# How to identify and characterize strongly correlated topological semimetals

Diana M. Kirschbaum∗, Monika Lužnik∗, Gwenvredig Le Roy, and Silke Paschen
Institute of Solid State Physics, Vienna University of Technology, Wiedner
Hauptstr. 8-10, 1040 Vienna, Austria

###### Abstract

How strong correlations and topology interplay is a topic of great current
interest. In this perspective paper, we focus on correlation-driven gapless
phases. We take the time-reversal symmetric Weyl semimetal as an example
because it is expected to have clear (albeit nonquantized) topological
signatures in the Hall response and because the first strongly correlated
representative, the noncentrosymmetric Weyl–Kondo semimetal Ce3Bi4Pd3, has
recently been discovered. We summarize its key characteristics and use them to
construct a prototype Weyl–Kondo semimetal temperature-magnetic field phase
diagram. This allows for a substantiated assessment of other Weyl–Kondo
semimetal candidate materials. We also put forward scaling plots of the
intrinsic Berry-curvature-induced Hall response vs the inverse Weyl velocity—a
measure of correlation strength, and vs the inverse charge carrier
concentration—a measure of the proximity of Weyl nodes to the Fermi level.
They suggest that the topological Hall response is maximized by strong
correlations and small carrier concentrations. We hope that our work will
guide the search for new Weyl–Kondo semimetals and correlated topological
semimetals in general, and also trigger new theoretical work.

##  1 Introduction

Heavy fermion compounds are materials where itinerant and localized (typically
$4f$ or $5f$) electrons coexist and, at low enough temperatures $T$, strongly
interact via the Kondo effect. They are best known for the heavy effective
masses of their charge carriers, the property that gave this class of
materials its name [1, 2]. They are also known for their ready tunability.
Small variations of an external (nonthermal) control parameter $\delta$ such
as pressure or magnetic field lead to strong changes in the effective mass.
Particularly drastic enhancements appear when approaching a quantum critical
point where, at a critical value $\delta_{\rm c}$ of the control parameter, a
second-order (typically antiferromagnetic) phase transition is just suppressed
to $T=0$ [3]. The standard method to experimentally determine effective masses
of heavy fermion metals is to measure a physical property at sufficiently low
temperatures such that it exhibits Fermi liquid behavior. The effective mass
can then be determined by comparison with the corresponding theoretical Fermi
liquid expression, e.g., $C(T)=\gamma T$ for the electronic specific heat,
$\Delta\rho(T)=AT^{2}$ for the electrical resistivity, or $\chi(T)=\chi_{0}$
for the magnetic susceptibility of the conduction electrons, where the
Sommerfeld coefficient $\gamma$, the resistivity $A$ coefficient, and the
Pauli susceptibility $\chi_{0}$ are all related to the effective mass [1, 2,
4, 5]. Upon approaching a quantum critical point, situated at $T=0$ and
$\delta=\delta_{\rm c}$, these temperature dependences hold in ever narrower
temperature ranges as they give way to non-Fermi liquid behavior emerging at
the quantum critical point and extending in a fan-like shape into the
$T(\delta)$ phase diagram [3, 6, 7, 8, 9, 10].

It is important to note that the above relations hold for metals. Whereas most
heavy fermion compounds are indeed metallic, there is a smaller subset of
materials that display semiconducting properties. They are typically referred
to as Kondo insulators [11, 12, 13, 14, 15, 16, 17, 18, 19]. In a simple mean-
field picture, the insulating state arises due to the hybridization of the
conduction electrons with the localized electrons, and the Fermi level lies
within this hybridization gap. The periodic Anderson and Kondo lattice models
are also known to exhibit such gaps [20, 21]; at half filling, where the lower
hybridized band is fully occupied and the upper hybridized band is empty, a
Kondo insulator results. The above Fermi liquid relations may still be
meaningful if effects such as doping or off-stoichiometry move the Fermi level
from within the gap into the conduction or valence band, or even into a
conductive impurity band. In that case, the knowledge of the charge carrier
concentration is needed to estimate the mass enhancement from experimental
values of $\gamma$, $A$, or $\chi_{0}$. An alternative measure of correlation
strength is the width of the gap (the narrower it is the stronger the
correlations), but experimentally determined gap magnitudes have typically
differed strongly depending on the quantity they were extracted from.

The field of Kondo insulators underwent a sudden revival as—with the advent of
topological insulators [22]—also topological Kondo insulators were proposed
[23]. In this first work, a topologically nontrivial insulating state was
found to result from the spin–orbit coupling associated with the hybridization
between the conduction and localized ($f$) electrons, in particular for
certain positions of the renormalized $f$ level relative to the bottom of the
conduction band and for certain crystal symmetries at the $f$ electron site.
This proposal raised great interest and triggered massive efforts [24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. Nevertheless, in spite of
considerable progress, there is no broad consensus yet on the topological
nature of the observed surface states. Part of the challenge derives from the
fact that the surface of a Kondo insulator is a delicate object. The formation
of the Kondo insulator gap requires the Kondo effect to operate, something
that naturally fails on a surface, where the Kondo screening cloud is cut off.
Secondly, the tools that have provided rapid progress in the field of
noninteracting topological insulators, most notably angle-resolved
photoemission spectroscopy (ARPES) in combination with density functional
theory (DFT), are of limited use for Kondo insulators, both due to their
narrow bandwidths and the absence of precise ab-initio methods. Finally,
predictions for robust and readily testable experimental signatures of the
expected topological surface states are scarce.

More recently, in a joint effort of experiment and theory, heavy fermion
compounds with metallic topology have been advanced, at first the Weyl–Kondo
semimetal [38, 39, 40, 41] and later the Weyl–Kondo nodal-line semimetal [42].
They result from the interplay of the Kondo effect, strong spin–orbit
coupling, and specific lattice symmetries, and are strongly correlated analogs
of the previously discovered noninteracting and weakly interacting Weyl
semimetals [43, 44]. The Weyl–Kondo semimetal, which has Weyl point nodes, was
theoretically demonstrated in a periodic Anderson model, with conduction
electrons on a zincblende lattice, a simple noncentrosymmetric structure [39,
40]. The Weyl–Kondo nodal-line semimetal, by contrast, was found for
conduction electrons on a 3D lattice of space group (SG) $Pmn2_{1}$ (No. 31)
[42]. For the considered commensurate filling, the nodes appear at the Fermi
energy as the Kondo effect develops. The linear dispersion near the Weyl nodes
is extremely flat, with the renormalized bandwidth given by the Kondo
temperature. Experimentally, Weyl–Kondo semimetal behavior was first found in
the heavy fermion compound Ce3Bi4Pd3 [38, 41], which crystallizes in a cubic,
noncentrosymmetric and nonsymmorphic structure of SG $I\bar{4}3d$ (No. 220).
Initial evidence for Weyl–Kondo nodal-line semimetal behavior was found in
Ce2Au3In5 [42], which forms in the orthorhombic, noncentrosymmetric, and
nonsymmorphic structure of SG $Pmn2_{1}$ (No. 31). Ce3Bi4Pd3 displays “giant”
signatures of nontrivial topology, which was attributed to the effect of
strong correlations due to the Kondo effect [38, 41].

