# Hedgehog orbital texture in $p-$type tellurium and the antisymmetric
nonreciprocal Hall response

Gabriele Maruggi  Instituto de Fisica, Universidade Federal do Rio de Janeiro,
Caixa Postal 68528, Rio de Janeiro, Brazil  Jaime Ferreira  Centro Brasileiro
de Pesquisas Físicas, Rua Dr. Xavier Sigaud, Rio de Janeiro, Brazil  Elisa
Baggio-Saitovitch  Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier
Sigaud, Rio de Janeiro, Brazil  Carsten Enderlein  Instituto de Fisica,
Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro,
Brazil  Marcello B. Silva Neto  Instituto de Fisica, Universidade Federal do
Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, Brazil

(August 28, 2024)

###### Abstract

Tellurium is a gyrotropic, $p-$type Weyl semiconductor with remarkable
electronic, optical, and transport properties. It has been argued that some of
these properties might stem from Weyl nodes at crossing points in the band
structure, and their nontrivial topological textures. However, Weyl nodes in
time-reversal invariant semiconductors are split up in energy, rather than in
momentum, and located deep below (far above) the top (bottom) of the valence
(conduction) band, challenging such an interpretation. Here, instead, we use a
$4-$band ${\bf k}\cdot{\bf p}$ Hamiltonian for $p-$type tellurium to show how
the ${\bf k}-$dependent spin-orbit interaction mixes up the top two (Weyl node
free) and bottom two (Weyl node containing) valence bands, generating a 3D
hedgehog orbital magnetic texture at the uppermost valence band, accessible to
transport already at the lowest doping. Hedgehog textures are important
signatures of Weyl fermion physics in general and in the context of condensed
matter physics arise form the carriers’ wave packet rotation being locked to
their propagation wavevector. For spatially dispersive media, such an induced
hedgehog texture/carrier rotation stabilizes two novel, nonreciprocal and
antisymmetric components to the Hall transport within different weak-
localization (antilocalization) relaxation regimes: the anomalous and planar
Hall effects, usually forbidden by time reversal symmetry. Our AC
magnetotransport measurements on Sn-doped tellurium confirm the theoretical
predictions and our work demonstrates how Weyl signatures generally appear in
transport on enantiomorphic materials with natural optical activity.

Anomalous Hall Effect, Berry curvature, Spin-Orbit coupling

##  I Introduction

The discovery of topological insulators (TIs) TopologicalInsulators  paved the
way for the emergence of a new class of remarkable quantum systems:
topological materials (TMs) TopologicalMaterials ; ReviewYoichi . One of the
earliest manifestations of topology in TMs is the integer quantum Hall effect
(IQHE) IQHE ; Halperin1982 , in which the Hall conductance plays the role of
an adiabatic curvature whose associated topological invariant corresponds to
the number of completely filled bands TKKN . When conduction and valence bands
linearly cross at the Fermi level one speaks of topological semimetals
TopologicalSemiMetals . Time-reversal symmetry breaking splits up the two
bands in momentum introducing a chiral shift BurkovPRL2014 , thus generating
two Weyl nodes that act as source and drain of nonzero Berry curvature. In
such Weyl semimetals (WSMs) the chiral anomaly gives rise to open Fermi
surfaces or Fermi arcs connecting the two Weyl nodes, leading to unique
signatures in transport WeylSemiMetals  such as the anomalous Hall effect
(AHE) which in WSMs is purely intrinsic and determined solely by the location
of the linearly dispersing Weyl nodes Burkov2018 . Conveniently, the
gauge/gravity duality provides a simple and elegant way to compute the AHE in
bulk topological WSMs because of its holographic equivalence to a nonzero
axial component for the horizon gauge field at the conformal boundary Gauge-
Gravity-Holography . The AHE is universal even for a doped Weyl metal when the
Fermi level occurs slightly away from the nodes BurkovPRL2014 .

Quite recently a novel class of TMs was discovered: Weyl semiconductors (WSCs)
Zhang2020 . The unique signatures of Weyl physics found in the enantiomorphic,
elemental semiconductor tellurium, include negative longitudinal
magnetoresistence, planar Hall effect (PHE), and logarithmically periodic
magneto-oscillations Zhang2020 . Following the discovery, tellurium
resurrected as a very promising material for both scientific research and the
design of novel technologies ResurrectionTe . Tellurium can be easily
synthesized either as 1D nanowires, with promising piezoelectric and
thermoelectric properties Te1D , grown as 2D films, fulfilling all the
necessary requirements to replace Silicon and Si-based semiconductor
technology Te2D , or as 3D bulk crystals, exhibiting quite remarkable chiral
thermoelectric properties Te3D  and pressure induced metal insulator
transition JaimeCommMat . Tellurium also exhibits several magnetoelectric
effects including current-induced magnetization CurrentInducedMagnetization ,
strong electrical magneto-chiral anisotropy, circular photogalvanic effect,
nonlinear (current induced) anomalous Hall effect, kinetic Faraday effect,
kinetic magnetoelectric effect Sahin2018 , and natural optical activity
Tsirkin2018 , that have all been interpreted as clear and unique signatures of
Weyl physics. Finally, the chiral-induced spin selectivity (CISS) in tellurium
Shalygin2012  builds an important bridge to systems that combine chiral
molecules, such as amino-acids, sugars or DNA, and magnetic moments of charge
carriers, therefore engineering new approaches to enantioselective chemistry
EnantioSelectivityChemistry .

Of particular relevance to understanding the origins of the Weyl physics in
WSCs is the Hall response. For a magnetic field, ${\bf B}$, at an angle
$\varphi$ with an applied current, ${\bf j}$, and both co-planar to the Hall
electric field, ${\bf E}$, a quadratic, reciprocal PHE,
$\sigma_{xy}^{ph}(B)\propto B^{2}\sin(2\varphi)$, is observed for $\varphi\neq
0,\pi/2$ in tellurium Zhang2020 . Although a symmetric PHE is ubiquitous in
WSMs PHEWeylSemiMetals , no signatures of an AHE BurkovPRL2014  has yet been
reported. The Weyl physics in WSCs must then come from pairs of nodes split up
in energy, not in momentum, that should also account for any antisymmetric,
nonreciprocal, anomalous or planar Hall responses OnsagerNonReciprocal , for
${\bf j}\perp{\bf B}$ ($\varphi=\pi/2$) and with ${\bf B}$ at an angle
$\pi/2-\theta$ with ${\bf E}$. However, even though tellurium is always
$p-$doped due to naturally occurring vacancies, such Weyl nodes are deep below
the top of the valence band Zhang2020 ; Hirayama2015 , making any contribution
to transport negligible.

In this work we use a $4-$band ${\bf k}\cdot{\bf p}$ Hamiltonian Doi1970 ;
Nakao1971  to demonstrate how the ${\bf k}-$dependent spin-orbit interaction
(${\bf k}-$SOI) induces nonzero Berry curvatures at the top two, topologically
trivial valence bands resulting from their mixing to the bottom two,
topologically nontrivial and Weyl-node containing valence bands. The resulting
form of wave-packet rotation undergone by the carriers XiaoRMP2010  generates
a 3D hedgehog orbital texture and contributes to transport. Using Boltzmann
transport equations Kim2014 ; Johansson2019  for spatially dispersive media
NagaosaNonreciprocal  and ${\bf j}\parallel\hat{\bf z}\perp{\bf
B}\not\parallel{\bf E}\parallel\hat{\bf x}$, we have found, in addition to the
reciprocal Hall response, $\sigma_{zx}^{h}(B)\propto B\cos\theta$, two novel,
antisymmetric, nonreciprocal phenomena: the AHE,
$\tilde{\sigma}_{zx}^{ah}(B)\propto B\cos\theta$ and the PHE
$\tilde{\sigma}_{zx}^{ph}(B)\propto B\sin\theta$. Our AC magnetotransport for
$p-$doped Te:Sn confirmed both the antisymmetry and phase relations amongst
$\sigma^{h}_{zx},\tilde{\sigma}^{ah}_{zx}$, and $\tilde{\sigma}^{ph}_{zx}$.

##  II Tellurium

Elemental tellurium is a chiral, non-centrosymmetric semiconductor composed by
helical chains arranged in a triangular lattice as shown in Fig. 1. The space
group of tellurium is either $D_{3}^{4}$ for right-handed or $D_{3}^{6}$ for
left-handed screw axis Firsov1957 . When choosing as set of primitive
translation vectors $\mathbf{t}_{1}=(a,0,0)$,
$\mathbf{t}_{2}=(-a/2,\sqrt{3}a/2,0)$, $\mathbf{t}_{3}=(0,0,c)$, where
$a=4.44\AA$ and $c=5.91\AA$, one can then write down the symmetry elements of
the factor group in the space group of $D_{3}^{6}$.

