See Eq. (IV)]

${\displaystyle\frac{\partial F}{\partial
z}}=-\epsilon_{\alpha\beta\gamma}\\!\iint\\!\mathop{\rm
tr}\nolimits\\!\left[G\star{\displaystyle\frac{\partial Q}{\partial
p_{\alpha}}}\star G\star{\displaystyle\frac{\partial Q}{\partial
p_{\beta}}}\star G\star{\displaystyle\frac{\partial Q}{\partial
p_{\gamma}}}\right]\\!\frac{d^{2}\bm{p}}{24\pi^{2}}.$ (32)

Obviously, the asymptotic values $F(\mbox{$\cal E$}_{F}\pm i\infty)$ do not
depend on the disorder and coincide with their values in the clean limit.

If the Fermi level lies outside the gap, then the contour of integration in
Eq. (IV) consists of two semi-infinite straight lines parallel to the
imaginary axis $\mathop{\rm Re}\nolimits z=\mbox{$\cal E$}_{F}-0$,
$-\infty<\mathop{\rm Im}\nolimits z<-0$ and $+0<\mathop{\rm Im}\nolimits
z<+\infty$ (See Fig. 3.3). In this case, a term proportional to the
discontinuity of $F(z)$ across the cut line corresponding to the continuous
spectrum of the electron is added to the expression for the Středa part of the
Hall conductance. So, we have

$\sigma_{H}^{\rm II}=\sigma_{0}\left\\{\begin{array}[]{ll}{\rm
Ch},&\mbox{$\cal E$}_{F}\in{\rm Gap},\\\\[4.0pt] {\rm Ch}+\Delta F(\mbox{$\cal
E$}_{F}),&\mbox{$\cal E$}_{F}\notin{\rm Gap},\end{array}\right.$ (33)

where $\Delta F(\mbox{$\cal E$}_{F})=F(\mbox{$\cal E$}_{F}-i0)-F(\mbox{$\cal
E$}_{F}+i0)$. The first term in the bottom line of (33) remains unchanged
since $\Delta F(\mbox{$\cal E$}_{F}\pm i\infty)$ does not depend on placing
the Fermi level. In the next section, we will illustrate the validity of (33)
using the massive Dirac electrons model as an example.

##  V  Intrinsic anomalous Hall conductance in SCBA

The intrinsic Hall conductance consists of two terms $\sigma_{H}^{\rm
int}=\sigma_{H}^{\rm Ib}+\sigma_{H}^{\rm II}$, where $\sigma_{H}^{\rm Ib}$ is
the bare bubble part of $\sigma_{H}^{\rm I}$ and $\sigma_{H}^{\rm II}$ is the
Středa term [see. Eq. (II)].

In the presence of an external magnetic field, the Hall conductance consists
of the normal $\sigma_{H}^{\rm n}$ and anomalous $\sigma_{H}^{\rm a}$ terms,
at that $\sigma_{H}^{\rm n}(-B)=-\sigma_{H}^{\rm n}(B)$ and $\sigma_{H}^{\rm
a}(-B)=\sigma_{H}^{\rm a}(B)$. Below, we restrict ourselves to calculating the
even in $B$ parts of $\sigma_{H}^{\rm Ib}$ and $\sigma_{H}^{\rm II}$. Of
course, the parity of the anomalous Hall conductance in the magnetic field
does not mean violation of the Onsager relation. The point is that this
relation is valid for the both normal and anomalous Hall conductances with
simultaneous reversal of the signs of both the magnetic field $B$ and the
Dirac mass $M$, since the introduction of each of them breaks the time
reversal symmetry. Thus, the Onsager relation for both parts of $\sigma_{H}$
must have the form

$\sigma_{H}(-B,-M)=-\sigma_{H}(B,M).$ (34)

The intrinsic anomalous Hall conductance calculated below satisfies the
relation (34) not only in the QAHE regimeboet_etal_2019 ; boet_etal_2020 , but
also outside it.

