# Quantum oscillations and three-dimensional quantum Hall effect in ZrTe5

Yi-Xiang Wang<EMAIL_ADDRESS>School of Science, Jiangnan
University, Wuxi 214122, China  School of Physics and Electronics, Hunan
University, Changsha 410082, China  Zhigang Cai  School of Science, Jiangnan
University, Wuxi 214122, China

###### Abstract

Recent experiments have reported a lot of spectacular transport properties in
topological materials, such as quantum oscillations and three-dimensional (3D)
quantum Hall effect (QHE) in ZrTe5. In this paper, by using a strong
topological insulator model to describe ZrTe5, we study the magnetotransport
property of the 3D system. With fixed carrier density, we find that there
exists a deferring effect in the chemical potential, which favors
distinguishing the saddle points of the inverted LLs. On the other hand, with
fixed chemical potential, the features of 3D QHE are demonstrated and we
attribute the underlying mechanisms to the interplay between Dirac fermions,
magnetic field and impurity scatterings.

##  I Introduction

In the past two decades, the search of various topological phases in materials
has ignited ongoing interest in condensed matter physics M.Z.Hasan ; X.L.Qi .
Compared to conventional trivial phases, the most exotic property of
topological phases is that they can support the surface states, which show
robustness to the weak disorder or defects. To detect the topological phases
in experiments, one may capture the surface states by tuning the chemical
potential inside the gap B.Q.Lv ; J.A.Sobota . This works well for wide-gap
topological insulators (TIs), such as Bi2Se3 and Bi2Te3, with a gap larger
than $200$ meV Y.Xia ; H.Zhang ; Y.L.Chen ; C.X.Liu . But for a narrow-gap
material, it is not easy to tune the chemical potential inside the gap.

Three-dimensional (3D) ZrTe5 is such a narrow-gap topological material H.Weng
. Recent experiments show that it can exhibit a variety of unique transport
properties, including chiral magnetic effect Q.Li , anomalous Hall effect
T.Liang2018 , saturating thermoelectric Hall effect J.L.Zhang ; W.Zhang , and
gigantic magnetochiral anisotropy Y.Wang . Instead of the surface states, an
alternative route to probe the topological phase in ZrTe5 was proposed through
detecting the saddle points of the inverted bands J.Wang . Under a magnetic
field, electrons are forced to move on quantized curved orbits-the Landau
levels (LLs), which, if inverted, can exhibit two saddle points in the
Brillouin zone (BZ) Y.Jiang ; L.You , the $\Gamma$ point and the $\zeta$
point. When the magnetic field changes, the chemical potential will cross the
LLs one by one, leading to quantum oscillations (QOs) in transport and
thermodynamics D.Shoeberg ; G.M.Monterio ; S.Kaushik , in which the signatures
of the saddle points need further study.

3D quantum Hall effect (QHE) was another intriguing phenomenon observed
recently in ZrTe5 F.Tang ; S.Galeski2020  and HfTe5 P.Wang ; S.Galeski2020 .
The Hall resistivity $\rho_{xy}$ was reported to exhibit a quasi-quantized
plateau in the quantum limit. The quantum limit means that only the zeroth LL
is occupied, which can be reached at a critical magnetic field of about 1.3 T
in ZrTe5 F.Tang  and 1.8 T in HfTe5 P.Wang , both indicating the low carrier
densities in these 3D crystals. More importantly, the plateau scales as
$\rho_{xy}\simeq\frac{\pi}{k_{F}}\frac{h}{e^{2}}$, with $k_{F}$ being the
Fermi wavevector. It was proposed that the enhanced interactions in the
quantum limit can induce the Fermi surface instability, which in turn drives
the charge density wave (CDW) that would open a gap around the Fermi level
F.Tang ; P.Wang . In a 3D system, applying a magnetic field can reduce the
energy spectrum to one dimension (1D). Since the 1D system features an almost
perfect nesting Fermi surface, its ground state favors the CDW states G.Gruner
, thus it seems that the CDW picture may be supported in explaining 3D QHE
F.Qin .

Nevertheless, there exist several facts against such a CDW scenario: (i) if
the Fermi energy lies in the gap, the quasi-quantized Hall plateaus should
correspond to the vanishing longitudinal resistivity $\rho_{xx}$, which,
however, remains finite as observed in ZrTe5 F.Tang ; S.Galeski2020  and HfTe5
P.Wang ; S.Galeski2020 ; (ii) the CDW picture can only account for the plateau
in the quantum limit, but not for the other plateaus in the QO regime F.Tang ;
S.Galeski2020 ; P.Wang ; and (iii) in a recent study S.Galeski2021 , Galeski
et al., corroborated that the CDW states were absent in the ZrTe5 crystal
through various measurements, based on which they suggested that 3D QHE
emerges from the interplay between the intrinsic properties of ZrTe5 and its
Dirac-type semimetal character. Thus, the physical mechanism behind 3D QHE
also requires further study.

Motivated by this progress, in this paper, we use the strong TI model to
describe the low-energy excitations in ZrTe5 and systematically investigate
its bulk magneotransport, with a focus on the problems of QOs and 3D QHE. We
will consider both conditions of fixed carrier density and fixed chemical
potential. The former condition was assumed in understanding the saturating
thermoelectric effect J.L.Zhang , anomalous resistivity at finite temperature
B.Fu ; C.Wang  in ZrTe5, whereas the latter one was used to explain the
optical spectroscopy B.Xu ; E.Martino  and magnetoinfared spectroscopy
measurements R.Y.Chen ; Y.Jiang  in ZrTe5. We will calculate the density of
states (DOS), the oscillating chemical potential or carrier density, as well
as the conductivities and resistivities in the 3D system. In the calculations,
the impurity scatterings are included in the system phenomenologically by
introducing the linewidth broadening to the LLs.

Our results show that under the condition of fixed carrier density, there
exists a deferring effect in the chemical potential, which favors
distinguishing the saddle points of the inverted LLs. At weak impurity
scatterings, the anomalous peak in the quantum limit and the fourfold peaks in
the QO regime can be captured in the DOS as well as the longitudinal
resistivity $\rho_{xx}$, which can help identify the band inversions of the
LLs. Under the condition of fixed chemical potential, the characteristics of
3D QHE are demonstrated, including the quasi-quantized plateau in the Hall
resistivity $\rho_{xy}$ and the nonvanishing dips in $\rho_{xx}$. We attribute
3D QHE to the interplay between Dirac fermions, magnetic field, and impurity
scatterings. Our paper could provide more insights to understand the peculiar
transport properties of ZrTe5 in experiments.

##  II Model

In the study of ZrTe5, a simple two-band model was proposed to calculate the
experimental observables Y.X.Wang2020a ; Z.Rukelj , and also a four-band model
based on the ab initio approach was used to investigate the optical
conductivity in the system C.Morice . Since it is generally believed that the
ground state of 3D ZrTe5 lies in the proximity of the phase boundary between a
strong TI and a weak TI H.Weng , here we adopt the strong TI model to describe
its low-energy excitations. The strong TI phase features the in-plane and out-
of-plane band inversions, and has been supported by many experimental
evidences in ZrTe5 G.Manzoni ; B.Xu ; Y.Jiang ; J.Wang ; Z.G.Chen .