###  II.1 ${\bf k}\cdot{\bf p}-$Hamiltonian

The valence band structure of tellurium has been derived using the ${\bf
k}\cdot{\bf p}$ perturbation theory Doi1970 ; Nakao1971 . One assumes the
energy spectrum, $E_{n}({\bf k}_{H})$, and wave-functions, $\psi_{n,{\bf
k}_{H}}({\bf r})$, for the $n-$th valence band at ${\bf
k}_{H}=(4\pi/3a,0,\pi/c)$, are solutions to the eigenvalue problem, ${\cal
H}_{0}\psi_{n,{\bf k}_{H}}({\bf r})=E_{n}({\bf k}_{H})\psi_{n,{\bf
k}_{H}}({\bf r})$, for an unperturbed ${\cal
H}_{0}=\mathbf{p}^{2}/2m+U(\mathbf{r})$, where $U(\mathbf{r})$ is the periodic
potential. Then one looks for solutions to the Schrödinger’s equation ${\cal
H}\psi_{n,{\bf k}}({\bf r})=E_{n}({\bf k})\psi_{n,{\bf k}}({\bf r})$ for ${\bf
k}$ away from the $H-$point where $E_{n}({\bf k})$ and $\psi_{n,{\bf k}}({\bf
r})$ can be obtained from a double perturbation theory in ${\bf k}\cdot{\bf
p}$ and the SOI. One inserts $\psi_{n,{\bf k}}({\bf r})=e^{i{\bf k}\cdot{\bf
r}}u_{n,{\bf k}_{H},{\bf k}}({\bf r})$ into Schrödinger’s equation to obtain
an eigenvalue problem for $u_{n,{\bf k}_{H},{\bf k}}({\bf r})$ as ${\cal
H}_{kp,so}u_{n,{\bf k}_{H},{\bf k}}({\bf r})=E^{\prime}_{n}({\bf k}_{H}+{\bf
k})u_{n,{\bf k}_{H},{\bf k}}({\bf r})$, where $E^{\prime}_{n}({\bf k}_{H}+{\bf
k})=E_{n}({\bf k}_{H}+{\bf k})-\hbar^{2}{\bf k}^{2}/2m$. The matrix elements
of ${\cal H}={\cal H}_{0}+{\cal H}_{1}+{\cal H}_{2}+{\cal H}_{3}$, with

$\displaystyle{\cal H}_{1}$ $\displaystyle=$ $\displaystyle\frac{\hbar}{m}{\bf
k}\cdot{\bf p},$ $\displaystyle{\cal H}_{2}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{4m^{2}c^{2}}(\mathbf{\sigma}\times\nabla
U(\mathbf{r}))\cdot\mathbf{p},$ $\displaystyle{\cal H}_{3}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{4m^{2}c^{2}}(\mathbf{\sigma}\times\nabla
U(\mathbf{r}))\cdot\mathbf{k},$ (1)

are calculated in terms of the unperturbed states $\psi_{n,{\bf k}_{H}}({\bf
r})$, around ${\bf k}={\bf k}_{H}$, and correspond, respectively, to
contributions from ${\bf k}\cdot{\bf p}$, from the ${\bf k}-$independent SOI,
and from the ${\bf k}-$SOI Doi1970 ; Nakao1971 .

Figure 1: $a-$ top and side views of the crystal structure of trigonal
tellurium showing the arrangement of the helices; $b-$ the Brillouin zone and
the $H$ and $H^{\prime}$ points; $c-$ valence band structure of tellurium and
degeneracy lifting by the spin-orbit interaction.

The doubly degenerate $H_{3}$ valence band state becomes split into
$H_{6}^{\pm},H_{4}^{+}$, and $H_{5}^{-}$ valence bands by the spin-orbit
interaction Doi1970 ; Nakao1971 . Neglecting ${\cal H}_{3}$ our starting-point
Hamiltonian reads Doi1970 ; Nakao1971

$\begin{bmatrix}\varepsilon_{++}({\bf
k})-\Delta_{2}&ak_{+}&-iRk_{-}&ib^{*}k_{-}\\\ ak_{-}&\varepsilon_{-+}({\bf
k})-\Delta_{2}&-ib^{*}k_{+}&iRk_{+}\\\ iRk_{+}&ibk_{-}&\varepsilon_{--}({\bf
k})+\Delta_{2}&-2\Delta_{1}\\\
-ibk_{+}&-iRk_{-}&-2\Delta_{1}&\varepsilon_{+-}({\bf
k})+\Delta_{2}\end{bmatrix}.$

The $k-$independent parameters $\Delta_{1,2}$ arise at first-order in ${\cal
H}_{2}$ and are responsible for the bands splitting. At second-order in ${\cal
H}_{1}$ we have $\varepsilon_{\pm,\pm}({\bf k})=\varepsilon({\bf k})\pm(S\pm
d)k_{z}$, with $\varepsilon({\bf k})=Ak_{\perp}^{2}+Bk_{z}^{2}$ Doi1970 ;
Nakao1971 . The $Sk_{z}$ and $Rk_{\pm}$ terms represent linear-in-$k$
contributions at first-order in ${\cal H}_{1}$ due to the lack of inversion
symmetry. Combining ${\cal H}_{1}$ and ${\cal H}_{2}$ gives rise to the
linear-in-$k$ band mixing elements $ak_{\pm}$ and $ibk_{\pm}$, as well as the
diagonal $\pm dk_{z}$ terms. We have discarded all trigonal warping terms
$k_{\pm}^{2}$ and $(k_{x}^{2}-k_{y}^{2})$, and all other second-order terms in
${\cal H}_{2}$ Doi1970 ; Nakao1971 .

###  II.2 Valence band structure

The band structure obtained from the above Hamiltonian, for $R=b=0$, at the
$H-$point is shown in Fig. 1$-c$ on the right, where

$\displaystyle\varepsilon_{H_{4}^{+},H_{5}^{-}}({\bf k})$ $\displaystyle=$
$\displaystyle\varepsilon({\bf
k})\pm\sqrt{(S-d)^{2}k_{z}^{2}+4\Delta_{1}^{2}}+\Delta_{2},$
$\displaystyle\varepsilon_{H_{6}^{\pm}}({\bf k})$ $\displaystyle=$
$\displaystyle\varepsilon({\bf
k})\pm\sqrt{(S+d)^{2}k_{z}^{2}+a^{2}k_{\perp}^{2}}-\Delta_{2}.$ (2)

The uppermost $H_{4}^{+}$ valence band exhibits a dumbbell-shaped structure at
$k_{0}^{\pm}=\pm\sqrt{(S-d)^{4}-16\Delta_{1}^{2}B^{2}}/2B|S-d|$, as long as
$|(S-d)^{2}/B|>4\Delta_{1}$, which is the case for tellurium. The normalized
eigenstates for $R=b=0$ are

$\displaystyle\left|u_{H_{4,5}^{\pm}}({\bf k})\right>$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2h({\bf k})(h({\bf k})\pm
h_{z}(k_{z}))}}\left(\begin{matrix}2\Delta_{1}\\\ -h_{z}(k_{z})\mp h({\bf
k})\end{matrix}\right),$ $\displaystyle\left|u_{H_{6}^{\pm}}({\bf k})\right>$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2\mathfrak{g}({\bf
k})(\mathfrak{g}({\bf
k})\mp\mathfrak{g}_{z}(k_{z}))}}\left(\begin{matrix}-ak_{+}\\\
\mathfrak{g}_{z}(k_{z})\mp\mathfrak{g}({\bf k})\end{matrix}\right),$

with $h_{z}(k_{z})=(S-d)k_{z}$ and $\mathfrak{g}_{z}(k_{z})=(S+d)k_{z}$, while
$h({\bf k})=\sqrt{h_{z}^{2}(k_{z})+4\Delta_{1}^{2}}$ and $\mathfrak{g}({\bf
k})=\sqrt{\mathfrak{g}_{z}^{2}(k_{z})+ak_{\perp}^{2}}$.

