# Non-Linear Hall Effect in Multi-Weyl Semimetals

Saswata Roy  Undergraduate Programme, Indian Institute of Science, Bangalore
560012, India  Awadhesh Narayan<EMAIL_ADDRESS>Solid State and
Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012,
India

###### Abstract

In the presence of time reversal symmetry, a non-linear Hall effect can occur
in systems without an inversion symmetry. One of the prominent candidates for
detection of such Hall signals are Weyl semimetals. In this article, we
investigate the Berry curvature induced second and third order Hall effect in
multi-Weyl semimetals with topological charges $n=1,2,3$. We use low energy
effective models to obtain general analytical expressions and discover the
presence of a large Berry curvature dipole in multi-Weyl semimetals. We also
study the Berry curvature dipole in a realistic tight-binding lattice model
and observe two different kinds of variation with increasing topological
charge – these can be attributed to different underlying Berry curvature
components. We provide estimates of the signatures of second harmonic of Hall
signal in multi-Weyl semimetals, which can be detected experimentally.
Furthermore, we predict the existence of a third order Hall signal in multi-
Weyl semimetals. We derive the analytical expressions of Berry connection
polarizability tensor, which is responsible for third order effects, using a
low energy model and estimate the measurable conductivity. Our work can help
guide experimental discovery of Berry curvature multipole physics in multi-
Weyl semimetals.

##  I Introduction

A geometric phase may be acquired by a classical or quantum system when
undergoing cyclic adiabatic processes. Independently discovered by Kato Kato
(1950), Pancharatnam Pancharatnam (1956), and Longuet-Higgins et al. Longuet-
Higgins  _et al._ (1958), in different settings, the notion was organized in a
general framework by Berry Berry (1984). Berry’s phase plays a key role in
various condensed matter phenomena such as quantum Hall effect, spin hall
effects, electric polarization and orbital magnetism Xiao  _et al._ (2010).
Berry curvature, which originates from the Berry flux, acts as the magnetic
field in the momentum space and is intimately connected to the celebrated
quantum Hall effect Thouless  _et al._ (1982).

Recently, Sodemann and Fu, building on earlier work Deyo  _et al._ (2009);
Moore and Orenstein (2010), have discovered the role of the _first order
moment_ of the Berry curvature – termed Berry curvature dipole (BCD) – in
transport properties of quantum systems Sodemann and Fu (2015). Surprisingly,
BCD can lead to a non-linear Hall effect in time-reversal invariant systems
Sodemann and Fu (2015). The conventional Hall effect can be thought of as
connected to the zeroth order moment of the Berry curvature while the first
order moment leads to the second harmonic generation in the Hall signal Du
_et al._ (2021a); Ortix (2021). A growing number of generalizations to other
non-linear phenomena have also been proposed Hamamoto  _et al._ (2017); Araki
(2018); König  _et al._ (2019); Papaj and Fu (2019); Yu  _et al._ (2019); Zeng
_et al._ (2019); Mandal  _et al._ (2020); Nakai and Nagaosa (2019); Zeng  _et
al._ (2020); Das  _et al._ (2021); Nandy and Sodemann (2019); Du  _et al._
(2019, 2021b); Xiao  _et al._ (2019); Resta (2021). Similarly, higher order
moments of the Berry curvature are predicted to produce higher harmonics of
the Hall signal Zhang  _et al._ (2020). To completely describe the third order
conductivity tensor, another intrinsic band geometric quantity known as Berry
connection polarizability (BCP) tensor has been proposed Liu  _et al._ (2021).

BCDs and the related non-linear Hall effect have been predicted for a spectrum
of materials ranging from two-dimensional materials Low  _et al._ (2015); You
_et al._ (2018); Zhang  _et al._ (2018a); Joseph and Narayan (2021); Samal
_et al._ (2021), tilted massive Dirac and Weyl cone systems Du  _et al._
(2018); Xiao  _et al._ (2020); Singh  _et al._ (2020), Weyl semimetals Zhang
_et al._ (2018b), large Rashba systems Facio  _et al._ (2018), and strained
monolayer and bilayer graphene Battilomo  _et al._ (2019), to name just a few.
In parallel to theoretical predictions, there have been noteworthy
developments along the experimental front. Second order non-linear Hall effect
was experimentally first observed in few layer transition metal dichalcogenide
WTe2 Ma  _et al._ (2019); Kang  _et al._ (2019). Following these pioneering
reports, signatures of non-linear Hall signals have been detected in a growing
number of materials, including Dirac semimetals Shvetsov  _et al._ (2019),
Kondo materials Dzsaber  _et al._ (2021), artificially corrugated bilayer
graphene Ho  _et al._ (2021), monolayer WSe2 Qin  _et al._ (2021), twisted
WSe2 Huang  _et al._ (2020), MoTe2 Tiwari  _et al._ (2021), organic Dirac
materials Kiswandhi and Osada (2021), topological insulator surfaces He  _et
al._ (2021) and Weyl semimetal TaIrTe4 Kumar  _et al._ (2021). Very recently,
third order non-linear signals have been experimentally detected in MoTe2 Lai
_et al._ (2021).

An important class of materials for the realization of non-linear Hall effect
are Weyl semimetals – topological semimetallic systems exhibiting a Weyl
fermion dispersion in the low energy regime Rao (2016); Yan and Felser (2017);
Armitage  _et al._ (2018). The point where two bands cross is known as the
Weyl point. These points act as a source or sink of Berry curvature in
momentum space and hence are associated with integer charged monopole and
chirality. Weyl points with opposite chirality can merge and annihilate each
other. On the other hand, Weyl nodes with same chirality can merge to form
Weyl points with higher topological charge, which are named multi-Weyl
semimetals. These systems are stabilised only when point group symmetry
protects such a merging Fang  _et al._ (2012), hence there are discrete
allowed values of topological charge, $n$ Huang  _et al._ (2016); Xu  _et al._
(2011). Multi-Weyl semimetals have been predicted to show many interesting
properties such as chiral effects, anomalous transport phenomena and distinct
optical signatures Ahn  _et al._ (2017, 2016); Sinha and Sengupta (2019);
Kulikov  _et al._ (2020); Park  _et al._ (2017); Huang  _et al._ (2017);
Dantas  _et al._ (2020); Lü  _et al._ (2019); Gorbar  _et al._ (2017);
Mukherjee and Carbotte (2018); Sun and Wang (2017); Soto-Garrido  _et al._
(2020); Chowdhury  _et al._ (2021); Wang  _et al._ (2017); Nag  _et al._
(2020).

Motivated by these exciting developments, here we study BCD in multi-Weyl
semimetals using a low energy model, as well as a suitable tight-binding
model. We obtain general analytical expressions (for arbitrary $n$) for the
Berry curvature using the low energy model and discover the presence of a
large BCD in multi-Weyl semimetals. To complement our analytical calculations,
we next turn to a three dimensional tight-binding model and study the
dependence of BCD on different model parameters for different monopole
charges. In general, we find that a higher monopole charge facilitates higher
magnitude of generated second harmonic of Hall signal. Further, we calculate
the BCP tensor components for our low energy model and predict the existence
of third-order Hall conductivity in multi-Weyl semimetals. For the existing
symmetries in our system the third order contribution is less than that of
second order contribution, but can be dominant if second order signal is
suppressed. Our work can guide experimental discovery of Berry curvature
multipole physics in multi-Weyl semimetals and also help to characterize new
classes of multi-Weyl semimetal materials with higher topological charges.