###  V.1  Středa like part of the anomalous Hall conductance

It is usually assumed that $\sigma_{H}^{\rm II}$ is weakly dependent on
disorder, and it is calculated without taking into account the scattering of
charge carriersdug_etal_2005 ; sin_etal_2006 ; sin_etal_2007 ; nun_etal_2007 ;
ado_etal_2015 ; ado_etal_2016 . It will be shown below that this approximation
is violated in a fairly wide vicinity of the Hall plateau. Let us rewrite the
expression (IV) for the Středa term in a more convenient for calculating
operator notation

$\sigma_{H}^{\rm
II}=\sigma_{0}\frac{\hbar^{2}v^{2}}{2}\\!\int_{C}\\!\\!\mathop{\rm
Tr}\nolimits\\!\left[\sigma_{y}G\sigma_{x}{\displaystyle\frac{\partial
G}{\partial z}}-(x\leftrightarrows y)\right]\\!dz.$ (35)

Here, it is taken into account that $\partial Q/\partial
p_{x(y)}=-v\sigma_{x(y)}$ in SCBA considered below. For simplicity, we assume
the temperature to be zero, the generalization to the case $T>0$ is obvious.

In the article of Dugaev et al.dug_etal_2005 , the Středa term of $\sigma_{H}$
was calculated for the case of a clean two-dimensional Rashba model in absence
of a magnetic field. We will apply their approach to calculate the Středa part
of the anomalous Hall conductance of the disordered Chern insulator (2) in
SCBA in the presence of an orthogonal magnetic field. Direct differentiation
of the averaged GF (11) with respect to $z$ gives the following result

${\displaystyle\frac{\partial G}{\partial
z}}=\left(1-{\displaystyle\frac{\partial\Sigma_{e}}{\partial
z}}+{\displaystyle\frac{\partial\Sigma_{m}}{\partial
z}}\sigma_{z}\right)\mbox{$\cal G$}-G\mbox{$\cal
G$}{\displaystyle\frac{\partial}{\partial z}}(\tilde{z}^{2}-\tilde{M}^{2}),$
(36)

where $\tilde{z}=z-\Sigma_{e}(z)$. In absence of a magnetic field, the second
term in this expression does not contribution to the Hall conductance
dug_etal_2005 . In the presence of a magnetic field, this is no longer the
case, but only the first term from Eq. (36) makes an even in $B$ contribution
to the Středa part of the Hall conductance. The substitution of this term into
the integrand of (35) gives, after some simple algebra, the following result

$\displaystyle\mathop{\rm Tr}\nolimits[\ldots]=$
$\displaystyle\,\frac{i}{mv^{2}\hbar\omega_{c}}\times$
$\displaystyle\,\times\mathop{\rm Tr}\nolimits(\bar{}\mbox{$\cal
G$}-\mbox{$\cal
G$})(\tilde{z}-\mbox{$\tilde{M}$}\sigma_{z})\left(1-{\displaystyle\frac{\partial\Sigma_{e}}{\partial
z}}+{\displaystyle\frac{\partial\Sigma_{m}}{\partial z}}\sigma_{z}\right),$
where $\bar{}\mbox{$\cal G$}=\sigma_{x(y)}\mbox{$\cal G$}\sigma_{x(y)}$, i.e.,
this is the result of transformation $\sigma_{z}\mapsto-\sigma_{z}$
$\displaystyle=$ $\displaystyle\,2i\frac{\mbox{$\cal
N$}_{F}}{mv^{2}}\frac{\mbox{$\tilde{M}$}\left(1-{\displaystyle\frac{\partial\Sigma_{e}}{\partial
z}}\right)-\tilde{z}{\displaystyle\frac{\partial\Sigma_{m}}{\partial
z}}}{\tilde{z}^{2}-\tilde{M}^{2}}$ $\displaystyle=$
$\displaystyle\,i\frac{\mbox{$\cal
N$}_{F}}{mv^{2}}{\displaystyle\frac{\partial}{\partial
z}}\ln\frac{\tilde{M}-\tilde{z}}{\tilde{M}+\tilde{z}}.$ (37)

Thus, we obtain the explicit expression for the antiderivative [see Eq. (32)]

$F(z)=\frac{1}{4\pi i}\ln\frac{\tilde{M}+\tilde{z}}{\tilde{M}-\tilde{z}}$ (38)

in the case of the model of disordered massive Dirac electrons. Obviously, its
increment along the straight line $\mathop{\rm Re}\nolimits z=\mbox{$\cal
E$}_{F}$ does not depend on the placing of the Fermi level $\mbox{$\cal
E$}_{F}$, nor on the external magnetic field $B$, nor on the electronic self-
energies $\Sigma_{e(m)}$, and is equal to the Chern number $\Delta
F(\mbox{$\cal E$}_{F}\pm i\infty)=1/2={\rm Ch}$ (31).