##  III Topological Properties

###  III.1 Berry curvature $-$ $\mathbf{\Omega}({\bf k})$

In this section we will demonstrate how a nonzero Berry curvature can be
induced at a Weyl node free valence band in second order perturbation theory
due to its coupling to Weyl node containing valence bands. The Berry curvature
of the $n-$th band can be calculated from the expression XiaoRMP2010

$\Omega_{n}^{\alpha}({\bf k})=i\sum_{n\neq
n^{\prime}}\sum_{\mu\nu}\epsilon^{\alpha\mu\nu}\frac{\langle
u_{n}\left|\partial_{\mu}{\cal H}\right|u_{n^{\prime}}\rangle\langle
u_{n^{\prime}}\left|\partial_{\nu}{\cal
H}\right|u_{n}\rangle}{(\varepsilon_{n}-\varepsilon_{n^{\prime}})^{2}},$ (3)

where $n,n^{\prime}=H_{4}^{+},H_{5}^{-},H_{6}^{\pm}$, derivaties are with
respect to $\partial_{\mu}\equiv\partial/\partial k^{\mu}$, and
$\epsilon^{\alpha\mu\nu}$ is the total anti-symmetric tensor with
$\alpha,\mu,\nu=x,y,z$. Using $R=b=0$ we find
$\mathbf{\Omega}_{H_{4}^{+},H_{5}^{-}}({\bf k})=(0,0,0)$, showing that, at the
$H$ or $H^{\prime}$ points, the $H_{4}^{+}$ and $H_{5}^{-}$ bands are
topologically trivial and carry zero Berry curvature. In contrast, the same
calculation for the lowest two bands yields
$\mathbf{\Omega}_{H_{6}^{\pm}}({\bf k})=(0,0,\Omega_{H_{6}^{\pm}}^{z}({\bf
k}))$, with

$\Omega_{H_{6}^{\pm}}^{z}({\bf
k})=\pm\frac{1}{2}a^{2}\frac{\mathfrak{g}_{z}(k_{z})}{\mathfrak{g}^{3}({\bf
k})},$ (4)

which is simply the field of a magnetic monopole at the origin in momentum
space XiaoRMP2010 . This shows that, at the $H$ or $H^{\prime}$ points, the
$H_{6}^{\pm}$ valence bands are indeed topological, a direct consequence of
the Weyl node at ${\bf k}_{H}$ Hirayama2015 .

For deviations from the $H$ point, however, the bands mixing may no longer be
ignored. By considering $b\neq 0$ and $R\neq 0$ the induced curvature at the
$H_{4}^{+}$ band due to the mixing with the $H_{6}^{\pm}$ bands reads
$\mathbf{\Omega}^{(0)}_{H_{4}^{+}}({\bf
k})=(0,0,\Omega_{H_{4}^{+}}^{(z,0)}({\bf k}))$, where

$\Omega_{H_{4}^{+}}^{(z,0)}({\bf
k})=\sum_{\sigma=\pm}\frac{R^{2}+|b|^{2}}{(h({\bf k})-\sigma\mathfrak{g}({\bf
k})+2\Delta_{2})^{2}}\frac{[h_{z}(k_{z})+h({\bf k})]^{2}[\mathfrak{g}^{2}({\bf
k})-\mathfrak{g}^{2}_{z}(k_{z})]-[h^{2}({\bf
k})-h_{z}^{2}(k_{z})][\mathfrak{g}_{z}(k_{z})-\sigma\mathfrak{g}({\bf
k})]^{2}}{2\mathfrak{g}({\bf k})h({\bf k})(h({\bf
k})+h_{z}(k_{z}))(\mathfrak{g}({\bf k})-\sigma\mathfrak{g}_{z}(k_{z}))}.$ (5)

The above result demonstrates that, by considering solely the ${\bf
k}-$independent SOI to the $H_{6,6^{\prime}}$ bands a $z-$component for the
Berry curvature can be induced at the $H_{4}^{+}$ valence band. Here the
superscript $(z,0)$ indicates that the $z-$component was calculated without
considering the ${\bf k}-$SOI.

However, so far, we have omitted all matrix elements associated to the ${\bf
k}-$SOI in ${\cal H}_{3}$ Doi1970 ; Nakao1971  and these are the central new
theoretical ingredient of the present work, since these will induce nonzero
contributions to the other two components of the Berry curvature. Combining
${\cal H}_{1}$ and ${\cal H}_{3}$ gives rise to the following perturbation
Hamiltonian

$\small\begin{bmatrix}0&0&(G-iu)k_{-}k_{z}&-vk_{z}k_{-}\\\
0&0&-vk_{z}k_{+}&(G+iu)k_{+}k_{z}\\\ (G+iu)k_{+}k_{z}&-v^{*}k_{z}k_{-}&0&0\\\
-v^{*}k_{z}k_{+}&(G-iu)k_{-}k_{z}&0&0\end{bmatrix}.$

We use again the general expression for the Berry curvature (3) to calculate
the fully induced curvature onto the $H_{4}^{+}$ band

$\mathbf{\Omega}^{(1)}_{H_{4}^{+}}({\bf k})=(\Omega_{H_{4}^{+}}^{(x,1)}({\bf
k}),\Omega_{H_{4}^{+}}^{(y,1)}({\bf k}),\Omega_{H_{4}^{+}}^{(z,1)}({\bf k})),$
(6)

where now

$\displaystyle\Omega_{H_{4}^{+}}^{(x,1)}({\bf k})$ $\displaystyle=$
$\displaystyle\sum_{\sigma=\pm}\frac{(G^{2}+|u|^{2})k_{x}k_{z}}{(h({\bf
k})-\sigma\mathfrak{g}({\bf k})-2\Delta_{2})^{2}}\frac{[h_{z}({\bf k})+h({\bf
k})]^{2}[\mathfrak{g}_{z}({\bf k})-\sigma\mathfrak{g}({\bf
k})]^{2}-[h^{2}({\bf k})-h_{z}^{2}(k_{z})][\mathfrak{g}^{2}({\bf
k})-\mathfrak{g}_{z}^{2}({\bf k})]}{2\mathfrak{g}({\bf k})h({\bf k})(h({\bf
k})+h_{z}(k_{z}))(\mathfrak{g}({\bf k})+\sigma\mathfrak{g}_{z}(k_{z}))},$
$\displaystyle\Omega_{H_{4}^{+}}^{(y,1)}({\bf k})$ $\displaystyle=$
$\displaystyle\sum_{\sigma=\pm}\frac{(G^{2}+|u|^{2})k_{y}k_{z}}{(h({\bf
k})-\sigma\mathfrak{g}({\bf k})-2\Delta_{2})^{2}}\frac{[h_{z}({\bf k})+h({\bf
k})]^{2}[\mathfrak{g}_{z}({\bf k})-\sigma\mathfrak{g}({\bf
k})]^{2}-[h^{2}({\bf k})-h_{z}^{2}(k_{z})][\mathfrak{g}^{2}({\bf
k})-\mathfrak{g}_{z}^{2}({\bf k})]}{2\mathfrak{g}({\bf k})h({\bf k})(h({\bf
k})+h_{z}(k_{z}))(\mathfrak{g}({\bf k})+\sigma\mathfrak{g}_{z}(k_{z}))},$
$\displaystyle\Omega_{H_{4}^{+}}^{(z,1)}({\bf k})$ $\displaystyle=$
$\displaystyle\sum_{\sigma=\pm}\frac{(G^{2}+|u|^{2})k_{z}^{2}}{(h({\bf
k})-\sigma\mathfrak{g}({\bf k})-2\Delta_{2})^{2}}\frac{[h_{z}({\bf k})+h({\bf
k})]^{2}[\mathfrak{g}_{z}({\bf k})-\sigma\mathfrak{g}({\bf
k})]^{2}-[h^{2}({\bf k})-h_{z}^{2}(k_{z})][\mathfrak{g}^{2}({\bf
k})-\mathfrak{g}_{z}^{2}({\bf k})]}{2\mathfrak{g}({\bf k})h({\bf k})(h({\bf
k})+h_{z}(k_{z}))(\mathfrak{g}({\bf k})+\sigma\mathfrak{g}_{z}(k_{z}))}.$ (7)

The above result finally demonstrates that, by including also the ${\bf
k}-$dependent SOI to the $H_{6,6^{\prime}}$ bands a full $3D$, radial Berry
curvature is induced at the $H_{4}^{+}$ valence band. Here the superscript
$(i,1)$ for all $i=x,y,z$ indicates that those components were calculated
including the ${\bf k}-$SOI.

