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# On the sample-dependent minimal conductivity in weakly disordered graphene

Weiwei Chen  Institute of Natural Sciences, Westlake Institute for Advanced
Study, 18 Shilongshan Road, Hangzhou 310024, China  School of Science,
Westlake University, 18 Shilongshan Road, Hangzhou 310024, China  Yedi Shen
Hefei National Laboratory for Physical Sciences at the Microscale, and
Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China  Bo
Fu  Department of Physics, The University of Hong Kong, Pokfulam Road, Hong
Kong, China  Qinwei Shi  Hefei National Laboratory for Physical Sciences at
the Microscale, and Synergetic Innovation Center of Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei, Anhui
230026, China  W. Zhu  Key Laboratory for Quantum Materials of Zhejiang
Province, Department of Physics, School of Science, Westlake University,
Hangzhou 310024, China

###### Abstract

We present a unified understanding of the experimentally observed minimal dc
conductivity in weakly disordered graphene. Firstly, based on linear response
theory, we reveal that randomness or disorder inevitably induces momentum
dependent corrections to the electron self-energy function, which naturally
yields a sample-dependent minimal conductivity. Taking the long-ranged
Gaussian and Coulomb potentials as examples, we derive the momentum dependent
self-energy function within the Born approximation, and further validate it
via numerical simulations using the large-scale Lanczos algorithm. The
explicit momentum dependences of the self-energy on the intensity,
concentration and range of potential are critically addressed. Therefore, our
results provide a reasonable interpretation of the sample-dependent minimal
conductivity observed in graphene samples.

Introduction
The earliest experimental measurements on graphene reported a finite dc
conductivity $\sigma_{\text{min}}\approx\frac{4e^{2}}{h}$ at the Dirac point
at low temperature Novoselov ; Zhang2005 ; Novoselov2005 , dubbed as the
minimal conductivity. The observed value is larger than the earlier prediction
of $\sigma_{\text{min}}=\frac{4e^{2}}{\pi h}$ Shon ; Ludwig ; Nersesyan ,
giving rise to the famous “missing $\pi$” problem Sarma2011 . To address this
problem, subsequent experiments have been carried out Bolotin ; Du ; Zhang2005
; Morozov ; Dean ; Zomer ; Mayorov2012 ; Nuno ; WangLujun ; Tan , which show
that the minimal conductivity is strongly sample-dependent, i.e.
$\sigma_{\text{min}}=C\frac{4e^{2}}{\pi h}$ with the factor $C$ varying from
1.7 to 10, indicating the crucial role of randomness or disorder.

The transport properties of two-dimensional disordered Dirac fermions have
been intensively studied for the d-wave superconductivity in the cuprate
superconductors Patrick ; Durst  and the plateau transition in the integer
quantum Hall effect Ludwig . The discovery of graphene Novoselov ; Zhang2005 ;
Novoselov2005  rejuvenated this problem. The previous theoretical results for
$\sigma_{\text{min}}$ can be mainly summarized as: (i) a scattering-
independent value $\sigma_{\text{min}}=\frac{4e^{2}}{\pi h}$ Peres2006 ; Shon
; Ludwig ; Katsnelson2006 ; Nersesyan ; Tworzydlo ; Ostrovsky ; Fedorenko ,
which is independent of the strength and nature of the disorder; (ii) a
universal value $\sigma_{\text{min}}=\frac{4e^{2}}{h}$, which is due to the
quantum criticality of graphene in the vicinity of the Dirac point
Ostrovsky2007 ; Ostrovsky2007prl ; Schuessler ; (iii) a universal value
$\sigma_{\text{min}}=\frac{\pi e^{2}}{2h}$ Ziegler ; Ryu , from the ac Kubo
formula, derived by removing the smearing of the single-particle Green’s
functions before taking the dc limit; (iv) a disorder-dependent
$\sigma_{\text{min}}$ Shon ; Trushin ; Adam2009 ; Sarma2011 ; Nomura ; Noro ;
Radchenko ; Bardarson ; Rycerz . In particular, although some previous studies
have yielded a disordered-dependent minimal conductivity, most of these are
based on semi-classical Boltzmann transport theory Shon ; Trushin ; Adam2009 ;
Sarma2011 , ignoring the disorder induced quantum corrections near the Dirac
point. Since Boltzmann theory is not applicable around the Dirac point
Sarma2011 , a fully quantum mechanical treatment is highly desirable in order
to address the minimal conductivity.

In parallel, numerical calculations have also addressed this problem using
various approaches Nomura ; Noro ; Radchenko ; Bardarson ; Rycerz . Nomura and
MacDonald numerically calculated $\sigma_{\text{min}}$ using the Kubo formula
and found that $\sigma_{\text{min}}$ is a few times larger for long-range
Coulomb scatterers than for short-range scatterers Nomura . Similar results
were also obtained by numerically calculating the transmission matrix
Bardarson ; Rycerz . Noro et al. find that the minimal conductivity at the
Dirac point remains universal in the clean limit, but increases with disorder
and becomes non-universal for long-range scatterers, by numerically solving
the self-energy and current vertex function Noro . These numerical results,
which are beyond the aforementioned theoretical descriptions, urgently call
for a reasonable analytical interpretation of the minimal conductivity.

In this paper, we offer a general picture for understanding the sample-
dependent minimal conductivity by studying the effect of long-ranged random
potentials. The long-ranged randomness could be realized by screened charges
in the substrate Rycerz ; Fan ; Chen2008 , local strain fluctuations Nuno ;
WangLujun , and other defects that vary smoothly on the atomic scale Tan ;
Klos . Recently, it has been confirmed that random strain fluctuations are the
dominant source of disorder for high-quality graphene on many different
substrates, from spatially resolved Raman spectroscopy measurements, where the
out-of-plane corrugation (stain) could induce a long-ranged scalar potential
Nuno ; WangLujun . We mainly consider the long-ranged Gaussian potential,
which is a simple model of atomic-scale random fluctuations, and the Gaussian
smoothing is chosen for computational convenience. We also checked that the
results are robust against the form of the random potentials by considering
the long-range Coulomb potential (shown in Supplemental Material Sec. S2 C).
In the presence of long-ranged random potentials, we elucidate that the self-
energy is momentum-dependent, which is overlooked in the existing literature.
Crucially, the electric minimal conductivity, evaluated via the standard Kubo
formula under disorder configuration average by taking into account the
corrected self-energy, obeys the relationship
$\sigma_{\text{min}}=\frac{1}{(1-\alpha)^{2}}\frac{4e^{2}}{\pi h}$ [Eq.(11)],
where the leading momentum-dependent correction is parameterized by a
dimensionless parameter $\alpha=K_{0}\frac{\xi^{2}k_{c}^{2}}{8\pi}$ [Eq.(19)],
which depends on the disorder strength and the spatial range of the potential
(see below for details). This finding, in sharp contrast to the disorder-
independent values in previous studies Nersesyan ; Ludwig ; Shon ; Peres2006 ;
Ostrovsky ; Katsnelson2006 ; Tworzydlo ; Fedorenko ; Ziegler , provides a
simple and physically appealing explanation for the observed minimum
conductivity in experiments.

Minimal conductivity with momentum dependent self-energy

The charge carriers of graphene near half filling can be modeled by a Dirac
Hamiltonian

$H=\hbar v_{f}\bm{\sigma}\cdot\bm{k}+U(\bm{r}),$ (1)

where $v_{f}\approx 10^{6}$ $m\ s^{-1}$ is the Fermi velocity,
$\bm{\sigma}=(\sigma_{x},\sigma_{y})$ are the Pauli matrices of pseudospin
(sublattice), $\bm{k}=(k_{x},k_{y})$ is a two-component wave vector, and
$U(\bm{r})$ describes the long-ranged random potential that is experienced by
the Dirac electrons. In the pure case, the eigenvalues and eigenstates of the
Hamiltonian Eq. (1) are

$E_{\bm{k}s}=s\hbar v_{f}k;\ \ \
\Psi_{\bm{k}s}(\bm{r})=\langle\bm{r}|\bm{k}s\rangle=\frac{e^{i\bm{k}\cdot\bm{r}}}{\sqrt{2\mathcal{V}}}\left(\begin{array}[]{c}1\\\
se^{i\theta_{\bm{k}}}\end{array}\right)$ (2)

where $s=\pm$ denotes the band index ($s=+1$ for the conduction band and
$s=-1$ for the valence band), $\mathcal{V}$ is the sample area, and
$\theta_{\bm{k}}=\arctan(k_{y}/k_{x})$ is the angle of the wave vector
$\bm{k}$ down from the positive $x$-axis. The velocity
$\frac{1}{\hbar}\frac{\partial H}{\partial k_{x}}$ along the x direction in
the eigenstates basis is

$v_{x}(\bm{k})=v_{f}(\cos\theta_{\bm{k}}\sigma_{z}+\sin\theta_{\bm{k}}\sigma_{y}).$
(3)

For simplicity, we model the random potential by
$U(\bm{r})=\sum_{i=1}^{N_{\text{imp}}}\pm
u_{0}e^{-|\bm{r}-\bm{R}_{i}|^{2}/\xi^{2}}$, where $\bm{R}_{i}$ and
$N_{\text{imp}}$ are the location and total number of impurities, $\xi$ is the
characteristic length scale and the strength $\pm u_{0}$ is randomly
distributed with equal probability. We also checked that the results shown
below are not sensitive to the form of the potentials. To do so, we checked
the long-range screened Coulomb potential and obtained similar results (shown
in Supplemental Material Sec. S2 C).

In general, the electron self-energy induced by the impurity scattering is
both momentum- and energy-dependent. For the short-ranged disorder potential,
the momentum dependence of the self-energy is negligible Shon ; Ludwig ;
Nersesyan ; Sarma2011 . For the long-range disorder potential, the self-energy
is usually treated in the on-shell approximation ($E=E_{\bm{k}s}$) Adam , i.e.
the momentum dependence of the self-energy is replaced by the single-particle
energy in the clean limit. One of motivations for this paper is to elucidate
that this on-shell approximation is not well justified around the Dirac point,
leading to a momentum-dependent self-energy function (We present both
analytical and numerical evidence to support this in the sections below).
Here, we first reveal the influence of the momentum-dependent self-energy
function on the minimal conductivity. Accordingly, for the discussion, we can
assume that the self-energy function contains a nonzero momentum-dependent
contribution,

$\Sigma(\bm{k}s,E)\approx\Sigma_{1}(E)-\alpha s\hbar v_{f}k+i\Sigma_{2}(E)$
(4)

where $s=\pm$ is the band index, while $\Sigma_{1}(E)$ and $\Sigma_{2}(E)$
describe the real and imaginary parts of the energy-dependent terms. Since the
long-ranged random scalar disorder we considered here does not on average
break the particle-hole symmetry Altland1997 ; Ostrovsky ; Fradkin ; Aleiner ,
the self-energy in the eigenstate basis satisfies the relations, ${\rm
Re}\Sigma(\bm{k}+,E)=-{\rm Re}\Sigma(\bm{k}-,-E)$ and ${\rm
Im}\Sigma(\bm{k}+,E)={\rm Im}\Sigma(\bm{k}-,-E)$, which preserves the
dispersion relation and broadens the width of the conduction and valence bands
symmetrically about the Dirac point Hu .

Now we study the influence of this momentum dependent self-energy function on
the minimal conductivity. Based on linear response theory, we can calculate
the longitudinal conductivity at zero temperature using the Kubo formula Bruus
(see Supplemental Material Sec. S1)

$\displaystyle\sigma_{xx}(E)=\sigma_{xx}^{RA}(E)-{\rm
Re}\left[\sigma_{xx}^{RR}(E)\right]$ (5)

with

$\displaystyle\sigma_{xx}^{RA}(E)$ $\displaystyle=$ $\displaystyle
4\frac{e^{2}\hbar}{2\pi\mathcal{V}}{\rm
Tr}\left[G^{R}(E)v_{x}G^{A}(E)v_{x}\right]_{c}$ (6)
$\displaystyle\sigma_{xx}^{RR}(E)$ $\displaystyle=$ $\displaystyle
4\frac{e^{2}\hbar}{2\pi\mathcal{V}}{\rm
Tr}\left[G^{R}(E)v_{x}G^{R}(E)v_{x}\right]_{c}$ (7)

where the factor 4 denotes the degeneracy of the real spin and valley, ${\rm
Tr}[\cdots]$ means the trace over both wave vector and pseudospin (sublattice)
spaces, and the subscript $c$ indicates a disorder configuration average. Here
the RA contribution is analogous to the transport of classical particles
within the relaxation time approximation and gives results close to
conventional Boltzmann theory, while the RR contribution comes from
calculations with a fully quantum mechanism and can be ignored in the limit
$E\tau\gg 1$ (but not when $E\tau\ll 1$). $G^{R/A}(E)$ is the full
retarded/advanced Green’s function. In the eigenstate basis, it is

$G^{R/A}(\bm{k},E)=\left(\begin{array}[]{cc}g^{R/A}_{+}(\bm{k},E)&0\\\
0&g^{R/A}_{-}(\bm{k},E)\end{array}\right)$ (8)

where

$g^{R/A}_{s}(\bm{k},E)=\frac{1}{a-(1-\alpha)s\hbar v_{f}k\pm
i\eta},\,\,\,\,\,\left\\{\begin{array}[]{l}a=E-\Sigma_{1}(E)\\\
\eta=-\Sigma_{2}(E)\\\ \end{array}\right.$ (9)

Plugging Eq. (3), Eq. (8) and Eq. (9) into the Kubo formula Eq. (5), we obtain
the conductivity

$\sigma_{xx}(E)=\frac{2e^{2}}{\pi
h}\frac{1}{(1-\alpha)^{2}}(1+\frac{a}{\eta}\arctan\frac{a}{\eta})+\frac{2e^{2}}{\pi
h}\frac{1}{(1-\alpha)^{2}}\frac{\eta}{a}\arctan\frac{a}{\eta}$ (10)

where the first and second terms denote the electron-hole incoherent and
coherent contributions, respectively. At the Dirac point ($E=0$), these two
parts contribute equally and the minimal conductivity is obtained as

$\sigma_{\text{min}}=\frac{1}{(1-\alpha)^{2}}\frac{4e^{2}}{\pi h}.$ (11)

Here we would like to give some remarks. First, Eq. (11) contains both
electron-hole coherent and incoherent contributions, which is beyond
traditional Boltzmann transport theory where the electron-hole incoherent
contribution from the retarded-retarded ($RR$) channel is usually discarded
Trushin . Second, the influence of the momentum-dependent self-energy on the
vertex correction is much smaller than the level of the bubble diagram (see
Supplemental Materials Sec. S5). This is consistent with previous works
showing that the vertex correction is negligible for calculations of the
minimal conductivity Durst ; Shon . Third, from the result Eq. (11), it is
clear that the minimal conductivity $\sigma_{\text{min}}$ is independent of
the energy-dependent part of the self-energy function, i.e. $\Sigma_{1}(E)$
and $\Sigma_{2}(E)$, if they satisfy the condition, $\frac{a}{\eta}\to 0$ when
$E\to 0$, but $\sigma_{\text{min}}$ is significantly influenced by the
momentum dependent term. This behavior explains why the majority of previous
theoretical studies predicted $\sigma_{\text{min}}=\frac{4e^{2}}{\pi h}$,
independent of the strength and nature of the disorder Ludwig ; Nersesyan ;
Ziegler . Fourth, for long range disorder, previous calculations within the
self-consistent Born approximation reported the momentum contribution to the
self-energy function, however, the role of this term on the transport was
unfortunately washed out in the widely-used on-shell approximation Ostrovsky .
Therefore, we claim that the application of the on-shell approximation around
the Dirac point is questionable. Finally, with all of these non-trivial
improvements, we point out that Eq. (11) shows that the minimal conductivity
is enhanced by disorder, in contrast to the results from Boltzmann theory Shon
; Adam2009 ; Trushin . One interpretation is that the presence of potential
fluctuations smooth on the scale of the graphene lattice spacing increases the
conductivity through quantum interference effects Adam ; Nomura ; Noro ;
Radchenko ; Bardarson ; Rycerz .

Estimation of $\alpha$ by the Born approximation
In order to theoretically incorporate insights gained from the momentum
dependent self-energy, we take the long-ranged Gaussian potential as an
example, with the correlation function as

$\mathcal{K}(\bm{r}-\bm{r}^{\prime})=\langle
U(\bm{r})U(\bm{r}^{\prime})\rangle=K_{0}\frac{(\hbar
v_{f})^{2}}{2\pi\xi^{2}}e^{-|\bm{r}-\bm{r}^{\prime}|^{2}/2\xi^{2}}$ (12)

where $\xi$ is the correlation length and $K_{0}$ is a dimensionless parameter
that parametrizes its magnitude. Its Fourier transformation form is

$\displaystyle\mathcal{K}(\bm{q})=$ $\displaystyle\int
d\bm{r}\mathcal{K}(\bm{r})e^{-i\bm{q}\cdot\bm{r}}=K_{0}(\hbar
v_{f})^{2}e^{-\frac{\xi^{2}\bm{q}^{2}}{2}}.$ (13)

Based on the Born approximation, the self-energy is

$\Sigma^{B}(\bm{k}s,E)=\sum_{s^{\prime}}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\mathcal{K}(\bm{k}-\bm{k}^{\prime})G^{R}_{0}(\bm{k}^{\prime}s^{\prime},E)\frac{1+ss^{\prime}\cos\theta}{2}$
(14)

where $G^{R}_{0}(\bm{k}^{\prime}s^{\prime},E)=(E-s^{\prime}\hbar
v_{f}k^{\prime}+i0^{+})^{-1}$ is the Green’s function of the clean system in
the eigenstate basis, $\theta=\theta_{\bm{k}}-\theta_{\bm{k}^{\prime}}$ is the
angle between the wave vectors $\bm{k}$ and $\bm{k}^{\prime}$. In the
following, we separate the calculation of the self-energy function into real
and imaginary parts, by using the Sokhotsky’s formula,
$\frac{1}{x+i0^{+}}=\mathcal{P}\frac{1}{x}-i\pi\delta(x)$, where $\mathcal{P}$
denotes the Cauchy principal value.

The imaginary part of the self-energy can be solved as

${\rm
Im}\Sigma^{B}(\bm{k}s,E)=-\frac{K_{0}}{4}|E|e^{-\frac{\xi^{2}}{2}\left(k^{2}+\frac{E^{2}}{\hbar^{2}v^{2}_{f}}\right)}\left[I_{0}(\frac{\xi^{2}kE}{\hbar
v_{f}})+sI_{1}(\frac{\xi^{2}kE}{\hbar v_{f}})\right]$ (15)

where $I_{0}(x)$ and $I_{1}(x)$ are the zero and first order modified Bessel
functions of the first kind Gradshteyn ; Abramowitz . Meanwhile, the real part
of the self-energy is obtained as

$\displaystyle{\rm Re}\Sigma^{B}(\bm{k}s,E)=$ $\displaystyle\frac{K_{0}(\hbar
v_{f})^{2}}{4\pi}\mathcal{P}\int^{k_{c}}_{-k_{c}}dk^{\prime}k^{\prime}e^{-\frac{\xi^{2}}{2}(k^{2}+k^{\prime
2})}\frac{I_{0}(\xi^{2}kk^{\prime})+sI_{1}(\xi^{2}kk^{\prime})}{E-\hbar
v_{f}k^{\prime}}.$ (16)

where $k_{c}$ is the ultraviolet momentum cutoff. Focusing on the Dirac
physics ($\frac{|E|}{\hbar v_{f}},k\ll\frac{1}{\xi}$), we obtain the self-
energy in powers of $k$ and $E$,

$\displaystyle{\rm Im}\Sigma^{B}(\bm{k}s,E)$ $\displaystyle=$
$\displaystyle-\frac{K_{0}}{4}|E|+\cdots$ (17) $\displaystyle{\rm
Re}\Sigma^{B}(\bm{k}s,E)$ $\displaystyle=$
$\displaystyle\frac{K_{0}}{2\pi}E\ln|\frac{E}{\hbar v_{f}k_{c}}|+\alpha
E-\alpha s\hbar v_{f}k+\cdots$ (18)

where $\cdots$ stands for terms in the order of $O(kE^{2},k^{2}E,E^{3})$ in
imaginary part and $O(kE^{2}\ln E,k^{2}E\ln E,E^{3}\ln E)$ in the real part.
From this result, a momentum dependent term appears in $Re\Sigma^{B}$,
controlled by a dimensionless coefficient

$\alpha=K_{0}\frac{\xi^{2}k^{2}_{c}}{8\pi}.$ (19)

Note that, if the on-shell approximation ($E=E_{\bm{k}s}=s\hbar v_{f}k$) is
allowed, the momentum dependent correction in Eq. (48) vanishes, recovering
the previous results Sarma2011 ; Shon ; Adam .

Here we stress that in the literature, the Born approximation is widely
applied together with the on-shell approximation, so that the momentum
dependence of the self-energy is ignored Trushin ; Noro ; Adam . Our findings
show that in the low energy regime the on-shell approximation is not well
justified, i.e. the momentum and energy dependence in the self-energy are
decoupled as shown in Eqs. (47) and (48). Moreover, we find that the momentum
dependent terms are less important in the imaginary part of self-energy than
those in the real part, since in the latter they are at least two orders
larger than the leading term in the imaginary part. Furthermore, our results
can recover the predictions in the literature made using the on-shell
approximation in the regime far away from the Dirac point Adam ; McCann ;
Nersesyan , where the imaginary part of the self-energy is a constant
independent of energy and momentum, and the real part of self-energy
disappears (details are shown in Supplemental Material Sec. S2 A). Third, it
should be noted that the Born approximation is essentially valid under weak
disorder conditions ($K_{0}\lesssim 1$) since it corresponds to the first
order of the self-energy perturbation expansion. In this respect, one may
wonder how the higher order expansions influence this calculation. We address
this by considering a self-consistent Born approximation, and we have also
confirmed a momentum dependent self-energy function, when the on-shell
approximation is abandoned (see Supplemental Material Sec. S2 B). Finally, to
further improve these perturbation calculations, we perform a renormalization
group analysis on our model (Eq. 1), and we confirm the momentum-dependent
self-energy function cannot be neglected near the Dirac point (supplementary
materials Sec. S3). Taken as a whole, different methods including the
perturbation theory based on the (self-consistent) Born’s approximation and
renormalization group calculation, all reach the same conclusion, which
strongly suggests our theoretical findings are robust and universal.