In this perspective, we will highlight these features to facilitate the
identification of other representatives of this new class of materials. We
will also examine the relationship between the size of the topological
responses and the strength of electronic correlations, which can be used to
verify experimental interpretations. The paper is organized as follows. We
summarize the key features of Weyl–Kondo semimetal phase in Section 2 and
discuss various candidate materials in Section 3. In Section 4 we explain how
the correlation strength can be quantified in these materials and in Section 5
we investigate the relationship between correlation strength and the size of
the topological responses. In Section 6 we summarize and discuss our findings,
and provide an outlook.

##  2 Characteristics of the Weyl–Kondo semimetal

The Weyl–Kondo semimetal is a new state of matter put forward in a joint
effort of experiment [38, 41] and theory [39, 40]. It may form in systems with
preserved time reversal symmetry but broken inversion symmetry. As it is the
currently best-established gapless topological state driven by strong electron
correlations, it is the focus of this perspective paper. The understanding
that results from the above works is that Weyl nodes, which are already
present in the noninteracting bandstructure, become part of the Kondo
resonance at low temperatures and thus appear in the immediate vicinity of the
Fermi energy. As a consequence, they play an important role in low-temperature
properties, including thermodynamics and transport. The resulting band is
extremely narrow (“flat”), corresponding to a Weyl (or Dirac) dispersion

$\varepsilon=\hbar vk$ (1)

with ultralow velocity $v$. $\varepsilon$ and $k$ are the energy and wave
vector counted from a Weyl (or Dirac) point. The heat capacity (for a sample
of volume $V$) resulting from this dispersion is [39]

$C=\frac{7\pi^{2}V}{30}k_{\rm B}\left(\frac{k_{\rm B}T}{\hbar
v}\right)^{3}=\Gamma T^{3}\;,$ (2)

which is indeed experimentally observed [38], as will be shown later.

Furthermore, a magnetic field-tuning experiment [45], also detailed below,
together with theoretical work on the field-tuning effect [46] revealed that,
with increasing magnetic field, Weyl nodes and their respective anti-nodes
move mostly (for details see [46]) at constant energy in momentum space until
they meet and annihilate. The theoretical work considers an Anderson lattice
model on a diamond crystal structure with an inversion-symmetry-breaking
sublattice potential and is solved in the strong-coupling (Kondo) limit using
the auxiliary boson method [46]. Torque magnetization measurements [45]
furthermore demonstrated that the Weyl nodes are positioned within a Kondo
insulator gap. For Ce3Bi4Pd3, this situation is expected in analogy with the
well-known Kondo insulator Ce3Bi4Pt3 [14, 47, 48, 49], which is an
isostructural and isoelectronic sibling of Ce3Bi4Pd3 [38, 45]. The topological
nodal states are situated within the gap because, apparently, they are robust
against being gapped out in the Kondo hybridization process [45]. The gapped
background, identified also in [50], is a fortuitous situation for experiments
because abundant topologically trivial states at the Fermi level might
otherwise cover the effect of the topological nodal states.

The key transport signature of a Weyl–Kondo semimetal is the “spontaneous”
Hall effect [41]. The term spontaneous refers to the situation that a
transverse voltage appears in response to a (longitudinal) electrical current
but in the absence of both internal and external magnetic fields. An
approximate formulation of the Hall response in a time-reversal symmetric but
inversion asymmetric setting is

$j_{y}=\sigma_{xy}\mathcal{E}_{x}=\frac{e^{3}\tau}{\hbar^{2}}\underbrace{\int\frac{d^{3}k}{(2\pi)^{3}}f_{0}(\bm{k})\frac{\partial\Omega^{\rm{odd}}_{z}(\bm{k})}{\partial
k_{x}}}_{D_{xz}}\mathcal{E}_{x}^{2}\;,$ (3)

where $\mathcal{E}_{x}$ is an electric field applied along $x$
(“longitudinal”), $\Omega^{\rm{odd}}$ the Berry curvature, which is odd in
momentum space, $f_{0}(\bm{k})$ the equilibrium Fermi–Dirac distribution
function, $D_{xz}$ the Berry curvature dipole, and $j_{y}$ the resulting
transverse (Hall) current density [51]. This is the first nonvanishing term in
an expansion in the longitudinal electric field. In this limit, the
spontaneous Hall conductivity $\sigma_{xy}$ is proportional to
$\mathcal{E}_{x}$; thus this response has also been called “nonlinear” Hall
effect. In Weyl–Kondo semimetals, however, the Weyl nodes can be situated so
close to the Fermi energy that, even for small applied electric fields, higher
order terms are needed to capture the experimentally observed behavior [41].
Indeed, in the first candidate material, Ce3Bi4Pd3, which is time reversal
invariant as demonstrated by zero-field muon spin rotation ($\mu$SR)
experiments [41] but has a noncentrosymmetric crystal structure, not only the
square-in-$\mathcal{E}_{x}$ spontaneous Hall current (or voltage) expected
from Eq. 3 but also a contribution that is linear in $\mathcal{E}_{x}$ was
observed and attributed to higher order terms that are neglected in Eq. 3
[41]. In applied magnetic fields (or magnetic induction $B=\mu_{0}H$), the
spontaneous Hall effect finds continuation as an even-in-$B$ Hall response.
The magnetic field can be ruled out to be the origin of this effect, as it
would necessarily always result in an odd-in-$B$ Hall effect.

In figure 1 we sketch these key signatures of a Weyl–Kondo semimetal.
According to Eq. 2 the Weyl contribution to the heat capacity $\Delta C$ shows
linear behavior on a $\Delta C/T$ vs $T^{2}$ plot (figure 1A), with a slope
$\Gamma$ that is inversely proportional to $v^{3}$. Because the Kondo
interaction can lead to bandwidth renormalizations of several orders of
magnitude, $v$ will be drastically reduced compared to the Fermi velocity of
simple noninteracting Schrödinger-like quasiparticles (e.g., $1.4\times
10^{6}$ m/s for gold) or, perhaps more significantly, noninteracting Dirac-
like quasiparticles (e.g., $1\times 10^{6}$ m/s for graphene [52]). This
reduction of $v$ boosts the heat capacity to the point that it may even
overshoot the low-temperature (Debye-like) phonon contribution [38]. The
temperature $T_{\rm W}$ up to which this law holds is a measure of the
stability of the Weyl–Kondo semimetal phase. It is plotted as full circles in
figure 1B. We note that, unlike broken-symmetry phases characterized by an
order parameter, this state is not bound by a phase transition but builds up
continuously as Kondo coherence sets in [39, 41]. This is symbolized by the
violet shading, which lacks a sharp boundary. With increasing applied magnetic
field, $T_{\rm W}$ is successively suppressed. This is because the Weyl and
anti-Weyl dispersions start to intersect as the Weyl nodes move towards each
other in momentum space [46] (figure 1C). The Weyl–Kondo semimetal phase
collapses when the Weyl and anti-Weyl nodes meet and annihilate. The slope of
the dispersions is, a priori, not expected to vary with $B$, as visualized by
the inverse Weyl velocity plotted as squares in figure 1B on the right $y$
axis. The magnitude of the even-in-$B$ Hall effect, by contrast, depends on
the momentum-space distance between a Weyl and the associated anti-Weyl node
[46]. As such it is expected to decrease with increasing field (see diamonds
plotted on the right $y$ axis in figure 1B).

