#  Diffusive transport in the lowest Landau level of disordered 2d semimetals:
the mean-square-displacement approach

Andreas Sinner1,2, and Gregor Tkachov1 1 Institut für Physik, Universität
Augsburg, D-86135 Augsburg, Germany
2 Instytut Fizyki, Uniwersytet Opolski, PL-45-052 Opole, Poland

###### Abstract

We study the electronic transport in the lowest Landau level of disordered
two-dimensional semimetals placed in a homogeneous perpendicular magnetic
field. The material system is modeled by the Bernevig-Hughes-Zhang
Hamiltonian, which has zero energy Landau modes due to the material’s
intrinsic Berry curvature. These turn out to be crucially important for the
density of states and the static conductivity of the disordered system. We
develop an analytical approach to the diffusion and conductivity based on a
self-consistent equation of motion for the mean squared displacement. The
obtained value of the zero mode conductivity is close to the conductivity of
disordered Dirac electrons without magnetic fields, which have zero energy
points in the spectrum as well. Our analysis is applicable in a broader
context of disordered two-dimensional electron gases in strong magnetic
fields.

##  I Introduction

The subjects of low dimensional semimetals and related topics of topological
states of matter and spintronics have witnessed an enormous boom during the
last decades [Guinea2009, ; Zhang2008, ; Hasan2010, ; Kotov2012, ; Qi2011, ;
Tkachov2020, ; Avsar2020, ]. In magnetic fields, the quantum mechanics of
charge carriers in these materials is governed by a spectacular interplay of
the intrinsic and magnetic-field-induced Berry curvatures. Several aspects of
this fascinating physics remain widely untouched, though. For instance,
relative little is known about the role of disorder and its interplay with the
magnetic field. The overall progress in this area has been slow, not least
because of the technical challenges, considerable even by the standards of the
community [Ando1974, ; Wegner1983, ; Brezin1984, ; Hikami1984a, ; Hikami1984b,
]. There is a number of issues which make the disordered electrons in the
homogeneous perpendicular magnetic field look differently as compared to the
situation without a magnetic field. Due to the freedom of the gauge choice,
the problem can be approached in a number of ways, which differ very much in
details. For instance, the choice of the central gauge has the advantage that
the solutions of the Schrödinger equation are states localized in the space.
Therefore one can do computations in the position space in an exact manner.
The envisaged problem is notoriously difficult because the model lacks a small
expansion parameter [Aoki1987, ]. This inevitably leads to divergent expansion
series. A powerful method to keep such divergences under control is the
renormalization group. In the past, our understanding of the physics of
disordered metals and semiconductors profited vastly from the various
combinations of variational and perturbative techniques with the
renormalization group, c.f. Refs. [Abrahams1979, ; Gorkov1979, ; Hikami1980, ;
Wegner1979a, ; Wegner1980, ; Hikami1981, ] and Refs. [Tkachov2015, ;
Sinner2014, ; Sinner2016, ].

However, in the central gauge picture, there is literary no continuous
variable to be sliced off by iterations in order to obtain the renormalization
group equations. Of course one can use a different gauge, which allows for
description in terms of states localized in one direction and propagating in
the other. The price to pay is the loss of exactness, which is too costly to
give up. In this paper we develop a diagrammatic approach to the conductivity
of the two-dimensional disordered electron gas in strong magnetic field in
central gauge picture. While these series can still be wrapped up exactly for
the single-particle propagators, as it was impressively demonstrated by F.
Wegner in Ref. [Wegner1983, ], additional technical issues make elusive every
attempt of applying these techniques to the two-particles propagators. The
available divergent series cannot be directly plugged into the Kubo formula
without some not a-priory obvious sort of regularization or resummation.
Hence, the usual way to approach the conductivity is via the Einstein relation
and correspondingly via the notion of diffusion [Goldenfeld, ; Huang, ;
Chaikin, ]. Because the corresponding statistical averages require
normalization with respect to the vacuum fluctuations [Ziegler2012, ], this
provides a tool of estimating the measurable quantities by means of some kind
of analytical continuation [Hikami1984a, ; Hikami1984b, ; Chakravarty1986, ;
Hikami1993, ].

Our approach may not differ much from the others in relaying on perturbative
expansions for the two-particles propagator. It too requires as much
information from the perturbative expansion as possible and so, we perform the
exact computations of perturbative series to the practical limits of doable.
Moreover, we even go beyond that. We classify diagrammatic channels according
to their behavior on large scales in the position space and identify the
dominant ones. In principle this gives a hint towards the exact asymptotics of
the correlation functions. What is different in our work is the way we
approach the mean squared displacement and from this the diffusion
coefficient. Instead of resumming the perturbative series, our departing point
is the individual behavior in time of each element of the perturbative series.
It turns out that the behavior at larger time is dominated by the higher order
elements and tends towards a stationary state. On the sublaying time scales
though, there is a large region with linear time dependence, characteristic of
the diffusion. To approach this regime we propose a self-consistent equation
of motion for the mean squared displacement and extract the diffusion
coefficient from there. With the obtained diffusion coefficient and density of
states we find via the Einstein relation a universal expression for the static
conductivity in the lowest Landau level.

For concreteness we chose the Hamiltonian proposed by Bernevig, Hughes and
Zhang in the context of the quantum spin Hall effect [Zhang2006, ; Zhang2008,
]. Its spectral properties are well known and thoroughly studied in the cited
literature. The distinctive feature of the BHZ-Hamiltonian in the lowest
Landau level is the appearance of states with zero energy at a certain
strength of the magnetic field. These Landau zero modes prove crucial to the
transport in presence of disorder. Both the density of states and the
conductivity obtained from the Einstein relation are peaked around these
modes. In all the system becomes metallic within a parameter window around
them, which becomes broader but also less expressed with increasing disorder.
The numerical value exactly at a zero point is close to that of Dirac
electrons in random potentials.

The paper is organized as follows: In Section II we specify the microscopic
BZH-Hamiltonian which we use for concrete discussions. We discuss the
specifics of its spectrum and eigenstates, and introduce the corresponding
Green’s function in Section III. In Section IV we discuss the effect of the
different types of disorder and the density of states. In Section V we discuss
the mean-squared displacement and its relation to the diffusion. We present an
explicit evaluation of the mean-squared displacement to any finite order in
perturbative expansion. For this we determine an exact asymptotic expression
for the two-particles propagators. In Section VI we propose a self-consistent
equation of motion for the mean-squared displacement and extract the diffusion
coefficient from that. This gives us an access to the static conductivity.
Lengthy auxiliary calculation pieces are moved into the Appendix.

##  II The model Hamiltonian

In the absence of magnetic field the BHZ-Hamiltonian reads

$H=\Delta_{0}\Sigma_{03}-\epsilon_{0}\Sigma_{00}-\left({\cal C}_{+}T_{-}+{\cal
C}_{-}T_{+}\right)\nabla_{+}\nabla_{-}-iv\left({\cal D}_{+}\nabla_{-}+{\cal
D}_{-}\nabla_{+}\right).$ (1)

the Hamiltonian is shown in its explicit form in Appendix A. The shorthands
used here are $\displaystyle\nabla_{\pm}=\partial_{x}\pm
i\partial_{y},\;\;\nabla_{+}\nabla_{-}=\partial^{2}_{x}+\partial^{2}_{y},\;\;{T_{\pm}}=\frac{1}{2m}\pm
B_{0}$. The $4\times 4$ matrix body of the Hamiltonian is spanned by some of
the 16 Dirac matrices $\displaystyle\Sigma_{ab}=\sigma_{a}\otimes\sigma_{b}$,
first index referring to the spin space, $a,b=0,1,2,3$, with
$\sigma_{a=1,2,3}$ denoting the Pauli matrices in their usual representation
and $\sigma_{a=0}$ being the two-dimensional unity matrix. With this the
matrices ${\cal C}_{\pm}$ and ${\cal D}_{\pm}$ read

${\cal C}_{\pm}=\frac{1}{2}\left[\Sigma_{00}\pm\Sigma_{03}\right],\;\;{\rm
and}\;\;\,{\cal D}_{\pm}=\frac{1}{2}\left[\Sigma_{01}\pm i\Sigma_{02}\right].$
(2)

The band gap $\Delta_{0}$ is supposed to be much smaller than the typical gap
in 2d semiconductors. For instance, for the BHZ - Hamiltonian adopted to HgTe
quantum wells $\Delta_{0}\sim 10$meV [Zhang2008, ].

We emphasize that even though we frequently use expressions like ’spin and its
projections’, we do not always mean the physical spin of the electron. For
graphene or other systems which host Weyl- or Dirac-fermions it is better to
think of the valley or sublattice degrees of freedom, rather than the electron
spin. It is mainly due to the established tradition that we use this
vocabulary.

In strong magnetic field we replace the usual derivatives by the covariant
ones $\displaystyle\partial_{\mu}\to\partial_{\mu}+iA_{\mu}$, with the vector
potential $A$ related to the magnetic field via $\displaystyle\nabla\times
A=B$. This condition can be realized by a number of gauges. We will use the
central gauge

$A=\frac{B}{2}\left(\begin{array}[]{c}-y\\\ x\\\ 0\end{array}\right),$ (3)

the choice which makes analytical calculations particularly convenient.
Introducing complex coordinates $\displaystyle z=x+iy$,
$\displaystyle\bar{z}=x-iy$, and corresponding derivatives
$\displaystyle\partial_{z}=(\partial_{x}-i\partial_{y})/2$,
$\displaystyle\partial_{\bar{z}}=(\partial_{x}+i\partial_{y})/2$, with the
properties
$\displaystyle\partial_{z}z=\partial_{\bar{z}}\bar{z}=1,\;\;\partial_{z}\bar{z}=\partial_{\bar{z}}z=0,$
we get $\displaystyle\nabla_{-}\to 2\partial_{z}+k^{2}\bar{z}=A$,
$\nabla_{+}\to 2\partial_{\bar{z}}-k^{2}z=A^{\dagger}$, and
$\displaystyle\nabla_{+}\nabla_{-}\to(2\partial_{\bar{z}}-k^{2}z)(2\partial_{z}+k^{2}\bar{z})-2k=A^{\dagger}A-2k$,
where

$k^{2}=\frac{eB}{2\hbar}=\frac{1}{\ell^{2}}.$ (4)

$\ell=1/k$ is referred to as the magnetic length. The operator $A$ annihilates
the functions

$\varphi_{n}(r)=\frac{k}{\sqrt{\pi}}\frac{(k\bar{z})^{n}}{\sqrt{n!}}e^{-\frac{k^{2}}{2}z\bar{z}}$
(5)

i.e. $\displaystyle A\varphi_{n}(r)=0$, for every positive integer $n$. Hence,
$\varphi_{n}(r)$ is the lowest Landau level eigenfunction of the conventional
operator of kinetic energy $\propto A^{\dagger}A$. The Hilbert space of the
lowest Landau level is infinitely degenerate, i.e. $n$ can assume every
positive integer value between zero and infinity.

