# Electronic transport in Weyl semimetals with a uniform concentration of
torsional dislocations

Daniel A. Bonilla<EMAIL_ADDRESS>Facultad de Física, Pontificia Universidad
Católica de Chile, Vicuña Mackenna 4860, Santiago, Chile  Enrique Muñoz
<EMAIL_ADDRESS>Facultad de Física, Pontificia Universidad Católica de Chile,
Vicuña Mackenna 4860, Santiago, Chile

###### Abstract

In this article, we consider a theoretical model for a type I Weyl semimetal,
under the presence of a diluted uniform concentration of torsional
dislocations. By a mathematical analysis for partial wave scattering (phase-
shift) for the T-matrix, we obtain the corresponding retarded and advanced
Green’s functions that include the effects of multiple scattering events with
the ensemble of randomly distributed dislocations. Combining this analysis
with the Kubo formalism, and including vertex corrections, we calculate the
electronic conductivity as a function of temperature and concentration of
dislocations. We further evaluate our analytical formulas to predict the
electrical conductivity of several transition metal monopnictides, i.e. TaAs,
TaP, NbAs and NbP.

##  I Introduction

Weyl semimetals (WSMs) constitute a remarkable example of three-dimensional,
gapless materials with nontrivial topological properties. First proposed
theoretically, Wan  _et al._ (2011); Fang  _et al._ (2012); Ruan  _et al._
(2016); Vafek and Vishwanath (2014); Yan and Felser (2017); Armitage  _et al._
(2018); Burkov (2018) and, more recently, discovered experimentally on TaAs
crystalsXu  _et al._ (2015).

In a WSM, the band structure possesses an even number of Weyl nodes with
linear dispersion, where the conduction and valence bands touch. These nodes
are monopolar sources of Berry curvature, and hence are protected from being
gapped since their charge (chirality) is a topological invariant Burkov
(2018). In the vicinity of these nodes, low energy conducting states behave as
Weyl fermions, i.e. massless quasi-particles with pseudo-relativistic Dirac
linear dispersion Vafek and Vishwanath (2014); Yan and Felser (2017); Armitage
_et al._ (2018, 2018); Burkov (2018). In Weyl fermions, conserved chirality
determines the projection of spin over their momentum direction, a condition
referred to as “spin-momentum locking”. While Type I WSMs are Lorentz
covariant, this symmetry is violated in Type II WSMs, where the Dirac cones
are strongly tilted Vanderbilt (2018).

The presence of Weyl nodes in the bulk spectrum determines the emergence of
Fermi arcsXu  _et al._ (2015), the chiral anomaly, and the chiral magnetic
effect, among other remarkable properties Vanderbilt (2018). Therefore,
considerable attention has been paid to understand the electronic transport
properties of WSMs Hosur and Qi (2013); Hu  _et al._ (2019); Nagaosa  _et al._
(2020). For instance, there are recent works on charge transport Hosur  _et
al._ (2012) in the presence of spin-orbit coupled impurities Liu  _et al._
(2017), electrochemicalFlores-Calderón and Martín-Ruiz (2021) and nonlinear
transport induced by Berry curvature dipoles Zeng  _et al._ (2021). Somewhat
less explored are the effects of mechanical strain and deformations in WSMs.
From a theoretical perspective, it has been proposed that different types of
elastic strains can be modeled as gauge fields in WSMs Cortijo  _et al._
(2015, 2016); Arjona and Vozmediano (2018). In previous works, we have studied
the combined effects of a single torsional dislocation and an external
magnetic field on the electronic Muñoz and Soto-Garrido (2019); Soto-Garrido
_et al._ (2020) and thermoelectric Muñoz and Soto-Garrido (2019); Bonilla  _et
al._ (2021) transport properties of WSMs, using the Landauer ballistic
formalism in combination with a mathematical analysis for the quantum
mechanical scattering cross-sectionsMuñoz and Soto-Garrido (2017).

In this work, we extend our previous analysis to study the case of a diluted,
uniform concentration of torsional dislocations and its effects on the
electrical conductivity of type I WSMs. In contrast to our former
studiesBonilla  _et al._ (2021); Muñoz and Soto-Garrido (2019); Soto-Garrido
_et al._ (2020), here we employ the Kubo linear-response formalism at finite
temperatures, that therefore requires to explicitly calculate the retarded and
advanced Green´s functions for the system, including the multiple scattering
events due to the random distribution of dislocation defects in the form of a
disorder-averaged self-energy term. For this purpose, we first analyze the
phase shift arising from a single torsional dislocation, and obtain the
corresponding (retarded and advanced) Green’s function in terms of the
T-matrix elements by solving analytically the Lippmann-Schwinger equation. We
further extend this analysis, incorporating the effect of a random
distribution of such dislocations, with a concentration $n_{d}$, in the form
of a disorder-averaged self-energy into the corresponding Dyson’s equation.
Finally, we analyze the correction to the scattering vertex, and by including
this additional contribution we calculate the electrical conductivity from the
Kubo formula, as a function of temperature and concentration of dislocations.
We present explicit evaluations of our analytical expressions for the
electrical conductivity as a function of temperature and concentration of
dislocations $n_{d}$, for several materials in the family of transition metals
monopnictides, i.e. TaAs, TaP, NbAs and NbP, where the corresponding
microscopic parameters, estimated by ab-initio methods, where reported in the
literatureSzot  _et al._ (2018); Lee  _et al._ (2015); Grassano  _et al._
(2018).

##  II Scattering by a single dislocation

As a continuum model for a type I WSM under the presence of a single
dislocation defect, as depicted in Fig. (1), we consider the
HamiltonianBonilla  _et al._ (2021)

$\hat{H}^{\xi}=\xi\hbar
v_{F}\bm{\sigma}\cdot\left(\mathbf{p}+e\mathbf{A}^{\xi}\right)+\sigma_{0}V_{0}\delta(r-a)\equiv\hat{H}^{\xi}_{0}+\hat{H}^{\xi}_{1},$
(1)

where

$\displaystyle\hat{H}^{\xi}_{0}$ $\displaystyle=$ $\displaystyle\xi
v_{F}\bm{\sigma}\cdot\mathbf{p},$ (2) $\displaystyle\hat{H}^{\xi}_{1}$
$\displaystyle=$ $\displaystyle\xi
ev_{F}\left(\bm{\sigma}\cdot\bm{\hat{\phi}}\right)\frac{1}{2}B^{\xi}r\Theta(a-r)+V_{0}\delta(r-a)\sigma_{0}.$
(3)

Here, $\xi=\pm$ labels each of the Weyl nodes located at
$\mathbf{K}_{\pm}=\pm\mathbf{b}/2$. The expression in Eq. (2) is the free-
particle Hamiltonian, whereas the expression in Eq. (3) represents the
interaction with the dislocation, where torsional strain is described as a
pseudo-magnetic field inside the cylinderMuñoz and Soto-Garrido (2017);
Bonilla  _et al._ (2021); Soto-Garrido  _et al._ (2020), as well as the
lattice mismatch effect at the boundary of the dislocation, modeled as
repulsive delta barrier on its surfaceBonilla  _et al._ (2021).

Figure 1: Pictorial description of the scattering of free incident Weyl
fermions coming from a left reservoir by a single cylindrical dislocation
defect.

The ”free” spinor eigenfunctions for the defect-free reference system satisfy

$\hat{H}_{0}^{\xi}\left|\Phi_{\mathbf{k},\lambda}\right\rangle=\mathscr{E}^{(0,\xi)}_{\lambda,\mathbf{k}}\left|\Phi_{\mathbf{k},\lambda}\right\rangle,$
(4)

where the energy spectrum is given by

$\mathscr{E}^{(0,\xi)}_{\lambda,\mathbf{k}}=\lambda\xi\hbar
v_{F}|\mathbf{k}|,$ (5)

and $\lambda=\pm 1$ is the band (helicity) index. When projected onto
coordinate space, these spinor eigenfunctions have the explicit form

$\Phi_{\lambda,\mathbf{k}}(\mathbf{r})=\sqrt{\frac{|\mathbf{k}|+\lambda
k_{z}}{2|\mathbf{k}|}}\begin{pmatrix}1\\\
\frac{\lambda|\mathbf{k}|-k_{z}}{k_{x}-ik_{y}}\end{pmatrix}e^{i\mathbf{k}\cdot\mathbf{r}},$
(6)

and constitute an orthonormal basis for the Hilbert space.

If we now consider the (elastic) scattering effects induced by the torsional
dislocation modeled by Eq. (3), we need to look for the eigenvectors
$\left|\Psi_{\lambda,\mathbf{k}}\right\rangle$ of the total Hamiltonian in Eq.
(1) with the same energy as in Eq. (5). The answer is provided by the solution
to the well known Lippmann-Schwinger equation

$\left|\Psi_{\mathbf{k},\lambda}\right\rangle=\left|\Phi_{\mathbf{k},\lambda}\right\rangle+\hat{G}^{\xi}_{R,0}(E)\hat{H}_{1}^{\xi}\left|\Psi_{\mathbf{k},\lambda}\right\rangle,$
(7)

where the free Green’s function can be expressed in a coordinate-independent
representation form via the resolvent,

$\hat{G}^{\xi}_{R/A,0}(E)=\left[E-\hat{H}^{\xi}_{0}\pm i\eta^{+}\right]^{-1}.$
(8)

Here, the index $R/A$ stands for retarded and advanced, respectively. As shown
in detail in the Appendix A, in the coordinate representation the
corresponding free Green’s function is given by the explicit matrix form
$\mathbf{G}^{\xi}_{R,0}\left(\mathbf{r},\mathbf{r^{\prime}};k\right)=\delta(z-z^{\prime})\mathbf{G}^{\xi}_{R,0}\left(\mathbf{x},\mathbf{x^{\prime}};k\right)$,
where $\mathbf{r}=(\mathbf{x},z)$ and

$\displaystyle\mathbf{G}^{\xi}_{R,0}\left(\mathbf{x},\mathbf{x^{\prime}};k\right)$
$\displaystyle=-\frac{\lambda\xi ik}{4\hbar v_{F}}$
$\displaystyle\times\begin{bmatrix}H^{(1)}_{0}\left(k|\mathbf{x}-\mathbf{x^{\prime}}|\right)&i\lambda
e^{-i\varphi}H^{(1)}_{1}\left(k|\mathbf{x}-\mathbf{x^{\prime}}|\right)\\\
i\lambda
e^{i\varphi}H^{(1)}_{1}\left(k|\mathbf{x}-\mathbf{x^{\prime}}|\right)&H^{(1)}_{0}\left(k|\mathbf{x}-\mathbf{x^{\prime}}|\right)\end{bmatrix}.$
(9)

Here, $H_{0}^{(1)}(z)$ and $H_{1}^{(1)}(z)$ are the Hankel functions and
$\mathbf{x}=(x,y)$ is the position vector on any plane perpendicular to the
cylinder’s axis.

For the scattering analysis, we need the retarded resolvent for the full
Hamiltonian, which is defined as the solution to the equation

$\left(E+i\eta^{+}-\hat{H}^{\xi}\right)\hat{G}^{\xi}_{R}(E)=\hat{I}.$ (10)

Figure 2: Pictorial description of the scattering event on a plane
perpendicular to the cylindrical defect axis.