##  II Berry curvature, Berry curvature dipole and Berry connection
polarizability tensor

Here we briefly review the notions of Berry curvature, its dipole (BCD) and
BCP tensor, and their relation to transport properties. In response to an
oscillating electric field,
$\mathbf{E}(t)=\mathrm{Re}[\boldsymbol{\varepsilon}e^{i\omega t}]$, a non-
linear current, $J_{a}=\mathrm{Re}[J_{a}^{(0)}+J_{a}^{(2)}e^{2i\omega t}]$
flows through the material with
$J_{a}^{(0)}=\chi_{abc}^{(0)}\varepsilon_{b}\varepsilon_{c}^{*}$ and
$J_{a}^{(2)}=\chi_{abc}^{(2)}\varepsilon_{b}\varepsilon_{c}$. In systems that
preserve time-reversal symmetry the coefficients are given by Sodemann and Fu
(2015)

$\chi_{abc}^{(0)}=\chi_{abc}^{(2)}=\frac{\epsilon^{acd}D_{bd}e^{3}\tau}{2\hslash^{2}(1+i\omega
t)},$ (1)

where $\epsilon^{acd}$ is the Levi-Civita symbol, $\tau$ is momentum
relaxation time, $\omega$ is the frequency and $-e$ is the charge of electron.
Here $D_{bd}$ is the BCD, which is formulated as

$D_{bd}=-\sum_{i}\int[d\boldsymbol{k}]\frac{\partial\epsilon_{\boldsymbol{k}}^{i}}{\partial
k_{b}}\boldsymbol{\Omega}_{i\boldsymbol{k}}^{d}\frac{\partial
f_{\boldsymbol{k}}}{\partial\epsilon_{\boldsymbol{k}}^{i}}.$ (2)

Here $a,b,c,d\in\\{x,y,z\\}$, $f_{\boldsymbol{k}}$ is the equilibrium Fermi-
Dirac distribution, $\epsilon_{\boldsymbol{k}}^{i}$ is the energy of the
$i$-th energy band and wavevector $\boldsymbol{k}$,
$\Omega_{i\boldsymbol{k}}^{d}$ is the Berry curvature component in direction
$d$ for the $i$-th energy band and wave vector $\boldsymbol{k}$ and
$[d\boldsymbol{k}]=d^{3}\boldsymbol{k}/{(2\pi)^{3}}$. The Berry curvature, in
turn, can be found as Xiao  _et al._ (2010)

$\Omega_{i\boldsymbol{k}}^{a}=-\epsilon^{abc}\text{Im}\sum_{j\neq
i}\frac{\langle i|\frac{\partial H}{\partial k_{b}}|j\rangle\langle
j|\frac{\partial H}{\partial k_{c}}|i\rangle-\langle i|\frac{\partial
H}{\partial k_{c}}|j\rangle\langle j|\frac{\partial H}{\partial
k_{b}}|i\rangle}{(\epsilon_{\boldsymbol{k}}^{i}-\epsilon_{\boldsymbol{k}}^{j})^{2}},$
(3)

where $|i\rangle$ is an eigenstate of the Hamiltonian $H$ that corresponds to
band $i$ with energy $\epsilon_{\boldsymbol{k}}^{i}$ for a wave vector
$\boldsymbol{k}$. Notably, the integral in Equation 2 can survive in time-
reversal symmetric systems, as long as the inversion symmetry is broken – this
enables a finite Berry curvature to obtain a non-zero value of the BCD. We
also define a BCD _density_ Du  _et al._ (2018)

$d_{bd}(\boldsymbol{k})=(\partial_{\boldsymbol{k}}^{b}\epsilon_{\boldsymbol{k}})\Omega_{\boldsymbol{k}}^{d},$
(4)

which is the kernel of the integral in Equation 2. As we will see, this
quantity, $d_{bd}$ gives further insights into the physics of the Berry
curvature dipole.

Going beyond the second order, third order contributions to the Hall effect
can be connected to the BCP tensor, which is an intrinsic band geometric
quantity Liu  _et al._ . It can be expressed in terms of the unperturbed
eigenstates $|i\rangle$ and band energy $\epsilon_{i}$ as Liu  _et al._

$G_{ab}=2\mathrm{Re}\sum_{i\neq
j}\frac{(\mathcal{A}_{a})_{ij}(\mathcal{A}_{b})_{ij}}{\epsilon_{i}-\epsilon_{j}}.$
(5)

where, $\boldsymbol{\mathcal{A}}$ is the Berry connection. The Berry curvature
needs to be corrected to first order in presence of an external electric field

$\tilde{\boldsymbol{\Omega}}(\boldsymbol{k})=\boldsymbol{\Omega}(\boldsymbol{k})+\boldsymbol{\Omega}^{(1)}(\boldsymbol{k}),$
(6)

which is given by

$\boldsymbol{\Omega}^{(1)}=\nabla_{\boldsymbol{k}}\times\boldsymbol{\mathcal{A}}^{(1)}.$
(7)

In turn, the first order correction of Berry connection,
$\boldsymbol{\mathcal{A}}^{(1)}$, is related to the BCP tensor as

$\mathcal{A}_{a}^{(1)}(\boldsymbol{k})=G_{ab}(\boldsymbol{k})E_{b}.$ (8)

Using Equations 6 and 8, we can also define a Berry curvature polarizability
as

$P_{ab}=\frac{\partial\tilde{\Omega}_{a}}{\partial
E_{b}}=\epsilon_{acd}\partial_{c}G_{db}.$ (9)

Using the components of the BCP tensor, we can calculate the third order
conductivity tensor $\chi$ that follows
$j_{a}^{(3)}=\chi_{abcd}E_{b}E_{c}E_{d}$, with Einstein summation convention
assumed. Further, $\chi$ can be broken into two parts – $\chi^{{I}}$ which is
linear in $\tau$ and $\chi^{{II}}$ which is proportional to $\tau^{3}$.

$\chi^{{I}}_{abcd}=\tau\left[2\int[d\boldsymbol{k}](\partial_{a}\partial_{b}G_{cd})f_{\boldsymbol{k}}-\int[d\boldsymbol{k}](\partial_{c}\partial_{d}G_{ab})f_{\boldsymbol{k}}-\frac{1}{2}\int[d\boldsymbol{k}](v_{a}v_{b}G_{cd})f_{\boldsymbol{k}}^{\prime\prime}\right],$
(10a)
$\chi^{{II}}_{abcd}=-\tau^{3}\int[d\boldsymbol{k}]v_{a}\partial_{b}\partial_{c}\partial_{d}f_{\boldsymbol{k}}.$
(10b)

Here $v_{i}=\frac{\partial E}{\partial{k_{i}}}$ and $\chi^{{I}}_{abcd}$ is the
major contributor to the third order non-linear Hall response as it has a
linear dependence on the relaxation time $\tau$.

##  III Results and Discussion

###  III.1 Low-energy model

Figure 1: Band structure, Berry curvature and Berry curvature dipole in multi-
Weyl semimetals. Band structures of the low energy Hamiltonian with tilt for
multi-Weyl semimetals along $k_{y}$ (setting $k_{x}=k_{z}=0$) with the
following choice of parameters (a1) $n=1$, $C_{s}=0$, (a2) $n=1$, $C_{s}=0.5$,
(a3) $n=1$, $C_{s}=1.5$, (b1) $n=2$, $C_{s}=0$, (b2) $n=2$, $C_{s}=0.5$, (b3)
$n=2$, $C_{s}=1.5$, (c1) $n=3$, $C_{s}=0$, (c2) $n=3$, $C_{s}=0.5$, (c3)
$n=3$, $C_{s}=1.5$. Value of the Berry curvature is superimposed on the band
structure and corresponding color scales are shown. The Berry curvature is
concentrated close to the multi-Weyl nodes, with opposite values for
conduction and valence bands. (d1-d3) Berry curvature dipole for the
corresponding tilt ($C_{s}$) values with $n=1$ in red, $n=2$ in green and
$n=3$ in blue. Berry curvature dipole becomes non-zero only when a tilt is
introduced, as expected. Note the increase in dipole with increasing
topological charge. The maximum value of dipole is observed in the vicinity of
the energy where Berry curvature is most concentrated and this maximum value
increases with an increase in $C_{s}$. Other parameters common to all plots
are $v_{z}=1$, $\alpha=2$, $s=-1$, and $Q=0.5$.