If $\mbox{$\cal E$}_{F}\in{\rm Gap}$, i.e., when $\mathop{\rm
Im}\nolimits\Sigma_{e(m)}=0$, then $F(z)$ is continuous across the real axis.
As a result, the anomalous part of the Středa term takes on quantized value
$\sigma_{H}^{\rm II\,a}=\sigma_{0}{\rm Ch}$ in this energy range [see Eq.
(33)]. Otherwise ($\mbox{$\cal E$}_{F}\notin{\rm Gap}$), $\sigma_{H}^{\rm
II\,a}$ acquires additional term $\propto\,\Delta F(\mbox{$\cal
E$}_{F})=F(\mbox{$\cal E$}_{F}-i0)-F(\mbox{$\cal E$}_{F}+i0)$ [see Eq. (33)].
Given the explicit form of the function $F(z)$ (38), we obtain the following
expression for the Středa term of the anomalous Hall conductance

$\sigma_{H}^{\rm II\,a}=\frac{\sigma_{0}}{2}\big{[}1+2\Delta F(\mbox{$\cal
E$}_{F})\big{]}=\frac{\sigma_{0}}{2\pi}\mathop{\rm
Im}\nolimits\ln\frac{\tilde{\cal E}^{A}+\tilde{M}^{A}}{\tilde{\cal
E}^{A}-\tilde{M}^{A}},$ (39)

Figure 4:  (color online) Left top panel: The energy dependence of the Středa
term of the anomalous Hall conductance $\sigma_{H}^{\rm II\,a}$ (39)
calculated in SCBA for various values of the disorder parameter
$\gamma_{0}=W\mbox{$\cal N$}_{F}/2mv^{2}$. Three curves are calculated for
$B=0$ and $\gamma_{0}=0.0~{}~{}(1),~{}0.02~{}~{}(2),~{}0.04~{}~{}(3)$.
Vertical dashed lines represent unperturbed edges of the electronic spectrum
$\mbox{$\cal E$}=\pm|M|$.
Left bottom panel: The same for the total intrinsic part of the anomalous Hall
conductance $\sigma_{H}^{\rm int}=\sigma_{H}^{\rm Ib\,a}+\sigma_{H}^{\rm
II\,a}$. The dashed line shows the energy dependence of $\sigma_{H}^{\rm
int}=\sigma_{0}|M|/2|\mbox{$\cal E$}|$ of the clean ($\gamma_{0}=0$) model of
massive Dirac fermions for $B=0$. The solid curves calculated for $B=0$ and
$\gamma_{0}\to 0~{}~{}(1),~{}=0.02~{}~{}(2),~{}=0.04~{}~{}(3)$ are practically
indistinguishable at this scale. The inset shows, on an enlarged scale, the
behavior of curves (1-3) in vicinity of the point $\mbox{$\cal E$}=|M|$
(circled in the main graph).
Central panel: The energy dependence of the Středa term of the anomalous Hall
conductance $\sigma_{H}^{\rm II\,a}$ (39) calculated in SCBA for
$\gamma_{0}=0.02$ and $\hbar\omega_{c}/M=0.50,\;0.33,\;0.25,\;0.17$ (from top
to bottom). The dashed curves depict the energy dependence of $\sigma_{H}^{\rm
II}$ for $\gamma_{0}=0.02$ and $B=0$. For clarity, the curves are shifted
relative to each other along the abscissa axis.
Right panel: The same for the total intrinsic part of the anomalous Hall
conductance $\sigma_{H}^{\rm int}=\sigma_{H}^{\rm Ib\,a}+\sigma_{H}^{\rm
II\,a}$.

which turns out to be valid both in the quantization regime and beyond it.
This expression for $\sigma_{H}^{\rm II\,a}$ has exactly the same form as in
absence of an external magnetic fieldnov_2019 . The entire dependence on the
magnetic field of (39) is contained in the electronic self-energies
$\Sigma_{e}$ and $\Sigma_{m}$ (9). The results of numerical analysis of Eq.
(39) are represent in Fig. 4 (upper left and central panels).

###  V.2  Bare bubble part of the anomalous Hall conductance

Now we turn to the calculation of the bare bubble contribution
$\sigma_{H}^{\rm Ib\,a}$ to the intrinsic anomalous Hall conductance. In the
case of the clean [$U(\bm{r})=0$] massive Dirac electrons, this term was
calculated by Sinitsyn and co-workerssin_etal_2007  in the basis of the
Hamiltonian $\cal H$ (2) in absence of a magnetic field. Here, we apply the
simple algebraic approach to calculating $\sigma_{H}^{\rm Ib\,a}$ of the
considered model in the same approximation as the Středa term, that is, in
SCBA.

