# Planar Hall effect in Weyl semimetals induced by pseudoelectromagnetic
fields

L. Medel Onofre<EMAIL_ADDRESS>Instituto de Ciencias
Nucleares, Universidad Nacional Autónoma de México, 04510 Ciudad de México,
México  A. Martín-Ruiz<EMAIL_ADDRESS>Instituto de Ciencias
Nucleares, Universidad Nacional Autónoma de México, 04510 Ciudad de México,
México

###### Abstract

The planar Hall effect (PHE), the appearance of an in-plane transverse voltage
in the presence of coplanar electric and magnetic fields, has been ascribed to
the chiral anomaly and Berry curvature effects in Weyl semimetals. In the
presence of position- and time-dependent perturbations, such as strain, Weyl
semimetals react as if they would be subjected to emergent electromagnetic
fields, kwnon as pseudo-fields. In this paper we investigate the possibility
of inducing nonlinear phenomena, including the PHE, in strained Weyl
semimetals. Using the chiral kinetic theory in the presence of pseudo-fields,
we derive general expressions for the magnetoconductivity tensor by
considering the simultaneous effects of the Berry curvature and orbital
magnetic moment of carriers, which are indeed of the same order of magnitude.
Since pseudo-fields couple to the Weyl fermions of opposite chirality with
opposite signs, we study chirality-dependent phenomena, including the
longitudinal magnetoconductivity and the planar Hall effect. We discuss our
results in terms of the chiral anomaly with pseudo-fields. These may open new
possibilities in chiralitytronics.

##  I Introduction

Weyl semimetals (WSMs) are topologically nontrivial conductors in which the
non-degenerate valence and conduction bands touch at isolated points (the so
called Weyl nodes) in the Brillouin zone Armitage  _et al._ (2018). Near these
touching-points, the electron spectrum can be described by the Weyl equation,
originally introduced in the particle physics context. The Weyl nodes occur in
pairs of opposite chirality which act as a source and sink of Berry curvature
in reciprocal space, and the WSM phase is topologically protected by a nonzero
Berry flux across the Fermi surface.

A distinguishing transport property of WSMs with broken time-reversal symmetry
is the anomalous Hall effect, which arises when the conduction (valence) band
is completely empty (filled) Zyuzin and Burkov (2012). Another intriguing
property of WSMs is the chiral anomaly, i.e., the nonconservation of the
chiral current in the presence of parallel electric and magnetic fields:
$\partial_{\mu}J^{\mu}_{5}=\frac{e^{2}}{2\pi^{2}\hbar^{2}}\bm{E}\cdot\bm{B}$.
The appearance of a positive longitudinal magnetoconductance has been regarded
as a manifestation of the chiral anomaly Son and Spivak (2013); Burkov (2014).
However, the magnetoconductivity tensor receives additional contributions, not
related with the chiral anomaly, which indeed reverses the overall sign of the
magnetoconductance.

The planar Hall effect (PHE), the appearance of an in-plane transverse voltage
in the presence of coplanar electric and magnetic fields, has been ascribed to
the chiral anomaly in WSMs as well as to Berry curvature effects Nandy  _et
al._ (2017, 2019); Ghosh  _et al._ (2020); Ma  _et al._ (2019); Sharma  _et
al._ (2020); Nag and Nandy (2020); Yadav  _et al._ (2022). However, as in the
case of the longitudinal magnetoconductance, within the semiclassical
Boltzmann transport theory, the PHE is not described solely by the Berry
curvature. In fact, as we show in this paper, the orbital magnetic moment
(OMM) of charge carriers contributes also to the PHE, in much the same order
of magnitude than the Berry curvature contribution, and therefore it cannot be
disregarded. In a similar footing, the electrochemical transport in WSMs has
been associated with the chiral anomaly, with the orbital magnetic moment
playing a fundamental role, since the statistical transport is sensitive to
the spatial gradients of the distribution function Flores-Calderón and Martín-
Ruiz (2021).

In conventional transport experiments, Weyl quasiparticles are coupled to the
electromagnetic fields $\bm{E}$ and $\bm{B}$, which cannot be used as probe
for chirality ($\chi=\pm 1$) since they do not differentiate the nodes.
However, an interesting phenomena arising in Dirac matter is the fact that
elastic deformations of the lattice couple to the electronic Hamiltonian as
pseudo-electromagnetic gauge potentials
$\tilde{\bm{A}}_{\chi}^{\mbox{\scriptsize el}}$ and
$\tilde{\Phi}_{\chi}^{\mbox{\scriptsize el}}$, which define pseudo-
electromagnetic fields as usual, $\bm{E}_{\chi}^{\mbox{\scriptsize
el}}=-\nabla\Phi_{\chi}^{\mbox{\scriptsize
el}}-\partial_{t}\bm{A}_{\chi}^{\mbox{\scriptsize el}}$ and
$\bm{B}_{\chi}^{\mbox{\scriptsize
el}}=\nabla\times\bm{A}_{\chi}^{\mbox{\scriptsize el}}$, known as pseudo-
fields or elastic-fields Ilan  _et al._ (2020). These pseudo-fields can be
expressed as the sum of two terms: $\bm{E}_{\chi}^{\mbox{\scriptsize
el}}=\bm{\mathcal{E}}+\chi\bm{E}_{5}$ and $\bm{B}_{\chi}^{\mbox{\scriptsize
el}}=\bm{\mathcal{B}}+\chi\bm{B}_{5}$. While the $\bm{\mathcal{E}}$ and
$\bm{\mathcal{B}}$ couple to the Weyl nodes in a similar fashion as
electromagnetic fields do (with the same sign), $\bm{E}_{5}$ and $\bm{B}_{5}$
couple to the nodes in an axial fasion (i.e. they couple opposite chiral
fermions with opposite signs). The notation for $\bm{E}_{5}$ and $\bm{B}_{5}$
in inherited from the high-energy physics literature, where axial fields
couple to the Dirac matrix $\gamma_{5}$. In a strained material, the hoping
parameters between atomic orbitals and on-site energies are both changed, and
the modifications are driven by the components of the strain-tensor $u_{ij}$.
Therefore, the axial fields become determined by the position- and/or time-
dependence of the the strain tensor. For example, in the case of strained
graphene, the induced pseudo-gauge fields couple to the Dirac fermions
oppositely in the two valleys $\bm{K}$ and $\bm{K}^{\prime}\,$ Guinea  _et
al._ (2010a). This gave rise to a new line of research called straintronics
Amorim  _et al._ (2016); Suzuura and Ando (2002). More recently, the study of
strain-induced gauge fields in Weyl semimetals has attracted great attention
Cortijo  _et al._ (2015, 2015), since it could opens a pathway for a prolific
industry associated with straintronics and chiraltronics.

In the case of WSMs, mechanical strain shifts the position of the Weyl nodes
in momentum and/or energy, which can be effectively described in terms of
pseudo-electromagnetic fields Cortijo  _et al._ (2015, 2015). This affects the
low-energy description of Weyl fermions, which now have to include the pseudo-
gauge potentials Liu  _et al._ (2013); Chernodub  _et al._ (2014); Pikulin
_et al._ (2016). Interestingly, axial pseudo-fields $\bm{E}_{5}$ and
$\bm{B}_{5}$ couple opposite chiral fermions with opposite signs, and hence
the chirality can be tested by using conventional experimental probes such as
electrical transport. It is worth to mention that whereas electromagnetic
potentials are gauge dependent and hence are not observables, the axial gauge
potentials are quantum expectation values and thus produce gauge-invariant and
observable effects. Indeed, an experimental realization of strain-induced
pseudo-magnetic fields was recently observed in strained crystals of Re-doped
MoTe2 Kamboj  _et al._ (2019). In this paper we aim to explore also nonlinear
transport phenomena induced by pseudo-fields, in particular the longitudinal
magnetoconductivity and the planar Hall effect. We clearly differentiate the
contributions arising from the Berry curvature from those arising from the OMM
of charge carriers.

The coupling of Weyl fermions with pseudo-gauge fields not only produces new
interesting transport phenomena in Weyl semimetals, but also affects the well-
known chiral anomaly. The inclusion of pseudo-fields in the semiclassical
derivation of the chiral anomaly produces an interesting generalization of the
anomaly equation, known in high-energy physics as the covariant anomaly Son
and Spivak (2013). In the presence of genuine and pseudo-electromagnetic
fields, the covariant anomaly equations reads

$\displaystyle\partial_{\mu}J^{\mu}_{5}$
$\displaystyle=\frac{e^{2}}{2\pi^{2}\hbar^{2}}(\bm{E}\cdot\bm{B}+\bm{E}_{5}\cdot\bm{B}_{5}),$
(1) $\displaystyle\partial_{\mu}J^{\mu}$
$\displaystyle=\frac{e^{2}}{2\pi^{2}\hbar^{2}}(\bm{E}\cdot\bm{B}_{5}+\bm{E}_{5}\cdot\bm{B}),$
(2)

where the axial current $J_{5}^{\mu}$ measures the difference between currents
of opposite chiralities and $J^{\mu}$ is the total current. The breaking of
charge conservation, as indicated by Eq. (2), implies that additional currents
mus exist in the system to restore local charge conservation. In fact, in the
case of WSMs, this problem is cured by including the conventional anomalous
Hall current. Nonlinear transport phenomena considered in this work also
provides a testing ground for the chiral anomaly induced transport. In fact,
we interpret our findings in terms of the chiral anomaly with pseudo-fields.

This paper is organized as follows. In Sec. II we use chiral kinetic theory to
investigate the planar Hall effect in Weyl semimetals in the presence of
pseudo-fields. We obtain a general expression for the magnetoconductivity
tensor, separating in a clear fashion the contributions arising from the Berry
curvature and the orbital magnetic moment. In Sec. III we evaluate such
contributions for a simple linearly dispersing model of a WSM. We discuss the
total and axial conductivities and interpret them in terms of the chiral
anomaly with pseudo-fields. Section IV is devoted to applications of our
results to strained Weyl semimetals. We conclude in Sec. V. All technical
calculations are relegated to the Appendices.

##  II Kinetic theory approach

We will now investigate the PHE in Weyl semimetals by using the chiral kinetic
theory, which is a topologically modified semiclassical Boltzmann formalism to
describe the behavior of Weyl fermions for a finite chemical potential. Within
this approach, the semiclassical equations of motion are extended to include
an anomalous velocity term arising from the Berry curvature, which acts as a
magnetic field in reciprocal space Xiao  _et al._ (2010). In the presence of
electromagnetic fields (${\bm{E}}$ and ${\bm{B}}$) and axial pseudo-fields
(${\bm{E}}_{5}$ and ${\bm{B}}_{5}$), the semiclassical equations of motion for
an electron wavepacket in a metal can be cast in the standard form Roy  _et
al._ (2018)

$\displaystyle\dot{{\bm{r}}}_{\alpha}$
$\displaystyle=\frac{1}{\hbar}\nabla_{{\bm{k}}}\mathcal{E}_{\alpha}({\bm{k}})-\dot{{\bm{k}}}_{\alpha}\times{\bm{\Omega}}_{\alpha}({\bm{k}}),$
(3) $\displaystyle\hbar\dot{{\bm{k}}}_{\alpha}$
$\displaystyle=-e{\bm{E}}_{\chi}-e\dot{{\bm{r}}}_{\alpha}\times{\bm{B}}_{\chi},$
(4)

where ${\bm{E}}_{\chi}={\bm{E}}+\chi{\bm{E}}_{5}$ and
${\bm{B}}_{\chi}={\bm{B}}+\chi{\bm{B}}_{5}$ are effective fields. Elastic
gauge fields $\bm{\mathcal{E}}$ and $\bm{\mathcal{B}}$, which couple to the
Weyl fermions in a similar manner than genuine electromagnetic fields, are
accounted by promoting ${\bm{E}}\to{\bm{E}}+\bm{\mathcal{E}}$ and
${\bm{B}}\to{\bm{B}}+\bm{\mathcal{B}}$. The presence of $\chi$ in the
definitions of the effective fields accounts for the fact that pseudo-fields
couple opposite chiral fermions with opposite signs. Here,
${\bm{\Omega}}_{\alpha}({\bm{k}})=i\bra{\nabla_{{\bm{k}}}u_{\alpha}({\bm{k}})}\times\ket{\nabla_{{\bm{k}}}u_{\alpha}({\bm{k}})}$
is the Berry curvature and
$\mathcal{E}_{\alpha}({\bm{k}})=\mathcal{E}_{\alpha}^{(0)}-{\bm{m}}_{\alpha}\cdot{\bm{B}}_{\chi}$
is the energy dispersion which includes a Zeeman-like correction due to the
orbital magnetic moment
${\bm{m}}_{\alpha}({\bm{k}})=-i\frac{e}{2\hbar}\bra{\nabla_{{\bm{k}}}u_{\alpha}({\bm{k}})}\times[\hat{H}({\bm{k}})-\mathcal{E}_{\alpha}^{(0)}({\bm{k}})]\ket{\nabla_{{\bm{k}}}u_{\alpha}({\bm{k}})}$
Chang and Niu (1996); Sundaram and Niu (1999). Here, the Bloch states
$\ket{u_{\alpha}({\bm{k}})}$ are defined by
$\hat{H}({\bm{k}})\ket{u_{\alpha}({\bm{k}})}=\mathcal{E}_{\alpha}^{(0)}({\bm{k}})\ket{u_{\alpha}({\bm{k}})}$
with $B_{\chi}=0$. The subindex $\alpha$ stands collectively for the band
index $s$ and the chirality index $\chi$.

