# Quantum-limit Hall effect with large carrier density in topological
semimetals

Guang Yang, Yi Zhang<EMAIL_ADDRESS>International Center for Quantum
Materials, School of Physics, Peking University, Beijing, 100871, China

###### Abstract

The quantum-limit Hall effect at $\nu=nh/eB\sim O(1)$ that hosts a variety of
exotic quantum phenomena requires demanding strong magnetic field $B$ and low
carrier density $n$. We propose to realize quantum-limit Hall effect even in
the presence of large carrier density residues $n_{e}$ and $n_{h}$ relative to
the magnetic field $B$ in topological semimetals, where a single Fermi surface
contour allow both electron-type and hole-type carriers and approaches charge
neutrality as $n_{e}\sim n_{h}$. The underlying filling factor
$\nu=|n_{e}-n_{h}|h/eB$ explicitly violates the Onsager’s relation for quantum
oscillations.

A two-dimensional electron system may display the integer quantum Hall effect
with exactly quantized Hall conductance $\sigma_{xy}=\nu e^{2}/h$ when
subjected to a strong magnetic field Klitzing  _et al._ (1980); Thouless  _et
al._ (1982); Laughlin (1981), where the filling factor $\nu$ takes integer
values. In the quantum limit $\nu=nh/eB\sim O(1)$, the discrete quantization
of the lowest Landau levels becomes essential, making ground for exotic
quantum phenomena such as the fractional quantum Hall effect Tsui  _et al._
(1982); Laughlin (1983); Read (1990); Haldane (1983); Jain (1989); Zhang  _et
al._ (1989), the chiral anomaly Nielsen and Ninomiya (1983); Ishikawa (1984);
Alekseev  _et al._ (1998); Zhang  _et al._ (2016a), etc. However, strong
magnetic fields $B$ and low carrier densities $n$ are among the prerequisites
to suppress the geometric phases for the quantum limit, restricting its
experimental realizations and widespread applications.

Semi-classically, the charge carriers in a magnetic field move along constant-
energy field-normal cyclotron orbits, whose overall phase’s Bohr-Sommerfeld
quantization $\phi=2\pi\nu+\gamma$ offers an intuitive perspective for the
discrete Landau levels Onsager (1952); Lifshitz and Kosevich (1956); Xiao  _et
al._ (2010). As the magnetic field changes, the highest occupied Landau level
follows, and hence a variety of properties oscillate periodically in $1/B$.
The frequency of quantum oscillations obeys the Onsager relation and is
proportional to the field-normal Fermi surface area, thus providing a useful
experimental probe to the material electronic structure and considered a
robust Fermi liquid behavior Chen and Lee (2009); Baym and Pethick (1991).
More recently, the cyclotron orbit extended its concept to the Weyl orbit
Potter  _et al._ (2014); Zhang  _et al._ (2016b); Moll  _et al._ (2016); Zhang
(2019) in topological semimetal Wan  _et al._ (2011); Xu  _et al._ (2015); Lv
_et al._ (2015); Armitage  _et al._ (2018); Weng  _et al._ (2015); Liu  _et
al._ (2014); Wang  _et al._ (2013), which consists of the Fermi arcs on the
surfaces and the chiral ($0^{th}$) Landau levels in the bulk (Fig. 1a) and
leads to quantum Hall effect in three spatial dimensions Zhang  _et al._
(2017a); Wang  _et al._ (2017); Zhang  _et al._ (2018); Li  _et al._ (2020).

Figure 1: (a) The Weyl orbit in a topological semimetal consists of the Fermi
arcs on the surfaces and the chiral Landau levels in the bulk (black dashed
lines). (b) The energy $E(k_{2})<\mu$ ($E(k_{1})>\mu$) inside an electron
(hole) pocket suggests that at least one point between $k_{1}$ and $k_{2}$ is
on the Fermi surface $E=\mu$. Even in this best-case scenario, the contour may
further divide into separate pockets up a magnetic field or inter-band
coupling. (c) The Fermi arcs on topological semimetal top and bottom surfaces
are stable and protected by spatial separation. (d) The topological semimetal
may have hole-type Fermi arcs and electron-type bulk Fermi surfaces, or vice
versa. The black arrows denote Fermi velocity directions.