Figure 1: (a-b) Numerical results of $E$ vs ${\rm Re}\Sigma$ and $E$ vs ${\rm
Im}\Sigma$ with different momenta $k$ and band index $s$. (c-d) Numerical
results of $k$ vs ${\rm Re}\Sigma$ and $k$ vs ${\rm Im}\Sigma$ with different
Fermi energies $E$ and band index $s$. Here we set an impurity concentration
$n_{\text{imp}}=5\%$, correlation length $\xi=3.6a$, and impurity strength
$u_{0}=0.16t$. Plots of $E$ vs $\Sigma$ ($k$ vs $\Sigma$) with more momentum
values (energy values) are shown in the Supplemental Material Sec. S3.

Numerical Simulations
Next we turn to numerical simulations using the Lanczos recursive method in
momentum space Chen2020 . The Green’s function $g(\bm{k}s,E)$ of the
disordered system in the eigenbasis is directly calculated using the Lanczos
recursive method, and the self-energy is obtained by the Dyson equation
$\Sigma(\bm{k}s,E)=g_{0}^{-1}(\bm{k}s,E)-g^{-1}(\bm{k}s,E)$. In order to reach
a high-energy resolution and reduce the finite-size errors, we consider a
large supercell with $N=2400\times 2400\times 2$ atoms. The data shown below
are averaged over $50$ random configurations. In the discrete lattice model,
the dimensionless parameter $K_{0}$ in the correlation function Eq. (12) has
different expressions for sharp ($\xi\ll a$) and smooth ($\xi\gg a$)
potentials Noro ; Fan ; Klos ; Rycerz

$K_{0}=\frac{n_{\text{imp}}u^{2}_{0}}{(\hbar
v_{f})^{2}}\times\left\\{\begin{array}[]{ll}\frac{A_{c}}{2};&\xi\ll a\\\
\frac{2\pi^{2}}{A_{c}}\xi^{4};&\xi\gg a\end{array}\right.$ (20)

where $a=1.42\mathring{\mathrm{A}}$ the carbon-carbon distance of graphene,
$A_{c}=\frac{3}{2}a^{2}$ is the area of the primitive cell, and
$n_{\text{imp}}=N_{\text{imp}}/N$ is the impurity concentration.

In Fig. 1, we display the self-energy functions versus Fermi energy and
momentum. Obviously, the self-energy depends not only on the Fermi energy, but
also on the momentum. As shown in the $E-{\rm Re}\Sigma$ plane of Fig. 1 (a),
the momentum dependence of the real part of the self-energy function is
significant in the low-energy region and gradually reduces with increasing
energy, while the situation is opposite for the imaginary part in the $E-{\rm
Im}\Sigma$ plane of Fig. 1 (b). From the curves in the $k-{\rm Re}\Sigma$
plane of Fig. 1 (c), we see that the real part of the self-energy function
near the Dirac point is linearly related to the momentum, and the coefficient
of the two eigenbands $s=\pm 1$ are opposite. These behaviors mean that the
Born estimation [Eq. (48)] without the on-shell approximation is reasonable
around the Dirac point, and therefore we can assume the self-energy as Eq.
(96).

Figures 2 (a)-(d) exhibit the effect of the correlation length $\xi$ on the
momentum dependence of the real part of the self-energy. When the disorder
strength is small, $\alpha$ has a positive correlation with $\xi$, which can
be fitted as $\alpha=1.6(\xi/a)^{3.3}\times 10^{-3}$. This result is roughly
consistent with the Born approximation which predicts: $\alpha\propto\xi^{2}$
in the limit $\xi\ll a$ and $\alpha\propto\xi^{6}$ in the limit $\xi\gg a$
according to Eqs. (19) and (20). Upon increasing $\xi$, the value of $\alpha$
tends to saturate and eventually decreases. This behavior indicates that the
Born approximation tends to be invalid away from the weak-scattering limit,
since the increase of $\xi$ also enhances the disorder strength $K_{0}$.

Figures 2 (f)-(h) show the effect of the impurity strength $u_{0}$. Similar to
the correlation length, we fit the relation of $\xi$ and $\alpha$ in the weak
disorder regime, and obtain $\alpha=2.9(u_{0}/t)^{2}$. This result is in
agreement with the prediction from the Born approximation: $\alpha\propto
K_{0}\propto u_{0}^{2}$.

In addition, from Fig. 2 (b)-(d) and (f)-(h), we find that the momentum
dependence of the self-energy gradually decreases at high energy, where the
on-shell approximation tends to be valid. The numerical simulation in the
limit $k,\frac{E}{\hbar v_{f}}\gg\frac{1}{\xi}$ is further addressed in the
Supplemental Material Sec. S3.

Figure 2:  (a) $\alpha$ versus $\xi$ with $n_{\text{imp}}=5\%$ and
$u_{0}=0.16t$. (b)-(d) Real part of the self-energy ${\rm Re}\Sigma$ for
different $\xi=3.6a$ (b), $4.4a$ (c), and $5.2a$ (d). (e) $\alpha$ versus
$u_{0}$ with $n_{\text{imp}}=5\%$ and $\xi=3.2a$. (f)-(h) Real part of the
self-energy ${\rm Re}\Sigma$ for different $u_{0}=0.24t$ (f), $0.32t$ (g), and
$0.44t$ (h). The dashed red lines (dotted blue lines) in (a,e) are fitting
curves based on the Born approximation, which have
$\alpha=1.6(\xi/a)^{3.3}\times 10^{-3}$ and $\alpha=2.9(u_{0}/t)^{2}$. The
dotted blue lines in (a,e) are fitting curves based on the self-consistent
Born approximation, which have
$\alpha=\frac{1}{2}-\sqrt{\frac{1}{4}-2.6(\xi/a)^{2.8}\times 10^{-3}}$ and
$\alpha=\frac{1}{2}-\sqrt{\frac{1}{4}-2.4(u_{0}/t)^{2}}$. In (b-d) and (f-h),
the colors correspond to various momenta: Dirac point $k=0$ (black),
$k=0.048/a$ and $s=-1$ (red), $k=0.096/a$ and $s=-1$ (light red), $k=0.048/a$
and $s=1$ (green), $k=0.096/a$ and $s=1$ (light green).

Discussion and Summary
We have shown that in the presence of a long-ranged disorder potential, the
self-energy function of the Dirac electrons has a peculiar dependence on the
momentum, and this momentum dependence is of paramount importance for the
transport properties of graphene, i.e. it produces a non-universal minimal
conductivity. Furthermore, we elaborate the origin of this momentum dependent
correction analytically and uncover its dependence on the disorder potential,
which is also verified by unbiased numerical simulations.

In closing, we would like to make several remarks. First, our findings offer
an intuitive way for understanding the existing numerical simulations Nomura ;
Klos ; Bardarson  where Coulomb impurities enhance the minimal conductivity.
We believe that the physics behind these numerical calculations is now clear,
from clarifying the self-energy function. Second, another main cautionary
message is that the on-shell approximation is not well justified around the
Dirac point. Third, both analytical and numerical results show that the on-
shell approximation tends to be valid in the limit $k,\frac{E}{\hbar
v_{f}}\gg\frac{1}{\xi}$, which means that in the high energy region the self-
energy function becomes momentum independent, in line with previous theories
Sarma2011 ; Shon ; Adam . Finally, this work implies that a consistent theory
for the quantum transport of graphene: Two main experimental observations,
i.e. the linear dependence of the conductivity on carrier density Nomura  and
the sample-dependent minimal conductivity, can be well understood by the
presence of long-ranged impurity potentials.

Acknowledgments.— We thank Michael Smidman for critical proof reading of the
final version of the current manuscript. This work was supported by “Pioneer”
and ”Leading Goose” R&D Program of Zhejiang (2022SDXHDX0005), National Natural
Science Foundation of China (No.12047544, No.11874337) and the foundation of
Westlake University.

Author contributions.— W.C., Q.W.S and W.Z. conceived the idea. W.C and Y.S
carried out the analytical derivation and numerical calculation. W.C., B.F.,
and W.Z drafted the manuscript with discussions.

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Supplemental Materials for “On the sample-dependent minimal conductivity in
weakly disordered graphene”

This supplemental material includes additional calculations to support the
discussion in the main text. In Sec. S1, we show the analytical derivation of
dc conductivity and analysis of the minimal conductivity. In Sec. S2, we
compute the self-energy function of graphene in the presence of long-ranged
Gaussian potential using the Born approximation and self-consistent Born
approximation. We also treat the screened Coulomb potential and obtain the
very similar expression for the self-energy function. In Sec. S3, we provide a
way to understand our main conclusion from a different angle based on the
renormalization group analysis. In Sec. S4, we display more numerical results
of the self-energy function for both long-range Gaussian and Coulomb
potentials. In Sec. S5, we exhibit the calculation of bubble diagram and
vertex correction of dc conductivity in detail.

###### Contents

  1. A Kubo formula for dc conductivity in terms of Green’s function
  2. B Analytical Derivation of Self-Energy Function
    1. B.1 Long-ranged Gaussian Potential (Born approximation)
    2. B.2 Long-ranged Gaussian Potential (SCBA)
    3. B.3 Long-range Screened Coulomb Potential
  3. C Renormalization Group Analysis
    1. C.0.1 Generating functional
    2. C.0.2 Momentum shell decomposition
    3. C.0.3 Corrections of the energy, momentum, and disorder coupling
    4. C.0.4 Renormalization group flow
  4. D Numerical Results of Self-Energy Function
    1. D.0.1 Long-ranged Gaussian Potential: $k,\frac{E}{\hbar v_{f}}\ll\frac{1}{\xi}$ regime
    2. D.0.2 Long-ranged Gaussian Potential: $k,\frac{E}{\hbar v_{f}}\gg\frac{1}{\xi}$ regime
    3. D.0.3 Long-range Screened Coulomb Potential: $k,\frac{E}{\hbar v_{f}}\ll q_{s}$ regime
  5. E Kubo Formalism for Bubble Diagram and Vertex Correction of Conductivity

##  Appendix A Kubo formula for dc conductivity in terms of Green’s function

The derivation of Eqs. (5-7) in the main text, Kubo formula for the dc
conductivity has been discussed in detail in some books of many-body quantum
theory Mahan; Bruus , so here we only briefly outline some significant steps.
Based on the linear response theory, the dc conductivity given by

$\sigma_{\alpha\beta}(E)=-\lim_{\omega\to 0}\lim_{q\to 0}\frac{{\rm
Im}\mathcal{C}^{R}_{\alpha\beta}(\bm{q},\omega)}{\omega}$ (21)

where $\mathcal{C}^{R}_{\alpha\beta}(\bm{q},\omega)$ is a retarded current-
current correlation function,

$C^{R}_{\alpha\beta}(\bm{q},\omega)=-\frac{i}{\hbar}\frac{1}{\mathcal{V}}\int
dt\theta(t-t^{\prime})\langle[\hat{J}_{\alpha}(\bm{q},t),\hat{J}_{\beta}(-\bm{q},t^{\prime})]\rangle
e^{i\omega(t-t^{\prime})}$ (22)

where $\hat{J}$ is the current operator, $\mathcal{V}$ is the area of sample
and the bracket $\langle\cdots\rangle$ means the thermodynamic average over
disorder. We use the Matsubara method to deal with the retarded correlation
function, which indicates that the retarded correlation function is equivalent
to the Matsubara function based on a analytical continuation:
$\mathcal{C}(iq_{n})\xrightarrow{iq_{n}=\omega+i\eta}C^{R}(\omega)$.

We start the calculation from the Matsubara current-current correlation
function,

$\displaystyle\mathcal{C}_{\alpha\beta}(\bm{q},iq_{n})=$
$\displaystyle-\frac{1}{\hbar\mathcal{V}}\int_{0}^{\beta}d(\tau-\tau^{\prime})\langle
T_{\tau}\hat{J}_{\alpha}(\bm{q},\tau)\hat{J}_{\beta}(-\bm{q},\tau^{\prime})\rangle
e^{iq_{n}(\tau-\tau^{\prime})}$ (23) $\displaystyle=$
$\displaystyle-\frac{1}{\hbar\mathcal{V}}\sum_{\bm{k}\bm{k}^{\prime}}\int_{0}^{\beta}d(\tau-\tau^{\prime})\langle
T_{\tau}J_{\alpha,\bm{k},\bm{k}+\bm{q}}c_{\bm{k}}^{\dagger}(\tau)c_{\bm{k}+\bm{q}}(\tau)J_{\beta,\bm{k}^{\prime},\bm{k}^{\prime}-\bm{q}}c_{\bm{k}^{\prime}}^{\dagger}(\tau^{\prime})c_{\bm{k}^{\prime}-\bm{q}}(\tau^{\prime})\rangle
e^{iq_{n}(\tau-\tau^{\prime})}$ $\displaystyle=$
$\displaystyle\frac{1}{\hbar\mathcal{V}}\sum_{\bm{k}\bm{k}^{\prime}}\int_{0}^{\beta}d(\tau-\tau^{\prime})e^{iq_{n}(\tau-\tau^{\prime})}{\rm
Tr}[J_{\alpha,\bm{k},\bm{k}+\bm{q}}\mathcal{G}_{\bm{k}+\bm{q}}(\tau-\tau^{\prime})J_{\beta,\bm{k}^{\prime},\bm{k}^{\prime}-\bm{q}}\mathcal{G}_{\bm{k}}(\tau^{\prime}-\tau)\delta_{\bm{k}^{\prime},\bm{k}+\bm{q}}]$
$\displaystyle=$
$\displaystyle\frac{1}{\hbar\mathcal{V}}\sum_{\bm{k}}\int_{0}^{\beta}d(\tau-\tau^{\prime})e^{iq_{n}(\tau-\tau^{\prime})}{\rm
Tr}[J_{\alpha,\bm{k},\bm{k}+\bm{q}}\mathcal{G}_{\bm{k}+\bm{q}}(\tau-\tau^{\prime})J_{\beta,\bm{k}+\bm{q},\bm{k}}\mathcal{G}_{\bm{k}}(\tau^{\prime}-\tau)]$
$\displaystyle=$
$\displaystyle\frac{1}{\hbar\mathcal{V}}\sum_{\bm{k}}\int_{0}^{\beta}d(\tau-\tau^{\prime})\int_{0}^{\beta}d\tau_{1}\frac{1}{\beta}\sum_{ik_{n}}{\rm
Tr}[J_{\alpha,\bm{k},\bm{k}+\bm{q}}\mathcal{G}_{\bm{k}+\bm{q}}(\tau-\tau^{\prime})e^{iq_{n}(\tau-\tau^{\prime})}e^{ik_{n}(\tau-\tau^{\prime})}J_{\beta,\bm{k}+\bm{q},\bm{k}}\mathcal{G}_{\bm{k}}(\tau_{1})e^{ik_{n}\tau_{1}}]$
$\displaystyle=$
$\displaystyle\frac{1}{\hbar\mathcal{V}\beta}\sum_{\bm{k},ik_{n}}{\rm
Tr}[J_{\alpha,\bm{k},\bm{k}+\bm{q}}\mathcal{G}_{\bm{k}+\bm{q},ik_{n}+iq_{n}}J_{\beta,\bm{k}+\bm{q},\bm{k}}\mathcal{G}_{\bm{k},ik_{n}}]$

where $\beta=1/k_{B}T$, a closed fermion loop always gives a factor $-1$ and
the trace of a product of Dirac matrices. $\mathcal{G}$ is Matsubara Green’s
function. The current operators are been expressed creation and annihilate
operators:
$\hat{J}_{\alpha}(\bm{q},\tau)=\sum_{\bm{k}}J_{\alpha,\bm{k},\bm{k}+\bm{q}}c_{\bm{k}}^{\dagger}(\tau)c_{\bm{k}+\bm{q}}(\tau)$.
In addition, the identities
$1=\int_{0}^{\beta}\delta(\tau_{1}-\tau^{\prime}+\tau)d\tau_{1}$ and
$\delta(\tau_{1}-\tau^{\prime}+\tau)=\frac{1}{\beta}\sum_{ik_{n}}e^{ik_{n}(\tau_{1}-\tau^{\prime}+\tau)}$
are applied. It is noticed that $ik_{n}=i(2n+1)\pi/\beta$ denotes Fermi
frequency and $iq_{n}=i2n\pi/\beta$ denotes Boson frequency.

Next, we treat the summations of Matsubara Green’s functions with known branch
cuts. According to the summation of $ik_{n}$, one can introduce a contour
integral

$I=\oint_{\mathcal{C}_{1}+\mathcal{C}_{2}+\mathcal{C}_{3}}\frac{dz}{2\pi
i}f(z)\mathcal{G}(z+iq_{n})\mathcal{G}(z)$ (24)

where $f(z)=\frac{1}{e^{\beta z}+1}$ and the contours are shown in Fig. 4,
since for variable $z$ the branch cut (also singularity line) of
$\mathcal{G}(z)$ is real axis, i.e. $z={\rm Re}z=\omega^{\prime}$, and the
branch cut of $\mathcal{G}(z+iq_{n})$ is $z=\omega-iq_{n}$. On the one hand,
we calculate $I$ by residue theorem

$I=\sum_{ik_{n}}\lim_{z\to
ik_{n}=i(2n+1)\pi/\beta}\frac{z-ik_{n}}{e^{ik_{n}\beta}+1}\mathcal{G}(ik_{n}+iq_{n})\mathcal{G}(ik_{n})=-\frac{1}{\beta}\sum_{ik_{n}}\mathcal{G}(ik_{n}+iq_{n})\mathcal{G}(ik_{n})$
(25)

On the other hand, we calculate $I$ by separating contour integrals

$\displaystyle I=$
$\displaystyle\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi
i}f(z)\mathcal{G}(z+iq_{n})\mathcal{G}(z)\bigg{|}_{z=\omega^{\prime}-iq_{n}+i0^{+}}+\int_{\infty}^{-\infty}\frac{d\omega^{\prime}}{2\pi
i}f(z)\mathcal{G}(z+iq_{n})\mathcal{G}(z)\bigg{|}_{z=\omega^{\prime}-iq_{n}-i0^{+}}$
(26) $\displaystyle+\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi
i}f(z)\mathcal{G}(z+iq_{n})\mathcal{G}(z)\bigg{|}_{z=\omega^{\prime}+i0^{+}}+\int_{\infty}^{-\infty}\frac{d\omega^{\prime}}{2\pi
i}f(z)\mathcal{G}(z+iq_{n})\mathcal{G}(z)\bigg{|}_{z=\omega^{\prime}-i0^{+}}$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi
i}f(\omega^{\prime})[\mathcal{G}(\omega^{\prime}+i0^{+})\mathcal{G}(\omega^{\prime}-iq_{n})-\mathcal{G}(\omega^{\prime}-i0^{+})\mathcal{G}(\omega^{\prime}-iq_{n})+\mathcal{G}(\omega^{\prime}+iq_{n})\mathcal{G}(\omega^{\prime}+i0^{+})-\mathcal{G}(\omega^{\prime}+iq_{n})\mathcal{G}(\omega^{\prime}-i0^{+})]$

Based on the comparison of the Eq. (25) and Eq. (26), we can rewrite Eq. (23)
as

$\displaystyle\mathcal{C}_{\alpha\beta}(\bm{q},iq_{n})=$
$\displaystyle\frac{-1}{\hbar\mathcal{V}}\sum_{\bm{k}}\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi
i}f(\omega^{\prime}){\rm
Tr}\left[J_{\alpha,\bm{k},\bm{k}+\bm{q}}\mathcal{G}_{\bm{k}+\bm{q},\omega^{\prime}+i0^{+}}J_{\beta,\bm{k}+\bm{q},\bm{k}}\mathcal{G}_{\bm{k},\omega^{\prime}-iq_{n}}-J_{\alpha,\bm{k},\bm{k}+\bm{q}}\mathcal{G}_{\bm{k}+\bm{q},\omega^{\prime}-i0^{+}}J_{\beta,\bm{k}+\bm{q},\bm{k}}\mathcal{G}_{\bm{k},\omega^{\prime}-iq_{n}}\right.$
(27)
$\displaystyle+\left.J_{\alpha,\bm{k},\bm{k}+\bm{q}}\mathcal{G}_{\bm{k}+\bm{q},\omega^{\prime}+iq_{n}}J_{\beta,\bm{k}+\bm{q},\bm{k}}\mathcal{G}_{\bm{k},\omega^{\prime}+i0^{+}}-J_{\alpha,\bm{k},\bm{k}+\bm{q}}\mathcal{G}_{\bm{k}+\bm{q},\omega^{\prime}+iq_{n}}J_{\beta,\bm{k}+\bm{q},\bm{k}}\mathcal{G}_{\bm{k},\omega^{\prime}-i0^{+}}\right]$