As they are, the equations of motion (3) and (4) are reminiscent of the
standard semiclassical equations by replacing the electromagnetic fields
${\bm{E}}$ and ${\bm{B}}$ by the pseudo-fields ${\bm{E}}_{\chi}$ and
${\bm{B}}_{\chi}$ solely; however, they do not immediately follow the
semiclassical theory. The wave-packet dynamics of electrons in crystals
subject to perturbations varying slowly in space and time yields to
generalized equations of motion that contain corrections which are accounted
by a generalized Berry curvature defined in terms of two derivatives of the
Bloch functions with respect to momentum (as the one defined above), position
and time Chang and Niu (1996); Sundaram and Niu (1999). From the generalized
equations of motion and assuming that the moment of the wavepacket is close to
a Weyl node, a change of coordinate frame produces the equation of motion (3)
and (4) Roy  _et al._ (2018).

In the presence of impurity scattering the phenomenological transport equation
can be written as Ashcroft and Mermin (1976)

$\displaystyle\left(\frac{\partial}{\partial
t}+\dot{{\bm{r}}}_{\alpha}\cdot\nabla_{{\bm{r}}}+\dot{{\bm{k}}}_{\alpha}\cdot\nabla_{{\bm{k}}}\right)f_{\alpha}({\bm{r}},{\bm{k}},t)=I_{\mbox{\scriptsize
coll }}[f_{\alpha}({\bm{r}},{\bm{k}},t)],$ (5)

where $f_{\alpha}({\bm{r}},{\bm{k}},t)$ is the electron distribution function.
The collision integral $I_{\text{coll}}$ accounts for the scattering
mechanisms of the conduction electrons (such as impurity scattering effects,
electron correlations, or scattering effects due to thermal vibrations of
lattice ions). In the relaxation time approximation, the collision integral
takes the simple form
$I_{\mbox{\scriptsize{coll}}}[f_{\alpha}]=-\frac{f_{\alpha}-f_{\alpha}^{\mbox{\tiny
eq}}}{\tau({\bm{k}})}$, where $\tau({\bm{k}})$ is the scattering time of
quasiparticles and $f_{\alpha}^{\mbox{\scriptsize eq}}$ is the equilibrium
Fermi-Dirac distribution to be evaluated at the modified dispersion
$\mathcal{E}_{\alpha}({\bm{k}})$. Although the momentum dependence of $\tau$
covers a wide range of scattering processes, taking it as a constant parameter
is still a good approximation that unveils interesting physics. In the
following we take it as constant.

Here we are interested in stationary and homogeneous solutions to the
Boltzmann equation (5). Using the equations of motion (3) and (4) we have

$\displaystyle-\tau\,\dot{{\bm{k}}}_{\alpha}\cdot\nabla_{{\bm{k}}}f_{\alpha}({\bm{k}})=f_{\alpha}({\bm{k}})-f_{\alpha}^{\mbox{\scriptsize
eq}}({\bm{k}}).$ (6)

Next, we expand the distribution function in powers of the electromagnetic
fields. Keeping only linear order dependence on the electric field, the
nonequilibrium distribution function becomes

$\displaystyle f_{\alpha}=f_{\alpha}^{\mbox{\scriptsize eq}}+e\tau
D_{\alpha}\left[{\bm{v}}_{\alpha}\cdot{\bm{E}}_{\chi}+\frac{e}{\hbar}({\bm{E}}_{\chi}\cdot{\bm{B}}_{\chi})({\bm{v}}_{\alpha}\cdot{\bm{\Omega}}_{\alpha})\right]\frac{\partial
f_{\alpha}^{\mbox{\scriptsize eq}}}{\partial\mathcal{E}_{\alpha}},$ (7)

where
${\bm{v}}_{\alpha}({\bm{k}})=\tfrac{1}{\hbar}\nabla_{{\bm{k}}}\mathcal{E}_{\alpha}({\bm{k}})$
is the band velocity of Bloch electrons and
$D_{\alpha}({\bm{k}})=[1+\tfrac{e}{\hbar}({\bm{B}}_{\chi}\cdot{\bm{\Omega}}_{\alpha})]^{-1}$
is the modification factor of the phase space volume element Xiao  _et al._
(2006); Duval  _et al._ (2006). In these expressions we have omitted all
momentum dependencies for simplicity.

In the absence of any thermal and chemical potential gradients, the charge
density current for a single $\alpha$ (i.e. band, chirality, etc.) can be
written as
${\bm{J}}_{\alpha}=-e\int\frac{d^{3}{\bm{k}}}{(2\pi)^{3}}D_{\alpha}^{-1}\dot{{\bm{r}}}_{\alpha}({\bm{k}})f_{\alpha}({\bm{k}})$,
accounting for the modified density of states due to the phase space factor
$D_{\alpha}$. Substituting the nonequilibrium distribution function (7) into
this equation and keeping the linear term in the electric field, but
neglecting the chiral magnetic and anomalous Hall effect contributions, we now
arrive at the expression for the magnetoconductivity tensor for a single
$\alpha$:

$\displaystyle\sigma^{(\alpha)}_{ij}({\bm{B}}_{\chi})$
$\displaystyle=-e^{2}\tau\int\frac{d^{3}{\bm{k}}}{(2\pi)^{3}}D_{\alpha}\left[v_{\alpha
i}+\frac{e}{\hbar}({\bm{v}}_{\alpha}\cdot{\bm{\Omega}}_{\alpha})B_{\chi
i}\right]$ $\displaystyle\phantom{===}\times\left[v_{\alpha
j}+\frac{e}{\hbar}({\bm{v}}_{\alpha}\cdot{\bm{\Omega}}_{\alpha})B_{\chi
j}\right]\frac{\partial f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha})}{\partial\mathcal{E}_{\alpha}},$ (8)

which includes the effects of the Berry curvature and the orbital magnetic
moment. Our main goal in this paper is to evince that the orbital magnetic
moment contributes in most the same fashion that the Berry curvature to the
PHE. To distinguish these contributions we write the conductivity tensor (8),
in the weak-field limit, as the sum of three terms:

$\displaystyle\sigma^{(\alpha)}_{ij}({\bm{B}}_{\chi})$
$\displaystyle=\sigma^{(0,\alpha)}_{ij}+\sigma^{(\Omega,\alpha)}_{ij}({\bm{B}}_{\chi})+\sigma^{(m,\alpha)}_{ij}({\bm{B}}_{\chi}),$
(9)

where the first term,

$\displaystyle\sigma^{(0,\alpha)}_{ij}=-e^{2}\tau\int\frac{d^{3}{\bm{k}}}{(2\pi)^{3}}v_{\alpha
i}^{(0)}v_{\alpha j}^{(0)}\,\frac{\partial f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)}},$ (10)

is the conductivity in the absence of the magnetic field (i.e. for
$\bm{B}=\bm{B}_{5}={\bf{0}}$) and the second term,

$\displaystyle\sigma^{(\Omega,\alpha)}_{ij}({\bm{B}}_{\chi})=-\frac{e^{4}\tau}{\hbar^{2}}\int\frac{d^{3}{\bm{k}}}{(2\pi)^{3}}Q_{\alpha
i}\,Q_{\alpha j}\,\frac{\partial f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)}},$ (11)

with
${\bm{Q}}_{\alpha}={\bm{\Omega}}_{\alpha}\times(\bm{v}_{\alpha}^{(0)}\times{\bm{B}}_{\chi})$,
is the contribution arising from the Berry curvature solely. In Eqs. (10) and
(11), $\mathcal{E}_{\alpha}^{(0)}({\bm{k}})$ is the band energy without the
Zeeman-like correction and
${\bm{v}}_{\alpha}^{(0)}({\bm{k}})=\tfrac{1}{\hbar}\nabla_{{\bm{k}}}\mathcal{E}_{\alpha}^{(0)}({\bm{k}})$
is the corresponding band velocity. In deriving Eq. (11) we have expanded the
phase space volume factor $D_{\alpha}({\bm{k}})$ up to second order in the
magnetic field.

The contribution from the orbital magnetic moment,
$\sigma^{(m,\alpha)}_{ij}({\bm{B}}_{\chi})$ , appears in various ways. In the
presence of a magnetic field, on the one hand, the energy dispersion is
corrected by
$\mathcal{E}_{\alpha}^{(m)}({\bm{k}})=-{\bm{m}}_{\alpha}({\bm{k}})\cdot{\bm{B}}_{\chi}$
and consequently the band velocity becomes corrected as
${\bm{v}}_{\alpha}^{(m)}({\bm{k}})=\tfrac{1}{\hbar}\nabla_{{\bm{k}}}\mathcal{E}_{\alpha}^{(m)}({\bm{k}})$.
On the other hand, for a weak magnetic field, the equilibrium distribution
function $f_{\alpha}^{\mbox{\scriptsize eq}}(\mathcal{E}_{\alpha})$ and the
phase space volume factor $D_{\alpha}({\bm{k}})$ can be Taylor expanded up to
the second power of the magnetic field (see Appendix A). Taking into account
these terms in Eq. (8) and substracting the contributions
$\sigma^{(0,\alpha)}_{ij}$ and
$\sigma^{(\Omega,\alpha)}_{ij}({\bm{B}}_{\chi})$, one gets

$\displaystyle\sigma_{ij}^{(\chi,m)}({\bm{B}}_{\chi})=\frac{2e^{3}\tau}{\hbar}\\!\int\\!\frac{d^{3}{\bm{k}}}{(2\pi)^{3}}\,\bigg{[}{Q}_{\alpha
i}v_{\alpha
j}^{(m)}+\frac{1}{e}\mathcal{E}_{\alpha}^{(m)}\nabla_{{\bm{k}}}\cdot\bm{T}_{\alpha
ij}+\frac{1}{2}\mathcal{E}_{\alpha}^{(m)}\bm{B}_{\chi}\cdot\bm{V}_{\alpha
ij}\frac{\partial}{\partial\mathcal{E}_{\alpha}^{(0)}}\bigg{]}\frac{\partial
f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)}},$ (12)

where we have defined the tensors

$\displaystyle\bm{T}_{\alpha ij}=\frac{e}{\hbar}\bm{\Omega}_{\alpha}B_{\chi
i}v_{\alpha j}^{(0)}+\frac{1}{2}\hat{\bm{e}}_{i}v_{\alpha
j}^{(m)},\qquad\bm{V}_{\alpha ij}=\bm{\Omega}_{\alpha}v_{\alpha
i}^{(0)}v_{\alpha
j}^{(0)}-\frac{1}{2e}\bm{m}_{\alpha}\partial_{k_{i}}v_{\alpha j}^{(0)}.$ (13)

A detailed derivation of the formula (12) is presented in the Appendix A.

##  III Planar Hall effect with pseudofields in Weyl semimetals

In this section we investigate the PHE in Weyl semimetals by using the
semiclassical approach developed in the previous section. To this end we
consider a simple model of a WSM consisting of two Weyl nodes of opposite
chiralities separated in momentum and energy, ignoring the nonuniversal
corrections due to band bending far away from the nodes. The low energy
Hamiltonian for each Weyl node can be expressed as

$\displaystyle\hat{H}_{\chi}({\bm{k}})=\chi\hbar
v_{F}\bm{\sigma}\cdot{\bm{k}}+b_{0\chi},$ (14)

where $v_{F}$ is the Fermi velocity, $\chi=\pm 1$ specifies the chirality,
$\bm{\sigma}$ is the vector of the Pauli matrices, ${\bm{k}}$ is the momentum
measured relative to the Weyl point and $b_{0\chi}$ denotes the energy shift
of the node with chirality $\chi$.

Figure 1: Low-energy spectrum of a Weyl semimetal with two bulk Weyl nodes of
different chiralities separated in momentum space by $2{\bm{b}}$ and shifted
in energy by $2b_{0}$.

The corresponding energy dispersion is
$\mathcal{E}^{(0)}_{\alpha}({\bm{k}})=b_{0\chi}+s\hbar v_{F}k$, where $s=\pm
1$ is the band index. As a result, the band velocity becomes
${\bm{v}}_{s}^{(0)}({\bm{k}})=sv_{F}\hat{{\bm{k}}}$, where $\hat{{\bm{k}}}$ is
the unit vector along ${\bm{k}}$. Using the Bloch states, it is
straightforward to obtain the Berry curvature and the orbital magnetic moment:

$\displaystyle{\bm{\Omega}}_{\alpha}({\bm{k}})=-s\chi\frac{\hat{{\bm{k}}}}{2k^{2}},\qquad{\bm{m}}_{\chi}({\bm{k}})=-\chi
ev_{F}\frac{\hat{{\bm{k}}}}{2k},$ (15)

respectively. In the presence of an effective magnetic magnetic field
${\bm{B}}_{\chi}$, the energy dispersion is corrected by
$\mathcal{E}_{\chi}^{(m)}({\bm{k}})=\frac{\chi
ev_{F}}{2k}\hat{{\bm{k}}}\cdot{\bm{B}}_{\chi}$, which implies a correction to
the band velocity as

$\displaystyle{\bm{v}}_{\chi}^{(m)}({\bm{k}})=\frac{\chi
ev_{F}}{2\hbar}\frac{{\bm{B}}_{\chi}-2\hat{{\bm{k}}}(\hat{{\bm{k}}}\cdot{\bm{B}}_{\chi})}{k^{2}}.$
(16)

For a given node of chirality $\chi$, the band index is determined by the sign
of the difference between the chemical potential and the energy shift of the
node, i.e. $s=\mbox{sgn}(\mu-b_{0\chi})$. This is so since $\mu>b_{0\chi}$
($\mu<b_{0\chi}$) implies $s=1$ ($s=-1$), as depicted in Fig. 1. So, we define
$\mu_{\chi}=s\mu_{0\chi}$, with $\mu_{0\chi}=|\mu-b_{0\chi}|>0$. Here we work
at finite temperature $T$.