A B

C

Figure 1: Weyl–Kondo semimetal characteristics. (A) Weyl contribution to the
heat capacity $\Delta C$, plotted as $\Delta C/T$ vs $T^{2}$, for different
magnetic fields (inductions) $B_{i}$. The linear behavior, corresponding to a
$\Delta C\propto T^{3}$ dependence, is a thermodynamic signature of bands with
linear dispersion (Eq. 1). For Weyl semimetals, its slope is related to the
Weyl velocity $v$ via Eq. 2. (B) Temperature–magnetic field phase diagram
displaying the region (violet shading) in which the Weyl–Kondo semimetal
signature in specific heat is observed. $T_{\rm W,max}$ is the temperature up
to which the $\Delta C\propto T^{3}$ dependence holds in zero field. The right
axes display the inverse Weyl velocity $1/v$ (squares) and the even-in-field
Hall resistivity $\rho_{\rm H}^{\rm even}$ (diamonds), both normalized to
their maximum values. (C) Sketch of the dispersions near a Weyl (W+) and its
anti-Weyl node (W-), in zero magnetic field (left) and in an applied magnetic
field (right). The dashed line indicates the Fermi energy $E_{\rm F}$, chosen
here to be positioned slightly below the Weyl nodes.

##  3 Weyl–Kondo semimetal candidate materials

The above-described experiments on Ce3Bi4Pd3 [38, 41] together with the
theoretical studies on nonsymmorphic Kondo lattice models [39, 40] have coined
the notion of the Weyl–Kondo semimetal. This sets the stage to consider
experimental results on other noncentrosymmetric compounds in this context. In
what follows we review the pertinent data and compare them with the behavior
seen in Ce3Bi4Pd3. In figure 2 we replot published specific heat data, in the
form of isofield $\Delta C/T$ vs $T^{2}$ curves, for Ce3Bi4Pd3 [45], the cubic
half-Heusler compound YbPtBi (SG $F\bar{4}3m$, No. 216) [53], the tetragonal
compound CeAlGe (SG $I4_{1}md$, No. 109) [54, 55, 56], and the cubic compound
Ce3Rh4Sn13 (SG $I2_{1}3$, No. 199) [57, 58], in panels A-D, respectively.
Details of the data analyses are explained in the caption.

A B

C D

Figure 2: $\Delta C/T$ vs $T^{2}$ at fixed magnetic fields for (A) Ce3Bi4Pd3,
(B) YbPtBi, (C) CeAlGe, and (D) Ce3Rh4Sn13. For Ce3Bi4Pd3, $\Delta
C/T=C/T-(\gamma+\beta T^{2})$, where $C$ is the total measured specific heat,
$\gamma$ a zero-temperature offset, which might originate from residual
topologically trivial “background” bands, and $\beta$ the prefactor of the
low-temperature phonon contribution as determined from the non-$f$ reference
compounds La3Bi4Pd3 [45] and La3Bi4Pt3 [38]. For the three other compounds, we
used $\Delta C/T=C/T-\gamma$, i.e., no phonon contribution was subtracted.
However, this is not expected to change the conclusions as in all three cases
$\beta$ is much smaller than (less than 2% of) the measured slopes [53, 54,
58], creating a maximum error of 0.6% on the extracted putative Weyl
velocities. Note that for CeAlGe, $\gamma$ is negative above 5 T, which is
unphysical and indicates that the temperature dependence has to change at
lower temperatures. The arrows mark the onset temperature $T_{\rm W}$ of the
$\Delta C\propto T^{3}$ behavior, defined here via a deviation of the data by
more than 5% from the low-temperature $\Delta C/T=\Gamma T^{2}$ fit. For
YbPtBi, the onset temperatures tabulated in [53] were taken, where the
definition criterion is not further specified. The data for the plots were
taken from [45] (Ce3Bi4Pd3, open symbols from Ref. [38]), [53] (YbPtBi), [54]
(CeAlGe), and [58] (Ce3Rh4Sn13).

For all four compounds these plots display ranges of linearity, as expected
for a Weyl–Kondo semimetal according to Eq. 2. However, a closer inspection
reveals distinct differences from the behavior of Ce3Bi4Pd3. Firstly, the
maximum temperature $T_{\rm W}$ up to which the linear behavior holds
increases with $B$ for YbPtBi, CeAlGe, and Ce3Rh4Sn13, whereas it decreases
with $B$ for Ce3Bi4Pd3. This would indicate that, in these other compounds,
magnetic field stabilizes the Weyl–Kondo semimetal phase as opposed to the
suppression predicted from Zeeman coupling tuning [46]. Secondly, the slopes
of the linear dependencies are sizably reduced with $B$, whereas for Ce3Bi4Pd3
all iso-$B$ curves have essentially the same slope. The (putative) Weyl
dispersions do thus not remain unchanged (as in the cartoon in figure 1) but
become steeper under magnetic field tuning. This sizable correlation tuning
effect may hint at the presence of a nearby quantum critical point.