In the four-component basis
$(|+\uparrow\rangle,|-\uparrow\rangle,|+\downarrow\rangle,|-\downarrow\rangle)^{T}$,
the Hamiltonian reads ($\hbar=1$) Y.Jiang ; J.Wang ; H.Zhang ; C.X.Liu

$\displaystyle H(\bm{k})=$ $\displaystyle
v(k_{x}\sigma_{z}\otimes\tau_{x}+k_{y}I\otimes\tau_{y})+v_{z}k_{z}\sigma_{x}\otimes\tau_{x}$
$\displaystyle+[M-\xi(k_{x}^{2}+k_{y}^{2})-\xi_{z}k_{z}^{2}]I\otimes\tau_{z},$
(1)

where $\sigma$ and $\tau$ are the Pauli matrices acting on the spin and orbit
degrees of freedom, respectively. $v$ and $v_{z}$ are the Fermi velocities,
$\xi$ and $\xi_{z}$ are the band inversion parameters, and $M$ denotes the
Dirac mass. In the following calculations, we take the parameters from the
magnetoinfrared measurements Y.Jiang : $(v,v_{z})=(6,0.5)\times 10^{5}$ m/s,
$(\xi,\xi_{z})=(100,200)$ meV nm2, $M=7.5$ meV. With these parameters, besides
the $\Gamma$ point, $H(\bm{k})$ owns the second saddle point$-\zeta$ point,
which is located at
$(k_{\parallel},k_{z})=\big{(}0,(\frac{M}{\xi_{z}}-\frac{v_{z}^{2}}{2\xi_{z}^{2}})^{\frac{1}{2}}\big{)}$,
with $k_{\parallel}=\sqrt{k_{x}^{2}+k_{y}^{2}}$. In addition, $H(\bm{k})$ can
support more topological phases when the parameters are taken in a broad
range, such as single Dirac point phase, Dirac ring phase Y.X.Wang2021 , weak
TI phase W.Wu  and Dirac semimetal phase.

Under a perpendicular magnetic field $\bm{B}=B\bm{e}_{z}$, the quantized LLs
are formed, with dispersions along the $z$ direction. The magnetic field is
included in the system with the help of the Peierls substitution,
$\bm{k}\rightarrow\bm{\pi}-e\bm{A}$, where the vector potential is chosen as
$\bm{A}=(0,-By,0)$ in the Landau gauge. We then introduce the ladder operators
$a^{\dagger}=\frac{l_{B}}{\sqrt{2}}(\pi_{x}-i\pi_{y})$ and
$a=\frac{l_{B}}{\sqrt{2}}(\pi_{x}+i\pi_{y})$, with the magnetic length
$l_{B}=\frac{1}{\sqrt{eB}}=\frac{25.6}{\sqrt{B}}$ nm. By using the trial
wavefunction
$\psi_{n}=(c_{n}^{1}|n\rangle,c_{n}^{2}|n-1\rangle,c_{n}^{3}|n-1\rangle,c_{n}^{4}|n\rangle)^{T}$,
in which the state $|n\rangle$ is defined as
$a^{\dagger}a|n\rangle=n|n\rangle$ and $c_{n}^{1,\cdots,4}$ denote the
coefficients, the energies for the zeroth and $n\geq 1$ LLs are obtained as
L.You

$\displaystyle\varepsilon_{0s}(k_{z})=$ $\displaystyle
s\Big{[}\Big{(}M-\frac{\xi}{l_{B}^{2}}-\xi_{z}k_{z}^{2}\Big{)}^{2}+v_{z}^{2}k_{z}^{2}\Big{]}^{\frac{1}{2}},$
(2) $\displaystyle\varepsilon_{ns\lambda}(k_{z})=$ $\displaystyle
s\Big{[}\Big{(}\sqrt{\Big{(}M-\frac{2n\xi}{l_{B}^{2}}-\xi_{z}k_{z}^{2}\Big{)}^{2}+\frac{2nv^{2}}{l_{B}^{2}}}+\frac{\lambda\xi}{l_{B}^{2}}\Big{)}^{2}$
$\displaystyle+v_{z}^{2}k_{z}^{2}\Big{]}^{\frac{1}{2}},$ (3)

respectively, where the index $s=\pm 1$ denotes the conduction/valence band,
and $\lambda=\pm 1$ the two branches representing the Zeeman splitting between
up spin and down spin L.You . With increasing magnetic field, we see that the
zeroth LL moves to zero energy, while the $n\geq 1$ LLs move away from zero
energy. The latter is because in Eq. (3), the second term
$\frac{2nv^{2}}{l_{B}^{2}}$ under the square root is much larger than the
first squared term.

The band inversions can also exist in LLs. The inverted $\zeta$ points are
located at

$\displaystyle
k_{z0}=\Big{(}\frac{M}{\xi_{z}}-\frac{\xi}{\xi_{z}l_{B}^{2}}-\frac{v_{z}^{2}}{2\xi_{z}^{2}}\Big{)}^{\frac{1}{2}},$
(4) $\displaystyle
k_{zn}\simeq\Big{(}\frac{M}{\xi_{z}}-\frac{2n\xi}{\xi_{z}l_{B}^{2}}-\frac{v_{z}^{2}}{2\xi_{z}^{2}}\Big{)}^{\frac{1}{2}},$
(5)

for the zeroth and $n\geq 1$ LL, respectively. Note that in $k_{zn}$, the weak
$\lambda$ term has been neglected.

At a strong magnetic field, the inverted LLs may become noninverted L.You ,
implying that the $\zeta$ point would be merged with the $\Gamma$ point. For
the zeroth LL, the critical magnetic field is
$B_{0}^{c}=\frac{2M\xi_{z}-v_{z}^{2}}{2e\xi\xi_{z}}=31.5$ T, and for the
$n\geq 1$ LL,
$B_{n}^{c}=\frac{2M\xi_{z}-v_{z}^{2}}{4ne\xi\xi_{z}}=\frac{15.8}{n}$ T. Below
we see that the chosen magnetic fields are much weaker than these critical
values and thus the LLs are always inverted.

##  III Method

###  III.1 Density of states and carrier density

We calculate the DOS of 3D ZrTe5 under a magnetic field. With the help of the
retarded Green’s function $G^{R}(\varepsilon)=(\varepsilon-H+i\eta)^{-1}$
Y.X.Wang2021 , the DOS is defined as

$\displaystyle D(\varepsilon)$ $\displaystyle=-\frac{g}{\pi
L_{z}}\sum_{k_{z}}\text{Tr}[\text{Im}G^{R}(\varepsilon,k_{z})]$
$\displaystyle=\frac{g}{\pi
L_{z}}\sum_{k_{z}}\sum_{ns\lambda}\frac{\eta}{[\varepsilon-\varepsilon_{ns\lambda}(k_{z})]^{2}+\eta^{2}},$
(6)

where $L_{z}$ is system size in the $z$ direction, $g=\frac{1}{2\pi
l_{B}^{2}}$ is the LL degeneracy in the $x$-$y$ plane and can be denoted as
the uniform DOS, and $\eta$ represents the linewidth broadening
phenomenologically that is induced by impurity scatterings. Consider the
short-range pointlike impurities, $U=u_{0}\sum_{i}\delta(\bm{r}-\bm{R}_{i})$,
and the impurity concentration $n_{i}$, then in the Born approximation, the
scattering time $\tau=\frac{1}{2\eta}=\frac{1}{2\pi\gamma D_{F}}$ P.Hosur ;
A.A.Burkov2014 ; J.Klier ; H.W.Wang , with $\gamma=\frac{1}{2}n_{i}u_{0}^{2}$
characterizing the strength of the impurity potential, and $D_{F}$ the DOS at
the Fermi energy. Note that when the band inversions are absent,
$\xi=\xi_{z}=0$, the divergence in the DOS is one over the square-root type
S.Kaushik ; J.Klier ; Y.X.Wang2019 . Now the LL energies show complicated
dependencies on $k_{z}$, leading to the intricated behavior of the DOS, as
shown below.

