## Introduction

Key to uncovering the richness of phenomena displayed by strongly correlated electron systems and to understanding the underlying mechanisms has been the great tunability of these systems, which enabled a systematic exploration of the landscape of their low-temperature phases1,2,3,4. As an example, such studies led to the discovery that high-temperature superconductivity is in many, if not all cases an emergent phase stabilized by quantum critical fluctuations5,6, and that critical electron delocalization transitions7,8 may play an important role therein9. Such deep insight is indispensable to tailoring properties at will, and ultimately exploiting them for applications.

Electrons are strongly correlated if their mutual Coulomb repulsion reaches or exceeds the same order as their kinetic energy. In bulk materials, this regime is typically realized by means of f and/or d orbitals with a suitable strength of orbital overlap. Flat bands, renormalized by orders of magnitude compared to simple metals and pinned to the Fermi energy, can be achieved via the Kondo effect, driven by a spin exchange interaction between localized (typically 4f) and itinerant (s, p, d) electrons10. Recently, it has been demonstrated that such very strongly correlated conductors may also exhibit extreme signatures of nontrivial topology11. This raises the question as to whether the excellent tunability in terms of correlation physics can also be exploited to control topology per se.

Our work shows that this can indeed be achieved. Upon tuning Ce3Bi4Pd312, a Weyl-Kondo semimetal11,12,13, by magnetic field we observe a continuous suppression of the giant topological response associated with the material’s Kondo-driven Weyl nodes, and the annihilation of the nodes. This transition is characterized by the suppression of singularities in the Berry curvature instead of a Landau order parameter; as such, it is a topological quantum phase transition. An important and surprising aspect is that this transition happens in an only smoothly varying correlated background; one could instead have expected topological states to be essentially inert to tuning parameter changes. This discovery opens up a new regime in the correlation–topology interplay, and will facilitate a systematic search for new correlation-driven topological phases.

## Results and discussion

Before presenting our results and explaining them in detail, we summarize the key signatures of the Weyl-Kondo semimetal Ce3Bi4Pd3 in zero magnetic field11,12 (see Supplementary Note 1 for details). They are (i) an electronic specific heat coefficient ΔC/T that is linear in T2, with a giant slope, evidencing ultraslow quasiparticles with linear electronic dispersion12; and (ii) a giant spontaneous Hall effect as a result of Weyl nodes—sources and sinks of Berry curvature—pinned to the immediate vicinity of the Fermi level11. We note that the semimetallic ground state of Ce3Bi4Pd3 is well documented (Supplementary Note 2). As will be shown in what follows, the momentum-space separation of the Weyl nodes, which are positioned in a Kondo insulating background, is successively reduced with increasing magnetic field, until the Weyl nodes meet in momentum space and annihilate. Only at considerably larger magnetic fields, the Kondo insulator gap is quenched and the system becomes a heavy fermion metal (see Supplementary Note 3 and Supplementary Fig. 2 for a cartoon of the correlated electronic bandstructure). The latter transition has also been observed in14.

Figure 1 gives an overview of our electrical transport and specific heat data. Salient transport features in zero magnetic field are an electrical resistivity that increases moderately with decreasing temperature (Fig. 1a) and a linear-response Hall coefficient with two ranges of thermally activated behavior and a saturation to a constant value at the lowest temperatures (Fig. 1b). A plausible interpretation, which will receive further support from the field-dependent data presented below, is that this behavior results from a (pseudo)gapped background density of states, within which a narrow (Kondo insulator) gap forms at lower temperatures, against which a small residual density of states associated with the Fermi-level-bound Weyl nodes becomes apparent at the lowest temperatures (see cartoon in Supplementary Fig. 2). The presence of a (pseudo)gap in the noninteracting density of states is in agreement with density functional theory (DFT) calculations, although the theoretical gap size is considerably larger15.