With the above information we are able to apply the semiclassical formulas
(10)-(12) to compute the different contributions to the planar Hall effect in
Weyl semimetals. Details of technical computations are relegated to the
Appendix B and here we present only the final results. Interestingly, all of
them become independent of the band index $s$, so we make the replacement
$\alpha\to\chi$ in the following expressions.

The $\bm{B}_{\chi}$-independent conductivity (10) takes the simple form

$\displaystyle\sigma^{(0,\chi)}_{ij}(T)=\frac{e^{2}\mu_{0\chi}^{2}\tau}{6\pi^{2}\hbar^{3}v_{F}}\delta_{ij}\,f_{2}(\Lambda_{\chi}),$
(17)

with $\Lambda_{\chi}\equiv k_{B}T/\mu_{0\chi}$ and where we have introduced
the function

$\displaystyle{f_{n}(y)\equiv\frac{1}{y}\int_{0}^{\infty}dx\,x^{n}\,\frac{e^{(x-1)/y}}{\left[1+e^{(x-1)/y}\right]^{2}}}.$
(18)

Interestingly, the conductivity (17) vanishes at the neutrality point
($\mu_{0\chi}=0$). Furthermore, at zero temperature one can further verify
that $\lim_{T\to 0}f_{n}(\Lambda_{\chi})=1$.

The Berry curvature contribution (11) becomes

$\displaystyle\sigma^{(\Omega,\chi)}_{ij}(\bm{B}_{\chi},T)$
$\displaystyle=\\!\frac{e^{4}v_{F}^{3}\tau}{120\pi^{2}\hbar\mu_{0\chi}^{2}}\left(\delta_{ij}B_{\chi}^{2}+7B_{\chi
i}B_{\chi j}\right)f_{-2}(\Lambda_{\chi}),$ (19)

while the result from orbital magnetic part is

$\displaystyle\sigma_{ij}^{(m,\chi)}(\bm{B}_{\chi},T)=\\!\frac{-e^{4}v_{F}^{3}\tau}{120\pi^{2}\hbar\mu_{0\chi}^{2}}\left(3\delta_{ij}B_{\chi}^{2}+B_{\chi
i}B_{\chi j}\right)f_{-2}(\Lambda_{\chi}),$ (20)

where $B_{\chi}^{2}=\bm{B}_{\chi}\cdot\bm{B}_{\chi}$. The conductivities of
the opposite chiralities are related by symmetry properties dictated by the
definition of the effective magnetic field $\bm{B}_{\chi}$, namely,
$\sigma^{(\xi,\chi)}_{ij}(\bm{B},\bm{B}_{5},T)=\sigma^{(\xi,-\chi)}_{ij}(-\bm{B},\bm{B}_{5},T)$
and
$\sigma^{(\xi,\chi)}_{ij}(\bm{B},\bm{B}_{5},T)=\sigma^{(\xi,-\chi)}_{ij}(\bm{B},-\bm{B}_{5},T)$,
where $\xi=\Omega,m$. If either $\bm{B}$ or $\bm{B}_{5}$ vanishes, the
contribution from both chiralities are the same. Besides, we observe that the
transverse conductivities does not satisfy the usual antisymmetry relation
($\sigma_{xy}=-\sigma_{yx}$) displayed by Hall effect systems, since in this
case the transverse conductivity does not stem from Lorentz force, but from
the chiral anomaly.

Figure 2: Prototypical experimental setup employed to measure the longitudinal
and planar Hall voltages generated by in-plane electromagnetic fields and
pseudo fields.

The Berry curvature induced conductivity (19) has been discussed recently in a
variety of papers and it has been regarded as a direct consequence of the
chiral anomaly Nandy  _et al._ (2017, 2019); Ghosh  _et al._ (2020); Ma  _et
al._ (2019); Sharma  _et al._ (2020). Our formula (19) generalizes the
previously reported results, where only some components were computed.
Besides, here we report a general formula (20) for the orbital magnetic moment
induced conductivity, which as far as we know, it has not been reported yet.
In order to elucidate the importance of the orbital magnetic moment
contribution, we next explore the angular dependence of both the longitudinal
magnetoconductivity and the planar Hall conductivity for the different
chiralities. To this end, we introduce the normalized conductivity tensor for
the Weyl node with chirality $\chi$:

$\displaystyle\Sigma_{ij}^{(\chi)}(\bm{B}_{\chi},T)=\frac{\sigma^{(\chi)}_{ij}(\bm{B}_{\chi},T)-\sigma_{0}^{(\chi)}(T)\delta_{ij}}{\sigma_{0}^{(\chi)}(T)},$
(21)

where $\sigma^{(\chi)}_{ij}(\bm{B}_{\chi},T)$ is the total conductivity given
by Eq. (9) and
$\sigma_{0}^{(\chi)}(T)\equiv\frac{e^{2}\mu_{0\chi}^{2}\tau}{6\pi^{2}\hbar^{3}v_{F}}\,f_{2}(\Lambda_{\chi})$
is the longitudinal conductivity. Note that $\Sigma_{ij}^{(\chi)}$ isolates
the joint contributions from the Berry curvature and the orbital magnetic
moment of charge carriers. The tensor (21) inherits the symmetries of
$\sigma^{(\Omega,\chi)}_{ij}$ and $\sigma^{(m,\chi)}_{ij}$. Now we assume an
electric field pointing along the $x$-axis, i.e. $\bm{E}=E\hat{\bm{e}}_{x}$,
and restrict the magnetic field to be in the $xy$-plane, i.e.
$\bm{B}=B(\cos\theta\hat{\bm{e}}_{x}+\sin\theta\hat{\bm{e}}_{y})$. We also
assume that the applied strain induces a pseudomagnetic field lying in the
$xy$-plane, i.e.
$\bm{B}_{5}=B_{5}(\cos\theta_{5}\hat{\bm{e}}_{x}+\sin\theta_{5}\hat{\bm{e}}_{y})$
and vanishing axial electric field $\bm{E}_{5}=\bm{0}$, as depicted in Fig. 2.
Later we will discuss the impact of a nonzero $\bm{E}_{5}$ upon the
longitudinal and planar Hall currents.

To discuss the longitudinal and planar Hall responses in a realistic WSM, it
is convenient to consider the precise values of the parameters appearing in
our expressions and verify first the validity of the chiral kinetic theory. We
take $B=B_{5}=0.5$T and use typical parameters for a Weyl semimetal such as
TaAs: $v_{F}=3\times 10^{5}$m/s, $b_{0\chi}=0$, $\mu=20$meV Weng  _et al._
(2015); Huang  _et al._ (2015), and $\tau\sim 10^{-13}$s Xiong  _et al._
(2016); Shekhar  _et al._ (2015). Therefore, the Boltzmann formalism is valid
because $\omega_{\mbox{\scriptsize C}}\tau\sim 0.08\ll 1$, where
$\omega_{\mbox{\scriptsize C}}=eB/m^{\ast}c$ is the cyclotron frequency and we
have used $m^{\ast}\sim 0.11m_{e}$ Weng  _et al._ (2015); Huang  _et al._
(2015) and $B\sim 0.5$T. One can further verify that near the nodes, the
corrections induced by the orbital magnetic moment to the energy satisfies
$\mathcal{E}_{\chi}^{(m)}\ll\mathcal{E}_{\alpha}^{(0)}$ and
$(e/\hbar)B\Omega\ll 1$, thus validating the expansions performed in Section
II. Here we take $T=2$K, which corresponds to the temperature at which the
angular dependence of the planar Hall conductivity was observed in topological
insulators Taskin  _et al._ (2017). This value, together with $\mu=20$meV for
TaAs, implies that $\Lambda_{\chi}=0.0086$ and hence
$f_{-2}(\Lambda_{\chi})=1.00073\approx 1$. Therefore, in the following we
safely take $f_{-2}(\Lambda_{\chi})=1$ for definiteness, which is appropriate
at low temperatures. In Fig. 3 we plot the normalized longitudinal (upper
panel) and planar Hall (lower panel) conductivities, given by Eq. (21), as a
function of the angle $\theta$ and fixed values $\theta_{5}=0$ (left panel)
and $\theta_{5}=\pi/2$ (right panel). In these plots, the dashed orange
(continuous red) line shows the Berry curvature contribution (19) normalized
by the longitudinal conductivity $\sigma_{0}^{(\chi)}$, i.e.
$\sigma^{(\Omega,\chi)}_{ij}/\sigma_{0}^{(\chi)}$, for the chirality $\chi=+1$
($\chi=-1$). The purple (blue) line with square (circle) markers shows the
normalized conductivity $\Sigma_{ij}^{(\chi)}$ as defined in Eq. (21) for the
chirality $\chi=+1$ ($\chi=-1$). These plots evince the importance of the
orbital magnetic moment upon the longitudinal and planar Hall conductivities.
In fact, as we can see in Fig. 3, the longitudinal conductivity is more
sensitive to the OMM than the planar Hall conductivity. However, this fact
does not mean that such contribution is zero in the planar Hall conductivity.
Clearly, for $\theta_{5}=0$, the OMM contribution is negligible near to
$\theta=0,\pi$, but becomes important near to some specific angles, namely, at
$\theta^{\ast}=\pi/3,\,5\pi/3$ for $\chi=+1$ and at
$\theta^{\ast}=2\pi/3,4\pi/3$ for $\chi=-1$. For $\theta_{5}=\pi/2$, however,
the OMM contribution approaches zero near to $\theta=\pi/2,\,3\pi/2$, and
becomes appreciable around $\theta^{\ast}=7\pi/6,\,11\pi/6$ for $\chi=+1$ and
around $\theta^{\ast}=\pi/6,\,5\pi/6$ for $\chi=-1$. The position of these
critical angles $\theta^{\ast}$ varies according to the ratio $B_{5}/B$ as
well as the direction of $\bm{B}_{5}$. In fact, it becomes determined by the
equation:

$\displaystyle\cos(2\theta^{\ast})+\chi(B_{5}/B)\cos(\theta^{\ast}+\theta_{5})=0.$
(22)

Figure 3: Angular dependence of the longitudinal (upper panel) and planar Hall
(lower panel) conductivities for $\theta_{5}=0$ (left panel) and
$\theta_{5}=\pi/2$ (right panel). The dashed orange (continuous red) line
shows the Berry curvature contribution (normalized by the longitudinal
conductivity) for the chirality $\chi=+1$ ($\chi=-1$), while the purple (blue)
line with square (circle) markers shows the full conductivities including the
orbital magnetic moment contribution.

Following the definition of the current for fermions with chirality $\chi$ in
the presence of effective electromagnetic fields,
$J_{i}^{(\chi)}=\sigma^{(\chi)}_{ij}E_{\chi j}$, with the conductivity given
by Eq. (8), we now define the total and axial currents by
$\bm{J}=\sum_{\chi=\pm 1}\bm{J}^{(\chi)}$ and $\bm{J}_{5}=\sum_{\chi=\pm
1}\chi\bm{J}^{(\chi)}$, respectively. These suggest the definition of the
total and the chiral conductivities as follows:

$\displaystyle\sigma_{ij}=\sum_{\chi=\pm
1}\sigma^{(\chi)}_{ij},\qquad\sigma_{5ij}=\sum_{\chi=\pm
1}\chi\sigma^{(\chi)}_{ij},$ (23)

each in order, such that the total and axial currents take the simple form

$\displaystyle
J_{i}=\sigma_{ij}({\bm{B}},{\bm{B}}_{5})\,E_{j}+\sigma_{5ij}({\bm{B}},{\bm{B}}_{5})\,E_{5j},$
(24) $\displaystyle
J_{5i}=\sigma_{5ij}({\bm{B}},{\bm{B}}_{5})\,E_{j}+\sigma_{ij}({\bm{B}},{\bm{B}}_{5})\,E_{5j},$
(25)

respectively.