In this notation, the Hamiltonian becomes

$H=\Delta\Sigma_{03}-\epsilon_{0}\Sigma_{00}+2k\left({\cal C}_{+}T_{-}+{\cal
C}_{-}T_{+}\right)-\left({\cal C}_{+}T_{-}+{\cal
C}_{-}T_{+}\right)A^{\dagger}A-iv\left({\cal D}_{+}A+{\cal
D}_{-}A^{\dagger}\right).$ (6)

The ground state (i.e. the eigenstate in the lowest Landau level) suffices the
condition

$\left[\left({\cal C}_{+}T_{-}+{\cal
C}_{-}T_{+}\right)A^{\dagger}A+iv\left({\cal D}_{+}A+{\cal
D}_{-}A^{\dagger}\right)\right]\psi=0,$ (7)

which suggests two solutions:

$\psi_{+,n}(r)=\varphi_{n}(r)\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
0\end{array}\right),\;\;\;\psi_{-,n}(r)=\varphi_{n}(r)\left(\begin{array}[]{c}0\\\
0\\\ 1\\\ 0\end{array}\right),$ (8)

which are unique up to a global phase and correspond to the two spin
polarizations. The respective eigenvalues of the Hamiltonian for each spin
projection in the lowest Landau level are found from the stationary
Schrödinger equation

$H\psi=E\psi,$ (9)

which yields for both spectral branches [Zhang2008, ]

$\displaystyle E_{+}=2{T_{+}}{k}-\epsilon_{0}-\Delta_{0}$ , $\displaystyle
E_{-}=2{T_{-}}{k}-\epsilon_{0}+\Delta_{0}.$ (10)

The existence of zero energy points is due to model’s intrinsic Berry
curvature [Zhang2006, ]. In general, both eigenvalues may become zero at
different magnetic fields:

$\displaystyle E_{\pm}=0$ $\displaystyle\rm for$ $\displaystyle
k_{\pm}=\frac{\epsilon_{0}\pm\Delta_{0}}{2T_{\pm}}.$ (11)

Below we refer to these points as the Landau zero points or modes. In the
generic case, there is a split between zero energy points as shown in Fig. 1.
For the material system of HgTe the Landau zero point is degenerate, i.e.
$E_{+}=E_{-}$ and the corresponding critical field is roughly $B\sim 6$T
[Zhang2008, ].

The distance between both critical fields on the field axis is given by

$\Delta
k_{0}=|k_{+}-k_{-}|=\left|\frac{\epsilon_{0}}{2T_{+}}-\frac{\epsilon_{0}}{2T_{-}}\right|+\left|\frac{\Delta_{0}}{2T_{+}}+\frac{\Delta_{0}}{2T_{-}}\right|.$
(12)

This model is also relevant for the case of graphene [Andrei2007, ;
Novoselov2005, ; Kim2007, ]. The Hamiltonian describing the chemically neutral
gapless graphene in the strong magnetic field follows from Eq. (1) by putting
the diagonal elements to zero. Therefore, the eigenstates Eq. (8) are
eigenstates of the graphene Hamiltonian too, i.e. the whole analysis is
applicable to graphene as well. The only substantial difference is that for
the gapless ($\Delta_{0}=0$) and chemically neutral ($\epsilon_{0}=0$)
graphene, the energy of the the lowest Landau level is zero [Andrei2007, ].

Figure 1:  The energy spectrum of the Hamiltonian Eq. (1) in the lowest Landau
level as function of the magnetic field. Red line shows $E_{-}$ and the blue
line $E_{+}$ from Eq. (10). In order to emphasize the split between the Landau
zero points, the model parameters are chosen as $(2T_{-})^{-1}=-0.4$,
$(2T_{+})^{-1}=0.8$, $\epsilon_{0}=0.1$, $\Delta_{0}=0.5$.

##  III Single-particle propagator in the lowest Landau level

The advanced (+) or retarded ($-$) Green’s function in the lowest Landau level
can be calculated using the spectral representation

$\displaystyle G^{\pm}_{r,r^{\prime}}$ $\displaystyle\sim$
$\displaystyle\sum^{\infty}_{n=0}\varphi_{n}(r)\bar{\varphi}_{n}(r^{\prime})\sum_{s=\pm}\frac{{\cal
P}_{s}}{E-E_{s}\pm 0^{+}},$ (13)

where $E_{s}$ are the eigenvalues of the Hamiltonian for each spin projection
in the lowest Landau level Eq. (10) and the normalization will be fixed later.
The projectors ${\cal P}_{\pm}$ on the spin space

${\cal P}_{+}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&1&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right)\;\;\;{\rm and}\;\;\;{\cal
P}_{-}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&1&0\\\
0&0&0&0\end{array}\right)$ (14)

are idempotent and orthogonal matrices with properties $\displaystyle{\cal
P}_{+}{\cal P}_{-}=0,\;\;{\cal P}_{+}{\cal P}_{+}={\cal P}_{+},\;\;\;{\cal
P}_{-}{\cal P}_{-}={\cal P}_{-}$. The summation over all $n$ yields

$\displaystyle\sum^{\infty}_{n=0}\varphi_{n}(r)\bar{\varphi}_{n}(r^{\prime})$
$\displaystyle=$
$\displaystyle\frac{k^{2}}{\pi}e^{-\frac{k^{2}}{2}(|z|^{2}+|z^{\prime}|^{2})}\sum_{n=0}^{\infty}\frac{(k^{2}\bar{z}z^{\prime})^{n}}{n!}=\frac{k^{2}}{\pi}e^{-\frac{k^{2}}{2}(|z|^{2}+|z^{\prime}|^{2}-2\bar{z}z^{\prime})},$
(15)

which then gives for the Green’s function [Wegner1983, ; Chakravarty2007, ]

$G^{\pm}_{r,r^{\prime}}(E)=\frac{k^{2}}{2\pi}e^{-\frac{k^{2}}{2}(|z|^{2}+|z^{\prime}|^{2}-2\bar{z}z^{\prime})}\sum_{s=\pm}\frac{{\cal
P}_{s}}{E-E_{s}\pm 0^{+}}.$ (16)

Notably, the local Green’s function ($r=r^{\prime}$) is a coordinate
independent constant. The propagator is normalized this way in order to
satisfy the usual sum rule

$\mp\intop^{\infty}_{-\infty}\frac{dE}{\pi}~{}{\rm Im}~{}{\rm
tr}G^{\pm}_{r,r}(E)=\frac{k^{2}}{\pi}=\frac{eB}{h},$ (17)

where the trace operator acts only on the spin space. Eq. (17) gives the
number of the elementary flux quanta $\phi_{0}=h/e$ per unit volume. In the
real time representation, the Green’s function represents a simple collection
of undamped harmonic functions with the period determined by the eigenenergies
of the lowest Landau level modes

$G^{\pm}_{r,r^{\prime}}(t)=\mp
i\frac{k^{2}}{2\pi}e^{-\frac{k^{2}}{2}(|z|^{2}+|z^{\prime}|^{2}-2\bar{z}z^{\prime})}\sum_{s=\pm}{\cal
P}_{s}e^{\pm iE_{s}t},$ (18)

the initial time is assumed to be at zero. The Green’s function is totally
separable on the space-time.

For the case of chemically neutral gapless graphene the Green’s function
becomes particularly simple [Chakravarty2007, ]:

$G^{\pm}_{rr^{\prime}}(E)=\frac{k^{2}}{2\pi}\frac{1}{E\pm
i0^{+}}e^{-\frac{k^{2}}{2}(|z|^{2}+|z^{\prime}|^{2}-2\bar{z}z^{\prime})}[{\cal
P}_{+}+{\cal P}_{-}],$ (19)

i.e. in the real time representation it is just a step function $\theta(t)$.

##  IV Dressing of the single-particle propagator due to the disorder

The disorder is introduced in the form of the fluctuating chemical potential
$v(r)$, which couples in the spin space to the unity matrix $\Sigma_{00}$,
with the white noise correlator:

$\langle v_{r}\rangle_{g}=0,\;\;\langle
v_{r_{1}}v_{r_{2}}\rangle_{g}=g\delta_{r_{1}r_{2}}.$ (20)

The averaged propagator reads

$\bar{G}^{\pm}_{r_{1}r_{2}}=\langle[(G^{\pm})^{-1}+v\Sigma_{00}]^{-1}_{r_{1}r_{2}}\rangle_{g}.$
(21)

To perform the disorder average perturbatively we expand the propagator in
powers of $v$:

$\displaystyle\displaystyle\bar{G}^{\pm}_{r_{1}r_{2}}=\langle
G^{\pm}_{r_{1}r_{2}}-G^{\pm}_{r_{1}x_{1}}v_{x_{1}}G^{\pm}_{x_{1}r_{2}}+G^{\pm}_{r_{1}x_{1}}v_{x_{1}}G^{\pm}_{x_{1}x_{2}}v_{x_{2}}G^{\pm}_{x_{2}r_{2}}-G^{\pm}_{r_{1}x_{1}}v_{x_{1}}G^{\pm}_{x_{1}x_{2}}v_{x_{2}}G^{\pm}_{x_{2}x_{3}}v_{x_{3}}G^{\pm}_{x_{3}r_{2}}+$
(22) $\displaystyle\displaystyle
G^{\pm}_{r_{1}x_{1}}v_{x_{1}}G^{\pm}_{x_{1}x_{2}}v_{x_{2}}G_{x_{2}x_{3}}v_{x_{3}}G^{\pm}_{x_{3}x_{4}}v_{x_{4}}G^{\pm}_{x_{4}r_{2}}-G^{\pm}_{r_{1}x_{1}}v_{x_{1}}G^{\pm}_{x_{1}x_{2}}v_{x_{2}}G_{x_{2}x_{3}}v_{x_{3}}G^{\pm}_{x_{3}x_{4}}v_{x_{4}}G^{\pm}_{x_{4}x_{5}}v_{x_{5}}G^{\pm}_{x_{5}r_{2}}\cdots\rangle_{g},$

where the summation over multiple indices is understood. Because of Eq. (20)
all terms with an odd number of potentials $v$ vanish. The series then becomes

$\displaystyle\displaystyle\bar{G}^{\pm}_{r_{1}r_{2}}=\langle
G^{\pm}_{r_{1}r_{2}}+G^{\pm}_{r_{1}x_{1}}v_{x_{1}}G^{\pm}_{x_{1}x_{2}}v_{x_{2}}G^{\pm}_{x_{2}r_{2}}+G^{\pm}_{r_{1}x_{1}}v_{x_{1}}G^{\pm}_{x_{1}x_{2}}v_{x_{2}}G^{\pm}_{x_{2}x_{3}}v_{x_{3}}G^{\pm}_{x_{3}x_{4}}v_{x_{4}}G^{\pm}_{x_{4}r_{2}}$
$\displaystyle\displaystyle+G^{\pm}_{r_{1}x_{1}}v_{x_{1}}G^{\pm}_{x_{1}x_{2}}v_{x_{2}}G^{\pm}_{x_{2}x_{3}}v_{x_{3}}G^{\pm}_{x_{3}x_{4}}v_{x_{4}}G^{\pm}_{x_{4}x_{5}}v_{x_{5}}G^{\pm}_{x_{5}x_{6}}v_{x_{6}}G^{\pm}_{x_{6}r_{2}}\cdots\rangle_{g}.$
(23)

The Green’s function Eq. (16) is spanned by the spin projectors ${\cal
P}_{s}$. Therefore, only the disorder diagonal in the spin space is of
importance. Besides the randomly fluctuating chemical potential considered
here, these might include the randomly fluctuating gap, which couples to
$\Sigma_{03}$, the random ”chiral” chemical potential ($\Sigma_{30}$), or the
random ”chiral” mass ($\Sigma_{33}$). Each product of these matrices with
${\cal P}_{s}$ projects them bar the sign back onto ${\cal P}_{s}$ again.
Therefore, the perturbative series Eq. (23) does not depend on a particular
disorder type and our analysis is generic and disorder type independent.

The exact Green’s function of disordered electrons in the lowest Landau level
was obtained by Wegner in Ref. [Wegner1983, ]. The constraint condition Eq.
(17) changes it to, cf. Appendix C.