We begin with a low energy model Hamiltonian Dantas  _et al._ (2018); Menon
and Basu (2020) that describes the multi-Weyl semimetals effectively. Setting
$\hbar=1$, the Hamiltonian reads

$H=C_{s}(k_{y}-sQ)\mathbb{I}+s\alpha\boldsymbol{\sigma}.\boldsymbol{n_{k}},$
(11)

where $\boldsymbol{n_{k}}$ =
[$k_{\perp}^{n}\cos(n\phi),k_{\perp}^{n}\sin(n\phi),v_{z}(k_{z}-sQ)/\alpha$],
$k_{\perp}=\sqrt{k_{x}^{2}+k_{y}^{2}}$,
$\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the triad of Pauli
matrices, $s=\pm 1$ characterizes the chirality of the Weyl point,
$\mathbb{I}$ is the identity matrix, $C_{s}$ is the tilt parameter, and
$\phi=\tan^{-1}{(k_{y}/k_{x})}$. Here $\alpha$ generalizes the Fermi velocity
in $k_{x}$-$k_{y}$ plane and is a dimensionless quantity (in our chosen
units). The multi-Weyl nodes are separated by $2Q$ along the $k_{z}$
direction. $v_{z}$ acts as the Fermi velocity. Finally, $n$ is the monopole
charge associated with the node. The dispersion relation for this model is
given by

$E_{\pm}(\boldsymbol{k})=C_{s}(k_{y}-sQ)\pm\sqrt{(k_{z}-sQ)^{2}v_{z}^{2}+\alpha^{2}(k_{x}^{2}+k_{y}^{2})^{n}}.$
(12)

Note the power dependence on $n$ with varying $k_{x}$ and $k_{y}$ – near the
Weyl point the energy bands follow a linear, quadratic and cubic behaviour,
respectively, for $n=$ 1, 2 and 3. The parameter $C_{s}$ controls the tilt
along $k_{y}$ direction. We have consistently used $s=-1$ and $Q=0.5$ in this
article. For these particular parameter values the bands will touch each other
at $(0,0,-0.5)$ point in the momentum space.

We begin our analysis by deriving analytical expressions for Berry curvature
for general topological charge $n$. Using Equations 3, 11, 12 we calculate the
Berry curvature components along the three directions. These read

$\Omega_{\mp}^{x}=\pm\frac{nv_{z}k_{x}\alpha^{2}(k_{x}^{2}+k_{y}^{2})^{n-1}}{2s[(k_{z}-sQ)^{2}v_{z}^{2}+\alpha^{2}(k_{x}^{2}+k_{y}^{2})^{n}]^{\frac{3}{2}}},$
(13a)
$\Omega_{\mp}^{y}=\mp\frac{nv_{z}k_{y}\alpha^{2}(k_{x}^{2}+k_{y}^{2})^{n-1}}{2s[(k_{z}-sQ)^{2}v_{z}^{2}+\alpha^{2}(k_{x}^{2}+k_{y}^{2})^{n}]^{\frac{3}{2}}},$
(13b)
$\Omega_{\mp}^{z}=\pm\frac{\alpha^{2}n^{2}(k_{z}-sQ)sv_{z}(k_{x}^{2}+k_{y}^{2})^{n-1}}{2[(k_{z}-sQ)^{2}v_{z}^{2}+\alpha^{2}(k_{x}^{2}+k_{y}^{2})^{n}]^{\frac{3}{2}}}.$
(13c)

It is worth noting that, $\Omega_{z}$ has $n^{2}$ dependence compared to the
$n$ dependence in $\Omega_{x}$ and $\Omega_{y}$. This plays a key role in
determining relative magnitudes of different BCD components. Also note that
these expressions reduce to the expected ones for $n=1$, i.e., the usual Weyl
semimetal case.

The band dispersions along $k_{y}$ (for $k_{x}=k_{z}=0$), with the value of
the $z$ component of the Berry curvature superimposed on them for various tilt
values are plotted in Fig. 1. We note that the Berry curvature is opposite for
the valence and conduction bands. In case of no tilt ($C_{s}=0$), the
inversion symmetry of the system is not broken and hence in a time-reversal
symmetric system, such as ours, there is no Berry dipole. This can be
attributed to the symmetric distribution of Berry curvature near the band
edges. With the introduction of tilt along the $k_{y}$ direction, the
inversion symmetry is broken and Berry curvature is now asymmetric at the band
edges for all values of topological charge $n$. Hence peaks in the BCD appear,
as shown in Fig. 1(d2)-(d3). We find that the maxima of the BCD is not exactly
at the same energy where the Berry curvature is concentrated as the group
velocity ($\frac{\partial E}{\partial k_{y}}$) vanishes at that point. So, an
optimal energy where the product of group velocity and Berry curvature is
maximum gives the maximum value of the dipole. As we will show, this optimal
energy window can be investigated by analysis of the BCD density, as defined
in Equation 4. Because of the absence of carriers in the band gap, the BCD is
zero for energies within the gap. However, with larger tilts, a finite density
of carriers is present invariably at all energy values for higher order band
crossings and hence a non-zero BCD is found at all energies (except precisely
at the band touching point). This can be seen clearly from Fig.1(d3). However,
for the Weyl semimetal with topological charge $n=1$, there is always a zero
BCD within the band gap. For higher order band crossings, Berry curvature is
concentrated further away from the Brillouin zone centre (i.e., $k_{y}$ = 0)
and the group velocity ($\frac{\partial E}{\partial k_{y}}$) is also higher
near those points owing to the higher order dispersion. As a result, we
discover that the BCD is highest for the case of $n=3$, followed by $n=2$ and
$1$. With further increase in tilt, the bands have an even more asymmetric
distribution of Berry curvature and larger group velocities, which lead to
further increase in the magnitude of BCD. Other than $D_{yz}$, only the
$D_{zy}$ component is non-zero amongst the off-diagonal ones in the BCD
tensor. This component is smaller than the $D_{yz}$ component but follows a
similar distribution where increasing tilt increases the magnitude of BCD.

Figure 2: Berry curvature dipole densities with tilt. Berry curvature dipole
densities plotted in the $k_{x}-k_{y}$ plane (setting $k_{z}=0$) for (a1)
$n=1$, $C_{s}=0$, (a2) $n=2$, $C_{s}=0$, (a3) $n=3$, $C_{s}=0$, (b1) $n=1$,
$C_{s}=1$, (b2) $n=2$, $C_{s}=1$, (b3) $n=3$, $C_{s}=1$. For plots (c1)-(c3)
we set $k_{z}=1$ for (c1) $n=1$, $C_{s}=1$, (c2) $n=2$, $C_{s}=1$, and (c3)
$n=3$, $C_{s}=1$. Color scale for panels (b) and (c) are kept same for a
better comparison. Dashed lines are contours for dipole density values close
to zero. The distribution of dipole densities is anti-symmetric along $k_{y}$
for zero tilt and hence no net Berry curvature dipole is obtained. With the
introduction of tilt this exact anti-symmetry disappears and as a result
finite dipole originates. As we go further away from $k_{z}=0$, the dipole
density declines and spreads out suggesting a lower contribution to Berry
curvature dipole from $k_{x}-k_{y}$ planes further away. Other parameters are
$s=-1$, $Q=0.5$, $v_{z}=1$, $\alpha=2$.