Thus, we start with the expression

$\sigma_{H}^{\rm Ib}=\frac{\hbar e^{2}v^{2}}{2\pi}\mathop{\rm
Tr}\nolimits\sigma_{y}G^{R}\sigma_{x}G^{A}.$ (40)

Given the properties of the Pauli matrices, it can be shown that the terms
$\propto(\mbox{\boldmath$\pi$}\cdot\mbox{\boldmath$\sigma$})$ in the numenator
of GF (11) do not contribute to $\mathop{\rm
Tr}\nolimits\sigma_{y}G^{R}\sigma_{x}G^{A}$ due to isotropy of the considered
system. So, substitution of the one-particle GFs (11), (12) in Eq. (40) gives
the result

$\displaystyle\sigma_{H}^{\rm Ib}=$ $\displaystyle\,\frac{\hbar
e^{2}v^{2}}{2\pi}\mathop{\rm Tr}\nolimits\sigma_{y}\mbox{$\cal
G$}^{R}(\tilde{\cal E}^{R}+\tilde{M}^{R}\sigma_{z})\sigma_{x}(\tilde{\cal
E}^{A}+\tilde{M}^{A}\sigma_{z})\mbox{$\cal G$}^{A}$ $\displaystyle=$
$\displaystyle\,\frac{\hbar e^{2}v^{2}}{2\pi i}\mathop{\rm
Tr}\nolimits\sigma_{z}(\tilde{\cal E}^{R}-\tilde{M}^{R}\sigma_{z})(\tilde{\cal
E}^{A}+\tilde{M}^{A}\sigma_{z})\bar{\cal G}^{R}\mbox{$\cal G$}^{A}.$

We are interested in the even in $B$ part of this expression

$\sigma_{H}^{\rm Ib\,a}=\frac{\hbar e^{2}v^{2}}{\pi}\mathop{\rm
Im}\nolimits(\tilde{\cal E}^{R}\tilde{M}^{A})\mathop{\rm Tr}\nolimits\bar{\cal
G}^{R}\mbox{$\cal G$}^{A}.$ (41)

Using the resolvent identity we write the product $\bar{\cal G}^{R}\mbox{$\cal
G$}^{A}$ in the following form

$\displaystyle\bar{\cal G}^{R}\mbox{$\cal
G$}^{A}=\frac{1}{i\hbar}\frac{\tau}{2mv^{2}}\frac{1+i\omega_{c}\tau\sigma_{z}}{1+\omega_{c}^{2}\tau^{2}}(\mbox{$\cal
G$}^{A}-\bar{\cal G}^{R}).$ (42)

The quantity $\tau$ that is defined by relation

$\frac{1}{\tau}=\frac{1}{mv^{2}}\left(\frac{\mbox{$\cal
E$}}{\tau_{e}}+\frac{M}{\tau_{m}}\right)$ (43)

plays the role of the transport time in the Drude-like denominator
$1+\omega_{c}^{2}\tau^{2}$. Let us introduce one more characteristic time

$\frac{1}{\tau^{\prime}}=\frac{2}{\hbar}\frac{\mathop{\rm
Im}\nolimits({\tilde{\cal
E}}^{R}{\tilde{M}}^{A})}{mv^{2}}=\frac{1}{mv^{2}}\left(\frac{\mbox{$\cal
E$}}{\tau_{m}}+\frac{M}{\tau_{e}}\right).$ (44)

Substitution of expressions (42)–(44) into Eq. (41) gives the following result

$\sigma_{H}^{\rm
Ib\,a}=\frac{\sigma_{0}}{2\pi}\frac{\tau}{\tau^{\prime}}\frac{\mathop{\rm
Im}\nolimits\Phi^{A}+\displaystyle\omega_{c}\tau\mathop{\rm
Re}\nolimits\frac{mv^{2}\hbar\omega_{c}}{\tilde{M}^{2}-\tilde{\cal
E}^{2}}}{1+\omega_{c}^{2}\tau^{2}},$ (45)

where the function $\Phi^{A}$ is defined in (17).