The application of magnetic fields gradually suppresses the low-temperature resistivity upturn. This is the case until, ultimately, metallic behavior is seen, albeit with higher resistivity and a different temperature dependence than in the nonmagnetic reference compound La3Bi4Pd3 (Fig. 1a). This indicates that Kondo physics is at play even at our largest field of 37 T. Isothermal magnetic-field-dependent measurements of the electrical resistivity ρxx(B) (Fig. 1c) and Hall resistivity ρxy(B) (Fig. 1d) reveal that this field-induced transformation occurs in two stages. The signatures thereof are most pronounced in the lowest-temperature data. Here, the electrical resistivity displays a shoulder at about 9 T and a crossover to almost field-independent behavior at about 14 T (see arrows labeled Bc1 and Bc2, respectively, and Supplementary Note 4 for further analyses). The corresponding signatures in the Hall resistivity are kinks at the same two fields. A quantitative analysis, presented further below (Fig. 2), reveals that this behavior reflects a two-stage Fermi surface reconstruction at two quantum phase transitions.

We first examine the effect the magnetic field has on the material’s topological characteristics. The most direct signature is a giant spontaneous as well as even-in-field nonlinear topological Hall effect, which evidences Berry curvature singularities from Weyl nodes in close vicinity to the Fermi energy11. Somewhat more indirect evidence is a temperature-dependent electronic specific heat that varies linearly in T3, with a slope that even surpasses the (Debye-like) phonon contribution, and evidences linearly-dispersing electronic bands with ultralow velocity12,13. Together they have established the inversion symmetry (IS)-broken (noncentrosymmetric and nonsymmorphic) but time reversal symmetry (TRS)-preserving heavy fermion compound Ce3Bi4Pd3 as a model case of a strongly correlated topological semimetal11,12 (see Supplementary Note 1 for further details).

In Fig. 1e we show how isothermal even-in-field (symmetrized and corrected for contact misalignment, see Supplementary Note 6 with Supplementary Fig. 5, and Ref. 11) topological Hall resistivities $${\rho }_{{{{{{{{\rm{x}}}}}}}}y}^{{{{{{{{\rm{even}}}}}}}}}$$ are successively suppressed by magnetic field. The apparent fine structure in this suppression, seen in the isotherms below 2 K, may reflect various regimes of Weyl node configurations in momentum space. Indeed, a rich sequence of Weyl node motion and annihilation under magnetic field tuning was found in Kondo model calculations on a diamond lattice16,17, which is an interesting topic for further investigations. Here, we focus on the ultimate total suppression of the effect, which occurs at $${B}_{{{{{{{{\rm{H}}}}}}}}}^{{{{{{{{\rm{even}}}}}}}}}$$ (see arrow in Fig. 1e for the lowest temperature isotherm). Also the linear-in-T3 electronic specific heat (corresponding to an electronic specific heat coefficient ΔC/T = ΓT2, see Supplementary Note 5 for details) is successively suppressed by magnetic fields, which we quantify by the parameter TC/T (see arrow in Fig. 1f on the zero-field curve). Interestingly, this suppression happens at constant Γ, indicating that the shape (slope and energy) of the Weyl dispersion remains unchanged as the Weyl nodes move in momentum space. Finally, we point to another feature that accompanies these two key signatures. It appears as an anomaly in the (normal, antisymmetrized and thus odd-in-field) Hall resistivity isotherms (see arrow denoting $${B}_{{{{{{{{\rm{H}}}}}}}}}^{{{{{{{{\rm{odd}}}}}}}}}$$ as upper end of the grey shading, marked on the lowest-temperature isotherm in Fig. 1d) and is known as the anomalous topological Hall effect in TRS-broken Weyl semimetals18. It is associated with a magnetic field-induced even-in-momentum Berry curvature, which is in addition to the intrinsic (zero-field) odd-in-momentum Berry curvature of Ce3Bi4Pd3 (see also Ref. 11).