$\bar{G}^{\pm}_{rr^{\prime}}(E)=\frac{k^{2}}{\pi}e^{-\frac{k^{2}}{2}(|r|^{2}+|r^{\prime}|^{2}-2\bar{r}r^{\prime})}\sum_{s=\pm}{\cal
F}^{\pm}_{s}(E){\cal P}_{s},$ (24)

where the frequency dependent part

${\cal F}^{\pm}_{s}(E)=\eta_{s}(E)\mp i\rho_{s}(E),$ (25)

has the following explicit expressions for the real

$\eta_{s}(E)=\frac{1}{E_{g}}\left(\frac{2}{\pi}\frac{\displaystyle
e^{\nu^{2}_{s}}\int^{\nu_{s}}_{0}dt~{}e^{t^{2}}}{\displaystyle
1+\left(\frac{2}{\sqrt{\pi}}\int^{\nu_{s}}_{0}dt~{}e^{t^{2}}\right)^{2}}-\nu_{s}\right),$
(26)

and imaginary part [Brezin1984, ; Hikami1984a, ]

$\rho_{s}(E)=\frac{1}{\sqrt{\pi}E_{g}}\frac{\displaystyle
e^{\nu^{2}_{s}}}{\displaystyle
1+\left(\frac{2}{\sqrt{\pi}}\int^{\nu_{s}}_{0}dt~{}e^{t^{2}}\right)^{2}}.$
(27)

They depend on the dimensionless energy

$\nu_{s}=\frac{E-E_{s}}{E_{g}},\;\;\,{\rm
where}\;\;E^{2}_{g}=\frac{gk^{2}}{4\pi}.$ (28)

In the chosen units the disorder related energy $E_{g}$ is a dimensionless
quantity. It represents essentially the ratio of two relevant lengths
$E_{g}\sim l_{\lambda}/\ell$: the magnetic length $\ell\sim 1/k$ and the
disorder related length $l_{\lambda}\sim\sqrt{g}$. The collective density of
states

$\rho(E)=\mp\frac{1}{\pi}{\rm Im}~{}{\rm
tr}G^{\pm}_{rr}(E)=\frac{1}{\pi^{5/2}}\frac{k^{2}}{E_{g}}\sum_{s=\pm}\frac{\displaystyle
e^{\nu^{2}_{s}}}{\displaystyle
1+\left(\frac{2}{\sqrt{\pi}}\int^{\nu_{s}}_{0}dt~{}e^{t^{2}}\right)^{2}}.$
(29)

is correctly normalized in accordance with Eq. (17). Fig. 2 shows Eq. (29) for
spectrum from Fig. 1. For weak disorder strength the density of states appears
in the form of two sharp peaks placed symmetrically around the Landau zeros.
It is plotted in units of $\frac{1}{\pi^{5/2}}\frac{k^{2}}{E_{g}}\sim(\ell
l_{\lambda})^{-1}$, ${\ell l_{\lambda}}$ being the parametric volume
constructed from the two specific lengths of the model. The peaks become
boarder with increasing disorder strength and overlap with each other until
they merge to a single structure.

Figure 2:  Density of states (DOS) Eq. (29) calculated for the spectrum shown
in Fig. 1 in units of $\frac{1}{\pi^{5/2}}\frac{k^{2}}{E_{g}}$. Top and bottom
left: Evolution of the DOS in the band center ($E=0$) with increasing disorder
strength as function of the magnetic field $k\sim\sqrt{B}$. Dashed lines
emphasize the contributions to the net density of states from each Landau zero
mode. The DOS is centered around the Landau zero points. Multiplying this
function by the factor given in Eq. (60) gives the dc conductivity according
to the Einstein relation Eq. (37). Bottom right: DOS as function of the
dimensionless energy $E/E_{g}$ for the magnetic fields adjusted to each of the
Landau zero points in Fig. 1. Here, the color scheme of Fig. 1 is preserved.

Performing the Fourier transformation we obtain the Wegner’s propagator in the
time representation. It comprises two parts, one periodically oscillating in
time and the second localized in the time with the maximum at zero time, cf.
Appendix C:

${\cal F}^{\pm}_{s}(t)=\mp\frac{i}{2}e^{\pm iE_{s}t}\Omega(E_{g}t).$ (30)

The real part of the Wegner’s propagator is antisymmetric in time, while the
imaginary part is symmetric. Notably, the period of oscillations is still
determined only by energies of the clean system, while the temporarily
localized part $\Omega(E_{g}t)$ depends only on the disorder related energy
$E_{g}$. It is shown in Fig. 3 and represents a smooth and strongly damped
oscillating function symmetrically placed around $t=0$. The series of
$\Omega(E_{g}t)$ contains only even powers of $E_{g}t$ and goes for small
times as $\Omega(E_{g}t)\sim 1-{\rm const}\cdot t^{2}$.

##  V Mean squared displacement of the disordered system

The access to the diffusion goes via the mean squared displacement

$\langle r^{2}_{\mu}(t)\rangle=\frac{\displaystyle{\rm
tr}\sum_{r}r^{2}_{\mu}P_{r0}(t)}{\displaystyle{\rm tr}\sum_{r}P_{r0}(t)},$
(31)

where $r_{\mu}$ is the position operator and $P_{rr^{\prime}}(t)$ is the
return probability density defined as

$P_{rr^{\prime}}(t)=\int\frac{dE}{2\pi}~{}e^{-iEt}P_{rr^{\prime}}(E),$ (32)

where

$P_{rr^{\prime}}(E)=\langle
G^{+}_{rr^{\prime}}(E)G^{-}_{r^{\prime}r}(E)\rangle_{g},$ (33)

is the disorder averaged two-particles propagator. The large time asymptotics
of the mean squared displacement is expected to be of the form [Goldenfeld, ;
Huang, ; Chaikin, ]

$\lim_{t\to\infty}\langle r^{2}_{\mu}(t)\rangle\to 2Dt^{1+\alpha}+{\rm
const},$ (34)

where $D$ is the diffusion coefficient and the exponent $\alpha$ is referred
to as the anomalous dimension and can be either positive (superdiffusion) or
negative (subdiffusion) [Goldenfeld, ]. For instance, for the ordinary
diffusion $\alpha=0$ with $P_{r0}\sim\exp[r^{2}/Dt]/Dt$, which can be easily
verified by evaluating Eq. (31) [Ziegler2012, ]. In the linear regime, the
relation between the mean squared displacement and diffusion is established
via

$\left.\frac{d}{dt}\right|_{t=0}\langle r^{2}_{\mu}(t)\rangle=2D.$ (35)

To start with, we evaluate Eq. (31) for the case of clean system. Because the
real time representation of the Green’s function of clean system Eq. (18) is
totally separable into the spatial and temporal parts, the same is also valid
for the return probability density
$P_{r,r^{\prime}}(t)=R({r,r^{\prime}})T(t)$. Hence, the time dependent parts
cancel each other in Eq. (31) exactly and we are left with a simple result

$\langle r^{2}_{\mu}(t)\rangle=\frac{\ell^{2}}{2},$ (36)

where we use explicitly the magnetic length defined in Eq. (4) and the factor
$1/2$ is attributed to the angular average. The result is transparent: it
gives the expected position of an electron to be a circle with radius $\ell$
around the position of the flux tube piercing the sample at the origin of
coordinates.

Our task is to determine the diffusion coefficient of the disordered system
through the direct evaluation of Eq. (31), in order to compute the
conductivity from the Einstein relation

$\sigma=\frac{e^{2}}{\hbar}D\rho(E),$ (37)

where $\rho(E)$ is the density of states discussed in the previous paragraph.

Figure 3:  Left: The shape of the function $\Omega(E_{g}t)$ from Eq. (91)
determined to the order $(E_{g}t)^{60}$. Middle: The second time convolution
of the Wegner’s function Eq. (40) (blue line) in comparison to
$\Omega^{2}(E_{g}t)$ (red line). Right: Same distributions as in the middle
panel plus the fourth time convolution (brown line) Eq. (42) for small times.
All curves are normalized to 1 at their maxima in respective units. The higher
is the order of the convolution, the broader is its distribution.

A rigorous evaluation of the full perturbative series for the two-particles
propagator $\langle G^{+}_{r,0}G^{-}_{0,r}\rangle_{g}$ along the lines of
Wegner’s calculations for the single-particle propagator is principally
impossible. The reason for this is the spatial decoherence in higher orders of
perturbative expansion, cf. Appendix B: While the disorder does not affect the
spatial dependence of the single-particle propagator, it does so for the two-
particles propagator. Thus, we need to consider the spatial averages. We
evaluate both expressions from the numerator and denominator of Eq. (31)
perturbatively. Evaluation of all perturbative diagrams to order $g^{3}$
yields for the numerator of Eq. (31)

$\displaystyle\displaystyle{\rm
tr}\sum_{r}r^{2}_{\mu}P_{r0}(E)=\frac{1}{4\pi}\frac{1}{E^{2}_{g}}\sum_{s=\pm}(2X_{s})^{2}\left[\frac{1}{2}+(2X_{s})^{2}+2(2X_{s})^{4}+\frac{167}{36}(2X_{s})^{6}+\cdots\right.$
$\displaystyle\displaystyle+\left.\left(\frac{3}{4}(2X_{s})^{4}+\frac{343}{72}(2X_{s})^{6}+\cdots\right)\cos
2\phi_{s}+\left(\frac{139}{72}(2X_{s})^{6}+\cdots\right)\cos
4\phi_{s}+\cdots\right],$ (38)

where $X^{2}_{s}(E)=E^{2}_{g}[\eta^{2}_{s}(E)+\rho^{2}_{s}(E)]$ and
$\phi_{s}(E)={\rm arctan}\left[\frac{\rho_{s}(E)}{\eta_{s}(E)}\right]$.
According to Eqs. (26) and (27), $X^{2}_{s}(E)$ and $\phi_{s}(E)$ are
dimensionless functions of the argument $\nu_{s}=(E-E_{s})/E_{g}$. The
analogous computation for the denominator of Eq. (31) yields

$\displaystyle\displaystyle{\rm
tr}\sum_{r}P_{r0}(E)=\frac{k^{2}}{4\pi}\frac{1}{E^{2}_{g}}\sum_{s=\pm}(2X_{s})^{2}\left[1+(2X_{s})^{2}+\frac{3}{2}(2X_{s})^{4}+\frac{13}{4}(2X_{s})^{6}+\cdots\right.$
$\displaystyle\displaystyle+\left.\left((2X_{s})^{4}+\frac{9}{2}(2X_{s})^{6}+\cdots\right)\cos
2\phi_{s}+\left(\frac{5}{2}(2X_{s})^{6}+\cdots\right)\cos
4\phi_{s}+\cdots\right].$ (39)

Technical details of the evaluation and results for each individual diagram
are summarized in Appendices B and D.

At first one can try to Fourier transform each individual term in both series.
Essentially, the Fourier transformed of Eqs. (38) and (39) are given in terms
of even time convolutions of the temporal part of the Wegner’s propagator Eq.
(30)

$\displaystyle\int^{\infty}_{-\infty}\frac{dE}{\pi}e^{iEt}X^{2}_{s}(E)$
$\displaystyle\sim$
$\displaystyle\int^{\infty}_{-\infty}d\tau~{}\Omega(E_{g}\tau)\Omega(E_{g}[\tau+t]),$
(40) $\displaystyle\int^{\infty}_{-\infty}\frac{dE}{\pi}e^{iEt}X^{4}_{s}(E)$
$\displaystyle\sim$
$\displaystyle\int^{\infty}_{-\infty}d\tau_{1}\int^{\infty}_{-\infty}d\tau_{2}\int^{\infty}_{-\infty}d\tau_{3}~{}\Omega(E_{g}\tau_{1})\Omega(E_{g}[\tau_{1}+\tau_{3}])$
(41) $\displaystyle\times$
$\displaystyle\Omega(E_{g}\tau_{2})\Omega(E_{g}[\tau_{2}+\tau_{3}+t]),$ (42)

and so on. In the practice though, this can be done to an extend only. To the
best of our knowledge, the function $\Omega(E_{g}\tau)$ itself is not
tabulated and its polynomial series has to be truncated at some order. For
practical calculations we use the polynomial with 31 terms (order
$(E_{g}t)^{60}$), which is sufficient to determine the second convolution Eq.
(40) only to the order $(E_{g}t)^{20}$, and the forth convolution Eq. (42)
only to the order $(E_{g}t)^{6}$, cf. Fig. 3. The convolutions have a typical
bell-curve shape. This result suffices to recognize that every higher order
convolution is broader and therefore, the large time asymptotics of the mean
squared displacement is determined by the higher order terms in the series
Eqs. (38) and (39). Another aspect is that since the Wegner’s propagator is
uniquely separable in temporal and spatial variables, the large time
asymptotics of the numerator and denominator of Eq. (31) differ only in
amplitudes but not in functional behavior. Hence, we can expect that for large
times the mean squared displacement approaches a time independent constant,
similarly to the case of the clean system Eq. (36) and the time evolution ends
up in a stationary state. In this state all electrons are again localized at
fixed orbits and no currents can flow through the sample. If there be a
diffusion in the system, i.e. the regime linear in time with a positive slope,
then it can only emerge due to the superposition of all convolutions at
intermediate time scales.