On the other hand, the carrier density $n_{0}$ is related to the DOS by Ashby

$\displaystyle n_{0}(\mu,B)=\int_{0}^{\infty}d\varepsilon
D(\varepsilon)f(\varepsilon)+\int_{-\infty}^{0}d\varepsilon
D(\varepsilon)[1-f(\varepsilon)],$ (7)

where
$f(\varepsilon)=\big{[}\text{exp}(\frac{\varepsilon-\mu}{k_{B}T})+1\big{]}^{-1}$
is the Fermi distribution function, with $\mu$ being the chemical potential,
$k_{B}$ the Boltzmann constant, and $T$ the temperature. In the following, we
focus on zero temperature.

###  III.2 Conductivity and resistivity

We also calculate the conductivity and resistivity of the system. The
conductivity tensors $\sigma_{\alpha\beta}$ can be derived from the linear-
response Kubo-Streda formula L.Smrcka ; G.D.Mahan ,

$\displaystyle\sigma_{\alpha\beta}=$ $\displaystyle\frac{1}{2\pi
V}\sum_{\bm{k}}\int_{-\infty}^{\infty}d\varepsilon
f(\varepsilon)\Big{[}\text{Tr}\big{(}J_{\alpha}\frac{dG^{R}}{d\varepsilon}J_{\beta}(G^{A}-G^{R})$
$\displaystyle-
J_{\alpha}(G^{A}-G^{R})J_{\beta}\frac{dG^{A}}{d\varepsilon}\big{)}\Big{]},$
(8)

where $V$ is the system volume, $J_{\alpha}=e\frac{\partial H}{\partial
k_{\alpha}}$ is the current density operator, and
$G^{A}(\varepsilon)=(\varepsilon-H-i\eta)^{-1}$ is the advanced Green’s
function. In the transport theory S.Datta , the relaxation time denotes the
time that an electron experiences from the undisturbed state to the
equilibrium state through impurity scatterings, while the inverse LL
broadening is related to the lifetime that an electron stays in the state.
Here we assume that both quantities are equal to the scattering time $\tau$
S.Galeski2021 . Such an assumption has the advantage that the conductivities
can be directly determined from the band structures.

After a straightforward calculation, we obtain the longitudinal conductivity
$\sigma_{xx}$ and transverse Hall conductivity $\sigma_{xy}$, which are
expressed as a summation over the LL index (see Appendix A),

$\displaystyle\sigma_{xx}=$
$\displaystyle\sigma_{0}\frac{4g\eta^{2}}{L_{z}}\sum^{\prime}\frac{M_{ns\lambda;n+1,s^{\prime}\lambda^{\prime}}^{2}}{[(\mu-\varepsilon_{ns\lambda})^{2}+\eta^{2}][(\mu-\varepsilon_{n+1,s^{\prime}\lambda^{\prime}})^{2}+\eta^{2}]},$
(9)

and

$\displaystyle\sigma_{xy}=$
$\displaystyle\sigma_{0}\frac{4g\pi}{L_{z}}\sum^{\prime}\frac{(\varepsilon_{ns\lambda}-\varepsilon_{n+1,s^{\prime}\lambda^{\prime}})^{2}-\eta^{2}}{[(\varepsilon_{ns\lambda}-\varepsilon_{n+1,s^{\prime}\lambda^{\prime}})^{2}+\eta^{2}]^{2}}M_{ns\lambda;n+1,s^{\prime}\lambda^{\prime}}^{2}$
$\displaystyle\times[\theta(\mu-\varepsilon_{ns\lambda})\theta(\varepsilon_{n+1,s^{\prime}\lambda^{\prime}}-\mu)$
$\displaystyle-\theta(\mu-\varepsilon_{n+1,s^{\prime}\lambda^{\prime}})\theta(\varepsilon_{ns\lambda}-\mu)].$
(10)

where $\sigma_{0}=\frac{e^{2}}{h}$ is the unit of the quantum conductivity,
$\theta(x)$ is the step function, the summation sign is

$\displaystyle\sum^{\prime}=\sum_{k_{z}}\sum_{n\geq
0,s\lambda}\sum_{s^{\prime}\lambda^{\prime}},$ (11)

and the matrix element $M_{ns\lambda;n+1,s^{\prime}\lambda^{\prime}}$ is

$\displaystyle
M_{ns\lambda;n+1,s^{\prime}\lambda^{\prime}}=v(c_{ns\lambda}^{1}c_{n+1,s^{\prime}\lambda^{\prime}}^{2}-c_{ns\lambda}^{4}c_{n+1,s^{\prime}\lambda^{\prime}}^{3})$
$\displaystyle+\frac{\sqrt{2(n+1)}\xi}{l_{B}}(-c_{ns\lambda}^{1}c_{n+1,s^{\prime}\lambda^{\prime}}^{1}+c_{ns\lambda}^{4}c_{n+1,s^{\prime}\lambda^{\prime}}^{4})$
$\displaystyle+\frac{\sqrt{2n}\xi}{l_{B}}(c_{ns\lambda}^{2}c_{n+1,s^{\prime}\lambda^{\prime}}^{2}-c_{ns\lambda}^{3}c_{n+1,s^{\prime}\lambda^{\prime}}^{3}).$
(12)

Note that the nonvanishing matrix element $\langle
ns\lambda|J_{x/y}|n^{\prime}s^{\prime}\lambda^{\prime}\rangle$ determines the
common selection rules $n\rightarrow n\pm 1$ Y.Jiang ; L.You .

In the transport experiments, since the measured quantities are the
resistivities $\rho_{\alpha\beta}$, we need to convert conductivities into
resistivities. Using the relation ${\bm{\sigma}}\cdot{\bm{\rho}}=I$, we obtain

$\displaystyle\rho_{xx}=\frac{\sigma_{xx}}{\sigma_{xx}^{2}+\sigma_{xy}^{2}},\quad\rho_{xy}=\frac{\sigma_{xy}}{\sigma_{xx}^{2}+\sigma_{xy}^{2}}.$
(13)

##  IV Main Results and Discussions

###  IV.1 Fixed carrier density

In this section, we study the magnetotransport in ZrTe5 under the condition of
fixed low carrier density, which is set as $n_{0}=5.2\times 10^{16}$ cm-3.

Figure 1: (Color online) (a) The DOS and chemical potential $\mu$ versus the
inverse magnetic field $B^{-1}$ with the fixed carrier density
$n_{0}=5.2\times 10^{16}$ cm-3. When the magnetic field is absent, the red
dotted line indicates the position of the chemical potential $\mu$, and the
inset shows the extremal Fermi surface shape in the $k_{x}$-$k_{y}$ plane.
(b), (c) The LL dispersions versus $k_{z}$, with the magnitude of $B^{-1}$
corresponding to the blue lines $b$ and $c$ labeled in (a), respectively. The
insets are enlarged plots of the dispersions around $\mu$. We set $\eta=0.01$
meV.