Obviously, expression (45) as a function of $\mbox{$\cal E$}_{F}$ vanishes
inside the gap of the one-electron energy spectrum. Thus, the quantization of
the $\sigma_{H}^{\rm int}=\sigma_{H}^{\rm Ib\,a}+\sigma_{H}^{\rm II\,a}$ (39),
(45) survives when an external magnetic field is turned on that is consistent
with recent results by Böttcher et al.boet_etal_2019 ; boet_etal_2020 . The
results of numerical analysis of $\sigma_{H}^{\rm int}$ are represented in
Fig. 4 (bottom left and right panels).

##  VI  Results and discussion

Let us summarize briefly the main results obtained in this work. The explicit
expressions in SCBA (16) are obtained for the total DOS and SDOS of the
minimal model of a two-dimensional disordered Chern insulator. The results of
a numerical analysis of these expressions are shown in Fig. 2. In the case
$B=0$, the curves in the left panels demonstrate a narrowing of the gap in the
spectrum of one-electron states as the disorder parameter increases. The
graphs presented in central and right panels in Fig. 2 show two effects caused
by an external magnetic field. First, the gap in the electronic spectrum is
shifted towards higher energies as a magnetic field increases. This is due to
the presence of the anomalous Landau level $\mbox{$\cal E$}_{-1,0}=-M$ at the
top of the valence band. Second, in the region of sufficiently strong magnetic
fields, the de Haas–van Alphen oscillations manifest themselves in the energy
dependencies of the DOSs.

Section IV provides the simple derivation of expressions [see Eq. (IV) for the
case $T>0$ and Eq. (IV) for the case $T=0$] for the Středa-like term
$\sigma_{H}^{\rm II}$ (7) of the anomalous Hall conductance in terms of the
averaged one-particle GFs in the momentum representation. These expressions
are valid both in absence of an external magnetic field and in its presence
and describe the dependence of $\sigma_{H}^{\rm II}$ on the location of the
Fermi level $\mbox{$\cal E$}_{F}$ both inside the gap of the one-electron
spectrum and outside it. In the first case ($\mbox{$\cal E$}_{F}\in{\rm
Gap}$), $\sigma_{H}$ is proportional to the Chern number (in the case $T=0$),
which takes on quantized values regardless of the presence or absence of
disorder and/or an external magnetic field. The half-integer ($\rm Ch=\pm
1/2$) quantization of the anomalous Hall conductance of massive Dirac
electronssin_etal_2006 ; sin_etal_2007 ; ado_etal_2015 ; ludw_etal_1994  is a
consequence of the fermionic numbers fractionalizationjack_reb_1976 ;
koen_etal_2014 ; koen_etal_2016 . It should be emphasized that the bare bubble
part of $\sigma_{H}^{\rm II}$, as well as the contributions to $\sigma_{H}$
due to extrinsic mechanisms, vanish under these conditions. Thus, the survival
of the gap in the electronic spectrum of the Chern insulator is the only
condition for quantizing its anomalous Hall conductance in the presence of
disorder, external magnetic field, and other perturbations.

As an illustration, we calculate in Section V the even in a magnetic field
intrinsic anomalous Hall conductance $\sigma_{H}^{\rm int}=\sigma_{H}^{\rm
Ib\,a}+\sigma_{H}^{\rm II\,a}$ of a two-dimensional disordered gas of massive
Dirac electrons in SCBA [see Eqs. (39) and (45)]. As noted above, the parity
of $\sigma_{H}^{\rm a}$ in a magnetic field does not mean the violation of the
Onsager relation, since it requires both normal and anomalous conductances be
odd in all parameters that break the time reversal symmetry of the considered
system [see Eq. (34)].

The results of a numerical analysis of the dependencies of $\sigma_{H}^{\rm
II\,a}$ (39), and $\sigma_{H}^{\rm int}=\sigma_{H}^{\rm Ib\,a}+\sigma_{H}^{\rm
II\,a}$ (39), (45) on the location of the Fermi level at various values of the
disorder parameter and magnetic field induction are shown in Fig. 4. As
expected, the quantization of the anomalous Hall conductance survives in the
presence of sufficiently weak perturbations that leave open the gap in the
electronic energy spectrum.