In the problem at hand, using the conductivity tensor (8), together with the
three contributions (17), (19) and (20), one can obtain general expressions
for the total and chiral conductivity tensors. Assuming $b_{0\chi}=0$ for
simplicity, and taking $T=0$K in view of the above discussion, the total
conductivity becomes

$\displaystyle\sigma_{ij}({\bm{B}},{\bm{B}}_{5})$
$\displaystyle=\sigma_{0}\delta_{ij}+\frac{\sigma_{0}}{10B_{0}^{2}}\Big{[}-\delta_{ij}(B^{2}+B^{2}_{5})$
$\displaystyle\hskip 72.55461pt+3(B_{i}B_{j}+B_{5i}B_{5j})\Big{]},$ (26)

where $\sigma_{0}\equiv\frac{e^{2}\mu^{2}\tau}{3\pi^{2}\hbar^{3}v_{F}}$ is the
field-independent longitudinal conductivity and
$B_{0}\equiv(e\hbar)^{-1}(\mu/v_{F})^{2}$ is a characteristic magnetic field.
Similarly, for the chiral conductivity we obtain

$\displaystyle\sigma_{5ij}({\bm{B}},{\bm{B}}_{5})=\frac{\sigma_{0}}{10B_{0}^{2}}\left[-2\delta_{ij}{\bm{B}}\cdot{\bm{B}}_{5}+3(B_{i}B_{5j}+B_{5i}B_{j})\right].$
(27)

Clearly, the chiral conductivity vanishes if either $\bm{B}$ or $\bm{B}_{5}$
are zero. Therefore, in such case, to probe the chiral current, a pseudo-
electric field is required. This is so because the axial gauge fields couple
opposite chiral fermions with opposite signs. To illustrate the angular
dependence of these conductivities we introduce the normalized total
conductivity

$\displaystyle\Sigma_{ij}({\bm{B}},{\bm{B}}_{5})=\frac{\sigma_{ij}({\bm{B}},{\bm{B}}_{5})-\sigma_{0}\delta_{ij}}{\sigma_{0}},$
(28)

and similarly one can define the normalized axial conductivity as
$\Sigma_{5ij}({\bm{B}},{\bm{B}}_{5})=\sigma_{5ij}({\bm{B}},{\bm{B}}_{5})/\sigma_{0}$.
Note that expressions (27) and (28) are valid for arbitrary orientations of
the fields ${\bm{B}}$ and ${\bm{B}}_{5}$. Now we take the same planar
configuration as before, depicted in Fig. 2. To elucidate the interplay of the
genuine and axial magnetic fields, we fix the direction of the effective
electric field to be along $+\hat{\bm{e}}_{x}$, i.e.
$\bm{E}_{\chi}=E_{\chi}\hat{\bm{e}}_{x}$, and rotate the magnetic field along
the $xy$-plane such that it makes an angle $\theta$ with respect to the
direction of the electric field, i.e.
$\bm{B}=B(\cos\theta\hat{\bm{e}}_{x}+\sin\theta\hat{\bm{e}}_{y})$. Also we
take the pseudo-magnetic field in the same plane, i.e.
$\bm{B}_{5}=B_{5}(\cos\theta_{5}\hat{\bm{e}}_{x}+\sin\theta_{5}\hat{\bm{e}}_{y})$.
In this case, the normalized total and axial longitudinal conductivities
become

$\displaystyle\Sigma_{xx}({\bm{B}},{\bm{B}}_{5})$
$\displaystyle=\frac{B^{2}(3\cos^{2}\theta-1)+B_{5}^{2}(3\cos^{2}\theta_{5}-1)}{10B_{0}^{2}},$
(29) $\displaystyle\Sigma_{5xx}({\bm{B}},{\bm{B}}_{5})$
$\displaystyle=\frac{BB_{5}}{10B_{0}^{2}}\left[3\cos(\theta+\theta_{5})+\cos(\theta-\theta_{5})\right],$
(30)

respectively. The corresponding expressions for the normalized total and axial
planar Hall conductivities are

$\displaystyle\Sigma_{xy}({\bm{B}},{\bm{B}}_{5})$
$\displaystyle=\frac{3}{20B_{0}^{2}}\left[B^{2}\sin(2\theta)+B_{5}^{2}\sin(2\theta_{5})\right],$
(31) $\displaystyle\Sigma_{5xy}({\bm{B}},{\bm{B}}_{5})$
$\displaystyle=\frac{3BB_{5}}{10B_{0}^{2}}\sin(\theta+\theta_{5}),$ (32)

respectively. We now plot these expressions. To this end, we take
$B=B_{5}=0.5$ T and use typical parameters for TaAs ($v_{F}=3\times 10^{5}$
m/s and $\mu=20$ meV) which yields $B_{0}=6.75$ T. In Fig. 4 we show the
longitudinal (upper panel) and planar Hall (lower panel) conductivities as a
function of the angle $\theta$ and fixed values $\theta_{5}=0$ (at left) and
$\theta_{5}=\pi/2$ (at right). Each panel displays both the normalized total
and axial conductivities. The dashed orange (continuous red) line corresponds
solely to the Berry curvature contribution to the total (axial)
conductivities. The purple (blue) line with square (circle) markers shows the
full contribution (i.e. including both Berry curvature and OMM) to the total
(axial) conductivities.

Figure 4: Longitudinal (upper panel) and planar Hall (lower panel)
conductivities for $\theta_{5}=0$ (left panel) and $\theta_{5}=\pi/2$ (right
panel). The dashed orange (continuous red) line shows the Berry curvature
contribution to the total (axial) conductivities, while the purple (blue) line
with square (circle) markers shows the full contribution (including the
orbital magnetic moment contribution) to the total (axial) conductivities.

(a) (b) (c) (d)

Figure 5: (a) and (c) show the total and axial planar Hall conductivities,
respectively, as a function of the magnetic field for $\theta_{5}=0$,
$B_{5}=0.5$ T and different values of the angle $\theta$. (b) and (d) show 3D
plots of the planar and axial Hall conductivities, respectively, as a function
of the angles $\theta$ and $\theta_{5}$ for $B=B_{5}=0.5$ T.

The importance of the OMM is quite evident in the case of the total and axial
longitudinal conductivities: the unmarked lines significantly differ from
those with markers. However, the total and axial planar Hall conductivities
exhibit slight changes which can be appreciated near to some specific angles.
For example, in the case of the total planar Hall conductivity $\Sigma_{yx}$,
the differences are important near to $\theta^{\ast}=n\pi/4$, where
$n=1,3,5,7$, in both situations $\theta_{5}=0$ and $\theta_{5}=\pi/2$. For an
arbitrary value of $\theta_{5}$ the position of these critical angles does not
change, but the value of the normalized conductivity is shifted by
$\frac{3}{20}(B_{5}/B_{0})^{2}\sin(2\theta_{5})$. In the case of the axial
conductivity, the main differences are near to $\theta^{\ast}=\pi/2,\,3\pi/2$
for $\theta_{5}=0$ and near to $\theta^{\ast}=0,\,\pi$ for $\theta_{5}=\pi/2$.
The behavior of the axial conductivity is quite the same for any value of
$\theta_{5}$; the only difference is that the critical angles are shifted by
$\theta_{5}$, i.e., they are given by $\theta^{\ast}=\theta_{5}+n\pi/2$.

Apart from the angular dependence of the longitudinal and planar Hall
conductivities shown in Fig. 4, the dependency on the magnitude of the
magnetic field would also be relevant for an experimental detection of our
results. In fact, as we read off from Eqs. (29)-(32), the total longitudinal
$\Sigma_{xx}$ and transverse $\Sigma_{xy}$ conductivities show $B^{2}$
dependence, while the axial conductivities $\Sigma_{5xx}$ and $\Sigma_{5xy}$
exhibit a linear dependency on $B$. In Figs. 5(a) and 5(c) we plot the total
and axial planar Hall conductivities, $\Sigma_{xy}({\bm{B}},{\bm{B}}_{5})$ and
$\Sigma_{5xy}({\bm{B}},{\bm{B}}_{5})$, respectively, as a function of the
(normalized) magnetic field $B/B_{0}$, for $\theta_{5}=0$, $B_{5}=0.5$ T and
different values of the angle $\theta$. In Fig. 5(a) the curves correspond to
angles ranging from $\theta=\pi/6$ to $\theta=\pi/2$, and they closes
progressively in such order, i.e. the focal length decreases as increasing the
angle $\theta$. The vertex of the parabola is located at
$\Sigma_{xy}({\bm{0}},{\bm{B}}_{5})=\frac{3}{20B_{0}^{2}}B_{5}^{2}\sin(2\theta_{5})$,
which for the chosen parameter values is zero. Different values of the angle
$\theta_{5}$ only shift the parabolas to the top for
$\theta_{5}\in[0,\pi/2]\cup[\pi,3\pi/2]$ or to the bottom for
$\theta_{5}\in[\pi/2,\pi]\cup[3\pi/2,2\pi]$. In Fig. 5(c) the straight lines
crosses the origin and the slop is given by
$\frac{3B_{5}}{10B_{0}^{2}}\sin(\theta+\theta_{5})$. The curves correspond to
angles ranging from $\theta=\pi/6$ to $\theta=\pi/2$, where the slope
increases. Figures 5(b) and 5(d) show the total and axial planar Hall
conductivities, $\Sigma_{xy}$ and $\Sigma_{5xy}$, as a function of the angles
$\theta$ and $\theta_{5}$, for $B=B_{5}=0.5$ T. Red (blue) shaded regions
display the maximum (minimum) of the corresponding functions.

Everything discussed thus far correspond, in a separate fashion, to the total
and axial conductivities appearing in the general formulas (24)-(25). In the
presence of a genuine electric field and vanishing axial electric field, the
plots presented in Figs. 4 and 5 correctly describes the behaviour of the
longitudinal and planar Hall conductivities. In the opposite case, for a
vanishing genuine electric field and nonzero axial electric field, the roles
of the total and axial conductivities becomes interchanged, as Eqs. (24) and
(25) suggest. However, in the presence of both genuine and axial electric
fields, the effects of the total $\Sigma_{ij}$ and axial $\Sigma_{5ij}$
conductivities become intertwined. To elucidate the interplay of the electric
fields, we now consider the configuration shown in Fig. 2 in the presence of
an electric field ${\bm{E}}=E_{x}\hat{{\bm{e}}}_{x}$ and an in-plane axial
electric field
${\bm{E}}_{5}=E_{5}(\cos\varphi\hat{{\bm{e}}}_{x}+\sin\varphi\hat{{\bm{e}}}_{y})$.
In this case, the total and axial currents become

$\displaystyle J_{i}/E_{x}$
$\displaystyle=\sigma_{ix}+\epsilon_{5}(\cos\varphi\,\sigma_{5ix}+\sin\varphi\,\sigma_{5iy}),$
(33) $\displaystyle J_{5i}/E_{x}$
$\displaystyle=\sigma_{5ix}+\epsilon_{5}(\cos\varphi\,\sigma_{ix}+\sin\varphi\,\sigma_{iy}),$
(34)

Figure 6: Angular dependence of the longitudinal (upper panel) and planar Hall
(lower panel) currents for ${\bm{E}}=E_{x}\hat{{\bm{e}}}_{x}$, $\theta_{5}=0$,
$\varphi=0$ (left panel) and $\varphi=\pi/2$ (right panel). The dashed orange
(continuous red) line shows the currents for $E_{5}=0$, while the purple
(blue) line with square (circle) markers shows the full currents including the
axial electric field.

where $\epsilon_{5}\equiv E_{5}/E_{x}$ is the ratio between the axial and
genuine electric field, and $\sigma_{ij}$ and $\sigma_{5ij}$ are the
conductivity tensors given by Eqs. (26) and (27), respectively. To plot these
functions we take appropriate values for the genuine and axial fields. Genuine
electromagnetic fields are extensively controlled in experiments. However,
strain-induced fields are more subtle. For example, for a WSM in the presence
of torsion, the maximum attainable axial magnetic field is $B_{5}\approx 0.5$T
Pikulin  _et al._ (2016). For simplicity we also take $B=0.5$T. An estimate
for the axial electric field can be obtained from the upper limit $E_{5}\leq
v_{F}B_{5}$, which is required to circumvent the collapsing of Landau levels
in WSMs Arjona  _et al._ (2017). In TaAs we find $E_{5}\leq 1.5\times
10^{5}$V/m, and then here we take $E_{5}=2\times 10^{4}$V/m, which vastly
fulfils the condition. In the next Section we deepen on the origin of strain-
induced pseudofields. If we take a moderate electric field of strength
$E_{x}=2\times 10^{4}$V/m we obtain $\epsilon_{5}=0.5$. In the following, to
better understand the interplay of the axial electric field, we fix the axial
magnetic field pointing parallel to the electric field, i.e.
${\bm{B}}=B\hat{{\bm{e}}}_{x}$, and explore the angular dependence of the full
currents (Berry curvature + OMM contributions) as a function of the angles
$\theta$ (defined by the magnetic field) and $\varphi$. In the upper panel of
Fig. 6 we plot the longitudinal total $J_{x}$ and axial $J_{5x}$ currents (in
units of $E_{x}$) as a function of the angle $\theta$ for $\theta_{5}=0$ and
two different values of $\varphi$, namely, $\varphi=0$ (left panel) and
$\varphi=\pi/2$ (right panel). The lower panel displays the total $J_{y}$ and
axial $J_{5y}$ planar Hall currents (in units of $E_{x}$). The dashed orange
and continuous red lines correspond to the currents in the absence of the
axial electric field, while the purple and blue lines with square and circle
markers correspond to the currents in the presence of $E_{5}$. We observe that
for an axial electric field in parallel configuration (i.e. with $\varphi=0$),
the longitudinal total and axial currents behave in a similar manner than in
the absence of $E_{5}$, changing sign approximately around the same angles.
However, in a perpendicular configuration (i.e. with $\varphi=\pi/2$), the
angular dependence of the current is slightly different, with the position of
the extrema shifted (indeed, the shifts can be directly calculated from Eqs.
(33)-(34)). We highlight the fact that the total conductivity (purple line
with square markers) reverses its sign near to $\theta=3\pi/2$, and this is a
direct consequence of the nonzero axial electric field. Therefore, this is a
direct signature of the effects of the $E_{5}$ upon the longitudinal
conductivities. The case of the planar Hall conductivities is similar. In
parallel configuration ($\varphi=0$), the planar Hall current both in the
presence and in the absence of the axial electric field display the same
angular dependence, flipping sign around the same angles, with the extrema
slightly shifted and with the amplitude controlled by the field strength
$E_{5}$. In perpendicular configuration, there are two aspects that could be
highlighted. On the one hand, in the case $E_{5}=0$ there are three angles at
which the current vanishes, i.e. $\theta=0,\pi,2\pi$, since $J_{y}\sim
3\sigma_{0}E_{x}\sin\theta\cos\theta$; however, in the presence of an axial
electric field the planar Hall current is of the form
$J_{y}\sim\sigma_{0}E_{x}(3\sin\theta-2\epsilon_{5})\cos\theta$, thus implying
that there is only one critical angle at which the current vanishes, given by
$\theta^{\ast}=\arcsin(2\epsilon_{5}/3)$, and take the constant value
$J_{y}\sim-2\sigma_{0}\epsilon_{5}E_{x}$ at $\theta=0,2\pi$. On the other
hand, as the lower-right panel of Fig. 6 shows, in the presence of the axial
electric field the periodicity of the current in the angle $\theta$ is broken,
which becomes clear from the previous equations for the currents with and
without $E_{5}$. These are distinguishing features of the presence of the
axial electric field. All these conclusions are also supported by the 3D plots
presented in Fig. 7, which display the total and axial planar Hall currents
(in units of $E_{x}$) as a function of the angles $\theta$ and $\varphi$. Red
(blue) shaded regions display the maximum (minimum) of the corresponding
functions.