The calculated DOS and chemical potential are plotted in Fig. 1(a). We see
that when $B^{-1}<0.58$ T-1, the system lies in the quantum limit, with an
anomalous peak in the DOS. When $B^{-1}>0.58$ T-1, the system lies in the QO
regime. In this regime, the DOS exhibits the fourfold peak structure and the
chemical potential oscillates with descending amplitudes. They both show a
period $\Delta_{1/B}=0.95$ T-1. When $B^{-1}$ is asymptotically large (not
shown here), the system will enter the semiclassical regime H.W.Wang , in
which the energy spacing between two adjacent LLs becomes smaller than the
linewidth broadening $\eta$.

The oscillation period can be understood with the Onsager’s relation L.Onsager
; D.Shoeberg . It tells us that the period is inversely proportional to the
extremal cross-sectional Fermi surface area $S_{e}$ in the plane perpendicular
to the magnetic field:

$\displaystyle\Delta_{1/B}^{O}=\frac{2\pi e}{S_{e}}.$ (14)

When the magnetic field is absent, the chemical potential for the fixed
$n_{0}$ is $\mu_{0}=23.41$ meV (see Appendix B), as indicated by the red
dotted line in Fig. 1(a). The corresponding extremal Fermi surface is plotted
in the inset of Fig. 1(a). The Fermi surface is almost an ellipse with the
area $S_{e}=0.01005$ nm-2. Then we obtain $\Delta_{1/B}^{O}=0.954$ T-1,
agreeing well with the period extracted from Fig. 1(a).

To illustrate the chemical potential oscillations, we show how $\mu$ crosses
the $n=3$ LLs. The LL dispersions are plotted in Figs. 1(b) and 1(c), with the
magnitudes of $B^{-1}$ corresponding to the blue lines $b$ and $c$ in Fig.
1(a), respectively. Actually with increasing $B^{-1}$, $\mu$ is determined by
the competition between the local DOS gaining due to the 1D LL dispersion and
the uniform DOS dropping induced by the magnetic field (i.e., the LL
degeneracy $g$). In Fig. 1(b), when $B^{-1}=2.5$ T-1, we have $\mu=24.39$ meV,
which meets the $\zeta$ point of the $(3+-)$ LL [see Fig. 1(b), inset]. Since
the local DOS gaining surpasses the uniform DOS dropping, to conserve the
carrier density, $\mu$ will decrease and thus $\mu=24.39$ meV behaves as a
peak. In Fig. 1(c), when $B^{-1}=3.078$ T-1, the chemical potential decreases
to $\mu=22.74$ meV, which lies above the $\Gamma$ point of the $(3++)$ LL [see
Fig. 1(c), inset]. With further increasing $B^{-1}$, the local DOS gaining is
inferior to the uniform DOS dropping and thus $\mu=22.74$ meV behaves as a
valley. After that, $\mu$ goes into the next oscillation. Importantly, in the
local region that includes four saddle points of the $n$-th LLs, the
decreasing chemical potential with $B^{-1}$ can defer its crossing over these
saddle points. Such a deferring effect favors distinguishing the saddle points
with weak energy difference.

In the DOS, since the peaks are related to the saddle points of the bands,
here the anomalous peak in the quantum limit is caused by the $\Gamma$ point
of the zeroth LL, and the fourfold peaks in the $n\geq 1$ LLs are attributed
to the fact that the $\lambda=\pm 1$ branches are splitted and each branch
owns two saddle points, the $\Gamma$ point and the $\zeta$ point.

Figure 2: (Color online) The longitudinal conductivity $\sigma_{xx}$ (a) and
Hall conductivity $\sigma_{xy}$ (b) versus the inverse magnetic field $B^{-1}$
for different linewidth $\eta$. In (b), the slope of the line when $\eta=0$ is
extracted as $k=0.0831$ m$\Omega^{-1}$cm-1T. The longitudinal resistivity
$\rho_{xx}$ and Hall resistivity $\rho_{xy}$ vs the magnetic field $B$ in (c)
and (d), with the linewidth $\eta=0.01$ meV and $\eta=5$ meV, respectively. In
(c), the green triangles indicate the fourfold peak structure and the inset
shows the enlarged plot framed by the dotted line. The legends are the same in
(a) and (b), as well as in (c) and (d). The fixed carrier density is
$n_{0}=5.2\times 10^{16}$ cm-3.

We discuss the saddle point sequence that the chemical potential sweeps with
increasing $B^{-1}$. If

$\displaystyle\varepsilon_{n+-}(\Gamma)>\varepsilon_{n++}(\zeta),$ (15)

the saddle point is swept in the sequence $\zeta$-$\zeta$-$\Gamma$-$\Gamma$.
By contrast, if

$\displaystyle\varepsilon_{n+-}(\Gamma)<\varepsilon_{n++}(\zeta),$ (16)

the sequence will be $\zeta$-$\Gamma$-$\zeta$-$\Gamma$. For example, in the
inset of Fig. 1(b), we have
$\varepsilon_{3+-}(\Gamma)>\varepsilon_{3++}(\zeta)$, thus the saddle points
are swept in the former sequence, which is also valid for other LLs. In a
previous work J.Wang , the carrier density of the ZrTe5 crystal is reported to
be above $10^{17}$ cm-3 and thus the quantum limit is reached at the critical
magnetic field $B_{c}>10$ T. Such a strong magnetic field results in the well-
separated $\lambda=\pm 1$ branches and the latter sequence
$\zeta$-$\Gamma$-$\zeta$-$\Gamma$.

If we include the spin Zeeman effect into the system, the Hamiltonian is

$\displaystyle H_{Z}=-\frac{1}{2}g\mu_{B}B\sigma_{z},$ (17)

where $g$ denotes the Landé $g$-factor and $\mu_{B}$ is the Bohr magneton.
Taking the value $g\simeq 10$ that is reported in ZrTe5 R.Y.Chen ; Z.G.Chen ,
the total Zeeman splittings will get strengthened, leading to the further
separation of the two $\lambda=\pm 1$ branches. Thus, for the lower LLs, the
saddle point sequence may become $\zeta$-$\Gamma$-$\zeta$-$\Gamma$, and for
the higher LLs, the sequence remains unchanged. This means that with
increasing LL index $n$, there exists a transition of the saddle point
sequence, which depends heavily on the carrier density and the $g$-factor of
the LLs.

Next we study the conductivities. The results are plotted as a function of the
inverse magnetic field $B^{-1}$ in Figs. 2(a) and 2(b). We analyze the
longitudinal conductivity $\sigma_{xx}$ in Fig. 2(a). In the limiting clean
case $\eta=0$, which corresponds to the infinite scattering time,
$\sigma_{xx}$ vanishes. The impurity-induced scatterings can give rise to the
drift current along the electric field direction, leading to the increasing of
$\sigma_{xx}$ with weak $\eta$. When $\eta=0.01$ meV, $\sigma_{xx}$ exhibits
clear oscillations and fourfold peaks that are similar to the DOS. When
$\eta=0.1$ meV, the fourfold peaks are smoothened and become broad. Further
increasing $\eta$ to be as strong as $\eta\geq 5$ meV, we observe that the
oscillations in $\sigma_{xx}$ are smeared out, implying that the LL structures
are broken. Moreover, in the quantum limit, $\sigma_{xx}$ will get further
increased, while in the QO regime, $\sigma_{xx}$ decreases. Such an effect of
impurity scatterings on $\sigma_{xx}$ is consistent with our self-consistent
Born approximation study of the diagonal disorder in a Weyl semimetal system
Y.X.Wang2020 .