###  III.2 Orbital texture $-$ ${\bf m}_{orb}({\bf k})$

Knowledge of the original eigenstates of the $4-$band ${\bf k}\cdot{\bf
p}-$Hamiltonian, obtained in eq. (II.2) allows us to calculate the induced
orbital moment at the Weyl node free $H_{4}^{+}$ valence band due to its
mixing to the two Weyl node containing $H_{6,6^{\prime}}$ valence bands. For
that we shall use ChangNiu

$m_{orb}^{\alpha}({\bf k})=\frac{ie}{2\hbar}\sum_{n\neq
n^{\prime}}\sum_{\mu\nu}\epsilon^{\alpha\mu\nu}\frac{\langle
u_{n}\left|\partial_{\mu}{\cal H}\right|u_{n^{\prime}}\rangle\langle
u_{n^{\prime}}\left|\partial_{\nu}{\cal
H}\right|u_{n}\rangle}{\varepsilon_{n}-\varepsilon_{n^{\prime}}},$ (8)

from which we can calculate the several contributions to the three components
of the induced orbital texture. In order to simplify our analysis, and without
loss of generality, we shall neglect the spin-splitting of the bands
$H_{6}^{\pm}$ and the dispersion of the $H_{4}^{+}$ band, when compared to
$\Delta_{2}$. Now if we consider both terms $b\neq 0$ and $R\neq 0$, we find a
collinear orbital magnetic moment, ${\bf
m}^{(0)}_{orb}=(0,0,m_{orb}^{(z,0)})$, with

$m_{orb}^{(z,0)}({\bf
k})\approx\frac{e}{\hbar}\frac{R^{2}+|b|^{2}}{2\Delta_{2}}\frac{(S-d)k_{z}}{\sqrt{(S-d)^{2}k_{z}^{2}+4\Delta_{1}^{2}}}.$
(9)

The existence of a nonzero $m_{orb}^{(z,0)}({\bf k})\sim k_{z}$ for the
$H_{4}^{+}$ band in tellurium has been previously obtained and accounts for
both the kinetic magnetoelectric effect observed in trigonal tellurium
Sahin2018  and the current induced spin polarization of holes in tellurium
Shalygin2012 . However, since at this stage $m_{orb}^{(x,0)}({\bf
k})=m_{orb}^{(y,0)}({\bf k})=0$ it does not correspond to a Weyl fermion and
we shall proceed with our analysis.

Figure 2: Left $-$ Dumbbel Fermi surfaces and the hedgehog texture. For
$k_{z}>0$ the Berry curvature is positive (yellow) while for $k_{z}<0$ the
Berry curvature is negative (blue). Right $-$ Helical motion and wave packet
rotation responsible for the hedgehog orbital texture.

Following the same reasoning used while discussing the Berry curvature, at
this point we need to include all matrix elements associated to the ${\bf
k}-$SOI in ${\cal H}_{3}$ Doi1970 ; Nakao1971 . The contributions from ${\cal
H}_{1}$ and ${\cal H}_{3}$ gives rise to a full $3D$ orbital magnetic moment,
${\bf m}^{(1)}_{orb}({\bf k})=(m_{orb}^{(x,1)}({\bf k}),m_{orb}^{(y,1)}({\bf
k}),m_{orb}^{(z,1)}({\bf k}))$, whose components are

$\displaystyle m_{orb}^{(x,1)}({\bf k})$ $\displaystyle\approx$
$\displaystyle\frac{e}{\hbar}\frac{G^{2}+|u|^{2}+|v|^{2}}{2\Delta_{2}}\frac{(S-d)k_{z}^{2}k_{x}}{\sqrt{(S-d)^{2}k_{z}^{2}+4\Delta_{1}^{2}}},$
$\displaystyle m_{orb}^{(y,1)}({\bf k})$ $\displaystyle\approx$
$\displaystyle\frac{e}{\hbar}\frac{G^{2}+|u|^{2}+|v|^{2}}{2\Delta_{2}}\frac{(S-d)k_{z}^{2}k_{y}}{\sqrt{(S-d)^{2}k_{z}^{2}+4\Delta_{1}^{2}}},$
$\displaystyle m_{orb}^{(z,1)}({\bf k})$ $\displaystyle\approx$
$\displaystyle\frac{e}{\hbar}\frac{G^{2}+|u|^{2}+|v|^{2}}{2\Delta_{2}}\frac{(S-d)k_{z}^{2}k_{z}}{\sqrt{(S-d)^{2}k_{z}^{2}+4\Delta_{1}^{2}}}.$
(10)

It is important to emphasize that the 3D hedgehog orbital texture shown in
Fig. 2 is not to be associated to sources or sinks of Berry curvatures at
$k_{z}=\pm k_{0}$, which are the positions of the two maxima of the uppermost
$H_{4}^{+}$ valence band. This would mean Weyl nodes split in momentum,
causing time-reversal symmetry to break down and it would be equivalent to the
case of Weyl nodes in Weyl semimetals. Here, instead, the Weyl nodes are split
up in energy, not in momentum, and the 3D hedgehog texture simply reflects the
existence of Weyl nodes within the valence and conduction bands Hirayama2015
at the same momentum ${\bf k}_{H}$ or ${\bf k}_{H^{\prime}}$, and are
therefore sources and sinks of Berry curvature at the same high-symmetry axis,
either $H-$ or $H^{\prime}-$, in the Brillouin zone.

##  IV Weyl Fermion Physics

In particle physics, a Weyl fermion is a massless, spin $1/2$ particle whose
spin is locked to its momentum, ${\bf s}^{\pm}({\bf p})\parallel\pm\vec{\bf
p}$, defining its chirality: right-handed ($+$) or left-handed ($-$). This can
be easily seen from the Weyl Hamiltonian

$H_{Weyl}=\pm\sigma\cdot{\bf p}.$ (11)

The solution to this Hamiltonian is straightforward. The eigenvalues define
the linearly dispersing Weyl cone, $E_{\pm}({\bf p})=\pm|{\bf p}|$, and the
eigenstates are

$\left|u_{\pm}({\bf p})\right>=\frac{1}{\sqrt{2|{\bf p}|(|{\bf p}|\mp
p_{z})}}\left(\begin{matrix}\pm p_{\perp}\\\ (|{\bf p}|\mp
p_{z})e^{i\phi}\end{matrix}\right),$ (12)

where as usual $p_{\perp}=\sqrt{p_{x}^{2}+p_{y}^{2}}$ and
$\phi=\arctan{(p_{y}/p_{x})}$. The $z-$component of the Berry curvature
associated to this Hamiltonian reads

$b^{z}({\bf p})=\pm\frac{1}{2}\frac{p_{z}}{|{\bf p}|^{3}},$ (13)

representing a magnetic monopole at the origin, ${\bf p}=(0,0,0)$. The spin of
the Weyl fermions can also be calculated from ${\bf s}^{\pm}({\bf
p})\equiv\langle u_{\pm}({\bf p})|\sigma|u_{\pm}({\bf p})\rangle$ and reads

${\bf s}^{\pm}({\bf p})=\pm\frac{1}{2}\frac{{\bf p}}{|{\bf p}|}.$ (14)

This result shows us that indeed ${\bf s}^{\pm}({\bf p})$ is locked to the
direction of the momentum $\pm{\bf p}$ with chirality right-handed ($+$) or
left-handed ($-$) depending on the sign, see Fig. 3.

Figure 3: $a-$ a right-handed Weyl fermion whose orbital angular momentum is
parallel to its wavevector, ${\bf m}^{+}_{orb}({\bf k})\parallel+{\bf k}$;
$b-$ a left-handed Weyl fermion whose orbital angular momentum is antiparallel
to its wavevector, ${\bf m}^{-}_{orb}({\bf k})\parallel-{\bf k}$.

In condensed matter physics, charge carriers in, for example, $p-$type doped
semiconductors are characterized by their nonzero effective masses,
$m_{\perp,\parallel}^{*}\neq 0$, and parabollic dispersing valence bands.
Nevertheless, Weyl fermion physics may arise from the interaction between
these carriers to quasiparticle excitations close to Weyl-cone-like, linearly
dispersing crossings in the band structure, in which case an intrinsic orbital
angular momentum, associated to a wave packet rotation, might become locked to
a propagation wavevector, ${\bf m}^{\pm}_{orb}({\bf k})\parallel\pm{\bf k}$,
see Fig. 3. Comparing eqs. (10) and (14) we conclude that, indeed, the
emergence of Weyl fermion physics in tellurium does occur and is a direct
consequence of the ${\bf k}-$SOI between Weyl node free and Weyl node
containing valence bands. The contribution from the ${\bf k}-$SOI to the
orbital momentum in eq. (10) can be written as

${\bf m}^{\pm}_{orb}({\bf
k})=\pm\frac{e}{\hbar}\left[\frac{G^{2}+|u|^{2}+|v|^{2}}{2\Delta_{2}}\frac{(S-d)k_{z}^{2}}{\sqrt{(S-d)^{2}k_{z}^{2}+4\Delta_{1}^{2}}}\right]{\bf
k},$ (15)

which has precisely the wavevector dependence expected for a Weyl fermion, as
discussed in eq. (14). Here, the $\pm$ signs are associated to the two
possible space groups of tellurium, either $D_{3}^{4}$ for right-handed $(+)$
or $D_{3}^{6}$ for left-handed $(-)$ screw axis Firsov1957 , and the $k_{z}$
dependence in the amplitude of ${\bf m}^{\pm}_{orb}({\bf k})$ is a consequence
of the ${\bf k}-$dependent spin-orbit interaction.