Contrary to conventional opinion, the presence of disorder has a significant
effect on the $\sigma_{H}^{\rm II}$ behavior beyond the QAHE-plateau (see the
upper left panel in the Fig. 4). But, as can be seen from the graphs on the
lower left panel of Fig. 4, the corrections to $\sigma_{H}^{\rm II}$ and
$\sigma_{H}^{\rm Ib}$ due to scattering of electrons in a random field of
impurities cancel out with a very high precision. In particular, the relative
deviation of the sum of these terms from its quantized value $\sigma_{0}/2$
for $|\mbox{$\cal E$}|<M$, and from its behavior in the clean limit

$\sigma_{H}^{\rm int}=\sigma_{0}\frac{|\mbox{$\cal E$}M|}{\mbox{$\cal
E$}^{2}+M^{2}}$ (46)

for $|\mbox{$\cal E$}|>M$ is at least an order of magnitude smaller than the
dimensionless disorder parameter $\gamma_{0}$.

As already mentioned, the quantization of $\sigma_{H}^{\rm int}$ survives in
an external magnetic field $B$. But, as can be seen from Fig. 4, the lower
boundary of the Hall plateau shifts towards higher energies due to the
broadening of the zero Landau level $\mbox{$\cal E$}_{0,-1}=-M$ as magnetic
field $B$ increases. At the same time, the upper boundary remains almost fixed
that leads to a narrowing of the Hall plateau.

In the region of sufficiently strong magnetic fields, the upper boundary of
the Hall plateau turns out to be inside the gap of the electron spectrum. In
this case, it represents the threshold at which the $\sigma_{H}^{\rm int}$ as
a function of the Fermi energy suddenly drops from its quantized value to zero
as in the clean limit. A similar dependence of the anomalous Hall conductance
on the magnetic field was predicted in Ref. boet_etal_2020, .

The main features of the behavior of the intrinsic anomalous Hall conductance
$\sigma_{H}^{\rm int}$ beyond the plateau are determined by the energy
dependence of the relaxation time $\tau$ (43). Indeed, on the one hand, this
parameter plays the role of transport time in the Drude denominator
$1+\omega_{c}^{2}\tau^{2}$ (45), and, on the other hand, it is related to the
Dingle temperature $1/\omega_{c}\tau\propto k_{B}T_{D}/\hbar\omega_{c}$.
Consequently, $\sigma_{H}^{\rm int}$ exhibits Shubnikov-de Haas (ShdH)
oscillations against the background of a pronounced dip in its energy
dependence in vicinity of the plateau if the condition of a strong magnetic
field, $\omega_{c}\tau>1$, is satisfied in this region. The relaxation time
$\tau$ (43) is proportional to $\mbox{$\cal E$}_{F}^{-2}$. As a result, the
amplitudes of ShdH oscillations rapidly tend to zero, and the conductance
asymptotically approaches its value in absence of a magnetic field as the
Fermi level increases (See right panel in Fig 4).

Of course, we are talking here about the behavior outside the plateau of only
intrinsic conductance $\sigma_{H}^{\rm int}$. Calculations of the
contributions of known external mechanisms to the anomalous Hall conductance
of a two-dimensional gas of massive Dirac fermions in the absence of a
magnetic field can be found in Refs. sin_etal_2006, ; sin_etal_2007,  (side
jump and skew scattering mechanisms) and in Ref. ado_etal_2015,  (coherent
skew scattering of electrons by impurity pairs).

In conclusion, we note that the SCBA used in this article is correct provided
that $k_{F}l\gg 1$, where $k_{F}$ is the Fermi momentum and $l$ is the mean
free path. This inequality is violated near the edges of the valence and
conduction bands $|\mbox{$\cal E$}|\simeq M$. In this energy region, SCBA
gives only a qualitative description of the behavior of the DOSs and Hall
conductance. Besides that, SCBA does not describe the so-called ”tails” of the
DOS within the energy gap $|\mbox{$\cal E$}|<M$. Therefore, the question of
how the presence of these ”tails” affects the quantization of anomalous Hall
conductance remains open.

###### Acknowledgements.

Author thanks V. V. Ustinov, I. I. Lyapilin, and N. G. Bebenin for helpful
discussions. This research was carried out within the state assignment of
Ministry of Science and Higher Education of the Russian Federation (theme
”Spin” No. AAAA-A18-118020290104-2).