As pointed out in [38], a $T^{3}$ contribution to the specific heat may
alternatively result from 3D antiferromagnetic (AFM) magnons, as seen for
instance in the heavy fermion antiferromagnets CeIn3 [59, 60], CePd2In
(between 3 and 6 T) [61], or CeGe1.76 [62] below the respective Néel
temperatures. A sizable reduction of the slope with increasing magnetic field,
as seen in YbPtBi, CeAlGe, and Ce3Rh4Sn13, would indeed be expected in this
situation [63, 64]. In fact, CeAlGe is known to order below 5 K [54], with a
complex structure of predominantly antiferromagnetic nature [65], suggesting
that AFM magnons may contribute to the observed $\Delta C\propto T^{3}$
dependence. For Ce3Rh4Sn13, there are conflicting reports on its magnetic
order. Whereas in [66] two antiferromagnetic phase transitions at 2 and 1.2 K
were reported from specific heat measurements, no clear evidence for magnetic
order was found in neutron diffraction experiments [67]. This calls for
further investigations, for instance by zero-field $\mu$SR experiments, which
ruled out even spurious magnetism in Ce3Bi4Pd3 [41]. YbPtBi is known to order
antiferromagnetically in zero field, but this order is suppressed to $T=0$ at
0.4 T and a Fermi liquid state is recovered at fields above 0.8 T [68]. The
$\Delta C\propto T^{3}$ dependence highlighted in [53] appears only at a much
larger field of 7 T, deep in the Fermi liquid region. This seems to rule out a
connection with the low-field AFM phase. On the other hand, it remains to be
understood why a compound with broken inversion symmetry (such as YbPtBi)
would not exhibit Weyl–Kondo semimetal features at smaller fields (including
$B=0$). It should also be clarified whether the $B$-induced increase of the
crystal electric field level splitting evidenced in [68] may underlie the
strong field dependence of the $\Delta C/T$ data (figure 2B).

A B

C D

Figure 3: Temperature–magnetic field phase diagrams for (A) Ce3Bi4Pd3, (B)
YbPtBi, (C) CeAlGe, and (D) Ce3Rh4Sn13, for comparison with the expectation
for a Weyl–Kondo semimetal sketched in figure 1B. The field-dependent onset
temperatures $T_{\rm W}$ (circles, left axes) of the $\Delta C\propto T^{3}$
behavior, defined as explained in figure 2 and normalized by the respective
maximum value $T_{\rm W,max}$, delineate the region of (putative) Weyl–Kondo
semimetal behavior (shading). The right axis displays the field dependence of
the inverse (putative) Weyl velocities $1/v$ extracted from the slopes
$\Gamma$ of the linear fits in figure 2 (squares) and, where available, the
even-in-field Hall resistivity $\rho_{\rm H}^{\rm even}$ (diamonds) and an
“anomalous” odd-in-field Hall resistivity $\rho_{\rm H}^{\rm odd}$ (crosses),
all normalized by the respective maximum values. The (putative) Weyl
velocities for Ce3Bi4Pd3 and YbPtBi are taken from [38] and [53],
respectively. For CeAlGe and Ce3Rh4Sn13, they were calculated in this work
using Eq. 2. The Hall data are the lowest-temperature isotherms available,
which were taken at 0.4 K for Ce3Bi4Pd3 [41, 45] and 0.3 K for YbPtBi [53].

In figure 3 we summarize the characteristics extracted for all four compounds
in temperature–magnetic field phase diagrams. As in figure 1B, the full
circles represent the onset temperatures $T_{\rm W}$ of the $\Delta C\propto
T^{3}$ behavior and the open squares the (putative) inverse Weyl velocities
extracted from the slopes $\Gamma$ of linear fits to $\Delta C/T$ vs $T^{2}$.
For Ce3Bi4Pd3 the Weyl velocity is approximately constant within the magnetic
field range where the Weyl–Kondo semimetal exists. For the other three
compounds, a pronounced field dependence is observed which, as discussed
above, may hint at alternative origins of the $\Delta C\propto T^{3}$
dependencies.

A spontaneous (nonlinear) Hall effect has so far only been observed for
Ce3Bi4Pd3 (diamond at $B=0$ in figure 3A). It is seen as a “smoking gun”
signature for Weyl nodes in a time reversal symmetric but inversion-symmetry-
broken semimetal, as the Berry curvature divergences at the Weyl nodes are its
only plausible origin. If they are placed very close to the Fermi energy, as
expected in a Weyl–Kondo semimetal [39], the resulting spontaneous Hall effect
may be giant. Also the corresponding finite field signature, the even-in-field
Hall effect as seen in Ce3Bi4Pd3 [41] (diamonds at $B>0$ in figure 3A),
remains to be discovered in the other Weyl–Kondo semimetal candidate
materials.

What has been analyzed and proposed as evidence for Weyl physics in YbPtBi is
an odd-in-field Hall effect (crosses in figure 3B) [53]. It represents a
magnetic-field induced effect, in contrast to the spontaneous Hall effect,
which exists in $B=0$, and the even-in-field Hall effect, which exists in
spite of the presence of a finite field (i.e., the field is not its origin).
As such, it is more ambiguous evidence for Weyl semimetal physics. In general,
the identification of intrinsic Berry curvature contributions in odd-in-field
Hall resistivity data is a nontrivial task, which has led to conflicting
results in particular in magnetic materials [69]. In Ce3Bi4Pd3, such a
contribution was identified as the deviation from a linear-in-field Hall
resistivity, which is observed only at low temperatures and fields, within the
Weyl–Kondo semimetal regime [41] (crosses in figure 3A). Note that in this
regime the magnetization is linear in field and can thus not be at the origin
of this effect. This contribution is necessarily zero for $B=0$, then
increases to its maximum value, and vanishes again as the Weyl–Kondo semimetal
is suppressed by magnetic field. For YbPtBi, the odd-in-field Hall signal
appears to exist outside the putative Weyl–Kondo semimetal regime identified
via the specific heat (red shading in figure 3B), which calls for measurements
at lower temperatures to verify whether the putative Weyl–Kondo semimetal
regime might persist to lower fields.

##  4 Quantifying the correlation strength of Weyl–Kondo semimetals

The Weyl–Kondo semimetal Ce3Bi4Pd3 was shown to exhibit “giant” topological
responses [38, 41]. This was attributed to the strong bandwidth
renormalization via the Kondo effect, which results in a flat Weyl dispersion
with very low Weyl velocity. It seems plausible that the Kondo effect leads to
similar renormalization effects for both Schrödinger and Dirac/Weyl-like
quasiparticles. Thus, a comparison between the respective renormalization
factors can serve as a consistency check.

To scrutinize the Weyl–Kondo semimetal interpretation discussed above, we use
experimental values of the Sommerfeld coefficient $\gamma$ (removed in the
plots in figure 2 by plotting $\Delta C/T=C/T-\gamma$) together with Hall
effect data for the charge carrier concentration $n$ to estimate the
renormalization in the effective (Schrödinger) mass via

$\frac{m}{m_{0}}=\frac{3\hbar^{2}}{m_{0}\cdot k_{\rm
B}^{2}\cdot(3\pi^{2})^{1/3}}\cdot\frac{\gamma}{n^{1/3}}\;,$ (4)

where $m_{0}$ and $m$ are the free electron mass and the mass renormalized by
correlations, respectively, and the other symbols have their usual meaning. As
renormalization factor for the Dirac/Weyl quasiparticles, we use
$(v/v_{0})^{-1}$, i.e. the inverse of the (putative) Weyl velocities $v$ from
figure 3 scaled by $v_{0}$ (for parameters and references, see table 1). The
inverse is taken because a larger renormalization of Dirac/Weyl-like bands is
reflected by smaller (not larger) velocities. For concreteness, we use
$v_{0}=10^{6}$ m/s, the Dirac velocity of graphene [52]. The expectation for
(correlated) Dirac or Weyl semimetals is that $m/m_{0}$ and $(v/v_{0})^{-1}$
have similar values. In the double-logarithmic plot in figure 4 this is
indicated by a straight line of slope 1. We see that only the data point for
Ce3Bi4Pd3 fulfills this expectation. For the other three materials, the
renormalization effect would be much larger for the Dirac/Weyl-like than for
the Schrödinger-like quasiparticles. This suggests that at least part of the
large slopes $\Gamma$ of the $\Delta C\propto T^{3}$ dependencies of YbPtBi,
CeAlGe, and Ce3Rh4Sn13 (figure 2) derive from effects other than a Weyl–Kondo
semimetal dispersion. In any case, evidence beyond the specific heat signature
should be sought to make a Weyl–Kondo semimetal assignment firm.