We note that electron pocket and hole pocket contribute opposite geometric
phases to their cyclotron orbits and their carrier densities compensate each
other, therefore a partially electron-type and partially hole-type Fermi
surface can suppress its overall filling factor
$\nu=\left|n_{e}-n_{h}\right|h/eB$, especially when $n_{e}\sim n_{h}$.
Conventionally, it seems unlikely for a pocket to be electron-type and hole-
type at the same time, see Fig. 1b 111An example is the Fermi surface at a
Lifshitz transition Lifshitz (1960).. Here, however, we demonstrate a simple
counterexample illustrated schematically in Fig. 1c, where the crossing
between the electron-type region and hole-type region of a Weyl orbit in
topological semimetal slab consists of spatially-protected Fermi arcs on
opposite surfaces. Further, we discuss the experimentally-accessible scenario
where the surface Fermi arcs and the bulk Weyl fermions have opposite carrier
types, as in Fig. 1d. In both cases, the system can approach the quantum limit
$\nu\sim O(1)$ despite large carrier density residues $n_{e}$ and $n_{h}$ in
relative comparison with the strength of magnetic field $B$.

_Model and method._ —For concreteness, let’s consider the following three-
dimensional Weyl semimetal tight-binding model with slab geometry of thickness
$L_{z}$ Hosur (2012):

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum_{z=1}^{L_{z}}\sum_{k_{\parallel}}\varepsilon_{k_{\parallel},z}c_{k_{\parallel},z}^{\dagger}c_{k_{\parallel},z}^{\vphantom{\dagger}}+\left(h_{k_{\parallel},z}c_{k_{\parallel},z+1}^{\dagger}c_{k_{\parallel},z}^{\vphantom{\dagger}}+\mbox{h.c.}\right)$
$\displaystyle\varepsilon_{k_{\parallel},z}=\left(2t\cos k_{x}+2t\cos
k_{y}-\epsilon_{0}\right)\times\left(-1\right)^{z-1}$ $\displaystyle
h_{k_{\parallel},z}=t_{z}+2t^{\prime}\sin(k_{y})\times\left(-1\right)^{z-1}$

where $k_{\parallel}=\left(k_{x},k_{y}\right)$ is the in-plane momentum.
Without loss of generality, we set $t=1.0$, $t_{z}=1.0$, and $t^{\prime}=0.5$
unless noted otherwise. Hole-type Fermi arcs appear on the top layer $z=L_{z}$
and bottom layers $z=1$, and center around $k_{\parallel}=(0,0)$ following the
condition $\varepsilon(k_{\parallel})=\mu$. On the other hand, the bulk states
are characterized by the momentum-space Hamiltonian:

$H_{0}(\vec{k})=\varepsilon(k_{\parallel})\sigma^{z}+2t_{z}\cos(k_{z}/2)\sigma^{x}+4t^{\prime}\sin(k_{z}/2)\sin(k_{y})\sigma^{y}$
(2)

where $k_{z}$ is the momentum in the $\hat{z}$ direction, and $\sigma$’s are
Pauli matrices with the first (second) row and column describing the odd
(even) layers. The two bulk bands $\pm E(\vec{k})$ meet at zero energy only at
the two Weyl nodes $(\pm k_{W},0,\pi)$, $k_{W}=\cos^{-1}(\epsilon_{0}/2t-1)$.
In the bulk, the chiral Landau levels at the Weyl nodes connect the Fermi arcs
on both surfaces, and together, they form a closed contour, namely, the Weyl
orbit (Fig.1a).