Then, we do the analytical continuity $iq_{n}\to\omega+i0^{+}$ and get

$\displaystyle C^{R}_{\alpha\beta}(\bm{q},\omega)=$
$\displaystyle\frac{-1}{\hbar\mathcal{V}}\sum_{\bm{k}}\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi
i}f(\omega^{\prime}){\rm
Tr}\left[J_{\alpha,\bm{k},\bm{k}+\bm{q}}\mathcal{G}_{\bm{k}+\bm{q},\omega^{\prime}+i0^{+}}J_{\beta,\bm{k}+\bm{q},\bm{k}}\mathcal{G}_{\bm{k},\omega^{\prime}-\omega-i0^{+}}-J_{\alpha,\bm{k},\bm{k}+\bm{q}}\mathcal{G}_{\bm{k}+\bm{q},\omega^{\prime}-i0^{+}}J_{\beta,\bm{k}+\bm{q},\bm{k}}\mathcal{G}_{\bm{k},\omega^{\prime}-\omega-i0^{+}}\right.$
(28)
$\displaystyle+\left.J_{\alpha,\bm{k},\bm{k}+\bm{q}}\mathcal{G}_{\bm{k}+\bm{q},\omega^{\prime}+\omega+i0^{+}}J_{\beta,\bm{k}+\bm{q},\bm{k}}\mathcal{G}_{\bm{k},\omega^{\prime}+i0^{+}}-J_{\alpha,\bm{k},\bm{k}+\bm{q}}\mathcal{G}_{\bm{k}+\bm{q},\omega^{\prime}+\omega+i0^{+}}J_{\beta,\bm{k}+\bm{q},\bm{k}}\mathcal{G}_{\bm{k},\omega^{\prime}-i0^{+}}\right]$
$\displaystyle=$
$\displaystyle\frac{i}{\hbar\mathcal{V}}\sum_{\bm{k}}\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi}f(\omega^{\prime}){\rm
Tr}\left[J_{\alpha,\bm{k},\bm{k}+\bm{q}}G^{R}_{\bm{k}+\bm{q}}(\omega^{\prime})J_{\beta,\bm{k}+\bm{q},\bm{k}}G^{A}_{\bm{k}}(\omega^{\prime}-\omega)-J_{\alpha,\bm{k},\bm{k}+\bm{q}}G^{A}_{\bm{k}+\bm{q}}(\omega^{\prime})J_{\beta,\bm{k}+\bm{q},\bm{k}}G^{A}_{\bm{k}}(\omega^{\prime}-\omega)\right.$
$\displaystyle+\left.J_{\alpha,\bm{k},\bm{k}+\bm{q}}G^{R}_{\bm{k}+\bm{q}}(\omega^{\prime}+\omega)J_{\beta,\bm{k}+\bm{q},\bm{k}}G^{R}_{\bm{k}}(\omega^{\prime})-J_{\alpha,\bm{k},\bm{k}+\bm{q}}G^{R}_{\bm{k}+\bm{q}}(\omega^{\prime}+\omega)J_{\beta,\bm{k}+\bm{q},\bm{k}}G^{A}_{\bm{k}}(\omega^{\prime})\right]$
$\displaystyle=$
$\displaystyle\frac{i}{\hbar\mathcal{V}}\sum_{\bm{k}}\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi}f(\omega^{\prime}+\omega){\rm
Tr}\left[J_{\alpha,\bm{k},\bm{k}+\bm{q}}G^{R}_{\bm{k}+\bm{q}}(\omega^{\prime}+\omega)J_{\beta,\bm{k}+\bm{q},\bm{k}}G^{A}_{\bm{k}}(\omega^{\prime})-J_{\alpha,\bm{k},\bm{k}+\bm{q}}G^{A}_{\bm{k}+\bm{q}}(\omega^{\prime}+\omega)J_{\beta,\bm{k}+\bm{q},\bm{k}}G^{A}_{\bm{k}}(\omega^{\prime})\right]$
$\displaystyle+f(\omega^{\prime})\left[J_{\alpha,\bm{k},\bm{k}+\bm{q}}G^{R}_{\bm{k}+\bm{q}}(\omega^{\prime}+\omega)J_{\beta,\bm{k}+\bm{q},\bm{k}}G^{R}_{\bm{k}}(\omega^{\prime})-J_{\alpha,\bm{k},\bm{k}+\bm{q}}G^{R}_{\bm{k}+\bm{q}}(\omega^{\prime}+\omega)J_{\beta,\bm{k}+\bm{q},\bm{k}}G^{A}_{\bm{k}}(\omega^{\prime})\right]$
$\displaystyle=$
$\displaystyle\frac{i}{\hbar\mathcal{V}}\sum_{\bm{k}}\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi}$
$\displaystyle[f(\omega^{\prime}+\omega)-f(\omega)]{\rm
Tr}\left[J_{\alpha,\bm{k},\bm{k}+\bm{q}}G^{R}_{\bm{k}+\bm{q}}(\omega^{\prime}+\omega)J_{\beta,\bm{k}+\bm{q},\bm{k}}G^{A}_{\bm{k}}(\omega^{\prime})-J_{\alpha,\bm{k},\bm{k}+\bm{q}}G^{A}_{\bm{k}+\bm{q}}(\omega^{\prime}+\omega)J_{\beta,\bm{k}+\bm{q},\bm{k}}G^{A}_{\bm{k}}(\omega^{\prime})\right]$
$\displaystyle+f(\omega^{\prime})\left[J_{\alpha,\bm{k},\bm{k}+\bm{q}}G^{R}_{\bm{k}+\bm{q}}(\omega^{\prime}+\omega)J_{\beta,\bm{k}+\bm{q},\bm{k}}G^{R}_{\bm{k}}(\omega^{\prime})-J_{\alpha,\bm{k},\bm{k}+\bm{q}}G^{A}_{\bm{k}+\bm{q}}(\omega^{\prime}+\omega)J_{\beta,\bm{k}+\bm{q},\bm{k}}G^{A}_{\bm{k}}(\omega^{\prime})\right]$

Figure 3: The temperature dependence of minimal conductivity obtained by
numerical and analytical methods.

Finally, the dc longitudinal conductivity is obtained under the conditions:
$\alpha=\beta=\hat{x}$, $\omega\to 0$, $\bm{q}\to 0$.

$\displaystyle\sigma_{xx}(E)=$ $\displaystyle-\lim\limits_{\omega\to
0}\frac{{\rm Im}C^{R}_{xx}(\bm{q}=0,\omega)}{\omega}$ (29) $\displaystyle=$
$\displaystyle\frac{e^{2}}{2\pi\hbar\mathcal{V}}\int_{-\infty}^{\infty}d\omega^{\prime}\left[-\frac{\partial
f(\omega^{\prime})}{\partial\omega^{\prime}}\right]{\rm
ReTr}\left[v_{x}G^{R}(\omega^{\prime})v_{x}G^{A}(\omega^{\prime})-v_{x}G^{R}(\omega^{\prime})v_{x}G^{R}(\omega^{\prime})\right]$

where the current is rewritten as $\bm{J}=e\bm{v}$. At the zero temperature,
where $\frac{\partial
f(\omega^{\prime})}{\partial\omega^{\prime}}=-\delta(\omega^{\prime}-\frac{E}{\hbar})$
for $f(\omega)=\frac{1}{e^{(\hbar\omega-E)/k_{B}T}+1}$, the above expression
can be further simplified to

$\sigma_{xx}(E)=\frac{e^{2}\hbar}{2\pi\mathcal{V}}{\rm
ReTr}\left[G^{R}(E)v_{x}G^{A}(E)v_{x}-G^{R}(E)v_{x}G^{R}(E)v_{x}\right]$ (30)

which corresponds to the Eqs. (5-7) in the main text.

While in the main text we only discuss the minimal conductivity at zero
temperature, here we do some analysis of finite temperature effects. According
to the Eq. (29), the finite-temperature minimal conductivity can be obtained
using the analytic Sommerfeld expansion or numerical integration. In the
vicinity of the Dirac point, the temperature dependent dc conductivity can be
evaluated as

$\displaystyle\sigma_{xx}(E\to 0,T)\approx\frac{2e^{2}}{\pi
h}\frac{1}{(1-\alpha)^{2}}\int_{-\infty}^{\infty}d\omega^{\prime}\left[-\frac{\partial
f(\omega^{\prime})}{\partial\omega^{\prime}}\right][2+(\frac{\omega^{\prime}}{\eta_{0}})^{2}]=\frac{4e^{2}}{\pi
h(1-\alpha)^{2}}+\frac{2e^{2}}{3h(1-\alpha)^{2}\eta_{0}^{2}}(k_{B}T)^{2}$ (31)

where we have used a Sommerfeld expansion generating function $\int
d\omega^{\prime}\omega^{\prime 2}\left[-\frac{\partial
f(\omega^{\prime})}{\partial\omega^{\prime}}\right]=E^{2}+\frac{\pi^{2}}{3}(k_{B}T)^{2}$.
This evaluation shows that the minimal conductivity at the finite temperature
varies as $\Delta\sigma_{\text{min}}\propto T^{2}$.

Meanwhile, we show the numerical results of low-temperature dependence of
minimal conductivity in Fig. 3 and compare it with the above analytical
results. In the numerical simulation, we consider the self-energy function
obtained by Born approximation: $a=\omega^{\prime}$,
$\eta=\frac{K_{0}}{4}|\omega^{\prime}|+\eta_{0}$ with the parameters
$\eta_{0}=0.005$ and $K_{0}=0.3$.

Figure 4: Contour integral with two branch cuts: $z={\rm Re}z=\omega$ and
$z=\omega-i\Omega_{n}$.

##  Appendix B Analytical Derivation of Self-Energy Function

###  B.1 Long-ranged Gaussian Potential (Born approximation)

In this section, we evaluate the self-energy function in the presence of long-
ranged Gaussian potential by the Born approximation:

$\displaystyle\Sigma^{B}(\bm{k},E)=$
$\displaystyle\frac{1}{\mathcal{V}^{2}}\sum_{\bm{k}^{\prime}}\overline{|V(\bm{k}-\bm{k}^{\prime})|^{2}}U^{\dagger}_{\bm{k}}U_{\bm{k}^{\prime}}G^{R}_{0}(\bm{k}^{\prime},E)U^{\dagger}_{\bm{k}^{\prime}}U_{\bm{k}}$
(32) $\displaystyle=$
$\displaystyle\frac{1}{2}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\mathcal{K}(\bm{k}-\bm{k}^{\prime})\left(\begin{array}[]{cc}(g_{0}^{+}+g_{0}^{-})+\cos\theta(g_{0}^{+}-g_{0}^{-})&i\sin\theta(g_{0}^{+}-g_{0}^{-})\\\
-i\sin\theta(g_{0}^{+}-g_{0}^{-})&(g_{0}^{+}+g_{0}^{-})-\cos\theta(g_{0}^{+}-g_{0}^{-})\end{array}\right)$

where $\overline{\cdots}$ represents the average of random configurations,
$\theta=\theta_{\bm{k}}-\theta_{\bm{k}^{\prime}}$ is the angle between
momentum $\bm{k}$ and $\bm{k}^{\prime}$, and
$g_{0}^{\pm}=G^{R}_{0}(\bm{k}^{\prime}\pm,E)=1/(E+i\eta-
E_{\bm{k}^{\prime}\pm})$ is the retarded Green’s function for eigenstate with
momentum $\bm{k}^{\prime}$ and chiral $s=\pm 1$. Based on the parity analysis,
we find the integral of off-diagonal term is vanishing so that the first-order
self-energy function in eigenstates basis can be expressed as

$\Sigma^{B}(\bm{k}s,E)=\sum_{s^{\prime}}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\mathcal{K}(\bm{k}-\bm{k}^{\prime})G^{R}_{0}(\bm{k}^{\prime}s^{\prime},E)\frac{1+ss^{\prime}\cos\theta}{2}$
(33)

which is the Eq.(14) in the main paper. After plugging the expression of
$\mathcal{K}(\bm{k}-\bm{k}^{\prime})$ into this equation, it is obtained

$\Sigma^{B}(\bm{k}s,E)=K_{0}(\hbar
v_{f})^{2}\sum_{s^{\prime}}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}e^{-\frac{\xi^{2}(\bm{k}-\bm{k}^{\prime})^{2}}{2}}G^{R}_{0}(\bm{k}^{\prime}s^{\prime},E)\frac{1+ss^{\prime}\cos\theta}{2}$
(34)

For $E>0$, the imaginary part of the self-energy is

$\displaystyle{\rm Im}\Sigma^{B}(\bm{k}s,E)=$ $\displaystyle K_{0}(\hbar
v_{f})^{2}\sum_{s^{\prime}}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}e^{-\frac{\xi^{2}(\bm{k}-\bm{k}^{\prime})^{2}}{2}}\frac{1+ss^{\prime}\cos\theta}{2}[-\pi\delta(E-s^{\prime}\hbar
v_{f}k^{\prime})]$ (35) $\displaystyle=$ $\displaystyle-\frac{\pi K_{0}\hbar
v_{f}}{2}\sum_{s^{\prime}}\int\frac{k^{\prime}dk^{\prime}d\theta}{(2\pi)^{2}}e^{-\frac{\xi^{2}(k^{2}+k^{\prime
2}-2kk^{\prime}\cos\theta)}{2}}(1+ss^{\prime}\cos\theta)\delta(\frac{E}{\hbar
v_{f}}-s^{\prime}k^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{K_{0}}{8\pi}Ee^{-\frac{\xi^{2}}{2}\left(k^{2}+\frac{E^{2}}{\hbar^{2}v^{2}_{f}}\right)}\int^{\pi}_{-\pi}d\theta
e^{\frac{\xi^{2}kE}{\hbar v_{f}}\cos\theta}(1+s\cos\theta)$ $\displaystyle=$
$\displaystyle-\frac{K_{0}}{4}Ee^{-\frac{\xi^{2}}{2}\left(k^{2}+\frac{E^{2}}{\hbar^{2}v^{2}_{f}}\right)}\left[I_{0}(\frac{\xi^{2}kE}{\hbar
v_{f}})+sI_{1}(\frac{\xi^{2}kE}{\hbar v_{f}})\right]$

where $\tilde{E}=\frac{E}{\hbar v_{f}}$. In the final expression above, we
have introduced modified Bessel functions of the first kind, whose integral
representation is

$I_{\alpha}(z)=\frac{1}{\pi}\int^{\pi}_{0}e^{z\cos\theta}\cos\alpha\theta
d\theta-\frac{\sin\alpha\pi}{\pi}\int^{\infty}_{0}e^{-z\cosh t-\alpha t}dt$
(36)

and has series expansion,

$I_{\alpha}(z)=\sum^{\infty}_{m=0}\frac{1}{m!\Gamma(m+\alpha+1)}\left(\frac{z}{2}\right)^{2m+\alpha},$
(37)

asymptotic expansion for large $z$,

$I_{\alpha}(z)=\frac{1}{\sqrt{2\pi
z}\sqrt[4]{1+\frac{\alpha^{2}}{z^{2}}}}\exp\left[-\alpha{\rm
arsinh}\left(\frac{\alpha}{z}\right)+z\sqrt{1+\frac{\alpha^{2}}{z^{2}}}\right]\left[1+o\left(\frac{1}{z\sqrt{1+\frac{\alpha^{2}}{z^{2}}}}\right)\right].$
(38)

Similarly, for $E<0$, the imaginary part of the self-energy is

$\displaystyle{\rm Im}\Sigma^{B}(\bm{k}s,E)=$ $\displaystyle K_{0}(\hbar
v_{f})^{2}\sum_{s^{\prime}}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}e^{-\frac{\xi^{2}(\bm{k}-\bm{k}^{\prime})^{2}}{2}}\frac{1+ss^{\prime}\cos\theta}{2}[-\pi\delta(E-s^{\prime}\hbar
v_{f}k^{\prime})]$ (39) $\displaystyle=$ $\displaystyle-\frac{\pi K_{0}\hbar
v_{f}}{2}\sum_{s^{\prime}}\int\frac{k^{\prime}dk^{\prime}d\theta}{(2\pi)^{2}}e^{-\frac{\xi^{2}(k^{2}+k^{\prime
2}-2kk^{\prime}\cos\theta)}{2}}(1+ss^{\prime}\cos\theta)\delta(\frac{E}{\hbar
v_{f}}-s^{\prime}k^{\prime})$ $\displaystyle=$
$\displaystyle\frac{K_{0}}{8\pi}Ee^{-\frac{\xi^{2}}{2}\left(k^{2}+\frac{E^{2}}{\hbar^{2}v^{2}_{f}}\right)}\int^{\pi}_{-\pi}d\theta
e^{-\frac{\xi^{2}kE\cos\theta}{\hbar v_{f}}}(1-s\cos\theta)$ $\displaystyle=$
$\displaystyle\frac{K_{0}}{4}Ee^{-\frac{\xi^{2}}{2}\left(k^{2}+\frac{E^{2}}{\hbar^{2}v^{2}_{f}}\right)}\left[I_{0}(-\frac{\xi^{2}kE}{\hbar
v_{f}})-sI_{1}(-\frac{\xi^{2}kE}{\hbar v_{f}})\right]$

Combining two conditions, we can get

${\rm
Im}\Sigma^{B}(\bm{k}s,E)=-\frac{K_{0}}{4}|E|e^{-\frac{\xi^{2}}{2}\left(k^{2}+\frac{E^{2}}{\hbar^{2}v^{2}_{f}}\right)}\left[I_{0}(\frac{\xi^{2}kE}{\hbar
v_{f}})+sI_{1}(\frac{\xi^{2}kE}{\hbar v_{f}})\right]$ (40)

For the limit $k,\frac{E}{\hbar v_{f}}\ll\frac{1}{\xi}$, the above result can
be expanded using Eq.(37) as

$\displaystyle{\rm Im}\Sigma^{B}(\bm{k}s,E)=$
$\displaystyle-\frac{K_{0}}{4}|E|\left[1-\frac{\xi^{2}}{2}\left(k^{2}+\frac{E^{2}}{\hbar^{2}v^{2}_{f}}\right)+O(k^{4},E^{4},k^{2}E^{2})\right]\left[1+s\frac{\xi^{2}kE}{2\hbar
v_{f}}+O(k^{2}E^{2})\right]$ (41) $\displaystyle=$
$\displaystyle-\frac{K_{0}|E|}{4}\left[1-\frac{\xi^{2}}{2}\left(k^{2}+\frac{E^{2}}{\hbar^{2}v^{2}_{f}}\right)+\frac{s\xi^{2}kE}{2\hbar
v_{f}}\right]+O(k^{4},E^{4},k^{2}E^{2})$ $\displaystyle=$
$\displaystyle-\frac{K_{0}|E|}{4}+O(kE^{2},k^{2}E,E^{3})$

which is the Eq.(17) in the main paper.

For the limit $k,\frac{E}{\hbar v_{f}}\gg\frac{1}{\xi}$, it has following
asymptotic expression by using Eq.(38)

$\displaystyle{\rm Im}\Sigma^{B}(\bm{k}s,E)=$
$\displaystyle-\frac{K_{0}}{4}|E|e^{-\frac{\xi^{2}}{2}\left(k^{2}+\frac{E^{2}}{\hbar^{2}v^{2}_{f}}\right)}e^{\frac{\xi^{2}k|E|}{\hbar
v_{f}}}\sqrt{\frac{\hbar v_{f}}{2\pi\xi^{2}k|E|}}\left[1+s{\rm sgn}(E)\right]$
(42) $\displaystyle=$
$\displaystyle-\frac{K_{0}}{4}|E|e^{-\frac{\xi^{2}}{2}\left(k-\frac{|E|}{\hbar
v_{f}}\right)^{2}}\sqrt{\frac{\hbar v_{f}}{2\pi\xi^{2}k|E|}}\left[1+s{\rm
sgn}(E)\right]$ $\displaystyle\approx$ $\displaystyle-\frac{K_{0}(\hbar
v_{f})^{2}}{4\xi^{2}}\delta(E-E_{\bm{k}s})$

where ${\rm sgn}(E)$ is sign function of $E$. The exponential decay term
$e^{-\frac{\xi^{2}}{2}\left(k-\frac{|E|}{\hbar v_{f}}\right)^{2}}$ and the
term $\left[1+s{\rm sgn}(E)\right]$ in the above at the second equation imply
that the on-shell approximation ($E=E_{\bm{k}s}=s\hbar v_{f}k$) is valid in
this condition based on the approximation $\lim\limits_{\sigma\to
0}\frac{1}{\sqrt{2\pi\sigma}}\exp(-x^{2}/2\sigma)=\delta(x)$. The value of
${\rm Im}\Sigma^{B}$ we obtained under the on-shell approximation is a
momentum and energy independent constant.

The corresponding real part of self-energy function can be calculated by the
Kramer-Kronig relation,

$\displaystyle{\rm Re}\Sigma^{B}(\bm{k}s,E)=$
$\displaystyle\frac{1}{\pi}\mathcal{P}\int^{E_{c}}_{-E_{c}}dE^{\prime}\frac{{\rm
Im}\Sigma^{B}(\bm{k}s,E^{\prime})}{E^{\prime}-E}$ (43) $\displaystyle=$
$\displaystyle\frac{1}{\pi}\mathcal{P}\int^{E_{c}}_{-E_{c}}dE^{\prime}\frac{-\frac{K_{0}}{4}|E^{\prime}|e^{-\frac{\xi^{2}}{2}\left(k^{2}+\frac{E^{\prime
2}}{\hbar^{2}v^{2}_{f}}\right)}\left[I_{0}(\frac{\xi^{2}kE^{\prime}}{\hbar
v_{f}})+sI_{1}(\frac{\xi^{2}kE^{\prime}}{\hbar v_{f}})\right]}{E^{\prime}-E}$

which is equivalent to the Eq.(16) in the main paper.