To gain further insights into the nature of the BCD in multi-Weyl semimetals,
we next turn to the BCD density. In Fig. 2, the dipole density, $d_{yz}$, as
defined by Equation 4 are plotted for all three values of $n$. When tilt is
zero the distribution of dipole density is symmetric in a $k_{x}-k_{y}$ plane
for all values of $k_{z}$ and $n$, hence cancels out to give a net zero
contribution to BCD, as we present in Fig. 2(a). As we introduce a finite
tilt, the dipole density distribution becomes asymmetric and gives a
contribution from all $k_{x}-k_{y}$ planes which is seen in Fig. 2(b). We also
note that the positive and negative values of $d_{yz}$ are also different,
thereby resulting in a finite net value. We also find that the extremum points
in the dipole density distribution are close to the multi-Weyl nodes. On the
other hand, the points in the momentum plane which are away from multi-Weyl
points contribute little to the BCD. For $n=3$, dipole density is highest in
magnitude and has a larger momentum space spread as well. As a result, $n=3$
multi-Weyl semimetal produces the highest BCD. In the case of $n=1$, maximum
value of dipole density is higher but it spans a very small momentum region,
when compared to $n=2$ and $n=3$. So, the total BCD value for $n=2$ surpasses
that of $n=1$, which produces the BCD variation that we show in Fig. 1. The
BCD densities in Fig.2(c) are plotted for the same tilt as Fig. 2(b), but for
$k_{z}=1$ which is located further away from the location of the multi-Weyl
points. We observe that the dipole density reduces. This decrease in dipole
density is specially pronounced for $n=1$ and $n=2$, as shown in Fig. 2(c).

It is worth noting here that the dipole density, $d_{yz}$, is an odd function
of $k_{z}$, while $d_{zy}$ is an odd function of $k_{y}$ for each of the bands
for a three-dimensional material. So, the full three-dimensional integral will
lead to net zero off-diagonal components of the BCD for this model, which is
consistent with the recent findings of Ref. Zeng  _et al._ , 2021. However, as
the authors have pointed out, this is not the case for many real materials and
first-principles calculations show a high value of off-diagonal components of
BCD in Weyl semimetal materials Zhang  _et al._ (2018b). Thus, low energy
models are insufficient to provide complete details of BCD in three-
dimensional systems. However, as we have seen, they can help in understanding
the general trends and their origin from two-dimensional subsystems. To get a
complete understanding of BCD in multi-Weyl semimetals, we move on to tight-
binding models in the next section, which can provide a more realistic
description of real materials.

###  III.2 Tight-binding model

Figure 3: Band structure, Berry curvature and Berry curvature dipole for
lattice models of multi-Weyl semimetals. Band structure with the values of
Berry curvature superimposed on it is plotted for the lattice Hamiltonian
along $k_{z}$ direction with $k_{y}=0.1\pi$ and $k_{x}=0.1\pi$ for (a) $n=1$,
(b) $n=2$, (c) $n=3$. Note the largest values of Berry curvature for highest
topological charge $n=3$. Non-zero Berry curvature dipole components (d)
$D_{yz}$ and (e) $D_{zy}$ are plotted for the same parameters. Relative peak
values of Berry curvature dipole for $n=1,2$, $3$ match our predictions from
the low-energy model for the $D_{yz}$ component. However, for the $D_{zy}$
component we observe a different trend because, for this case, the dominant
factor for determining magnitude of Berry dipole is the $y$ component of the
Berry curvature, which follows a different dependence on the topological
charge. Other parameters are chosen to be $t_{C}=1.5$, $t=1$ and $t_{z}=1$.

Motivated by the intriguing behavior of BCD in low energy models of multi-Weyl
semimetals, we next study a lattice model for these systems. The tight-binding
Hamiltonian reads Menon  _et al._ (2021)

$H^{n}=d_{0}^{n}+\boldsymbol{d}^{n}.\boldsymbol{\sigma},$ (14)

where $\boldsymbol{\sigma}$ = ($\sigma_{x},\sigma_{y},\sigma_{z}$) are the
Pauli matrices, $d_{0}^{n}$ and $\boldsymbol{d}^{n}$ are lattice periodic
functions and $n$ is the topological charge of the multi-Weyl semimetal. The
following parameters control the behaviour of the Hamiltonian: $t_{C}$
represents the tilt of the Weyl spectrum, while $t$ and $t_{z}$ are the
hopping strengths. The lattice constant is taken to be unity. The components
of $D^{n}=[d_{0}^{n},\boldsymbol{d}^{n}]$ for $n=1,2,3$ are the following

$D^{1}=[t_{C}(\cos{k_{z}}+\cos{k_{x}}-1),t\sin{k_{x}},t\sin{k_{y}},t_{z}\cos{k_{z}}],$
(15a)
$D^{2}=[t_{C}(\cos{k_{z}}+\cos{k_{x}}-1),t(\cos{k_{x}}-\cos{k_{y}}),2t\sin{k_{x}}\sin{k_{y}},t_{z}\cos{k_{z}}],$
(15b)
$D^{3}=[t_{C}(\cos{k_{z}}+\cos{k_{x}}-1),t\sin{k_{x}}(1-\cos{k_{x}}-3(1-\cos{k_{y}})),-t\sin{k_{y}}(1-\cos{k_{y}}-3(1-\cos{k_{x}})),t_{z}\cos{k_{z}}].$
(15c)

In all the three cases the Weyl points are found at ($0,0,\pm\pi/2$). The
magnitude of the tilt ($t_{C}$) determines whether the system is a type-I or a
type-II Weyl semimetal. The dispersion relations for the different topological
charge $n$ are as follows

$E^{n=1}_{\pm}=t_{C}(\cos{k_{x}}+\cos{k_{z}}-1)\pm[(t^{2}(2-\cos{2k_{x}}-\cos{2k_{y}})+t_{z}^{2}(1+\cos{2k_{z}}))/2]^{1/2},$
(16a) $\displaystyle E^{n=2}_{\pm}=$ $\displaystyle
t_{C}(\cos{k_{x}}+\cos{k_{z}}-1)\pm[(t^{2}(4-\cos{2k_{x}}-\cos{2k_{y}}+\cos{(2(k_{x}+k_{y}))}+\cos{(2(k_{x}-k_{y}))}$
(16b)
$\displaystyle-2(\cos{k_{x}-k_{y}}+\cos(k_{x}+k_{y})))+t_{z}^{2}(\cos{2k_{z}}+1))/2]^{1/2},$
$\displaystyle E^{n=3}_{\pm}=$ $\displaystyle
t_{C}(\cos{k_{x}}+\cos{k_{z}}-1)\pm[t^{2}((\sin{k_{x}}(1-\cos{k_{x}}-3(1-\cos{k_{y}})))^{2}$
(16c)
$\displaystyle+(-t\sin{k_{y}}(1-\cos{k_{y}}-3(1-\cos{k_{x}})))^{2})+(t_{z}\cos{k_{z}})^{2}]^{1/2}.$

For our numerical computations using the above tight-binding models, we employ
the PythTB pyt  code to generate the lattice system. Then we use the Wannier-
Berri package Tsirkin (2021); Destraz  _et al._ (2020) for calculating the
band structures, densities of states, Berry curvature and BCD. For the
calculation of BCD a $100\times 100\times 100$ $k$-grid was used and
convergence was checked.