Figure 4: Weyl vs Schrödinger renormalization. (Putative) inverse Weyl velocity $v^{-1}$, scaled by the inverse of the Dirac quasiparticle velocity of graphene $v_{0}=1\times 10^{6}$ m/s [52], as extracted from the linear-in-$T^{3}$ electronic (or, more generally, nonphononic) specific heat of Ce3Bi4Pd3, YbPtBi, CeAlGe, and Ce3Rh4Sn13 (see figure 2) vs the effective (Schrödinger) mass renormalization as calculated via Eq. 4, using the published Sommerfeld coefficients $\gamma$ and charge carrier concentrations $n$ given in the table 1. For the open symbol of CeAlGe, $m/m_{0}=49$ given in [56] was used, where it was determined using the plasma frequency instead of $n$. Table 1: Parameters used for the data in figure 4, as extracted from the cited publications. The (putative) Weyl velocities $v$ of CeAlGe and Ce3Rh4Sn13 were determined within this work from the slopes $\Gamma$ of the linear fits in figure 2 using Eq. 2; this is indicated by the $\ast$ after the reference. Because the specific heat of CeAlGe exhibits a phase transition anomaly due to the magnetic ordering, a reliable extraction of $\gamma$ is nontrivial. The values we used at 0 T and 14 T correspond to the lowest value of $C/T(B=0)$ above the transition and the lowest measured $C/T(B=14$ T) value in the entire $T$ range, respectively [54]. To obtain the renormalization of the effective mass from Eq. 4 the $\gamma$ values from this table must be converted to SI units (J/(K2m3)) by dividing them by the respective molar volume $V_{M}$. The carrier density $n$ of Ce3Bi4Pd3 was determined in the region where the Weyl nodes are gapped out (between about 9 and 14 T) [45]. This is needed for consistency with the Sommerfeld coefficient $\gamma$, which also counts only the Schrödinger-like carriers. compound |  $v$ (m/s) |  $\gamma$ (J/(mol K2)) |  $n$ (1/cm3)
---|---|---|---
Ce3Bi4Pd3 |  885 [38] |  0.627 [45] |  8.2$\cdot 10^{19}$ [41, 45]
YbPtBi (7 T) |  213 [53] |  0.244 [53] |  $5.2\cdot 10^{20}$ [70]
YbPtBi (9 T) |  292 [53] |  0.182 [53] |  $5.2\cdot 10^{20}$ [70]
YbPtBi (13 T) |  394 [53] |  0.089 [53] |  $5.2\cdot 10^{20}$ [70]
CeAlGe (0 T) |  288 [54]∗ |  0.05 [54] |  $1.4\cdot 10^{20}$ [54]
CeAlGe (14 T) |  496 [54]∗ |  0.041 [54] |  $1.4\cdot 10^{20}$ [54]
Ce3Rh4Sn13 (0 T) |  121 [58]∗ |  3.44 [58] |  $5\cdot 10^{22}$ [71]
Ce3Rh4Sn13 (3 T) |  138 [58]∗ |  1.06 [58] |  $5\cdot 10^{22}$ [71]
Ce3Rh4Sn13 (6 T) |  176 [58]∗ |  0.57 [58] |  $5\cdot 10^{22}$ [71]
Ce3Rh4Sn13 (9 T) |  200 [58]∗ |  0.43 [58] |  $5\cdot 10^{22}$ [71]

##  5 Topological response vs correlation strength

As discussed above, the giant spontaneous Hall effect of Ce3Bi4Pd3 may
represent such firm evidence. To the best of our knowledge, it has so far not
been reported in any other strongly correlated nonmagnetic (time-reversal-
symmetry-preserving) Weyl semimetal candidate material, including the three
above-discussed heavy fermion compounds. To nevertheless examine whether its
magnitude depends on the correlation strength, we resort to a comparison with
noninteracting/weakly interacting reference materials. In studies of these
compounds, the term nonlinear Hall effect (NLHE) is used, and reference is
made to the Berry curvature dipole $D_{xz}$ (see Eq. 3). As explained in
section 2, this is only part of the Berry-curvature-related response observed
in Ce3Bi4Pd3. Further terms arise when expanding the out-of-equilibrium
distribution function around a finite-current setpoint (instead of around
$j_{x}=0$ as done to obtain Eq. 3), which is deemed necessary in Weyl–Kondo
semimetals [41]. To discriminate this fully nonequilibrium response from the
Berry curvature dipole effect (the lowest-order term), the expression
“spontaneous Hall effect” was used instead of “NLHE” [41]. For the purpose of
comparison, we adopt the NLHE terminology in what follows.

NLHE studies have been carried out in various (non- or weakly interacting)
materials [72, 73, 74, 75, 76, 77, 78, 79, 80, 81], but the identification of
intrinsic Berry curvature contributions has been challenging. It involves the
separation from extrinsic contributions due to effects such as side jump and
skew scattering [82]. For Ce3Bi4Pd3, the Hall angle is constant in the
Weyl–Kondo semimetal regime, i.e.,

$\tan\Theta=\frac{\sigma_{xy}}{\sigma_{xx}}=\mbox{const}\;,$ (5)

as seen from the approximate linear $\sigma_{xy}$ vs $\sigma_{xx}$ dependence
below about 3 K (Fig. 2B in [41]). Interestingly, this holds for both the dc
(and, by extension, the $2\omega$) response and the (fully out-of-equilibrium)
$1\omega$ response. In the context of the NLHE, only the $2\omega$ response is
considered and it is investigated how $\tan\Theta$, typically scaled by the
applied longitudinal electric field $\mathcal{E}_{x}$, depends on the
longitudinal conductivity, i.e.