Figure 4:  The initial mean squared displacement $\langle
r^{2}_{\mu}(0)\rangle$ Eq. (44) in units of $k^{-2}$ evaluated to different
order ($n=0,1,2,3$) of perturbative expansion.

To give these general arguments a more firm footing we propose a resummation
procedure leaned on the general idea of the renormalization group. We start
with the considerations of the mean squared displacement at time zero

$\langle r^{2}_{\mu}(0)\rangle=\frac{\displaystyle{\rm
tr}\int^{\infty}_{-\infty}dE\sum_{r}r^{2}_{\mu}P_{r0}(E)}{\displaystyle{\rm
tr}\int^{\infty}_{-\infty}dE\sum_{r}P_{r0}(E)}.$ (43)

Changing to the integration over $\nu_{s}$ we then have

$k^{2}\langle
r^{2}_{\mu}(0)\rangle=\frac{\displaystyle\sum_{s=\pm}~{}\int^{\infty}_{-\infty}d\nu_{s}~{}f^{(2)}(\nu_{s})}{\displaystyle\sum_{s=\pm}\int^{\infty}_{-\infty}d\nu_{s}~{}f^{(0)}(\nu_{s})}=\frac{\langle
f^{(2)}\rangle}{\langle f^{(0)}\rangle},$ (44)

where

$\displaystyle\langle f^{(2)}\rangle={\rm
tr}\int^{\infty}_{-\infty}d\nu_{s}\sum_{r,s=\pm}r^{2}_{\mu}P_{r0}(\nu_{s}),$
$\displaystyle\;{\rm and}\;$ $\displaystyle\langle f^{(0)}\rangle={\rm
tr}\int^{\infty}_{-\infty}d\nu_{s}\sum_{r,s=\pm}P_{r0}(\nu_{s}).$ (45)

Each of both spin projections contribute equally in $t=0$ limit and we can
skip the summation over $s$. In Fig. 4 we plot the right hand side of Eq. (44)
evaluated for the increasing order $n$ of perturbative expansion.

With larger $n$ the statical mean squared displacement approaches a linear
asymptotics as function of $n$. Among all possible diagrammatic channels,
there is a unique channel which suffices this property. This includes all
diagrams of the so-called ladder channel, cf. Appendix D. Assembling all
ladder diagrams yields the asymptotics of the two-particles propagator in the
form of an infinite series

$P^{\rm
lad}_{r0}(E)\approx\frac{1}{4E^{2}_{g}}\left(\frac{k^{2}}{\pi}\right)^{2}\sum_{s=\pm}~{}\sum^{\infty}_{n=1}\frac{(2X_{s})^{2n}}{n}\exp\left[-\frac{k^{2}r^{2}}{n}\right].$
(46)

This expression gives rise to the series element of the denominator of Eq.
(44) in the form

$\langle f^{(0)}_{n}\rangle^{\rm lad}=\frac{k^{2}}{4\pi
E^{2}_{g}}\sum_{s=\pm}\int^{\infty}_{-\infty}d\nu~{}(2X_{s})^{2n},$ (47)

and for the numerator

$\langle f^{(2)}_{n}\rangle^{\rm lad}=\frac{1}{4\pi
E^{2}_{g}}\sum_{s=\pm}\int^{\infty}_{-\infty}d\nu~{}\frac{n}{2}(2X_{s})^{2n}.$
(48)

Because the argument $2X_{s}$ is larger than 1 at its maximum the series do
not converge. Hence, at every finite expansion order, the behavior of both
numerator and denominator of the mean squared displacement are dominated by
the respective last terms in the series, which then cancel within the formula
for $\langle r^{2}_{\mu}(0)\rangle$ Eq. (43). In this way the linear
dependence of $\langle r^{2}_{\mu}(0)\rangle\sim n/2$ visible in Fig. 4
emerges. All remaining channels exhibit either slower logarithmic grow or
decay with increasing $n$, which allows to neglect them.

##  VI Equation of motion for the mean squared displacement

Figure 5:  Integral $I_{N}$ Eq. (51) (left) and product $NI_{N}$ (right) .

To arrive at the equation of motion for the mean squared displacement, we
assume that it is possible to determine the perturbative series up to a large
order $N$. In this limit, using of the approximate formulas Eqs. (47) and (48)
is justified. If necessary, the limit $N\to\infty$ can be carried out at the
end of calculations. We write Eq. (44) as

$k^{2}\langle r^{2}_{\mu}(0)\rangle=\frac{\langle
f^{(2)}_{0}\rangle+\langle\delta f^{(2)}\rangle}{\langle
f^{(0)}_{0}\rangle+\langle\delta f^{(0)}\rangle}.$ (49)

In Eq. (49) $\langle f^{(i)}_{0}\rangle$ include all terms of the respective
series with exception of the $N$th. The latter are contained in the
fluctuation terms $\langle\delta f^{(i)}\rangle$ and are treated as the
generators of the time evolution of $\langle r^{2}_{\mu}\rangle$ at small
positive/negative times $\Delta t$. After some algebra presented in Appendix E
we obtain a self-consistent equation of motion for $\langle
r^{2}_{\mu}(t)\rangle$ in the form of the second order ordinary differential
equation

$k^{2}\frac{\partial^{2}}{\partial t^{2}}\langle
r^{2}_{\mu}(t)\rangle=-E^{2}_{g}I_{N}\left[\frac{N}{2}-k^{2}\langle
r^{2}_{\mu}(t)\rangle\right].$ (50)

The integral $I_{N}$ in the expression for the time-dependent diffusion
coefficient is defined as

$\displaystyle I_{N}$ $\displaystyle=$ $\displaystyle\frac{1}{\langle
f^{(0)}\rangle}\sum_{s=\pm}\int^{\infty}_{-\infty}d\nu_{s}~{}\nu^{2}_{s}~{}(2X_{s})^{2N}\approx\frac{\displaystyle\int^{\infty}_{-\infty}d\nu~{}\nu^{2}X^{2N}(\nu)}{\displaystyle\int^{\infty}_{-\infty}d\nu~{}X^{2N}(\nu)},$
(51)

where we approximated $\langle f^{(0)}\rangle$ by the largest power in the
series. Integral $I_{N}$ shows a very slow decay for large $N$, cf. Fig. 5,
and seems to approach a finite limit for infinitely large $N$. Numerical
routines struggle to compute it much beyond $N=600$ and it remains an open
question whether the $N\to\infty$ bound exists. In turn, the product $NI_{N}$
grows very slowly, which can also be interpreted as approaching a finite value
in the limit $N\to\infty$.

The large-time asymptotics ($t\gg t_{c}$, $t_{c}$ being some crossover time
far in the past, which can be chosen zero) is given by the solution of the
self-consistent equation

$\langle r^{2}_{\mu}(t)\rangle^{>}\approx\frac{N}{2}\ell^{2}+\ell^{2}{\cal
C}\exp\left[-\frac{t}{\tau}\right],$ (52)

where we introduced the scattering time as

$\tau\approx\frac{1}{\sqrt{I_{N}}E_{g}}.$ (53)

In the limit $t\to\infty$ the mean squared displacement approaches its upper
bound

$\lim_{t\to\infty}\langle r^{2}_{\mu}(t)\rangle^{>}\to\frac{N}{2}\ell^{2},$
(54)

which for $N\to\infty$ lies in the infinity and is therefore never reached.
Hence it can be only approached from below, which requires $\cal C$ to be
negative. If $\sqrt{I_{N}}$ is small, then $\tau$ is large and the regime with
linear time dependence should be broad. The diffusion coefficient is then
obtained from

$\left.\frac{\partial}{\partial t}\langle
r^{2}_{\mu}(t)\rangle^{>}\right|_{t=t_{c}}=|{\cal C}|\frac{\ell^{2}}{\tau}.$
(55)

Formally, $|{\cal C}|$ should follow from the initial condition at $t=t_{c}$,
but for this we need to know $\langle r^{2}_{\mu}(t_{c})\rangle^{>}$, which
lies far in the past and therefore forgotten. In order to be a physical
quantity, we demand for $D$ an invariance with respect to $N$. This is similar
to the version of the renormalization group typically used in the high energy
physics. This implies

$\frac{\partial}{\partial N}(\sqrt{I_{N}}|{\cal C}|)=0,$ (56)

from where then follows

$\sqrt{I_{N}}|{\cal C}|={\rm const}.$ (57)

Even though this constrain might appear not entirely transparent it has a
natural analogy in the case of disordered electron gas without magnetic field.
Here, the diffusion coefficient is determined from the self-consistent Born
approximation and appears unchanged in the partial series, e.g. cooperon or
diffuson [Altshuler1994, ; Efetov1997, ]. Comparison with Eq. (34) suggests
this constant to be $2$. Then the physical diffusion coefficient becomes

$D\approx\frac{E_{g}}{k^{2}}\sim\ell l_{\lambda},$ (58)

i.e. it is proportional to the parametric volume of the model.

Inserting the density of states from Eq. (29) and the diffusion coefficient
Eq. (58) into the Einstein relation Eq. (37) yields the conductivity. The
system is conducting within a parametric window located around each of the
Landau zeros. The width of the conducting window is determined by the
parameters of the microscopic model and by the disorder. The transition $g\to
0$ is smooth and the conductivity degenerates to two sharp peaks at the Landau
zeros. With increasing disorder, the peaks becomes broader and merge at some
point to an amorphous structure. Simultaneously the amplitude becomes smaller,
signaling the suppression of the conductivity in the strong disorder limit.

We can compute the conductivity at an arbitrary Landau zero point. For weak
disorder, the contribution from the other mode is negligible and we get for
the density of states

$\rho^{\rm LZ}\approx\frac{1}{\pi^{5/2}}\frac{k^{2}}{E_{g}}.$ (59)

Using the units of $e^{2}/h$ instead of $e^{2}/\hbar$ adds an extra factor of
$2\pi$, i.e.

$\sigma^{\rm LZ}\approx\frac{2\pi}{\pi^{5/2}}\frac{e^{2}}{h}\approx
0.36\frac{e^{2}}{h}.$ (60)

Numerically it is close to the dc limit of the optical conductivity of a
single Dirac electron (which also has a zero energy point in the spectrum) in
clean and even weakly disordered systems, which depending on the way it is
calculated is either $\frac{e^{2}}{\pi h}\approx 0.32\frac{e^{2}}{h}$ or
$\frac{\pi e^{2}}{8h}\approx 0.39\frac{e^{2}}{h}$ [Fradkin1986, ; Ludwig1994,
; Gusynin2006, ; Ziegler2006, ; Ziegler2007, ; Sinner2018, ]. If compared to
the established value evaluated for conventional electron gas with the
spectrum without zero points, then the Padé-Borel resummation [Hikami1984a, ;
Hikami1984b, ; Chakravarty1986, ] gives the value $0.46\frac{e^{2}}{h}$, while
the numerical value of Ref. [Bhatt1993, ] is $0.5\frac{e^{2}}{h}$ within a
statistical uncertainty of 10% . However, we are not aware of any further
results for the static conductivity at the Landau zero points.