Combining Eq. (10) with Eq. (8), we readily obtain

$\displaystyle\hat{G}^{\xi}_{R}(E)$ $\displaystyle=$
$\displaystyle\hat{G}^{\xi}_{R,0}(E)+\hat{G}^{\xi}_{R,0}(E)\hat{H}^{\xi}_{1}\hat{G}^{\xi}_{R}(E)$
(11) $\displaystyle=$
$\displaystyle\hat{G}^{\xi}_{R,0}(E)+\hat{G}^{\xi}_{R,0}(E)\hat{T}^{\xi}(E)\hat{G}^{\xi}_{R,0}(E),$

where we introduced the standard definition of the T-matrix operator
$\hat{T}^{\xi}(E)$, that can be formally expressed in closed form by

$\displaystyle\hat{T}^{\xi}(E)$ $\displaystyle=$
$\displaystyle\hat{H}_{1}^{\xi}+\hat{H}_{1}^{\xi}\hat{G}^{\xi}_{R,0}(E)\hat{T}^{\xi}(E)$
(12) $\displaystyle=$
$\displaystyle\hat{H}_{1}^{\xi}\left(\hat{I}-\hat{G}^{\xi}_{R,0}(E)\hat{H}_{1}^{\xi}\right)^{-1}.$

Using this definition, along with the property
$\hat{H}_{1}^{\xi}|\Psi_{\mathbf{k},\lambda}\rangle=\hat{T}^{\xi}|\Phi_{\mathbf{k},\lambda}\rangle$,
we obtain the Lippmann-Schwinger Eq. (7) in the coordinate representation

$\displaystyle\Psi_{\mathbf{k},\lambda}(\mathbf{r})=\Phi_{\mathbf{k},\lambda}(\mathbf{r})\,+$
$\displaystyle\int d^{3}r^{\prime}\int
d^{3}r^{\prime\prime}\left\langle\mathbf{r}\middle|\hat{G}^{\xi}_{R,0}\left(E\right)\middle|\mathbf{r^{\prime}}\right\rangle$
(13)
$\displaystyle\times\left\langle\mathbf{r^{\prime}}\middle|\hat{T}^{\xi}(E)\middle|\mathbf{r^{\prime\prime}}\right\rangle\Phi_{\mathbf{k},\lambda}(\mathbf{r^{\prime\prime}}).$

As shown in detail in Appendix B, by considering the asymptotic behavior of
the Hankel functions, $H^{(1)}_{\nu}(x)\sim\sqrt{\frac{2}{\pi
x}}e^{i\left(x-\frac{\nu\pi}{2}-\frac{\pi}{4}\right)}$ (for
$x\rightarrow\infty.$), Eq. (13) can be reduced to the $x$-$y$ plane and takes
the explicit asymptotic expression

$\Psi_{\mathbf{k}_{\parallel},\lambda}(\mathbf{x})\sim\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
\lambda\end{pmatrix}e^{ikx}-\frac{\lambda\xi}{2\hbar
v_{F}}\sqrt{\frac{ik}{\pi}}T^{(\lambda,\xi)}_{\mathbf{k^{\prime}}_{\parallel}\mathbf{k}_{\parallel}}\begin{pmatrix}1\\\
\lambda e^{i\phi}\end{pmatrix}\frac{e^{ikr}}{\sqrt{r}},$ (14)

where as we explain in the Appendix, the particles have only momenta
perpendicular to the defect’s axis, i.e.,
$\mathbf{k}_{\parallel}=(k_{x},k_{y})$. Comparing this last result with our
previous reported expression for the scattering amplitude Soto-Garrido and
Muñoz (2018)

$\begin{bmatrix}f_{1}(\phi)\\\
f_{2}(\phi)\end{bmatrix}=\frac{e^{-\frac{i\pi}{4}}}{\sqrt{4\pi
k}}\sum_{m=-\infty}^{\infty}\begin{bmatrix}e^{im\phi}\\\ \lambda
e^{i(m+1)\phi}\end{bmatrix}\left(e^{2i\delta_{m}}-1\right),$ (15)

we identify
$T^{(\lambda,\xi)}_{\mathbf{k^{\prime}}\mathbf{k}}=-2\lambda\xi\hbar
v_{F}\sqrt{\pi/ik}f_{1}(\phi)$. Therefore, we arrived at an explicit
analytical expression for the T-matrix elements in terms of the phase shift
$\delta_{m}(k)$ for each angular momentum channel $m\in\mathbb{Z}$

$T^{(\lambda,\xi)}_{\mathbf{k^{\prime}}_{\parallel}\mathbf{k}_{\parallel}}=-\frac{2\lambda\xi\hbar
v_{F}}{k}\sum_{m=-\infty}^{\infty}e^{i\delta_{m}(k)}\sin\delta_{m}(k)e^{im\phi},$
(16)

where $\phi$ is the angle between $\mathbf{k}_{\parallel}$ and
$\mathbf{k^{\prime}}_{\parallel}$, and the analytical expression for the phase
shift is given in Appendix B by Eq. (80).

##  III Scattering by a uniform concentration of dislocations

Figure 3: Random distribution of torsional dislocations seen from a plane
perpendicular to the cylinders axis.

Let us now consider a uniform concentration $n_{d}=N_{d}/A$ (per unit
transverse surface) of identical cylindrical dislocations, as depicted in FIG.
3, represented by the density function

$\rho(\mathbf{x})=\sum_{j=1}^{N_{d}}\delta(\mathbf{x}-\mathbf{X}_{j}),$ (17)

where $\mathbf{X}_{j}$ is the position of the $j$-defect’s axis. The Fourier
transform of this density function is thus given by the expression

$\tilde{\rho}(\mathbf{k}_{\parallel})=\int
d^{2}xe^{-i\mathbf{k}_{\parallel}\cdot\mathbf{x}}\rho(\mathbf{x})=\sum_{j=1}^{N_{d}}e^{-i\mathbf{k}_{\parallel}\cdot\mathbf{X}_{j}}.$
(18)

The operator that plays the role of a scattering potential for this
distribution of dislocation defects is

$V(\mathbf{x})=\int
d^{2}x^{\prime}\rho(\mathbf{x^{\prime}})H^{\xi}_{1}(\mathbf{x}-\mathbf{x}^{\prime})=\sum_{j=1}^{N_{d}}H^{\xi}_{1}(\mathbf{x}-\mathbf{X}_{j}),$
(19)

where $H^{\xi}_{1}$ is defined in Eq. (3) as the contribution from a single
dislocation. The matrix elements of the scattering operator Eq. (19) in the
free spinor basis defined by Eq. (4) are

$\displaystyle\left\langle\Phi_{\mathbf{k}_{\parallel},\lambda}\middle|V(\mathbf{x})\middle|\Phi_{\mathbf{k^{\prime}}_{\parallel},\lambda^{\prime}}\right\rangle=\left[\tilde{V}(\mathbf{k}_{\parallel}-\mathbf{k^{\prime}}_{\parallel})\right]_{\lambda\lambda^{\prime}},$
(20)

where $\tilde{V}(\mathbf{k}_{\parallel})$ is the Fourier transform

$\displaystyle\tilde{V}(\mathbf{k}_{\parallel})$
$\displaystyle=\int_{\mathbb{R}^{2}}d^{2}xe^{-i\mathbf{k}_{\parallel}\cdot\mathbf{x}}V(\mathbf{x})=\sum_{j=1}^{N_{d}}\int_{\mathbb{R}^{2}}d^{2}xe^{-i\mathbf{k}_{\parallel}\cdot\mathbf{x}}H^{\xi}_{1}(\mathbf{x}-\mathbf{X}_{j})$
$\displaystyle=\tilde{H}^{\xi}_{1}(\mathbf{k}_{\parallel})\tilde{\rho}(\mathbf{k}_{\parallel}).$
(21)

Then, the matrix elements of the potential in Eq. (20) become

$\left[\tilde{V}(\mathbf{k}_{\parallel})\right]_{\lambda\lambda^{\prime}}=\left[\tilde{H}^{\xi}_{1}(\mathbf{k}_{\parallel})\right]_{\lambda\lambda^{\prime}}\tilde{\rho}(\mathbf{k}_{\parallel}).$
(22)

Let us also introduce the configurational average of some quantity
$f(\mathbf{X}_{j})$ over the distributed dislocations as

$\left\langle
f\right\rangle=\int_{\mathbb{R}^{2}}d^{2}X_{j}\,\,P(\mathbf{X}_{j})f(\mathbf{X}_{j}),$
(23)

where $P(\mathbf{X}_{j})$ is the normalized distribution function for the
defects in the sample. In particular, for a uniform distribution we have
$P(\mathbf{X}_{j})=1/A$, where $A$ is the area of the plane normal to each
cylinder’s axis. Now, the full retarded Green’s function satisfies for the
potential of several dislocations $\hat{V}$ given in Eq. (19) is

$\hat{G}^{\xi}_{R}(E)=\hat{G}^{\xi}_{R,0}(E)+\hat{G}^{\xi}_{R,0}(E)\hat{V}\hat{G}^{\xi}_{R}(E).$
(24)

The configurational average, as defined in Eq. (23), of the complete Green’s
function in this last equation can be written as

$\left\langle\hat{G}^{\xi}_{R}(E)\right\rangle=\hat{G}^{\xi}_{R,0}(E)+\hat{G}^{\xi}_{R,0}(E)\Sigma_{R}^{\lambda,\xi}(E)\left\langle\hat{G}^{\xi}_{R}(E)\right\rangle.$
(25)

This is the Dyson’s equation with the retarded self-energy
$\Sigma_{R}^{\lambda,\xi}(E)$, that can be explicitly solved to yield

$\displaystyle\left\langle
G_{R}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle=\frac{1}{E-\lambda\xi\hbar
v_{F}|\mathbf{k}_{\parallel}|-\Sigma_{R}^{\lambda,\xi}(\mathbf{k}_{\parallel})}.$
(26)

The effect of the statistical distribution of dislocations’ is entirely
dictated by the function $\tilde{\rho}(\mathbf{k}_{\parallel})$. In the
perturbative expansion of the complete Green’s function, we encounter
$n^{th}$-products of the form
$\tilde{\rho}(\mathbf{k_{1}})\tilde{\rho}(\mathbf{k_{2}})\cdots\tilde{\rho}(\mathbf{k}_{n})$.
The configurational average of these products are