We begin our investigation by examining the band structures for different
multi-Weyl systems, as shown in Fig. 3(a)-(c) along $k_{z}$ with
$k_{x}=k_{y}=0.1\pi$. The superimposed values of Berry curvature confirm that
the Berry curvature is indeed concentrated near the Weyl points. The $z$
component of the Berry curvature is highest in magnitude for $n=3$ by some
margin, which is the case for our low energy model as well. Further, the Berry
curvature is opposite in sign for valence and conduction bands and also
changes sign at the point at which the bands nearly touch.

The Hamiltonian described above has certain symmetry features that are
important to understand which components of BCD will be non-zero. First of all
it is time-reversal symmetric. Note that in this case anti-Hermitian complex
conjugation is the time reversal operator. Inversion symmetry is broken for
this system. It has mirror symmetries along $k_{y}$ and $k_{z}$. In presence
of these mirror symmetries Sodemann and Fu (2015), only two components of BCD
are expected to be non-zero, $D_{yz}$ and $D_{zy}$, and this is confirmed by
our numerical calculations.

Figure 4: Maximum of Berry curvature dipole with tilt parameter. Variation of
maximum magnitude of Berry curvature dipole component (a) $D_{yz}$ and (b)
$D_{zy}$ with tilt ($t_{C}$) for the lattice models of multi-Weyl semimetals
for $n=1,2$ and $3$. The maximum of Berry curvature dipole variation with tilt
is different for the two components. However, both components start to
saturate after a certain tilt value. This indicates that several factors come
in play in determining the magnitude of Berry curvature dipole in multi-Weyl
semimetals. Other fixed parameters are $t=1$ and $t_{z}=1$.

The two non-zero components for all three topological charges are plotted in
Fig.3(d)-(e). The two components have an opposite relative magnitude of peaks
of Berry dipole for the three topological charges. For $D_{yz}$, highest
magnitude is observed for $n=3$, whereas magnitude is highest for $n=1$ for
$D_{zy}$ component. These differences occur because of the difference in
origin of the BCD in these two cases. For $D_{yz}$ the involved Berry
curvature component is $\Omega_{z}$, which is perpendicular to the
$k_{x}-k_{y}$ planes, in which multi-Weyl dispersion properties are present in
our model. In other words, $\Omega_{z}$ is directly sensitive to the multi-
Weyl nature of the dispersion as well as the higher topological charge.
Besides $\Omega_{z}$, the group velocity plays a major role in determining the
magnitude of Berry dipole here. It is different for the three topological
charges and largest for $n=3$, again owing to the different dispersion. On the
other hand, for $D_{zy}$, the relevant Berry curvature component is
$\Omega_{y}$, which decreases with increase in topological charge. Since the
$\sigma_{x}$ and $\sigma_{y}$ coefficients of Hamiltonian change for the three
topological charges, behaviour of $\Omega_{y}$ is different than $\Omega_{z}$.
This plays the major role in determining the BCD magnitude, as the relevant
group velocity ($v_{z}$) is the same for all three values of $n$. Thus we
uncover an unexpected variation in BCD sign and magnitude with topological
charge, depending on the tensor component. In Fig. 4(a), we plot the variation
of the maximum of $D_{yz}$ with tilt and observe that for topological charge
$n=3$, the dipole is greatest in magnitude for all tilt values. Topological
charges $n=1$ and $n=2$ are close in value to each other. On the other hand,
for $D_{zy}$ component, shown in Fig. 4(b), we find that $n=1$ has the highest
maximum BCD for all tilt values followed by $n=2$ and $n=3$. Our striking
prediction of this anisotropy in the BCD should be directly verifiable in non-
linear Hall measurements, as we will discuss later.

##  IV Third order Hall response

Figure 5: Berry connection polarizability tensor components. Distribution of
Berry curvature polarizability tensor components in the $k_{x}-k_{y}$ plane.
Here $G_{xx}$ is plotted for (a1) $n=1$, (a2) $n=2$, (a3) $n=3$, and $G_{xy}$
is shown for (b1) $n=1$, (b3) $n=2$, (b3) $n=3$ for $s=-1$, $Q=0.5$ and
$k_{z}=0$. These are independent of the tilt of the system. Here $G_{xx}$
behaves like a monopole, while $G_{xy}$ exhibits a quadrupole-like pattern.
Magnitudes of both components are peaked at positions where the Berry
curvature peaks, i.e., where the band gap is minimal.

After studying the second order non-linear Hall response in multi-Weyl
semimetals, a natural question arises: can these systems show a higher order
Hall response? To answer this question, we next investigate the third order
Hall response in these systems. As we discussed before, the third order Hall
response can be understood in the framework of the BCP tensor Liu  _et al._
(2021); Lai  _et al._ (2021). We begin by deriving analytical expressions for
the BCP components, for a general $n$, using our low energy Hamiltonian (see
Equation 11). Using Equation 9, we find that the components of BCP are as
follows

$G_{xx}=-\frac{n^{2}\alpha^{2}(k_{x}^{2}+k_{y}^{2})^{n-2}[k_{x}^{2}k_{z}^{2}v_{z}^{2}+k_{y}^{2}(k_{z}^{2}v_{z}^{2}+(k_{x}^{2}+k_{y}^{2})^{n}\alpha^{2})]}{4[k_{z}^{2}v_{z}^{2}+(k_{x}^{2}+k_{y}^{2})^{n}\alpha^{2}]^{5/2}},$
(17a)
$G_{yy}=-\frac{n^{2}\alpha^{2}(k_{x}^{2}+k_{y}^{2})^{n-2}[k_{y}^{2}k_{z}^{2}v_{z}^{2}+k_{x}^{2}(k_{z}^{2}v_{z}^{2}+(k_{x}^{2}+k_{y}^{2})^{n}\alpha^{2})]}{4[k_{z}^{2}v_{z}^{2}+(k_{x}^{2}+k_{y}^{2})^{n}\alpha^{2}]^{5/2}},$
(17b)
$G_{zz}=-\frac{v_{z}^{2}\alpha^{2}(k_{x}^{2}+k_{y}^{2})^{n}}{4[k_{z}^{2}v_{z}^{2}+(k_{x}^{2}+k_{y}^{2})^{n}\alpha^{2}]^{5/2}},$
(17c)
$G_{xy}=G_{yx}=\frac{n^{2}k_{x}k_{y}\alpha^{4}(k_{x}^{2}+k_{y}^{2})^{2(n-1)})}{4[k_{z}^{2}v_{z}^{2}+(k_{x}^{2}+k_{y}^{2})^{n}\alpha^{2}]^{5/2}},$
(17d)
$G_{xz}=G_{zx}=\frac{k_{x}k_{z}nv_{z}^{2}\alpha^{2}(k_{x}^{2}+k_{y}^{2})^{(n-1)}}{4[k_{z}^{2}v_{z}^{2}+(k_{x}^{2}+k_{y}^{2})^{n}\alpha^{2}]^{5/2}},$
(17e)
$G_{yz}=G_{zy}=\frac{k_{y}k_{z}nv_{z}^{2}\alpha^{2}(k_{x}^{2}+k_{y}^{2})^{(n-1)}}{4[k_{z}^{2}v_{z}^{2}+(k_{x}^{2}+k_{y}^{2})^{n}\alpha^{2}]^{5/2}}.$
(17f)

Note that all nine components of the BCP tensor are, in general, non-zero for
multi-Weyl semimetals and depend on the topological charge $n$. Further, the
off-diagonal components are symmetric, i.e., $G_{xy}=G_{yx}$, $G_{xz}=G_{zx}$,
and $G_{yz}=G_{zy}$. In Fig. 5, we plot the different representative
components of the BCP tensor in the $k_{x}-k_{y}$ plane. We notice several
interesting features of the BCP tensor. The diagonal components $G_{xx}$ and
$G_{yy}$ behave like a monopole for topological charge $n=1$ and are peaked at
$k_{x}=k_{y}=0$, i.e., where the band gap is minimal. For topological charges
$n=2$ and $3$, these components still behave like a monopole, but are not
peaked at the origin, but rather at a position slightly away from it. These
peaks are in the same momentum region where the Berry curvature shows peaks
for the corresponding topological charge, i.e., where the band gaps are
minimal. Strikingly, the off-diagonal component, $G_{xy}$, has a quadrupole-
like structure and shows a similar peak behaviour, where the peaks move
further from $k_{x}=k_{y}=0$ with increase in the topological charge.