We also analyze the Hall conductivity $\sigma_{xy}$ in Fig. 2(b). When
$\eta=0$, $\sigma_{xy}$ is proportional to the inverse magnetic field
$B^{-1}$, which retrieves the classical Hall conductivity expression V.Konye ,

$\displaystyle\sigma_{xy}=\frac{n_{0}e}{B}.$ (18)

With the extracted slope $k=8.31\times 10^{-2}$ m$\Omega^{-1}$ cm-1 T, the
carrier density is obtained as $n_{0}=\frac{k}{e}=5.19\times 10^{16}$ cm-3,
which is consistent with the chosen carrier density. For the weak impurity
scattering $\eta<0.1$ meV, $\eta$ can be ignored in Eq. (10), thus
$\sigma_{xy}$ shows certain robustness to weak $\eta$. When $\eta$ increases,
the robustness disappears and $\sigma_{xy}$ will get suppressed from the
higher LLs to lower ones Y.X.Wang2019 . At strong $\eta\geq 5$ meV,
$\sigma_{xy}$ even becomes negative at large $B^{-1}$. This is because when
the chemical potential lies between the $n$-th and $(n+1)$-th LLs, the
dominate contributions to $\sigma_{xy}$ come from the LL transition
$(n,1,\lambda)\rightarrow(n+1,1,\lambda)$, which is similar to two-dimensional
(2D) Dirac fermion in graphene V.P.Gusynin . Since the energy spacing between
the higher LLs is smaller, the associated $\sigma_{xy}$ will get suppressed at
first, which will occur successively for the lower LLs. In the case when the
energy spacing is less than the strong $\eta$, $\sigma_{xy}$ becomes negative.

Figure 3: (Color online) The DOS and carrier density $n_{0}$ versus the
inverse magnetic field $B^{-1}$ with the fixed chemical potential $\mu=30$
meV. When the magnetic field is absent, the red dotted line indicates the
position of the carrier density, and the inset shows the extremal Fermi
surface in the $k_{x}$-$k_{y}$ plane. (b), (c) The LL dispersions versus
$k_{z}$, with the magnitude of $B^{-1}$ corresponding to the blue lines $b$
and $c$ labeled in (a), respectively. The insets are the enlarged plots of the
dispersions around $\mu$. We set $\eta=0.01$ meV.

Then we study the resistivity. To make direct comparisons with experiments,
the resistivities are plotted as a function of the magnetic field $B$ in Figs.
2(c) and 2(d). With weak $\eta=0.01$ meV, the conductivities satisfy
$\sigma_{xx}\ll\sigma_{xy}$, and correspondingly, the longitudinal resistivity
and Hall resistivity are given as

$\displaystyle\rho_{xx}=\sigma_{xx}\sigma_{xy}^{-2},\quad\rho_{xy}=\sigma_{xy}^{-1},$
(19)

respectively. Thus in Fig. 2(c), $\rho_{xx}$ exhibits the SdH oscillations on
an increasing background, which keeps the features of $\sigma_{xx}$. The
fourfold peak structure of $\rho_{xx}$ is more clearly seen in the inset of
Fig. 2(c). On the other hand, $\rho_{xy}$ shows a linear dependence on $B$.
The linear $\rho_{xy}$ was derived by Abrikosov for Dirac electrons occupying
the lowest zeroth LL Abrikosov , where the large and nonsaturated
magnetoresistivity was understood with fixed carrier density. In experiment,
such a linear $\rho_{xy}$ was observed in the Weyl semimetal NbP C.Shekhar
and the Dirac semimetal Cd3As2 L.P.He ; T.Liang2015  even at a very low
magnetic field.

At strong $\eta=5$ meV, $\sigma_{xx}$ becomes comparable to $\sigma_{xy}$.
When the chemical potential crosses the $n$th LLs, the fourfold peaks in
$\sigma_{xx}$ merge into the single peak structure. Now in Fig. 2(d), for
$\rho_{xx}$, the SdH oscillations are blurred but the kinks are seen. For
$\rho_{xy}$, it exhibits a ramp in the quantum limit that gradually increases
with $B$. Note that $\rho_{xy}$ shows a plateau in a broad range $0.604$
T$<B<1.186$ T. In fact, such a plateau originates from the plateaus in
$\sigma_{xx}$ and $\sigma_{xy}$ that are induced by impurity scatterings, as
indicated by the arrows in Figs. 2(a) and 2(b). These results strongly suggest
that 3D QHE cannot be explained with fixed carrier density.

###  IV.2 Fixed chemical potential

Figure 4: (Color online) The longitudinal conductivity $\sigma_{xx}$ and Hall
conductivity $\sigma_{xy}$ versus the inverse magnetic field $B^{-1}$ for
different linewidths $\eta$ in (a) and (b). The longitudinal resistivity
$\rho_{xx}$ and Hall resistivity $\rho_{xy}$ versus the magnetic field $B$ in
(c) and (d), with the linewidth $\eta=0.01$ meV and $\eta=1$ meV,
respectively. In (c), the green triangles indicate the fourfold peak structure
and the inset shows enlarged plot framed by the dotted line. The legends are
the same in (a) and (b), as well as in (c) and (d). The fixed chemical
potential is $\mu=30$ meV.

In this section, we study the magnetotransport of ZrTe5 under the condition of
fixed chemical potential, which is set as $\mu=30$ meV.

The calculated DOS and carrier density are plotted in Fig. 3(a). The quantum
limit is reached at the critical magnetic field $B_{c}=1.88$ T. Now as the
$\Gamma$ point of the zeroth LL is always below the chemical potential,

$\displaystyle\varepsilon_{0+}(\Gamma)=M-\frac{\xi}{l_{B}^{2}}<\mu,$ (20)

the anomalous peak that appears in Fig. 1(a) is absent. In the QO regime, we
can see that the fourfold peaks of the DOS are overlapping and thus are
difficult to be distinguished. On the other hand, the carrier density
oscillates with a period $\Delta_{1/B}=0.55$ T, implying that the charging
energies exist in the system. The period can also be understood with the
Onsager’s relation. Without a magnetic field, the extremal Fermi surface in
the $k_{x}$-$k_{y}$ plane is plotted in the inset of Fig. 3(a), with its area
$S_{e}=0.01724$ nm-2. Then, according to Eq. (14), we have
$\Delta_{1/B}^{O}=0.556$ T-1, which shows good consistency.

The competition between the local DOS and the uniform DOS that was used to
analyze the chemical potential oscillations in the previous section can be
extended to illustrate the carrier density oscillations here. It is worth
pointing out that the oscillations in Figs. 1(a) and 3(a) are out of phase.
For example, in the insets of Figs. 1(b) and 3(b), although both the chemical
potentials $\mu$ meet the $\zeta$ point of the $(3+-)$ LL, in Fig. 1(a), $\mu$
behaves as a peak, whereas in Fig. 3(a), the corresponding carrier density
behaves as a valley.