All these results together establish that magnetic field quenches the topological response, apparently via a process that moves the Weyl nodes at equal energy in momentum space. This raises the following questions: Which mechanism underlies this effect? Does magnetic field suppress the Kondo effect just as increasing the temperature above the Kondo coherence scale does11, thereby removing the correlated electrons and thus the basis for the Weyl-Kondo semimetal formation? Or did we succeed to annihilate the Weyl nodes in an intact Kondo coherent system? The Kondo-like electrical resistivity in 37 T suggests that some form of Kondo correlations persist. To show that the magnetic field indeed controls Kondo-driven Weyl nodal excitations, however, what needs to be established is that the Kondo effect as realized in zero magnetic field operates over the entire field range with topological response. We now turn to the search for such evidence.

We start with a quantitative analysis of the (normal, antisymmetrized) Hall resistivity isotherms ρxy(B) of Fig. 1d. As established previously7, when magnetic field drives transitions between ground states with different Fermi volumes, the resulting (finite temperature) crossovers manifest as (broadened) kinks in ρxy(B), and (broadened) steps in the differential Hall coefficient $${\tilde{R}}_{{{{{{{{\rm{H}}}}}}}}}(B)=\partial {\rho }_{xy}(B)/\partial B$$. Such behavior has been observed in a number of heavy fermion metals7,19,20,21 driven by magnetic field across quantum critical points. From fits with a phenomenological crossover function7 (see Supplementary Note 7 for details) one can extract not only the $${\tilde{R}}_{{{{{{{{\rm{H}}}}}}}}}$$ values associated with the different phases, but also the crossover fields B* and sharpnesses, quantified by the full width at half maximum (FWHM). In Fig. 2a, b we show two representative fits, for data at 0.5 K (for which we have subtracted the above-discussed anomalous topological Hall effect contribution) and 1.9 K, respectively. Fits of similar quality are obtained at all temperatures up to 10 K (Supplementary Fig. 6). At higher temperatures, we lose track of the two-stage nature and thus this model does no longer give meaningful results. Note that anomalous (nontopological) Hall contributions and multiband effects do not play significant roles here (see Supplementary Notes 8 and 9). The temperature-dependent fit parameters are shown in Fig. 2c–f. The two crossover fields $${B}_{1}^{*}$$ and $${B}_{2}^{*}$$ (Fig. 2b), determined for all available isotherms, are plotted as characteristic temperatures $${T}_{1}^{*}(B)$$ and $${T}_{2}^{*}(B)$$ in Fig. 2c. The extrapolations to T = 0 of these curves identify Bc1 and Bc2 (see arrows in Fig. 2c). Both crossovers sharpen considerably with decreasing temperature (Fig. 2e, f), indicating that the phase diagram of magnetic field-tuned Ce3Bi4Pd3 comprises three phases with distinct Fermi volumes: a phase below Bc1 with a small hole-like Fermi volume, an intermediate-field phase between Bc1 and Bc2 with an even smaller electron-like Fermi volume, and a high-field phase beyond Bc2 with a much larger Fermi volume.

To understand this behavior and elucidate the character of these phases we have carried out torque magnetometry measurements. At low magnetic fields, and in particular across $${B}_{1}^{*}(T)$$, no sizable torque signal is detected (Fig. 3a), thus ruling out that a magnetic phase transition occurs at this field (see Supplementary Note 10 and Supplementary Fig. 10). A pronounced torque signal appears only above about 14 T; it corresponds to the onset of nonlinearity in the magnetization14. Similar behavior, albeit with a much larger magnetic field scale, is also seen in the canonical Kondo insulator Ce3Bi4Pt3, which we have studied for comparison (Fig. 3b). The corresponding characteristics in the first and second derivative with respect to the magnetic field are a step-like increase and a maximum, respectively. For Ce3Bi4Pt3, the step in the lowest temperature isotherm occurs at 38.9 T (blue arrow in Fig. 3c), which is close to the field where a Kondo insulator to metal transition has previously been evidenced by a jump of the Sommerfeld coefficient22. The very similar feature seen for Ce3Bi4Pd3 (red urve in Fig. 3c) then suggests that also in this system a Kondo insulator to metal transition takes place, albeit at much lower fields. As characteristic field Bτ of this transition, which is well-defined for all isotherms, we use the middle field at half height of the second derivative curves (Fig. 3d).