Figure 6: Third order conductivity tensor for multi-Weyl semimetals. Angular
dependence of the third order conductivity tensor. The different curves are
for $n=1$ (red), $n=2$ (green) and $n=3$ (blue). (b) Zoomed-in plot of the
curve for $n=1$ in panel (a). Third order conductivity increases substantially
with increasing monopole charge. (c) The variation of the maximum conductivity
with energy. It is near zero at very low and high Fermi energies and peaks at
a moderate Fermi energy similar to the Berry curvature dipole distribution.
Other parameters are chosen to be $C_{s}=0.5$, $v_{z}=1$, $\alpha=2$, $s=-1$,
$Q=0.5$, $E-E_{F}=1.1$, and $k_{z}=0$.

Calculating the full conductivity tensor analytically for a three-dimensional
material is a formidable task with many components of $G$ and $\chi$
contributing to it and leading to cumbersome expressions. Rather, to gain more
insight, we calculate the conductivity tensor for the special case considering
$k_{z}=0$. For simplifying the symmetry discussions, we assume $Q=0$. In this
case, the multi-Weyl points are located on the mirror line $M_{x}$ =
$\sigma_{x}$, such that $M_{x}H(k_{x},k_{y})M_{x}^{-1}=H(-k_{x},k_{y})$. When
the applied electric field is along or perpendicular to this mirror line,
there will not be any third order Hall current generated. This means that
$\chi_{yyyx}$, $\chi_{xxxy}$ and other such permutations with three $y$ and
one $x$ components or three $x$ and one $y$ components will be zero and will
not contribute to the conductivity tensor. On the other hand, an in-plane
electric field, defined as $\boldsymbol{E}=E(\cos\theta,\sin\theta)$ will
produce a third order response
$j_{H}^{(3)}=\boldsymbol{j}^{(3)}\cdot\hat{\boldsymbol{n}}$, where
$\hat{\boldsymbol{n}}=(-\sin\theta,\cos\theta)$ is the normal to
$\boldsymbol{E}$. The third order contribution to the Hall conductivity is
then determined by the conductivity tensor
$\chi_{H}=\frac{j_{H}^{(3)}}{E^{3}}$. For our system, with the prescribed
symmetries we find the third order conductivity tensor as

$\chi_{H}(\theta)=(-\chi_{11}+3\chi_{21})\sin^{3}\theta\cos\theta+(\chi_{22}-3\chi_{12})\sin\theta\cos^{3}\theta,$
(18)

where $\theta$ is measured from the mirror line, and the shorthand notations
stand for $\chi_{11}=\chi_{xxxx}$, $\chi_{22}=\chi_{yyyy}$,
$\chi_{12}=(\chi_{xxyy}+\chi_{xyxy}+\chi_{xyyx})/3$ and
$\chi_{21}=(\chi_{yyxx}+\chi_{yxyx}+\chi_{yxxy})/3$. We have evaluated the
third order conductivity tensor numerically for different multi-Weyl
semimetals and the results are shown in Fig. 6. For the special case of no
tilt, $\chi_{H}$ is zero for all values of $\theta$ due to symmetry reasons
discussed above. For a finite tilt, we observe a non-zero value of the
conductivity tensor and the sign for $n=1$ is opposite to the cases of $n=2$
and $n=3$. Remarkably, we find that the magnitude of $\chi$ is highest for
$n=3$ followed by $n=2$ and $n=1$. This is due to higher group velocity, less
symmetry and sharper features in $G_{ab}$ for higher topological charges. For
all values of topological charge, $\chi_{H}$ varies with the direction of
applied field (i.e., with $\theta$) with a periodicity of $\pi$. It vanishes
when $\theta$ is a multiple of $\pi/2$, which is consistent with our symmetry
analysis. In Fig. 6(c), we have presented the maximum value of the
conductivity tensor with the Fermi energy. It turns out to show a variation
similar to the BCD, where there is a peak at moderate Fermi energy but becomes
nearly zero at higher and lower values of Fermi energy. Further, multi-Weyl
semimetals with $n=3$ have the highest third order response followed by $n=2$
and $n=1$. Therefore, our calculations show that multi-Weyl semimetals with
higher topological charges can be suitable platforms for observing enhanced
higher order Hall response.

##  V Experimental considerations

Here, we estimate the magnitude of second order non-linear Hall response that
can be measured in experiments. First, we need to calculate the conductivity
tensors in Equation 1. In typical experimental setups Ma  _et al._ (2019);
Kang  _et al._ (2019), the relaxation time, $\tau$, is of the order of
picoseconds and the ac frequency can be varied between 10-1000 Hz. So, the
frequency dependence in denominator can be neglected as $\omega\tau\ll 1$.
Using these values, the non zero components of the conductivity tensor are

$\chi_{xyy}=-\chi_{yyx}=0.1846D_{yz},$ (19a)
$\chi_{xzz}=-\chi_{zzx}=-0.1846D_{zy}.$ (19b)

Using the previously mentioned relation,
$J_{a}^{2\omega}=\chi_{abc}E_{b}E_{c}$, we can find the second order Hall
current. For simplicity, we consider three cases where the current is confined
in a cartesian plane, namely $xy$, $yz$ or $zx$. This gives us the following
expressions for the current

$J_{y}^{2\omega}=\chi_{yyx}E_{y}E_{x}\quad\textrm{when}\quad\boldsymbol{E}=(E_{x},E_{y},0),$
(20a)
$J_{x}^{2\omega}=\chi_{xyy}E_{y}^{2}+\chi_{xzz}E_{z}^{2}\quad\textrm{when}\quad\boldsymbol{E}=(0,E_{y},E_{z}),$
(20b)
$J_{z}^{2\omega}=\chi_{zzx}E_{z}E_{x}\quad\textrm{when}\quad\boldsymbol{E}=(E_{x},0,E_{z}).$
(20c)

Consider a typical sample dimension to be $10\times 6\times 2$ $\mu$m3 and the
resistivity to be isotropic in all directions such that
$\rho_{x}=\rho_{y}=\rho_{z}=50$ $\mu\Omega$cm. Let us consider $V_{x}=1$ V and
$V_{y}=1$ V. This choice gives us the current density,
$J_{y}^{2\omega}=-0.1846D_{yz}\frac{V_{y}V_{x}}{L_{x}L_{y}}=-3.077\times
10^{9}D_{yz}$ A/m2. This, in turn, produces a voltage,
$V_{y}^{2\omega}=-1.8462\times 10^{4}\rho_{y}D_{yz}$ V. Therefore, we obtain
the ratio to be $\frac{V_{y}^{2\omega}}{(V_{x}^{\omega})^{2}}=-9.231\times
10^{-3}D_{yz}$ V-1. This ratio is $16.3\times 10^{-6}$ V-1 for $n=1$,
$31.9\times 10^{-6}$ V-1 for $n=2$, and $112.8\times 10^{-6}$ V-1 for $n=3$.
These are well within the reach of current experimental techniques and can
clearly distinguish our proposal of topological charge dependence of the non-
linear Hall signal. It can also be noted that, for the typical parameters we
used, the current $J_{x}^{2\omega}$ (Equation 20b), will have a overall higher
magnitude of response. The above discussion also makes it clear that we can
orient the applied electric fields to measure second harmonics that are
dependent only on one of the BCD components or both the components.