For the limit $k,\frac{E}{\hbar v_{f}}\ll\frac{1}{\xi}$, the corresponding
real part of self-energy function is

$\displaystyle{\rm Re}\Sigma^{B}(\bm{k}s,E)\approx$
$\displaystyle\frac{1}{\pi}\mathcal{P}\int^{E_{c}}_{-E_{c}}dE^{\prime}\frac{-\frac{K_{0}}{4}|E^{\prime}|[1-\frac{\xi^{2}}{2}\left(k^{2}+\frac{E^{\prime
2}}{\hbar^{2}v^{2}_{f}}\right)]\left[1+\frac{1}{4}(\frac{\xi^{2}kE^{\prime}}{\hbar
v_{f}})^{2}+s\frac{\xi^{2}kE^{\prime}}{2\hbar v_{f}}\right]}{E^{\prime}-E}$
(44) $\displaystyle\approx$
$\displaystyle\frac{1}{\pi}\mathcal{P}\int^{E_{c}}_{-E_{c}}dE^{\prime}\frac{-\frac{K_{0}}{4}|E^{\prime}|\left[1+s\frac{\xi^{2}kE^{\prime}}{2\hbar
v_{f}}-\frac{\xi^{2}}{2}\left(k^{2}+\frac{E^{\prime
2}}{\hbar^{2}v^{2}_{f}}\right)\right]}{E^{\prime}-E}$ $\displaystyle=$
$\displaystyle-\frac{K_{0}\hbar
v_{f}}{4\pi}\mathcal{P}\left[\int^{0}_{-k_{c}}d\tilde{E}^{\prime}\frac{-\tilde{E}^{\prime}-\frac{s\xi^{2}k\tilde{E}^{\prime
2}}{2}+\frac{\xi^{2}}{2}\tilde{E}^{\prime}(k^{2}+\tilde{E}^{\prime
2})}{\tilde{E}^{\prime}-\tilde{E}}+\int_{0}^{k_{c}}d\tilde{E}^{\prime}\frac{\tilde{E}^{\prime}+\frac{s\xi^{2}k\tilde{E}^{\prime
2}}{2}-\frac{\xi^{2}}{2}\tilde{E}^{\prime}(k^{2}+\tilde{E}^{\prime
2})}{\tilde{E}^{\prime}-\tilde{E}}\right]$ $\displaystyle=$
$\displaystyle-\frac{K_{0}\hbar
v_{f}}{4\pi}\mathcal{P}\int_{0}^{k_{c}}d\tilde{E}^{\prime}\left[\frac{-\tilde{E}^{\prime}+\frac{s\xi^{2}k\tilde{E}^{\prime
2}}{2}+\frac{\xi^{2}}{2}\tilde{E}^{\prime}(k^{2}+\tilde{E}^{\prime
2})}{\tilde{E}^{\prime}+\tilde{E}}+\frac{\tilde{E}^{\prime}+\frac{s\xi^{2}k\tilde{E}^{\prime
2}}{2}-\frac{\xi^{2}}{2}\tilde{E}^{\prime}(k^{2}+\tilde{E}^{\prime
2})}{\tilde{E}^{\prime}-\tilde{E}}\right]$ $\displaystyle=$
$\displaystyle-\frac{K_{0}\hbar
v_{f}}{4\pi}\mathcal{P}\int_{0}^{k_{c}}d\tilde{E}^{\prime}\left(\frac{\tilde{E}^{\prime}}{\tilde{E}^{\prime
2}-\tilde{E}^{2}}2\tilde{E}+\frac{\tilde{E}^{\prime 3}}{\tilde{E}^{\prime
2}-\tilde{E}^{2}}s\xi^{2}k-\frac{\tilde{E}^{\prime}(k^{2}+\tilde{E}^{\prime
2})}{\tilde{E}^{\prime 2}-\tilde{E}^{2}}\xi^{2}\tilde{E}\right)$
$\displaystyle=$ $\displaystyle\frac{K_{0}\hbar
v_{f}}{2\pi}\left[\tilde{E}(\ln|\tilde{E}|-\ln|k_{c}|)-s\left(\frac{\xi^{2}k_{c}^{2}}{4}\right)k+\left(\frac{\xi^{2}k_{c}^{2}}{4}\right)\tilde{E}\right]+O(kE^{2}\ln
E,k^{2}E\ln E,E^{3}\ln E)$ $\displaystyle=$
$\displaystyle\frac{K_{0}}{2\pi}E\ln|\frac{E}{\hbar v_{f}k_{c}}|+\alpha
E-s\alpha\hbar v_{f}k+O(kE^{2}\ln E,k^{2}E\ln E,E^{3}\ln E)$

with

$\alpha=K_{0}\frac{\xi^{2}k_{c}^{2}}{8\pi}$ (45)

so that we obtain the Eqs.(18) and (19) in the main paper. Here,
$\tilde{E}=\frac{E}{\hbar v_{f}}$, $\tilde{E}^{\prime}=\frac{E^{\prime}}{\hbar
v_{f}}$, and $k_{c}=\frac{E_{c}}{\hbar v_{f}}$. The modified Bessel functions
are expanded as Eq. (37), and we keep the leading orders up to:
$I_{0}(x)=1+\frac{x^{2}}{4}+o(x^{3})$ and $I_{1}(x)=\frac{x}{2}+o(x^{3})$.

Then combining Eq.(42) and (43), we can obtain ${\rm Re}\Sigma^{B}$ at the
limit $k,\frac{E}{\hbar v_{f}}\gg\frac{1}{\xi}$ as

${\rm Re}\Sigma^{B}(\bm{k}s,E)\approx\frac{K_{0}(\hbar
v_{f})^{2}}{4\pi}\frac{E-E_{\bm{k}s}}{\xi^{2}(E-E_{\bm{k}s})^{2}+(\hbar
v_{f})^{2}}$ (46)

To sum up, we obtain the self-energy function for $\frac{|E|}{\hbar
v_{f}},k\ll\frac{1}{\xi}$

$\displaystyle{\rm Im}\Sigma^{B}(\bm{k}s,E)$ $\displaystyle=$
$\displaystyle-\frac{K_{0}}{4}|E|+\cdots$ (47) $\displaystyle{\rm
Re}\Sigma^{B}(\bm{k}s,E)$ $\displaystyle=$
$\displaystyle\frac{K_{0}}{2\pi}E\ln|\frac{E}{\hbar v_{f}k_{c}}|+\alpha
E-\alpha s\hbar v_{f}k+\cdots$ (48)

and for $k,\frac{E}{\hbar v_{f}}\gg\frac{1}{\xi}$

$\displaystyle{\rm Im}\Sigma^{B}(\bm{k}s,E)$ $\displaystyle=$
$\displaystyle-\frac{K_{0}(\hbar v_{f})^{2}}{4\xi^{2}}\delta(E-E_{\bm{k}s})$
(49) $\displaystyle{\rm Re}\Sigma^{B}(\bm{k}s,E)$ $\displaystyle=$
$\displaystyle\frac{K_{0}(\hbar
v_{f})^{2}}{4\pi}\frac{E-E_{\bm{k}s}}{\xi^{2}(E-E_{\bm{k}s})^{2}+(\hbar
v_{f})^{2}}$ (50)

###  B.2 Long-ranged Gaussian Potential (SCBA)

Going beyond the Born approximation, we try to calculate the self-energy
function using the self-consistent Born approximation (SCBA). It is expected
that SCBA contains partial higher-order corrections in the perturbative
expansion. Inspired by the Born approximation, we assume the self-energy
function at the limit $k,\frac{E}{\hbar v_{f}}\ll\frac{1}{\xi}$ can be
expressed in the form of

$\Sigma(\bm{k}s,E)=\Sigma_{1}(E)-s\alpha\hbar
v_{f}k+i\Sigma_{2}(E)=\tilde{\Sigma}(E)-s\alpha\hbar v_{f}k$ (51)

where $\tilde{\Sigma}(E)=\Sigma_{1}(E)+i\Sigma_{2}(E)$ is the part that
depends only on energy. Thus, the Feynman diagram of SCBA is given by

$\displaystyle\Sigma(\bm{k}s,E)=$
$\displaystyle\sum_{s^{\prime}}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\mathcal{K}(\bm{k}-\bm{k}^{\prime})G(\bm{k}^{\prime}s^{\prime},E)\frac{1+ss^{\prime}\cos\theta}{2}$
(52) $\displaystyle=$ $\displaystyle K_{0}(\hbar
v_{f})^{2}\sum_{s^{\prime}}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}e^{-\frac{\xi^{2}(\bm{k}-\bm{k}^{\prime})^{2}}{2}}\frac{1}{E-E_{\bm{k}^{\prime}s^{\prime}}-\Sigma(\bm{k}^{\prime}s^{\prime},E)}\frac{1+ss^{\prime}\cos\theta}{2}$
$\displaystyle=$ $\displaystyle K_{0}(\hbar
v_{f})^{2}\sum_{s^{\prime}}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}e^{-\frac{\xi^{2}(\bm{k}-\bm{k}^{\prime})^{2}}{2}}\frac{1}{E-s^{\prime}(1-\alpha)\hbar
v_{f}k^{\prime}-\tilde{\Sigma}}\frac{1+ss^{\prime}\cos\theta}{2}$
$\displaystyle=$ $\displaystyle\frac{K_{0}(\hbar
v_{f})^{2}}{2}\int\frac{k^{\prime}dk^{\prime}d\theta}{(2\pi)^{2}}e^{-\frac{\xi^{2}(\bm{k}-\bm{k}^{\prime})^{2}}{2}}\left[\frac{1+s\cos\theta}{(E-\tilde{\Sigma})-(1-\alpha)\hbar
v_{f}k^{\prime}}+\frac{1-s\cos\theta}{(E-\tilde{\Sigma})+(1-\alpha)\hbar
v_{f}k^{\prime}}\right]$ $\displaystyle=$ $\displaystyle\frac{K_{0}(\hbar
v_{f})^{2}}{2\pi^{2}}\int_{0}^{k_{c}}k^{\prime}dk^{\prime}\int_{0}^{\pi}d\theta
e^{-\frac{\xi^{2}(k^{2}+k^{\prime
2}-2kk^{\prime}\cos\theta)}{2}}\frac{(E-\tilde{\Sigma})+s\cos\theta(1-\alpha)\hbar
v_{f}k^{\prime}}{(E-\tilde{\Sigma})^{2}-(1-\alpha)^{2}(\hbar
v_{f}k^{\prime})^{2}}$ $\displaystyle=$ $\displaystyle\frac{K_{0}(\hbar
v_{f})^{2}}{2\pi}\int_{0}^{k_{c}}k^{\prime}dk^{\prime}e^{-\frac{\xi^{2}(k^{2}+k^{\prime
2})}{2}}\left[\frac{(E-\tilde{\Sigma})}{(E-\tilde{\Sigma})^{2}-(1-\alpha)^{2}(\hbar
v_{f}k^{\prime})^{2}}I_{0}(kk^{\prime}\xi^{2})+\frac{s(1-\alpha)\hbar
v_{f}k^{\prime}}{(E-\tilde{\Sigma})^{2}-(1-\alpha)^{2}(\hbar
v_{f}k^{\prime})^{2}}I_{1}(kk^{\prime}\xi^{2})\right]$ $\displaystyle=$
$\displaystyle\frac{K_{0}(\hbar
v_{f})^{2}}{2\pi}\int_{0}^{k_{c}}k^{\prime}dk^{\prime}[1+o(k^{2}\xi^{2})]e^{-\frac{\xi^{2}k^{\prime
2}}{2}}\frac{(E-\tilde{\Sigma})[1+o(k^{2}\xi^{2})]+s(1-\alpha)\hbar
v_{f}k^{\prime}[\frac{kk^{\prime}\xi^{2}}{2}+o(k^{2}\xi^{2})]}{(E-\tilde{\Sigma})^{2}-(1-\alpha)^{2}(\hbar
v_{f}k^{\prime})^{2}}$ $\displaystyle\approx$ $\displaystyle\frac{K_{0}(\hbar
v_{f})^{2}}{2\pi}\int_{0}^{k_{c}}k^{\prime}dk^{\prime}\frac{(E-\tilde{\Sigma})e^{-\frac{\xi^{2}k^{\prime
2}}{2}}}{(E-\tilde{\Sigma})^{2}-(1-\alpha)^{2}(\hbar
v_{f}k^{\prime})^{2}}+s(1-\alpha)\hbar v_{f}k\frac{\frac{\xi^{2}k^{\prime
2}}{2}e^{-\frac{\xi^{2}k^{\prime
2}}{2}}}{(E-\tilde{\Sigma})^{2}-(1-\alpha)^{2}(\hbar v_{f}k^{\prime})^{2}}$
$\displaystyle=$
$\displaystyle\frac{K_{0}}{4\pi(1-\alpha)^{2}}\int_{0}^{\frac{\xi^{2}k_{c}^{2}}{2}}dx\frac{(E-\tilde{\Sigma})e^{-x}}{(E-\tilde{\Sigma})^{2}\tilde{\xi}^{2}-x}+s(1-\alpha)\hbar
v_{f}k\frac{xe^{-x}}{(E-\tilde{\Sigma})^{2}\tilde{\xi}^{2}-x}$
$\displaystyle=$
$\displaystyle\frac{K_{0}}{4\pi(1-\alpha)^{2}}\left[(E-\tilde{\Sigma})e^{-(E-\tilde{\Sigma})^{2}\tilde{\xi}^{2}}\left({\rm
Ei}[(E-\tilde{\Sigma})^{2}\tilde{\xi}^{2}]-{\rm
Ei}[(E-\tilde{\Sigma})^{2}\tilde{\xi}^{2}-\Lambda^{2}\tilde{\xi}^{2}]\right)-s(1-\alpha)\hbar
v_{f}k(1-e^{-\Lambda^{2}\tilde{\xi}^{2}})\right.$
$\displaystyle\left.+s(1-\alpha)\hbar
v_{f}k(E-\tilde{\Sigma})^{2}\tilde{\xi}^{2}e^{-(E-\tilde{\Sigma})^{2}\tilde{\xi}^{2}}\left({\rm
Ei}[(E-\tilde{\Sigma})^{2}\tilde{\xi}^{2}]-{\rm
Ei}[(E-\tilde{\Sigma})^{2}\tilde{\xi}^{2}-\Lambda^{2}\tilde{\xi}^{2}]\right)\right]$
$\displaystyle\approx$
$\displaystyle\frac{K_{0}}{4\pi(1-\alpha)^{2}}\left\\{(E-\tilde{\Sigma})\left[\ln\frac{(E-\tilde{\Sigma})^{2}}{(E-\tilde{\Sigma})^{2}-\Lambda^{2}}+\Lambda^{2}\tilde{\xi}^{2}\right]-s(1-\alpha)\hbar
v_{f}k\left[\Lambda^{2}-(E-\tilde{\Sigma})^{2}\ln\frac{(E-\tilde{\Sigma})^{2}}{(E-\tilde{\Sigma})^{2}-\Lambda^{2}}\right]\tilde{\xi}^{2}\right\\}$

where $\tilde{\xi}^{2}=\frac{\xi^{2}}{2(\hbar v_{f})^{2}(1-\alpha)^{2}}$,
$\tilde{k}=s(1-\alpha)E_{c}$, $E_{c}=\hbar v_{f}k_{c}$, and
$\Lambda=(1-\alpha)(\hbar v_{f}k_{c})$. ${\rm Ei}(x)$ is exponential integral
function and can be expanded as ${\rm Ei}(x)=\gamma+\ln x+x+o(x^{2})$. The
first term in square brackets is consistent with the short-range disorder when
setting $\alpha=0$ and $\xi=0$, which can be solved in two limits
$|E|\ll\Gamma_{0}$ and $|E|\gg\Gamma_{0}$, where
$\Gamma_{0}=E_{c}e^{-2\pi/K_{0}}$ is an exponential small energy scale defined
by the imaginary part of self-energy at Dirac point Ostrovsky . Similarly, we
can simplify the first term in square brackets in the last line of Eq.(52) at
two limits.

At the limit $|E|\ll\Gamma_{0}$, we estimate

$\displaystyle(E-\tilde{\Sigma})\ln\frac{(E-\tilde{\Sigma})^{2}}{(E-\tilde{\Sigma})^{2}-\Lambda^{2}}\approx$
$\displaystyle(E-\Sigma_{1}-i\Sigma_{2})\ln\frac{\Sigma_{2}^{2}+2i\Sigma_{2}(E-\Sigma_{1})}{\Lambda^{2}}=(E-\Sigma_{1}-i\Sigma_{2})\left[\ln\frac{\Sigma_{2}^{2}}{\Lambda^{2}}+\ln(1+2i\frac{E-\Sigma_{1}}{\Sigma_{2}})\right]$
(53) $\displaystyle\approx$
$\displaystyle(E-\Sigma_{1}-i\Sigma_{2})\left[\ln\frac{\Sigma_{2}^{2}}{\Lambda^{2}}+2i\frac{E-\Sigma_{1}}{\Sigma_{2}}\right]$
$\displaystyle\approx$
$\displaystyle(E-\Sigma_{1})\ln\frac{\Sigma_{2}^{2}}{\Lambda^{2}}+2(E-\Sigma_{1})-i\Sigma_{2}\ln\frac{\Sigma_{2}^{2}}{\Lambda^{2}}$

and

$\displaystyle-(E-\tilde{\Sigma})^{2}\ln\frac{(E-\tilde{\Sigma})^{2}}{(E-\tilde{\Sigma})^{2}-\Lambda^{2}}\approx$
$\displaystyle[2i(E-\Sigma_{1})\Sigma_{2}+\Sigma_{2}^{2}]\left[\ln\frac{\Sigma_{2}^{2}}{\Lambda^{2}}+2i\frac{E-\Sigma_{1}}{\Sigma_{2}}\right]\approx\Sigma_{2}^{2}\ln\frac{\Sigma_{2}^{2}}{\Lambda^{2}}$
(54)

In this condition, the Eq. (52) can be further reduced as

$\displaystyle\Sigma(\bm{k}s,E)\approx\frac{K_{0}}{4\pi(1-\alpha)^{2}}\left\\{(E-\Sigma_{1})\ln\frac{\Sigma_{2}^{2}}{\Lambda^{2}}+2(E-\Sigma_{1})-i\Sigma_{2}\ln\frac{\Sigma_{2}^{2}}{\Lambda^{2}}-s(1-\alpha)\hbar
v_{f}k\Lambda^{2}\tilde{\xi}^{2}\right\\}$ (55)

The above self-consistent equation can be separated into three parts
corresponding to the $\Sigma_{1}$, $i\Sigma_{2}$, and $-s\alpha\hbar v_{f}k$
parts in the self-energy function,

$\displaystyle\Sigma_{1}=$
$\displaystyle\frac{K_{0}}{4\pi(1-\alpha)^{2}}\left[(E-\Sigma_{1})\ln\frac{\Sigma_{2}^{2}}{\Lambda^{2}}+2(E-\Sigma_{1})\right]$
(56) $\displaystyle i\Sigma_{2}=$
$\displaystyle\frac{K_{0}}{4\pi(1-\alpha)^{2}}(-i\Sigma_{2})\ln\frac{\Sigma_{2}^{2}}{\Lambda^{2}}$
$\displaystyle-s\alpha\hbar v_{f}k=$
$\displaystyle\frac{K_{0}}{4\pi(1-\alpha)^{2}}[-s(1-\alpha)\hbar
v_{f}k]\Lambda^{2}\tilde{\xi}^{2}$

Solving the above self-consistent equations, we can approximately obtain

$\displaystyle\Sigma_{1}=$ $\displaystyle-\frac{2\pi(1-\alpha)^{2}}{K_{0}}E$
(57) $\displaystyle\Sigma_{2}=$ $\displaystyle-\Lambda
e^{-\frac{2\pi(1-\alpha)^{2}}{K_{0}}}$ $\displaystyle\alpha=$
$\displaystyle\frac{1}{2}-\sqrt{\frac{1}{4}-K_{0}\frac{\xi^{2}k_{c}^{2}}{8\pi}}$

At the limit $|E|\gg|{\rm Im}\Sigma|$, we estimate

$(E-\tilde{\Sigma})\ln\frac{(E-\tilde{\Sigma})^{2}}{(E-\tilde{\Sigma})^{2}-\Lambda^{2}}\approx
E\ln\frac{E^{2}e^{-i\pi{\rm
sgn}(E)}}{\Lambda^{2}}=E\left[\ln\frac{E^{2}}{\Lambda^{2}}-i\pi{\rm
sgn}(E)\right]=E\ln\frac{E^{2}}{\Lambda^{2}}-i\pi|E|$ (58)
$\displaystyle-(E-\tilde{\Sigma})^{2}\ln\frac{(E-\tilde{\Sigma})^{2}}{(E-\tilde{\Sigma})^{2}-\Lambda^{2}}\approx$
$\displaystyle-[E^{2}-2iE\Sigma_{2}]\left[\ln\frac{E^{2}}{\Lambda^{2}}-i\pi{\rm
sgn}(E)\right]\approx-E^{2}\ln\frac{E^{2}}{\Lambda^{2}}+i\pi E|E|$ (59)

In this condition, the Eq. (52) can be further reduced as

$\displaystyle\Sigma(\bm{k}s,E)\approx\frac{K_{0}}{4\pi(1-\alpha)^{2}}\left\\{E\ln\frac{E^{2}}{\Lambda^{2}}-i\pi|E|+(E-i\Sigma_{2})\Lambda^{2}\tilde{\xi}^{2}-s(1-\alpha)\hbar
v_{f}k\Lambda^{2}\tilde{\xi}^{2}\right\\}$ (60)

Similarly, this self-consistent equation can be solved by separating into
three parts

$\displaystyle\Sigma_{1}=$
$\displaystyle\frac{K_{0}}{4\pi(1-\alpha)^{2}}(E\ln\frac{E^{2}}{\Lambda^{2}}+E\Lambda^{2}\tilde{\xi}^{2})$
(61) $\displaystyle i\Sigma_{2}=$
$\displaystyle\frac{K_{0}}{4\pi(1-\alpha)^{2}}(-i\pi|E|-i\Sigma_{2}\Lambda^{2}\tilde{\xi}^{2})$
$\displaystyle s\alpha\hbar v_{f}k=$
$\displaystyle\frac{K_{0}}{4\pi(1-\alpha)^{2}}[-s(1-\alpha)\hbar
v_{f}k]\Lambda^{2}\tilde{\xi}^{2}$

Solving the above self-consistent equations, we can approximately obtain

$\displaystyle\Sigma_{1}=$
$\displaystyle\frac{K_{0}}{4\pi(1-\alpha)^{2}}E\ln\frac{E^{2}}{\Lambda^{2}}+\frac{K_{0}\xi^{2}k_{c}^{2}}{8\pi(1-\alpha)^{2}}E$
(62) $\displaystyle\Sigma_{2}=$ $\displaystyle-\frac{\pi
K_{0}}{4\pi(1-\alpha)^{2}}|E|$ $\displaystyle\alpha=$
$\displaystyle\frac{1}{2}-\sqrt{\frac{1}{4}-K_{0}\frac{\xi^{2}k_{c}^{2}}{8\pi}}$

Combining Eq. (57 and Eq. (62, the resulting self-energy function obtained
from SCBA is

$\Sigma(\bm{k}s,E)=\left\\{\begin{aligned}
&-\frac{2\pi(1-\alpha)^{2}}{K_{0}}E-s\alpha\hbar v_{f}k-i\Lambda
e^{-\frac{2\pi(1-\alpha)^{2}}{K_{0}}};&|E|\ll\Gamma_{0}\\\
&\frac{K_{0}}{4\pi(1-\alpha)^{2}}E\ln\frac{E^{2}}{\Lambda^{2}}+\frac{\alpha}{1-\alpha}E-s\alpha\hbar
v_{f}k-i\frac{\pi
K_{0}}{4\pi(1-\alpha)^{2}}|E|;&|E|\gg\Gamma_{0}\end{aligned}\right.$ (63)

with

$\alpha=\frac{1}{2}-\sqrt{\frac{1}{4}-K_{0}\frac{\xi^{2}k_{c}^{2}}{8\pi}}.$
(64)

From the SCBA’s results, we find that the dimensionless parameter $\alpha$,
which controls the momentum linear dependent term in self-energy, has a upper
limit $\frac{1}{2}$. Please note that this result naturally revert to the
Born’s result at weak disorder limit,
$\alpha\xrightarrow{K_{0}\frac{\xi^{2}k_{c}^{2}}{8\pi}\ll\frac{1}{4}}K_{0}\frac{\xi^{2}k_{c}^{2}}{8\pi}$.
To sum up, we prove that the self-energy function is momentum-dependent via
the SCBA. More Feynman diagrams beyond the SCBA is hard to calculate, which
will be further validated by the numerical simulations.