$\displaystyle\left\langle\tilde{\rho}(\mathbf{k}_{\parallel})\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\sum_{j=1}^{N_{d}}e^{-i\mathbf{k}_{\parallel}\cdot\mathbf{X}_{j}}\right\rangle=\sum_{j=1}^{N_{d}}\int_{\mathbb{R}^{2}}d^{2}X_{j}\,\frac{1}{A}\,e^{-i\mathbf{k}_{\parallel}\cdot\mathbf{X}_{j}}$
(27) $\displaystyle=$
$\displaystyle\frac{N_{d}}{A}(2\pi)^{2}\delta^{(2)}(\mathbf{k}_{\parallel}),$

for a single factor. For the product of two factors, we obtain

$\displaystyle\left\langle\tilde{\rho}(\mathbf{k}_{\parallel}^{1})\tilde{\rho}(\mathbf{k}_{\parallel}^{2})\right\rangle=\left\langle\sum_{j=1}^{N_{d}}\sum_{l=1}^{N_{d}}e^{-i\mathbf{k}_{\parallel}^{1}\cdot\mathbf{X}_{j}-i\mathbf{k}_{\parallel}^{2}\cdot\mathbf{X}_{l}}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\sum_{j=l}e^{-i(\mathbf{k}_{\parallel}^{1}+\mathbf{k}_{\parallel}^{2})\cdot\mathbf{X}_{j}}+\sum_{j\neq
l}e^{-i\mathbf{k}_{\parallel}^{1}\cdot\mathbf{X}_{j}-i\mathbf{k}_{\parallel}^{2}\cdot\mathbf{X}_{l}}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{N_{d}}{A}(2\pi)^{2}\delta^{(2)}(\mathbf{k}_{\parallel}^{1}+\mathbf{k}_{\parallel}^{2})+\frac{N_{d}(N_{d}-1)}{A^{2}}(2\pi)^{4}\delta^{(2)}(\mathbf{k}_{\parallel}^{1})\delta^{(2)}(\mathbf{k}_{\parallel}^{2}),$

and we have a similar behavior for higher order products. Now, notice that for
$N_{d}\gg 1$ we have $N_{d}(N_{d}-1)\approx N_{d}^{2}$,
$N_{d}(N_{d}-1)(N_{d}-3)\approx N_{d}^{3}$ and so on. We define the
concentration of defects, i.e., the number of dislocations per unit of area
perpendicular to the cylinder’s axis as $n_{d}=N_{d}/A$. As discussed in
standard references Hewson (1993); Mahan (2000), for small concentrations
$n_{d}\ll 1$ the scaling discussed before ensures that the total Green’s
function in Eq. (25) can be calculated accurately by the sequence of diagrams
for the retarded self-energy in momentum space as given in FIG. 4, an approach
well known as the non-crossing approximation (NCA). This series of diagrams
corresponds to the configurational average of the $T$-matrix over the random
distribution of dislocations after Eq. (23)

$\Sigma_{R}^{\lambda,\xi}(E)=\left\langle\hat{T}^{\xi}(E)\right\rangle=n_{d}T^{(\lambda,\xi)}_{\mathbf{k}_{\parallel}\mathbf{k}_{\parallel}}.$
(29)

Figure 4: Diagrams contributing to the retarded self-energy $\Sigma_{R}$. The
solid line corresponds to the free retarded Green’s function, the dashed line
the scattering perturbation $H_{1}$, and the $\times$ a factor of $n_{d}$.

Using the expression in Eq. (16) for the $T$-matrix elements, for
$\mathbf{k}_{\parallel}=\mathbf{k^{\prime}}_{\parallel}$ then $\phi=0$ and we
have that the real part of the self-energy

$\text{Re}\,\,\Sigma_{R}^{\lambda,\xi}(\mathbf{k}_{\parallel})=-\frac{2\lambda\xi
n_{d}\hbar
v_{F}}{k}\sum_{m=-\infty}^{\infty}\cos\delta^{(\lambda,\xi)}_{m}(k)\sin\delta^{(\lambda,\xi)}_{m}(k)$
(30)

contains an infinite sum over highly oscillatory terms, that converges to
zero. So no contribution comes from the real part of the self-energy; The
imaginary part, on the other hand, defines the relaxation time,

$\frac{1}{\tau^{(\lambda,\xi)}(k)}=-\frac{2\lambda\xi}{\hbar}n_{d}\,\,\text{Im}\,\,T^{(\lambda,\xi)}_{\mathbf{k}_{\parallel}\mathbf{k}_{\parallel}}.$
(31)

###  III.1 Electrical Conductivity in the linear-response

Now, we consider a single Fourier mode for an external electric field in the
gauge

$\mathbf{E}=-\frac{\partial}{\partial t}\mathbf{A}(\mathbf{r},t)$ (32)

where $\mathbf{A}(\mathbf{r},t)=\mathbf{A}(\mathbf{r},\omega)e^{-i\omega t}$
is the vector potential. Then, $\mathbf{E}=i\omega\mathbf{A}$. In the linear
response formalism, the current is given by the Kubo expression

$\displaystyle j_{\alpha}(\mathbf{r},\omega)$ $\displaystyle=$
$\displaystyle\int
d^{3}r^{\prime}\,\sigma_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega)E_{\beta}(\mathbf{r^{\prime}}.\omega),$
(33)

Here, the conductivity tensor is given by

$\sigma_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega)=\frac{1}{i\omega}K_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega).$
(34)

The tensor $K_{\alpha\beta}$ is defined, in the Kubo formalism, in terms of
the retarded current-current correlator as follows

$\displaystyle
K_{\alpha\beta}(\mathbf{r},t;\mathbf{r^{\prime}},t^{\prime})=i\hbar^{-1}\theta(t-t^{\prime})\rm{Tr}\left\\{\hat{\rho}\left[\hat{j}_{\alpha}(\mathbf{r},t),\hat{j}_{\beta}(\mathbf{r^{\prime}},t^{\prime})\right]\right\\},$
(35)

where $\hat{\rho}$ is the statistical density matrix operator. As shown in
detail in the Appendix D, the Fourier transform to the frequency domain of
this tensor can be expressed by

$\displaystyle K^{\xi}_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega)$
$\displaystyle=$ $\displaystyle
e^{2}v_{F}^{2}\int_{-\infty}^{\infty}\frac{dE^{\prime}}{2\pi}\int_{-\infty}^{\infty}\frac{dE}{2\pi}\frac{f_{0}\left(E^{\prime}\right)-f_{0}\left(E\right)}{\hbar\omega+E-E^{\prime}+i\eta^{+}}$
(36)
$\displaystyle\times\rm{Tr}\left[\sigma_{\alpha}\bm{\mathscr{A}}^{\xi}(\mathbf{r},\mathbf{r^{\prime}};E^{\prime})\sigma_{\beta}\bm{\mathscr{A}}^{\xi}(\mathbf{r^{\prime}},\mathbf{r};E)\right],$

where $f_{0}(E)=\left[e^{(E-\mu)/kT}+1\right]^{-1}$ is the Fermi distribution,
and we introduced the (disorder-averaged) spectral function

$\displaystyle\mathscr{A}^{\lambda,\xi}(k)$ $\displaystyle=$ $\displaystyle
i\left[\left\langle
G_{R}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle-\left\langle
G_{A}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle\right]$ (37)
$\displaystyle=$
$\displaystyle\frac{2\left(\frac{\hbar}{2\tau^{(\lambda,\xi)}(k)}\right)}{\left(E-\mathscr{E}^{\lambda,\xi}_{k}\right)^{2}+\left(\frac{\hbar}{2\tau^{(\lambda,\xi)}(k)}\right)^{2}},$

that clearly reduces to a Lorentzian distribution whose spectral width is
defined by the inverse of the relaxation time. See Appendix C for the details.
After some algebraic manipulations, we obtain the conductivity tensor at
finite frequency and temperature

$\displaystyle\text{Re}\,\sigma^{\xi}_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega)=-\frac{e^{2}\hbar
v_{F}^{2}}{2\pi}\int_{-\infty}^{\infty}dE\left\\{\frac{f_{0}(E+\hbar\omega)-f_{0}(E)}{\hbar\omega}\right\\}$
$\displaystyle\times\rm{Tr}\left[\sigma_{\alpha}\bm{\mathscr{A}}^{\xi}(\mathbf{r},\mathbf{r^{\prime}};E+\hbar\omega)\sigma_{\beta}\bm{\mathscr{A}}^{\xi}(\mathbf{r^{\prime}},\mathbf{r};E)\right].$
(38)

Using the coordinates representation of the spectral function given in
Appendix C, after Eq. (86), we can read off the Fourier transform to momentum
space of the conductivity

$\displaystyle\sigma^{\xi}_{\alpha\beta}(\mathbf{q};\omega)=-\frac{e^{2}\hbar
v_{F}^{2}}{2\pi}\int\frac{d^{3}k}{(2\pi)^{3}}\int_{-\infty}^{\infty}dE\left\\{\frac{f_{0}(E+\hbar\omega)-f_{0}(E)}{\hbar\omega}\right\\}$
$\displaystyle\quad\times\sum_{\lambda,\lambda^{\prime}}\operatorname{Tr}\left\\{\sigma_{\alpha}\left(\sigma_{0}+\lambda\frac{\bm{\sigma\cdot}(\mathbf{k}_{\parallel}+\mathbf{q})}{|\mathbf{k}_{\parallel}+\mathbf{q}|}\right)\sigma_{\beta}\left(\sigma_{0}+\lambda^{\prime}\frac{\bm{\sigma\cdot}\mathbf{k}_{\parallel}}{|\mathbf{k}_{\parallel}|}\right)\right\\}$
$\displaystyle\quad\times\mathscr{A}^{\lambda,\xi}(|\mathbf{k}_{\parallel}+\mathbf{q}|;E+\hbar\omega)\mathscr{A}^{\lambda^{\prime},\xi}(|\mathbf{k}_{\parallel}|;E).$
(39)

We are interested in the DC conductivity, so we take the limit
$\mathbf{q}\rightarrow\mathbf{0}$ first and then the limit $\omega\rightarrow
0$. After a long calculation (details in the Appendix D), the result is

$\displaystyle\sigma_{\alpha\beta}^{(\lambda,\xi)}(T)$
$\displaystyle=\delta_{\alpha\beta}\frac{e^{2}\hbar
v_{F}^{2}}{\pi^{3}}\int_{0}^{\infty}dk\,\,\int_{-\infty}^{\infty}dE\,\,\left(-\frac{\partial
f_{0}(E)}{\partial E}\right)$ $\displaystyle\times\left\langle
G_{R}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle\left\langle
G_{A}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle\mathbf{k}_{\parallel}\cdot\mathbf{k}_{\parallel}.$
(40)

###  III.2 Vertex corrections

The self-energy contribution modifies the definition of the retarded and
advanced Green´s functions in Eq. (40), as depicted by the double lines in
Fig.5(b). However, there are also scattering processes involving links between
the two internal Green function lines, as depicted in FIG. 5(a). When
considering such diagrams with cross-links, as in FIG. 5(a), we must include
the vertex correction as depicted in FIG. 5(b).

Figure 5: (a) A typical diagram contributing to the conductivity in Eq (40),
involving the configurational average of the two internal GF with cross-links
among them. The upper line corresponds to the retarded GF and the lower to the
advanced GF. (b) Diagrammatic representation of the two complete averaged GF
(double lines) corresponding to the sum of all diagrams of the kind in (a)
with the vertex correction $\bm{\Gamma}(\mathbf{k}_{\parallel})$.