We also performed temperature-dependent electrical resistivity measurements at fields around this Kondo insulator to metal transition. On the high-field side of the transition, we observe Fermi liquid behavior, ρ = ρ0 + AT2 (see Fig. 3e, bottom and left axes, for data at 15 T). This confirms that Ce3Bi4Pd3 has indeed metallized. The A coefficient measures the strength of electronic correlations. Values in the range of several μΩcm/K2, as observed here, are typical of heavy fermion metals23 (see Supplementary Note 11 for details). Thus, the quenching of the Kondo insulator gap by the magnetic field has indeed (as already indicated by the high-field resistivity curves in Fig. 1a) still not suppressed the Kondo interaction. Thus, what happens at Bc2 is a field-induced Kondo insulator to heavy fermion metal transition, which was also studied in Ref. 14. In Fig. 3f we plot the A coefficient determined also for other fields (see Supplementary Fig. 11) as function of the magnetic field. Upon approaching the transition from the high-field side, the A coefficient increases, and is well described by a divergence, A 1/(B − Bc2), with the critical field Bc2 = 13.8 T from the Hall resistivity analysis (see caption of Fig. 3 for details). Such behavior is known from heavy fermion metals tuned by a magnetic field to a quantum critical point (QCP)21,24. In this case, also non-Fermi liquid (NFL) behavior should develop25,26, which we indeed observe in the form of a linear-in-temperature resistivity at 15 T, at temperatures above the Fermi liquid behavior seen at the lowest temperatures (Fig. 3e, top and right axes). This confirms that, at 15 T, Ce3Bi4Pd3 is slightly away from a QCP. Closer to the expected quantum critical field Bc2, the resistivity appears to be influenced by the nearby Kondo insulator phase, as seen by a rapid suppression of the A coefficient and a crossover to T2 behavior with a negative slope (Supplementary Fig. 11h), as well as an increase of the residual resistivity ρ0 not only towards but even across Bc2 (Supplementary Fig. 11i). Whether these (nonmetallic) characteristics are generic to field-induced Kondo insulator to heavy fermion metal transitions is an interesting topic for future studies (we note that in a related pressure-induced transition in SmB6 no such effects were seen27,28, but this may reflect the need of extending those measurements down to the temperature range of dilution refrigerators). Clearly, they do not represent normal (metallic) behavior and thus we define the upper boundary of Fermi liquid behavior TFL (see arrows in Fig. 3e and Supplementary Fig. 11) only in the field range where these effects have minor influence. We conclude that a heavy Fermi liquid phase exists at fields above Bc2. In turn, this implies that the Kondo effect as operating at B = 0 remains intact across Bc1.

We are now in the position to construct a temperature-magnetic field phase diagram that assembles the above-discussed characteristics (Fig. 4a). At low fields, these are TC/T(B), $${T}_{{{{{{{{\rm{H}}}}}}}}}^{{{{{{{{\rm{even}}}}}}}}}(B)$$, and $${T}_{{{{{{{{\rm{H}}}}}}}}}^{{{{{{{{\rm{odd}}}}}}}}}(B)$$, all denoting temperatures up to which, at a given field, a certain signature of Weyl-Kondo semimetal behavior is detected. These scales all collapse at a critical field Bc1 of 9 T. Importantly, the correlated background remains essentially unchanged across this field and no magnetic phase transition takes place. At high fields, we plot the scales Tτ(B) and TFL(B) associated with the torque anomaly and Fermi liquid behavior, respectively, and also include the inverse of the Fermi liquid resistivity coefficient, 1/A, which hits zero at the effective mass divergence. In addition, we show the $${T}_{1}^{*}(B)$$ and $${T}_{2}^{*}(B)$$ scales extracted from our quantitative Hall resistivity analysis. Note that all these temperature scales are crossovers; they do not represent phase boundaries in the thermodynamic sense as none of the phases is associated with an order parameter.