In the presence of an external magnetic field $\vec{B}$, we employ the Landau
gauge $\vec{A}=\left(0,\Phi_{z}x-\Phi_{x}z,-\Phi_{y}x\right)$, where
$\Phi_{d}$ is the magnetic flux quantum $\Phi_{0}=h/e$ per plaquette
perpendicular to the $\hat{d}$ direction, $d=x,y,z$. Since $k_{y}$ remains a
good quantum number, we can calculate the properties of this Hamiltonian
efficiently with the recursive Green’s function method, such as the density of
states (DOS) $\rho(\mu)=-\frac{1}{\pi
L_{x}L_{z}}\sum_{x,z}\text{Im}G(x,z;x,z;\mu+i\delta)$ at the chemical
potential $\mu$, where the imaginary part $\delta=0.001$ gives a small level
broadening.

_Heuristic surface-state example._ —We first illustrate the scenario in Fig.
1c by including an extra layer $H=H_{0}+H_{1}$ and altering the Fermi arc on
the top surface:

$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle\sum_{k_{\parallel}}\varepsilon^{\prime}_{k_{\parallel}}c_{k_{\parallel},L_{z}+1}^{\dagger}c_{k_{\parallel},L_{z}+1}+\Delta\left(c_{k_{\parallel},L_{z}+1}^{\dagger}c_{k_{\parallel},L_{z}}+\mbox{h.c.}\right)$
(3)
$\displaystyle\varepsilon^{\prime}_{k_{\parallel}}=-2t\cos\left(k_{x}-k_{0}\right)-2t\cos
k_{y}+\epsilon^{\prime}_{0}$

The dispersion $\epsilon^{\prime}_{k_{\parallel}}$ gives an electron pocket
centered at $k_{\parallel}=(k_{0},0)$, which interacts with the top-surface
Fermi arc upon a coupling $\Delta=0.25$. The Fermi arc on the bottom surface
is spatially separated and remains intact Zhang (2019). As shown in Fig. 2a,
the reconstructed Fermi surface on the $k_{\parallel}$ plane encloses both an
electron-type region and a hole-type region. We set $\epsilon_{0}=3.0$,
$L_{z}=65$, $\mu=0$ at the energy of the Weyl node, and
$k_{0}=\cos^{-1}(\epsilon^{\prime}_{0}/2t-1)$.

Figure 2: (a) The red curve denotes the reconstructed Fermi surface contour in
the $k_{x}-k_{y}$ plane for $H=H_{0}+H_{1}$. The white dots are locations of
the bulk Weyl nodes where the electrons tunnel between the surfaces. The white
arrows are the directions electron move semi-classically along the contour.
$\epsilon_{0}=3.0$, $\epsilon^{\prime}_{0}=2.75$. (b) A schematic diagram of
the Weyl orbit for $H=H_{0}+H_{1}$ in slab geometry. The red and blue region
represents the hole-type pocket and the electron-type pocket, respectively,
which now coexist in a single contour. (c) The periods of the quantum
oscillations $\Delta(1/B)$ versus $\epsilon^{\prime}_{0}$ is consistent with
the absolute difference between the area $S_{elec}$ of the electron-type
region and the area $S_{hole}$ of the hole-type region in (a). The inset
illustrates the quantum oscillations in the range $1/B\in[300,700]$ for
$\epsilon_{0}=3.0$, $\epsilon^{\prime}_{0}=2.75$.

As we vary the magnetic field $B$ in the $\hat{z}$ direction, we observe clear
signatures of quantum oscillations in $\rho(\mu)$ corresponding to the Weyl
orbit, see Fig. 2c inset. Interestingly, the frequency of quantum oscillations
$f=1/\Delta(1/B)=\left|S_{elec}-S_{hole}\right|/S_{BZ}$ is consistent with the
_absolute difference_ between the area of the electron-type region $S_{elec}$
and the area of the hole pocket $S_{hole}$ (Fig. 2b), instead of the total
area $S_{hole}+S_{elec}$ enclosed by the Fermi surface. $S_{BZ}$ is the area
of the Brillouin zone. For instance, when $\epsilon^{\prime}_{0}=2.75$ and
$\epsilon_{0}=3.0$, $S_{elec}=7.30\%S_{BZ}$, $S_{hole}=4.92\%S_{BZ}$, whose
sum $S_{elec}+S_{hole}$ is more than four times larger than their difference
$S_{elec}-S_{hole}$. Consequently, this example constitutes an explicit
violation of the Onsager relation. Semi-classically, the electron moves
counter-clockwise (clockwise) around the electron-type (hole-type) region, see
Fig. 2a. Therefore, while proportional to their respective areas, the
contribution of these two segments to the overall geometric phase differs in
sign.