Taking into account the vertex correction, the conductivity becomes

$\displaystyle\sigma_{\alpha\beta}^{(\lambda,\xi)}(T)$
$\displaystyle=\delta_{\alpha\beta}\frac{e^{2}\hbar
v_{F}^{2}}{\pi^{3}}\int_{0}^{\infty}dk\,\,\int_{-\infty}^{\infty}dE\,\,\left(-\frac{\partial
f_{0}(E)}{\partial E}\right)$ $\displaystyle\quad\times\left\langle
G_{R}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle\left\langle
G_{A}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle\mathbf{k}_{\parallel}\cdot\bm{\Gamma}_{RA}(\mathbf{k}_{\parallel},E),$
(41)

Figure 6: The Bethe-Salpeter integral equation for the vertex function
$\bm{\Gamma}_{RA}(\mathbf{k}_{\parallel})$.

where the vertex function $\bm{\Gamma}_{RA}(\mathbf{k}_{\parallel},E)$ is
given as the solution to the Bethe-Salpeter equation as depicted in FIG. 6.
Then, we have

$\displaystyle\bm{\Gamma}_{RA}(\mathbf{k}_{\parallel},E)$ $\displaystyle=$
$\displaystyle\mathbf{k}_{\parallel}+n_{d}\int\frac{d^{2}k^{\prime}}{(2\pi)^{2}}\left\langle
G_{R}^{\lambda,\xi}(\mathbf{k^{\prime}}_{\parallel})\right\rangle\left\langle
G_{A}^{\lambda,\xi}(\mathbf{k^{\prime}}_{\parallel})\right\rangle$ (42)
$\displaystyle\times\left|T^{(\lambda,\xi)}_{\mathbf{k^{\prime}}_{\parallel}\mathbf{k}_{\parallel}}\right|^{2}\bm{\Gamma}_{RA}(\mathbf{k^{\prime}}_{\parallel},E).$

The iterative solution of the Eq. (42) for
$\bm{\Gamma}_{RA}(\mathbf{k}_{\parallel},E)$ shows that the vertex function
must be of the form

$\bm{\Gamma}_{RA}(\mathbf{k}_{\parallel},E)=\gamma(\mathbf{k}_{\parallel},E)\mathbf{k}_{\parallel}.$
(43)

Then we obtain an integral equation for the scalar function
$\gamma(\mathbf{k}_{\parallel},E)$ that in the low concentration limit becomes

$\displaystyle\gamma(\mathbf{k}_{\parallel},E)$ $\displaystyle=$
$\displaystyle
1+n_{d}\frac{2\pi}{\hbar}\int\frac{d^{2}k^{\prime}}{(2\pi)^{2}}\tau^{(\lambda,\xi)}(k^{\prime})\left|T^{(\lambda,\xi)}_{\mathbf{k^{\prime}}_{\parallel}\mathbf{k}_{\parallel}}\right|^{2}$
(44) $\displaystyle\times\delta(E-\lambda\xi\hbar
v_{F}k^{\prime})\,\,\gamma(\mathbf{k^{\prime}}_{\parallel},E)\frac{\mathbf{k}_{\parallel}\cdot\mathbf{k^{\prime}}_{\parallel}}{k^{2}}.$

In the limit of low concentrations, we use the result in Appendix D, Eq.(99),
to obtain

$\displaystyle\sigma_{\alpha\alpha}^{(\lambda,\xi)}(T)$
$\displaystyle=\frac{2e^{2}v_{F}^{2}}{\pi^{2}}\int_{0}^{\infty}dk\,k^{2}\,\left(-\frac{\partial
f_{0}(E)}{\partial E}\right)_{E=\lambda\xi\hbar v_{F}k}$
$\displaystyle\quad\times\tau^{(\lambda,\xi)}(k)\,\,\gamma(\mathbf{k}_{\parallel},\lambda\xi\hbar
v_{F}k).$ (45)

At low temperatures, an exact solution is possible since the Fermi
distribution derivative takes a compact support at the Fermi energy. Then we
can evaluate $\gamma(k)$ and $\tau^{(\lambda,\xi)}(k)$ at the Fermi momentum
$k_{F}^{\xi}$, to obtain

$\gamma(k_{F}^{\xi})=\frac{\tau^{(\lambda,\xi)}_{1}(k_{F}^{\xi})}{\tau^{(\lambda,\xi)}_{1}(k_{F}^{\xi})-\tau^{(\lambda,\xi)}(k_{F}^{\xi})},$
(46)

where we defined (for
$\cos\phi^{\prime}=\mathbf{k}_{\parallel}\cdot\mathbf{k^{\prime}}_{\parallel}/k^{2}$)

$\frac{1}{\tau^{(\lambda,\xi)}_{1}(k_{F}^{\xi})}=n_{d}\frac{2\pi}{\hbar}\int\frac{d^{2}k^{\prime}}{(2\pi)^{2}}\left|T^{(\lambda,\xi)}_{\mathbf{k^{\prime}}_{\parallel}\mathbf{k}_{\parallel}}\right|^{2}\cos\phi^{\prime}\,\,\delta(\hbar
v_{F}k_{F}^{\xi}-\hbar v_{F}k^{\prime}).$ (47)

After the substitution in Eq. (45) of $\gamma(\mathbf{k}_{\parallel},E)$ given
in Eq. (44), we get

$\displaystyle\sigma_{\alpha\alpha}^{(\lambda,\xi)}(T)$ $\displaystyle=$
$\displaystyle\frac{2e^{2}v_{F}^{2}}{\pi^{2}k_{B}T}\tau^{(\lambda,\xi)}_{\text{tr}}(k_{F}^{\xi})\int_{0}^{\infty}dk\,k^{2}\,f_{0}\left(\mathscr{E}^{\lambda,\xi}_{\mathbf{k}_{\parallel}}\right)\left[1-f_{0}\left(\mathscr{E}^{\lambda,\xi}_{\mathbf{k}_{\parallel}}\right)\right]$
(48) $\displaystyle=$
$\displaystyle-\frac{4}{\pi^{2}v_{F}}\left(\frac{e^{2}}{\hbar}\right)\left(\frac{k_{B}T}{\hbar}\right)^{2}\tau^{(\lambda,\xi)}_{\text{tr}}(k_{F}^{\xi})\,\text{Li}_{2}\left(-e^{\frac{\hbar
v_{F}k_{F}}{k_{B}T}}\right),$

where $\text{Li}_{2}(x)$ is the polylogarithm of order 2. Here, the total
transport relaxation time is defined by

$\displaystyle\frac{1}{\tau^{(\lambda,\xi)}_{\text{tr}}(k_{F}^{\xi})}=\frac{1}{\tau^{(\lambda,\xi)}(k_{F}^{\xi})}-\frac{1}{\tau^{(\lambda,\xi)}_{1}(k_{F}^{\xi})}$
$\displaystyle=$ $\displaystyle\frac{2\pi
n_{d}}{\hbar}\int\frac{d^{2}k^{\prime}}{(2\pi)^{2}}\delta(\hbar
v_{F}k_{F}^{\xi}-\hbar
v_{F}k^{\prime})\left|T^{(\lambda,\xi)}_{\mathbf{k^{\prime}}_{\parallel}\mathbf{k}_{\parallel}}\right|^{2}(1-\cos\phi^{\prime})$

Using the form of the $T$-matrix elements in Eq. (16), we get a closed
expression in terms of the scattering phase shifts $\delta_{m}(k)$

$\frac{1}{\tau^{(\lambda,\xi)}_{\text{tr}}(k_{F}^{\xi})}=\frac{2n_{d}v_{F}}{k_{F}^{\xi}}\sum_{m=-\infty}^{\infty}\sin^{2}\left[\delta_{m}(k_{F}^{\xi})-\delta_{m-1}(k_{F}^{\xi})\right].$
(50)

From Eq. (48), we can investigate the zero temperature $T\rightarrow 0$ and
high temperature $T\gg\hbar v_{F}k_{F}/k_{B}$ limits, respectively. In the
zero temperature limit, we obtain

$\sigma_{\alpha\alpha}^{(\lambda,\xi)}(T\rightarrow
0)=\frac{2}{\pi^{2}}k_{F}^{\xi
2}\left(\frac{e^{2}}{\hbar}\right)v_{F,\alpha}\tau^{(\lambda,\xi)}_{\text{tr}}(k_{F}^{\xi}),$
(51)

a constant that depends on the miscroscopic material properties (such as
$v_{F}$), as well as on the concentration of dislocations $n_{d}$ through the
relaxation time.

On the other hand, in the high-temperature limit $T\gg\hbar v_{F}k_{F}/k_{B}$,
we obtain a quadratic dependence on temperature

$\sigma_{\alpha\alpha}^{(\lambda,\xi)}(T\gg\hbar
v_{F}k_{F}/k_{B})=\frac{1}{3v_{F,\alpha}}\left(\frac{e^{2}}{\hbar}\right)\left(\frac{k_{B}T}{\hbar}\right)^{2}\tau^{(\lambda,\xi)}_{\text{tr}}(k_{F}^{\xi}),$
(52)

where the overall constant depends on the microscopic parameters for each
material, as well as on the concentration of dislocations through the
relaxation time.

##  IV RESULTS

In this section, we apply the theory and analytical expressions obtained in
the previous section to calculate the electrical conductivity of several
materials in the family of transition metals monopnictides, i.e. TaAs, TaP,
NbAs and NbP. For an estimation of the concentration of defects $n_{d}$ in
real crystal systems, the Ref. Szot  _et al._ (2018) reports that the native
concentration of dislocations in the lattice of the materials TiO2 and SrTiO3,
vary in the range $n_{d}\sim 10^{5}-10^{7}\,\text{cm}^{-2}$. These
concentrations can be enhanced using different treatments up to
$10^{13}\,\text{cm}^{-2}$, nearly to the rendering amorphous limit. The
microscopic/atomistic parameters involved in our theory are obtained from ab-
initio studies for WSM materials, as reported in Ref. Lee  _et al._ (2015) and
Ref. Grassano  _et al._ (2018). In particular, the later reference identifies
anisotropies in the Fermi velocities and density of charge carries at
different Weyl nodes and bands. Using these results for the densities of
carriers, we compute the Fermi momentum at each Weyl node, i.e.,
$k_{F}^{\xi}$, as displayed in TABLE 1.

Material |  $k_{F}^{+}$ [nm-1] |  $k_{F}^{-}$ [nm-1]
---|---|---
TaAs | 0.23 | 0.05
TaP | 0.50 | 0.09
NbAs | 0.46 | 0.03
NbP | 1.04 | 0.15
Table 1: Values of $k_{F}^{\xi}$ computed from the carrier densities reported
in Ref. Grassano  _et al._ (2018).

In what follows, for definiteness we shall assume that the axis of the defects
is along the crystallographic $z$-direction and that we are measuring the
conductivity along the $x$-direction. Then, we use the reported $x$-components
of the Fermi velocitiesLee  _et al._ (2015); Grassano  _et al._ (2018). We
have different Fermi velocities $v^{(\lambda,\xi)}_{F,x}$, for the conduction
band ($\lambda=+1$) and for the valence band ($\lambda=-1$), and for each of
the Weyl nodes ($\xi=\pm$). Actually, for the valence band Refs. Lee  _et al._
(2015); Grassano  _et al._ (2018) report the hole velocity. Their results are
presented in the TABLE 2.

Material | $v^{(+,+)}_{F,x}$ | $v^{(-,+)}_{F,x}$ | $v^{(+,-)}_{F,x}$ | $v^{(-,-)}_{F,x}$
---|---|---|---|---
TaAs | 3.2 | -5.3 | 2.6 | -4.3
TaP | 3.7 | -5.4 | 2.0 | -3.9
NbAs | 3.0 | -4.8 | 2.5 | -3.2
NbP | 3.0 | -5.1 | 1.7 | -2.4
Table 2: Values of the Fermi velocity $v^{(\lambda,\xi)}_{F,x}$ in units of
$10^{5}$ m/s, as reported in Ref. Grassano  _et al._ (2018). Notice that for
the valence bands ($\lambda=-1$) they report the hole velocity.