##  VII Discussions

The electronic transport properties of disordered electronic systems in strong
magnetic fields represents a technically very demanding problem, which evades
a conclusive solution even despite several decades of active research. In the
past there have been impressive break troughs, though. At the single-particle
level, the problem has been effectively resolved by Ando [Ando1974, ] and
Wegner [Wegner1983, ]. The latter work provided an exact expression for the
single-particle propagator. The resulting Green’s function does not reveal any
singularities and describes a state of the matter without pronounced
resonances and consequently without clearly defined quasiparticles.

On the technical side the difficulties aggregate if one goes beyond the
single-particle picture and considers processes involving two or more
particles. While the single-particle propagator retains its form in the
position space in every order of perturbative expansion, which suggests a kind
of invariance under the operation of perturbative expansion, the two-particles
propagator does not show up this property. Instead, the disorder washes out
the expansion series in the position space, making higher order terms broader
in comparison to the clean system. This is the reason, why the Wegner’s
technique cannot be applied to the two-particles propagator with the same
success. From studying previous works available to us, it is not clear,
whether this simple fact has been realized so far.

In our work we perform the perturbative expansion in powers of disorder
strength and classify all scattering processes into two main groups: Firstly
those which dress the single-particle propagators. To order $g^{3}$ all of
these diagrams are shown in Fig. 7. They are absorbed into the fully dressed
single-particle propagators, for which exact results are known [Wegner1983, ;
Brezin1984, ]. Secondly, there are processes which cannot be accounted for at
the single-particle level. They are represented by loop diagrams, which
involve at least one scattering process between the advanced and retarded
sides. To order $g^{3}$ all of them are shown in Fig. 6. We take all diagrams
to third order to into account, without truncating the perturbative series.
Ideally, one would compute all diagrams in all orders of expansion. The number
of diagrams grows rapidly with increasing order of expansion though, while the
computational effort for the evaluation of every individual diagram increases
considerably. However, it is possible to extract the diagrammatic channel
which dominates the large-scale behavior of the two-particles propagator.
Still, due to absence of a small expansion parameter, the obtained series do
not show any sign of convergence [Aoki1987, ].

Our main intention is to calculate the transport coefficients, in particular
the static conductivity. Because the series diverge, a naive use of the Kubo
formula fails. Therefore we approach the conductivity via the Einstein
relation, which requires the knowledge of the density of states and of the
diffusion coefficient. While the former is known from the Wegner’s solution,
the latter is not. We acquire it from the mean-squared displacement, which at
large times should be proportional to the diffusion coefficient. For infinite
times the mean-squared displacement approaches an infinite time independent
value. The processes linear in time emerge at the time scales determined by
the combined action of disorder and magnetic field. The former being weak and
the latter strong puts this characteristic time to considerable absolute
values, which can indeed appear as the large time limit in an experiment.
Whatever happens at larger times can evoke an impression of the onset
subdiffusion. The infinite times asymptotics might be inaccessible in
realistic measurements. With this we can compute the conductivity from the
Einstein relation with the extracted diffusion coefficient.

For discussions of a concrete physical system we consider the BZH-Hamiltonian
[Zhang2006, ]. Its lowest Landau level energy spectrum has two modes,
corresponding to the two particle species related to each other by the time-
reversal symmetry. Both modes exhibit zero energy points at critical magnetic
fields, the so-called Landau zero modes. It is possible to lift the degeneracy
of the Landau zero points parametrically, i.e. to move them apart from each
other on the magnetic field axis. The analytical expressions for the
conductivity adopted for this case suggest that the system becomes metallic in
the parametric window around the Landau zeros. The weaker the disorder, the
narrower is the metallic window. This picture appears rather intuitive and
hopefully likely to be confirmed experimentally with a modest effort.
Adjusting the magnetic field exactly to the zero Landau energy gives for the
conductivity a universal value $\sim 0.36e^{2}/h$, which is surprisingly close
to the established results for the conductivity of the disordered Dirac
electrons without magnetic fields, which also have zero energy points in the
spectrum.

The most obvious avenue leading beyond the scope of the present work is the
challenge of reconciling the Einstein relation with the Kubo-Greenwood
formalism [Wegner1979b, ; McKane1981, ; Ludwig1994, ; Sinner2018, ]. In the
Kubo-Greenwood formalism, the computation of the conductivity requires the
knowledge of the two-particles propagators as well. However, usually the Kubo-
Greenwood formula does not involve any normalization and the issue of series
convergence becomes much more urgent. Curiously though, the static
conductivity of the clean system can be straightforwardly evaluated from the
Kubo-Greenwood formula, giving the value of $e^{2}/2h$ for each Landau mode.
At the first superficial glance, once the disorder is brought into the system
this conductivity is destroyed. Another possible direction is an extension to
the higher Landau levels. These problems are left for the future activities.

##  VIII Acknowledgments

A.S. acknowledges the support by the grants of the Julian Schwinger Foundation
for Physics Research and expresses his gratitude to Prof. C. Weber for
hospitality during the later phase of manuscript preparation. The work was
concluded at IMDEA Nanotechnologia Madrid with the support from the personal
research grant (A.S.) PCI2021-122057-2B (MSCA IF EF-ST 2020) of the Agencia
Estatal de Investigacion, Spain.

##  Appendix A BHZ-Hamiltonian in explicit form

Without magnetic field, the BHZ-Hamiltonian Eq. (1) reads:

$\displaystyle H=$ (61)
$\displaystyle\displaystyle\left(\begin{array}[]{cccc}\displaystyle-{T_{-}}{\nabla_{+}\nabla_{-}}-\epsilon_{0}+\Delta_{0}&-iv\nabla_{-}&0&0\\\
-iv\nabla_{+}&\displaystyle-{T_{+}}{\nabla_{+}\nabla_{-}}-\epsilon_{0}-\Delta_{0}&0&0\\\
0&0&\displaystyle-{T_{-}}{\nabla_{+}\nabla_{-}}-\epsilon_{0}+\Delta_{0}&-iv\nabla_{+}\\\
0&0&-iv\nabla_{-}&\displaystyle-{T_{+}}{\nabla_{+}\nabla_{-}}-\epsilon_{0}-\Delta_{0}\end{array}\right).$
(66)

All quantities are explained in the Section II. The model does not presume any
coupling or scattering mechanisms between spin projections (or valleys), which
would appear in the off-diagonal blocs. To introduce the magnetic field we
replace $\displaystyle\nabla_{-}\to 2\partial_{z}+k^{2}\bar{z}=A$,
$\nabla_{+}\to 2\partial_{\bar{z}}-k^{2}z=A^{\dagger}$, and
$\displaystyle\nabla_{+}\nabla_{-}\to(2\partial_{\bar{z}}-k^{2}z)(2\partial_{z}+k^{2}\bar{z})-2k=A^{\dagger}A-2k$.

Figure 6: Perturbative processes contributing to the dressing of the two-
particles propagator up to the third order in disorder strength. Solid lines
denote the fully dressed Wegner’s propagators and the dashed lines the
disorder correlators.

##  Appendix B Technical preliminary

In Section V we emphasize that, while the position dependence of the single-
particle propagator $G^{\pm}_{rr^{\prime}}$ of an electron in lowest Landau
level is preserved in every order of perturbative expansion, this is not so
for the two-particles propagator $G^{+}_{rr^{\prime}}G^{-}_{r^{\prime}r}$. We
demonstrate this by evaluating respective leading order terms. The
calculations are easy to perform using the following integration formula

$\frac{1}{\pi}\int
d\bar{x}dx~{}e^{\displaystyle-[c\bar{x}x-a\bar{y}x-b\bar{x}z]}=\frac{1}{c}\exp\left[\frac{ab}{c}\bar{y}z\right],\;\;\;(a,b,c)>0.$
(67)

After performing the disorder average in the lowest order of the single-
particle propagator we have

$\langle
G^{\pm}_{rx}v_{x}G^{\pm}_{xy}v_{y}G^{\pm}_{yr^{\prime}}\rangle_{g}=g\sum_{xy}\delta_{xy}G^{\pm}_{rx}G^{\pm}_{xy}G^{\pm}_{yr^{\prime}}=g\sum_{x}G^{\pm}_{rx}G^{\pm}_{xx}G^{\pm}_{xr^{\prime}}.$
(68)

The local propagator $G^{\pm}_{xx}$ is a position independent constant.
Because of the idempotence of projectors, the product of Green’s functions is
separable in the spin space (for sake of simplicity we skip below the
convergence factor $\pm i0^{+}$, unless the opposite is necessary):

$\displaystyle
g\sum_{x}G^{\pm}_{rx}G^{\pm}_{xx}G^{\pm}_{xr^{\prime}}=g\left(\frac{k^{2}}{2\pi}\right)^{3}\sum_{s=\pm}\frac{{\cal
P}_{s}}{[E-E_{s}]^{3}}\int
d\bar{x}dx~{}e^{-\frac{k^{2}}{2}(|r|^{2}+|x|^{2}-2\bar{r}x)}e^{-\frac{k^{2}}{2}(|x|^{2}+|r^{\prime}|^{2}-2\bar{x}r^{\prime})}.$
(69)

We evaluate the integral as follows:

$\displaystyle\int
d\bar{x}dx~{}e^{-\frac{k^{2}}{2}(|r|^{2}+|x|^{2}-2\bar{r}x)}e^{-\frac{k^{2}}{2}(|x|^{2}+|r^{\prime}|^{2}-2\bar{x}r^{\prime})}$
$\displaystyle=$ $\displaystyle
e^{-\frac{k^{2}}{2}(|r|^{2}+|r^{\prime}|^{2})}\int
d\bar{x}dx~{}e^{-k^{2}(\bar{x}x-\bar{r}x-\bar{x}r^{\prime})}$ (70)
$\displaystyle=$ $\displaystyle
e^{-\frac{k^{2}}{2}(|r|^{2}+|r^{\prime}|^{2}-2\bar{r}r^{\prime})},$

where in the last line we used the formula Eq. (67). Putting all terms
together we get

$g\sum_{x}G_{rx}G_{xx}G_{xr^{\prime}}=\frac{g}{2}\left(\frac{k^{2}}{2\pi}\right)^{2}e^{-\frac{k^{2}}{2}(|r|^{2}+|r^{\prime}|^{2}-2\bar{r}r^{\prime})}\sum_{s=\pm}\frac{{\cal
P}_{s}}{[E-E_{s}]^{3}}.$ (71)

Indeed, the spatial part of this expression is the same as that of the Green’s
function of the clean system. This property can be traced back in every order
of perturbative expansion, irrespective of the diagram’s topology. This means
that the disorder does not affect the coherence of the single-particle Green’s
function and therefore, the series can be summed up exactly [Wegner1983, ;
Brezin1984, ].