##  V Onsager’s relations in linear response

Ordinary matter, when unperturbed, flows towards an equilibrium state Hertel .
Such equilibrium state depends on the temperature, pressure, as well as
several other external parameters such as: regions of space, containing a
certain concentration of particles, mechanical stress, intensity of applied
electric and magnetic fields, among others. When such parameters vary slowly,
the system can return to its equilibrium state almost instantaneously and such
process is said to be reversible, see for example Fig. 4$-$a. When, instead,
the variation is so fast that the system fails to adapt, it will remain out of
equilibrium and the process is said to be irreversible. For example, when an
external AC electric field varies so rapidly that the charge carriers
contained inside micrometric regions in the sample fail to relax
instantaneously, through some relaxation process such as scattering by
impurities or by the lattice, then retardation will occur, see for example
Fig. 4$-$b. This is most commonly achieved in the presence of dissipation or
strong correlations MorimotoNagaosa2018 . Retardation can be calculated in
linear approximation and several relations for generalized susceptibilities
can be derived, irrespective of the specific type of Hamiltonian, such as
Kramers-Kronig relations, fluctuation-dissipation theorem, second law of
thermodynamics and Onsager’s relations. These relations in turn allow us to
describe several transport, electro-optical, and magneto-optical effects:
including Pockels effect, Faraday effect, Kerr effect, chiral magnetic effect,
anomalous Hall effect Hertel , and if we include also spatial dispersion,
these relations can be readly generalized to include also natural optical
activity, gyrotropic birrefringence, and several other non reciprocal
phenomena.

The response of a system in equilibrium to external perturbations can be
expressed in terms of generalized susceptibilities, $\sigma_{ij}(\omega)$,
which are matrices connecting the response $i$ to the perturbation $j$ Hertel
. In the case of adiabatic processes, when the external parameters vary very
slowly, the system will return to equilibrium. In these static, $\omega=0$,
cases we have

$\sigma_{ij}(0)=\sigma_{ji}(0).$ (16)

This is Onsager’s original relation for the case of generalized, static
susceptibilities, as the one depicted in Fig. 4$-$a. In order to extend such
relation to the case of fast processes, $\omega\neq 0$, one needs the concept
of time reversal Hertel .

###  V.1 Reciprocity and space homogeneity

Let us discuss the behavior of the generalized susceptibilities,
$\sigma_{ij}(\omega)$, for the electric field

${\bf E}(\omega)={\bf E}_{0}\;e^{-i\omega t},$ (17)

describing a homogeneous perturbation in space. This approximation is valid
when atomic dimensions, $a$, are negligible when compared to the wavelength,
$\lambda$, of the electric field, $a\ll\lambda$. The requirement that the
Hamiltonian should be even under time reversal Hertel  implies that, for fast
processes and in the presence of an external magnetic field, one should have

$\sigma_{ij}(\omega,{\bf B})=\sigma_{ji}(\omega,-{\bf B}).$ (18)

We see that the symmetries restrict the components of the generalized
susceptibility matrices and that, in the Ohmic regime of an isotropic,
homogeneous system, they can be written as an expansion in powers of the
applied magnetic field

$\sigma_{ij}(\omega,{\bf
B})=\sigma(\omega)\delta_{ij}+h(\omega)\epsilon_{ijk}B_{k}+\dots,$ (19)

where $\delta_{ij}$ Kronecker’s delta and $\epsilon_{ijk}$ is totally anti-
symmetric Levi-Civita symbol. The first term describes the electrical
conductivity inside the material in the absence of external magnetic fields.
The second term needs to be anti-symmetric in $i,j$ due to the linear coupling
to $B$, so that the Hamiltonian remains even under time reversal symmetry. The
resulting electric current is thus necessarily perpendicular to both the
applied magnetic field and to the applied external current, and this is the
geometry that defines the conventional Hall effect.

Figure 4: a $-$ Reciprocal response: the electric field varies slowly and the
system follows adiabatically the perturbation via several relaxation processes
such as impurity or lattice scattering. b $-$ Nonreciprocal response: the
electric field varies rapidly and the system fails to follow adiabatically the
perturbation, independent of the various relaxation processes available.
Nonreciprocal responses are very common in dissipative or strongly correlated
systems, as well as in the case of gyrotropic media with natural optical
activity, and lead to microscopic charge accumulation and spatial dispersion
MorimotoNagaosa2018 .

###  V.2 Nonreciprocity and spatial dispersion

Up to this point Onsager’s relations have been discussed in the regime where
atomic dimensions are negligible when compared to the wavelength of the
electric field, $a\ll\lambda$, with

${\bf E}({\bf q},\omega)={\bf E}_{0}\;e^{i({\bf q}\cdot{\bf x}-\omega t)}.$
(20)

The homogeneous limit, ${\bf q}=0$, corresponds to the zero-th order in a
$a/\lambda$ expansion, with $\lambda=2\pi/|{\bf q}|$ LandauLifshitz . However,
in the macroscopic theory of electrodynamics the power expansion in
$a/\lambda$ corresponds to an expansion in the displacement vector, ${\bf D}$,
and not only in the electric field, ${\bf E}$, but also in its derivatives
LandauLifshitz

$D_{i}=\epsilon_{ik}E_{k}+\gamma_{ik\ell}\frac{\partial E_{k}}{\partial
x_{\ell}},$ (21)

where $\epsilon_{ik}$ and $\gamma_{ik\ell}$ are functions of the frequency,
$\omega$, and these tensors follow the symmetry principles of the kinetic
coefficients in such a way that LandauLifshitz

$\epsilon_{ik}=\epsilon_{ki},\quad\quad\gamma_{ik\ell}=-\gamma_{ki\ell}.$ (22)

This expansion can be extended to the conductivity tensor and should satisfy
the extended Onsager’s relations Hertel

$\sigma_{ij}({\bf q},\omega,{\bf B})=\sigma_{ji}(-{\bf q},\omega,-{\bf B}).$
(23)

Once again power expansion of the external magnetic field and electromagnetic
wavevector gives us

$\displaystyle\sigma_{ij}({\bf q},\omega,{\bf B})$ $\displaystyle=$
$\displaystyle\sigma^{D}({\bf
q},\omega)\delta_{ij}+\chi_{ijk}(\omega)q_{k}+h(\omega)\epsilon_{ijk}B_{k}$
(24) $\displaystyle+$
$\displaystyle\underbrace{g_{ijk\ell}(\omega)q_{k}B_{\ell}}_{\text{nonreciprocal
gyrotropy}}+\dots,$

where $\sigma^{D}({\bf q},\omega)$ is Drude’s conductivity for an isotropic
system and the tensors $\chi_{ijk}$ and $\epsilon_{ijk}$ are anti-symmetric in
$i,j$. We have seen before that the term $h(\omega)\epsilon_{ijk}B_{k}$
describes the conventional Hall effect while the new term
$\chi_{ijk}(\omega)q_{k}$ describes the phenomenon of natural optical
activity.

Most remarkable is the nonreciprocal gyrotropy contribution appearing in the
second line of eq. (24). Here the tensor $g_{ijk\ell}$ is symmetric in the
first two indices, $i,j$, in such a way that the new contribution
$g_{ijk\ell}(\omega)q_{k}B_{\ell}$ allows for a linear term in the magnetic
field contributing to the conductivity without the necessity of being
simultaneously perpendicular to the applied current and applied magnetic
field. This contribution was already known to give rise to the phenomenon of
the gyrotropic birefringence, the nonreciprocal propagation of waves inside
the material. As we shall soon see, this is precisely the ${\bf q}-$dependent
term that also makes room for a nonreciprocal linear planar Hall effect in
tellurium.

##  VI Transport Properties

###  VI.1 General inhomogeneous Weyl systems

As we have learned from the previous discussion, nonreciprocity allows for the
observation of a range of phenomena that would be otherwise forbidden in the
reciprocal regime. Two of such phenomena are precisely the anomalous and the
linear, antisymmetric planar Hall effects. Both are forbidden by time reversal
symmetry in the reciprocal response, but may arise in the nonreciprocal
response for spatially dispersive or inhomogeneous media. In order to see how
this generally comes about, let us revisit the problem of Weyl fermions in the
presence of inhomogeneities. This is usually done by minimally coupling Weyl
fermions to spatially dispersive background gauge fields
ChiralFermionsEWiMatter , $A({\bf r})$ for the electromagnetic and ${\cal
A}({\bf r})$ for the chiral, and whose ${\bf r}$ dependence is characterized
by a modulation wavevector, ${\bf q}\neq 0$ AHEiQCD . By moving from the
laboratory to a local reference frame, through a local gauge transformation,
one ends up with a Weylq Hamiltonian AHEiQCD

$H_{Weyl_{\bf q}}=\begin{bmatrix}(\tau_{z}/2)\sigma\cdot{\bf
q}&\sigma\cdot{\bf p}\\\ \sigma\cdot{\bf p}&(\tau_{z}/2)\sigma\cdot{\bf
q}\end{bmatrix},$ (25)

where $\tau_{z}$ is a pseudospin Pauli matrix. Once again the spectrum and
eingenstates can be calculated straightforwardly from which the $z-$component
of the Berry curvature reads

$b^{z}({\bf p})=\pm\frac{1}{2}\frac{p_{z}\mp q/2}{|E({\bf p})|^{3}},$ (26)

with $E({\bf p})=\pm\sqrt{p^{2}+q^{2}/4\pm|{\bf p}\cdot{\bf q}|}$ AHEiQCD  and
we have chosen, for simplicity, ${\bf q}=(0,0,q)$.