Now, in order to study the additional effect of the torsional dislocations, we
follow our previous work Bonilla  _et al._ (2021). We assume that the
dislocations are cylindrical regions along the $z$-axis with radius $a$. Here,
we further assume that the defects possess an average radius of $a=15$ nm. The
simple relation between the torsional angle $\theta$ (in degrees) and the
pseudo-magnetic field representing strain is
$B_{S}a^{2}=1.36\,\theta\,\tilde{\phi}_{0}$ Bonilla  _et al._ (2021), where
the modified flux quantum in this material is approximately
$\displaystyle\tilde{\phi}_{0}\equiv\frac{\hbar
v_{F}}{e}=\frac{1}{2\pi}\frac{v_{F}}{c}\frac{hc}{e}=\frac{1}{2\pi}\frac{1.5}{300}\cdot
4.14\times 10^{5}$ T$\mathring{\text{A}}^{2}\approx 330$
T$\mathring{\text{A}}^{2}$. In this work, we have chosen a torsion angle
$\theta=15^{\circ}$. The lattice mismatch effect at the surface of the
dislocation cylinders is modeled by a repulsive delta-potential, with strength
$V_{0}$, expressed in terms of the “spinor rotation” angle $\alpha=V_{0}/\hbar
v_{F}$. According to our previous work Bonilla  _et al._ (2021), a realistic
choice is $\alpha=3\pi/4$.

With all of these parameters fixed, we can compute the total relaxation time
for each material. Our results are presented in TABLE 3.

Material |  $\tau$ [$10^{-13}$ s] |  $\tau_{\text{tr}}$ [$10^{-13}$ s]
---|---|---
TaAs | 2.2 | 2.6
TaP | 2.4 | 3.2
NbAs | 2.2 | 3.1
NbP | 2.4 | 4.2
Table 3: Computed values for the total relaxation time and the total transport
relaxation time for each material. We consider a concentration of dislocations
$n_{d}=10^{11}$ cm-2.

Now, we compute the conductivity along the $x$-direction $\sigma_{xx}$. In
what follows, we simply call it $\sigma(T)$, as a function of temperature. The
total conductivity is the sum over nodes and bands

$\sigma(T)=\sum_{\xi=\pm 1}\sum_{\lambda=\pm
1}\sigma_{xx}^{(\lambda,\xi)}(T),$ (53)

where $\sigma_{xx}^{(\lambda,\xi)}(T)$ is given in Eq. (48), including the
vertex correction. Our results for $T=0$ are presented in the TABLE 4.

Material |  $\sigma_{0}$ [$10^{3}$ $\Omega^{-1}$ cm-1]
---|---
TaAs | 1.5
TaP | 7.9
NbAs | 7.5
NbP | 34.6
Table 4: Computed values for the total conductivity $\sigma_{0}=\sigma(T=0)$
at zero temperature for each material. We consider a value of $n_{d}=10^{11}$
cm-2.

The conductivity as a function of temperature, for the transition metals
monopnictides TaAs, TaP, NbAs and NbP, is presented in FIG. 7 for all of them
compared, and individually in the pannel FIG. 8 (a)-(d).

Figure 7: A comparison of the total conductivity versus temperature behavior
for the transition metals monopnictides TaAs, TaP, NbAs and NbP. We use a
value of $n_{d}=10^{11}$ cm-2.

Figure 8: The total conductivity vs. temperature behavior for the transition
metals monopnictides TaAs, TaP, NbAs and NbP. We use a value of
$n_{d}=10^{11}$ cm-2..

Now, let us study the conductivity behavior with respect to the density of
dislocations $n_{d}$. In FIG. 9, we present a plot of the natural logarithm of
the conductivity versus temperature for three different concentrations of
dislocations.

Figure 9: Natural logarithm of the conductivity versus temperature for 3
different concentrations of dislocations. The graphs were computed at zero
temperature.

The total conductivity as a function of the concentration of defects and at
zero temperature is presented in FIG. 10.

Figure 10: Plot of total conductivity versus defects’ concentration. The
graphs were computed at zero temperature.

Finally, a plot of the resistance, defined as the inverse of conductivity, as
a function of the dislocations’ density is presented in FIG. 11.

Figure 11: Total resistance, $R=1/G$, as a function of the concentration of
defects $n_{d}$ for the family of materials TaAs, TaP, NbAs and NbP. The
graphs were computed at zero temperature.

##  V Conclusions

In this work, we have studied the effect of a distribution of mechanical
defects, i.e. torsional dislocations, over the electrical conductivity of the
family of transition metals monopnictides TaAs, TaP, NbAs and NbP. Our theory
is based on the mathematical analysis of the scattering phase shifts from a
single defect, as stated in our previous work Bonilla  _et al._ (2021); Muñoz
and Soto-Garrido (2019, 2017); Soto-Garrido and Muñoz (2018); Soto-Garrido
_et al._ (2020). We extended this previous analysis to develop a Green´s
function formalism, in order to represent the scattering due to a finite
concentration of randomly distributed defects. Within the non-crossing
approximation for the self-energy, we solved explicitly for the disorder-
averaged retarded Green´s function, that allows us to calculate the electrical
conductivity in the Kubo linear-response formalism. We obtained general
analytical expressions in terms of the parameters involved in the low-energy
model representing the family of materials, and using the ab-initio
estimations for such parameters, we provided a characterization of the
conductivity as a function of temperature and concentration of defects for the
transition metal monopnictides TaAs, TaP, NbAs and NbP. As a universal
feature, we identified a $\sim T^{2}$ temperature dependence for $T\gg\hbar
v_{F}k_{F}/k_{B}$, where the pre-factor depends on material-specific
microscopic parameters as well as in the concentration of dislocations $n_{d}$
through the scattering relaxation time. Our results do not involve the
electron-phonon scattering effects, that will presumably contribute at higher
temperatures, which is a subject of further study.

##  Appendix A  Calculation of the retarded free Green’s function

The free retarded Green’s function (GF) is represented in the coordinate basis
as follows

$\mathbf{G}^{\xi}_{R,0}(\mathbf{r},\mathbf{r^{\prime}};E)=\left\langle\mathbf{r}\middle|\frac{1}{E-\hat{H}^{\xi}_{0}+i\eta^{+}}\middle|\mathbf{r^{\prime}}\right\rangle.$
(54)

In this basis, $\mathbf{G}^{\xi}_{R,0}(\mathbf{r},\mathbf{r^{\prime}};E)$
satisfies the differential equation

$\left(E+i\eta^{+}-\xi
v_{F}\bm{\sigma}\cdot\frac{\hbar}{i}\bm{\nabla}\right)\mathbf{G}^{\xi}_{R,0}(\mathbf{r},\mathbf{r^{\prime}};E)=\sigma_{0}\delta^{(3)}(\mathbf{r}-\mathbf{r^{\prime}}).$
(55)

where $\sigma_{0}$ is the $2\times 2$ unit matrix. Let us introduce the scalar
GF, $\mathscr{G}^{\xi}_{R,0}(\mathbf{r},\mathbf{r^{\prime}})$, by means of the
expression

$\mathbf{G}^{\xi}_{R,0}(\mathbf{r},\mathbf{r^{\prime}};E)=\left(E+i\eta^{+}+\xi
v_{F}\bm{\sigma}\cdot\frac{\hbar}{i}\bm{\nabla}\right)\mathscr{G}^{\xi}_{R,0}(\mathbf{r},\mathbf{r^{\prime}}).$
(56)

Bearing in mind that we are treating the elastic scattering problem, the
energy of the out-state must be the same as those of the incident-free-
particle state, i.e., $E=\lambda\xi\hbar v_{F}|\mathbf{k}|$. Then, the scalar
GF satisfies the Helmholtz equation

$\left(\nabla^{2}+k^{2}+i\eta^{+}\right)\mathscr{G}^{\xi}_{R,0}(\mathbf{r},\mathbf{r^{\prime}})=\frac{1}{\hbar^{2}v_{F}^{2}}\delta^{(3)}(\mathbf{r}-\mathbf{r^{\prime}}).$
(57)

Due to the symmetry along the $z$-axis we can decouple it from its
perpendicular plane as follows

$\mathscr{G}^{\xi}_{R,0}(\mathbf{r},\mathbf{r^{\prime}};k)=\int_{-\infty}^{\infty}\frac{dq_{z}}{2\pi}e^{iq_{z}(z-z^{\prime})}\mathscr{G}^{\xi}_{R,0}(\mathbf{x},\mathbf{x^{\prime}};q_{z},k),$
(58)

where $\mathscr{G}^{\xi}_{R,0}(\mathbf{x},\mathbf{x^{\prime}};q_{z},k)$ is a
reduced GF and $\mathbf{x}=(x,y)$ is the position vector on the plane. Then,
the Helmholtz equation for the reduced GF on the plane takes the form

$\left(\nabla_{\parallel}^{2}-q_{z}^{2}+k^{2}+i\eta^{+}\right)\mathscr{G}^{\xi}_{R,0}(\mathbf{x},\mathbf{x^{\prime}};q_{z},k)=\frac{1}{\hbar^{2}v_{F}^{2}}\delta^{(2)}(\mathbf{x}-\mathbf{x^{\prime}}),$
(59)

where $\nabla_{\parallel}^{2}=\partial_{x}^{2}+\partial_{y}^{2}$. As we can be
seen from the FIG.1, the free incident particle’s propagation is normal to the
cylinder’s axis. We assume that the incident particles have negligible
momentum along the $z$-axis, and by momentum conservation, they remain with
negligible momentum along that direction during the transport process. Then,
we can write $\mathbf{k}=(\mathbf{k}_{\parallel},0)$ where
$\mathbf{k}_{\parallel}=(k_{x},k_{y})$. Hence, the system is reduced to an
effective two-dimensional description and we can consider the reduced GF on
the plane as independent of the Fourier mode $q_{z}$. Then, from Eq.(58) we
have
$\mathscr{G}^{\xi}_{R,0}(\mathbf{r},\mathbf{r^{\prime}};k)=\delta(z-z^{\prime})\mathscr{G}^{\xi}_{R,0}(\mathbf{x},\mathbf{x^{\prime}};k)$,
and we can expand the reduced GF on the plane in the traverse Fourier space

$\mathscr{G}^{\xi}_{R,0}(\mathbf{x},\mathbf{x^{\prime}};k)=\int\frac{d^{2}q_{\parallel}}{(2\pi)^{2}}e^{i\mathbf{q}_{\parallel}\cdot(\mathbf{x}-\mathbf{x^{\prime}})}\tilde{\mathscr{G}}^{\xi}_{R,0}(\mathbf{q}_{\parallel}),$
(60)

where $\mathbf{q}_{\parallel}=(q_{x},q_{y})$. Replacing in the Eq.(59), we
obtain in the traverse Fourier space

$\tilde{\mathscr{G}}^{\xi}_{R,0}(\mathbf{q}_{\parallel})=-\frac{1}{\hbar^{2}v_{F}^{2}}\frac{1}{q_{\parallel}^{2}-k^{2}-i\eta^{+}}.$
(61)