To illustrate the effect of the disorder on the two-particles propagator we
start with the expression of the clean system:

$G^{+}_{rr^{\prime}}G^{-}_{r^{\prime}r}=\left(\frac{k^{2}}{2\pi}\right)^{2}e^{-k^{2}(\bar{r}-\bar{r}^{\prime})(r-r^{\prime})}\sum_{s=\pm}\frac{{\cal
P}_{s}}{(E-E_{s}-i0^{+})(E-E_{s}+i0^{+})}.$ (72)

The two-particles propagator is a real valued quantity and decays
exponentially in the position space with the Gaussian exponent 1. The leading
order perturbative vertex correction reads

$\displaystyle
g\sum_{x}G^{+}_{rx}G^{+}_{xr^{\prime}}G^{-}_{r^{\prime}x}G^{-}_{xr}$
$\displaystyle=$ $\displaystyle
g\left(\frac{k^{2}}{2\pi}\right)^{4}\sum_{s=\pm}\frac{{\cal
P}_{s}}{(E-E_{s}-i0^{+})^{2}(E-E_{s}+i0^{+})^{2}}$ (73) $\displaystyle\times$
$\displaystyle\int
d\bar{x}dx~{}e^{-k^{2}(\bar{r}-\bar{x})(r-x)}e^{-k^{2}(\bar{x}-\bar{r}^{\prime})(x-r^{\prime})}$
(74)

At first, the integral can be rewritten as

$\int
d\bar{x}dx~{}e^{-k^{2}(\bar{r}-\bar{x})(r-x)}e^{-k^{2}(\bar{x}-\bar{r}^{\prime})(x-r^{\prime})}=e^{-k^{2}(|r|^{2}+|r^{\prime}|^{2})}\int
d\bar{x}dx~{}e^{-k^{2}[2|x|^{2}-\bar{x}(r+r^{\prime})-(\bar{r}+\bar{r}^{\prime})x]},$
(75)

which gives with the help of Eq. (67)

$g\sum_{x}G^{+}_{rx}G^{+}_{xr^{\prime}}G^{-}_{r^{\prime}x}G^{-}_{xr}=\frac{g}{2^{2}}\left(\frac{k^{2}}{2\pi}\right)^{3}e^{-\frac{k^{2}}{2}(\bar{r}-\bar{r}^{\prime})(r-r^{\prime})}\sum_{s=\pm}\frac{{\cal
P}_{s}}{(E-E_{s}-i0^{+})^{2}(E-E_{s}+i0^{+})^{2}}.$ (76)

Comparing this with Eq. (72) we recognize the different Gaussian exponent 1/2,
i.e. the spatial part of the first correction is broader than that of the
clean system. This effect takes place in all orders but differently for
topologically different diagrammatic channels, i.e. the coherence of the two-
particles propagator is violated by the disorder. This is the ultimate reason
why the adoption of the Wegner’s technique to the two-particles propagator is
difficult. Table 1 shows the Gaussian exponents for each diagram from Fig. 6.
They reveal certain patterns for particular diagrammatic channels. For
instance, one recognizes that the large scale behavior in every order of
perturbative expansion is dominated by the diagrams belonging to the so-called
ladder channel. Those are the diagrams no. 1,2,3 and 4. The dominant
oscillating channel is represented by the diagrams no. 5, 7, etc., which we
call the menora diagrammatic channel. The diagrams belonging to the so-called
fan channel, e.g. no. 4, 17, etc., retain the same Gaussian exponent in every
perturbative order. In Appendix D we argue that they contribute the least to
the amplitudes in every order of expansion and therefore are clearly
subdominant in comparison to the ladder channel.

No. | Diagram | Exponent |  | No. | Diagram | Exponent |  | No. | Diagram | Exponent
---|---|---|---|---|---|---|---|---|---|---
1 |  | $1$ |  | 6 |  | $\frac{3}{4}$ |  | 12 |  | $\frac{2}{3}$
2 |  | $\frac{1}{2}$ |  | 7 |  | $\frac{3}{4}$ |  | 13 |  | $\frac{1}{2}$
3 |  | $\frac{1}{3}$ |  | 8 |  | $\frac{2}{3}$ |  | 14 |  | $\frac{1}{4}$
4 |  | $\frac{1}{2}$ |  | 9 |  | $\frac{3}{5}$ |  | 15 |  | $\frac{1}{2}$
5 |  | $\frac{2}{3}$ |  | 10 |  | $\frac{3}{5}$ |  | 16 |  | $\frac{1}{3}$
|  |  |  | 11 |  | $\frac{2}{5}$ |  | 17 |  | $\frac{1}{2}$
Table 1:  Gaussian exponents $\chi_{i}$ of all perturbative processes
contributing to the two-particles Green’s function shown in Fig. 6.

##  Appendix C Exact Wegner’s propagator of the disordered system

We start the discussion of the dressing effect for the single-particle
propagator by evaluating all perturbative contributions to order $g^{2}$ and
$g^{3}$. The number of diagrams to order $g^{n}$, $n$ being a positive integer
number and zero is $\frac{(2n)!}{2^{n}n!}$, hence, there are in total 3
diagrams to the order $g^{2}$ and 15 diagrams to order $g^{3}$, Fig. 7. In
Table 2 we summarize their amplitudes. Next order $g^{4}$ would require the
evaluation of 105 diagrams with a larger computational effort for each
individual diagram. To this end though, we have an asymptotically exact
expression:

$\bar{G}^{\pm}_{rr^{\prime}}(E)=\frac{k^{2}}{\pi}e^{-\frac{k^{2}}{2}(|r|^{2}+|r^{\prime}|^{2}-2\bar{r}r^{\prime})}\sum_{s=\pm}{\cal
P}_{s}{\cal F}^{\pm}_{s}(E),$ (77)

where the frequency dependent part of the Green’s function reads

$\displaystyle{\cal F}^{\pm}_{s}(E)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{1}{E-E_{s}}\left[1+\frac{E^{2}_{g}}{[E-E_{s}]^{2}}+\frac{5}{2}\frac{E^{4}_{g}}{[E-E_{s}]^{4}}+\frac{37}{4}\frac{E^{6}_{g}}{[E-E_{s}]^{6}}\cdots\right],$
(78)

where $E^{2}_{g}=\frac{gk^{2}}{4\pi}$. Not surprisingly, the expansion
coefficients 1, 1, 5/2, 37/4.. are precisely those of the exact solution by
Wegner Ref. [Wegner1983, ], extended by Brézin et al. with a different
technique in Ref. [Brezin1984, ].

Figure 7:  Perturbative processes contributing to the dressing of the single-particle propagator due to the disorder to order $g^{1}$ (one diagram), $g^{2}$ (three diagrams), and $g^{3}$ (fifteen diagrams).  No. | Diagram | Amplitude |  | No. | Diagram | Amplitude |  | No. | Diagram | Amplitude
---|---|---|---|---|---|---|---|---|---|---
1 |  | ${\displaystyle\sum_{s}}E_{g}F^{3}_{s}$ |  | 6 |  | $\frac{1}{3}{\displaystyle\sum_{s}}E^{3}_{g}F^{7}_{s}$ |  | 11 |  | ${\displaystyle\sum_{s}}E^{3}_{g}F^{7}_{s}$
2 |  | ${\displaystyle\sum_{s}}E^{2}_{g}F^{5}_{s}$ |  | 7 |  | ${\displaystyle\sum_{s}}E^{3}_{g}F^{7}_{s}$ |  | 12 |  | $\frac{1}{2}{\displaystyle\sum_{s}}E^{3}_{g}F^{7}_{s}$
3 |  | ${\displaystyle\sum_{s}}E^{2}_{g}F^{5}_{s}$ |  | 8 |  | $\frac{1}{2}{\displaystyle\sum_{s}}E^{3}_{g}F^{7}_{s}$ |  | 13 |  | $\frac{2}{3}{\displaystyle\sum_{s}}E^{3}_{g}F^{7}_{s}$
4 |  | $\frac{1}{2}{\displaystyle\sum_{s}}E^{2}_{g}F^{5}_{s}$ |  | 9 |  | ${\displaystyle\sum_{s}}E^{3}_{g}F^{7}_{s}$ |  | 14 |  | $2{\displaystyle\sum_{s}}E^{3}_{g}F^{7}_{s}$
5 |  | ${\displaystyle\sum_{s}}E^{3}_{g}F^{7}_{s}$ |  | 10 |  | ${\displaystyle\sum_{s}}E^{3}_{g}F^{7}_{s}$ |  | 15 |  | $\frac{1}{4}{\displaystyle\sum_{s}}E^{3}_{g}F^{7}_{s}$
Table 2: Amplitudes of each perturbative process contributing to the dressing
of the single-particle propagator up to the order $g^{3}$ in multiples of
$k^{2}/2\pi$.

Following the line of Wegner’s reasoning, the series in the rectangular
brackets in Eq. (78) is related to the zero dimensional functional integral

$\displaystyle\frac{1}{a}\sum^{\infty}_{n=0}f_{n}\left(-\frac{b}{a^{2}}\right)^{n}$
$\displaystyle=$ $\displaystyle\frac{\displaystyle\int
d\phi^{\ast}d\phi~{}\phi^{\ast}\phi~{}e^{-a\phi^{\ast}\phi-\frac{b}{4}(\phi^{\ast}\phi)^{2}}}{\displaystyle\int
d\phi^{\ast}d\phi~{}e^{-a\phi^{\ast}\phi-\frac{b}{4}(\phi^{\ast}\phi)^{2}}},$
(79)

with $f_{n}$ representing the combinatorial factor. In order to insure the
convergence, the real part of $a$ has to be positive. The integral can be
carried out exactly by quadratic complement, giving

$\displaystyle\frac{1}{a}\sum^{\infty}_{n=0}f_{n}\left(-\frac{b}{a^{2}}\right)^{n}$
$\displaystyle=$ $\displaystyle-\frac{\partial}{\partial
a}\log\left[\frac{2\pi}{\sqrt{b}}e^{-\frac{a^{2}}{b}}\int^{\infty}_{\frac{a}{\sqrt{b}}}dt~{}e^{-t^{2}}\right].$
(80)

The integral under the logarithm is rewritten as

$\int^{\infty}_{\frac{a}{\sqrt{b}}}dt~{}e^{-t^{2}}=\frac{\sqrt{\pi}}{2}\left[1-\frac{2}{\sqrt{\pi}}\int^{\frac{a}{\sqrt{b}}}_{0}dt~{}e^{-t^{2}}\right],$
(81)

and therefore

$\displaystyle\frac{1}{a}\sum^{\infty}_{n=0}f_{n}\left(-\frac{b}{a^{2}}\right)^{n}$
$\displaystyle=$ $\displaystyle-\frac{\partial}{\partial
a}\log\left[e^{-\frac{a^{2}}{b}}\left(1-\frac{2}{\sqrt{\pi}}\int^{\frac{a}{\sqrt{b}}}_{0}dt~{}e^{-t^{2}}\right)\right]$
(82) $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{b}}\left[\frac{a}{\sqrt{b}}+\frac{1}{\sqrt{\pi}}\frac{\displaystyle
e^{-\frac{a^{2}}{b}}}{\displaystyle
1-\frac{2}{\sqrt{\pi}}\int^{\frac{a}{\sqrt{b}}}_{0}dt~{}e^{-t^{2}}}\right].$
(83)

Plugging into this expression

$a=\mp i\left[E-E_{s}\pm i0^{+}\right],\;\;\;{\rm and}\;\;\;b=E^{2}_{g},$ (84)

we get the frequency dependent part of the propagator for each spectral mode
of the lowest Landau level

${\cal
F}^{\pm}_{s}(E)=-\frac{1}{2E_{g}}\left[2\nu_{s}\pm\frac{i}{\sqrt{\pi}}\frac{e^{\nu^{2}_{s}}}{\displaystyle
1\mp i\frac{2}{\sqrt{\pi}}\int^{\nu^{2}_{s}}_{0}dt~{}e^{t^{2}}}\right],$ (85)

where $\nu_{s}=(E-E_{s})/E_{g}$. From here one obtains the real and imaginary
part of the Green’s function Eqs. (26) and (27) shown in Fig. 8. The imaginary
part, which is essentially the contribution to the net density of states from
each spectral mode in the lowest Landau level, is placed symmetrically around
the Landau zero point with the maximum at the band center.

Figure 8:  The shape of the real (left) and imaginary (right) part of the
Wegner’s propagator.