##  Appendix A  Momentum representation in a magnetic field

An averaged one-electron GF in the spatially uniform disordered system subject
to an external magnetic field has in coordinate representation
$G(\bm{r},\bm{r}^{\prime})=\left\langle{\bm{r}}\left|{G}\right|{\bm{r}^{\prime}}\right\rangle$
the following general structure

$G(\bm{r},\bm{r}^{\prime})=e^{i\Phi(\bm{r},\bm{r}^{\prime})}G(\mbox{\boldmath$\varrho$})=\exp\left[i\frac{e}{c\hbar}\\!\int_{\bm{r}^{\prime}}^{\bm{r}}\\!\bm{A}(\bm{x})\cdot
d\bm{x}\right]\\!G(\mbox{\boldmath$\varrho$}),$ (47)

where $G(\mbox{\boldmath$\varrho$})=G(\bm{r}-\bm{r}^{\prime})$ is a
translation invariant and gauge-independent multiplier,
$\Phi(\bm{r},\bm{r}^{\prime})$ is the gauge-dependent phase, and integral in
the exponent is calculated along the straight line connecting points
$\bm{r}^{\prime}$ and $\bm{r}$. For example, in symmetric gauge, this phase is
equal to
$\Phi(\bm{r},\bm{r}^{\prime})=e\bm{B}\cdot(\bm{r}^{\prime}\times\bm{r})/2c\hbar$.
Obviously, any two-point function invariant in magnetic translations must have
such factorization. For example, the electron self-energy operator in
coordinate representation can be written in the form
$\Sigma(\bm{r},\bm{r}^{\prime})=\exp[i\Phi(\bm{r},\bm{r}^{\prime})]\Sigma(\mbox{\boldmath$\varrho$})$.
Therefore, given the gauge-independent identity

$e^{-i\Phi(\bm{r},\bm{r}^{\prime})}\left[\bm{p}-\frac{e}{c}\bm{A}(\bm{r})\right]e^{i\Phi(\bm{r},\bm{r}^{\prime})}=\bm{p}-\frac{e}{2c}\bm{B}\times\mbox{\boldmath$\varrho$},$
(48)

we obtain equation for the translation invariant multiplier of the one-
particle GF:

$[\mbox{$\cal E$}-\mbox{$\cal
H$}(\mbox{\boldmath$\pi$})]G(\mbox{\boldmath$\varrho$})-\int
e^{i\Theta(\mbox{\boldmath\scriptsize$\varrho$},\mbox{\boldmath\scriptsize$\varrho$}^{\prime})}\Sigma(\mbox{\boldmath$\varrho$}-\mbox{\boldmath$\varrho$}^{\prime})G(\mbox{\boldmath$\varrho$}^{\prime})d\mbox{\boldmath$\varrho$}^{\prime}=\delta(\mbox{\boldmath$\varrho$}),$
(49)

where $\mbox{$\cal H$}(\mbox{\boldmath$\pi$})$ is the Hamiltonian of an
electron moving in an external orthogonal $(\perp\rm OXY)$ magnetic field
$(\bm{B}=\rotatebox[origin={c}]{180.0}{$\varDelta$}\times\bm{A})$ ,
$\mbox{\boldmath$\pi$}=-i\hbar\rotatebox[origin={c}]{180.0}{$\varDelta$}_{\\!\\!\mbox{\boldmath\scriptsize$\varrho$}}-e(\bm{B}\times\mbox{\boldmath$\varrho$})/2c$
is the operator of gauge-invariant mechanical momentum,
$\Theta(\mbox{\boldmath$\varrho$},\mbox{\boldmath$\varrho$}^{\prime})=e\bm{B}\cdot(\mbox{\boldmath$\varrho$}^{\prime}\times\mbox{\boldmath$\varrho$})/2c\hbar$
is the phase proportional to the magnetic flux across the area of triangle
built on vectors $\varrho$ and $\mbox{\boldmath$\varrho$}^{\prime}$.

The equation (49) is translation invariant that allows us to passage to the
momentum representation with help of Fourier transform

$G_{\bm{p}}=\\!\int\\!e^{-i\bm{p}\cdot\mbox{\boldmath\scriptsize$\varrho$}/\hbar}G(\mbox{\boldmath$\varrho$})\,d\bm{r}\,,\quad
G(\mbox{\boldmath$\varrho$})=\\!\int\\!e^{i\bm{p}\cdot\mbox{\boldmath\scriptsize$\varrho$}/\hbar}G_{\bm{p}}\frac{d\bm{p}}{(2\pi\hbar)^{2}}.$
(50)