The above Berry curvature corresponds to magnetic monopoles shifted from the
origin towards ${\bf p}=(0,0,\pm q/2)$, with opposite signs for the two chiral
screw axis, see Fig. 5. This simple and general result demonstrates that
spatial dispersion does indeed produce Weyl nodes that become shifted in
momentum, breaking time-reversal symmetry, and thus allowing for both a ${\bf
q}-$dependent anomalous Hall effect, as well as a ${\bf q}-$dependent chiral
anomaly, which has already been acknowledged to be responsible for the planar
Hall effect PHEWeylSemiMetals .

Figure 5: a $-$ For a homogeneous background, ${\bf q}=0$, the Weyl cones
associated to the two topological charges, $Q=+1$ (right-handed screw axis)
and $Q=-1$ (left-handed screw axis), are located at the ${\bf
k}_{H,H^{\prime}}$ high-symmetry points in the Brillouin zone. Here,
reciprocity forbids any anti-symmetric anomalous or planar Hall responses, as
imposed by time-reversal symmetry. b $-$ For an inhomogeneous background,
${\bf q}\neq 0$, however, the two Weyl cones become shifted in momentum, ${\bf
k}_{\pm}={\bf k}_{H}\pm\frac{{\bf q}}{2}$. Nonreciprocity thus allows for
nonzero, anti-symmetric anomalous and planar Hall effects.

###  VI.2 Spatially dispersive, gyrotropic tellurium

In order to check whether or not the above general argument applies to
tellurium we must do a proper calculation, taking into account the full
$4-$fold valence band structure calculated in ${\bf k}\cdot{\bf
p}-$perturbation theory. Differently from the previous argument, we shall
remain in the laboratory reference frame, in which both the Berry curvature
and hedgehog texture remain are centered at the high-symmetry points, and
introduce the spatial dispersion by extending the traditional, semiclassical
approach of Boltzmann transport equations to the case of nonreciprocal
transport.

The current density, ${\bf j}({\bf r})$, the carrier distribution, $f({\bf
k},{\bf r})$, and the velocity, $\dot{\bf r}$, satisfy ${\bf j}({\bf
r})=-(e/V)\sum_{\bf k}f({\bf k},{\bf r})\dot{\bf r}$. Boltzmann’s equation in
the steady-state, relaxation time, $\tau_{\bf k}$, approximation for $f({\bf
k},{\bf r})$ reads PHEWeylSemiMetals ; Johansson2019 ; XiaoRMP2010 ; Kim2014

$\left(\dot{\bf r}\nabla_{\bf r}+\dot{\bf k}\nabla_{\bf k}\right)f({\bf
k},{\bf r})=-(\tau_{\bf k}^{-1}\pm\tau_{\phi}^{-1})[f({\bf k},{\bf
r})-f_{eq}({\bf k},{\bf r})],$ (27)

where $f_{eq}({\bf k},{\bf r})$ is the equilibrium distribution and
$+\tau_{\phi}^{-1}$ ($-\tau_{\phi}^{-1}$) represents the contribution from
weak-localization (antilocalization) due to multiple impurity collisions
Kim2014 . The equations of motion in the presence of Berry curvature are

$\dot{\bf r}=\frac{1}{\hbar}\nabla_{\bf k}\epsilon_{H_{4}^{+}}({\bf
k})+\hbar\dot{\bf k}\times{\bf\Omega}_{\bf k},\quad\dot{\bf
k}=\frac{e}{\hbar}{\bf E}+\frac{e}{\hbar c}\dot{\bf r}\times{\bf B}.$ (28)

For a homogeneous system and as discussed in detail in the previous section,
eq. (27) can be expanded in powers of ${\bf E}$ and ${\bf B}$ to provide us
with

${\bf j}=-\frac{e}{V}\sum_{\bf k}\eta_{\bf k}^{-1}\left(\tilde{\bf v}_{\bf
k}+\frac{e}{\hbar}{\bf E}\times{\bf\Omega}_{\bf
k}+\frac{e}{\hbar}({\bf\Omega}_{\bf k}\cdot\tilde{\bf v}_{\bf k}){\bf
B}\right)\times\left\\{\tilde{f}_{eq}({\bf
k})+\frac{\partial\tilde{f}_{eq}({\bf k})}{\partial\tilde{\cal E}({\bf
k})}\left[1+\frac{e(\tilde{\bf\Lambda}_{\bf k}\times{\bf B})\cdot\nabla_{\bf
k}}{\hbar}\right](e\tilde{\bf\Lambda}_{\bf k}\cdot{\bf E})\right\\},$ (29)

where $\eta_{\bf k}=1+e{\bf\Omega}\cdot{\bf B}/\hbar$ is the curvature-
modified density of states accounting for Fermi surface volume changes
XiaoRMP2010 , and $\tilde{f}_{eq}({\bf k})$ is the Fermi-Dirac equilibrium
distribution function for $\tilde{\cal E}({\bf k})=\epsilon_{H_{4}^{+}}({\bf
k})-{\bf m}_{orb}({\bf k})\cdot{\bf B}$.

####  VI.2.1 Chiral velocity shift - Berry curvature

The induced Berry curvature, ${\bf\Omega}_{\bf k}$, calculated in eq. (7) will
provide a nontrivial contribution to the mean free path

$\tilde{\bf\Lambda}_{\bf k}=\frac{\eta^{-1}_{\bf k}[\tilde{\bf v}_{\bf
k}+e({\bf\Omega}_{\bf k}\cdot\tilde{\bf v}_{\bf k}){\bf B}/\hbar]}{(\tau_{\bf
k}^{-1}\pm\tau_{\phi}^{-1})}$ (30)

which is given in terms of the relaxation time $\tau_{\bf k}$ and the chiral
velocity shift $e({\bf\Omega}_{\bf k}\cdot\tilde{\bf v}_{\bf k}){\bf
B}/\hbar$. This is the mechanism behind several chiral anomaly related
phenomena, such as the chiral-magnetic and planar Hall effects
PHEWeylSemiMetals .

####  VI.2.2 Group velocity shift - hedgehog texture

The $3D$ hedgehog orbital texture calculated in eq. (10) will provide a
contribution to the group velocity shift

$\tilde{\bf v}_{\bf k}=\frac{1}{\hbar}\nabla_{\bf
k}\left(\epsilon_{H_{4}^{+}}({\bf k})-{\bf m}_{orb}({\bf k})\cdot{\bf
B}\right)={\bf v}_{\bf k}+\delta{\bf v}_{\bf k},$ (31)

for the charge carriers at the valence band. The group velocity shift
$\delta{\bf v}_{\bf k}=-\nabla_{\bf k}({\bf m}_{orb}\cdot{\bf B})$ has been
previously considered in the description of the magnetochiral anisotropy in
$p-$type tellurium Rikken2019 . For ${\bf B}\perp{\bf j}\parallel\hat{\bf z}$
there is an energy splitting arising from the breakdown of the degeneracy
between valence band states with ${\bf k}$ and $-{\bf k}$ Doi1970 ; Nakao1971
, that needs to be enantioselective in order to be symmetry allowed. For this
reason, in ref. Rikken2019  it was proposed, heuristically, that such energy
splitting should be $\Delta{\epsilon}({\bf k})=\chi^{D/L}{\bf k}\cdot{\bf B}$,
with $\chi^{D}=-\chi^{L}$. Our eq. (31), together with eqs. (9) and (10), thus
places such heuristic argument on solid theoretical grounds since the coupling
${\bf m}_{orb}\cdot{\bf B}$ is inherently enantioselective.