We perform the integration in Eq.(60) in polar coordinates

$\displaystyle\mathscr{G}^{\xi}_{R,0}(\mathbf{x},\mathbf{x^{\prime}};k)$
$\displaystyle=-\frac{1}{\hbar^{2}v_{F}^{2}}\frac{2\pi}{(2\pi)^{2}}\int_{0}^{\infty}dq_{\parallel}\,q_{\parallel}\frac{J_{0}(q_{\parallel}R)}{q_{\parallel}^{2}-k^{2}-i\eta^{+}},$
(62)

where $R=|\mathbf{x}-\mathbf{x^{\prime}}|$ and we have used the integral
representation of the Bessel functions $\int_{0}^{2\pi}e^{iz\cos\phi\pm
in\phi}d\phi=2\pi i^{n}J_{n}(z)$. In order to perform the last integration we
need the result

$\int_{0}^{\infty}\frac{x^{\nu+1}J_{\nu}(bx)}{\left(x^{2}+a^{2}\right)^{\mu+1}}dx=\frac{a^{\nu-\mu}b^{\mu}}{2^{\mu}\Gamma(\mu+1)}K_{\nu-\mu}(ab),$
(63)

together with the relation
$K_{n}(z)=\frac{\pi}{2}i^{n+1}H^{(1)}_{n}\left(iz\right)$. The result is

$\mathscr{G}^{\xi}_{R,0}(\mathbf{x},\mathbf{x^{\prime}};k)=-\frac{i}{4\hbar^{2}v_{F}^{2}}H^{(1)}_{0}\left(k|\mathbf{x}-\mathbf{x^{\prime}}|\right).$
(64)

This form is adequate because in the asymptotic form for large
$|\mathbf{x}-\mathbf{x^{\prime}}|$ it produces outgoing cylindrical waves as
it is desired for the retarded GF. Now, to obtain the final form for the free
GF matrix we apply the definition in Eq.(56) with $E=\lambda\xi\hbar
v_{F}|\mathbf{k}|$, taking into account that we have reduced to a two
dimensional system on the plane $x$-$y$

$\displaystyle\mathbf{G}^{\xi}_{R,0}\left(\mathbf{r},\mathbf{r^{\prime}};k\right)$
$\displaystyle=-\frac{i\lambda\xi}{4\hbar v_{F}}\delta(z-z^{\prime})$
$\displaystyle\quad\quad\times\left(k\sigma_{0}-i\lambda\bm{\sigma}\cdot\bm{\nabla}_{\parallel}\right)H^{(1)}_{0}\left(k|\mathbf{x}-\mathbf{x^{\prime}}|\right).$
(65)

In plane polar coordinates

$\bm{\sigma}\cdot\bm{\nabla}_{\parallel}=\left(\bm{\sigma}\cdot\mathbf{\hat{r}}\right)\frac{\partial}{\partial
r}+\left(\bm{\sigma}\cdot\bm{\hat{\varphi}}\right)\frac{1}{r}\frac{\partial}{\partial\varphi}$
(66)

where $r=|\mathbf{x}-\mathbf{x^{\prime}}|$, $\varphi$ is the angle the vector
$\mathbf{x}-\mathbf{x^{\prime}}$ makes with the $x$ axis and

$\bm{\sigma}\cdot\mathbf{\hat{r}}=\begin{pmatrix}0&e^{-i\varphi}\\\
e^{i\varphi}&0\end{pmatrix},\quad\bm{\sigma}\cdot\bm{\hat{\varphi}}=\begin{pmatrix}0&-ie^{-i\varphi}\\\
ie^{i\varphi}&0\end{pmatrix}.$ (67)

The final form for the retarded Green’s function matrix in the coordinates
representation is

$\displaystyle\mathbf{G}^{\xi}_{R,0}\left(\mathbf{r},\mathbf{r^{\prime}};k\right)$
$\displaystyle=-\frac{\lambda\xi ik}{4\hbar v_{F}}\delta(z-z^{\prime})$
$\displaystyle\times\begin{bmatrix}H^{(1)}_{0}\left(k|\mathbf{x}-\mathbf{x^{\prime}}|\right)&i\lambda
e^{-i\varphi}H^{(1)}_{1}\left(k|\mathbf{x}-\mathbf{x^{\prime}}|\right)\\\
i\lambda
e^{i\varphi}H^{(1)}_{1}\left(k|\mathbf{x}-\mathbf{x^{\prime}}|\right)&H^{(1)}_{0}\left(k|\mathbf{x}-\mathbf{x^{\prime}}|\right)\end{bmatrix},$
(68)

which produces Eq.(9).

##  Appendix B Scattering by a single cylindrical defect

We can represent the Lippmann-Schwinger Eq.(7) in the coordinate basis as
follows

$\displaystyle\Psi_{\mathbf{k},\lambda}(\mathbf{r})=\Phi_{\mathbf{k},\lambda}(\mathbf{r})+$
$\displaystyle\int d^{3}r^{\prime}\int
d^{3}r^{\prime\prime}\left\langle\mathbf{r}\middle|\hat{G}^{\xi}_{R,0}\left(E\right)\middle|\mathbf{r^{\prime}}\right\rangle$
(69)
$\displaystyle\times\left\langle\mathbf{r^{\prime}}\middle|\hat{T}^{\xi}(E)\middle|\mathbf{r^{\prime\prime}}\right\rangle\Phi_{\mathbf{k},\lambda}(\mathbf{r^{\prime\prime}}).$

The form of free spinors in Eq.(6) with momentum $\mathbf{k}_{\parallel}$ on
the $x$-$y$ plane is

$\Phi_{\lambda,\mathbf{k}}(\mathbf{r})=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
\lambda
e^{i\phi}\end{pmatrix}e^{i\mathbf{k}\cdot\mathbf{r}}\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
\lambda
e^{i\phi}\end{pmatrix}e^{i\mathbf{k}_{\parallel}\cdot\mathbf{x}}=\Phi_{\lambda,\mathbf{k}_{\parallel}}(\mathbf{x}).$
(70)

where $k_{x}=k\cos\phi$ and $k_{y}=k\sin\phi$. The incident spinors are
assumed to enter the scattering region with momentum along the $x$ axis, and
they are represented by

$\Phi^{\text{inc}}_{\lambda,\mathbf{k}_{\parallel}}(\mathbf{x})=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
\lambda\end{pmatrix}e^{ikx}.$ (71)

Now, $\hat{H}_{1}^{\xi}$ is a local potential independent of the $z$
coordinate as can be seen from Eq.(3). Then, the $T$-matrix is diagonal in the
coordinate basis and depends only on vectors $\mathbf{x}$ on the plane. Thus,
$\left\langle\mathbf{r^{\prime}}\middle|\hat{T}^{\xi}(E)\middle|\mathbf{r^{\prime\prime}}\right\rangle=\mathbf{T}^{\xi}(\mathbf{x^{\prime}},E)\delta^{(3)}(\mathbf{r^{\prime}}-\mathbf{r^{\prime\prime}})$,
where $\mathbf{T}^{\xi}(\mathbf{x^{\prime}},E)$ is a $2\times 2$ matrix. The
incident spinor is given in Eq.(71), and using the retarded GF in Eq.(68), the
Lippmann-Schwinger equation in Eq.(69) is reduced to the $x$-$y$ plane as
follows

$\displaystyle\Psi_{\mathbf{k}_{\parallel},\lambda}(\mathbf{x})$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
\lambda\end{pmatrix}e^{ikx}$ $\displaystyle\quad+\int
d^{2}x^{\prime}\mathbf{G}^{\xi}_{R,0}\left(\mathbf{x},\mathbf{x^{\prime}};k\right)\mathbf{T}^{\xi}(\mathbf{x^{\prime}},E)\Phi_{\mathbf{k}_{\parallel},\lambda}(\mathbf{x^{\prime}}),$
(72)

where $\mathbf{G}^{\xi}_{R,0}\left(\mathbf{x},\mathbf{x^{\prime}};k\right)$ is
given in Eq.(9). The asymptotic form for large argument of the Hankel’s
functions are

$\displaystyle H^{(1)}_{0}(k|\mathbf{x}-\mathbf{x^{\prime}}|)$
$\displaystyle\sim\sqrt{\frac{2}{i\pi
k|\mathbf{x}-\mathbf{x^{\prime}}|}}e^{ik|\mathbf{x}-\mathbf{x^{\prime}}|},$
(73) $\displaystyle H^{(1)}_{1}(k|\mathbf{x}-\mathbf{x^{\prime}}|)$
$\displaystyle\sim-i\sqrt{\frac{2}{i\pi
k|\mathbf{x}-\mathbf{x^{\prime}}|}}e^{ik|\mathbf{x}-\mathbf{x^{\prime}}|},$
(74)

where we have used the known limiting form

$H^{(1)}_{\nu}(x)\sim\sqrt{\frac{2}{\pi
x}}e^{i\left(x-\frac{\nu\pi}{2}-\frac{\pi}{4}\right)},\quad
x\rightarrow\infty.$ (75)

Now, recall the geometry of the scattering process as depicted in FIG.2. We
expand $|\mathbf{x}-\mathbf{x^{\prime}}|$ for large $|\mathbf{x}|$ as follows

$|\mathbf{x}-\mathbf{x^{\prime}}|\sim
r-\mathbf{\hat{n}}\cdot\mathbf{x^{\prime}}+\mathscr{O}\left((r^{\prime}/r)^{2}\right),$
(76)

where $r=|\mathbf{x}|$, $r^{\prime}=|\mathbf{x^{\prime}}|$ and
$\mathbf{\hat{n}}$ is the unit vector in the direction of $\mathbf{x}$, i.e.,
$\mathbf{\hat{n}}=\mathbf{x}/r$. Noting that in this asymptotic form the
direction of $\mathbf{k^{\prime}}_{\parallel}$ coincides with that of
$\mathbf{x}$ and is practically the same of $\mathbf{x}-\mathbf{x^{\prime}}$,
i.e., $\mathbf{k^{\prime}}_{\parallel}=k\mathbf{\hat{n}}$ and that the angle
$\phi$ the vector $\mathbf{k^{\prime}}_{\parallel}$ makes with the
$\mathbf{k}_{\parallel}$ incident momentum is approximately the same angle
$\mathbf{x}-\mathbf{x^{\prime}}$ does, i.e., $\phi\sim\varphi$, we have the
asymptotic form for the free Green’s function in Eq.(9)

$\mathbf{G}^{\xi}_{R,0}\left(\mathbf{x},\mathbf{x^{\prime}};k\right)\sim-\frac{\lambda\xi
k}{4\hbar v_{F}}\sqrt{\frac{2i}{\pi k}}\begin{bmatrix}1&\lambda e^{-i\phi}\\\
\lambda
e^{i\phi}&1\end{bmatrix}e^{-i\mathbf{k^{\prime}}_{\parallel}\cdot\mathbf{x^{\prime}}}\frac{e^{ikr}}{\sqrt{r}}.$
(77)

Replacing the asymptotic form in Eq.(77) in the Eq.(72) we obatain Eq.(14)