Figure 7: 3D plots of the total and axial planar Hall current (in units of
$E_{x}$) as a function of the angles $\theta$ (defined by the magnetic field)
and $\varphi$ (defined by the axial electric field)

As pointed out, anomaly related transport phenomena have attracted great
attention in condensed matter physics. For example, in graphene, fermionic
excitations near the Dirac cones are described by a (2+1)-dimensional quantum
field theory exhibiting the parity anomaly. This gives rise to the valley Hall
effect, observed experimentally Gorbachev  _et al._ (2014); Shimazaki  _et
al._ (2015) and extensively explored due to possible applications in
valleytronics Rycerz  _et al._ (2007). On the same footing, Weyl semimetals
provide an electronic route for realizing the chiral anomaly in condensed
matter. The chiral anomaly induces a number of novel phenomena in WSMs,
including the chiral magnetic effect Fukushima  _et al._ (2008) that manifests
itself through the negative longitudinal magnetoresistance Son and Spivak
(2013); Burkov (2014), and it was confirmed in transport experiments in the
TaAs family Huang  _et al._ (2015); Zhang  _et al._ (2016). The planar Hall
effect has been ascribed also to the chiral anomaly; however, as shown in this
paper, there are other contributions that affect the PHE in a similar order of
magnitude than the Berry curvature contribution, namely, the orbital magnetic
moment of charge carriers.

WSMs in the presence of pseudofields give unique opportunities to probe the
different contributions to the covariant anomaly equations (1)-(2). Along this
line, our results provide a testing ground for the chiral and charge anomalies
by means of the planar Hall effect. Experimentally, pseudofields are induced
by applying strain on the crystal. For example, a pseudomagnetic field
${\bm{B}}_{5}$ can be created by applying a static torsion or bending the
sample. A nonzero pseudoelectric field ${\bm{E}}_{5}$ is generated, for
instance, by dynamically stretching or compressing the sample. Therefore,
properly combining genuine and pseudo-electromagnetic fields, we are able to
test the four terms in the right-hand side of Eqs. (1) and (2). On the one
hand, the two terms in the chiral anomaly equation (1),
${\bm{E}}\cdot{\bm{B}}$ and ${\bm{E}}_{5}\cdot{\bm{B}}_{5}$, can be tested by
using nonorthogonal electromagnetic fields or pseudo-electromagnetic fields,
respectively. The latter can be achieved by a simultaneous application of
torsion and time-dependent unidirectional strain. Physically, in both cases,
this can be understood as pumping of charge from one node to the other. On the
other hand, the two terms in the charge anomaly equation (2),
${\bm{E}}\cdot{\bm{B}}_{5}$ and ${\bm{E}}_{5}\cdot{\bm{B}}$, can be tested by
the combined application of electromagnetic and pseudo-electromagnetic fields.
These nonconserving charge terms should be interpreted with caution, since in
a real solid, charge is strictly conserved. Indeed, they can be understood as
pumping of charge between the bulk and the boundary of the system Liu  _et
al._ (2013); Chernodub  _et al._ (2014); Pikulin  _et al._ (2016). This
implies that additional currents must exist in the system to restore charge
conservation. This problem is solved by adding Bardeen polynomials Behrends
_et al._ (2019); Bardeen and Zumino (1984); Gorbar  _et al._ (2017), which in
the present case, corresponds to the topological Chern-Simons charge and
current densities, in the order given by $\rho_{\mbox{\scriptsize
CS}}=-\frac{e^{2}}{2\pi^{2}\hbar^{2}}{\bm{b}}\cdot{\bm{B}}$ and
${\bm{J}}_{\mbox{\scriptsize
CS}}=\frac{e^{2}}{2\pi^{2}\hbar^{2}}(-b_{0}{\bm{B}}+{\bm{b}}\times{\bm{E}})$,
being $2{\bm{b}}$ and $2b_{0}$ the separation between the Weyl nodes in
momentum and energy, respectively, as depicted in Fig. 1. To understand this
result physically, we must recall that pseudo-fields are induced by
inhomogeneous strain, which results in a position and time dependent
separation between the nodes in momentum and energy, i.e.
${\bm{b}}({\bm{r}},t)$ and $b_{0}({\bm{r}},t)$, such that the emerging pseudo-
fields are determined by ${\bm{B}}_{5}=\nabla\times{\bm{b}}$ and
${\bm{E}}_{5}=-\nabla b_{0}-\partial_{t}{\bm{b}}$. It is now straightforward
to check that the Chern-Simons 4-current $J^{\mu}_{\mbox{\scriptsize
CS}}=(c\rho_{\mbox{\scriptsize CS}},{\bm{J}}_{\mbox{\scriptsize CS}})$
satisfies $\partial_{\mu}J^{\mu}_{\mbox{\scriptsize
CS}}=\frac{e^{2}}{2\pi^{2}\hbar^{2}}(\bm{E}\cdot\bm{B}_{5}+\bm{E}_{5}\cdot\bm{B})$.
This fact suggests the definition of consistent total and axial currents
$J_{\mbox{\scriptsize cons}}^{\mu}=J^{\mu}+J^{\mu}_{\mbox{\scriptsize CS}}$
and $J_{5\mbox{\scriptsize
cons}}^{\mu}=J^{\mu}_{5}+J^{\mu}_{5\mbox{\scriptsize CS}}$, respectively,
restoring charge conservation. In this manner, the consistent version of the
anomaly equations (1) and (2) are

$\displaystyle\partial_{\mu}J^{\mu}_{5\mbox{\scriptsize cons}}$
$\displaystyle=\frac{e^{2}}{2\pi^{2}\hbar^{2}}(\bm{E}\cdot\bm{B}+\frac{1}{3}\bm{E}_{5}\cdot\bm{B}_{5}),$
(35) $\displaystyle\partial_{\mu}J^{\mu}_{\mbox{\scriptsize cons}}$
$\displaystyle=0,$ (36)

respectively. Therefore, charge nonconservation stated in the anomaly
equations (1) and (2) not to worry us. In the next Section we will back to
this point in particular examples.

##  IV Strain-induced nonlinear transport phenomena in WSMs

So far we have discussed in general the effects of pseudo-fields upon
nonlinear transport phenomena in Weyl semimetals and remarked in passing that
they emerge due to position and time dependent deformations. In this Section
we deepen on the physical origin of the pseudo-fields and apply our results to
particular strain configurations.

Figure 8: Strain-induced shift of a single Weyl node in momentum and energy,
defining the pseudo-gauge potentials
$\tilde{{\bm{A}}}_{\chi}^{\mbox{\scriptsize el}}$ and
$\tilde{\Phi}_{\chi}^{\mbox{\scriptsize el}}$, respectively.

In a strained material, the modifications in the hoping parameters (between
atomic orbitals) and on-site energies are determined by the components of the
strain-tensor $u_{ij}=\tfrac{1}{2}\left(\partial u_{i}/\partial x^{j}+\partial
u_{j}/\partial x^{i}\right)$, where $u_{i}$ is the local displacement vector
of the strained lattice. In Weyl semimetals the deformation of the crystal
lattice shifts the Weyl nodes in momentum and energy, as shown in Fig. 8, and
such node shifts can be described in terms of pseudo-gauge potentials
$\tilde{{\bm{A}}}_{\chi}^{\mbox{\scriptsize el}}({\bm{r}},t)$ and
$\tilde{\Phi}_{\chi}^{\mbox{\scriptsize el}}({\bm{r}},t)$ Cortijo  _et al._
(2015, 2015). These couple to the electronic degrees of freedom as the
electromagnetic vector and scalar potentials do. Therefore, the effects of
strain on a single Weyl node of chirality $\chi$, as depicted in Fig. 8, are
captured by the Hamiltonian

$\displaystyle\hat{H}_{\chi}({\bm{k}})=\chi
v_{F}\bm{\sigma}\cdot\big{(}\hbar{\bm{k}}+\tilde{{\bm{A}}}_{\chi}^{\mbox{\scriptsize
el}}\big{)}+\tilde{\Phi}_{\chi}^{\mbox{\scriptsize el}}+b_{0\chi},$ (37)

where the pseudo-gauge potentials are expressed in terms of the strain-tensor
as $\tilde{A}_{\chi i}^{\mbox{\scriptsize el}}=h_{ijk}u_{jk}$ and
$\tilde{\Phi}_{\chi}^{\mbox{\scriptsize el}}=g_{ij}u_{ij}$. Here, $h_{ijk}$ is
the Weyl node shift per unit strain and $g_{ij}$ is the energy shift per unit
strain, which have to be computed with a microscopic model (e.g. tight-
binding, ab initio) or determined by experiments Lee  _et al._ (2022). These
tensors contain all material details, such as anisotropy and elastic
parameters. The corresponding pseudo-fields
$\tilde{\bm{E}}_{\chi}^{\mbox{\scriptsize
el}}=-\nabla\tilde{\Phi}_{\chi}^{\mbox{\scriptsize
el}}-\partial_{t}\tilde{{\bm{A}}}_{\chi}^{\mbox{\scriptsize el}}$ and
$\tilde{\bm{B}}_{\chi}^{\mbox{\scriptsize
el}}=\nabla\times\tilde{{\bm{A}}}_{\chi}^{\mbox{\scriptsize el}}$ may couple
opposite chiral fermions with opposite signs, in much the same fashion than
the case of strained graphene, in which pseudo-fields couple to the Dirac
fermions oppositely in the two valleys. To account for the two possibilities,
that pseudo-fields couple axially or not to the Weyl nodes, we introduce the
notation $\tilde{\bm{E}}_{\chi}^{\mbox{\scriptsize
el}}=\bm{\mathcal{E}}+\chi\bm{E}_{5}$ and
$\tilde{\bm{B}}_{\chi}^{\mbox{\scriptsize
el}}=\bm{\mathcal{B}}+\chi\bm{B}_{5}$, where $\bm{\mathcal{E}}$ and
$\bm{\mathcal{B}}$ are strain-induced pseudo-fields which couple to the Weyl
nodes in the same manner as electromagnetic fields do, while $\bm{E}_{5}$ and
$\bm{B}_{5}$ are the axial parts of the pseudo-fields, which couple opposite
chiral fermions with opposite signs. All these fields are fully determined by
the strain-tensor. Note that our results of the previous Section holds in this
case; however, the elastic (non-axial) electric and magnetic fields should be
considered as genuine electromagnetic fields.

In order to probe strain-induced nonlinear phenomena in Weyl semimetals, we
have to consider suitable non-uniform strain tensors. In the following we
shall discuss two interesting cases with experimental possibilities. To be
precise, we study the effects of position-dependent strain tensors: (i)
bending the WSM into a circular arc and (ii) twisting a wire-shaped WSM. These
configurations produce uniform pseudo-fields, such that the pseudo-magnetic
field couples axially to the Weyl fermions, and the pseudo-electric field is
non-axial and then couples to the Weyl fermions in the usual manner Lee  _et
al._ (2022). With the help of an additional magnetic field, these can be used
to test the covariant anomaly equations.

###  IV.1 Bending of WSM thin films

Films and wires realizations of Weyl semimetals are excellent probes to test
strain-induced phenomena. In fact, the best experimentally accessible geometry
of applied strain is obtained by bending thin films of WSMs, which is a 3D
generalization of the configuration suggested for graphene sheets in Ref.
Guinea  _et al._ (2010b). As we shall see, bending of WSM thin films is an
excellent test bed for the covariant anomaly equations.

Figure 9: Sketch of the bending geometry that generates pseudo-fields. The
applied strain bends the original configuration (upper panel) into a circular
arc in the $x$-$y$ plane (lower panel).