###  B.3 Long-range Screened Coulomb Potential

For simplicity, most of our discussions in this paper are based on the random
Gaussian potential. Here, we present a calculation using the long-ranged
screened Coulomb potential. We will elucidate that, the main findings shown in
this paper are robust, independent of the specific form of impurity potential.

The long-range screened Coulomb potential has the form in real space as

$V(\bm{r})=\pm\frac{e^{2}}{\kappa r}e^{-q_{s}r}$ (65)

where $\kappa$ is the static dielectric constant, scatters of $\pm$ are
randomly distributed with equal probability, and $q_{s}$ is the Thomas-Fermi
screening constant. It can be written in momentum space as

$\displaystyle V(\bm{q})=$ $\displaystyle\int
d\bm{r}V(\bm{r})e^{-i\bm{q}\cdot\bm{r}}=\pm\int d\bm{r}\frac{e^{2}}{\kappa
r}e^{-q_{s}r}e^{-i\bm{q}\cdot\bm{r}}=\pm\int rdrd\theta\frac{e^{2}}{\kappa
r}e^{-(q_{s}+iq\cos\theta)r}=\pm\frac{e^{2}}{\kappa}\int drd\theta
e^{-(q_{s}+iq\cos\theta)r}$ (66) $\displaystyle=$
$\displaystyle\pm\frac{e^{2}}{\kappa}\int
d\theta\frac{e^{-(q_{s}+iq\cos\theta)r}}{q_{s}+iq\cos\theta}\bigg{|}^{0}_{+\infty}=\pm\frac{e^{2}}{\kappa}\int
d\theta\frac{1}{q_{s}+iq\cos\theta}=\pm\frac{e^{2}}{\kappa}\frac{2}{\sqrt{q^{2}_{s}+q^{2}}}\arctan(\sqrt{\frac{q_{s}-iq}{q_{s}+iq}}\tan\frac{\theta}{2})\bigg{|}^{\pi}_{-\pi}$
$\displaystyle=$
$\displaystyle\pm\frac{e^{2}}{\kappa}\frac{2\pi}{\sqrt{q_{s}^{2}+q^{2}}}$

Similar to the long-ranged Gaussian potential, we obtain the self-energy of
long-range screened Coulomb potential by Born approximation

$\Sigma(\bm{k}s,E)=\sum_{s^{\prime}}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\frac{V^{2}(\bm{k}-\bm{k}^{\prime})}{\mathcal{V}}G^{R}_{0}(\bm{k}^{\prime}s^{\prime},E)\frac{1+ss^{\prime}\cos\theta}{2}$
(67)

After plugging the expression of $V(\bm{k}-\bm{k}^{\prime})$ into this
equation, it is obtained

$\displaystyle\Sigma(\bm{k}s,E)=$
$\displaystyle\sum_{s^{\prime}}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\frac{N_{i}}{\mathcal{V}}(\frac{2\pi
e^{2}}{\kappa})^{2}\frac{1}{q_{s}^{2}+|\bm{k}-\bm{k}^{\prime}|^{2}}G^{R}_{0}(\bm{k}^{\prime}s^{\prime},E)\frac{1+ss^{\prime}\cos\theta}{2}$
(68) $\displaystyle=$
$\displaystyle\sum_{s^{\prime}}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\frac{n_{i}V_{0}^{2}}{q_{s}^{2}+k^{2}+k^{\prime
2}-2kk^{\prime}\cos\theta}G^{R}_{0}(\bm{k}^{\prime}s^{\prime},E)\frac{1+ss^{\prime}\cos\theta}{2}$

where $N_{i}$ and $n_{i}=N_{i}/\mathcal{V}$ are the number and concentration
of impurities and $V_{0}=\frac{2\pi e^{2}}{\kappa}$. Then we the above
equation under the domain $\frac{E}{\hbar v_{f}},k\ll q_{s}$.

For $E>0$, the imaginary part of the self-energy is

$\displaystyle{\rm Im}\Sigma(\bm{k}s,E)=$
$\displaystyle\sum_{s^{\prime}}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\frac{n_{i}V_{0}^{2}}{q_{s}^{2}+k^{2}+k^{\prime
2}-2kk^{\prime}\cos\theta}\frac{1+ss^{\prime}\cos\theta}{2}[-\pi\delta(E-s^{\prime}\hbar
v_{f}k^{\prime})]$ (69) $\displaystyle=$ $\displaystyle-\frac{\pi
n_{i}V_{0}^{2}}{2}\int\frac{k^{\prime}dk^{\prime}d\theta}{(2\pi)^{2}}\frac{1+s\cos\theta}{q_{s}^{2}+k^{2}+k^{\prime
2}-2kk^{\prime}\cos\theta}\delta(E-\hbar v_{f}k^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{n_{i}V_{0}^{2}\tilde{E}}{8\pi\hbar v_{f}}\int
d\theta\frac{1+s\cos\theta}{q_{s}^{2}+k^{2}+\tilde{E}^{2}-2k\tilde{E}\cos\theta}$
$\displaystyle=$ $\displaystyle-\frac{n_{i}V_{0}^{2}\tilde{E}}{8\pi\hbar
v_{f}}\left(-\frac{1}{2k\tilde{E}}\right)\int
d\theta\frac{s(q_{s}^{2}+k^{2}+\tilde{E}^{2}-2k\tilde{E}\cos\theta)-[s(q_{s}^{2}+k^{2}+\tilde{E}^{2})+2k\tilde{E}]}{q_{s}^{2}+k^{2}+\tilde{E}^{2}-2k\tilde{E}\cos\theta}$
$\displaystyle=$ $\displaystyle\frac{n_{i}V^{2}_{0}}{16\pi\hbar
v_{f}k}\left\\{2\pi s-[s(q_{s}^{2}+k^{2}+\tilde{E}^{2})+2k\tilde{E}]\int
d\theta\frac{1}{q_{s}^{2}+k^{2}+\tilde{E}^{2}-2k\tilde{E}\cos\theta}\right\\}$
$\displaystyle=$ $\displaystyle\frac{n_{i}V^{2}_{0}}{8\hbar
v_{f}k}\left[s-\frac{s(q_{s}^{2}+k^{2}+\tilde{E}^{2})+2k\tilde{E}}{\sqrt{(q_{s}^{2}+k^{2}+\tilde{E}^{2}-2k\tilde{E})(q_{s}^{2}+k^{2}+\tilde{E}^{2}+2k\tilde{E})}}\right]$
$\displaystyle=$ $\displaystyle\frac{n_{i}V^{2}_{0}}{8\hbar
v_{f}k}s\left[1-\sqrt{\frac{q^{2}_{s}+(k+s\tilde{E})^{2}}{q^{2}_{s}+(k-s\tilde{E})^{2}}}\right]=\frac{n_{i}V^{2}_{0}}{8\hbar
v_{f}k}s\left[1-\sqrt{1+\frac{4sk\tilde{E}}{q^{2}_{s}+(k-s\tilde{E})^{2}}}\right]$
$\displaystyle\approx$ $\displaystyle-\frac{n_{i}V^{2}_{0}}{4\hbar
v_{f}}\frac{\tilde{E}}{q^{2}_{s}+(k-s\tilde{E})^{2}}$

where $\tilde{E}=\frac{E}{\hbar v_{f}}$.

For $E<0$, the imaginary part of the self-energy is

$\displaystyle{\rm Im}\Sigma(\bm{k}s,E)=$
$\displaystyle\sum_{s^{\prime}}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\frac{n_{i}V_{0}^{2}}{q_{s}^{2}+k^{2}+k^{\prime
2}-2kk^{\prime}\cos\theta}\frac{1+ss^{\prime}\cos\theta}{2}[-\pi\delta(E-s^{\prime}\hbar
v_{f}k^{\prime})]$ (70) $\displaystyle=$ $\displaystyle-\frac{\pi
n_{i}V_{0}^{2}}{2}\int\frac{k^{\prime}dk^{\prime}d\theta}{(2\pi)^{2}}\frac{1-s\cos\theta}{q_{s}^{2}+k^{2}+k^{\prime
2}-2kk^{\prime}\cos\theta}\delta(E+\hbar v_{f}k^{\prime})$ $\displaystyle=$
$\displaystyle\frac{n_{i}V_{0}^{2}\tilde{E}}{8\pi\hbar v_{f}}\int
d\theta\frac{1-s\cos\theta}{q_{s}^{2}+k^{2}+\tilde{E}^{2}+2k\tilde{E}\cos\theta}$
$\displaystyle\approx$ $\displaystyle\frac{n_{i}V^{2}_{0}}{4\hbar
v_{f}}\frac{\tilde{E}}{q^{2}_{s}+(k-s\tilde{E})^{2}}$

Combining two conditions, we can get

$\displaystyle{\rm Im}\Sigma(\bm{k}s,E)=-\frac{n_{i}V^{2}_{0}}{4\hbar
v_{f}}\frac{|\tilde{E}|}{q^{2}_{s}+(k-s\tilde{E})^{2}}$ (71)

Then the corresponding real part of self-energy function can be calculated by
the Kramer-Kronig relation

$\displaystyle{\rm Re}\Sigma(\bm{k}s,E)=$
$\displaystyle\frac{1}{\pi}\mathcal{P}\int^{E_{c}}_{-E_{c}}dE^{\prime}\frac{{\rm
Im}\Sigma(\bm{k}s,E^{\prime})}{E^{\prime}-E}=-\frac{n_{i}V^{2}_{0}}{4\pi\hbar
v_{f}}\int^{\tilde{E}_{c}}_{-\tilde{E}_{c}}d\tilde{E}^{\prime}\frac{1}{\tilde{E}^{\prime}-\tilde{E}}\frac{|\tilde{E}^{\prime}|}{q^{2}_{s}+(k-s\tilde{E}^{\prime})^{2}}$
(72) $\displaystyle=$ $\displaystyle-\frac{n_{i}V^{2}_{0}}{4\pi\hbar
v_{f}}\left\\{\int^{\tilde{E}_{c}}_{0}d\tilde{E}^{\prime}\frac{\tilde{E}^{\prime}}{(\tilde{E}^{\prime}-\tilde{E})[q^{2}_{s}+(k-s\tilde{E}^{\prime})^{2}]}+\int_{-\tilde{E}_{c}}^{0}d\tilde{E}^{\prime}\frac{-\tilde{E}^{\prime}}{(\tilde{E}^{\prime}-\tilde{E})[q^{2}_{s}+(k-s\tilde{E}^{\prime})^{2}]}\right\\}$
$\displaystyle=$ $\displaystyle-\frac{n_{i}V^{2}_{0}}{4\pi\hbar
v_{f}}\left\\{\int^{\tilde{E}_{c}}_{0}d\tilde{E}^{\prime}\frac{\tilde{E}^{\prime}}{(\tilde{E}^{\prime}-\tilde{E})[q^{2}_{s}+(k-s\tilde{E}^{\prime})^{2}]}+\int^{\tilde{E}_{c}}_{0}d\tilde{E}^{\prime}\frac{\tilde{E}^{\prime}}{(-\tilde{E}^{\prime}-\tilde{E})[q^{2}_{s}+(k+s\tilde{E}^{\prime})^{2}]}\right\\}$
$\displaystyle=$ $\displaystyle-\frac{n_{i}V^{2}_{0}}{4\pi\hbar
v_{f}}\int^{\tilde{E}_{c}}_{0}d\tilde{E}^{\prime}\left\\{\frac{\tilde{E}^{\prime}}{(\tilde{E}^{\prime}-\tilde{E})[q^{2}_{s}+(k-s\tilde{E}^{\prime})^{2}]}-\frac{\tilde{E}^{\prime}}{(\tilde{E}^{\prime}+\tilde{E})[q^{2}_{s}+(k+s\tilde{E}^{\prime})^{2}]}\right\\}$
$\displaystyle\approx$ $\displaystyle-\frac{n_{i}V^{2}_{0}}{4\pi\hbar
v_{f}q^{2}_{s}}\int^{\tilde{E}_{c}}_{0}d\tilde{E}^{\prime}\left\\{\frac{\tilde{E}^{\prime}}{\tilde{E}^{\prime}-\tilde{E}}\left[1-\frac{(k-s\tilde{E}^{\prime})^{2}}{q^{2}_{s}}\right]-\frac{\tilde{E}^{\prime}}{\tilde{E}^{\prime}+\tilde{E}}\left[1-\frac{(k+s\tilde{E}^{\prime})^{2}}{q^{2}_{s}}\right]\right\\}$
$\displaystyle=$ $\displaystyle-\frac{n_{i}V^{2}_{0}}{4\pi\hbar
v_{f}q^{2}_{s}}\int^{\tilde{E}_{c}}_{0}d\tilde{E}^{\prime}\left[\left(\frac{\tilde{E}^{\prime}}{\tilde{E}^{\prime}-\tilde{E}}-\frac{\tilde{E}^{\prime}}{\tilde{E}^{\prime}+\tilde{E}}\right)(1-\frac{k^{2}}{q^{2}_{s}}-\frac{\tilde{E}^{\prime
2}}{q^{2}_{s}})+\left(\frac{\tilde{E}^{\prime}}{\tilde{E}^{\prime}-\tilde{E}}+\frac{\tilde{E}^{\prime}}{\tilde{E}^{\prime}+\tilde{E}}\right)\frac{2sk\tilde{E}^{\prime}}{q^{2}_{s}}\right]$
$\displaystyle=$ $\displaystyle-\frac{n_{i}V^{2}_{0}}{2\pi\hbar
v_{f}q^{2}_{s}}\int^{\tilde{E}_{c}}_{0}d\tilde{E}^{\prime}\left[\frac{\tilde{E}^{\prime}}{\tilde{E}^{\prime
2}-\tilde{E}^{2}}\tilde{E}(1-\frac{k^{2}}{q^{2}_{s}})+\frac{\tilde{E}^{\prime
3}}{\tilde{E}^{\prime
2}-\tilde{E}^{2}}\frac{2sk}{q^{2}_{s}}-\frac{\tilde{E}^{\prime
3}}{\tilde{E}^{\prime 2}-\tilde{E}^{2}}\frac{\tilde{E}}{q^{2}_{s}}\right]$
$\displaystyle=$ $\displaystyle-\frac{n_{i}V^{2}_{0}}{2\pi\hbar
v_{f}q^{2}_{s}}\left\\{\tilde{E}(\ln\tilde{E}_{c}-\ln\tilde{E})(1-\frac{k^{2}}{q^{2}_{s}})+\left[\frac{1}{2}\tilde{E}^{2}_{c}+\tilde{E}^{2}(\ln\tilde{E}_{c}-\ln\tilde{E})\right]\left(\frac{2sk}{q^{2}_{s}}-\frac{\tilde{E}}{q^{2}_{s}}\right)\right\\}$

where $E_{c}=\hbar v_{f}k_{c}$.

Thus, if we consider the domain $\frac{E}{\hbar v_{f}},k\ll q_{s}$, the self-
energy function can be estimated as

$\displaystyle{\rm Re}\Sigma(\bm{k}s,E)$ $\displaystyle=$
$\displaystyle\frac{n_{i}V^{2}_{0}}{2\pi(\hbar
v_{f})^{2}q^{2}_{s}}E\ln|\frac{E}{E_{c}}|+\alpha E-s\alpha\hbar
v_{f}k+O(kE^{2}\ln E,k^{2}E\ln E,E^{3}\ln E)$ (73) $\displaystyle{\rm
Im}\Sigma(\bm{k}s,E)$ $\displaystyle=$
$\displaystyle-\frac{n_{i}V^{2}_{0}}{4(\hbar
v_{f})^{2}q^{2}_{s}}|E|+O(kE^{2}\ln E,k^{2}E\ln E,E^{3}\ln E)$ (74)

where

$\alpha=\frac{n_{i}V_{0}^{2}E^{2}_{c}}{2\pi(\hbar v_{f}q_{s})^{2}}$ (75)

It can be seen from the above derivation that, the self-energy function of
long-range Screened Coulomb potential has same expression as that of long-
ranged Gaussian potential expect for the value of $\alpha$. Physically, it can
be understood from the following picture: Around the Dirac point, the form of
random potential is not important, because the wave-length of electron is
longer than the spatial range of random potentials. So the different forms of
random potentials give the similar results, as we shown in the paper. However,
away from the Dirac point, the form of disorder potential is relevant.

##  Appendix C Renormalization Group Analysis

In this section, we perform a Wilson’s renormalization group calculation on
the model that we studied in the main text. The purpose is two-fold. First,
the renormalization group analysis could help to clarify some effect beyond
SCBA to the momentum dependent self-energy function. Second, physically, the
obtained flow equations clearly demonstrate the renormalized velocity together
with the energy and disorder coupling constant, which provide a different
angle to understand our main conclusion.

####  C.0.1 Generating functional

At first, we expand the generating functional with cutoff prescription and
disorder averaging. For simplicity, we set sources $\bar{\eta}=0$ and
$\eta=0$.

$\displaystyle\langle Z\rangle=$ $\displaystyle\left\langle\int
D[\bar{\Psi},\Psi]_{\Lambda}\exp\left[i\int
dtd^{2}\bm{x}\bar{\Psi}_{\bm{x},t}(i\partial_{t}+iv_{f}\nabla\cdot\bm{\sigma})\Psi_{\bm{x},\tau}\right]\exp\left[-i\int
dtd^{2}\bm{x}V(\bm{x})\bar{\Psi}_{\bm{x},t}\Psi_{\bm{x},t}\right]\right\rangle$
(76) $\displaystyle=$ $\displaystyle\int
D[\bar{\Psi},\Psi]_{\Lambda}\exp\left[\int d\tau
d^{2}\bm{x}\bar{\Psi}_{\bm{x},\tau}(-\partial_{\tau}+iv_{f}\nabla\cdot\bm{\sigma})\Psi_{\bm{x},\tau}\right]$
$\displaystyle\left\langle 1-\int d\tau
d^{2}\bm{x}V(\bm{x})\bar{\Psi}_{\bm{x},\tau}\Psi_{\bm{x},\tau}+\frac{1}{2!}\int
d\tau
d^{2}\bm{x}d\tau^{\prime}d^{2}\bm{x}^{\prime}V(\bm{x})V(\bm{x}^{\prime})\bar{\Psi}_{\bm{x},\tau}\Psi_{\bm{x},\tau}\bar{\Psi}_{\bm{x}^{\prime},\tau^{\prime}}\Psi_{\bm{x}^{\prime},\tau^{\prime}}\right.$
$\displaystyle-\frac{1}{3!}\int
d\tau_{1}d^{2}\bm{x}_{1}d\tau_{2}d^{2}\bm{x}_{2}d\tau_{3}d^{2}\bm{x}_{3}V(\bm{x}_{1})V(\bm{x}_{2})V(\bm{x}_{3})\bar{\Psi}_{\bm{x}_{1},\tau_{1}}\Psi_{\bm{x}_{1},\tau_{1}}\bar{\Psi}_{\bm{x}_{2},\tau_{2}}\Psi_{\bm{x}_{2},\tau_{2}}\bar{\Psi}_{\bm{x}_{3},\tau_{3}}\Psi_{\bm{x}_{3},\tau_{3}}$
$\displaystyle+\left.\frac{1}{4!}\int
d\tau_{1}d^{2}\bm{x}_{1}d\tau_{2}d^{2}\bm{x}_{2}d\tau_{3}d^{2}\bm{x}_{3}d\tau_{4}d^{2}\bm{x}_{4}V(\bm{x}_{1})V(\bm{x}_{2})V(\bm{x}_{3})V(\bm{x}_{4})\bar{\Psi}_{\bm{x}_{1},\tau_{1}}\Psi_{\bm{x}_{1},\tau_{1}}\bar{\Psi}_{\bm{x}_{2},\tau_{2}}\Psi_{\bm{x}_{2},\tau_{2}}\bar{\Psi}_{\bm{x}_{3},\tau_{3}}\Psi_{\bm{x}_{3},\tau_{3}}\bar{\Psi}_{\bm{x}_{4},\tau_{4}}\Psi_{\bm{x}_{4},\tau_{4}}+\cdots\right\rangle$
$\displaystyle=$ $\displaystyle\int
D[\bar{\Psi},\Psi]_{\Lambda}\exp\left[\int\frac{d\omega}{2\pi}\frac{d^{2}\bm{k}}{(2\pi)^{2}}\bar{\Psi}_{\bm{k}\omega}(i\omega-
v_{f}\bm{k}\cdot\bm{\sigma})\Psi_{\bm{k}\omega}\right]\left\\{1+\frac{1}{2}\int
d\tau
d\tau^{\prime}\frac{d^{2}\bm{q}d^{2}\bm{k}d^{2}\bm{k}^{\prime}}{(2\pi)^{6}}\mathcal{K}(\bm{q})\bar{\Psi}_{\bm{k},\tau}\Psi_{\bm{k}-\bm{q},\tau}\bar{\Psi}_{\bm{k}^{\prime},\tau^{\prime}}\Psi_{\bm{k}^{\prime}+\bm{q},\tau^{\prime}}\right.$
$\displaystyle+\left.\frac{1}{8}\int
d\tau_{1}d\tau_{2}d\tau_{3}d\tau_{4}\frac{d^{2}\bm{q}d^{2}\bm{k}_{1}d^{2}\bm{k}_{2}}{(2\pi)^{6}}\frac{d^{2}\bm{q}^{\prime}d^{2}\bm{k}_{3}d^{2}\bm{k}_{4}}{(2\pi)^{6}}\mathcal{K}(\bm{q})\mathcal{K}(\bm{q}^{\prime})\bar{\Psi}_{\bm{k}_{1},\tau_{1}}\Psi_{\bm{k}_{1}-\bm{q},\tau_{1}}\bar{\Psi}_{\bm{k}_{2},\tau_{2}}\Psi_{\bm{k}_{2}+\bm{q},\tau_{2}}\bar{\Psi}_{\bm{k}_{3},\tau_{3}}\Psi_{\bm{k}_{3}-\bm{q}^{\prime},\tau_{3}}\bar{\Psi}_{\bm{k}_{4},\tau_{4}}\Psi_{\bm{k}_{4}+\bm{q}^{\prime},\tau_{4}}+\cdots\right\\}$

Notice that there are 3 combinations of $\langle
V(\bm{x}_{1})V(\bm{x}_{2})\rangle\langle V(\bm{x}_{3})V(\bm{x}_{4})\rangle$.
Here, we have transformed real time into imaginary time through Wick’s
rotation, $it\to\tau$, since it is convenient to use Matsubara Green functions
in the following calculations.