In Fig. 4b we plot the charge carrier concentration n (in units of carriers per Ce atom) of the three phases, extracted in a single-band model from the lowest-temperature values of the differential Hall coefficients (Fig. 2d). It changes from about 0.01 holes  below Bc1, via 0.007 electrons between Bc1 and Bc2, to 0.18 electrons beyond Bc2. The 25-fold increase across Bc2 is independent evidence for the above-discussed Kondo insulator to heavy fermion metal transition. The more modest reduction of the absolute value of the carrier concentration across Bc1 indicates a Fermi surface reconstruction that involves only a fraction of momentum space. The annihilation and associated gapping-out of Weyl nodes is exactly such a phenomenon. Also the sign change of n across Bc1 can be understood in this scenario (see Supplementary Note 12 and Supplementary Fig. 12). That n is field independent below Bc1 (as seen from the linear-in-field behavior of the Hall resistivity in this regime, Fig. 2b) confirms that magnetic field moves the Weyl nodes at constant energy. The only process that can then remove them is their mutual annihilation, which we thus propose to happen at Bc1. Importantly, this takes place in a background of unchanged symmetry and with the Kondo effect continuing to operate. In other words, we have realized a controlled suppression of the Kondo-driven topological semimetal. A corollary is that we have isolated the tuning of topology from the tuning of correlation physics as such.

In Fig. 4c we summarize these findings in a schematic zero-temperature phase diagram. The low-field phase is a Weyl-Kondo semimetal (WKSM)11,12,13 that, as evidenced here, consists of Weyl nodes situated within the narrow energy gap of a Kondo insulator. As function of magnetic field a sequence of two Fermi volume-changing quantum phase transitions is observed. At Bc1, all signatures of the Weyl-Kondo semimetal disappear as the Weyl nodes annihilate in a topological quantum phase transition. At Bc2, the Kondo insulator with gapped-out Weyl nodes (denoted by the symbol Δ) transforms into a heavy fermion (HF) metal, in a transition that displays signatures of quantum criticality and must thus be at least nearly continuous.

This raises the question of why the phenomena of genuine topology tuning and Weyl node annihilation have remained elusive in the much more extensively studied noninteracting regime29? A magnetic field can either act on the spin (via the Zeeman effect) or the charge of an electron (via the orbital effect). In Kondo systems the former dominates, in high-mobility semimetals the latter. The momentum space motion of Weyl nodes requires sizable Zeeman coupling16,17. In its absence30, field-driven changes of topological signatures31,32 can arise from orbital effects such as the tunneling between zeroth Landau level states of adjacent Weyl nodes31, at finite Weyl node separation. Of course, also changes in a material’s broken symmetry state can be accompanied by changes of topology33, but this is not the topic of interest to us here (see Supplementary Notes 13 and 14 for further details).

In summary, we have demonstrated the genuine control of Weyl nodes, with clarity and ease. The clarity is attributed to the fact that in Ce3Bi4Pd3 the Weyl nodes form within a (strongly correlated, Kondo insulating) gapped state and are positioned in close vicinity of the Fermi energy. Because topologically trivial states are gapped out, there is no need to disentangle Weyl fermions from topologically trivial carriers, which hampers the field of weakly interacting Weyl semimetals. The ease of control—namely that a rather modest field of 9 T was not only enough to manipulate the positions of the Weyl nodes in momentum space, but even drive their annihilation—shows that the well-known excellent tunability of (topologically trivial) strongly correlated electron systems1,2,3,4 holds also for topological features in such systems. As such, our study lays the ground for establishing a global phase diagram for strongly correlated topological materials. Key open questions to address include whether the phases and transitions discovered here exist also in other materials and are thus universal, and whether quantum criticality plays an important role in stabilizing them. The latter is hinted at by recent inelastic neutron scattering experiments34.