##  VI Summary and outlook

In summary, we studied Berry curvature multipole physics in multi-Weyl
semimetals. Using a low-energy model, we calculated the general expressions
for Berry curvature and its dependence on topological charge, $n$. We
discovered the occurrence of Berry curvature dipole in multi-Weyl semimetals.
Our low-energy model predicted a general trend of increasing BCD magnitude
with topological charge. With inspiration from low-energy model results, we
used a tight-binding lattice model to study the dipole in a more realistic
setting. We found two different variations in the its components with
topological charges. While one component ($D_{yz}$) increases with topological
charge the other one ($D_{zy}$) decreases with it. From studying their
origins, we concluded that the magnitude of the Berry curvature dipole depends
on size of the involved Berry curvature component (which, in turn, is
dependent on the topological charge), the extent to which it is distributed in
the momentum space and the group velocity of the relevant bands. Further, we
analytically obtained the Berry connection polarizability tensor – responsible
for the third harmonic generation of the Hall signal – for a low energy model
and found all its components to be non-zero in multi-Weyl semimetals. We
calculated the third order conductivity and showed that its magnitude
increases with the underlying topological charge. We hope that our work can
motivate exploration of Berry curvature multipole physics in multi-Weyl
semimetals and help in characterizing new classes of multi-Weyl semimetal
materials with higher topological charges.

## Acknowledgments

We acknowledge useful discussions with N. B. Joseph, D. Varghese, A
Bandyopadhyay, H. Liu, S. A. Yang, S. Bhowal and N. A. Spaldin. S. R. thanks
the Kishore Vaigyanik Protsahan Yojana (KVPY) for a fellowship. A. N.
acknowledges support from the startup grant of the Indian Institute of Science
(SG/MHRD-19-0001) and DST-SERB (project number SRG/2020/000153).