####  C.0.2 Momentum shell decomposition

According to Wilson’s approach, we divide the integration variables
$\bar{\Psi}(\bm{k})$ and $\Psi(\bm{k})$ into two groups by a dimensionless
variable $b>1$,

$\bar{\Psi}(\bm{k})=\left\\{\begin{aligned}
&\bar{\Psi}^{<}(\bm{k});&0\leq|\bm{k}|<\Lambda/b;\\\
&\bar{\Psi}^{>}(\bm{k});&\Lambda/b\leq|\bm{k}|<\Lambda;\end{aligned}\right.\ \
\ \Psi(k)=\left\\{\begin{aligned}
&\Psi^{<}(\bm{k});&0\leq|\bm{k}|<\Lambda/b;\\\
&\Psi^{>}(\bm{k});&\Lambda/b\leq|\bm{k}|<\Lambda;\end{aligned}\right.$ (77)

We replace the old $\bar{\Psi}$ and $\Psi$ with
$\bar{\Psi}^{<}+\bar{\Psi}^{>}$ and $\Psi^{<}+\Psi^{>}$, and rewrite the
generating functional as

$\displaystyle\langle Z\rangle=$ $\displaystyle\int
D[\bar{\Psi}^{<},\Psi^{<}]e^{S_{0}[\bar{\Psi}^{<},\Psi^{<}]}\left\\{\int
D[\bar{\Psi}^{>},\Psi^{>}]e^{S_{0}[\bar{\Psi}^{>},\Psi^{>}]}\right\\}\left\\{1+\int
d\tau
d\tau^{\prime}\frac{d^{2}\bm{q}d^{2}\bm{k}d^{2}\bm{k}^{\prime}}{(2\pi)^{6}}\mathcal{K}(\bm{q})\bar{\Psi}^{<}_{\bm{k},\tau}\Psi^{<}_{\bm{k}-\bm{q},\tau}\Psi^{<}_{\bm{k}^{\prime}+\bm{q},\tau^{\prime}}\bar{\Psi}^{<}_{\bm{k}^{\prime},\tau^{\prime}}\right.$
(78) $\displaystyle+\int d\tau
d\tau^{\prime}\frac{d^{2}\bm{q}d^{2}\bm{k}d^{2}\bm{k}^{\prime}}{(2\pi)^{6}}\mathcal{K}(\bm{q})\bar{\Psi}^{<}_{\bm{k},\tau}\Psi^{<}_{\bm{k}^{\prime}+\bm{q},\tau^{\prime}}\langle\Psi^{>}_{\bm{k}-\bm{q},\tau}\bar{\Psi}^{>}_{\bm{k}^{\prime},\tau^{\prime}}\rangle$
$\displaystyle+\frac{1}{2}\int
d\tau_{1}d\tau_{2}d\tau_{3}d\tau_{4}\frac{d^{2}\bm{q}d^{2}\bm{k}_{1}d^{2}\bm{k}_{2}}{(2\pi)^{6}}\frac{d^{2}\bm{q}^{\prime}d^{2}\bm{k}_{3}d^{2}\bm{k}_{4}}{(2\pi)^{6}}\mathcal{K}(\bm{q})\mathcal{K}(\bm{q}^{\prime})\bar{\Psi}^{<}_{\bm{k}_{1},\tau_{1}}\Psi^{<}_{\bm{k}_{3}-\bm{q}^{\prime},\tau_{3}}\bar{\Psi}^{<}_{\bm{k}_{4},\tau_{4}}\Psi^{<}_{\bm{k}_{2}+\bm{q},\tau_{2}}\langle\Psi^{>}_{\bm{k}_{1}-\bm{q},\tau_{1}}\bar{\Psi}^{>}_{\bm{k}_{3},\tau_{3}}\rangle\langle\Psi^{>}_{\bm{k}_{4}+\bm{q}^{\prime},\tau_{4}}\bar{\Psi}^{>}_{\bm{k}_{2},\tau_{2}}\rangle$
$\displaystyle+\frac{1}{2}\int
d\tau_{1}d\tau_{2}d\tau_{3}d\tau_{4}\frac{d^{2}\bm{q}d^{2}\bm{k}_{1}d^{2}\bm{k}_{2}}{(2\pi)^{6}}\frac{d^{2}\bm{q}^{\prime}d^{2}\bm{k}_{3}d^{2}\bm{k}_{4}}{(2\pi)^{6}}\mathcal{K}(\bm{q})\mathcal{K}(\bm{q}^{\prime})\bar{\Psi}^{<}_{\bm{k}_{1},\tau_{1}}\Psi^{<}_{\bm{k}_{4}+\bm{q}^{\prime},\tau_{4}}\bar{\Psi}^{<}_{\bm{k}_{2},\tau_{2}}\Psi^{<}_{\bm{k}_{3}-\bm{q}^{\prime},\tau_{3}}\langle\Psi^{>}_{\bm{k}_{1}-\bm{q},\tau_{1}}\bar{\Psi}^{>}_{\bm{k}_{4},\tau_{4}}\rangle\langle\Psi^{>}_{\bm{k}_{2}+\bm{q},\tau_{2}}\bar{\Psi}^{>}_{\bm{k}_{3},\tau_{3}}\rangle$
$\displaystyle+\left.\int
d\tau_{1}d\tau_{2}d\tau_{3}d\tau_{4}\frac{d^{2}\bm{q}d^{2}\bm{k}_{1}d^{2}\bm{k}_{2}}{(2\pi)^{6}}\frac{d^{2}\bm{q}^{\prime}d^{2}\bm{k}_{3}d^{2}\bm{k}_{4}}{(2\pi)^{6}}\mathcal{K}(\bm{q})\mathcal{K}(\bm{q}^{\prime})\bar{\Psi}^{<}_{\bm{k}_{1},\tau_{1}}\Psi^{<}_{\bm{k}_{1}-\bm{q},\tau_{1}}\bar{\Psi}^{<}_{\bm{k}_{4},\tau_{4}}\Psi^{<}_{\bm{k}_{3}-\bm{q}^{\prime},\tau_{3}}\langle\Psi^{>}_{\bm{k}_{2}+\bm{q},\tau_{2}}\bar{\Psi}^{>}_{\bm{k}_{3},\tau_{3}}\rangle\langle\Psi^{>}_{\bm{k}_{4}+\bm{q}^{\prime},\tau_{4}}\bar{\Psi}^{>}_{\bm{k}_{2},\tau_{2}}\rangle+\cdots\right\\}$
$\displaystyle=\int
D[\bar{\Psi}^{<},\Psi^{<}]e^{S_{0}[\bar{\Psi}^{<},\Psi^{<}]}\left\\{\int
D[\bar{\Psi}^{>},\Psi^{>}]e^{S_{0}[\bar{\Psi}^{>},\Psi^{>}]}\right\\}\left\\{1+\int
d\tau
d\tau^{\prime}\frac{d^{2}\bm{q}d^{2}\bm{k}d^{2}\bm{k}^{\prime}}{(2\pi)^{6}}\mathcal{K}(\bm{q})\bar{\Psi}^{<}_{\bm{k},\tau}\Psi^{<}_{\bm{k}-\bm{q},\tau}\Psi^{<}_{\bm{k}^{\prime}+\bm{q},\tau^{\prime}}\bar{\Psi}^{<}_{\bm{k}^{\prime},\tau^{\prime}}\right.$
$\displaystyle-\int d\tau
d\tau^{\prime}\frac{d^{2}\bm{k}}{(2\pi)^{2}}\bar{\Psi}^{<}_{\bm{k},\tau}\Psi^{<}_{\bm{k},\tau^{\prime}}\int\frac{d^{2}\bm{p}}{(2\pi)^{2}}\mathcal{K}(\bm{k}-\bm{p})\mathcal{G}^{>,0}(\bm{p},\tau-\tau^{\prime})$
$\displaystyle+\frac{1}{2}\int
d\tau_{1}d\tau_{2}d\tau_{3}d\tau_{4}\frac{d^{2}\bm{q}d^{2}\bm{k}d^{2}\bm{k}^{\prime}}{(2\pi)^{6}}\bar{\Psi}^{<}_{\bm{k},\tau_{1}}\Psi^{<}_{\bm{k}-\bm{q},\tau_{3}}\bar{\Psi}^{<}_{\bm{k}^{\prime},\tau_{4}}\Psi^{<}_{\bm{k}^{\prime}+\bm{q},\tau_{2}}\int\frac{d^{2}\bm{p}}{(2\pi)^{2}}\mathcal{K}(\bm{k}-\bm{p})\mathcal{K}(\bm{p}+\bm{q}-\bm{k})\mathcal{G}^{>,0}_{\bm{p},\tau_{1}-\tau_{3}}\mathcal{G}^{>,0}_{\bm{k}^{\prime}+\bm{q}-\bm{k}+\bm{p},\tau_{4}-\tau_{2}}$
$\displaystyle+\frac{1}{2}\int
d\tau_{1}d\tau_{2}d\tau_{3}d\tau_{4}\frac{d^{2}\bm{q}d^{2}\bm{k}d^{2}\bm{k}^{\prime}}{(2\pi)^{6}}\bar{\Psi}^{<}_{\bm{k},\tau_{1}}\Psi^{<}_{\bm{k}-\bm{q},\tau_{4}}\bar{\Psi}^{<}_{\bm{k}^{\prime},\tau_{2}}\Psi^{<}_{\bm{k}^{\prime}+\bm{q},\tau_{3}}\int\frac{d^{2}\bm{p}}{(2\pi)^{2}}\mathcal{K}(\bm{k}-\bm{p})\mathcal{K}(\bm{p}-\bm{k}+\bm{q})\mathcal{G}^{>,0}_{\bm{p},\tau_{1}-\tau_{4}}\mathcal{G}^{>,0}_{\bm{k}+\bm{k}^{\prime}-\bm{p},\tau_{2}-\tau_{3}}$
$\displaystyle+\left.\int
d\tau_{1}d\tau_{2}d\tau_{3}d\tau_{4}\frac{d^{2}\bm{q}d^{2}\bm{k}d^{2}\bm{k}^{\prime}}{(2\pi)^{6}}\bar{\Psi}^{<}_{\bm{k},\tau_{1}}\Psi^{<}_{\bm{k}-\bm{q},\tau_{1}}\bar{\Psi}^{<}_{\bm{k}^{\prime},\tau_{4}}\Psi^{<}_{\bm{k}^{\prime}+\bm{q},\tau_{3}}\int\frac{d^{2}\bm{p}}{(2\pi)^{2}}\mathcal{K}(\bm{q})\mathcal{K}(\bm{k}^{\prime}-\bm{p})\mathcal{G}^{>,0}_{\bm{p},\tau_{2}-\tau_{3}}\mathcal{G}^{>,0}_{\bm{p}+\bm{q},\tau_{4}-\tau_{2}}+\cdots\right\\}$

where
$S_{0}[\bar{\Psi}^{<},\Psi^{<}]=\int\frac{d\omega}{2\pi}\frac{d^{2}\bm{k}}{(2\pi)^{2}}\bar{\Psi}^{<}_{\bm{k},\omega}(i\omega-
v_{f}\bm{k}\cdot\bm{\sigma})\Psi^{<}_{\bm{k},\omega}$ is the unperturbed
action in the momentum shell $|\bm{k}|<\Lambda/b$,
$S_{0}[\bar{\Psi}^{>},\Psi^{>}]=\int\frac{d\omega}{2\pi}\frac{d^{2}\bm{k}}{(2\pi)^{2}}\bar{\Psi}^{>}_{\bm{k},\omega}(i\omega-
v_{f}\bm{k}\cdot\bm{\sigma})\Psi^{>}_{\bm{k},\omega}$ is the unperturbed
action in the momentum shell $\Lambda/b<|\bm{k}|<\Lambda$, and
$\langle\bar{\Psi}_{1}^{>}\Psi_{2}^{>}\rangle=\frac{\int
D[\bar{\Psi}^{>},\Psi^{>}]\Psi_{1}^{>}\bar{\Psi}_{2}^{>}e^{S_{0}[\bar{\Psi}^{>},\Psi^{>}]}}{\int
D[\bar{\Psi}^{>},\Psi^{>}]e^{S_{0}[\bar{\Psi}^{>},\Psi^{>}]}}=-\mathcal{G}^{>,0}_{12}$
denotes the unperturbed correction function for the fields $\bar{\Psi}^{>}$
and $\Psi^{>}$. The four terms containing integral of $\mathcal{G}^{>,0}$ in
the brace correspond in turn to the four one-loop RG diagrams shown in Fig. 5.
The diagram (a) is responsible for the renormalization of the energy and
velocity, while others are for disorder coupling. Additionally, diagram (a)
has 2 degenerates due to the exchange $\tau\leftrightarrow\tau^{\prime}$.
Diagram (b) and (c) have 4 degenerates due to the exchanges
$\tau_{1}\leftrightarrow\tau_{2}$ and
$(\tau_{1},\tau_{2})\leftrightarrow(\tau_{3},\tau_{4})$. Diagram (d) has 8
degenerates due to the exchanges $\tau_{1}\leftrightarrow\tau_{2}$,
$\tau_{3}\leftrightarrow\tau_{4}$ and
$(\tau_{1},\tau_{2})\leftrightarrow(\tau_{3},\tau_{4})$.

Figure 5: One loop RG diagrams responsible for the renormalization of (a) the
energy and velocity, (b), (c), (d) the disorder coupling. Red lines denote
external legs.

####  C.0.3 Corrections of the energy, momentum, and disorder coupling

According to the Eq. (78), we can get the corrections of the energy, momentum,
and disorder coupling after performing the integral over
$\bar{\Psi}^{>}(\Psi^{>})$. The corresponding four RG diagrams in Fig. 5 are
calculated one by one. When calculating the diagrams of the renormalization of
disorder coupling, we assume that the momenta of the external lines are zero.

$\displaystyle I^{(a)}=$
$\displaystyle-\int_{[\Lambda/b,\Lambda]}\frac{d^{2}\bm{p}}{(2\pi)^{2}}\mathcal{K}(\bm{k}-\bm{p})\mathcal{G}^{0}(\bm{p},\omega)$
(79) $\displaystyle=$ $\displaystyle-K_{0}(\hbar
v_{f})^{2}\int_{[\Lambda/b,\Lambda]}\frac{d^{2}\bm{p}}{(2\pi)^{2}}e^{-\frac{\xi^{2}|\bm{k}-\bm{p}|^{2}}{2}}\frac{1}{i\omega-\hbar
v_{f}\bm{p}\cdot\bm{\sigma}}=-K_{0}(\hbar
v_{f})^{2}\int_{[\Lambda/b,\Lambda]}\frac{d^{2}\bm{p}}{(2\pi)^{2}}e^{-\frac{\xi^{2}|\bm{k}-\bm{p}|^{2}}{2}}\frac{i\omega+\hbar
v_{f}\bm{p}\cdot\bm{\sigma}}{\omega^{2}+(\hbar v_{f}p)^{2}}$ $\displaystyle=$
$\displaystyle\frac{K_{0}(\hbar
v_{f})^{2}}{4\pi^{2}}\int_{-\pi}^{\pi}d\theta\int_{\Lambda/b}^{\Lambda}pdpe^{-\frac{\xi^{2}(k^{2}+p^{2}-2kp\cos\theta)}{2}}\left[\frac{i\omega}{\omega^{2}+(\hbar
v_{f}p)^{2}}+\frac{\hbar v_{f}p\cos\theta}{\omega^{2}+(\hbar
v_{f}p)^{2}}\frac{\bm{k}\cdot\bm{\sigma}}{k}\right]$ $\displaystyle\approx$
$\displaystyle\frac{K_{0}}{4\pi^{2}}\int_{-\pi}^{\pi}d\theta\int_{\Lambda/b}^{\Lambda}dpe^{-\frac{\xi^{2}(k^{2}+p^{2}-2kp\cos\theta)}{2}}\left[\frac{i\omega}{p}+\cos\theta\frac{\hbar
v_{f}\bm{k}\cdot\bm{\sigma}}{k}\right]$ $\displaystyle=$
$\displaystyle\frac{K_{0}}{2\pi}\int_{\Lambda/b}^{\Lambda}dpe^{-\frac{\xi^{2}(k^{2}+p^{2})}{2}}\left[I_{0}(kp\xi^{2})\frac{i\omega}{p}+I_{1}(kp\xi^{2})\frac{\hbar
v_{f}\bm{k}\cdot\bm{\sigma}}{k}\right]\approx\frac{K_{0}}{2\pi}\int_{\Lambda/b}^{\Lambda}dp\left[\frac{i\omega}{p}+\frac{p\xi^{2}}{2}\hbar
v_{f}\bm{k}\cdot\bm{\sigma}\right]$ $\displaystyle=$
$\displaystyle\frac{K_{0}\ln
b}{2\pi}i\omega+\frac{K_{0}\xi^{2}\Lambda^{2}}{8\pi}(1-b^{-2})\hbar
v_{f}\bm{k}\cdot\bm{\sigma}$

Here, we have assumed $k\xi,p\xi<1$, and the modified Bessel functions are
expanded as $I_{0}(x)=1+\frac{x^{2}}{4}+o(x^{3})$ and
$I_{1}(x)=\frac{x}{2}+o(x^{3})$. Meanwhile, $\theta$ is the angle between the
momenta $\bm{k}$ and $\bm{p}$. The $\bm{p}\cdot\bm{\sigma}$ in the above
derivation is transformed by

$\displaystyle\bm{p}\cdot\bm{\sigma}=$ $\displaystyle
p\cos\theta_{\bm{p}}\sigma_{x}+p\sin\theta_{\bm{p}}\sigma_{y}=p\cos(\theta_{\bm{k}}+\theta)\sigma_{x}+p\sin(\theta_{\bm{k}}+\theta)\sigma_{y}$
(80) $\displaystyle\to$
$\displaystyle\frac{p}{k}[k\cos\theta_{\bm{k}}\cos\theta\sigma_{x}+k\sin\theta_{\bm{k}}\cos\theta\sigma_{y}]$
$\displaystyle=$ $\displaystyle\frac{p}{k}\cos\theta\bm{k}\cdot\bm{\sigma}$

where the terms proportional to $\sin\theta$ are omitted based on the parity
analysis of the integral.

$\displaystyle I^{(b)}=$
$\displaystyle\int_{[\Lambda/b,\Lambda]}\frac{d^{2}\bm{p}}{(2\pi)^{2}}\mathcal{K}(-\bm{p})\mathcal{K}(\bm{p})\mathcal{G}^{0}(\bm{p},\omega)\mathcal{G}^{0}(\bm{p},\omega)$
(81) $\displaystyle=$ $\displaystyle K_{0}^{2}(\hbar
v_{f})^{4}\int_{[\Lambda/b,\Lambda]}\frac{d^{2}\bm{p}}{(2\pi)^{2}}e^{-\xi^{2}p^{2}}\frac{1}{i\omega-\hbar
v_{f}\bm{p}\cdot\bm{\sigma}}\frac{1}{i\omega-\hbar
v_{f}\bm{p}\cdot\bm{\sigma}}=K_{0}^{2}(\hbar
v_{f})^{4}\int_{[\Lambda/b,\Lambda]}\frac{d^{2}\bm{p}}{(2\pi)^{2}}e^{-\xi^{2}p^{2}}\frac{(i\omega+\hbar
v_{f}\bm{p}\cdot\bm{\sigma})^{2}}{[\omega^{2}+(\hbar v_{f}p)^{2}]^{2}}$
$\displaystyle\approx$ $\displaystyle K_{0}^{2}(\hbar
v_{f})^{2}\int_{\Lambda/b}^{\Lambda}\frac{pdp}{2\pi}e^{-\xi^{2}p^{2}}\frac{1}{p^{2}}=\frac{K_{0}^{2}(\hbar
v_{f})^{2}}{4\pi}[{\rm Ei}(-\xi^{2}\Lambda^{2})-{\rm
Ei}(-\xi^{2}\Lambda^{2}/b^{2})$ $\displaystyle\approx$
$\displaystyle\frac{K_{0}^{2}(\hbar v_{f})^{2}}{4\pi}[2\ln
b+(1-b^{-2})\xi^{2}\Lambda^{2}]$ $\displaystyle I^{(c)}=$
$\displaystyle\int_{[\Lambda/b,\Lambda]}\frac{d^{2}\bm{p}}{(2\pi)^{2}}\mathcal{K}(-\bm{p})\mathcal{K}(\bm{p})\mathcal{G}^{0}(\bm{p},\omega)\mathcal{G}^{0}(-\bm{p},\omega)$
(82) $\displaystyle=$ $\displaystyle K_{0}^{2}(\hbar
v_{f})^{4}\int_{[\Lambda/b,\Lambda]}\frac{d^{2}\bm{p}}{(2\pi)^{2}}e^{-\xi^{2}p^{2}}\frac{1}{i\omega-\hbar
v_{f}\bm{p}\cdot\bm{\sigma}}\frac{1}{i\omega+\hbar
v_{f}\bm{p}\cdot\bm{\sigma}}=K_{0}^{2}(\hbar
v_{f})^{4}\int_{[\Lambda/b,\Lambda]}\frac{d^{2}\bm{p}}{(2\pi)^{2}}e^{-\xi^{2}p^{2}}\frac{(i\omega+\hbar
v_{f}\bm{p}\cdot\bm{\sigma})(i\omega-\hbar
v_{f}\bm{p}\cdot\bm{\sigma})}{[\omega^{2}+(\hbar v_{f}p)^{2}]^{2}}$
$\displaystyle\approx$ $\displaystyle-K_{0}^{2}(\hbar
v_{f})^{2}\int_{\Lambda/b}^{\Lambda}\frac{pdp}{2\pi}e^{-\xi^{2}p^{2}}\frac{1}{p^{2}}=-\frac{K_{0}^{2}(\hbar
v_{f})^{2}}{4\pi}[{\rm Ei}(-\xi^{2}\Lambda^{2})-{\rm
Ei}(-\xi^{2}\Lambda^{2}/b^{2})$ $\displaystyle\approx$
$\displaystyle-\frac{K_{0}^{2}(\hbar v_{f})^{2}}{4\pi}[2\ln
b+(1-b^{-2})\xi^{2}\Lambda^{2}]$ $\displaystyle I^{(d)}=$
$\displaystyle\int_{[\Lambda/b,\Lambda]}\frac{d^{2}\bm{p}}{(2\pi)^{2}}\mathcal{K}(0)\mathcal{K}(-\bm{p})\mathcal{G}^{0}(\bm{p},\omega)\mathcal{G}^{0}(\bm{p},\omega)$
(83) $\displaystyle=$ $\displaystyle K_{0}^{2}(\hbar
v_{f})^{4}\int_{[\Lambda/b,\Lambda]}\frac{d^{2}\bm{p}}{(2\pi)^{2}}e^{-\xi^{2}p^{2}/2}\frac{1}{i\omega-\hbar
v_{f}\bm{p}\cdot\bm{\sigma}}\frac{1}{i\omega-\hbar
v_{f}\bm{p}\cdot\bm{\sigma}}=K_{0}^{2}(\hbar
v_{f})^{4}\int_{[\Lambda/b,\Lambda]}\frac{d^{2}\bm{p}}{(2\pi)^{2}}e^{-\xi^{2}p^{2}/2}\frac{(i\omega+\hbar
v_{f}\bm{p}\cdot\bm{\sigma})^{2}}{[\omega^{2}+(\hbar v_{f}p)^{2}]^{2}}$
$\displaystyle\approx$ $\displaystyle K_{0}^{2}(\hbar
v_{f})^{2}\int_{\Lambda/b}^{\Lambda}\frac{pdp}{2\pi}e^{-\xi^{2}p^{2}/2}\frac{1}{p^{2}}=\frac{K_{0}^{2}(\hbar
v_{f})^{2}}{4\pi}[{\rm Ei}(-\xi^{2}\Lambda^{2}/2)-{\rm
Ei}(-\xi^{2}\Lambda^{2}/2b^{2})$ $\displaystyle\approx$
$\displaystyle\frac{K_{0}^{2}(\hbar v_{f})^{2}}{4\pi}[2\ln
b+(1-b^{-2})\xi^{2}\Lambda^{2}/2]$

where ${\rm Ei}(x)$ is exponential integral function and can be expanded as
${\rm Ei}(-x)=\gamma+\ln x-x+o(x^{2})$. The results of RG diagrams (b)
[$I^{(b)}$] and (c) [$I^{(c)}$] cancel each other out.