Figure 3: The Landau fan diagrams depict the dependence of the lowest Landau
levels of the model $H=H_{0}+H_{1}$ in Eqs. Quantum-limit Hall effect with
large carrier density in topological semimetals and 3 on the magnetic field
$B$ and the chemical potential $\mu$ at (a) $\epsilon^{\prime}_{0}=3.0$ and
(b) $\epsilon^{\prime}_{0}=2.98$. The color scale represents the DOS
$\rho(\mu)$, and the magenta dashed lines are linear guides to the eye.

Such cancellation effect between the electron carriers and hole carriers is
also in effect for the quantum Hall effects in general, such as the conditions
for the quantum limit $\nu=\left|n_{e}-n_{h}\right|h/eB$ at relatively larger
magnetic fields. The Landau fan diagrams in Fig. 3 demonstrate the entrance of
the quantum limit. Importantly, as the filling factor is dominated by the
absolute difference between the electron density $n_{e}$ and the hole density
$n_{h}$, the necessary magnetic field $B$ to reach a target $\nu$ can be
suppressed by orders of magnitude, especially when the electron and hole
densities are close $n_{e}\sim n_{h}$. For example, the quantum limit
occupying only the first landau level is approached at
$B(\Phi_{z}/\Phi_{0})=0.0048$ for $H=H_{0}+H_{1}$ with
$\epsilon_{0}^{\prime}=3.02$ and $\mu=0.0$, while the required magnetic field
is $B(\Phi_{z}/\Phi_{0})=0.199$ for a fully hole-type system $H=H_{0}$ with
similar settings to reach the quantum limit - the cancellation effect leads to
small effective carrier density despite large carrier density residues and
reduced the required magnetic field by over 97%. Therefore, cyclotron orbits
with both electron-type and hole-type carriers is indeed physically possible
and can potentially allow quantum-limit Hall effect despite large carrier
density residues, as we have showcased with the Weyl orbit in topological
semimetal.

_Practical example with surface-bulk dichotomy._ —Next, we discuss the
scenario in Fig. 1d where the different carrier types amongst the Weyl orbit
descend from material surface and bulk, respectively. Let’s consider the
original Weyl semimetal model $H_{0}$ in Eq. Quantum-limit Hall effect with
large carrier density in topological semimetals at a Fermi energy sufficiently
away from the Weyl nodes. While the surface Fermi arcs amount to a hole-type
pocket, the bulk states are two electron (hole) pockets around the Weyl nodes
for $\mu>0$ ($\mu<0$), see examples of the Fermi surface contours in the
$k_{\parallel}$ plane in Fig. 4. The area enclosed by the Fermi arcs is
$S_{FA}=7.71\%S_{BZ}$ ($S_{FA}=20.04\%S_{BZ}$) at $\epsilon_{0}=2.5$ and
$\mu=0.6$ ($\mu=-0.6$), and $0.76\%S_{BZ}$ for each of the bulk Fermi
surfaces. On the other hand, the DOS $\rho(\mu)$ shows clear signatures of
quantum oscillations in a magnetic field $B$ in the $\hat{z}$ direction that
correspond to the Weyl orbits, yet their periods at $\Delta(1/B)=16.77\pm
0.25$ for $\mu=0.6$ and $\Delta(1/B)=3.89\pm 0.03$ for $\mu=-0.6$ are
inconsistent with the expectations ($\Delta(1/B)=12.97$ at $\mu=0.6$ and
$\Delta(1/B)=4.99$ at $\mu=-0.6$) from the Onsager relation by non-negligible
margins.