In order to Fourier transform the Wegner’s propagator with respect to the
energy we start with Eq. (78) rewritten in accord with Eq. (79)

$\displaystyle{\cal F}^{\pm}_{s}(E)$ $\displaystyle=$
$\displaystyle\mp\frac{i}{2}\frac{1}{\mp
i(E-E_{s})}\left[1-\frac{E^{2}_{g}}{[\mp
i(E-E_{s})]^{2}}+\frac{5}{2}\frac{E^{4}_{g}}{[\mp
i(E-E_{s})]^{4}}-\frac{37}{4}\frac{E^{6}_{g}}{[\mp
i(E-E_{s})]^{6}}\cdots\right].$ (86)

Then, the leading order term is easily Fourier transformed with the help of
the Cauchy theorem

$\mp\int\frac{dE}{2\pi i}\frac{e^{\pm iEt}}{E-E_{s}\pm i0^{+}}=e^{\pm
iE_{s}t},$ (87)

with the contour integration stretching over the entire complex plane, hence
necessary enveloping the pole. All higher terms follow essentially by taking
derivatives of the oscillating function on the right hand side with respect to
$E_{s}$, e.g.

$\displaystyle\int\frac{dE}{2\pi}\frac{e^{\pm iEt}}{[\mp i(E-E_{s}\pm
i0^{+})]^{2n+1}}$ $\displaystyle=$
$\displaystyle\mp\frac{(-1)^{n}}{2n!}\frac{\partial^{2n}}{\partial
E^{2n}_{s}}\int\frac{dE}{2\pi i}\frac{e^{\pm iEt}}{E-E_{s}\pm i0^{+}}$ (88)
$\displaystyle=$
$\displaystyle\frac{(-1)^{n}}{2n!}\frac{\partial^{2n}}{\partial
E^{2n}_{s}}e^{\pm iE_{s}t}=\frac{t^{2n}}{2n!}e^{\pm iE_{s}t}.$ (89)

This gives for the Fourier transformed Wegner’s function the following series

${\cal F}^{\pm}_{s}(t)=\mp\frac{i}{2}e^{\pm
iE_{s}t}\left[1-\frac{(E_{g}t)^{2}}{2!}+\frac{5}{2}\frac{(E_{g}t)^{4}}{4!}-\frac{37}{4}\frac{(E_{g}t)^{6}}{6!}\cdots\right].$
(90)

The combinatorial factors of the series in the rectangular brackets can be
determined to every order using the integral representation Eq. (79). We
therefore can write it as

${\cal F}^{\pm}_{s}(t)=\mp\frac{i}{2}e^{\pm
iE_{s}t}\Omega(E_{g}t),\;\;\;\Omega(E_{g}t)=\sum^{\infty}_{n=0}(-1)^{n}f_{n}\frac{(E_{g}t)^{2n}}{(2n)!},$
(91)

where $f_{n}$ denotes the combinatorial factors from Eq. (79).

##  Appendix D Evaluation of the mean squared displacement: Technical issues

After accounting for all processes which dress the single-particle Green’s
propagators, the disorder effect for the two-particles propagator and
correspondingly for the mean squared displacement comes from the loop
diagrams, which involve at least one scattering process between advanced and
retarded sides. All such diagrams to order $g^{3}$ are shown in Fig. 6. Every
single diagram can be evaluated exactly in a very efficient way using the
master formula Eq. (67). The results for each diagram are summarized in the
Table 3. The appearance of the trigonometric cosine functions in some of the
terms is due to the fact that the corresponding loop diagrams contain unequal
numbers of advanced/retarded propagators. Each diagram in the respective class
is then complex, but the sum of all complex conjugate diagrams is real and
proportional to the cosine. Even numbers in the argument of the cosine are
equal to the difference between the number of propagators in each side.
Because our series contain only cosine functions they might seem different in
comparison to Refs. [Hikami1984a, ; Hikami1984b, ; Chakravarty1986, ;
Hikami1993, ]. There is no discrepancy though, since these authors use
extensively the addition formulae of trigonometric functions, cf. especially
Ref. [Hikami1984b, ].

Diagrams in every order organize in distinct subgroups by the topology of the
diagrams. The sum of all terms in a subgroup is a real function. We summarize
the results for every subgroup in expansion of the second moment of the
position operator $\sum_{r}r^{2}_{\mu}P_{r,0}$ in Table 3 and for the
corresponding normalization $\sum_{r}P_{r,0}$ in Table 4. Each subgroup
provides in the $n$th order of $\sum_{r}r^{2}_{\mu}P_{r,0}$ for a contribution

${\cal
R}^{(n)}_{i}=\frac{1}{4\pi}\frac{1}{E^{2}_{g}}\sum_{s}(2X_{s})^{2n+2}\cos\left[({\cal
N}^{+}_{i}-{\cal N}^{-}_{i})\phi_{s}\right]{\cal C}^{(n)}_{2,i},$ (92)

where $X^{2}_{s}=E^{2}_{g}[\eta^{2}_{s}(E)+\rho^{2}_{s}(E)]$, $\phi_{s}={\rm
arctan}\left[\frac{\rho_{s}}{\eta_{s}}\right]$, and ${\cal N}^{\pm}_{i}$ is
the number of advanced (+) and retarder ($-$) propagators in the propagator,
obeying ${\cal N}^{+}_{i}+{\cal N}^{-}_{i}=2n$, and $i$ is the index of the
subgroup as it appears in Tables 3 and Table 4. The corresponding expression
for the normalization $\sum_{r}P_{r,0}$ reads

${\cal
Z}^{(n)}_{i}=\frac{k^{2}}{4\pi}\frac{1}{E^{2}_{g}}\sum_{s}(2X_{s})^{2n+2}\cos\left[({\cal
N}^{+}_{i}-{\cal N}^{-}_{i})\phi_{s}\right]{\cal C}^{(n)}_{0,i}.$ (93)

The factors ${\cal N}^{+}_{i}-{\cal N}^{-}_{i}$ can be read off directly from
the Fig. 6 giving ${\cal N}^{+}_{i}-{\cal N}^{-}_{i}=0$ for
$i=1,2,3,4,6,14,15,16,17$, ${\cal N}^{+}_{i}-{\cal N}^{-}_{i}=2$ for
$i=5,10,11,12,13$, and finally ${\cal N}^{+}_{i}-{\cal N}^{-}_{i}=4$ for
$i=7,8,9$.

The combinatorial factors ${\cal C}^{(n)}_{2,i}$ for each subgroup represent
rational numbers which appear as the result of integration in the position
space (all variables rescaled in units of $k$)

${\cal
C}^{(n)}_{2,i}=\int\frac{d^{2}r}{2\pi}~{}r^{2}e^{-r^{2}}\underbrace{\int\frac{d^{2}x}{\pi}\cdots\int\frac{d^{2}y}{\pi}\cdots,}_{\rm
n\,\,integrals}$ (94)

where the additional factor $1/2$ in the very first integral is due to the
angular average $\langle r^{2}_{\mu}\rangle_{\varphi}=1/2\langle
r^{2}\rangle_{\varphi}$. These integral chains are unique for each diagram.
Correspondingly, the combinatorial factors ${\cal C}^{(n)}_{0,i}$ for the
normalization are defined as

${\cal
C}^{(n)}_{0,i}=\int\frac{d^{2}r}{\pi}~{}e^{-r^{2}}\underbrace{\int\frac{d^{2}x}{\pi}\cdots\int\frac{d^{2}y}{\pi}\cdots.}_{\rm
n\,\,integrals}$ (95)

There is a simple relation between ${\cal C}^{(n)}_{2,i}$ and ${\cal
C}^{(n)}_{0,i}$

${\cal C}^{(n)}_{0,i}=2\chi_{i}{\cal C}^{(n)}_{2,i},$ (96)

where $\chi_{i}$ are taken from the Table 1. The combinatorial factors ${\cal
C}^{(n)}_{2,i}$ and ${\cal C}^{(n)}_{0,i}$ for every term are summarized in
Tables 3 and Table 4 respectively.

No. | Diagram | Amplitude |  | No. | Diagram | Amplitude |  | No. | Diagram | Amplitude
---|---|---|---|---|---|---|---|---|---|---
1 |  | $\frac{1}{2}$ |  | 6 |  | $\frac{2}{9}$ |  | 12 |  | $\frac{3}{8}$
2 |  | $1$ |  | 7 |  | $\frac{4}{9}$ |  | 13 |  | $\frac{4}{3}$
3 |  | $\frac{3}{2}$ |  | 8 |  | $\frac{3}{8}$ |  | 14 |  | $2$
4 |  | $\frac{1}{2}$ |  | 9 |  | $\frac{10}{9}$ |  | 15 |  | $\frac{2}{3}$
5 |  | $\frac{3}{4}$ |  | 10 |  | $\frac{5}{9}$ |  | 16 |  | $\frac{3}{2}$
|  |  |  | 11 |  | $\frac{5}{2}$ |  | 17 |  | $\frac{1}{4}$
Table 3: Amplitudes ${\cal C}^{(n)}_{2,i}$ of every term of $\sum_{r}r^{2}_{\mu}P_{r,0}$ up to the order $g^{3}$, $i$ is the number of the diagram. The net contribution from each diagram is acquired by inserting the elements of the Table into Eq. (92). In every order, the ladder diagrams weight substantially heavier as all other non-oscillating terms (no. 4,6,15,16,17) combined. No. | Diagram | Amplitude |  | No. | Diagram | Amplitude |  | No. | Diagram | Amplitude
---|---|---|---|---|---|---|---|---|---|---
1 |  | $1$ |  | 6 |  | $\frac{1}{3}$ |  | 12 |  | $\frac{1}{2}$
2 |  | $1$ |  | 7 |  | $\frac{2}{3}$ |  | 13 |  | $\frac{4}{3}$
3 |  | $1$ |  | 8 |  | $\frac{1}{2}$ |  | 14 |  | $1$
4 |  | $\frac{1}{2}$ |  | 9 |  | $\frac{4}{3}$ |  | 15 |  | $\frac{2}{3}$
5 |  | 1 |  | 10 |  | $\frac{2}{3}$ |  | 16 |  | $1$
|  |  |  | 11 |  | $2$ |  | 17 |  | $\frac{1}{4}$
Table 4: Amplitudes ${\cal C}^{(n)}_{0,i}$ of every term of $\sum_{r}P_{r,0}$
up to the order $g^{3}$, $i$ is the number of the diagram. The net
contribution for the respective expansion order from each diagrams is acquired
by inserting the elements of the Table into Eq. (93).