## References

  * Kato (1950) T. Kato, Journal of the Physical Society of Japan 5, 435 (1950).
  * Pancharatnam (1956) S. Pancharatnam, in  _Proceedings of the Indian Academy of Sciences-Section A_ , Vol. 44 (Springer, 1956) pp. 398–417.
  * Longuet-Higgins  _et al._ (1958) H. C. Longuet-Higgins, U. Öpik, M. H. L. Pryce, and R. Sack, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 244, 1 (1958).
  * Berry (1984) M. V. Berry, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 392, 45 (1984).
  * Xiao  _et al._ (2010) D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
  * Thouless  _et al._ (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
  * Deyo  _et al._ (2009) E. Deyo, L. Golub, E. Ivchenko, and B. Spivak, arXiv preprint arXiv:0904.1917 (2009).
  * Moore and Orenstein (2010) J. E. Moore and J. Orenstein, Physical review letters 105, 026805 (2010).
  * Sodemann and Fu (2015) I. Sodemann and L. Fu, Phys. Rev. Lett. 115, 216806 (2015).
  * Du  _et al._ (2021a) Z. Du, H.-Z. Lu, and X. Xie, arXiv preprint arXiv:2105.10940 (2021a).
  * Ortix (2021) C. Ortix, arXiv preprint arXiv:2104.06690 (2021).
  * Hamamoto  _et al._ (2017) K. Hamamoto, M. Ezawa, K. W. Kim, T. Morimoto, and N. Nagaosa, Physical Review B 95, 224430 (2017).
  * Araki (2018) Y. Araki, Scientific reports 8, 1 (2018).
  * König  _et al._ (2019) E. König, M. Dzero, A. Levchenko, and D. Pesin, Physical Review B 99, 155404 (2019).
  * Papaj and Fu (2019) M. Papaj and L. Fu, Physical review letters 123, 216802 (2019).
  * Yu  _et al._ (2019) X.-Q. Yu, Z.-G. Zhu, J.-S. You, T. Low, and G. Su, Physical Review B 99, 201410 (2019).
  * Zeng  _et al._ (2019) C. Zeng, S. Nandy, A. Taraphder, and S. Tewari, Physical Review B 100, 245102 (2019).
  * Mandal  _et al._ (2020) D. Mandal, K. Das, and A. Agarwal, Physical Review B 102, 205414 (2020).
  * Nakai and Nagaosa (2019) R. Nakai and N. Nagaosa, Physical Review B 99, 115201 (2019).
  * Zeng  _et al._ (2020) C. Zeng, S. Nandy, and S. Tewari, Physical Review Research 2, 032066 (2020).
  * Das  _et al._ (2021) S. K. Das, T. Nag, and S. Nandy, Phys. Rev. B 104, 115420 (2021).
  * Nandy and Sodemann (2019) S. Nandy and I. Sodemann, Phys. Rev. B 100, 195117 (2019).
  * Du  _et al._ (2019) Z. Du, C. Wang, S. Li, H.-Z. Lu, and X. Xie, Nature communications 10, 1 (2019).
  * Du  _et al._ (2021b) Z. Du, C. Wang, H.-P. Sun, H.-Z. Lu, and X. Xie, Nature communications 12, 1 (2021b).
  * Xiao  _et al._ (2019) C. Xiao, Z. Du, and Q. Niu, Physical Review B 100, 165422 (2019).
  * Resta (2021) R. Resta, arXiv preprint arXiv:2101.10949 (2021).
  * Zhang  _et al._ (2020) C.-P. Zhang, X.-J. Gao, Y.-M. Xie, H. C. Po, and K. T. Law, arXiv preprint arXiv:2012.15628 (2020).
  * Liu  _et al._ (2021) H. Liu, J. Zhao, Y. Huang, X. Feng, X. Cong, W. Wu, S. Lai, W.-b. Gao, and S. A. Yang, arXiv preprint arXiv:2106.04931 (2021).
  * Low  _et al._ (2015) T. Low, Y. Jiang, and F. Guinea, Physical Review B 92, 235447 (2015).
  * You  _et al._ (2018) J.-S. You, S. Fang, S.-Y. Xu, E. Kaxiras, and T. Low, Phys. Rev. B 98, 121109 (2018).
  * Zhang  _et al._ (2018a) Y. Zhang, J. van den Brink, C. Felser, and B. Yan, 2D Materials 5, 044001 (2018a).
  * Joseph and Narayan (2021) N. B. Joseph and A. Narayan, Journal of Physics: Condensed Matter 33, 465001 (2021).
  * Samal  _et al._ (2021) S. S. Samal, S. Nandy, and K. Saha, Physical Review B 103, L201202 (2021).
  * Du  _et al._ (2018) Z. Z. Du, C. M. Wang, H.-Z. Lu, and X. C. Xie, Phys. Rev. Lett. 121, 266601 (2018).
  * Xiao  _et al._ (2020) R.-C. Xiao, D.-F. Shao, W. Huang, and H. Jiang, Physical Review B 102, 024109 (2020).
  * Singh  _et al._ (2020) S. Singh, J. Kim, K. M. Rabe, and D. Vanderbilt, Physical review letters 125, 046402 (2020).
  * Zhang  _et al._ (2018b) Y. Zhang, Y. Sun, and B. Yan, Physical Review B 97, 041101 (2018b).
  * Facio  _et al._ (2018) J. I. Facio, D. Efremov, K. Koepernik, J.-S. You, I. Sodemann, and J. Van den Brink, Physical review letters 121, 246403 (2018).
  * Battilomo  _et al._ (2019) R. Battilomo, N. Scopigno, and C. Ortix, Phys. Rev. Lett. 123, 196403 (2019).
  * Ma  _et al._ (2019) Q. Ma, S.-Y. Xu, H. Shen, D. MacNeill, V. Fatemi, T.-R. Chang, A. M. Mier Valdivia, S. Wu, Z. Du, C.-H. Hsu, S. Fang, Q. D. Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, E. Kaxiras, H.-Z. Lu, H. Lin, L. Fu, N. Gedik, and P. Jarillo-Herrero, Nature 565, 337 (2019).
  * Kang  _et al._ (2019) K. Kang, T. Li, E. Sohn, J. Shan, and K. F. Mak, Nature materials 18, 324 (2019).
  * Shvetsov  _et al._ (2019) O. O. Shvetsov, V. D. Esin, A. V. Timonina, N. N. Kolesnikov, and E. Deviatov, JETP Letters 109, 715 (2019).
  * Dzsaber  _et al._ (2021) S. Dzsaber, X. Yan, M. Taupin, G. Eguchi, A. Prokofiev, T. Shiroka, P. Blaha, O. Rubel, S. E. Grefe, H.-H. Lai, _et al._ , Proceedings of the National Academy of Sciences 118 (2021).
  * Ho  _et al._ (2021) S.-C. Ho, C.-H. Chang, Y.-C. Hsieh, S.-T. Lo, B. Huang, T.-H.-Y. Vu, C. Ortix, and T.-M. Chen, Nature Electronics 4, 116 (2021).
  * Qin  _et al._ (2021) M.-S. Qin, P.-F. Zhu, X.-G. Ye, W.-Z. Xu, Z.-H. Song, J. Liang, K. Liu, and Z.-M. Liao, Chinese Physics Letters 38, 017301 (2021).
  * Huang  _et al._ (2020) M. Huang, Z. Wu, J. Hu, X. Cai, E. Li, L. An, X. Feng, Z. Ye, N. Lin, K. T. Law, _et al._ , arXiv preprint arXiv:2006.05615 (2020).
  * Tiwari  _et al._ (2021) A. Tiwari, F. Chen, S. Zhong, E. Drueke, J. Koo, A. Kaczmarek, C. Xiao, J. Gao, X. Luo, Q. Niu, _et al._ , Nature communications 12, 1 (2021).
  * Kiswandhi and Osada (2021) A. Kiswandhi and T. Osada, arXiv preprint arXiv:2103.00300 (2021).
  * He  _et al._ (2021) P. He, H. Isobe, D. Zhu, C.-H. Hsu, L. Fu, and H. Yang, Nature communications 12, 1 (2021).
  * Kumar  _et al._ (2021) D. Kumar, C.-H. Hsu, R. Sharma, T.-R. Chang, P. Yu, J. Wang, G. Eda, G. Liang, and H. Yang, Nature Nanotechnology 16, 421 (2021).
  * Lai  _et al._ (2021) S. Lai, H. Liu, Z. Zhang, J. Zhao, X. Feng, N. Wang, C. Tang, Y. Liu, K. S. Novoselov, S. A. Yang, and W.-b. Gao, Nature Nanotechnology 16, 869 (2021).
  * Rao (2016) S. Rao, arXiv preprint arXiv:1603.02821 (2016).
  * Yan and Felser (2017) B. Yan and C. Felser, Annual Review of Condensed Matter Physics 8, 337 (2017).
  * Armitage  _et al._ (2018) N. Armitage, E. Mele, and A. Vishwanath, Reviews of Modern Physics 90, 015001 (2018).
  * Fang  _et al._ (2012) C. Fang, M. J. Gilbert, X. Dai, and B. A. Bernevig, Phys. Rev. Lett. 108, 266802 (2012).
  * Huang  _et al._ (2016) S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, T.-R. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane, _et al._ , Proceedings of the National Academy of Sciences 113, 1180 (2016).
  * Xu  _et al._ (2011) G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Physical review letters 107, 186806 (2011).
  * Ahn  _et al._ (2017) S. Ahn, E. Mele, and H. Min, Physical Review B 95, 161112 (2017).
  * Ahn  _et al._ (2016) S. Ahn, E. Hwang, and H. Min, Scientific reports 6, 1 (2016).
  * Sinha and Sengupta (2019) D. Sinha and K. Sengupta, Physical Review B 99, 075153 (2019).
  * Kulikov  _et al._ (2020) K. Kulikov, D. Sinha, Y. M. Shukrinov, and K. Sengupta, Physical Review B 101, 075110 (2020).
  * Park  _et al._ (2017) S. Park, S. Woo, E. Mele, and H. Min, Physical Review B 95, 161113 (2017).
  * Huang  _et al._ (2017) Z.-M. Huang, J. Zhou, and S.-Q. Shen, Physical Review B 96, 085201 (2017).
  * Dantas  _et al._ (2020) R. M. Dantas, F. Peña-Benitez, B. Roy, and P. Surówka, Physical Review Research 2, 013007 (2020).
  * Lü  _et al._ (2019) H.-F. Lü, Y.-H. Deng, S.-S. Ke, Y. Guo, and H.-W. Zhang, Physical Review B 99, 115109 (2019).
  * Gorbar  _et al._ (2017) E. Gorbar, V. Miransky, I. Shovkovy, and P. Sukhachov, Physical Review B 96, 155138 (2017).
  * Mukherjee and Carbotte (2018) S. Mukherjee and J. Carbotte, Physical Review B 97, 045150 (2018).
  * Sun and Wang (2017) Y. Sun and A. Wang, Journal of Physics: Condensed Matter 29, 435306 (2017).
  * Soto-Garrido  _et al._ (2020) R. Soto-Garrido, E. Muñoz, and V. Juričić, Physical Review Research 2, 012043 (2020).
  * Chowdhury  _et al._ (2021) D. Chowdhury, A. Banerjee, and A. Narayan, Physical Review A 103, L051101 (2021).
  * Wang  _et al._ (2017) J.-R. Wang, G.-Z. Liu, and C.-J. Zhang, Physical Review B 96, 165142 (2017).
  * Nag  _et al._ (2020) T. Nag, A. Menon, and B. Basu, Physical Review B 102, 014307 (2020).
  * (73) H. Liu, J. Zhao, Y. Huang, X. Feng, C. Xiao, W. Wu, S. Lai, W. bo Gao, and S. A. Yang,  http://arxiv.org/abs/2106.04931v2 .
  * Dantas  _et al._ (2018) R. M. A. Dantas, F. Peña-Benitez, B. Roy, and P. Surõwka, Journal of High Energy Physics 2018, 69 (2018).
  * Menon and Basu (2020) A. Menon and B. Basu, Journal of Physics: Condensed Matter 33, 045602 (2020).
  * Zeng  _et al._ (2021) C. Zeng, S. Nandy, and S. Tewari, Phys. Rev. B 103, 245119 (2021).
  * Menon  _et al._ (2021) A. Menon, S. Chattopadhay, and B. Basu, Phys. Rev. B 104, 075129 (2021).
  * (78) “Pythtb code,” http://www.physics.rutgers.edu/pythtb/, accessed: 2021-07-31.
  * Tsirkin (2021) S. S. Tsirkin, npj Computational Materials 7, 33 (2021).
  * Destraz  _et al._ (2020) D. Destraz, L. Das, S. S. Tsirkin, Y. Xu, T. Neupert, J. Chang, A. Schilling, A. G. Grushin, J. Kohlbrecher, L. Keller, P. Puphal, E. Pomjakushina, and J. S. White, npj Quantum Materials 5, 5 (2020).