Figure 4: The contours of surface Fermi arcs and bulk Fermi surfaces in the
$k_{x}-k_{y}$ plane at Fermi energy (a) $\mu=0.6$ and (b) $\mu=-0.6$,
respectively. $\epsilon_{0}=2.5$, and $L_{z}=145$. The Fermi arcs are hole-
type in both cases, whereas the bulk states are two electron pockets or two
hole pockets around the Weyl nodes, respectively. The directions of the
electron semiclassical motion and Fermi velocities are also shown. (c) and (d)
are schematic plots of the Weyl orbit in the presence of a magnetic field,
which consists of Fermi arcs on the surfaces, the chiral Landau levels in the
bulk, and the transition regions in between, where the directions of the
semiclassical motions of the charge carriers are (c) opposite and (d)
identical to those along the Fermi arcs, respectively.

To understand the discrepancies, we revisit the Weyl orbit components for
Fermi energy away from the Weyl nodes. Previous studies have established that
the components localized on the surface and deep inside the bulk are the Fermi
arcs and the chiral Landau levels descending from the Weyl nodes, respectively
Potter  _et al._ (2014); Zhang  _et al._ (2016b). However, intermediate
transitions necessarily exist for a nonzero $\mu$ around the four turning
points, where the electron ventures a finite separation between the the Fermi
arc’ endpoint and the $k_{\parallel}$ of the Weyl node, while its wave packet
gradually evolves from the surface Fermi arc states localized in the real
space $z=1,L_{z}$ to the chiral Landau level states localized in the momentum
space $k_{z}=\pm k_{F}$, or vice versa 222We discuss more details on the
property of the transition regions in Supplementary Materials, and work on a
consistent, quantitative semiclassical theory that also includes the
transitions is in preparation.. The winding directions at the transitions
depend on the carrier type of the bulk states and are opposite (identical) to
that of the hole-type Fermi arcs at $\mu=0.6$ ($\mu=-0.6$), see Figs.4c and 4d
for illustration, enclosing magnetic flux and accumulating Berry phase during
the process, therefore contributing an opposite (additional) geometric phase
to the overall quantization condition:

$\displaystyle\frac{h}{e}\frac{S_{T}-S_{FA}}{B_{z}}+\gamma_{T}+\left(2k_{F}+2k_{W}\sin\theta\right)L_{z}\sec\theta=2\pi\nu$
(4)

where $k_{F}$ is the Fermi vector of the chiral Landau levels, $\theta$ is the
magnetic field’s tilting angle from $\hat{z}$ to $\hat{x}$ direction, and
$2k_{W}\sin\theta=\left(\vec{k}_{W1}-\vec{k}_{W2}\right)\cdot\hat{B}$ is the
displacement between the Weyl nodes projected along the magnetic field. The
contributions $-hS_{FA}/eB_{z}$ and
$\left(2k_{F}+2k_{W}\sin\theta\right)L_{z}\sec\theta$ to the geometric phase
are from the surface Fermi arcs and the bulk chiral Landau levels,
respectively Potter  _et al._ (2014); Zhang  _et al._ (2016b), where we have
adopted the convention that contributions of electron-type (hole-type)
sections are positive (negative). In addition, we note that the four
transition regions along the Weyl orbit also contribute to the geometric phase
and give rise to the violation of the Onsager relation. We attribute their
contributions $hS_{T}/eB_{z}+\gamma_{T}$ to two origins: (1) an orbital piece
$hS_{T}/eB_{z}$ as the trajectory encloses the magnetic flux, sometimes
multiple times, and (2) a Berry phase $\gamma_{T}$ as the (even-odd layer)
pseudo-spin rotates during the transitions. In particular, the effective
momentum-space area $S_{T}>0$ ($S_{T}<0$) for $\mu=0.6$ ($\mu=-0.6$) negates
(adds to) $S_{FA}$, therefore increasing (decreasing) the period of quantum
oscillations.