$\frac{\tan\Theta}{\mathcal{E}_{x}^{\omega}}=\frac{\sigma_{xy}}{\sigma_{xx}\mathcal{E}_{x}^{\omega}}=\frac{\mathcal{E}_{xy}^{2\omega}}{(\mathcal{E}_{x}^{\omega})^{2}}=f(\sigma_{xx})\;.$
(6)

The influence of disorder scattering was studied in a 2D tilted massive
(gapped) Dirac model as a minimal symmetry-allowed model for a NLHE [82].
$f(\sigma_{xx})=\mbox{const}$, as observed for Ce3Bi4Pd3, was found only for
the intrinsic contribution (due to the Berry curvature dipole), and side-jump
and skew-scattering terms from dynamic (e.g. phonon-induced) disorder. These
latter should depend on temperature and disappear in the zero-temperature
limit. The fact that, for Ce3Bi4Pd3, $f(\sigma_{xx})=\mbox{const}$ holds over
the entire temperature range of Weyl–Kondo semimetal behavior is strong
evidence for dynamic disorder effects playing a minor role and thus for the
intrinsic nature of the spontaneous (or nonlinear) Hall effect. We note that
also the linear-response anomalous Hall effect from skew scattering and side-
jump scattering was shown to be negligibly small in Ce3Bi4Pd3 (see SI, part B
of [41]). As extrinsic scattering effects in the linear-response and nonlinear
regimes are related [72, 82, 83], this is a further confirmation for their
absence in the NLHE in Ce3Bi4Pd3. In general, the situation is considerably
more complex. Here we focus on investigations of (Pb1-xSnx)1-yInyTe [80],
MoTe2 [78], WTe2 [72], and TaIrTe2 [73], where—via stoichiometry optimization
in (Pb1-xSnx)1-yInyTe to reach a ferroelectric state with extremely low
carrier concentration and via exfoliation in the other three
noninteracting/weakly interacting reference compounds—a Berry curvature dipole
contribution to the NLHE became sufficiently large to be identified with some
confidence.

(Pb1-xSnx)1-yInyTe is an In-doped alloy of two rock salt-type compounds: the
normal insulator (NI) PbTe and the topological crystalline insulator SnTe. For
certain compositions ($x$ and $y$ values), ferroelectric order appears, which
breaks the inversion symmetry of the undeformed system (SG $Fm\bar{3}m$, No.
225), thereby enabling the formation of a Weyl semimetal [84]. An “optimally
doped” sample shows an electrical conductivity that decreases with decreasing
temperature [80]. Using this and the temperature dependent
$\mathcal{E}_{xy}^{2\omega,\rm{intr}}/(\mathcal{E}_{x}^{\omega})^{2}$,
$f(\sigma_{xx})$ can be obtained. An extrapolation of the low-$\sigma_{xx}$
(low-$T$) values to $\sigma_{xx}=0$ leads to an intrinsic NLHE of $4.35\times
10^{-4}$ m/V. Bulk MoTe2 crystallizes in the noncentrosymmetric $T_{\rm
d}$-MoTe2 structure of SG $Pmn2_{1}$ (No. 31) [85, 86], but the exfoliated
films of interest here have a lower $Pm$ symmetry [87]. $f(\sigma_{xx})$ shows
a pronounced dependence on $\sigma_{xx}$, with a functional form that changes
with temperature. There is also a pronounced thickness dependence. Thinner
films have larger residual resistivity (due to surface scattering), which tips
the balance between different (extrinsic) scattering processes. The best
estimate of the intrinsic Berry curvature contribution comes from the thinnest
samples because they have the smallest conductivity and thus the lowest skew-
scattering contribution (which is the dominant extrinsic scattering effect at
high temperatures). The extrapolation of
$f(\sigma_{xx})\propto\sigma_{xx}^{2}$ to $\sigma_{xx}=0$ (at
$T\rightarrow\infty$) gives $1.2\times 10^{-6}$ m/V. This is one order of
magnitude larger than the upper bound estimated from DFT calculations of the
Berry curvature dipole, so presumably it is still dominated by extrinsic
scattering [78]. The situation is similar in $T_{\rm d}$-WTe2 [72]. Its SG
$Pmn2_{1}$ [88] is again reduced to $Pm$ in exfoliated multilayer films [87].
For three films of 5-6 layer thickness, $f(\sigma_{xx})$ was found to be
proportional to $\sigma_{xx}^{2}$ in temperature ranges between 2 and 100 K.
Again, the Berry curvature dipole contribution is estimated by extrapolating
this dependence to $\sigma_{xx}=0$. That the values obtained for the three
films vary by almost an order of magnitude ($0.15-1\times 10^{-9}$ m/V) is
attributed to the different carrier concentrations and mobilities, though no
systematic dependence is seen. Finally, also exfoliated samples of $T_{\rm
d}$-TaIrTe2 (again SG $Pmn2_{1}$ for bulk) reveal such behavior [73]. Using
the same procedure for the thinnest and thus most resistive film yields
$1.8\times 10^{-8}$ m/V as an estimate for the intrinsic Berry curvature
dipole contribution to the NLHE.

A B

Figure 5: Intrinsic Berry-curvature-dipole-induced nonlinear Hall effect
(NLHE), quantified by
$\mathcal{E}_{xy}^{2\omega,\rm{intr}}/(\mathcal{E}_{x}^{\omega})^{2}$, of
(candidate) Weyl semimetals as function of (A) the inverse scaled Weyl
velocity $(v/v_{0})^{-1}$ as a measure of correlation strength. The full
symbols show the experimentally extracted values, the open symbols DFT
results. $T_{\rm d}$-TaIrTe4, $T_{\rm d}$-MoTe2, and Ce3Bi4Pd3, which have
similar charge carrier concentrations (of order $10^{20}$ cm-3), lie on a
universal curve $\sim(v/v_{0})^{-2}$, as seen from the dashed guide-to-the-
eyes curve. For the two shaded lines at higher and lower $n$, we used the same
slope; (B) the inverse charge carrier concentration $n^{-1}$ (of hole-like
charge carriers for consistency with [80]). The charge carrier concentrations
of the quasi-2D material $T_{\rm d}$-WTe2 were calculated using
$n=n_{\rm{2D}}/d$, where $d$ is the interlayer distance of (2.7-2.8) Å [87].
The noninteracting/weakly interacting materials lie on a universal curve $\sim
n^{-1.3}$, which was determined by fitting (dashed line). A curve with the
same slope is plotted through the data point of Ce3Bi4Pd3 (shaded line).