###  VI.3 Reciprocal conventional Hall effect

The Lorentz force generates a reciprocal Hall current corresponding to the
totally anti-symmetric, linear in $B$ term, $h(\omega)\epsilon_{ijk}B_{k}$
from eq. (19) for the Onsager’s reciprocal relations

${\bf j}=-\frac{e^{3}}{V\hbar}\sum_{\bf k}\frac{\eta_{\bf k}^{-3}\tilde{\bf
v}_{\bf k}}{(\tau_{\bf
k}^{-1}\pm\tau_{\phi}^{-1})^{2}}\frac{\partial\tilde{f}_{eq}({\bf
k})}{\partial\tilde{\cal E}({\bf k})}(\tilde{\bf v}_{\bf k}\times{\bf
B})\cdot\nabla_{\bf k}(\tilde{\bf v}_{\bf k}\cdot{\bf E}),$ (32)

and a Hall conductivity tensor

$\sigma_{zx}^{h}\sim i\epsilon_{zxy}B_{y}\sim B\cos\theta.$ (33)

Because of time reversal symmetry, however, a reciprocal anomalous Hall
current,

${\bf j}=-\frac{e}{V}\sum_{\bf k}\eta_{\bf k}^{-1}({\bf
E}\times{\bf\Omega}_{\bf k})\tilde{f}_{eq}({\bf k})=0,$ (34)

vanishes. The reason for the vanishing of eq. (34) is very simple. Time
reversal symmetry ensures that the Berry curvature is odd under ${\bf
k}\rightarrow-{\bf k}$

${\bf\Omega}_{-{\bf k}}=-{\bf\Omega}_{{\bf k}}.$ (35)

Since the sum in eq. (34) is for the entire Brillouin zone, including positive
as well as negative values for the Berry curvature, then eq. (34) vanishes
identically. The same holds for the reciprocal PHE Johansson2019 , or simply

${\bf j}=-\frac{e}{V}\sum_{\bf k}\eta_{\bf k}^{-2}\tilde{\bf v}_{\bf
k}({\bf\Omega}_{\bf k}\cdot\tilde{\bf v}_{\bf k})\partial\tilde{f}_{eq}({\bf
k})/\partial\tilde{\cal E}({\bf k})({\bf B}\cdot{\bf E})=0.$ (36)

Once again, not only ${\bf\Omega}_{\bf k}$ is odd under time reversal, but so
is the velocity, $\tilde{\bf v}_{-{\bf k}}=-\tilde{\bf v}_{{\bf k}}$. As a
result, the overall parity of the term $\tilde{\bf v}_{\bf k}({\bf\Omega}_{\bf
k}\cdot\tilde{\bf v}_{\bf k})$ is odd and for that reason eq. (36) also
vanishes identically. Time reversal symmetry forbids both reciprocal AHE and
reciprocal PHE.

It is important to emphasize, at this point, that the absence of a reciprocal,
${\bf q}=0$, AHE occurs only at the linear order in the electric field, ${\bf
E}$, see eq. (34). However, reciprocal, ${\bf q}=0$, anomalous Hall-like
currents do indeed arise as quantum nonlinear effects, ${\bf E}^{2}$, induced
by Berry curvature dipole tensor, $D_{ab}=\int_{\bf k}f_{eq}({\bf
k})\partial_{a}{\bf\Omega}_{b}({\bf k})\neq 0$, in noncentrosymmetric
materials, even in the presence of time reversal symmetry Sodemann2015 . This
remarkable reciprocal, nonlinear Hall effect has a purely quantum origin and
produces an anomalous velocity shift when the system is in a current carrying
state Sodemann2015 .

###  VI.4 Nonreciprocal Response

In what follows, we shall demonstrate that both the anomalous and planar Hall
effects arise in the linear electric field regime, ${\bf E}$, and
nonreciprocal response, ${\bf q}\neq 0$. For gyrotropic media such as
tellurium, two novel, nonreciprocal Hall responses are possible. The
electrical acceleration $\delta\dot{\bf k}$ imposed to the carriers by an AC
current follows the same $e^{i({\bf q}\cdot{\bf r}-\omega t)}$ dependence
DresselGruener , so

$\delta\dot{\bf k}=\frac{e{\bf E}/\hbar}{1-i\omega\tau_{\bf k}+i(\tilde{\bf
v}_{\bf k}\cdot{\bf q})\tau_{\bf k}}.$ (37)

####  VI.4.1 AHE - group velocity shift - hedgehog texture

This leads to an anomalous Hall contribution at zero-th order in the expansion
of the distribution function Zhong2015  originated from the group velocity
shift $\delta{\bf v}_{\bf k}=-\nabla_{\bf k}({\bf m}_{orb}\cdot{\bf B})/\hbar$

${\bf j}=-\frac{e^{2}}{V\hbar}\sum_{\bf k}\eta_{\bf
k}^{-1}\left[\frac{\delta{\bf v}_{\bf k}\cdot{\bf q}}{\omega}\right]\left({\bf
E}\times{\bf\Omega}_{\bf k}\right)\tilde{f}_{eq}({\bf k}),$ (38)

whose linear in $B$ dependence at small fields is stabilized by the wave-
vector components ${q}_{m}=\omega{n}_{m}/c$ for the spatially dispersive,
anisotropic media with refraction index ${\bf n}$. Differently from the
vanishing reciprocal AHE in eq. (34), now the ${\bf q}-$dependent term inside
the brackets in eq. (38) brings a second odd quantity, the group velocity
shift, $\delta{\bf v}_{-{\bf k}}=-\delta{\bf v}_{\bf k}$, to the sum in the
Brillouin zone, ensuring that the nonreciprocal anomalous Hall response does
not vanish in the presence of the Berry curvature, ${\bf\Omega}_{\bf k}$,
which is also odd under ${\bf k}\rightarrow-{\bf k}$.

We shall neglect corrections from the density of states, $\eta_{\bf k}$, and
drop all integral over occupied states of $({\bf\Omega}_{\bf k}\cdot\tilde{\bf
v}_{\bf k}){\bf B}$, because these are known to vanish (absense of an
equilibrium chiral magnetic effects) even in the presence of monopole
singularities Zhong2015 . For gyrotropic, spatially dispersive media Eq. (38)
can be rewritten in terms of the constitutive equations
${j}_{i}=\gamma_{ik\ell}\partial_{\ell}{E}_{k}=-i\epsilon_{ik\ell}g_{\ell
m}({\bf B}){E}_{k}{q}_{m}$ LandauLifshitz , Because the tensor
$\gamma_{ik\ell}$ is antisymmetric, $\gamma_{ik\ell}=-\gamma_{ki\ell}$, one
often makes use of the dual pseudotensor,
$\epsilon_{ikm}g_{m\ell}=\omega\gamma_{ik\ell}/c$, where

$g_{m\ell}({\bf B})=\frac{ie^{2}}{V\hbar\omega}\sum_{\bf k}\eta_{\bf
k}^{-1}f_{eq}({\bf k})\delta{v}^{\ell}_{\bf k}({\bf B}){\Omega}^{m}_{\bf k},$
(39)

is only nonzero for systems lacking an inversion center Zhong2015 .

For ${\bf j}\parallel\hat{\bf z}\perp{\bf B}\not\parallel{\bf
E}\parallel\hat{\bf x}$ as in Fig. 6$-$a the novel, nonreciprocal AHE requires
that both the induced hedgehog orbital texture, $m_{orb}^{y}({\bf k})\neq 0$,
as well as the Berry curvature, ${\Omega}^{y}_{\bf k}\neq 0$, have nonzero $y$
components, which can only be the case due to the presence of a ${\bf
k}-$dependent spin-orbit interaction. In this case, for small $B$ the novel,
nonreciprocal AHE reads

$\tilde{\sigma}^{ah}_{zx}\sim B\cos\theta,$ (40)

and adds up to the reciprocal contribution from the conventional (Lorentz
force) Hall effect given in eq. (33).

Figure 6: a $-$ Hall geometry ${\bf j}\parallel\hat{\bf z}\perp{\bf
B}\not\parallel{\bf E}\parallel\hat{\bf x}$; b $-$ theoretical curves for the
conventional $+$ anomalous and planar Hall responses. Inset: effects of $B$ at
$\theta=0$, top left, and $\theta=\pi/2$, bottom right, to the orbital
texture.