Figure 5: (a) The Landau fan diagram shows the dependence of the lowest Landau
levels of the model in Eq. Quantum-limit Hall effect with large carrier
density in topological semimetals on the magnetic field $B_{z}$ and the Fermi
energy $\mu$ sufficiently above the Weyl node. $\epsilon_{0}=3.4$,
$L_{z}=145$, and $\theta=0$. The color scale represents the DOS $\rho(\mu)$,
and sharp Landau levels are visible. The slope changes sign at $\mu\sim 0.51$
suggesting the full cancellation of $S_{FA}$ and $S_{T}$, and the $B_{z}$
dependence vanishes. The blur on the bottom right is due to the hybridization
with the conventional bulk Landau levels, which kick in at
$\mu\propto\sqrt{B}$. The black lines are Eq. 4 with polynomial fits to the
transition region’s contributions $hS_{T}/eB_{z}+\gamma_{T}$. (b) The
effective momentum-space area of the transition regions $S_{T}$ extracted from
the fit increase rapidly as the Fermi energy moves further away from the Weyl
node and surpasses $S_{FA}$ \- the hole-pocket area of the Fermi arcs - at
around $\mu\sim 0.50$.

The Landau fan diagram in Fig. 5a at relatively larger magnetic fields
indicates quantum-limit Hall effect behaviors despite the large carrier
densities both on the surface and in the bulk at Fermi energy $\mu\sim 0.5$.
Thankfully, the electron carrier density $n_{elec}$ of the surface states
proportional to $S_{FA}$, and the hole carrier density $n_{hole}$ of the
transition regions proportional to $S_{T}$, cancel due to their opposite
carrier types within the same Fermi surface contour. We set
$\epsilon_{0}=3.4$, $L_{z}=145$ and $\theta=0$ in these calculations. We note
that both $S_{T}$ and $\gamma_{T}$ may depend explicitly on the transition
trajectory and thus the Fermi energy $\mu$ and the magnetic field $B_{z}$, and
perform a simple polynomial fit to $hS_{T}/e+\gamma_{T}B_{z}$ with all other
quantities in Eq. 4 derived analytically. More details on the model fitting
are available in Supplemental Materials. The effective $B_{z}$-independent
momentum-space area $S_{T}$ from the transition region are illustrated in Fig.
5b. $S_{T}$ increases rapidly as a function of the Fermi energy $\mu$,
especially above $\mu\sim 0.3$, consistent with the fact that both the
$k_{\parallel}$ cross-section and the extents of winding of the transitions
increase with $\mu$. Therefore, even a moderate transition region can
significantly impact the overall geometric phase as long as the Fermi energy
is sufficiently far away from the Weyl nodes.

We note that while the fan diagram in Fig. 5 exhibits a near cancellation of
the $B_{z}$-dependent surface ($S_{FA}$) and transition ($S_{T}$)
contributions, there remains a non-negligible constant geometric phase
$2k_{F}L_{z}$ from the bulk chiral Landau levels at finite $\mu$. To make our
‘quantum limit’ live up to its name, we can tilt the magnetic field away from
$\hat{z}$ to a so-called magnetic angle Zhang  _et al._ (2016b), so that
$\left(\vec{k_{W1}}-\vec{k_{W2}}\right)\cdot\hat{B}=-2k_{F}$ and the thickness
contribution $(2k_{W}\sin\theta+2k_{F})L_{z}sec\theta$ vanishes. We include
detailed results and analysis for a tilted magnetic field to Supplemental
Materials.