The inspection of the Gaussian exponents in Table 1 suggests that the dominant
contribution on large scales in every order of expansion arises from the so-
called ladder diagrams, i.e. diagrams no. 1,2,3, 14 etc. According to Table 3
they provide more weight in every order of expansion of
$\sum_{r}r^{2}_{\mu}P_{r0}$ than all other non-oscillating diagrams (without
cosine) combined. In particular the so-called fan diagrams, e.g. no. 4, 17,
etc. contribute the least in every order of expansion. This is the ultimate
reason for neglecting the contributions from the latter. Four lowest order
ladder diagrams are

$\displaystyle\includegraphics[height=8.53581pt]{D0.eps}=\frac{1}{4E^{2}_{g}}\left(\frac{k^{2}}{\pi}\right)^{2}\sum_{s=\pm}(2X_{s})^{2}\exp\left[-k^{2}r^{2}\right],$
$\displaystyle\includegraphics[height=8.53581pt]{D1.eps}=\frac{1}{4E^{2}_{g}}\left(\frac{k^{2}}{\pi}\right)^{2}\sum_{s=\pm}~{}\frac{(2X_{s})^{4}}{2}\exp\left[-\frac{k^{2}r^{2}}{2}\right],$
(97)
$\displaystyle\includegraphics[height=8.53581pt]{D2t1.eps}=\frac{1}{4E^{2}_{g}}\left(\frac{k^{2}}{\pi}\right)^{2}\sum_{s=\pm}~{}\frac{(2X_{s})^{6}}{3}\exp\left[-\frac{k^{2}r^{2}}{3}\right],$
$\displaystyle\includegraphics[height=8.53581pt]{D3t9.eps}=\frac{1}{4E^{2}_{g}}\left(\frac{k^{2}}{\pi}\right)^{2}\sum_{s=\pm}~{}\frac{(2X_{s})^{8}}{4}\exp\left[-\frac{k^{2}r^{2}}{4}\right],$
(98)

which suggests the following approximate expression for the two-particles
propagator:

$P^{\rm
lad}_{r0}(E)\approx\frac{1}{4E^{2}_{g}}\left(\frac{k^{2}}{\pi}\right)^{2}\sum_{s=\pm}~{}\sum^{\infty}_{n=0}\frac{(2X_{s})^{2n+2}}{n+1}\exp\left[-\frac{k^{2}r^{2}}{n+1}\right].$
(99)

The dominant oscillating correction comprises the menora diagrams no. 5, 7
etc., Table 1. The contribution from the diagrams 5 to the two-particles
propagator reads

$\includegraphics[height=8.53581pt]{D2t3.eps}=\frac{1}{4E^{2}_{g}}\left(\frac{k^{2}}{\pi}\right)^{2}\sum_{s=\pm}~{}\frac{2}{3}\exp\left[-\frac{2}{3}k^{2}r^{2}\right]\left(2X_{s}\right)^{6}\cos
2\phi_{s},$ (100)

from the diagrams 7

$\includegraphics[height=8.53581pt]{D3t2.eps}=\frac{1}{4E^{2}_{g}}\left(\frac{k^{2}}{\pi}\right)^{2}\sum_{s=\pm}~{}\frac{2}{4}\exp\left[-\frac{3}{4}k^{2}r^{2}\right]\left(2X_{s}\right)^{8}\cos
4\phi_{s},$ (101)

and so on. The full series accounting for the contribution from the menora
channel then becomes

$P^{\rm
men}_{r0}(E)=\frac{2}{4E^{2}_{g}}\left(\frac{k^{2}}{\pi}\right)^{2}\sum_{s=\pm}\sum^{\infty}_{n=2}\frac{(2X_{s})^{2n+2}}{n+1}\cos[(2n-2)\phi_{s}]\exp\left[-\frac{n}{n+1}k^{2}r^{2}\right].$
(102)

Both ladder and menora channels combined represent a reasonable exact large
scale asymptotics for the dressed two-particles propagator.

##  Appendix E Details of the derivation of Eq. (50)

We begin with Eq. (49) and use the fluctuations $\langle\delta f^{(i)}\rangle$
as the generators of the time evolution of the mean squared displacement at
small positive/negative times $\Delta t$:

$k^{2}\langle r^{2}_{\mu}(\pm\Delta t)\rangle\approx\frac{\displaystyle\langle
f^{(2)}\rangle+\sum_{s=\pm}\int^{\infty}_{-\infty}d\nu_{s}\left[e^{\pm
i\nu_{s}E_{g}\Delta t}-1\right]\frac{N}{2}(2X_{s})^{2N}}{\displaystyle\langle
f^{(0)}\rangle+\sum_{s=\pm}\int^{\infty}_{-\infty}d\nu_{s}\left[e^{\pm
i\nu_{s}E_{g}\Delta t}-1\right](2X_{s})^{2N}},$ (103)

where we carry out the Fourier transform in the energy units specific for the
time representation of the Wegner’s propagator Eq. (30). Obviously, the case
$\Delta t=0$ restores Eq. (103). For small $(e^{\pm i\nu_{s}E_{g}\Delta t}-1)$
we have

$\displaystyle k^{2}[\langle r^{2}_{\mu}(\pm\Delta t)\rangle-\langle
r^{2}_{\mu}(0)\rangle]\approx\frac{1}{\langle
f^{(0)}\rangle}\sum_{s=\pm}\int^{\infty}_{-\infty}d\nu_{s}\left[e^{\pm
i\nu_{s}E_{g}\Delta t}-1\right](2X_{s})^{2N}\left[\frac{N}{2}-k^{2}\langle
r^{2}_{\mu}(0)\rangle\right].$ (104)

Summing both sides yields

$\displaystyle k^{2}[\langle r^{2}_{\mu}(\Delta t)\rangle-2\langle
r^{2}_{\mu}(0)\rangle+\langle r^{2}_{\mu}(-\Delta t)\rangle]$
$\displaystyle\approx$ $\displaystyle\frac{2}{\langle
f^{(0)}\rangle}\sum_{s=\pm}\int^{\infty}_{-\infty}d\nu_{s}\left[\cos(\nu_{s}E_{g}\Delta
t)-1\right]$ (105) $\displaystyle\times$
$\displaystyle(2X_{s})^{2N}\left[\frac{N}{2}-k^{2}\langle
r^{2}_{\mu}(0)\rangle\right].$ (106)

Taking the limit $\Delta t\to 0$ and using the self-consistent approximation
yields Eq. (50).

## References

  * (1) A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009).
  * (2) M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  * (3) V.N. Kotov, B. Uchoa, V.M. Pereira, F. Guinea, and A.H. Castro Neto, Electron-Electron Interactions in Graphene: Current Status and Perspectives, Rev. Mod. Phys. 84, 1067 (2012).
  * (4) X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  * (5) D. Culcer, A.C. Keser, Y. Li and G. Tkachov, Transport in two-dimensional topological materials: recent developments in experiment and theory, 2D Materials, 7, 022007 (2020).
  * (6) A. Avsar, H. Ochoa, F. Guinea, B. Özyilmaz, B.J. van Wees, and I.J. Vera-Marun, Colloquium: Spintronics in graphene and other two-dimensional materials, Rev. Mod. Phys. 92, 021003 (2020).
  * (7) M. König, H. Buhmann, L.W. Molenkamp, T. Hughes, C.-X. Liu, X.-L. Qi, and S.-C. Zhang, The quantum spin Hall effect: Theory and experiment, J. Phys. Soc. Jpn. 77, 031007 (2008).
  * (8) T. Ando, Theory of quantum transport in a two-dimensional electron system under magnetic field. III. Many-site approximation, J. Phys. Soc. Jpn. 37, 622 (1974).
  * (9) F.J. Wegner, Exact density of states for lowest Landau level in white noise potential. Superfield representation for interacting systems, Z. Phys. B - Cond. Mat. 51, 279 (1983).
  * (10) E. Brézin, D.J. Gross, and C. Itzykson, Density of states in the presence of a strong magnetic field and random impurities, Nucl. Phys. B 235, 24 (1984).
  * (11) S. Hikami, Borel-Padé analysis for the two-dimensional electron in a random potential under a strong magnetic field, Phys. Rev. B 29, 3726(R) (1984).
  * (12) S. Hikami, Anderson Localization of the two-dimensional electron in a random potential under a strong magnetic field, Prog. Theor. Phys 72, 722 (1984).
  * (13) H. Aoki, Quantised Hall effect, Rep. Prog. Phys. 50 655 (1987).
  * (14) E. Abrahams, P.W. Anderson, D.C. Licciardello, and T.V. Ramakrishnan, Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions, Phys. Rev. Lett. 42, 673 (1979).
  * (15) L.G. Gor’kov, A.I. Larkin, and D.E. Khmel’nitskii, Particle conductivity in a two-dimensional random potential, Pis’ma Zh. Eksp. Teor. Fiz. 30, 248 (1979) [JETP Lett. 30, 228 (1979)].
  * (16) S. Hikami, A. Larkin, and Y. Nagaoka, Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System, Prog. Theor. Phys. 63, 707 (1980).
  * (17) F. Wegner, The mobility edge problem: Continuous symmetry and a conjecture, Z. Physik B 35, 207 (1979).
  * (18) L. Schäfer and F. Wegner, Disordered system withn orbitals per site: Lagrange formulation, hyperbolic symmetry, and Goldstone modes, Z. Physik B 38, 113 (1980).
  * (19) S. Hikami, Anderson localization in a nonlinear-$\sigma$-model representation, Phys. Rev. B 24, 2671 (1981).
  * (20) G. Tkachov, Topological insulators: The physics of spin helicity in quantum transport, Pan Stanford, (2015).
  * (21) A. Sinner and K. Ziegler, Two-parameter scaling theory of transport near a spectral node, Phys. Rev. B 90, 174207 (2014).
  * (22) A. Sinner and K. Ziegler, Finite-size scaling in a 2D disordered electron gas with spectral nodes, J. Phys.: Cond. Mat. 28, 305701 (2016).
  * (23) N. Goldenfeld, Lectures on phase transitions and the renormalization group, Perseus books, Reading, Massachusetts (1992).
  * (24) K. Huang, Statistical Mechanics, Second edition, John Wiley, New York (1987).
  * (25) P.M. Chaikin and T.C. Lubenski, Principles of condensed matter physics, Cambridge University Press, Cambridge UK (1995).
  * (26) K. Ziegler, Quantum diffusion in two-dimensional random systems with particle–hole symmetry, J. Phys. A: Math. Theor. 45, 335001 (2012).
  * (27) R.R.P. Singh and S. Chakravarty, A disordered two-dimensional system in a magnetic field: Borel-Padé analysis, Nucl. Phys. B 265, 265 (1986).
  * (28) S. Hikami, M. Shirai, and F.J. Wegner, Anderson localization in the lowest Landau level for a two-subband model, Nucl. Phys. B 408, 415 (1993).
  * (29) B.A. Bernevig, T.L. Hughes, S.-C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314, 1757 (2006).
  * (30) G. Li and E.Y. Andrei, Observation of Landau levels of Dirac fermions in graphite, Nat. Phys. 3, 623 (2007).
  * (31) K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, and A.A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature 438, 197 (2005).
  * (32) Z. Jianga, Y. Zhanga, Y.-W. Tana, H.L. Stormer, and P. Kim, Quantum Hall effect in graphene, Sol. State Com. 143, 14 (2007).
  * (33) P. Goswami, X. Jia, and S. Chakravarty, Quantum Hall plateau transition in the lowest Landau level of disordered graphene, Phys. Rev. B 76, 205408 (2007).
  * (34) F.J. Wegner, Disordered system with n orbitals per site: $n=\infty$ limit, Phys. Rev. B 19, 783 (1979).
  * (35) Y. Huo, R. E. Hetzel, and R. N. Bhatt Universal conductance in the lowest Landau level, Phys. Rev. Lett. 70, 481 (1993).
  * (36) A.J. McKane and M. Stone, Localization as an alternative to Goldstone’s theorem, Ann. Phys. 131, 36 (1981).
  * (37) A.W.W. Ludwig, M.P.A. Fisher, R. Shankar, and G. Grinstein, Integer quantum Hall transition: An alternative approach and exact results, Phys. Rev. B 50, 7526 (1994).
  * (38) B.L. Altshuler, B.D. Simons, Universalities: From Anderson localization to quantum chaos. In Mesoscopic quantum physics, Les Houches 1994, Akkermans, E.; Montambaux, G.; Pichard, J.-L.; Zinn-Justin, J. North Holland, Amsterdam, Netherlands, pp. 1-98, (1995).
  * (39) K. Efetov, Supersymmetry in disorder and chaos, Cambridge University Press, Cambridge UK, (1997).
  * (40) E. Fradkin, Critical behavior of disordered degenerate semiconductors. II. Spectrum and transport properties in mean-field theory, Phys. Rev. B 33, 3263 (1986).
  * (41) V.P. Sharapov and S.G. Gusynin, Transport of Dirac quasiparticles in graphene: Hall and optical conductivities, Phys. Rev. B 73, 245411 (2006).
  * (42) K. Ziegler, Robust transport properties in graphene, Phys. Rev. Lett. 97, 266802 (2006).
  * (43) K. Ziegler, Minimal conductivity of graphene: Nonuniversal values from the Kubo formula, Phys. Rev. B 75, 233407 (2007).
  * (44) A. Sinner and K. Ziegler, Conductivity of disordered 2d binodal Dirac electron gas: Effect of internode scattering, Phil. Mag. 98, 1799 (2018).