Plugging Eqs. (79)-(83) into Eq. (78), therefore, we can get the effective
generating functional in the shell $|k|<\Lambda/b$ as

$\displaystyle\langle Z\rangle_{\text{eff}}=$ $\displaystyle\int
D[\bar{\Psi}^{<},\Psi^{<}]e^{S_{0}[\bar{\Psi}^{<},\Psi^{<}]}\left\\{1+\int
d\tau
d\tau^{\prime}\frac{d^{2}\bm{q}d^{2}\bm{k}d^{2}\bm{k}^{\prime}}{(2\pi)^{6}}\mathcal{K}(\bm{q})\bar{\Psi}^{<}_{\bm{k},\tau}\Psi^{<}_{\bm{k}-\bm{q},\tau}\Psi^{<}_{\bm{k}^{\prime}+\bm{q},\tau^{\prime}}\bar{\Psi}^{<}_{\bm{k}^{\prime},\tau^{\prime}}+\int\frac{d\omega}{2\pi}\frac{d^{2}\bm{k}}{(2\pi)^{2}}\bar{\Psi}^{<}_{\bm{k},\omega}\Psi^{<}_{\bm{k},\omega}(\Delta_{E}i\omega+\alpha\hbar
v_{f}\bm{k}\cdot\bm{\sigma})\right.$ (84) $\displaystyle+\Delta_{K}\int
d\tau_{1}d\tau_{2}\frac{d^{2}\bm{q}d^{2}\bm{k}d^{2}\bm{k}^{\prime}}{(2\pi)^{6}}\bar{\Psi}^{<}_{\bm{k},\tau_{1}}\Psi^{<}_{\bm{k}-\bm{q},\tau_{1}}\bar{\Psi}^{<}_{\bm{k}^{\prime},\tau_{2}}\Psi^{<}_{\bm{k}^{\prime}+\bm{q},\tau_{2}}$
$\displaystyle\approx$ $\displaystyle\int D[\bar{\Psi},\Psi]_{\Lambda/b}$
$\displaystyle\exp\left\\{\int\frac{d\omega}{2\pi}\frac{d^{2}\bm{k}}{(2\pi)^{2}}\bar{\Psi}_{\bm{k},\omega}[(1+\Delta_{E})i\omega-(1-\alpha)\hbar
v_{f}\bm{k}\cdot\bm{\sigma}]\Psi_{\bm{k},\omega}+\int d\tau
d\tau^{\prime}\frac{d^{2}\bm{q}d^{2}\bm{k}d^{2}\bm{k}^{\prime}}{(2\pi)^{6}}[\mathcal{K}(\bm{q})+\Delta_{K}]\bar{\Psi}_{\bm{k},\tau}\Psi_{\bm{k}-\bm{q},\tau}\Psi_{\bm{k}^{\prime}+\bm{q},\tau^{\prime}}\bar{\Psi}_{\bm{k}^{\prime},\tau^{\prime}}\right\\}$

with

$\Delta_{E}=\frac{K_{0}\ln b}{2\pi};\ \ \
\alpha=\frac{K_{0}\xi^{2}\Lambda^{2}}{8\pi}(1-b^{-2});\ \ \ \
\Delta_{K}=\frac{K_{0}^{2}(\hbar v_{f})^{2}}{4\pi}[2\ln
b+(1-b^{-2})\xi^{2}\Lambda^{2}/2]$ (85)

The coefficient $\left\\{\int
D[\bar{\Psi}^{>},\Psi^{>}]e^{S_{0}[\bar{\Psi}^{>},\Psi^{>}]}\right\\}$ is
eliminated since it will be absorbed into the normalization of generating
function.

####  C.0.4 Renormalization group flow

Let us now rescale momenta and fields in the effective generating functional
according to

$\bm{k}^{\prime}=b\bm{k};\ \ \
\bar{\Psi}^{\prime}(\Psi^{\prime})=b^{-3/2}\bar{\Psi}(\Psi)$ (86)

so that the momentum $\bm{k}^{\prime}$ is integrated over
$|k^{\prime}|<\Lambda$. The rescaling of fields is to keep the free propagator
unchanged. The rescaled effective generating functional is

$\displaystyle\langle Z\rangle_{\text{eff}}$ $\displaystyle=\int
D[\bar{\Psi}^{\prime},\Psi^{\prime}]_{\Lambda}$ (87)
$\displaystyle\exp\left\\{\int\frac{d\omega}{2\pi}\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\bar{\Psi}^{\prime}_{\bm{k}^{\prime},\omega}[b(1+\Delta_{E})i\omega-(1-\alpha)\hbar
v_{f}\bm{k}^{\prime}\cdot\bm{\sigma}]\Psi_{\bm{k}^{\prime},\omega}+\int d\tau
d\tau^{\prime}\frac{d^{2}\bm{q}d^{2}\bm{k}d^{2}\bm{k}^{\prime}}{(2\pi)^{6}}[\mathcal{K}(\bm{q})+\Delta_{K}]\bar{\Psi}^{<}_{\bm{k},\tau}\Psi^{<}_{\bm{k}-\bm{q},\tau}\Psi^{<}_{\bm{k}^{\prime}+\bm{q},\tau^{\prime}}\bar{\Psi}^{<}_{\bm{k}^{\prime},\tau^{\prime}}\right\\}$

which gives the transformation laws of energy, velocity and disorder

$\left\\{\begin{aligned} E^{\prime}=&b(1+\Delta_{E})E=b(1+\frac{K_{0}\ln
b}{2\pi})E\\\
v_{f}^{\prime}=&(1-\alpha)v_{f}=\left[1-\frac{K_{0}\xi^{2}\Lambda^{2}}{8\pi}(1-b^{-2})\right]v_{f}\\\
K_{0}^{\prime}=&K_{0}+\frac{K_{0}^{2}}{8\pi}[4\ln
b+(1-b^{-2})\xi^{2}\Lambda^{2}]\end{aligned}\right.$ (88)

where we only consider the leading term of disorder coupling
$\mathcal{K}(\bm{q})\to K_{0}(\hbar v_{f})^{2}$ and do the analytic
continuation for energy, $i\omega\to E+i0^{+}$.

Then we renormalization parameter $b=e^{l}\approx 1+l$ and get the
renormalization flows of energy, velocity and disorder

$\left\\{\begin{aligned} \frac{dE}{dl}=&(1+\frac{K_{0}}{2\pi})E\\\
\frac{dv_{f}}{dl}=&-\frac{K_{0}\xi^{2}\Lambda^{2}}{4\pi}v_{f}\\\
\frac{dK_{0}}{dl}=&\frac{K_{0}^{2}}{4\pi}(2+\xi^{2}\Lambda^{2})\end{aligned}\right.$
(89)

Here we would like to provide some remarks. First, by setting $\xi\rightarrow
0$, the flow equations Eq. 89 goes back to the existing results (e.g. Ref.
Ostrovsky ). That is, the renormalization of velocity is zero for short-ranged
disorder potential. This is the main reason why the previous work overlooked
the renormalization of velocity. Second, as we shown here under the low energy
condition ($k\xi,E\xi<1$), a long-ranged fluctuation $\xi\neq 0$ leads to
renormalization of the velocity. Importantly, the effective velocity is
reduced under the renormalization group flow. Third, the renormalization group
calculation is consistent with the calculation of self-energy function,
because the momentum-dependent part of self-energy function, i.e. $\alpha\hbar
v_{f}k$ in Eq. 48 or Eq. 18 in the main text, effectively reduces velocity in
energy dispersion of electron. Taken all together, the independent
renormalization group calculation further supports our conclusion shown in the
main text.

##  Appendix D Numerical Results of Self-Energy Function

Figure 6: (a-e) Numerical results of ${\rm Re}\Sigma$ vs $E$ with different
momentum. Insets in (a-e) are results but in a large energy range. (f-j)
Numerical results of ${\rm Im}\Sigma$ vs $E$ with different momentum. (k-o)
Numerical results of ${\rm Re}\Sigma$ vs $k$ with different Fermi energy.
(p-t) Numerical results of ${\rm Im}\Sigma$ vs $k$ with different Fermi
energy. Blue lines denotes subband with $s=1$ while orange lines denotes
$s=-1$. Other parameters are the same as the FIG. 1 in the main paper:
impurity concentration $n_{\text{imp}}=5\%$, correlation length $\xi=3.6a$,
impurity strength $u_{0}=0.16t$.

####  D.0.1 Long-ranged Gaussian Potential: $k,\frac{E}{\hbar
v_{f}}\ll\frac{1}{\xi}$ regime

In the main paper, we have shown the numerical results of ${\rm Re}\Sigma/{\rm
Im}\Sigma$ vs $E$ with momentum $k=0$, $0.048/a$ and $0.096/a$ and ${\rm
Re}\Sigma/{\rm Im}\Sigma$ vs $k$ with Fermi energy $E=0t$, $0.025t$ and
$0.050t$. Here, we show the evolution with more momentum and Fermi energy
values in Fig. 6.

Figure 7: (Color online). (a-e) Numerical results of ${\rm Re}\Sigma$ vs $E$
with different momentum. (f-j) Numerical results of ${\rm Im}\Sigma$ vs $E$.
Blue lines denotes subband with $s=1$ while orange lines denotes $s=-1$. Other
parameters: impurity concentration $n_{\text{imp}}=1\%$, correlation length
$\xi=10a$, impurity strength $u_{0}=0.04t$.

####  D.0.2 Long-ranged Gaussian Potential: $k,\frac{E}{\hbar
v_{f}}\gg\frac{1}{\xi}$ regime

Since we have shown in the main paper the numerical self-energy function that
satisfies $k,\frac{E}{\hbar v_{f}}\ll\frac{1}{\xi}$, here, we will show the
corresponding result under the condition $k,\frac{E}{\hbar
v_{f}}\gg\frac{1}{\xi}$. In the Fig. 7, we display the self-energy functions
versus Fermi energy with different momentum and versus momentum with different
energy with impurity concentration $n_{\text{imp}}=1\%$, correlation length
$\xi=10a$, impurity strength $u_{0}=0.04t$. For the very small Fermi energy
and momentum in Figs. 7(a) and (f), where the condition $k,E\ll\frac{1}{\xi}$
still satisfied, the behavior of the real and imaginary parts of self-energy
is qualitatively consistent with the results shown in Figs. 6(a) and (f). On
the contrary, for the large Fermi energy and momentum, the imaginary part of
self-energy tends to be a delta function which is same with the prediction
from Born approximation Eq. (42). Similarly, the numerical result of real part
of self-energy is also consistent with the analytical expression Eq. (46).

Figure 8: (Color online). (a) Numerical results of ${\rm Re}\Sigma$ vs $E$.
(b) Numerical results of ${\rm Re}\Sigma$ vs $k$. Inset in (a) is same as (a)
but in a large energy range and only keeping the data of $k=0$ and
$k=0.072/a$. Other parameters: $n_{i}=0.05$, $V_{0}=0.1t$, and $q_{s}=0.1a$.
In order to avoid the divergence at $r\to 0$ in the discrete simulation, we
set a minimum radius $r_{c}=0.01a$.

####  D.0.3 Long-range Screened Coulomb Potential: $k,\frac{E}{\hbar v_{f}}\ll
q_{s}$ regime

In the Sec. B.3, we have derived the self-energy function in the presence of
long-range screened Coulomb potential based on the Born’s approximation. We
also calculate it by numerical simulation. In Fig. 8, we show the numerical
results of ${\rm Re}\Sigma$ vs $E$ and ${\rm Re}\Sigma$ vs $k$ near the Dirac
point with parameters: $n_{i}=0.05$, $V_{0}=0.1t$, and $q_{s}=0.1a$.
Additionally, in order to avoid the divergence at $r\to 0$ in the discrete
simulation, we set a minimum radius $r_{c}=0.01a$.

##  Appendix E Kubo Formalism for Bubble Diagram and Vertex Correction of
Conductivity

Figure 9: (a) The self-energy diagram (bubble diagram) of dc conductivity. (b)
The self-energy modified Green’s function, where the first wigwam diagram
represents for the Born approximation. (c) The vertex corrected diagram of dc
conductivity. (d) The “dressed” vertex.

The self-energy diagram (bubble diagram) of dc conductivity as shown in Fig.
9(a) can be calculated by

$\displaystyle\sigma_{xx}^{0}(E)=\sigma_{xx}^{0,RA}(E)-{\rm
Re}\left[\sigma_{xx}^{0,RR}(E)\right]$ (90)

with

$\displaystyle\sigma_{xx}^{0,RA}(E)$ $\displaystyle=$ $\displaystyle
4\frac{e^{2}\hbar}{2\pi}\int\frac{d^{2}\bm{k}}{(2\pi)^{2}}{\rm
Tr}\left[G^{R}(\bm{k},E)v_{x}(\bm{k})G^{A}(\bm{k},E)v_{x}(\bm{k})\right]_{c}$
(91) $\displaystyle\sigma_{xx}^{0,RR}(E)$ $\displaystyle=$ $\displaystyle
4\frac{e^{2}\hbar}{2\pi}\int\frac{d^{2}\bm{k}}{(2\pi)^{2}}{\rm
Tr}\left[G^{R}(\bm{k},E)v_{x}(\bm{k})G^{R}(\bm{k},E)v_{x}(\bm{k})\right]_{c}$
(92)

where the factor 4 denotes degeneracy of spin and valley, ${\rm Tr}[\cdots]$
means the trace in chiral basis, and the subscript $c$ indicates a disorder
configuration average. We do the calculation in the eigen basis of pseudospin
so that the velocity operator and Green’s function are expressed as

$v_{x}(\bm{k})=v_{f}(\cos\theta_{\bm{k}}\sigma_{z}+\sin\theta_{\bm{k}}\sigma_{y})=v_{f}\left(\begin{array}[]{cc}\cos\theta_{\bm{k}}&-i\sin\theta_{\bm{k}}\\\
i\sin\theta_{\bm{k}}&-\cos\theta_{\bm{k}}\end{array}\right)$ (93)

and

$G^{L}(\bm{k},E)=\left(\begin{array}[]{cc}\frac{1}{E-E_{\bm{k}+}-\Sigma(\bm{k}+,E)}&0\\\
0&\frac{1}{E-E_{\bm{k}-}-\Sigma(\bm{k}-,E)}\end{array}\right)=\left(\begin{array}[]{cc}g^{L}_{+}(\bm{k},E)&0\\\
0&g^{L}_{-}(\bm{k},E)\end{array}\right)$ (94)

where $L=R,A$ denotes the retarded or advanced Green’s function and
$E_{\bm{k}\pm}=\pm\hbar v_{f}k$ is the eigenvalue. Plugging Eq.(93) and
Eq.(94) into Eq.(91) and Eq.(92), the value of
$\sigma_{xx}^{0,RA}(\sigma_{xx}^{0,RR})$ is evaluated

$\displaystyle\sigma^{0,LM}_{xx}(E)=$ $\displaystyle\frac{2e^{2}\hbar
v^{2}_{f}}{\pi}\int\frac{d^{2}\bm{k}}{(2\pi)^{2}}{\rm
Tr}\left[\left(\begin{array}[]{cc}g^{L}_{+}&0\\\
0&g^{L}_{-}\end{array}\right)\left(\begin{array}[]{cc}\cos\theta_{\bm{k}}&-i\sin\theta_{\bm{k}}\\\
i\sin\theta_{\bm{k}}&-\cos\theta_{\bm{k}}\end{array}\right)\left(\begin{array}[]{cc}g^{M}_{+}&0\\\
0&g^{M}_{-}\end{array}\right)\left(\begin{array}[]{cc}\cos\theta_{\bm{k}}&-i\sin\theta_{\bm{k}}\\\
i\sin\theta_{\bm{k}}&-\cos\theta_{\bm{k}}\end{array}\right)\right]$ (95)
$\displaystyle=$ $\displaystyle\frac{2e^{2}\hbar
v^{2}_{f}}{\pi}\int\frac{d^{2}\bm{k}}{(2\pi)^{2}}\left[\cos^{2}\theta_{\bm{k}}(g^{L}_{+}g^{M}_{+}+g^{L}_{-}g^{M}_{-})+\sin^{2}\theta_{\bm{k}}(g^{L}_{+}g^{M}_{-}+g^{L}_{-}g^{M}_{+})\right]$
$\displaystyle=$ $\displaystyle\frac{e^{2}\hbar
v^{2}_{f}}{\pi}\int\frac{kdk}{2\pi}\left[g^{L}_{+}g^{M}_{+}+g^{L}_{-}g^{M}_{-}+g^{L}_{+}g^{M}_{-}+g^{L}_{-}g^{M}_{+}\right]$
$\displaystyle=$ $\displaystyle\frac{e^{2}\hbar
v^{2}_{f}}{\pi}\int\frac{kdk}{2\pi}(g^{L}_{+}+g^{L}_{-})(g^{M}_{+}+g^{M}_{-})$

where $L,M=R,A$. Since we assume the self-energy function depending on both
energy and magnitude of wave vector as

$\Sigma(\bm{k}s,E)=\Sigma_{1}(E)-s\alpha\hbar v_{f}k+i\Sigma_{2}(E),$ (96)

the full Green’s functions can be written as

$\displaystyle g^{R}_{+}(\bm{k},E)$ $\displaystyle=$
$\displaystyle\frac{1}{E-\hbar v_{f}k-{\rm Re}\Sigma(E,k+)-i{\rm
Im}\Sigma(E,k+)}=\frac{1}{a-(1-\alpha)v_{f}\hbar k+i\eta}$ (97) $\displaystyle
g^{R}_{-}(\bm{k},E)$ $\displaystyle=$ $\displaystyle\frac{1}{E+\hbar
v_{f}k-{\rm Re}\Sigma(E,k-)-i{\rm
Im}\Sigma(E,k-)}=\frac{1}{a+(1-\alpha)v_{f}\hbar k+i\eta}$ (98) $\displaystyle
g^{A}_{+}(\bm{k},E)$ $\displaystyle=$ $\displaystyle\frac{1}{E-\hbar
v_{f}k-{\rm Re}\Sigma(E,k+)+i{\rm
Im}\Sigma(E,k+)}=\frac{1}{a-(1-\alpha)v_{f}\hbar k-i\eta}$ (99) $\displaystyle
g^{A}_{-}(\bm{k},E)$ $\displaystyle=$ $\displaystyle\frac{1}{E+\hbar
v_{f}k-{\rm Re}\Sigma(E,k-)+i{\rm
Im}\Sigma(E,k-)}=\frac{1}{a+(1-\alpha)v_{f}\hbar k-i\eta}$ (100)

with

$\left\\{\begin{array}[]{l}a=E-\Sigma_{1}(E)\\\ \eta=-\Sigma_{2}(E)\\\
\end{array}\right.$ (102)

Therefore we can obtain $\sigma^{0,RA}_{xx}(E)$ and ${\rm
Re}\sigma^{0,RR}_{xx}(E)$ as