Let us consider a rectangular lattice model of a WSM, as depicted in the upper
panel of Fig. 9, with two nodes separated by a distance $b$ in the $k_{x}$
direction. Bending the system into a circular arc in the $x$-$y$ plane, as
sketched in the lower panel of Fig. 9, is described by the deformation Liu
_et al._ (2017):

$\displaystyle u_{x}$ $\displaystyle=u_{0}(2xy+Cx),$ (38) $\displaystyle
u_{y}$ $\displaystyle=u_{0}\left[-x^{2}-Dy(y+C)\right],$ (39) $\displaystyle
u_{z}$ $\displaystyle=0,$ (40)

where $u_{0}$, $C$ and $D$ are constants that depend on the material. This
yields the axial vector potential $A_{5x}=u_{xx}b=u_{0}(2y+C)b$, i.e. it
couples to the Weyl nodes with opposite signs. The corresponding axial pseudo-
magnetic field is then $B_{5z}=-2u_{0}b$. On the other hand, this strain
configuration produces also a scalar (deformation) potential
$\tilde{\Phi}_{\chi}=u_{xx}+u_{yy}=u_{0}(1-D)(2y+C)$, which produces the
elastic (i.e. non-axial) pseudo-electric field
$\mathcal{E}_{y}=-2u_{0}(1-D)g$, perpendicular to the pseudo-magnetic field,
as shown in Fig. 9. The orthogonality between the strain-induced pseudo-fields
makes this configuration appropriate to investigate the planar Hall effect in
the presence of an external magnetic field in the $y$-$z$ plane, i.e.
${\bm{B}}=B(\cos\theta\hat{{\bm{e}}}_{y}+\sin\theta\hat{{\bm{e}}}_{z})$,
coplanar to the pseudo-fields. Using equations (24) and (25) one determines
directly the total and axial currents. Normalizing the currents by
$\sigma_{0}\mathcal{E}_{y}$ one finds the following expressions for the total
${\bm{j}}$ and axial ${\bm{j}}_{5}$ currents

$\displaystyle{\bm{j}}(\theta)$
$\displaystyle=\frac{1}{10}\left(\begin{array}[]{c}0\\\
\tilde{B}^{2}(3\cos^{2}\theta-1)-\tilde{B}^{2}_{5}\\\
3\tilde{B}^{2}\sin\theta\cos\theta\end{array}\right),$ (44)
$\displaystyle{\bm{j}}_{5}(\theta)$
$\displaystyle=\frac{\tilde{B}_{5}\tilde{B}}{10}\left(\begin{array}[]{c}0\\\
-2\sin\theta\\\ 3\cos\theta\end{array}\right)$ (48)

where we have subtracted the field-independent current from the total current
${\bm{j}}$. Here, $\tilde{B}=B/B_{0}$ and $\tilde{B}_{5}=B_{5}/B_{0}$.
Interestingly, these are planar Hall currents., i.e. they are in the $y$-$z$
plane, where the pseudo-fields and the magnetic field lie. The angular
dependences of the components of ${\bm{j}}$ and ${\bm{j}}_{5}$ are plotted in
Fig. 10. We have taken $B=1$T, $B_{5}=0.5$T, $B_{0}=6.75$T (characteristic of
TaAs) and normalized the currents by its maximum values $j_{\mbox{\scriptsize
max}}=j_{y}(\pi)$ and $j_{5\mbox{\scriptsize max}}=j_{5z}(\pi/2)$. The left
panel shows the total current components: the blue dashed line corresponds to
$j_{y}(\theta)/j_{\mbox{\scriptsize max}}$ and the red continuous line is
$j_{z}(\theta)/j_{\mbox{\scriptsize max}}$. The right panel shows the axial
current components: the blue dashed line corresponds to
$j_{5y}(\theta)/j_{5\mbox{\scriptsize max}}$ and the red continuous line is
$j_{5z}(\theta)/j_{5\mbox{\scriptsize max}}$. The effects of pseudo-fields can
be increased according to the strain. In the numerical calculations we have
used $B_{5}=0.5$ T; however, strain-induced pseudo-magnetic fields (on the
order of 3 Tesla) were recently observed in strained crystals of Re-doped
MoTe2 Kamboj  _et al._ (2019).

The analysis of this subsection can be extended to situations involving more
deformation profiles. In particular, it would be interesting to analyse the
effect of an axial pseudo-electric field, which unlike the deformation
potential considered above, couple with opposite sign to the nodes of opposite
chirality.

Figure 10: Angle dependence of the (normalized) total ${\bm{j}}$ (at left) and
axial ${\bm{j}}_{5}$ (at right) currents for a WSM strained into a circular
arc in the $x$-$y$ plane, as depicted in Fig. 9. The blue dashed lines show
the $y$-components, and the red continuous line are the $z$-components.

###  IV.2 Rotational strain

In three dimensions, the antisymmetric part of the strain tensor,
$\omega_{ij}=\tfrac{1}{2}\left(\partial u_{j}/\partial x^{i}-\partial
u_{i}/\partial x^{j}\right)$, also gives rise to pseudo-fields in Weyl
semimetals. Physically, it is related to infinitesimal rotations by a vector
$\bm{\Omega}$ with the components given by
$\Omega_{k}=\tfrac{1}{2}\epsilon_{ijk}\omega_{ij}$. One can further see that
this vector is related to the deformation vector $\bm{u}$ by

$\displaystyle\bm{\Omega}(\bm{r})=\frac{1}{2}\,\nabla\times\bm{u}(\bm{r}).$
(49)

A full discussion on the effects of rotational strain in Dirac matter is
presented in Ref. Arjona and Vozmediano (2018). There, the authors derive the
low-energy effective Hamiltonians for the electron-strain interactions around
Weyl nodes associated with the antisymmetric part of the strain tensor. A
distinguishing physical example, which we consider here, is a wire-shaped Weyl
semimetal of length $L$ with an axis along the $z$-direction and the nodes
separated by a distance $b$ along the axis. The displacement vector $\bm{u}$
that derives from twisting the sample an angle $\theta$ is given by:

$\displaystyle\bm{u}(\bm{r})=\theta\frac{z}{L}(\bm{r}\times\hat{\bm{e}}_{z}),$
(50)

where $\bm{r}$ is the position relative to the origin located on the axis of
the wire. In Fig. 11 we show the effect of the deformation (50). The strain
tensor associated with the deformation (50) is traceless, and therefore the
deformation potential generated from $u_{ij}$ is zero. The corresponding axial
vector potential is given by
$\bm{A}_{5}(\bm{r})=\theta\frac{b}{2L}(y\hat{\bm{e}}_{x}-x\hat{\bm{e}}_{y})$,
producing the uniform pseudo-magnetic field
$\bm{B}_{5}=-\theta\frac{b}{L}\hat{\bm{e}}_{z}$. According to Eq. (49), the
rotation vector becomes
$\bm{\Omega}(\bm{r})=\frac{\theta}{2L}(\bm{r}-3z\hat{\bm{e}}_{z})$. This
produces a deformation potential
$\tilde{\Phi}(\bm{r})=\bm{b}\cdot\bm{\Omega}(\bm{r})=-\frac{\theta}{L}bz$,
which is non-axial. The corresponding pseudo-electric field is
$\bm{\mathcal{E}}=-\nabla\bm{\Omega}(\bm{r})=\frac{\theta}{L}b\hat{\bm{e}}_{z}$,
which is antiparallel to the pseudo-magnetic field $\bm{B}_{5}$, as depicted
in Fig. 11. It is interesting that these pseudo-fields induce a longitudinal
current along the direction of the pseudo-fields:

$\displaystyle
J_{z}=\sigma_{0}\left(1+\frac{B_{5}^{2}}{5B_{0}^{2}}\right)\mathcal{E}.$ (51)

Note that the direction of the torsion-induced current can be controlled by
the direction of the rotation: it is positive (negative) when the twisting is
clockwise (anticlockwise). This configuration, supplemented by an external
magnetic field, could also be used to test the covariant anomaly equations.

Figure 11: Schematic representation of a cross-section of the wire-shaped
nanowire twisted by an angle $\theta$.

##  V Discussion

In recent years, the study of strain-induced transport has attracted great
attention due to its potential applications for the development of
straintronic devices. The most salient example is perhaps graphene, in which
strain induces pseudo-fields which couples to the Dirac fermions oppositely in
the two valleys. Other well-known examples are carbon nanotubes Kane and Mele
(1997), bilayer graphene Mariani  _et al._ (2012); Rostami and Asgari (2013)
and transition metal dichalcogenides Cazalilla  _et al._ (2014); Rostami  _et
al._ (2015). More recently, the study of strain-induced pesudo-fields in Dirac
materials has garnered a lot of attention since it could open a pathway to the
development of the strain-induced chiralitytronics. In Weyl semimetals, for
example, these chirality selective pseudo-fields has lead to interesting
strain-induced phenomena, such as quantum oscillations Liu  _et al._ (2017),
the chiral magnetic effect and the negative resistivity Cortijo  _et al._
(2016); Grushin  _et al._ (2016), the chiral torsional effect Sumiyoshi and
Fujimoto (2016), the acoustogalvanic effect Sukhachov and Rostami (2020),
among others.

In this paper we used the chiral kinetic theory approach, at a finite
temperature, to investigate nonlinear transport phenomena in Weyl semimetals
induced by electromagnetic fields and strain-induced pseudo-fields. Our main
focus was the study of the planar Hall effect, the appearance of an in-plane
transverse voltage in the presence of coplanar electromagnetic fields. Using
the relaxation time approximation for the collision integral, we first derived
general expressions for the nonlinear conductivity tensor and clearly
differentiate the contributions arising from the Berry curvature and the
orbital magnetic moment of charge carries, since the latter has been
continuously disregarded in the analysis of the PHE, and we showed that it is
as important as the Berry curvature. Next, using the simplest linearly
dispersing model for a Weyl semimetal with two nodes of opposite chiralities
separated in momentum and energy, we obtained analytical expressions for the
contributions to the magnetoconductivity tensor at a finite temperature. A
rough estimation of the finite temperature effect reveals that, at low
temperatures ($T\sim 2$K, which is the temperature used to detect the PHE in
topological insulators), the amplitude of the currents differ by a factor of
$10^{-4}$ with respect to the result at $T=0$. Our numerical calculations,
presented in Figs. 3-7, reveal that the OMM contribution is more pronounced in
the longitudinal magentoconductivity than in the planar Hall conductivity. Our
general expressions for the total $\bm{J}$ and axial $\bm{J}_{5}$ currents,
(24) and (25) respectively, include the effects of usual electromagnetic
fields $\bm{E}$ and $\bm{B}$, besides the contributions arising from the
pseudo-electric $\bm{E}_{5}$ and the pseudo-magnetic $\bm{B}_{5}$ fields. In
the absence of the axial electric field the longitudinal and planar Hall
currents exhibit periodicity in the angle $\theta$, defined by the magnetic
field as $\cos\theta=\bm{\hat{B}}_{5}\cdot\bm{\hat{E}}$; however, when the
axial electric field is turned on, such periodicity goes down and the position
of the extrema in the conductivities change. Interestingly, these expressions
show the possibility of inducing nonlinear transport phenomena by using only
strain-induced pseudo-fields without needing real electromagnetic fields,
which opens up new avenues for manipulating and controlling this effect.
Finally, we apply our results to two strained configurations with experimental
possibilities: (i) bending a WSM thin film into a circular arc and (ii)
applying torsion to a WSM wire. These configurations could also be useful in
the investigation of the covariant anomaly equations (1) and (2) of Weyl
fermions.

The planar Hall effect in multilayer structures of NiFe/IrMn and NiFe/Cu/NiFe
has been widely used in the development of high-performance magnetic field
sensors, including those based on spin valves, giant magnetoresistance and
tunneling magnetoresistance Luong  _et al._ (2018); Wang  _et al._ (2019);
Quynh  _et al._ (2016). On the other hand, in Dirac materials, for example,
recent magneto-transport measurements on single crystals of the magnetic Weyl
semimetal Co3Sn2S2 reported the observation of the PHE Shama  _et al._ (2020).
Along these lines, our results create opportunities for the design of novel
devices or sensors that exploit pseudo-fields to detect the PHE in Weyl
semimetals. Finally, we note that the study of strain-induced PHE can be
extended straightforwardly to include dynamical local deformations of the
crystal, giving rise to an acoustic-induced PHE, thus enlarging the
possibilities for the design of devices for chiralitytronics.

###### Acknowledgements.

L.M.O. was supported by the CONACyT PhD fellowship No. 834773. A.M.-R. has
been partially supported by DGAPA-UNAM Project No. IA102722 and by Project
CONACyT (México) No. 428214. We are also indebted to the reviewers for their
valuable comments and suggestions to improve the quality of the paper.