In figure 5 we compare the magnitudes of these intrinsic Berry curvature
dipole contributions to the NLHE by plotting
$\mathcal{E}_{xy}^{2\omega,\rm{intr}}/(\mathcal{E}_{x}^{\omega})^{2}$ as a
function of the respective reciprocal Weyl velocities $v^{-1}$ (scaled by
$v_{0}^{-1}$, panel A) and charge carrier concentrations $n^{-1}$ (panel B).
All values are also given in table 2, and the caption contains details on how
they were obtained. The data points in figure 5A fall into three groups of
similar $n$. In particular, Ce3Bi4Pd3 has roughly the same $n$ as $T_{\rm
d}$-TaIrTe4 and $T_{\rm d}$-MoTe2 ($\sim$ $10^{20}$ cm-3), which is
highlighted by the dashed guide-to-the-eyes line which represents a $v^{-2}$
dependence (for the other data points, shaded lines with the same slope are
plotted). At constant $n$,
$\mathcal{E}_{xy}^{2\omega,\rm{intr}}/(\mathcal{E}_{x}^{\omega})^{2}$ thus
appears to be boosted by strong correlations, which flatten the Weyl bands
(smaller slope $v$ of the Weyl dispersion, Eq. 1) and enhance the electronic
density of states at the Fermi level [which scales as $D(E_{\rm F})\sim
v^{-3}$ in a 3D material with Weyl dispersion]. A second trend that becomes
clear from this plot is that, at constant $v$,
$\mathcal{E}_{xy}^{2\omega,\rm{intr}}/(\mathcal{E}_{x}^{\omega})^{2}$ is
enhanced by reducing $n$. This dependence is explicitly revealed in figure 5B.
All data of the noninteracting/weakly interacting Weyl semimetals fall on a
universal curve,
$\mathcal{E}_{xy}^{2\omega,\rm{intr}}/(\mathcal{E}_{x}^{\omega})^{2}\sim
n^{-1.3}$ (dashed line), evidencing a strong dependence on the proximity of
the Weyl nodes to the Fermi energy ($E_{\rm F}\sim n^{1/3}$ in a 3D material
with Weyl dispersion). Ce3Bi4Pd3 lies orders of magnitude above this line.
Again, we also include a line of the same slope for Ce3Bi4Pd3 (shaded line),
which makes a strategy for further enhancing
$\mathcal{E}_{xy}^{2\omega,\rm{intr}}/(\mathcal{E}_{x}^{\omega})^{2}$
explicit: to reduce the charge carrier concentration in a strongly correlated
Weyl semimetal such as Ce3Bi4Pd3. Whether, at constant $n$, the correlation-
induced $v$ reduction is the only cause of the drastic enhancement of
$\mathcal{E}_{xy}^{2\omega,\rm{intr}}/(\mathcal{E}_{x}^{\omega})^{2}$ or
whether also other ingredients—such as the multiplicity of Weyl nodes near the
Fermi energy, the $k$ space separation of node and anti-node, or the tilting
of the nodes [40]—contribute, should be clarified by future work.

Table 2: Estimates of the intrinsic NLHE contribution due to the Berry curvature dipole, $\mathcal{E}_{xy}^{2\omega,\rm{intr}}/(\mathcal{E}_{x}^{\omega})^{2}$ (obtained as explained in the text), the 3D/quasi-2D charge carrier concentration of hole-like charge carriers, $n_{\rm(h)}$, and the inverse ratio of the Weyl velocity $v$ and the velocity of graphene (taken as $v_{0}=1\times 10^{6}$ m/s), for the selected Weyl semimetal (candidate) materials. These data are used in figure 5. The values of $v$ were obtained as follows. Ce3Bi4Pd3: from the slope $\Gamma$ of $\Delta C/T$ vs $T^{2}$, with the phonon contribution subtracted [38]. (Pb1-xSnx)1-yInyTe: from a linear fit of the optical conductivity vs photon energy, yielding $v=1.7\times 10^{5}$ m/s [84], and from a linear-in-$T$ fit of the electrical conductivity, leading to $v=1.2\times 10^{5}$ m/s [80]. MoTe2: from $1/v=m/[\hbar(3\pi^{2}n)^{1/3}]$, using the charge carrier concentrations $n=0.70\times 10^{20}$ and $0.93\times 10^{20}$ 1/cm3 of two orbits in Shubnikov–de Haas (SdH) oscillations, and effective masses of $m=(1.0-1.2)m_{0}$ extracted in a Liftshitz–Kosevich analysis [89]. WTe2: from slopes of linearly dispersing (surface) bands in ARPES [90], giving $(1.5-2.1)\times 10^{5}$ m/s and from a Weyl orbit in SdH oscillations [91] on 14-layer thick exfoliated WTe2, giving $v=3.09\times 10^{5}$ m/s. TaIrTe4: from ARPES revealing linearly dispersing surface states with a slope of 2 eVÅ (or 1 eVÅ as given in the text) [92], yielding a Dirac/Weyl velocity of $v=3.04\times 10^{5}$ m/s (or $1.52\times 10^{5}$ m/s). Compound |  $\mathcal{E}_{xy}^{2\omega,\rm{intr}}/(\mathcal{E}_{x}^{\omega})^{2}$ (m/V) | $n_{(h)}$ | $(v/v_{0})^{-1}$
---|---|---|---
Ce3Bi4Pd3 |  3E-3 [41] |  8.2E19 (cm-3) [41, 45] |  1130 [38]
(Pb1-xSnx)1-yInyTe |  4.35E-4 [80] |  2E16 (cm-3) [80] |  5.9 [84], 8.3 [80]
$T_{\rm d}$-MoTe2 |  1.2E-6 [78] |  7.7E19 (cm-3) [78] |  6.2-8.1 [89]
$T_{\rm d}$-WTe2 |  (0.15-1)E-9 [72] |  (1.49-2.18)E13 (cm-2) [72] |  3.2 [91], 4.8-6.7 [90]
$T_{\rm d}$-TaIrTe4 |  1.8E-8 [73] |  6.3E19 (cm-3) [73] |  3.3-6.6 [92]

##  6 Discussion and outlook

We have investigated the role of strong correlations in topological
semimetals. As a starting point, we used the recently discovered time-
reversal-invariant but inversion-symmetry-broken Weyl–Kondo semimetal
Ce3Bi4Pd3 [38, 39, 40, 41]. We reviewed its topological signatures in both
thermodynamic and transport measurements, namely (i) a “giant” value of the
electronic specific heat coefficient $\Gamma=\Delta C/T^{3}$ of Dirac-like
quasiparticles, which is associated with ultraslow quasiparticle velocities
$v\propto\Gamma^{-1/3}$ and thus ultraflat linearly-dispersing bands [38];
(ii) an equally giant value of the intrinsic nonlinear (spontaneous, i.e.
$B=0$) Hall effect arising from the Berry curvature monopoles at the Weyl
nodes [41]; (iii) a continuation of this zero-field Hall effect as an even-
in-$B$ component, confirming that magnetic field is not the cause of the
effect [41]; (iv) a clearly identified odd-in-$B$ anomalous Hall effect due to
the Berry curvature induced by a magnetic field [41]. We have explained the
understanding of these effects in terms of a Weyl–Kondo semimetal model [39,
40] where, at the appropriate filling, Weyl nodes appear in the immediate
vicinity of the Fermi level and are associated with Weyl bands with ultraflat
dispersion [38, 39, 40, 41].