Next we turn to the conductivity and resistivity. Figures 4(a) and 4(b) show
that the effects of the increasing impurity scatterings $\eta$ on the
conductivities $\sigma_{xx}$ and $\sigma_{xy}$ are the same as those in Figs.
2(a) and 2(b), respectively. Figure 4(c) shows that with weak linewidth
$\eta=0.01$ meV, in the longitudinal resistivity $\rho_{xx}$, the SdH
oscillations are evident, but the fourfold peaks can only be distinguished for
the $n=1$ LLs and are overlapping for other LLs, with details seen in the
inset of Fig. 4(c). This means that the saddle points of the inverted LLs
cannot be well distinguished with fixed chemical potential L.You , even when
the spin Zeeman effect is included. In the Hall resistivity $\rho_{xy}$, the
quasi-quantized plateaus are clearly observable.

Actually, a 3D Hall system under a magnetic field can be regarded as the 2D
slice stacking A.A.Burkov2011 ; Y.X.Wang2019 . Then the Hall conductivity (in
unit of $\sigma_{0}$) is a summation of the Chern numbers of the occupied LLs
(see Appendix C),

$\displaystyle\sigma_{xy}=\sigma_{0}\sum_{ns\lambda}\int_{-\pi}^{\pi}\frac{dk_{z}}{2\pi}C_{ns\lambda}^{k_{z}}\theta[\varepsilon_{ns\lambda}(k_{z})<\mu],$
(21)

in which the Chern number of the $ns\lambda$ LL at wave vector $k_{z}$ is
$C_{ns\lambda}^{k_{z}}=1$ for Dirac fermions. In the quantum limit, the Fermi
wave vector $k_{F0}$ is obtained as

$\displaystyle
k_{F0}\simeq\sqrt{\frac{\mu+M}{\xi_{z}}-\frac{\xi}{\xi_{z}l_{B}^{2}}}\simeq\sqrt{\frac{\mu+M}{\xi_{z}}}=0.43\text{
nm}^{-1}.$ (22)

in which the $B$-dependent term $\frac{\xi}{\xi_{z}l_{B}^{2}}$ is much weaker
than $\frac{\mu+M}{\xi_{z}}$ and can be neglected. Then we have
$\sigma_{xy}=\sigma_{0}\frac{k_{F0}}{\pi}$, leading to the quasi-quantized
plateau in $\rho_{xy}$ that is estimated to be
$\rho_{xy}=\sigma_{xy}^{-1}=18.9$ m$\Omega$ cm, as shown in Fig. 4(c). For the
$n\geq 1$ LL, if occupied, the Fermi wavevector $k_{Fn}$ is obtained as

$\displaystyle
k_{Fn}\simeq\Big{(}\frac{1}{\xi_{z}}\sqrt{\mu^{2}-\frac{2nv^{2}}{l_{B}^{2}}}+\frac{M}{\xi_{z}}\Big{)}^{\frac{1}{2}}.$
(23)

Here, as the $B$-dependent term cannot be neglected, $k_{Fn}$ decreases with
$B$. Thus in Fig. 4(c), the $n\geq 1$ plateau of $\rho_{xy}$, which is given
as $\rho_{xy}=\pi\sigma_{0}^{-1}(k_{F0}+2\sum_{n}k_{Fn})^{-1}$, with the
factor $2$ accounting for the two LL branches, increases slowly with $B$. For
the kink of $\rho_{xy}$ around $B=1.8$ T in Figs. 4(c) and 4(d), it occurs
when the $\lambda=-1$ branch is occupied and the $\lambda=1$ branch is
unoccupied, with the magnitude
$\rho_{xy}=\pi\sigma_{0}^{-1}(k_{F0}+k_{F1})^{-1}$.

In Fig. 4(d) when $\eta$ increases to 1 meV, we observe that in $\rho_{xx}$,
the fourfold peaks no longer exist, but the SdH oscillations are still
discernible, while in $\rho_{xy}$, the Hall plateaus are preserved, with the
neighboring plateau transitions becoming smooth. More importantly, the
nonvanishing dips in $\rho_{xx}$ correspond to the plateaus in $\rho_{xy}$,
and the peaks in $\rho_{xx}$ correspond to the plateau transition regions in
$\rho_{xy}$. Although the peak heights in $\rho_{xx}$ and the plateau
magnitudes in $\rho_{xy}$ depend heavily on the model parameters, the
numerical results are qualitatively consistent with the behaviors of
$\rho_{xx}$ and $\rho_{xy}$ observed experimentally in ZrTe5 F.Tang  and HfTe5
P.Wang , which thus capture the main features of 3D QHE.

In a 2D electron system, the magnetotransport is often described by fixed
chemical potential, resulting in the well-defined Hall plateaus V.P.Gusynin ;
D.N.Sheng . Here also under the condition of fixed chemical potential, we can
understand 3D QHE as a close analog to 2D QHE, in which the quasi-quantized
plateaus in $\rho_{xy}$ are determined by the dependence of the Fermi
wavevectors on the magnetic field. Moreover, the impurity scatterings play
indispensable roles in driving 3D QHE, since they can enhance the peak heights
in $\rho_{xx}$ as well as smoothen the plateau transitions in $\rho_{xy}$.
Although our results based on a non-interacting model are quite different from
the interaction-induced CDW scenario F.Tang ; P.Wang ; F.Qin , but are
supported by a recent study S.Galeski2021 , in which the CDW states are
demonstrated to be absent in ZrTe5.

Note that in a Dirac semimetal $(\xi=\xi_{z}=0)$ with fixed chemical
potential, the Fermi wavevectors are solved as

$\displaystyle k_{F0}=\frac{1}{v_{z}}\sqrt{\mu^{2}-M^{2}},$ (24)

for the $n=0$ LL, and

$\displaystyle
k_{Fn}=\frac{1}{v_{z}}\sqrt{\mu^{2}-M^{2}-\frac{2nv^{2}}{l_{B}^{2}}},$ (25)

for the $n\geq 1$ LL. We see that the Fermi wavevectors in Eqs. (24) and (25)
show similar dependence on the magnetic field as those in Eqs. (22) and (23).
Based on these results, we suggest that the proposed mechanism for 3D QHE is
justified for a 3D Dirac fermion system, no matter the ground state is a
strong TI ($\xi>0$ and $\xi_{z}>0$), a weak TI ($\xi>0$ and $\xi_{z}<0$), or
even a Dirac semimetal.

##  V Conclusions

To summarize, we have numerically studied QOs and 3D QHE in ZrTe5 by using the
strong TI model. Under the conditions of fixed carrier density and fixed
chemical potential, although both QOs can be explained in a similar way and
the impurity scatterings show a similar effects on the conductivities, the
main conclusions are totally different.

With fixed carrier density, we find that there exists a deferring effect in
the chemical potential, thus the saddle points of the inverted LLs are well
distinguished by tuning the magnetic field magnitude. This can provide an
effective route beyond the surface states to identify the nontrivial
topological phase. On the other hand, with fixed chemical potential, we
demonstrate that 3D QHE originates from the interplay between Dirac fermions,
magnetic field and impurity scatterings, which may represent a universal
mechanism of the 3D Dirac fermion systems and needs more studies in the
future.