##  Appendix A Derivation of the orbital magnetic moment contribution

In this Section we derive the formula for the orbital magnetic moment
contribution $\sigma^{(m,\alpha)}_{ij}$ to the magnetoconductivity tensor
$\sigma^{(\alpha)}_{ij}$. To this end, we start with the general expression
for the magnetoconductivity tensor, given by Eq. (8) and expand the factor
$D_{\alpha}({\bm{k}})$ and the equilibrium distribution function
$f_{\alpha}^{\mbox{\scriptsize eq}}(\mathcal{E}_{\alpha})$ in powers of the
pseudo-magnetic field. Keeping terms up to quadratic order in the pseudo-
magnetic field we obtain

$\displaystyle D_{\alpha}({\bm{k}})$ $\displaystyle\approx
1-\frac{e}{\hbar}\left[{\bm{B}}_{\chi}\cdot{\bm{\Omega}}_{\alpha}({\bm{k}})\right]+\frac{e^{2}}{\hbar^{2}}\left[{\bm{B}}_{\chi}\cdot{\bm{\Omega}}_{\alpha}({\bm{k}})\right]^{2}$
$\displaystyle\equiv 1+D_{\alpha}^{(1)}({\bm{k}})+D_{\alpha}^{(2)}({\bm{k}})$
(52)

and

$\displaystyle f_{\alpha}^{\mbox{\scriptsize eq}}(\mathcal{E}_{\alpha})\approx
f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})+\mathcal{E}_{\alpha}^{(m)}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})+\frac{1}{2}\mathcal{E}_{\alpha}^{(m)2}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime}(\mathcal{E}_{\alpha}^{(0)}),$ (53)

where $f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})=\frac{\partial
f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)}}$ and
$f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime}(\mathcal{E}_{\alpha}^{(0)})=\frac{\partial^{2}f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)2}}$. Now
we substitute these expansions into Eq. (8) and separate the velocity as
${\bm{v}}_{\alpha}={\bm{v}}_{\alpha}^{(0)}+{\bm{v}}_{\alpha}^{(m)}$. Keeping
the terms quadratic in the pseudo-magnetic field we obtain:

$\displaystyle\sigma^{(m,\alpha)}_{ij}$
$\displaystyle=-e^{2}\tau\int\frac{d^{3}{\bm{k}}}{(2\pi)^{3}}\Bigg{\\{}\
\\!\\!\left[v_{\alpha i}^{(m)}v_{\alpha j}^{(m)}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})+(v_{\alpha i}^{(0)}v_{\alpha
j}^{(m)}+v_{\alpha i}^{(m)}v_{\alpha
j}^{(0)})\mathcal{E}_{\alpha}^{(m)}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime}(\mathcal{E}_{\alpha}^{(0)})+\frac{1}{2}v_{\alpha
i}^{(0)}v_{\alpha
j}^{(0)}\mathcal{E}_{\alpha}^{(m)2}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime\prime}(\mathcal{E}_{\alpha}^{(0)})\right]$
$\displaystyle\phantom{=}+\frac{e}{\hbar}\left[\Omega_{\alpha k}\left(B_{\chi
i}(v_{\alpha j}^{(0)}v_{\alpha k}^{(m)}+v_{\alpha j}^{(m)}v_{\alpha
k}^{(0)})+B_{\chi j}(v_{\alpha i}^{(0)}v_{\alpha k}^{(m)}+v_{\alpha
i}^{(m)}v_{\alpha k}^{(0)})-B_{\chi k}(v_{\alpha i}^{(0)}v_{\alpha
j}^{(m)}+v_{\alpha i}^{(m)}v_{\alpha
j}^{(0)})\right)f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})\right.$
$\displaystyle\phantom{=}\left.+\;\Omega_{\alpha k}\left(B_{\chi i}v_{\alpha
j}^{(0)}v_{\alpha k}^{(0)}+B_{\chi j}v_{\alpha i}^{(0)}v_{\alpha
k}^{(0)}-B_{\chi k}v_{\alpha i}^{(0)}v_{\alpha
j}^{(0)}\right)\mathcal{E}_{\alpha}^{(m)}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime}(\mathcal{E}_{\alpha}^{(0)})\right]\,\,\Bigg{\\}}\ ,$ (54)

where we have subtracted the field-independent and Berry curvature
contributions, given by equations (10) and (11) respectively. To simplify this
expression, on the one hand, we observe that:

$\displaystyle\partial_{k_{i}}\partial_{k_{j}}\left(\frac{1}{2\hbar^{2}}\mathcal{E}_{\alpha}^{(m)2}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})\right)$
$\displaystyle=\frac{1}{\hbar}\left(\partial_{k_{j}}v_{\alpha
i}^{(m)}\right)\mathcal{E}_{\alpha}^{(m)}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})+\frac{1}{2\hbar}\left(\partial_{k_{j}}v_{\alpha
i}^{(0)}\right)\mathcal{E}_{\alpha}^{(m)2}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime}(\mathcal{E}_{\alpha}^{(0)})+\left[v_{\alpha
i}^{(m)}v_{\alpha j}^{(m)}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})\right.$
$\displaystyle\phantom{=}\left.\;+\;(v_{\alpha i}^{(0)}v_{\alpha
j}^{(m)}+v_{\alpha i}^{(m)}v_{\alpha
j}^{(0)})\mathcal{E}_{\alpha}^{(m)}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime}(\mathcal{E}_{\alpha}^{(0)})+\frac{1}{2}v_{\alpha
i}^{(0)}v_{\alpha
j}^{(0)}\mathcal{E}_{\alpha}^{(m)2}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime\prime}(\mathcal{E}_{\alpha}^{(0)})\right],$ (55)

where we have used that $\nabla_{\bm{k}}f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})=\hbar\bm{v}_{\alpha}^{(0)}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})$. The term in square brackets is
exactly the same that appears in the first line of Eq. (54). Besides, upon
integration of Eq. (55) over the Brillouin zone, the left-hand side vanishes
due to the periodic boundary conditions. Hence, the first line of Eq. (54) can
be replaced by the two first terms in the right-hand side of Eq. (55). On the
other hand, the second term in square brakets in Eq. (54) can be further
simplified in terms of the vector
$\bm{Q}_{\alpha}\equiv\bm{\Omega}_{\alpha}\times(\bm{v}^{(0)}_{\alpha}\times\bm{B}_{\chi})$.
We obtain

$\displaystyle\sigma^{(m,\alpha)}_{ij}$
$\displaystyle=-e^{2}\tau\int\frac{d^{3}{\bm{k}}}{(2\pi)^{3}}\Bigg{\\{}\
\\!\\!-\left[\frac{1}{\hbar}\left(\partial_{k_{j}}v_{\alpha
i}^{(m)}\right)\mathcal{E}_{\alpha}^{(m)}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})+\frac{1}{2\hbar}\left(\partial_{k_{j}}v_{\alpha
i}^{(0)}\right)\mathcal{E}_{\alpha}^{(m)2}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime}(\mathcal{E}_{\alpha}^{(0)})\right]$
$\displaystyle\phantom{=}+\frac{e}{\hbar}\left[-\left({Q}_{\alpha i}v_{\alpha
j}^{(m)}+{Q}_{\alpha j}v_{\alpha i}^{(m)}\right)f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})+\left(B_{\chi i}v_{\alpha
j}^{(0)}+B_{\chi j}v_{\alpha
i}^{(0)}\right)\bm{\Omega}_{\alpha}\cdot\left(\bm{v}^{(m)}_{\alpha}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})+\bm{v}^{(0)}_{\alpha}\mathcal{E}_{\alpha}^{(m)}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime}(\mathcal{E}_{\alpha}^{(0)})\right)\right]$
$\displaystyle\phantom{=}-\frac{e}{\hbar}(\bm{\Omega}_{\alpha}\cdot\bm{B}_{\chi})v_{\alpha
i}^{(0)}v_{\alpha
j}^{(0)}\mathcal{E}_{\alpha}^{(m)}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime}(\mathcal{E}_{\alpha}^{(0)})\,\,\Bigg{\\}}\ .$ (56)

Now, using the fact that
$\frac{1}{\hbar}\nabla_{\bm{k}}\left[\mathcal{E}_{\alpha}^{(m)}\,f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})\right]=\bm{v}^{(m)}_{\alpha}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})+\bm{v}^{(0)}_{\alpha}\mathcal{E}_{\alpha}^{(m)}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime}(\mathcal{E}_{\alpha}^{(0)})$ and integrating by parts we
get

$\displaystyle\sigma^{(m,\alpha)}_{ij}$
$\displaystyle=\frac{e^{3}\tau}{\hbar}\int\frac{d^{3}{\bm{k}}}{(2\pi)^{3}}\Bigg{\\{}\
\\!\\!\left({Q}_{\alpha i}v_{\alpha j}^{(m)}+{Q}_{\alpha j}v_{\alpha
i}^{(m)}\right)f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})+\mathcal{E}_{\alpha}^{(m)}\left[(\bm{\Omega}_{\alpha}\cdot\bm{B}_{\chi})v_{\alpha
i}^{(0)}v_{\alpha
j}^{(0)}+\frac{1}{2e}\mathcal{E}_{\alpha}^{(m)}\left(\partial_{k_{j}}v_{\alpha
i}^{(0)}\right)\right]f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime}(\mathcal{E}_{\alpha}^{(0)})$
$\displaystyle\phantom{=}+\frac{1}{e}\mathcal{E}_{\alpha}^{(m)}\left[\left(\partial_{k_{j}}v_{\alpha
i}^{(m)}\right)+\frac{e}{\hbar}\nabla_{\bm{k}}\cdot\left[\bm{\Omega}_{\alpha}(B_{\chi
i}v_{\alpha j}^{(0)}+B_{\chi j}v_{\alpha
i}^{(0)})\right]\right]f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})\,\,\Bigg{\\}}\ ,$ (57)

where we have used the periodicity of the Brillouin zone. Finally, using the
symmetry of the tensor under the interchange $i\leftrightarrow j$ and the
expression $\mathcal{E}_{\alpha}^{(m)}=-{\bm{m}}_{\alpha}\cdot{\bm{B}}_{\chi}$
we obtain

$\displaystyle\sigma^{(m,\alpha)}_{ij}$
$\displaystyle=\frac{e^{3}\tau}{\hbar}\int\frac{d^{3}{\bm{k}}}{(2\pi)^{3}}\Bigg{\\{}\
\\!\\!{Q}_{\alpha i}v_{\alpha j}^{(m)}f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})+\frac{1}{2}\mathcal{E}_{\alpha}^{(m)}{\bm{B}}_{\chi}\cdot\left[\bm{\Omega}_{\alpha}v_{\alpha
i}^{(0)}v_{\alpha
j}^{(0)}-{\bm{m}}_{\alpha}\frac{1}{2e}\left(\partial_{k_{j}}v_{\alpha
i}^{(0)}\right)\right]f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime\prime}(\mathcal{E}_{\alpha}^{(0)})$
$\displaystyle\phantom{=}+\frac{1}{e}\,\mathcal{E}_{\alpha}^{(m)}\left[\frac{e}{\hbar}\nabla_{\bm{k}}\cdot(\bm{\Omega}_{\alpha}B_{\chi
i}v_{\alpha j}^{(0)})+\frac{1}{2}\left(\partial_{k_{j}}v_{\alpha
i}^{(m)}\right)\right]f_{\alpha}^{\mbox{\scriptsize
eq}\,\prime}(\mathcal{E}_{\alpha}^{(0)})\,\,\Bigg{\\}}\ +(i\leftrightarrow
j).$ (58)

This result suggests the definition of the tensors in Eq. (13) and yields the
final expression for the orbital magnetic moment contribution, given by Eq.
(12).

##  Appendix B Computation of the conductivity tensors

Here we evaluate in detail the different contributions to the
magnetoconductivity tensor defined in Section II for a Weyl semimetal. Owing
to the rotational symmetry of the problem, the required integrals can be
performed straightforwardly by using the spherical coordinate system
${\bm{k}}=k(\sin\theta\cos\phi\hat{{\bm{e}}}_{x}+\sin\theta\sin\phi\hat{{\bm{e}}}_{y}+\cos\theta\hat{{\bm{e}}}_{z})$
and the volume element $d^{3}{\bm{k}}=k^{2}dkd\Omega$, being $d\Omega$ the
differential solid angle. Note that the difference between the chemical
potential and the energy shift of the node determines the band index, i.e.
$s=\mbox{sgn}(\mu-b_{0\chi})$, as evinced in figure 1. This is so since
$\mu>b_{0\chi}$ ($\mu<b_{0\chi}$) implies $s=1$ ($s=-1$). Therefore we take
$\mu_{\chi}=s\mu_{0\chi}$, with $\mu_{0\chi}=|\mu-b_{0\chi}|>0$. Besides, we
work at finite temperature.

Let us first consider the $B$-independent conductivity, given by Eq. (10).
Substituting the required components of the band velocity
${\bm{v}}_{s}^{(0)}=sv_{F}\hat{{\bm{k}}}$ one gets

$\displaystyle\sigma^{(0,\alpha)}_{ij}(T)=-\frac{e^{2}v_{F}^{2}\tau}{8\pi^{3}}\int
d^{3}{\bm{k}}\,\hat{k}_{i}\hat{k}_{j}\,\frac{\partial
f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)}}.$ (59)

Owing to the rotational symmetry, the angular integration becomes $\int
d\Omega\,\hat{k}_{i}\hat{k}_{j}=\int
d\Omega\,\frac{1}{3}\delta_{ij}=\frac{4\pi}{3}\delta_{ij}$. Therefore we
obtain

$\displaystyle\sigma^{(0,\alpha)}_{ij}(T)=\frac{e^{2}v_{F}^{2}\tau}{6\pi^{2}}\delta_{ij}\int_{0}^{\infty}dk\,k^{2}\,\frac{1}{k_{B}T}\frac{e^{\frac{\mathcal{E}_{\alpha}^{(0)}-\mu_{\chi}}{k_{B}T}}}{\left(1+e^{\frac{\mathcal{E}_{\alpha}^{(0)}-\mu_{\chi}}{k_{B}T}}\right)^{2}},$
(60)

where we have used the Fermi-Dirac distribution. Using the fact that
$\mathcal{E}^{(0)}_{\alpha}({\bm{k}})=b_{0\chi}+s\hbar v_{F}k$ and
$\mu_{\chi}=s\mu_{0\chi}$, with $\mu_{0\chi}=|\mu-b_{0\chi}|>0$, this result
can be written in the simple form

$\displaystyle\sigma_{ij}^{(\chi,0)}(T)=\frac{e^{2}\tau\mu_{0\chi}^{2}}{6\pi^{2}\hbar^{3}v_{F}}\delta_{ij}\,f_{2}(\Lambda_{\chi}),$
(61)

with $\Lambda_{\chi}\equiv k_{B}T/\mu_{0\chi}$ and where we have introduced
the function

$\displaystyle{f_{n}(y)\equiv\frac{1}{y}\int_{0}^{\infty}dx\,x^{n}\,\frac{e^{(x-1)/y}}{\left[1+e^{(x-1)/y}\right]^{2}}.}$
(62)

It is interesting that $\lim_{y\to 0}f_{n}(y)=1$, which applied to our result
of Eq. (61) corresponds to the zero temperature limit.