We have produced a temperature–magnetic field phase diagram that delineates
the region of Weyl–Kondo semimetal signatures, using the (high-temperature)
onset temperature $T_{\rm W}$ of the $\Delta C\propto T^{3}$ dependence as the
“phase” boundary (note that a Weyl semimetal is not a phase in the
thermodynamic sense). With increasing field, this boundary is suppressed to
zero at a critical field $B_{\rm c}$, which is understood in terms of a
Zeeman-coupling induced motion of Weyl nodes in momentum space until a Weyl
and its anti-Weyl node meet and annihilate [45, 46]. We have also included the
magnitudes of the topological signatures (i)-(iv), scaled to their maximum
values, in this phase diagram. Whereas $\Gamma$ remains essentially constant
within the boundary, the Hall signatures get successively suppressed towards
$B_{\rm c}$. This behavior indicates that, with increasing field, the Weyl
nodes move at constant energy in momentum space, without an appreciable change
of the slope of the Weyl bands, until they meet and annihilate at $B_{\rm c}$
[45], in good agreement with theoretical expectations [46].

The key aspects that make the Weyl–Kondo semimetal Ce3Bi4Pd3 a prime example
for correlation-driven metallic topology are summarized as follows:

  * •

Its Weyl–Kondo semimetal phase is well delineated: It emerges only at low
temperatures as the material becomes fully Kondo coherent, and is suppressed
at a readily accessible magnetic field as the Weyl nodes annihilate.

  * •

Its Weyl–Kondo bands reside within a Kondo insulating gap: This eliminates
contributions from topologically trivial “background” bands to a large extent,
aiding the identification of topological signatures; in addition, it pins the
Fermi level to the immediate vicinity of the Weyl nodes.

  * •

Its Weyl–Kondo semimetal signatures are “giant”: The orders of magnitude mass
renormalization of Schrödinger-like quasiparticles known from heavy fermion
compounds is inherited by the Weyl quasiparticles in terms of a corresponding
band flattening, Weyl velocity suppression, and Weyl density of states
enhancement.

We have searched the literature for other candidate Weyl–Kondo semimetals and
considered the noncentrosymmetric compounds YbPtBi, CeAlGe, and Ce3Rh4Sn13 as
promising candidates because they all exhibit temperature and field ranges
with $\Delta C/T\propto T^{2}$ behavior with large slopes [53, 54, 58]. The
phase diagrams that we constructed, however, show several differences from the
one of Ce3Bi4Pd3, namely: (i) the (putative) phase boundaries are stabilized
as opposed to suppressed with increasing magnetic field; (ii) the (putative)
Weyl velocities are significantly increased with field as opposed to
essentially unchanged in Ce3Bi4Pd3; (iii) no spontaneous or even-in-field Hall
effect is detected; (iv) the odd-in-field Hall effect detected in one of the
materials [53] seems to appear outside the (putative) phase boundary. As a
further consistency check, we estimated effective (Schrödinger) masses and
Dirac (or Weyl) velocities of the candidate materials. Whereas the expected
renormalization ratio of order unity was found for Ce3Bi4Pd3, much stronger
Weyl than Schrödinger renormalizations would have to be at play in the three
other compounds. This calls for further studies, to pin down whether and to
which extent other effects (e.g., antiferromagnetic magnons of CEF splitting
in large fields) intervene.

Finally, as Ce3Bi4Pd3 is so far the only Weyl–Kondo semimetal in which a
spontaneous or even-in-field Hall response has been identified, we resorted to
a comparison with noninteracting systems. To the best of our knowledge, the
only candidate time-reversal symmetric noninteracting Weyl semimetals that
have shown evidence for an intrinsic (Berry-curvature-related) NLHE are
exfoliated thin films of $T_{\rm d}$-MoTe2 [78], $T_{\rm d}$-WTe2 [72], and
$T_{\rm d}$-TaIrTe4 [73], as well as carrier-concentration-optimized
ferroelectric (Pb1-xSnx)1-yInyTe [80]. The quantity that conveniently
benchmarks the size of this effect is
$\mathcal{E}_{xy}^{2\omega,\rm{intr}}/(\mathcal{E}_{x}^{\omega})^{2}$, where
$\mathcal{E}_{x}^{\omega}$ is the applied electric field and
$\mathcal{E}_{xy}^{2\omega,\rm{intr}}$ the intrinsic part of the resulting
transverse electric field at double frequency. A comparison of all available
data reveals that
$\mathcal{E}_{xy}^{2\omega,\rm{intr}}/(\mathcal{E}_{x}^{\omega})^{2}$ is
drastically enhanced by strong correlations. Furthermore, as it also increases
with decreasing charge carrier concentration, a strategy for further boosting
the intrinsic topological Hall response is to reduce the carrier concentration
of strongly correlated Weyl semimetals. We propose gating experiments on thin
films as a promising strategy to explore this route.

An interesting topic for further studies across the correlation spectrum are
nonlinear optical responses, as seen in several noninteracting/weakly
interacting Weyl semimetals and discussed also in terms of their potential for
applications [73, 93, 94]. Strongly correlated Weyl semimetals might amplify
such responses and reduce the pertinent energies, thereby enabling e.g. non-
reciprocal devices and rectification in the microwave regime.

We hope that our comparison of the key characteristics of the Weyl–Kondo
semimetal Ce3Bi4Pd3 with features of other candidate materials provides
valuable guidance to discover new strongly correlated Weyl semimetals. This
would allow the determination of universal aspects in Weyl–Kondo semimetals,
such as the dependence of the magnitude of the nonlinear Hall response with
the Weyl velocity (correlation strengths), charge carrier concentration
(distance of the nodes from the Fermi energy), and potentially other factors
such as the node vs anti-node separation in momentum space, tilting, and
multiplicity of the Weyl nodes. This, in turn, may motivate further
theoretical development and, more generally, boost progress toward a broader
understanding of correlation-driven topological semimetals across different
materials classes, as well as the development of technological applications.

## Acknowledgements

We thank J. Cano, J. Checkelsky, G. Eguchi, S. Grefe, A. Prokofiev, Q. Si, X.
Yan, and D. Zocco for fruitful discussions, which were in part conducted at
the Kavli Institute for Theoretical Physics at UC Santa Barbara. This work was
supported by the Austrian Science Fund (I5868 - FOR 5249 QUAST, F86 - SFB
Q-M&S), the European Union’s Horizon 2020 Research and Innovation Programme
(824109, EMP), and the European Research Council (ERC Advanced Grant
101055088, CorMeTop), and in part by the US National Science Foundation (Grant
No. NSF PHY-1748958). For the purpose of open access, the authors have applied
a CC BY public copyright licence to the Author Accepted Manuscript version
arising from this submission.

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