The above proposal relies upon the proper tuning of the Fermi energy and the
magnetic field angle and is generally more applicable for experiments and
realizable with existing Weyl semimetals. The separation between the Weyl
nodes should be appropriate: a large separation may lead to Fermi arcs and
surface carrier density too large for a proper chemical potential in the bulk
to compensate; a small separate may lead to the pair-wise annihilation of Weyl
nodes in a magnetic field Chan and Lee (2017); Zhang  _et al._ (2017b);
Ramshaw  _et al._ (2018). It further helps the material not to possess
irrelevant pockets in the explored energy window to avoid potential
intervention. For instance, the Weyl semimetal material HfCuP was predicted to
possess a hole-type surface Fermi arcs, and electron-type bulk states only
around the Weyl nodes Meng  _et al._ (2020). A signature of the cancellation
of the geometric phase is the sign change of $n_{e}{\sim}n_{h}$ thus the Hall
number $n_{H}=B/e\rho_{xy}$ upon tuning or gating, while properties concerning
the overall charge carrier density $n_{e}+n_{h}$, e.g. the longitudinal
conductivity $\sigma=(n_{e}+n_{h})e^{2}\tau/m$, remain little changed. Our
discussions also generalize straightforwardly to Dirac semimetals, which
possess Weyl orbits below a threshold value of the magnetic field Potter  _et
al._ (2014).

_Conclusion._ —We discover that electron-type and hole-type carriers may
coexist in the same Fermi surface contour and contribute oppositely to the
geometric phase and quantum Hall phenomena, which can approach the quantum
limit $\nu=|n_{e}-n_{h}|h/eB$ with a much smaller magnetic field than commonly
needed with properly-tuned despite large carrier density residues
$n_{e}{\sim}n_{h}$. We illustrate our conclusions with the Weyl orbit in
topological semimetals and note that the physics also holds for quantum
systems where a single cyclotron orbit traverses different segments with
opposite carrier types, such as surface states and bulk states.

_Acknowledgements._ —We thank Ryuichi Shindou for helpful discussions and
comments. G.Y. and Y.Z. are supported by the start-up grant at Peking
University. The computation was supported by High-Performance Computing
Platform of Peking University.