We now turn to the Berry curvature contribution, given by Eq. (11). Using the
Berry curvature ${\bm{\Omega}}_{\alpha}$ and band velocity
$\bm{v}_{\alpha}^{(0)}$, given in Eq. (15), one finds
${\bm{Q}}_{\alpha}=-\chi\frac{v_{F}}{2k^{2}}\hat{{\bm{k}}}\times(\hat{{\bm{k}}}\times{\bm{B}}_{\chi})$.
Therefore, Eq. (11) yields

$\displaystyle\sigma_{ij}^{(\alpha,\Omega)}(\bm{B}_{\chi},T)$
$\displaystyle=-\frac{e^{4}v_{F}^{2}\tau}{32\pi^{3}\hbar^{2}}\int
d^{3}{\bm{k}}\,\frac{1}{k^{4}}\left[\hat{{\bm{k}}}\times(\hat{{\bm{k}}}\times{\bm{B}}_{\chi})\right]_{i}$
$\displaystyle\phantom{=}\times\left[\hat{{\bm{k}}}\times(\hat{{\bm{k}}}\times{\bm{B}}_{\chi})\right]_{j}\,\frac{\partial
f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)}}.$ (63)

Algebraic manipulation of the integrand produces

$\displaystyle\sigma_{ij}^{(\alpha,\Omega)}(\bm{B}_{\chi},T)$
$\displaystyle=-\frac{e^{4}v_{F}^{2}\tau}{32\pi^{3}\hbar^{2}}\\!\int\\!\frac{dk}{k^{2}}\,\frac{\partial
f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)}}\int
d\Omega\Big{[}B_{\chi i}B_{\chi j}$
$\displaystyle-(\hat{{\bm{k}}}\cdot{\bm{B}}_{\chi})(\hat{k}_{i}B_{\chi
j}+\hat{k}_{j}B_{\chi
i})+(\hat{{\bm{k}}}\cdot{\bm{B}}_{\chi})^{2}\hat{k}_{i}\hat{k}_{j}\Big{]}.$
(64)

These integrals are simple, but not straightforward. The necessary angular
integral over products of rectangular components of $\hat{{\bm{k}}}$ is
readily found to be

$\displaystyle\int
d\Omega\,\hat{k}_{i}\hat{k}_{j}\hat{k}_{l}\hat{k}_{m}=\frac{4\pi}{15}\left(\delta_{ij}\delta_{lm}+\delta_{il}\delta_{jm}+\delta_{im}\delta_{jl}\right).$
(65)

On the other hand, the required radial integral can be expressed in terms of
the function $f_{n}(\Lambda_{\chi})$, defined by Eq. (62), with $n=-2$, i.e.

$\displaystyle\int_{0}^{\infty}dk\,\frac{1}{k^{2}}\,\frac{\partial
f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)}}$
$\displaystyle=-\frac{\hbar v_{F}}{\mu_{0\chi}^{2}}f_{-2}(\Lambda_{\chi}).$
(66)

Substituting these results into Eq. (64) we establish Eq. (19).

We now evaluate the orbital magnetic moment contribution
$\sigma_{ij}^{(\chi,m)}({\bm{B}}_{\chi},T)$, given by Eq. (12). To this end,
we require the corrections to the energy $\mathcal{E}_{\alpha}^{(m)}$ and band
velocity
${\bm{v}}_{\alpha}^{(m)}=\frac{1}{\hbar}\nabla_{{\bm{k}}}\mathcal{E}_{\alpha}^{(m)}$
due to the orbital magnetic moment. Using the OMM ${\bm{m}}_{\alpha}$, given
in Eq. (15), one finds $\mathcal{E}_{\alpha}^{(m)}=\frac{\chi
ev_{F}}{2k}\hat{{\bm{k}}}\cdot{\bm{B}}_{\chi}$, and

$\displaystyle{\bm{v}}_{\alpha}^{(m)}({\bm{k}})=\frac{\chi
ev_{F}}{2\hbar}\frac{{\bm{B}}_{\chi}-2\hat{{\bm{k}}}(\hat{{\bm{k}}}\cdot{\bm{B}}_{\chi})}{k^{2}}.$
(67)

Now we compute separately the three integrals in Eq. (12) in the order they
appear, i.e.
$\sigma_{ij}^{(\chi,m)}=\sigma_{1ij}^{(\chi,m)}+\sigma_{2ij}^{(\chi,m)}+\sigma_{3ij}^{(\chi,m)}$.
Let us consider first:

$\displaystyle\sigma_{1ij}^{(\chi,m)}({\bm{B}}_{\chi},T)=\frac{2e^{3}\tau}{\hbar}\\!\int\\!\frac{d^{3}{\bm{k}}}{(2\pi)^{3}}{Q}_{\alpha
i}v_{\alpha j}^{(m)}\frac{\partial f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)}}.$ (68)

Substituting the function ${Q}_{\alpha i}$ and the velocity (67) we have

$\displaystyle\sigma_{1ij}^{(\chi,m)}({\bm{B}}_{\chi},T)$
$\displaystyle=-\frac{e^{4}v_{F}^{2}\tau}{16\pi^{3}\hbar^{2}}\int
d^{3}{\bm{k}}\,\frac{1}{k^{4}}\,[\hat{{\bm{k}}}\times(\hat{{\bm{k}}}\times{\bm{B}}_{\chi})]_{i}$
$\displaystyle\phantom{=}\times[{\bm{B}}_{\chi}-2\hat{{\bm{k}}}(\hat{{\bm{k}}}\cdot{\bm{B}}_{\chi})]_{j}\,\frac{\partial
f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)}}.$ (69)

Manipulating the integrand we obtain

$\displaystyle\sigma_{1ij}^{(\chi,m)}$
$\displaystyle=\frac{e^{4}v_{F}^{2}\tau}{16\pi^{3}\hbar^{2}}\int\frac{dk}{k^{2}}\,\frac{\partial
f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)}}\int
d\Omega\,\Big{[}B_{\chi i}B_{\chi j}$
$\displaystyle\phantom{=}+2\hat{k}_{i}\hat{k}_{j}(\hat{{\bm{k}}}\cdot{\bm{B}}_{\chi})^{2}-(\hat{{\bm{k}}}\cdot{\bm{B}}_{\chi})(\hat{k}_{i}B_{\chi
j}+2\hat{k}_{j}B_{\chi i})\Big{]}.$ (70)

The angular integration can be performed by using the formula (65), and the
radial integration is given by Eq. (66). These results imply:

$\displaystyle\sigma_{1ij}^{(\chi,m)}({\bm{B}}_{\chi},T)$
$\displaystyle=-\frac{e^{4}v_{F}^{3}\tau}{30\pi^{2}\hbar\mu_{0\chi}^{2}}(\delta_{ij}B_{\chi}^{2}+2B_{\chi
i}B_{\chi j})f_{-2}(\Lambda_{\chi}).$ (71)

We now consider the second term in Eq. (12), namely

$\displaystyle\sigma_{2ij}^{(\chi,m)}({\bm{B}}_{\chi})=\frac{2e^{3}\tau}{\hbar}\\!\int\\!\frac{d^{3}{\bm{k}}}{(2\pi)^{3}}\,\frac{1}{e}\mathcal{E}_{\alpha}^{(m)}\nabla_{{\bm{k}}}\cdot\bm{T}_{\alpha
ij}\frac{\partial f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)}},$ (72)

where $\bm{T}_{\alpha ij}$ is defined in Eq. (13). Using the Berry curvature
${\bm{\Omega}}_{\alpha}$ and the contributions to the band velocity
$\bm{v}_{\alpha}^{(0)}$ and $\bm{v}_{\alpha}^{(m)}$, one finds

$\displaystyle\bm{T}_{\alpha ij}=\frac{\chi ev_{F}}{4\hbar
k^{2}}\left[\hat{\bm{e}}_{i}B_{\chi j}-2\hat{k}_{j}\left(B_{\chi
i}\hat{{\bm{k}}}+\hat{\bm{e}}_{i}\,\hat{{\bm{k}}}\cdot{\bm{B}}_{\chi}\right)\right],$
(73)

wherefrom we obtain

$\displaystyle\nabla_{{\bm{k}}}\cdot\bm{T}_{\alpha ij}$
$\displaystyle=\frac{\chi ev_{F}}{2\hbar
k^{3}}\Big{[}(4\hat{k}_{i}\hat{k}_{j}-\delta_{ij})\hat{\bm{k}}\cdot{\bm{B}}_{\chi}-\hat{k}_{i}B_{\chi
j}-\hat{k}_{j}B_{\chi i}\Big{]}.$ (74)

Inserting this result into Eq. (72) and manipulating the integral we have

$\displaystyle\\!\\!\sigma_{2ij}^{(\chi,m)}\\!$
$\displaystyle=-\frac{e^{4}v_{F}^{2}\tau}{16\pi^{3}\hbar^{2}}\int\frac{dk}{k^{2}}\frac{\partial
f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)}}\,\int
d\Omega\,\hat{{\bm{k}}}\cdot{\bm{B}}_{\chi}$
$\displaystyle\phantom{=}\times\Big{[}(\delta_{ij}-4\hat{k}_{i}\hat{k}_{j})\hat{\bm{k}}\cdot{\bm{B}}_{\chi}+\hat{k}_{i}B_{\chi
j}+\hat{k}_{j}B_{\chi i}\Big{]}.$ (75)

Finally, using the integrals (65) and (66) one gets

$\displaystyle\sigma_{2ij}^{(\chi,m)}({\bm{B}}_{\chi},T)=\frac{e^{4}v_{F}^{3}\tau}{60\pi^{2}\hbar\mu_{0\chi}^{2}}(\delta_{ij}B_{\chi}^{2}+2B_{\chi
i}B_{\chi j})f_{-2}(\Lambda_{\chi}).$ (76)

The last term we have to evaluate is:

$\displaystyle\sigma_{3ij}^{(\chi,m)}({\bm{B}}_{\chi},T)=\frac{e^{3}\tau}{\hbar}\\!\int\\!\frac{d^{3}{\bm{k}}}{(2\pi)^{3}}\mathcal{E}_{\alpha}^{(m)}\bm{B}_{\chi}\cdot\bm{V}_{\alpha
ij}\frac{\partial^{2}f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)2}},$ (77)

where $\bm{V}_{\alpha ij}$ is defined in Eq. (13). Using the Berry curvature
${\bm{\Omega}}_{\alpha}$ and the orbital magnetic moment ${\bm{m}}_{\alpha}$,
given by Eq. (15), we find

$\displaystyle\bm{V}_{\alpha ij}=s\chi
v_{F}^{2}\frac{\hat{{\bm{k}}}}{4k^{2}}(\delta_{ij}-3\hat{k}_{i}\hat{k}_{j}),$
(78)

and the integral to be solved is

$\displaystyle\sigma_{3ij}^{(\chi,m)}({\bm{B}}_{\chi},T)$
$\displaystyle=\frac{e^{4}v_{F}^{3}\tau}{64\pi^{3}\hbar}\int
d^{3}{\bm{k}}\,\frac{s}{k^{3}}(\bm{B}_{\chi}\cdot\hat{{\bm{k}}})^{2}(\delta_{ij}-3\hat{k}_{i}\hat{k}_{j})$
$\displaystyle\hskip
71.13188pt\times\frac{\partial^{2}f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)2}}.$ (79)

Further algebraic manipulations yield

$\displaystyle\sigma_{3ij}^{(\chi,m)}({\bm{B}}_{\chi},T)$
$\displaystyle=\frac{e^{4}v_{F}^{3}\tau}{64\pi^{3}\hbar}\int
d\Omega\,(\bm{B}_{\chi}\cdot\hat{{\bm{k}}})^{2}(\delta_{ij}-3\hat{k}_{i}\hat{k}_{j})$
$\displaystyle\hskip
19.91684pt\times\left[s\int\frac{dk}{k}\frac{\partial^{2}f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)2}}\right].$
(80)

The angular integration is directly evaluated with the help of the result
(65). For the radial integration, starting from the function $f_{n}(y)$ given
by Eq. (66), one can spot the identity

$\displaystyle
s\int\frac{dk}{k}\frac{\partial^{2}f_{\alpha}^{\mbox{\scriptsize
eq}}(\mathcal{E}_{\alpha}^{(0)})}{\partial\mathcal{E}_{\alpha}^{(0)2}}$
$\displaystyle=-\frac{1}{\mu_{0\chi}^{2}}f_{-2}(\Lambda_{\chi}).$ (81)

All in all, the final result is

$\displaystyle\sigma_{3ij}^{(\chi,m)}({\bm{B}}_{\chi},T)$
$\displaystyle=\frac{e^{4}v_{F}^{3}\tau}{120\pi^{2}\hbar\mu_{0\chi}^{2}}\left(3B_{\chi
i}B_{\chi j}-\delta_{ij}B_{\chi}^{2}\right)f_{-2}(\Lambda_{\chi}).$ (82)

Summing up the three contributions, (71), (76) and (82), we establishes the
result of Eq. (20) for the orbital magnetic moment contribution
$\sigma_{ij}^{(\chi,m)}({\bm{B}}_{\chi},T)$.

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