##  VI Acknowledgment

This work was supported by the Natural Science Foundation of China (Grant No.
11804122), the China Postdoctoral Science Foundation (Grant No. 2021M690970).

##  VII Appendix

###  VII.1 Derivation of Eqs. (9) and (10) in the main text

The conductivity tensors can be derived from the Kubo-Streda formula in Eq.
(8). At zero temperature, the longitudinal conductivity $\sigma_{xx}$ is
written as

$\displaystyle\sigma_{xx}$ $\displaystyle=-\frac{1}{\pi V}\sum_{\bm{k}}\int
d\varepsilon\frac{df(\varepsilon)}{d\varepsilon}\text{Tr}\Big{[}J_{x}\text{Im}G^{R}(\varepsilon)J_{x}\text{Im}G^{R}(\varepsilon)\Big{]}$
$\displaystyle=\frac{1}{\pi
V}\sum_{\bm{k}}\text{Tr}\Big{[}J_{x}\text{Im}G^{R}(\mu)J_{x}\text{Im}G^{R}(\mu)\Big{]},$
(A1)

in which the identity
$\frac{df(\varepsilon)}{d\varepsilon}=-\delta(\varepsilon-\mu)$ is used in the
second line. Note that the expression of $\sigma_{xx}$ in Eq. (A1) is the same
as those used in previous studies H.W.Wang ; Y.X.Wang2020 . Consider the LL
degeneracy $g=\frac{1}{2\pi l_{B}^{2}}$, and insert the energies and
wavefunctions of the LLs, then we have

$\displaystyle\sigma_{xx}$ $\displaystyle=\frac{g\eta^{2}}{\pi
L_{z}}\sum_{k_{z}}\sum_{ns\lambda}\sum_{n^{\prime}s^{\prime}\lambda^{\prime}}|\langle
ns\lambda|J_{x}|n^{\prime}s^{\prime}\lambda^{\prime}\rangle|^{2}$
$\displaystyle\times[(\mu-\varepsilon_{ns\lambda})^{2}+\eta^{2}]^{-1}[(\mu-\varepsilon_{n^{\prime}s^{\prime}\lambda^{\prime}})^{2}+\eta^{2}]^{-1}.$
(A2)

Figure 5: (Color online) (a) The DOS versus the energy $\varepsilon$, and (b)
the carrier density $n_{0}$ vs the chemical potential $\mu$. The linewidth
broadening is set as $\eta=0.01$ meV. The kinks in (a) are related to the
saddle points, the $\Gamma$ point and the $\zeta$ point. The red dashed lines
in (b) show that $n_{0}=5.2\times 10^{16}$ cm-3 corresponds to $\mu_{0}=23.41$
meV.

Similarly, the Hall conductivity $\sigma_{xy}$ is

$\displaystyle\sigma_{xy}=$ $\displaystyle\frac{g}{2\pi
L_{z}}\sum_{k_{z}}\sum_{ns\lambda}\sum_{n^{\prime}s^{\prime}\lambda^{\prime}}\int
d\varepsilon f(\varepsilon)\Big{[}\langle
ns\lambda|J_{x}\frac{dG^{R}}{d\varepsilon}|n^{\prime}s^{\prime}\lambda^{\prime}\rangle\langle
n^{\prime}s^{\prime}\lambda^{\prime}|J_{y}(G^{A}-G^{R})|ns\lambda\rangle-\langle
ns\lambda|J_{x}(G^{A}-G^{R})|n^{\prime}s^{\prime}\lambda^{\prime}\rangle$
$\displaystyle\times\langle
n^{\prime}s^{\prime}\lambda^{\prime}|J_{y}\frac{dG^{A}}{d\varepsilon}|ns\lambda\rangle\Big{]}$
$\displaystyle=$
$\displaystyle\frac{g}{L_{z}}\sum_{k_{z}}\sum_{ns\lambda}\sum_{n^{\prime}s^{\prime}\lambda^{\prime}}\text{Im}\Big{[}\frac{\langle
ns\lambda|J_{x}|n^{\prime}s^{\prime}\lambda^{\prime}\rangle\langle
n^{\prime}s^{\prime}\lambda^{\prime}|J_{y}|ns\lambda\rangle}{(\varepsilon_{ns\lambda}-\varepsilon_{n^{\prime}s^{\prime}\lambda^{\prime}}+i\eta)^{2}}-\frac{\langle
ns\lambda|J_{y}|n^{\prime}s^{\prime}\lambda^{\prime}\rangle\langle
n^{\prime}s^{\prime}\lambda^{\prime}|J_{x}|ns\lambda\rangle}{(\varepsilon_{ns\lambda}-\varepsilon_{n^{\prime}s^{\prime}\lambda^{\prime}}-i\eta)^{2}}\Big{]}\theta(\mu-\varepsilon_{ns\lambda})\theta(\varepsilon_{n^{\prime}s^{\prime}\lambda^{\prime}}-\mu).$
(A3)

In previous literatures, the expressions of $\sigma_{xy}$ are much more
complicated H.W.Wang ; B.Fu ; C.Wang  and include a normal part and an
anomalous part, where the former is determined by the states around the Fermi
level and the latter reflects the thermodynamic property of the system. Here
Eq. (A3) is quite concise and thus are facilitate to the numerical
calculations.

After calculating the current density matrix elements $\langle
ns\lambda|J_{x/y}|n^{\prime}s^{\prime}\lambda^{\prime}\rangle$, we obtain

$\displaystyle\sigma_{xx}=$
$\displaystyle\sigma_{0}\frac{2g\eta^{2}}{L_{z}}\sum^{\prime}\frac{M_{ns\lambda;n+1,s^{\prime}\lambda^{\prime}}^{2}}{[(\mu-\varepsilon_{ns\lambda})^{2}+\eta^{2}][(\mu-\varepsilon_{n+1,s^{\prime}\lambda^{\prime}})^{2}+\eta^{2}]}+\sigma_{0}\frac{2g\eta^{2}}{L_{z}}\sum^{\prime\prime}\frac{M_{ns\lambda;n-1,s^{\prime}\lambda^{\prime}}^{2}}{[(\mu-\varepsilon_{ns\lambda})^{2}+\eta^{2}][(\mu-\varepsilon_{n-1,s^{\prime}\lambda^{\prime}})^{2}+\eta^{2}]},$
(A4)

and

$\displaystyle\sigma_{xy}=$
$\displaystyle\sigma_{0}\frac{4g\pi}{L_{z}}\sum^{\prime}\frac{(\varepsilon_{ns\lambda}-\varepsilon_{n+1,s^{\prime}\lambda^{\prime}})^{2}-\eta^{2}}{[(\varepsilon_{ns\lambda}-\varepsilon_{n+1,s^{\prime}\lambda^{\prime}})^{2}+\eta^{2}]^{2}}M_{ns\lambda;n+1,s^{\prime}\lambda^{\prime}}^{2}\theta(\mu-\varepsilon_{ns\lambda})\theta(\varepsilon_{n+1,s^{\prime}\lambda^{\prime}}-\mu)$
$\displaystyle-\sigma_{0}\frac{4g\pi}{L_{z}}\sum^{\prime\prime}\frac{(\varepsilon_{ns\lambda}-\varepsilon_{n-1,s^{\prime}\lambda^{\prime}})^{2}-\eta^{2}}{[(\varepsilon_{ns\lambda}-\varepsilon_{n-1,s^{\prime}\lambda^{\prime}})^{2}+\eta^{2}]^{2}}M_{ns\lambda;n-1,s^{\prime}\lambda^{\prime}}^{2}\theta(\mu-\varepsilon_{ns\lambda})\theta(\varepsilon_{n-1,s^{\prime}\lambda^{\prime}}-\mu),$
(A5)

where the summation signs are

$\displaystyle\sum^{\prime}=\sum_{k_{z}}\sum_{n\geq
0,s\lambda}\sum_{s^{\prime}\lambda^{\prime}},\quad\sum^{\prime\prime}=\sum_{k_{z}}\sum_{n\geq
1,s\lambda}\sum_{s^{\prime}\lambda^{\prime}},$ (A6)

and the matrix element $M_{ns\lambda;n+1,s^{\prime}\lambda^{\prime}}$ is given
in Eq. (12).