$\displaystyle\sigma^{0,RA}_{xx}(E)=$ $\displaystyle\frac{e^{2}\hbar
v^{2}_{f}}{\pi}\int\frac{kdk}{2\pi}\left[\frac{1}{a-(1-\alpha)\hbar
v_{f}k+i\eta}+\frac{1}{a+(1-\alpha)\hbar v_{f}k+i\eta}\right]$ (103)
$\displaystyle\left[\frac{1}{a-(1-\alpha)\hbar
v_{f}k-i\eta}+\frac{1}{a+(1-\alpha)\hbar v_{f}k-i\eta}\right]$
$\displaystyle=$ $\displaystyle\frac{e^{2}\hbar
v^{2}_{f}}{\pi}\int\frac{kdk}{2\pi}\frac{2(a+i\eta)}{(a+i\eta)^{2}-(1-\alpha)^{2}(\hbar
v_{f}k)^{2}}\frac{2(a-i\eta)}{(a-i\eta)^{2}-(1-\alpha)^{2}(\hbar v_{f}k)^{2}}$
$\displaystyle=$
$\displaystyle\frac{2e^{2}}{\pi\hbar}\frac{1}{(1-\alpha)^{2}}(a^{2}+\eta^{2})\int^{\infty}_{\eta^{2}-a^{2}}dx\frac{1}{x^{2}+4a^{2}\eta^{2}}$
$\displaystyle=$ $\displaystyle\frac{2e^{2}}{\pi
h}\frac{1}{(1-\alpha)^{2}}(\frac{a}{\eta}+\frac{\eta}{a})\arctan\frac{a}{\eta}$

and

$\displaystyle{\rm Re}\sigma^{0,RR}_{xx}(E)=$ $\displaystyle{\rm
Re}\frac{e^{2}\hbar
v^{2}_{f}}{\pi}\int\frac{kdk}{2\pi}\left[\frac{1}{a-(1-\alpha)\hbar
v_{f}k+i\eta}+\frac{1}{a+(1-\alpha)\hbar v_{f}k+i\eta}\right]^{2}$ (104)
$\displaystyle=$ $\displaystyle{\rm Re}\frac{2e^{2}}{\pi
h}\frac{1}{(1-\alpha)^{2}}\int
dx\frac{(a+i\eta)^{2}}{\left[(a+i\eta)^{2}-x\right]^{2}}$ $\displaystyle=$
$\displaystyle\frac{2e^{2}}{\pi
h}\frac{1}{(1-\alpha)^{2}}\frac{(a+i\eta)^{2}}{(a+i\eta)^{2}-x}\bigg{|}^{\infty}_{0}$
$\displaystyle=$ $\displaystyle-\frac{2e^{2}}{\pi h}\frac{1}{(1-\alpha)^{2}}$

Thus the total bubble conductivity is

$\sigma^{0}_{xx}(E)=\sigma^{0,RA}_{xx}(E)-{\rm
Re}[\sigma^{0,RR}_{xx}(E)]=\frac{2e^{2}}{\pi
h}\frac{1}{(1-\alpha)^{2}}\left[1+(\frac{a}{\eta}+\frac{\eta}{a})\arctan\frac{a}{\eta}\right]$
(105)

from which we can obtain minimal conductivity at the limit $E\to 0$

$\sigma^{0}_{\text{min}}=\frac{2e^{2}}{\pi
h}\frac{1}{(1-\alpha)^{2}}\left[1+\frac{\eta}{a}\frac{a}{\eta}\right]=\frac{4}{\pi}\frac{e^{2}}{h}\frac{1}{(1-\alpha)^{2}}$
(106)

The vertex correction of dc conductivity as shown in Fig. 9(c)(d), can be
calculated by

$\displaystyle\sigma_{xx}^{v}(E)=\sigma_{xx}^{v,RA}(E)-{\rm
Re}\left[\sigma_{xx}^{v,RR}(E)\right]$ (107)

with

$\displaystyle\sigma_{xx}^{v,RA}(E)$ $\displaystyle=$ $\displaystyle
4\frac{e^{2}\hbar}{2\pi}\int\frac{d^{2}\bm{k}}{(2\pi)^{2}}{\rm
Tr}\left[G^{R}(\bm{k},E)v_{x}(\bm{k})G^{A}(\bm{k},E)\tilde{v}^{RA}_{x}(\bm{k})\right]$
(108) $\displaystyle\sigma_{xx}^{v,RR}(E)$ $\displaystyle=$ $\displaystyle
4\frac{e^{2}\hbar}{2\pi}\int\frac{d^{2}\bm{k}}{(2\pi)^{2}}{\rm
Tr}\left[G^{R}(\bm{k},E)v_{x}(\bm{k})G^{R}(\bm{k},E)\tilde{v}^{RR}_{x}(\bm{k})\right]$
(109)

Here, the “dressed” vertex function $\tilde{v}^{LM}_{x}$ is defined by self-
consistent Bethe-Salpeter equation

$\tilde{v}^{LM}_{x}(\bm{k},E)=v_{x}(\bm{k})+\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\mathcal{K}(\bm{k}-\bm{k}^{\prime})U^{\dagger}_{\bm{k}}U_{\bm{k}^{\prime}}G^{L}(\bm{k}^{\prime},E)\tilde{v}^{LM}_{x}(\bm{k},E)G^{M}(\bm{k}^{\prime},E)U^{\dagger}_{\bm{k}^{\prime}}U_{\bm{k}}$
(110)

where $U^{\dagger}_{\bm{k}}U_{\bm{k}^{\prime}}$ denotes the spin rotation
while momentum changing

$U_{\bm{k}}^{\dagger}U_{\bm{k}^{\prime}}=\frac{1}{2}\left(\begin{array}[]{cc}1+e^{i\theta}&1-e^{i\theta}\\\
1-e^{i\theta}&1+e^{i\theta}\end{array}\right)$ (111)

with $\theta=\theta_{\bm{k}^{\prime}}-\theta_{\bm{k}}$. In order to solve this
self-consistent equation, we at first consider the first order approximation
in the following

$\displaystyle\tilde{v}^{(1),LM}_{x}(\bm{k},E)=$ $\displaystyle
v_{x}(\bm{k})+\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\mathcal{K}(\bm{k}-\bm{k}^{\prime})U^{\dagger}_{\bm{k}}U_{\bm{k}^{\prime}}G^{L}(\bm{k}^{\prime},E)v_{x}(\bm{k})G^{M}(\bm{k}^{\prime},E)U^{\dagger}_{\bm{k}^{\prime}}U_{\bm{k}}$
(112) $\displaystyle=$ $\displaystyle
v_{x}(\bm{k})+v_{f}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\frac{\mathcal{K}(\bm{k}-\bm{k}^{\prime})}{4}\left(\begin{array}[]{cc}1+e^{i\theta}&1-e^{i\theta}\\\
1-e^{i\theta}&1+e^{i\theta}\end{array}\right)\left(\begin{array}[]{cc}g^{L}_{+}&\\\
&g^{L}_{-}\end{array}\right)\left(\begin{array}[]{cc}\cos\theta_{\bm{k}^{\prime}}&-i\sin\theta_{\bm{k}^{\prime}}\\\
i\sin\theta_{\bm{k}^{\prime}}&-\cos\theta_{\bm{k}^{\prime}}\end{array}\right)$
$\displaystyle\left(\begin{array}[]{cc}g^{M}_{+}&\\\
&g^{M}_{-}\end{array}\right)\left(\begin{array}[]{cc}1+e^{-i\theta}&1-e^{-i\theta}\\\
1-e^{-i\theta}&1+e^{-i\theta}\end{array}\right)$ $\displaystyle=$
$\displaystyle
v_{x}(\bm{k})+v_{f}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\frac{\mathcal{K}(\bm{k}-\bm{k}^{\prime})}{4}\left(\begin{array}[]{cc}M_{11}&M_{12}\\\
M_{21}&M_{22}\end{array}\right)$

where the expressions of $M_{ss^{\prime}}$ are in the following

$\displaystyle M_{11}=$ $\displaystyle
2\cos\theta(g^{L}_{+}g^{M}_{+}-g^{L}_{-}g^{M}_{-})\cos\theta_{\bm{k}}+\left[2\cos^{2}\theta(g^{L}_{+}g^{M}_{+}+g^{L}_{-}g^{M}_{-})+2\sin^{2}\theta(g^{L}_{+}g^{M}_{-}+g^{L}_{-}g^{M}_{+})\right]\cos\theta_{\bm{k}}$
(113) $\displaystyle M_{22}=$ $\displaystyle
2\cos\theta(g^{L}_{+}g^{M}_{+}-g^{L}_{-}g^{M}_{-})\cos\theta_{\bm{k}}+\left[2\cos^{2}\theta(g^{L}_{+}g^{M}_{+}+g^{L}_{-}g^{M}_{-})+2\sin^{2}\theta(g^{L}_{+}g^{M}_{-}+g^{L}_{-}g^{M}_{+})\right](-\cos\theta_{\bm{k}})$
$\displaystyle M_{12}=$ $\displaystyle
2i\cos\theta(g^{L}_{+}g^{M}_{-}-g^{L}_{-}g^{M}_{+})\sin\theta_{\bm{k}}+\left[2\cos^{2}\theta(g^{L}_{+}g^{M}_{+}+g^{L}_{-}g^{M}_{-})+2\sin^{2}\theta(g^{L}_{+}g^{M}_{-}+g^{L}_{-}g^{M}_{+})\right](-i\sin\theta_{\bm{k}})$
$\displaystyle M_{21}=$ $\displaystyle
2i\cos\theta(g^{L}_{+}g^{M}_{-}-g^{L}_{-}g^{M}_{+})\sin\theta_{\bm{k}}+\left[2\cos^{2}\theta(g^{L}_{+}g^{M}_{+}+g^{L}_{-}g^{M}_{-})+2\sin^{2}\theta(g^{L}_{+}g^{M}_{-}+g^{L}_{-}g^{M}_{+})\right](i\sin\theta_{\bm{k}})$

Based on the expressions of $M_{ss^{\prime}}$, we can rewrite
$\tilde{v}^{(1),LM}_{x}(\bm{k},E)$ as

$\displaystyle\frac{\tilde{v}^{(1),LM}_{x}(\bm{k},E)}{v_{f}}=$ $\displaystyle
f^{LM}_{0}(\bm{k},E)\cos\theta_{\bm{k}}\sigma_{0}+f^{LM}_{x}(\bm{k},E)\sin\theta_{\bm{k}}\sigma_{x}+f^{LM}_{y}(\bm{k},E)\sin\theta_{\bm{k}}\sigma_{y}+f^{LM}_{z}(\bm{k},E)\cos\theta_{\bm{k}}\sigma_{z}$
(114)

For $\tilde{v}^{(1),RA}_{x}(\bm{k},E)$, the functions $f^{RA}_{i}(\bm{k},E)$
with $(i=0,x,y,z)$ are

$\displaystyle f^{RA}_{0}(\bm{k},E)=$
$\displaystyle\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\frac{\mathcal{K}(\bm{k}-\bm{k}^{\prime})}{2}\cos\theta(g^{R}_{+}g^{A}_{+}-g^{R}_{-}g^{A}_{-})$
(115) $\displaystyle\approx$ $\displaystyle\frac{K_{0}(\hbar
v_{f})^{2}}{8\pi^{2}}\int
k^{\prime}dk^{\prime}d\theta\left[1-\frac{\xi^{2}}{2}(k^{2}+k^{\prime
2})+\xi^{2}kk^{\prime}\cos\theta\right]\cos\theta(g^{R}_{+}g^{A}_{+}-g^{R}_{-}g^{A}_{-})$
$\displaystyle\approx$
$\displaystyle\frac{K_{0}\xi^{2}a}{4\pi(1-\alpha)^{3}\hbar
v_{f}}\ln\left(\frac{E^{2}_{c}}{a^{2}+\eta^{2}}\right)k$ $\displaystyle
f^{RA}_{x}(\bm{k},E)=$
$\displaystyle\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\frac{\mathcal{K}(\bm{k}-\bm{k}^{\prime})}{2}i\cos\theta(g^{R}_{0,+}g^{A}_{0,-}-g^{R}_{0,-}g^{A}_{0,+})$
(116) $\displaystyle\approx$ $\displaystyle\frac{K_{0}(\hbar
v_{f})^{2}}{8\pi^{2}}\int
k^{\prime}dk^{\prime}d\theta\left[1-\frac{\xi^{2}}{2}(k^{2}+k^{\prime
2})+\xi^{2}kk^{\prime}\cos\theta\right]i\cos\theta(g^{R}_{+}g^{A}_{-}-g^{R}_{-}g^{A}_{+})$
$\displaystyle\approx$
$\displaystyle\frac{K_{0}\xi^{2}\eta}{4\pi(1-\alpha)^{3}\hbar
v_{f}}\ln\left(\frac{E^{2}_{c}}{a^{2}+\eta^{2}}\right)k$ $\displaystyle
f^{RA}_{y}(\bm{k},E)=f^{RA}_{z}(\bm{k},E)=$ $\displaystyle
1+\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\frac{\mathcal{K}(\bm{k}-\bm{k}^{\prime})}{2}\left[\cos^{2}\Delta\theta(g^{L}_{+}g^{M}_{+}+g^{L}_{-}g^{M}_{-})+\sin^{2}\Delta\theta(g^{L}_{+}g^{M}_{-}+g^{L}_{-}g^{M}_{+})\right]$
(117) $\displaystyle\approx$ $\displaystyle 1+\frac{K_{0}(\hbar
v_{f})^{2}}{8\pi}\int
k^{\prime}dk^{\prime}(g^{L}_{+}g^{M}_{+}+g^{L}_{-}g^{M}_{-}+g^{L}_{+}g^{M}_{+}+g^{L}_{-}g^{M}_{-})$
$\displaystyle\approx$ $\displaystyle
1+\frac{K_{0}}{4\pi(1-\alpha)^{2}}(\frac{a}{\eta}+\frac{\eta}{a})\arctan\frac{a}{\eta}$

For $\tilde{v}^{RR}_{x}(\bm{k},E)$, the functions $f^{RR}_{i}(\bm{k},E)$ with
$(i=0,x,y,z)$ are

$\displaystyle f^{RR}_{0}(\bm{k},E)=$
$\displaystyle\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\frac{\mathcal{K}(\bm{k}-\bm{k}^{\prime})}{2}\cos\theta(g^{R}_{+}g^{R}_{+}-g^{R}_{-}g^{R}_{-})$
(118) $\displaystyle\approx$ $\displaystyle\frac{K_{0}(\hbar
v_{f})^{2}}{8\pi^{2}}\int
k^{\prime}dk^{\prime}d\theta\left[1-\frac{\xi^{2}}{2}(k^{2}+k^{\prime
2})+\xi^{2}kk^{\prime}\cos\theta\right]\cos\theta(g^{R}_{+}g^{R}_{+}-g^{R}_{-}g^{R}_{-})$
$\displaystyle\approx$
$\displaystyle\frac{K_{0}\xi^{2}(a+i\eta)}{4\pi(1-\alpha)^{3}\hbar
v_{f}}\ln\left(\frac{E_{c}^{2}}{a^{2}+\eta^{2}}\right)k$ $\displaystyle
f^{RR}_{x}(\bm{k},E)=$
$\displaystyle\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\frac{\mathcal{K}(\bm{k}-\bm{k}^{\prime})}{2}i\cos\theta(g^{R}_{+}g^{R}_{-}-g^{R}_{-}g^{R}_{+})=0$
(119)

$\displaystyle f^{RR}_{y}(\bm{k},E)=f^{RR}_{z}(\bm{k},E)=$ $\displaystyle
1+\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}\frac{\mathcal{K}(\bm{k}-\bm{k}^{\prime})}{2}\left[\cos^{2}\theta(g^{R}_{+}g^{R}_{+}+g^{R}_{-}g^{R}_{-})+\sin^{2}\theta(g^{R}_{+}g^{R}_{-}+g^{R}_{-}g^{R}_{+})\right]$
(120) $\displaystyle\approx$ $\displaystyle 1+\frac{K_{0}(\hbar
v_{f})^{2}}{8\pi}\int
k^{\prime}dk^{\prime}(g^{R}_{+}g^{R}_{+}+g^{R}_{-}g^{R}_{-}+g^{R}_{+}g^{R}_{+}+g^{R}_{-}g^{R}_{-})$
$\displaystyle\approx$ $\displaystyle 1-\frac{K_{0}}{4\pi(1-\alpha)^{2}}$

Plugging Eqs.(115)-(120) into Eqs.(114), (108), and (109), we can get the
first-order $\sigma_{xx}^{v,RA}$ and ${\rm Re}(\sigma_{xx}^{v,RR})$ as

$\displaystyle\sigma^{(1),RA}_{xx}(E)=$ $\displaystyle\frac{e^{2}\hbar
v^{2}_{f}}{2\pi^{2}}\int
kdk\left[1+\frac{K_{0}}{4\pi(1-\alpha)^{2}}(\frac{a}{\eta}+\frac{\eta}{a})\arctan\frac{a}{\eta}\right](g^{R}_{+}g^{A}_{+}+g^{R}_{-}g^{A}_{-}+g^{R}_{+}g^{A}_{-}+g^{R}_{-}g^{A}_{+})$
(121) $\displaystyle+\frac{K_{0}\xi^{2}a}{4\pi(1-\alpha)^{3}\hbar
v_{f}}\ln\left(\frac{E^{2}_{c}}{a^{2}+\eta^{2}}\right)k(g^{R}_{+}g^{A}_{+}-g^{R}_{-}g^{A}_{-})-i\frac{K_{0}\xi^{2}\eta}{4\pi(1-\alpha)^{3}\hbar
v_{f}}\ln\left(\frac{E^{2}_{c}}{a^{2}+\eta^{2}}\right)k(g^{R}_{+}g^{A}_{-}-g^{R}_{-}g^{A}_{+})$
$\displaystyle=$ $\displaystyle\frac{2e^{2}}{\pi
h}\left\\{(\frac{a}{\eta}+\frac{\eta}{a})\arctan\frac{a}{\eta}\left[1+\frac{K_{0}}{4\pi(1-\alpha)^{4}}(\frac{a}{\eta}+\frac{\eta}{a})\arctan\frac{a}{\eta}\right]+\frac{K_{0}\xi^{2}(a^{2}-\eta^{2})}{4\pi(\hbar
v_{f})^{2}}\left[\frac{1}{(1-\alpha)^{3}}\ln\left(\frac{E^{2}_{c}}{a^{2}+\eta^{2}}\right)\right]^{2}\right\\}$

and

$\displaystyle{\rm Re}[\sigma^{(1),RR}_{xx}(E)]=$
$\displaystyle\frac{e^{2}\hbar v^{2}_{f}}{2\pi^{2}}{\rm Re}\int
kdk\left[1-\frac{K_{0}}{4\pi(1-\alpha)^{2}}\right](g^{R}_{+}g^{R}_{+}+g^{R}_{-}g^{R}_{-}+g^{R}_{+}g^{R}_{-}+g^{R}_{-}g^{R}_{+})$
(122) $\displaystyle+\frac{K_{0}\xi^{2}(a+i\eta)}{4\pi(1-\alpha)^{3}\hbar
v_{f}}\ln\left(\frac{E_{c}^{2}}{a^{2}+\eta^{2}}\right)k(g^{R}_{+}g^{R}_{+}-g^{R}_{-}g^{R}_{-})$
$\displaystyle=$ $\displaystyle\frac{2e^{2}}{\pi
h}\left\\{-1+\frac{K_{0}}{4\pi(1-\alpha)^{4}}+\frac{K_{0}\xi^{2}(a^{2}-\eta^{2})}{4\pi(\hbar
v_{f})^{2}}\left[\frac{1}{(1-\alpha)^{3}}\ln\left(\frac{E^{2}_{c}}{a^{2}+\eta^{2}}\right)\right]^{2}\right\\}$

where $E_{c}=\hbar v_{f}k_{c}$. Thus the first-order vertex correction of dc
conductivity is obtained as

$\displaystyle\sigma_{xx}^{(1)}(E)=$
$\displaystyle\sigma_{xx}^{(1),RA}(E)-{\rm
Re}\left[\sigma_{xx}^{(1),RR}(E)\right]$ (123) $\displaystyle=$
$\displaystyle\frac{2e^{2}}{\pi
h}\left[1+(\frac{a}{\eta}+\frac{\eta}{a})\arctan\frac{a}{\eta}\right]+\frac{K_{0}}{4\pi(1-\alpha)^{4}}\left\\{\left[(\frac{a}{\eta}+\frac{\eta}{a})\arctan\frac{a}{\eta}\right]^{2}-1\right\\}$

from which we can see that coefficients $f^{LM}_{0}$ and $f^{LM}_{x}$ can be
ignored in the “dressed” vertex. Thus, we can find that “dressed” vertex has
same matrix structure with the bare velocity and can be solved as

$\displaystyle\tilde{v}^{RA}_{x}(\bm{k},E)=$
$\displaystyle\frac{1}{1-\frac{K_{0}}{4\pi(1-\alpha)^{2}}(\frac{a}{\eta}+\frac{\eta}{a})\arctan\frac{a}{\eta}}v_{f}(\cos\theta_{\bm{k}}\sigma_{z}+\sin\theta_{\bm{k}}\sigma_{y})$
(124) $\displaystyle\tilde{v}^{RR}_{x}(\bm{k},E)=$
$\displaystyle\frac{1}{1+\frac{K_{0}}{4\pi(1-\alpha)^{2}}}v_{f}(\cos\theta_{\bm{k}}\sigma_{z}+\sin\theta_{\bm{k}}\sigma_{y})$

Based on this corrected vertex, the total dc conductivity will arrive at

$\sigma^{v}_{xx}(E)=\frac{2e^{2}}{\pi
h}\frac{1}{(1-\alpha)^{2}}\left[\frac{1}{1+\frac{K_{0}}{4\pi(1-\alpha)^{2}}}+\frac{(\frac{a}{\eta}+\frac{\eta}{a})\arctan\frac{a}{\eta}}{1-\frac{K_{0}}{4\pi(1-\alpha)^{2}}(\frac{a}{\eta}+\frac{\eta}{a})\arctan\frac{a}{\eta}}\right]$
(125)

and the corresponding vertex corrected minimal conductivity is

$\sigma^{v}_{\text{min}}=\frac{4e^{2}}{\pi
h}\frac{1}{(1-\alpha)^{2}}\frac{1}{1-\left[\frac{K_{0}}{4\pi(1-\alpha)^{2}}\right]^{2}}\stackrel{{\scriptstyle
K_{0}\lesssim 1}}{{\longrightarrow}}\frac{4e^{2}}{\pi
h}\frac{1}{(1-\alpha)^{2}}$ (126)

Since typical experiment conditions of high mobility graphene correspond to
$K_{0}\lesssim 1$, the influence of vertex correction to the minimal
conductivity is much smaller than those induced by the linear momentum
dependent self-energy in the bubble diagram.