As a result, we obtain the equation

$\left[\mbox{$\cal E$}-\mbox{$\cal
H$}\left(\bm{p}-i\frac{e\hbar}{2c}\bm{B}\times\rotatebox[origin={c}]{180.0}{$\varDelta$}_{\\!\\!\bm{p}}\right)\right]G_{\bm{p}}-\Sigma_{\bm{p}}\star
G_{\bm{p}}=1,$ (51)

where symbol $F\star G$ denotes the Moyal $\star$-productmoyal_1949 ;
zach_etal_2005  of the functions dependent on the momentum $\bm{p}$

$F_{\bm{p}}\star
G_{\bm{p}}=\frac{l_{B}^{4}}{\pi^{2}}\\!\iint\\!F_{\bm{p}+\hbar\bm{k}}e^{-i2l_{B}^{2}\bm{b}\cdot(\bm{k}\times\bm{k}^{\prime})}G_{\hbar\bm{k}^{\prime}+\bm{p}}\,d\bm{k}\,d\bm{k}^{\prime},$
(52)

where $l_{B}=\sqrt{c\hbar/|e|B}$ is the magnetic length and $\bm{b}=\bm{B}/B$
is the unit vector parallel to the magnetic field $\bm{B}$. In the general
case, the $\star$-product (52) is non-commutative, but it satisfies the
requirements of associativity and distributivity. An useful representation of
the $\star$-product can be obtained by rewriting (52) with help of the
argument shift operator
$F_{\bm{p}+\hbar\bm{k}}=\exp(\hbar\bm{k}\cdot\rotatebox[origin={c}]{180.0}{$\varDelta$}_{\\!\\!\bm{p}})F_{\bm{p}}$.
After calculating the resulting Gaussian integral, we obtain the definition of
the $\star$-product (20), which is equivalent to (52). Since $\star$-product
(20) includes an exponential function dependent on the gradient operator, it
can be represented through shift of the argument in one of its multipliers,
for example,

$F_{\bm{p}}\star
G_{\bm{p}}=F\left(\bm{p}-i\frac{e\hbar}{2c}\bm{B}\times\stackrel{{\scriptstyle\rightarrow}}{{\rotatebox[origin={c}]{180.0}{$\varDelta$}}}_{\\!\\!\bm{p}}\right)G_{\bm{p}}.$
(53)

Given the last definition of the $\star$-product, we can rewrite the equation
(51) in the form used in main text [See Eq. (19)]. It looks almost like an
equation for the averaged one-particle GF in absence of a magnetic field. The
only difference is that (19) contains the $\star$-product instead of usual
multiplication. The momentum determined by the Fourier transform (50) has the
meaning of gauge-invariant mechanical momentum, which is proportional to the
electron velocity, i.e., $\bm{p}=m\bm{v}$. Indeed the turning on a magnetic
field results in this representation to replacing conventional products by the
$\star$-products of momentum dependent functions $F_{\bm{p}}G_{\bm{p}}\mapsto
F_{\bm{p}}\star G_{\bm{p}}$. In particularly,
$p_{\alpha}p_{\beta}\,\mapsto\,p_{\alpha}\star
p_{\beta}=p_{\alpha}p_{\beta}-i\epsilon_{\alpha\beta}\hbar^{2}/2l_{B}^{2}$
($\alpha(\beta)=x,y$). It follows that the $\star$-commutator of the Cartesian
components of the momentum is equal to $\big{[}p_{x}\star
p_{y}\big{]}=p_{x}\star p_{y}-p_{y}\star p_{x}=-i\hbar^{2}/l_{B}^{2}$. Thus,
the algebra of the momentum dependent $c$-numerical functions with respect to
$\star$-multiplication is equivalent to the algebra of the mechanical momentum
operators of an electron in a magnetic field.

The transition to the Fourier transform in a magnetic field is useful, since
$G_{\bm{p}}$ possesses some properties of the one-electron GF in the momentum
representation. In particularly, the trace over spatial degrees of freedom is
expressed through integral with respect to momentum

$\mathop{\rm Sp}\nolimits
G=\int\\!G_{\bm{p}}\frac{d\bm{p}}{(2\pi\hbar)^{2}}\,.$ (54)

A similar representation also holds for the trace of the two-particle
correlation function $\mathop{\rm
Sp}\nolimits\big{\langle}V_{\alpha}\mbox{$\sf G$}^{R}V_{\beta}\mbox{$\sf
G$}^{A}\big{\rangle}$ through which the components of the electrical
conductivity tensor are expressed.

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