We note that the components of the conductivity satisfy the following
relations:

$\displaystyle\sigma_{xx}(ns\lambda\rightarrow
n+1,s^{\prime}\lambda^{\prime})=\sigma_{xx}(n+1,s^{\prime}\lambda^{\prime}\rightarrow
ns\lambda),$ (A7) $\displaystyle\sigma_{xy}(ns\lambda\rightarrow
n+1,s^{\prime}\lambda^{\prime})=-\sigma_{xy}(n+1,s^{\prime}\lambda^{\prime}\rightarrow
ns\lambda),$ (A8)

implying that the contributions to $\sigma_{xx}$ from the LL transition
$ns\lambda\rightarrow(n+1,s^{\prime}\lambda^{\prime})$ and from
$(n+1,s^{\prime}\lambda^{\prime})\rightarrow ns\lambda$ are equal, while those
to $\sigma_{xy}$ are opposite. According to these properties, we can change
the index $n\rightarrow n+1$, $s\leftrightarrow s^{\prime}$, and
$\lambda\leftrightarrow\lambda^{\prime}$ in the second summation of Eqs. (A4)
and (A5), and obtain the final expressions in Eqs. (9) and (10) in the main
text.

###  VII.2 Density of states and chemical potential with no magnetic field

When the magnetic field is absent, the energies for the Hamiltonian
$H(\bm{k})$ in Eq. (1) can be obtained directly, which are given as

$\displaystyle\varepsilon_{s\bm{k}}=s\varepsilon_{\bm{k}}=$ $\displaystyle
s\Big{[}v^{2}(k_{x}^{2}+k_{y}^{2})+v_{z}^{2}k_{z}^{2}+\big{(}M-\xi(k_{x}^{2}+k_{y}^{2})$
$\displaystyle-\xi_{z}k_{z}^{2}\big{)}^{2}\Big{]}^{\frac{1}{2}},$ (A9)

where $s=\pm 1$ denotes the conduction/valence band. Note that both the
conduction and valence bands own twofold degeneracy.

The DOS is calculated as,

$\displaystyle D(\varepsilon>0)=\frac{g_{b}\eta}{8\pi^{4}}\iiint
dk_{x}dk_{y}dk_{z}\frac{1}{(\varepsilon-\varepsilon_{\bm{k}})^{2}+\eta^{2}},$
(A10)

where $g_{b}=2$ denotes the twofold degeneracy. To do the integrations, we
make the following substitutions Y.X.Wang2020 : $x=vk_{x}$, $y=vk_{y}$,
$z=v_{z}k_{z}$, and $\xi^{\prime}=\frac{\xi}{v^{2}}$,
$\xi_{z}^{\prime}=\frac{\xi_{z}}{v_{z}^{2}}$, and change the Cartesian into
the Cylindrical system: $x=\rho\text{cos}\theta$, $y=\rho\text{sin}\theta$.
After completing the integration over the azimuth angle $\theta$, we have

$\displaystyle
D(\varepsilon)=\frac{g_{b}\eta}{4\pi^{3}v^{2}v_{z}}\int_{-\infty}^{\infty}dz\int_{0}^{\infty}d\rho\rho$
$\displaystyle\times\Big{[}\Big{(}\varepsilon-\sqrt{\rho^{2}+z^{2}+(M-\xi^{\prime}\rho^{2}-\xi_{z}^{\prime}z^{2})^{2}}\Big{)}^{2}+\eta^{2}\Big{]}^{-1}.$
(A11)

On the other hand, the carrier density is defined in Eq. (7), which, at zero
temperature, is written as

$\displaystyle n_{0}$ $\displaystyle=\int_{0}^{\mu}d\varepsilon
D(\varepsilon).$ (A12)

Evidently, the integrands in Eqs. (A11) and (A12) show complicated
dependencies on $\rho$ and $z$, thus the integration cannot be solved
analytically. But we can resort to the numerical calculations. In Figs. 5(a)
and 5(b), we plot the calculated DOS as a function of the energy $\varepsilon$
and the carrier density $n_{0}$ as a function of the chemical potential $\mu$,
respectively. We see that the DOS increase with $\varepsilon$ and $n_{0}$
increases with $\mu$, both in a monotonous tendency. In Fig. 5(b), the red
dashed lines denote that the chosen carrier density $n_{0}=5.2\times 10^{16}$
cm-3 in Sec. IVA corresponds to the chemical potential $\mu=23.41$ meV.

###  VII.3 Derivation of Eq. (21) in the main text

Here we give a detailed derivation of Eq. (21). The 2D Chern number of the
$n$-th band is defined as an integration over the BZ,

$\displaystyle C_{n}$ $\displaystyle=-\frac{1}{2\pi
i}\int_{\text{BZ}}d^{2}k\Omega_{n},$ (A13)

where $\Omega_{n}(\bm{k})=\nabla_{\bm{k}}\times A_{n}(\bm{k})\big{|}_{z}$ is
the Berry curvature and $A_{n}(\bm{k})=\langle
u_{n}|\nabla_{\bm{k}}|u_{n}\rangle$ is the Berry connection. Then we have

$\displaystyle C_{n}$
$\displaystyle=-\frac{1}{\pi}\text{Im}\sum_{n^{\prime}}\int_{\text{BZ}}d^{2}k\langle\frac{\partial
u_{n}}{\partial k_{x}}|u_{n^{\prime}}\rangle\langle
u_{n^{\prime}}|\frac{\partial u_{n}}{\partial k_{y}}\rangle$
$\displaystyle=-\frac{1}{\pi}\text{Im}\sum_{n^{\prime}}\int_{\text{BZ}}d^{2}k\frac{\langle
u_{n}|v_{x}|u_{n^{\prime}}\rangle\langle
u_{n^{\prime}}|v_{y}|u_{n}\rangle}{(\varepsilon_{n}-\varepsilon_{n^{\prime}})^{2}},$
(A14)

where $\varepsilon_{n}$ and $u_{n}$ are the energy and wave function of the
$n$-th band. Note that in the second line of Eq. (A14), the identity
$\langle\frac{\partial u_{n}}{\partial
k_{\alpha}}|u_{n^{\prime}}\rangle=\frac{\langle
u_{n}|v_{\alpha}|u_{n^{\prime}}\rangle}{\varepsilon_{n}-\varepsilon_{n^{\prime}}}$
is used. Comparing Eq. (A14) with Eq. (A3), we obtain Eq. (21) in the main
text.

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