## References

  * Klitzing  _et al._ (1980) K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
  * Thouless  _et al._ (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
  * Laughlin (1981) R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).
  * Tsui  _et al._ (1982) D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).
  * Laughlin (1983) R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
  * Read (1990) N. Read, Phys. Rev. Lett. 65, 1502 (1990).
  * Haldane (1983) F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983).
  * Jain (1989) J. K. Jain, Phys. Rev. Lett. 63, 199 (1989).
  * Zhang  _et al._ (1989) S. C. Zhang, T. H. Hansson, and S. Kivelson, Phys. Rev. Lett. 62, 82 (1989).
  * Nielsen and Ninomiya (1983) H. Nielsen and M. Ninomiya, Physics Letters B 130, 389 (1983).
  * Ishikawa (1984) K. Ishikawa, Phys. Rev. Lett. 53, 1615 (1984).
  * Alekseev  _et al._ (1998) A. Y. Alekseev, V. V. Cheianov, and J. Fröhlich, Phys. Rev. Lett. 81, 3503 (1998).
  * Zhang  _et al._ (2016a) C. L. Zhang, S. Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. Tong, G. Bian, N. Alidoust, C. C. Lee, and S. M. a. Huang, Nature Communications 7, 10735 (2016a).
  * Onsager (1952) L. Onsager, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 43, 1006 (1952).
  * Lifshitz and Kosevich (1956) I. Lifshitz and A. Kosevich, Journal of Experimental and Theoretical Physics 2, 636 (1956).
  * Xiao  _et al._ (2010) D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
  * Chen and Lee (2009) K.-T. Chen and P. A. Lee, Phys. Rev. B 79, 180510 (2009).
  * Baym and Pethick (1991) G. Baym and C. Pethick,  _Landau Fermi-Liquid Theory: Concepts and Applications_ (Wiley-VCH, Weinheim, 1991).
  * Potter  _et al._ (2014) A. C. Potter, I. Kimchi, and A. Vishwanath, Nature Communications 5 (2014), 10.1038/ncomms6161.
  * Zhang  _et al._ (2016b) Y. Zhang, D. Bulmash, P. Hosur, A. C. Potter, and A. Vishwanath, Scientific Reports 6 (2016b), 10.1038/srep23741.
  * Moll  _et al._ (2016) P. J. W. Moll, N. L. Nair, T. Helm, A. C. Potter, I. Kimchi, A. Vishwanath, and J. G. Analytis, Nature 535, 266 (2016).
  * Zhang (2019) Y. Zhang, Phys. Rev. Research 1, 022005 (2019).
  * Wan  _et al._ (2011) X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011).
  * Xu  _et al._ (2015) S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, Science 349, 613 (2015).
  * Lv  _et al._ (2015) B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, Phys. Rev. X 5, 031013 (2015).
  * Armitage  _et al._ (2018) N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018).
  * Weng  _et al._ (2015) H. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai, Phys. Rev. X 5, 011029 (2015).
  * Liu  _et al._ (2014) Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng, D. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain, and Y. L. Chen, Science 343, 864 (2014), https://science.sciencemag.org/content/343/6173/864.full.pdf .
  * Wang  _et al._ (2013) Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Phys. Rev. B 88, 125427 (2013).
  * Zhang  _et al._ (2017a) C. Zhang, A. Narayan, S. Lu, J. Zhang, H. Zhang, Z. Ni, X. Yuan, Y. Liu, J.-H. Park, E. Zhang, W. Wang, S. Liu, L. Cheng, L. Pi, Z. Sheng, S. Sanvito, and F. Xiu, Nature Communications 8 (2017a), 10.1038/s41467-017-01438-y.
  * Wang  _et al._ (2017) C. M. Wang, H.-P. Sun, H.-Z. Lu, and X. C. Xie, Phys. Rev. Lett. 119, 136806 (2017).
  * Zhang  _et al._ (2018) C. Zhang, Y. Zhang, X. Yuan, S. Lu, J. Zhang, A. Narayan, Y. Liu, H. Zhang, Z. Ni, R. Liu, E. S. Choi, A. Suslov, S. Sanvito, L. Pi, H.-Z. Lu, A. C. Potter, and F. Xiu, Nature 565, 331 (2018).
  * Li  _et al._ (2020) H. Li, H. Liu, H. Jiang, and X. C. Xie, Phys. Rev. Lett. 125, 036602 (2020).
  * Note (1) An example is the Fermi surface at a Lifshitz transition Lifshitz (1960).
  * Hosur (2012) P. Hosur, Phys. Rev. B 86, 195102 (2012).
  * Note (2) We discuss more details on the property of the transition regions in Supplementary Materials, and work on a consistent, quantitative semiclassical theory that also includes the transitions is in preparation.
  * Chan and Lee (2017) C.-K. Chan and P. A. Lee, Phys. Rev. B 96, 195143 (2017).
  * Zhang  _et al._ (2017b) C.-L. Zhang, S.-Y. Xu, C. M. Wang, Z. Lin, Z. Z. Du, C. Guo, C.-C. Lee, H. Lu, Y. Feng, S.-M. Huang, G. Chang, C.-H. Hsu, H. Liu, H. Lin, L. Li, C. Zhang, J. Zhang, X.-C. Xie, T. Neupert, M. Z. Hasan, H.-Z. Lu, J. Wang, and S. Jia, Nature Physics 13, 979 (2017b).
  * Ramshaw  _et al._ (2018) B. J. Ramshaw, K. A. Modic, A. Shekhter, Y. Zhang, E.-A. Kim, P. J. W. Moll, M. D. Bachmann, M. K. Chan, J. B. Betts, F. Balakirev, A. Migliori, N. J. Ghimire, E. D. Bauer, F. Ronning, and R. D. McDonald, Nature Communications 9, 2217 (2018).
  * Meng  _et al._ (2020) W. Meng, X. Zhang, T. He, L. Jin, X. Dai, Y. Liu, and G. Liu, Journal of Advanced Research 24, 523 (2020).
  * Lifshitz (1960) I. Lifshitz, Journal of Experimental and Theoretical Physics 11, 1130 (1960).