####  VI.4.2 PHE - chiral velocity shift - Berry curvature

Since we are looking for the possibility of a linear in $B$ planar Hall
effect, from start we have intentionally chosen ${\bf j}\perp{\bf B}$ (or
$\varphi=\pi/2$) to remove any possible symmetric $B^{2}\sin(2\varphi)$ PHE
contribution from the Hall response Zhang2020 . In this case, a nonreciprocal,
antisymmetric PHE can also be stabilized in spatially dispersive media by the
chiral velocity shift $\delta{\bf v}_{\bf k}=e({\bf\Omega}_{\bf
k}\cdot\tilde{\bf v}_{\bf k}){\bf B}/\hbar$

${\bf j}=-\frac{e^{3}}{V\hbar}\sum_{\bf k}\frac{\eta_{\bf k}^{-2}\tilde{\bf
v}_{\bf k}}{\tau_{\bf k}^{-1}\pm\tau_{\phi}^{-1}}\left[\frac{\tilde{\bf
v}_{\bf k}\cdot{\bf q}}{\omega}\right]\frac{\partial\tilde{f}_{eq}({\bf
k})}{\partial\tilde{\cal E}({\bf k})}({\bf\Omega}_{\bf k}\cdot\tilde{\bf
v}_{\bf k})({\bf B}\cdot{\bf E}),$ (41)

and provides us with $j_{i}=g_{ik\ell m}B_{\ell}q_{m}E_{k}=D_{im}\delta_{\ell
k}B_{\ell}q_{m}E_{k}$, where the even-rank chiral diffusion pseudotensor
OnsagerNonReciprocal

$D_{im}=\frac{e^{3}}{V\hbar\omega}\sum_{\bf k}\frac{\eta_{\bf
k}^{-2}}{\tau_{\bf
k}^{-1}\pm\tau_{\phi}^{-1}}\frac{\partial\tilde{f}_{eq}({\bf
k})}{\partial\tilde{\cal E}({\bf k})}v^{i}_{\bf k}({\bf\Omega}_{\bf
k}\cdot{\bf v}_{\bf k})v^{m}_{\bf k}.$ (42)

Once again the ${\bf q}-$dependent term in eq. (41) ensures the sum over ${\bf
k}$ in the Brillouin zone not to vanish, because even though ${\bf v}_{\bf k}$
and ${\bf\Omega}_{\bf k}$ individually are odd, the four-product $v^{i}_{\bf
k}({\bf\Omega}_{\bf k}\cdot{\bf v}_{\bf k})v^{m}_{\bf k}$ is even under time
reversal. In this case, for ${\bf j}\parallel\hat{\bf z}\perp{\bf
B}\not\parallel{\bf E}\parallel\hat{\bf x}$ as in Fig. 6$-$a a nonreciprocal
PHE requires that ${\Omega}^{(x,y)}_{\bf k}\neq 0$ and is therefore our second
evidence of a hedgehog texture with

$\tilde{\sigma}^{ph}_{zx}\sim B\sin\theta.$ (43)

Most importantly, the nonreciprocal PHE has a different phase than both the
reciprocal Hall and nonreciprocal AHE. The PHE is distinguishable from the
other two and will leave unambiguous signatures in transport, to be discussed
below.

Figure 7: a $-$ Experimental and b $-$ Theoretical curves for the
antysimmetric AC magnetoconductance as a function of the magnetic field, $B$,
for various angles $0\leq\theta\leq 2\pi$.

##  VII Comparison with Experiments

In order to test our predictions, we performed magnetoconductance (MC)
measurements at room temperature on $p-$type Sn-doped tellurium and extracted
the antisymmetric component $G_{AS}$ from the data. The sample has been grown
by slow cooling from a fine tellurium powder with 0.04% added antimony. We
have measured the response for different orientations of the external field,
which has been rotated within the $xy$ plane. Contacts with platinum wires
were glued with silver ink and prepared in a classical 4-point set-up. As a
measurement device, we use a commercial PPMS Dynacool with a maximum field of
9 T and a minimum temperature of 1.8 K from Quantum Design. The samples have
been installed on the horizontal rotator stage. The current direction is
always along the $z$-axis direction and the Quantum Design ETO-option was used
for AC-transport data acquisition. Naturally, the antisymmetric MC-component
is proportional to the Hall response as in a Hall geometry shown in Fig.
6$-$a.

In 6$-$b, we show our predictions for the conventional, anomalous and planar
Hall responses. For $\theta=0$, the almost linear behavior originates from the
conventional (reciprocal) and anomalous (nonreciprocal) Hall signals (both
indistinguishable) in the regime $\omega_{c}\tau\ll 1$, where
$\omega_{c}=eB/m^{*}c$ is the cyclotron frequency and
$\tau^{-1}=\tau_{0}^{-1}+\tau_{\phi}^{-1}$ is a lower bound for the multiple-
impurity collisions, weak-localization relaxation time. For $\theta=\pi/2$,
however, the non-monotonic behavior corresponds to the planar (nonreciprocal)
Hall effect in the regime $\omega_{c}\tilde{\tau}\sim 1$, where
$\tilde{\tau}\approx\tau_{0}$ is an upper bound for the single-impurity,
$\tau_{\phi}^{-1}\rightarrow 0$, relaxation time Kim2014 . Clearly the
alignment of the orbital magnetic moments, ${\bf m}_{orb}\parallel\hat{\bf
x}$, promoted by an applied field ${\bf B}\parallel\hat{\bf x}$, results in
the phase-coherence among the scattering states with ${\bf k},{\bf
k}^{\prime}$ in the relevant $zx$ Hall geometry of Fig. 6$-$a), suppressing
multiple-impurity scattering and enlarging the relaxation time. Obviously, the
Hall response at planar field orientation should be zero, without any non-
trivial effects. This becomes even clearer, when taking the derivative of
$G_{AS}$ for different external field strength and plot it vs. the applied
field angle, as shown in Fig. 6 b). As the magnetic field rotates from
$0\leq\theta\leq 2\pi$, the antisymmetric contribution to the $zx$ Hall
conductivity, $\sigma_{zx}^{AS}$, shown in Figs. 7 and 8 can be remarkably
well described by

$\sigma_{zx}^{AS}=(\sigma_{zx}^{h}+\tilde{\sigma}_{zx}^{ah})\frac{\omega_{c}\tau\,\cos\theta}{1+(\omega_{c}\tau)^{2}}+\tilde{\sigma}_{zx}^{ph}\frac{\omega_{c}\tilde{\tau}\,\sin\theta}{1+(\omega_{c}\tilde{\tau})^{2}},$
(44)

where the ratios
$\alpha=(\sigma_{zx}^{h}+\tilde{\sigma}_{zx}^{ah})/\tilde{\sigma}_{zx}^{ph}$
and $\beta=\tilde{\tau}/\tau$ are the same for all curves. For
$\tan{\theta}>0$ we find $\sigma_{zx}^{AS}=0$ only at $\omega_{c}=0$, that is,
at zero field, $B=0$. For $\tan{\theta}<0$, however, besides $\omega_{c}=0$,
the conductivity $\sigma_{zx}^{AS}=0$ also at
$\omega_{c}\approx\pm\sqrt{\beta|\tan{\theta}|/\alpha-1}/\beta\tau$ for
$\omega_{c}\tau\ll 1$, see Fig. 8.

It is remarkable that the quantum corrections to conductivity, encoded in the
weak-localization relaxation time $\tau_{\phi}$, $\beta\gg 1$, are accessible
to our AC measurements even at $T=300$K. By looking at the graph, we see that
for high fields the phase is located around zero degrees, proving that within
this field range the perpendicular Hall signal dominates. Unfortunately the
contributions from the conventional Hall and nonreciprocal anomalous Hall
effects are not distinguishable within the sample set we used. In the low
field, on the other hand, the phase is shifted by about $50^{\circ}$, proving
the existence of the planar component in the data. Here the existence of an
anti-symmetric component for the planar Hall current is evident. The field
dependence of the phase shift is not due to any misalignment of the contacts,
that would produce an overall constant phase shift irrespective of the field
intensity. Finally, the two novel, nonreciprocal Hall responses reported in
this work are linear in $B$ for small fields, since for carrier concentrations
of the order $n\sim 10^{16}/cm^{3}$ the Fermi energy calculated from the free
electron model gives us $E_{F}\sim 0.1$meV, which corresponds to a Zeeman
field of $\leavevmode\nobreak\ 10$T.

Figure 8: a $-$ Theoretical and b $-$ Experimental curves for the
antysimmetric AC magnetoconductance as a function of the angle, $\theta$, for
various intensities of the applied magnetic field, $0\leq B\leq 9$T.

##  VIII Conclusions

In conclusion, our work unveils the true origin of the Weyl physics observed
in gyrotropic, $p-$type tellurium by demonstrating how a $3D$ hedgehog chiral
magnetic texture is induced at the topologically trivial top valence bands by
their $k-$SOI to two Weyl node containing bands. Therewith, our findings not
only clarify the origin of previous experimental observations of Weyl
signatures in tellurium Zhang2020 , even though Weyls nodes are not directly
accessible Hirayama2015 , but also bring into light two novel, nonreciprocal
and antisymmetric Hall responses allowed in spatially dispersive media with
natural optical activity, and accessed via AC measurements. Finally, our work
opens up new venues for scientific and technological research by including
nonreciprocity into the holography dictionary, as a candidate route for
generating axial components to dual gauge fields Gauge-Gravity-Holography ,
lays the foundations for tellurium based, nonreciprocal topological FETs
TelluriumFET , and also introduces alternative controlling mechanisms for AC
electron transfer in enantioselective molecules EnantioSelectivityChemistry .

##  IX Acknowledgements

The authors acknowledge the financial support from CAPES and CNPq. EBS
acknowledges CNPq and several grants from FAPERJ, including Professor Emeritus
fellowship. JF acknowledges FAPERJ for his PDR 10.

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