$\Psi_{\mathbf{k}_{\parallel},\lambda}(\mathbf{x})\sim\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
\lambda\end{pmatrix}e^{ikx}-\frac{\lambda\xi}{2\hbar
v_{F}}\sqrt{\frac{ik}{\pi}}T^{(\lambda,\xi)}_{\mathbf{k^{\prime}}_{\parallel}\mathbf{k}_{\parallel}}\begin{pmatrix}1\\\
\lambda e^{i\phi}\end{pmatrix}\frac{e^{ikr}}{\sqrt{r}},$ (78)

where the $T$-matrix elements are

$T^{(\lambda,\xi)}_{\mathbf{k^{\prime}}_{\parallel}\mathbf{k}_{\parallel}}\left(E\right)=\int
d^{2}x^{\prime}\Phi^{\dagger}_{\mathbf{k^{\prime}}_{\parallel},\lambda}(\mathbf{x^{\prime}})\mathbf{T}^{\xi}(\mathbf{x^{\prime}},E)\Phi_{\mathbf{k}_{\parallel},\lambda}(\mathbf{x^{\prime}}).$
(79)

In order to compute the $T$-matrix elements we need the phase shifts, whose
analytical expression is presented in Eq.(32) of the supplemental material of
our previous work Ref.Bonilla  _et al._ (2021). Here we reproduce the final
result

$\tan\delta_{m}(k)=\frac{\beta J_{m+1}(ka)-\varrho_{n}^{\xi}J_{m}(ka)\cdot
z_{a}^{\frac{|m+1|-|m|}{2}}\frac{L_{n^{\prime}_{\rho}}^{|m+1|}\left(z_{a}\right)}{L_{n_{\rho}}^{|m|}\left(z_{a}\right)}+\tan\alpha\left[J_{m}(ka)+\beta\varrho_{n}^{\xi}J_{m+1}(ka)\cdot
z_{a}^{\frac{|m+1|-|m|}{2}}\frac{L_{n^{\prime}_{\rho}}^{|m+1|}\left(z_{a}\right)}{L_{n_{\rho}}^{|m|}\left(z_{a}\right)}\right]}{\beta
Y_{m+1}(ka)-\varrho_{n}^{\xi}Y_{m}(ka)\cdot
z_{a}^{\frac{|m+1|-|m|}{2}}\frac{L_{n^{\prime}_{\rho}}^{|m+1|}\left(z_{a}\right)}{L_{n_{\rho}}^{|m|}\left(z_{a}\right)}+\tan\alpha\left[Y_{m}(ka)+\beta\varrho_{n}^{\xi}Y_{m+1}(ka)\cdot
z_{a}^{\frac{|m+1|-|m|}{2}}\frac{L_{n^{\prime}_{\rho}}^{|m+1|}\left(z_{a}\right)}{L_{n_{\rho}}^{|m|}\left(z_{a}\right)}\right]},$
(80)

where $z_{a}=|B_{\xi}|a^{2}/2\tilde{\phi}_{0}$ ($a$ is the cylinder’s radius).

##  Appendix C The spectral function

The spectral function can be defined as follows

$\displaystyle\hat{\mathscr{A}}^{\xi}(E)$ $\displaystyle=$ $\displaystyle
i\left[\hat{G}^{\xi}_{R}(E)-\hat{G}^{\xi}_{A}(E)\right],$ (81)

in terms of the complete retarded and advanced Green’s functions. Then, the
spectral function is Hermitian
$\left[\hat{\mathscr{A}}^{\xi}(E)\right]^{\dagger}=\hat{\mathscr{A}}^{\xi}(E)$.
Given the averaged complete retarded Green’s function in Eq.(26), the form of
the spectral function in momentum space is

$\displaystyle\mathscr{A}^{\lambda,\xi}(\mathbf{k}_{\parallel})=i\left[\left\langle
G_{R}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle-\left\langle
G_{A}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle\right]$
$\displaystyle=i\left[\frac{1}{E-\lambda\xi\hbar
v_{F}k-\Sigma_{R}^{\lambda,\xi}(\mathbf{k}_{\parallel})}-\frac{1}{E-\lambda\xi\hbar
v_{F}k-\Sigma_{A}^{\lambda,\xi}(\mathbf{k}_{\parallel})}\right].$ (82)

Clearly, it takes the form of a Lorentzian distribution with compact support
around the free particle’s energy

$\mathscr{A}^{\lambda,\xi}(k)=\frac{2\left(\frac{\hbar}{2\tau^{(\lambda,\xi)}(k)}\right)}{\left(E-\mathscr{E}^{\lambda,\xi}_{k}\right)^{2}+\left(\frac{\hbar}{2\tau^{(\lambda,\xi)}(k)}\right)^{2}},$
(83)

where $\tau^{(\lambda,\xi)}(k)$ is the relaxation time and
$\mathscr{E}^{\lambda,\xi}_{k}=\lambda\xi\hbar v_{F}k$. In the limit of low
concentration of defects, i.e., large relaxation time because of Eq.(31), the
spectral function becomes a delta distribution

$\lim_{\tau\rightarrow\infty}\mathscr{A}^{\lambda,\xi}(k)=2\pi\delta\left(E-\mathscr{E}^{\lambda,\xi}_{k}\right).$
(84)

Due to its behavior as a Lorentzian, the spectral function has the important
property Mahan (2000)

$\mathscr{A}^{\lambda,\xi}(k)\mathscr{A}^{\lambda^{\prime},\xi}(k^{\prime})=\mathscr{A}^{\lambda,\xi}(k)\mathscr{A}^{\lambda,\xi}(k)\delta_{\lambda\lambda^{\prime}}.$
(85)

Representing the spectral function Eq.(81) in the coordinate basis using the
complete set of eigenstates of the full Hamiltonian we have

$\bm{\mathscr{A}}^{\xi}(\mathbf{r},\mathbf{r^{\prime}})=\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\mathbf{k}\cdot(\mathbf{r}-\mathbf{r^{\prime}})}\sum_{\lambda}\left(\sigma_{0}+\lambda\frac{\bm{\sigma\cdot}\mathbf{k}_{\parallel}}{|\mathbf{k}_{\parallel}|}\right)\mathscr{A}^{\lambda,\xi}(k).$
(86)

Notice that because $\mathbf{k}_{\parallel}=(k_{x},k_{y})$, when we perform
the integration the spectral function takes the form
$\bm{\mathscr{A}}^{\xi}(\mathbf{r},\mathbf{r^{\prime}})=\delta(z-z^{\prime})\bm{\mathscr{A}}^{\xi}(\mathbf{x},\mathbf{x^{\prime}})$,
which looks similar to the decoupled form of the GF in Eq.(68).

##  Appendix D Linear Response Theory

The tensor $K_{\alpha\beta}$ is defined using the retarded current-current
correlator in Eq.(35). Introducing in Eq.(35) the complete and orthonormal
basis $\left\\{\left|\Psi_{\lambda,\mathbf{k}}\right\rangle\right\\}$ of the
total Hamiltonian, such that
$\hat{H}^{\xi}\left|\Psi_{\lambda,\mathbf{k}}\right\rangle=\mathscr{E}^{\lambda,\xi}_{\mathbf{k}}\left|\Psi_{\lambda,\mathbf{k}}\right\rangle$
and
$\hat{\rho}\left|\Psi_{\lambda,\mathbf{k}}\right\rangle=\rho\left(\mathscr{E}^{\lambda,\xi}_{\mathbf{k}}\right)\left|\Psi_{\lambda,\mathbf{k}}\right\rangle$,
we obtain

$\displaystyle
K^{\xi}_{\alpha\beta}(\mathbf{r},t;\mathbf{r^{\prime}},t^{\prime})=i\hbar^{-1}\theta(t-t^{\prime})$
$\displaystyle\quad\times\int\frac{d^{3}k}{(2\pi)^{3}}\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\sum_{\lambda,\lambda^{\prime}}\left[\rho\left(\mathscr{E}^{\lambda,\xi}_{\mathbf{k}}\right)-\rho\left(\mathscr{E}^{\lambda^{\prime},\xi}_{\mathbf{k^{\prime}}}\right)\right]
$
$\displaystyle\quad\times\left\langle\Psi_{\lambda,\mathbf{k}}\middle|\hat{j}_{\alpha}(\mathbf{r})\middle|\Psi_{\lambda^{\prime},\mathbf{k^{\prime}}}\right\rangle\left\langle\Psi_{\lambda^{\prime},\mathbf{k^{\prime}}}\middle|\hat{j}_{\beta}(\mathbf{r^{\prime}})\middle|\Psi_{\lambda,\mathbf{k}}\right\rangle
e^{\frac{i}{\hbar}\left(\mathscr{E}^{\lambda,\xi}_{\mathbf{k}}-\mathscr{E}^{\lambda^{\prime},\xi}_{\mathbf{k^{\prime}}}\right)(t-t^{\prime})}.$
(87)

Using the Fourier representation of the Heaviside step function we obtain the
correlator in the frequency domain

$\displaystyle K^{\xi}_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega)$
$\displaystyle=\int\frac{d^{3}k}{(2\pi)^{3}}\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\sum_{\lambda,\lambda^{\prime}}\frac{\left[\rho\left(\mathscr{E}^{\lambda^{\prime},\xi}_{\mathbf{k^{\prime}}}\right)-\rho\left(\mathscr{E}^{\lambda,\xi}_{\mathbf{k}}\right)\right]}{\hbar\omega+\mathscr{E}^{\lambda,\xi}_{\mathbf{k}}-\mathscr{E}^{\lambda^{\prime},\xi}_{\mathbf{k^{\prime}}}+i\eta^{+}}$
$\displaystyle\times\left\langle\Psi_{\lambda,\mathbf{k}}\middle|\hat{j}_{\alpha}(\mathbf{r})\middle|\Psi_{\lambda^{\prime},\mathbf{k^{\prime}}}\right\rangle\left\langle\Psi_{\lambda^{\prime},\mathbf{k^{\prime}}}\middle|\hat{j}_{\beta}(\mathbf{r^{\prime}})\middle|\Psi_{\lambda,\mathbf{k}}\right\rangle.$
(88)

The electric current density operator for the Weyl equation is
$\mathbf{\hat{j}}^{\xi}(\mathbf{r})=-e\xi
v_{F}\left|\mathbf{r}\right\rangle\bm{\sigma}\left\langle\mathbf{r}\right|$.
Then

$\displaystyle K^{\xi}_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega)$
$\displaystyle\quad=e^{2}v_{F}^{2}\int\frac{d^{3}k}{(2\pi)^{3}}\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\sum_{\lambda,\lambda^{\prime}}\frac{\left[\rho\left(\mathscr{E}^{\lambda^{\prime},\xi}_{\mathbf{k^{\prime}}}\right)-\rho\left(\mathscr{E}^{\lambda,\xi}_{\mathbf{k}}\right)\right]}{\hbar\omega+\mathscr{E}^{\lambda,\xi}_{\mathbf{k}}-\mathscr{E}^{\lambda^{\prime},\xi}_{\mathbf{k^{\prime}}}+i\eta^{+}}$
$\displaystyle\quad\phantom{=}\times\operatorname{Tr}\left[\sigma_{\alpha}\Psi_{\lambda^{\prime},\mathbf{k^{\prime}}}(\mathbf{r})\otimes\Psi_{\lambda^{\prime},\mathbf{k^{\prime}}}^{\dagger}(\mathbf{r^{\prime}})\sigma_{\beta}\Psi_{\lambda,\mathbf{k}}(\mathbf{r^{\prime}})\otimes\Psi_{\lambda,\mathbf{k}}^{\dagger}(\mathbf{r})\right].$
(89)

We can rewrite this last expression in terms of the spectral function and the
retarded/advanced GFs as follows