Our findings may also guide investigations in related materials classes. Much effort is currently devoted to artificial materials such as twisted bilayer systems where correlations can be enhanced via a moiré potential35,36,37,38. Additional tuning knobs in such heterostructures are the dielectric displacement and electrostatic doping, which may become powerful if further advances towards highly reproducible structures can be accomplished. Finally, we point to the potential of strongly correlated bulk materials such as Ce3Bi4Pd3 for quantum devices4,39,40,41, where the robust and giant topological response—together with the high level of topology control demonstrated here—opens new opportunities. As an example we name microwave nonreciprocity at zero magnetic field, a key functionality needed in circuit quantum electrodynamics systems42, that could be realized via the spontaneous Hall response of Ce3Bi4Pd3.

## Methods

### Synthesis

Single crystals of Ce3Bi4Pd3, Ce3Bi4Pt3, and of the nonmagnetic reference compound La3Bi4Pd3 were grown using the flux method. For Ce3Bi4Pt3, elementary Ce, Pt, and Bi in an atomic ratio of 1:1:7 were placed in an alumina crucible, and heated to 1100C in a vacuum-sealed quartz tube, using a box furnace. The melt was then slowly cooled to 600C after a dwell time of 12 h at 1100C, with a cooling rate of 1C/h, and then left annealing for 12 h. Crystals of typically 1 mm in diameter were then extracted from the melt using a centrifuge. For Ce3Bi4Pd3 and La3Bi4Pd3, as the primary stable phases at Bi excess are CeBi2Pd and LaBi2Pd, the Bi content was strongly reduced to about 1:1:1.5 (see also Ref. 12). The chemical composition and crystal structure of the samples were determined by energy dispersive x-ray spectroscopy and powder x-ray diffraction. Laue diffraction was utilized to determine the crystallographic orientation of selected samples.

### Measurement setups

#### High-field experiments

Magnetoresistance, Hall effect, and torque magnetization measurements in DC fields up to 37 T were performed at the HFML-EMFL facility at Nijmegen. Magnetotransport data were measured using Stanford Research SR830 lock-in amplifiers, with the measured voltage signal pre-amplified 100 times using Princeton Applied Research low-noise transformers. Electrical contacts where made by either spot welding or gluing with silver paint 12 μm diameter gold wires to the samples in a 5-wire configuration. All displayed Hall resistivity curves were obtained by the standard antisymmetrizing procedure of the resistivity $${\rho }_{xy}^{{{{{{{{\rm{meas}}}}}}}}}$$ measured across the Hall contacts, i.e., $${\rho }_{xy}(B)=[{\rho }_{xy}^{{{{{{{{\rm{meas}}}}}}}}}(+B)-{\rho }_{xy}^{{{{{{{{\rm{meas}}}}}}}}}(-B)]/2$$. This cancels out both the spontaneous Hall effect (zero-field signal) and any even-in-field component. Torque and magnetization measurements in pulsed fields up to 65 T were performed at the NHMFL-LANL facility at Los Alamos. Magnetization data were obtained only for Ce3Bi4Pt3, using an extraction magnetometer and the "sample-in/sample-out” technique to separate the sample signal from the background. In this case, the magnetic field was not aligned to any particular crystallographic axis. In all torque experiments, piezo-cantilevers were used for enhanced sensitivity. Samples were attached to the levers with Dow Corning high vacuum grease. The measured torque signal was obtained after balancing a Wheatstone bridge containing the resistance of the sample’s cantilever and a "dummy” (empty lever) resistor. Due to the small samples required for this technique, the magnetic field was initially not aligned to any particular crystallographic axis. A sample rotator was used to scan the torque signal across different orientations. In all cases, temperatures down to 0.35 K were obtained using a 3He cryostat.

#### Low-temperature experiments

Additional magnetoresistance, Hall effect, and specific heat measurements were obtained in Vienna using a Quantum Design Physical Property Measurement System, equipped with 3He options. Low-temperature electrical resistivity measurements down to 70 mK and in magnetic fields up to 15 T were performed in an Oxford dilution refrigerator.