$\displaystyle
K^{\xi}_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega)=-2e^{2}v_{F}^{2}\int_{-\infty}^{\infty}\frac{dE}{2\pi}f_{0}(E)$
$\displaystyle\quad\times\operatorname{Tr}\left[\sigma_{\alpha}\bm{\mathscr{A}}^{\xi}(\mathbf{r},\mathbf{r^{\prime}};E)\sigma_{\beta}\mathbf{G}^{\xi}_{A}(\mathbf{r^{\prime}},\mathbf{r};E-\hbar\omega)\right.$
$\displaystyle\quad\quad\quad\quad\left.+\sigma_{\alpha}\mathbf{G}^{\xi}_{R}(\mathbf{r},\mathbf{r^{\prime}};E+\hbar\omega)\sigma_{\beta}\bm{\mathscr{A}}^{\xi}(\mathbf{r^{\prime}},\mathbf{r};E)\right],$
(90)

where the additional factor of 2 is due to the spin degeneracy and $f_{0}(E)$
is the Fermi distribution. In the first term, we can shift energy variable
$E\rightarrow E+\hbar\omega$, such that

$\displaystyle
K^{\xi}_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega)=-2e^{2}v_{F}^{2}\int_{-\infty}^{\infty}\frac{dE}{2\pi}$
$\displaystyle\quad\times\left[f_{0}(E+\hbar\omega)\operatorname{Tr}\sigma_{\alpha}\bm{\mathscr{A}}^{\xi}(\mathbf{r},\mathbf{r^{\prime}};E+\hbar\omega)\sigma_{\beta}\mathbf{G}^{\xi}_{A}(\mathbf{r^{\prime}},\mathbf{r};E)\right.$
$\displaystyle\quad\quad\quad\left.+f_{0}(E)\operatorname{Tr}\sigma_{\alpha}\mathbf{G}^{\xi}_{R}(\mathbf{r},\mathbf{r^{\prime}};E+\hbar\omega)\sigma_{\beta}\bm{\mathscr{A}}^{\xi}(\mathbf{r^{\prime}},\mathbf{r};E)\right].$
(91)

We are interested in the real part of the conductivity tensor. Then

$\displaystyle\text{Re}\,\sigma^{\xi}_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega)=\text{Re}\,\left(\frac{1}{i\omega}K^{\xi}_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega)\right)$
$\displaystyle\quad=-\frac{i}{2\omega}\left[K^{\xi}_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega)-K^{\xi\dagger}_{\beta\alpha}(\mathbf{r},\mathbf{r^{\prime}};\omega)\right],$
(92)

where we have used that $K^{*}_{\alpha\beta}=K^{\dagger}_{\beta\alpha}$. Then,
taking the Hermitian conjugate of Eq.(91) and using the cyclic property of the
trace we can write

$\displaystyle\text{Re}\,\sigma^{\xi}_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega)$
$\displaystyle=-\frac{e^{2}\hbar
v_{F}^{2}}{2\pi}\int_{-\infty}^{\infty}dE\left\\{\frac{f_{0}(E+\hbar\omega)-f_{0}(E)}{\hbar\omega}\right\\}$
$\displaystyle\quad\times\operatorname{Tr}\sigma_{\alpha}\bm{\mathscr{A}}^{\xi}(\mathbf{r},\mathbf{r^{\prime}};E+\hbar\omega)\sigma_{\beta}\bm{\mathscr{A}}^{\xi}(\mathbf{r^{\prime}},\mathbf{r};E).$
(93)

Introducing the spectral density in Eq.(86) in the DC conductivity expression
in Eq.(93) we have

$\displaystyle\text{Re}\,\sigma^{\xi}_{\alpha\beta}(\mathbf{r},\mathbf{r^{\prime}};\omega)=\int\frac{d^{3}q}{(2\pi)^{3}}e^{i\mathbf{q}\cdot(\mathbf{r}-\mathbf{r^{\prime}})}\left[-\frac{e^{2}\hbar
v_{F}^{2}}{2\pi}\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\right.$
$\displaystyle\quad\times\int_{-\infty}^{\infty}dE\,\left\\{\frac{f_{0}(E+\hbar\omega)-f_{0}(E)}{\hbar\omega}\right\\}$
$\displaystyle\quad\times\sum_{\lambda,\lambda^{\prime}}\operatorname{Tr}\left\\{\sigma_{\alpha}\left(\sigma_{0}+\lambda\frac{\bm{\sigma\cdot}(\mathbf{k}_{\parallel}+\mathbf{q})}{|\mathbf{k}_{\parallel}+\mathbf{q}|}\right)\sigma_{\beta}\left(\sigma_{0}+\lambda^{\prime}\frac{\bm{\sigma\cdot}\mathbf{k}_{\parallel}}{|\mathbf{k}_{\parallel}|}\right)\right\\}$
$\displaystyle\quad\left.\phantom{-\frac{e^{2}\hbar
v_{F}^{2}}{2\pi}}\times\mathscr{A}^{\lambda,\xi}(|\mathbf{k}_{\parallel}+\mathbf{q}|;E+\hbar\omega)\mathscr{A}^{\lambda^{\prime},\xi}(|\mathbf{k}_{\parallel}|;E)\right].$
(94)

Then, the Fourier transform to the momentum space of the conductivity is the
result in Eq.(39). We are computing the DC conductivity, so we take the limit
$\mathbf{q}\rightarrow\mathbf{0}$ first and then the limit $\omega\rightarrow
0$. The result is

$\displaystyle\sigma^{\xi}_{\alpha\beta}(T)=-\frac{e^{2}\hbar
v_{F}^{2}}{2\pi}\int\frac{d^{3}k}{(2\pi)^{3}}\int_{-\infty}^{\infty}dE\frac{\partial
f_{0}(E)}{E}$
$\displaystyle\quad\times\sum_{\lambda}\operatorname{Tr}\left\\{\sigma_{\alpha}\left(\sigma_{0}+\lambda\frac{\bm{\sigma\cdot}\mathbf{k}_{\parallel}}{|\mathbf{k}_{\parallel}|}\right)\sigma_{\beta}\left(\sigma_{0}+\lambda\frac{\bm{\sigma\cdot}\mathbf{k}_{\parallel}}{|\mathbf{k}_{\parallel}|}\right)\right\\}$
$\displaystyle\quad\times\mathscr{A}^{\lambda,\xi}(|\mathbf{k}_{\parallel}|;E)\mathscr{A}^{\lambda,\xi}(|\mathbf{k}_{\parallel}|;E).$
(95)

where we have used Eq.(85). Now, we perform the trace

$\displaystyle\operatorname{Tr}\left\\{\sigma_{\alpha}\left(\sigma_{0}+\lambda\frac{\bm{\sigma\cdot}\mathbf{k}_{\parallel}}{|\mathbf{k}_{\parallel}|}\right)\sigma_{\beta}\left(\sigma_{0}+\lambda\frac{\bm{\sigma\cdot}\mathbf{k}_{\parallel}}{|\mathbf{k}_{\parallel}|}\right)\right\\}$
$\displaystyle\quad=\operatorname{Tr}\left\\{\sigma_{\alpha}\sigma_{\beta}+\lambda\left(\sigma_{\alpha}\sigma_{\beta}\sigma_{\gamma}+\sigma_{\alpha}\sigma_{\gamma}\sigma_{\beta}\right)\frac{k_{\gamma}}{|\mathbf{k}_{\parallel}|}\right.$
$\displaystyle\quad\quad\quad\quad\left.+\lambda^{2}\sigma_{\alpha}\sigma_{\gamma}\sigma_{\beta}\sigma_{\gamma^{\prime}}\frac{k_{\gamma}k_{\gamma^{\prime}}}{|\mathbf{k}_{\parallel}|^{2}}\right\\}$
$\displaystyle\quad=4\frac{k_{\alpha}k_{\beta}}{|\mathbf{k}_{\parallel}|^{2}},$
(96)

where we have used the Pauli matrices trace technology and that
$\lambda^{2}=1$. The angular integration is performed immediately as follows

$\int
d\Omega\,\,k_{\alpha}k_{\beta}=k^{2}\int_{0}^{\pi}d\theta\sin\theta\int_{0}^{2\pi}d\phi\,\,n_{\alpha}n_{\beta}=2\pi
k^{2}\delta_{\alpha\beta},$ (97)

where $n_{\alpha}$ is the component of the unit vector $\mathbf{n}$ along the
$\alpha$-direction (on the plane) and $\mathbf{k}_{\parallel}=k\mathbf{n}$.
Then, we have

$\displaystyle\sigma^{\xi}_{\alpha\beta}(T)$
$\displaystyle=-\delta_{\alpha\beta}\frac{4e^{2}\hbar
v_{F}^{2}}{(2\pi)^{3}k_{B}T}\sum_{\lambda}\int_{-\infty}^{\infty}dE\,\,f_{0}(E)\left[1-f_{0}(E)\right]$
$\displaystyle\quad\times\int_{0}^{\infty}dk\left[\left\langle
G_{R}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle-\left\langle
G_{A}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle\right]$
$\displaystyle\quad\times\left[\left\langle
G_{R}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle-\left\langle
G_{A}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle\right]\mathbf{k}_{\parallel}\cdot\mathbf{k}_{\parallel},$
(98)

where we have used the definition of the spectral function in terms of the
retarded and advanced GFs. In the limit of low concentration of defects, i.e.,
$n_{d}\rightarrow 0$, we have that the unique leading contribution to the
conductivity are given by the combination

$\displaystyle\left\langle
G_{R}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle\left\langle
G_{A}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle=\frac{1}{\left(E-\lambda\xi\hbar
v_{F}k\right)^{2}+\left(\frac{\hbar}{2\tau^{(\lambda,\xi)}(k)}\right)^{2}}$
$\displaystyle\quad\rightarrow\frac{2\pi\tau^{(\lambda,\xi)}(k)}{\hbar}\delta(E-\lambda\xi\hbar
v_{F}k),$ (99)

where we have used that the self-energy is purely imaginary and its relation
with the relaxation time. The other contributions are negligible because they
are not singular in $n_{d}$. For instance, in the low concentration limit, we
have a contribution for two retarded GFs of the form

$\displaystyle\left\langle
G_{R}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle\left\langle
G_{R}^{\lambda,\xi}(\mathbf{k}_{\parallel})\right\rangle$
$\displaystyle=\frac{\left(E-\lambda\xi\hbar
v_{F}k\right)^{2}-\left(\frac{\hbar}{2\tau^{(\lambda,\xi)}(k)}\right)^{2}}{\left\\{\left(E-\lambda\xi\hbar
v_{F}k\right)^{2}+\left(\frac{\hbar}{2\tau^{(\lambda,\xi)}(k)}\right)^{2}\right\\}^{2}}$
$\displaystyle+i\left(E-\lambda\xi\hbar
v_{F}k\right)2\pi\delta(E-\lambda\xi\hbar v_{F}k).$ (100)

The second term is zero and the first term is a function centered at
$E-\lambda\xi\hbar v_{F}k$, but when integrated over $k$ it gives zero.
Something similar occurs with the contribution of two advanced GFs. The result
is the diagonal conductivity tensor given in Eq.(40).

_Acknowledgments:_ This research was funded by Fondecyt grant number 1190361
and by ANID PIA Anillo ACT/192023.

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