Field manipulation of Weyl modes in an ideal Dirac semimetal

Domain: physics
Published: 2025-11-28
Editorial outcome: major_revision

Abstract
The emergent Weyl modes with the broken time-reversal symmetry or inversion symmetry provide large Berry curvature and chirality to carriers, offering the realistic platforms to explore topology of electrons in three-dimensional systems. However, the reversal transition between different types of Weyl modes in a single material, which is of particular interest in the fundamental research in Weyl physics and potential application in spintronics, is scarcely achieved due to restriction of inborn symmetry in crystals. Here, by tuning the direction and strength of magnetic field in an ideal Dirac semimetal, Bi4(Br0.27I0.73)4, we report the realization of multiple Weyl modes, including gapped Weyl mode, Weyl nodal ring, and coupled Weyl mode by the magnetoresistivity measurements and electronic structure calculations. Specifically, under a magnetic field with broken mirror symmetry, anomalous Hall effect with step feature results from the large Berry curvature for the gapped Weyl mode. A prominent negative magnetoresistivity is observed at low magnetic field with preserved mirror symmetry and disappears at high magnetic field, which is correlated to the chiral anomaly and its annihilation of Weyl nodal ring, respectively. Our findings reveal distinct Weyl modes under the intertwined crystal symmetry and time-reversal breaking, laying the foundation of manipulating multiple Weyl modes in chiral spintronic network.

Introduction
Symmetry and topology are symbiotic in condensed matter physics, and their interplay is a fundamental aspect classifying the different topological phases such as topological insulator, topological nodal point/line/surface semimetal, and topological crystalline insulator, etc1–5. The topological phase transitions are expected to be generated when the symmetry of a system is broken by varying the ordering parameters or introducing the external perturbation. For example, Weyl physics can be induced by breaking inversion symmetry (IS) or time-reversal symmetry (TRS) in a Dirac semimetal, along with the abundant intriguing phenomena like anomalous Hall effect (AHE), unconventional thermal magnetic responses, negative magnetoresistivity (NMR) by chiral anomaly, and three-dimensional (3D) quantum Hall effect6–13. Despite numerous Weyl materials revealed in non-centrosymmetric or magnetic systems so far10–14, a symmetry guideline for searching for various Weyl modes and designing their evolution is still unrevealed, which is crucial for the manipulation of Weyl fermions towards the chiral device application.

Among various modulation ways, the magnetic field is regarded as an effective and reversible method to evoke topological phase transitions, which has been theoretically proposed in the 3D warped semimetals15 and experimentally verified in HfTe5 (from topological insulator to quasi-one-dimensional (quasi-1D) gapped Weyl mode)10,14,16,17. Here, we propose a strategy to realize the magnetic field manipulation of various Weyl modes from a Dirac semimetal (DSM). As shown in Fig. 1a, a DSM with IS and TRS possesses the band structure with four-fold degenerate Dirac point (DP). Introducing magnetic field B can break TRS and lift the spin degeneracy, engendering the split of Dirac bands into two Weyl bands with opposite spins. The chiral symmetry for Weyl fermions can be also dictated by the crystalline symmetries. If B breaks the symmetry maintaining the chiral feature, the disturbance of spin-orbital coupling (SOC) leads to a gapped Weyl mode with the energy gap \Delta_W as shown in Fig. 1b. When B preserves some crystal symmetry, e.g., along the normal direction of a mirror plane M, the gapless Weyl points (WPs) or Weyl nodal ring (WNR) can be formed, as displayed in Fig. 1c. Noticeably, magnetic field tends to circulate and localize electrons in real space, which broadens the spreading of wavefunctions (denoted by the gray peaks) in momentum space18–20. These wavefunctions can overlap with each other in momentum space when the B is large enough. Consequently, the chiral charges are able to tunnel between chirality-opposite WPs to open the magnetic tunneling gap \Delta_T without breaking chiral symmetry in Fig. 1d. As a result, the magnetic field manipulation of various Weyl modes can be realized based on this symmetry guideline. Nevertheless, these topological phase transitions cannot be directly measured by angle-resolved photoemission spectroscopy (ARPES) due to the influence of magnetic field. As an alternate, the magnetoresistivity measurement has been applied to identify the different Weyl modes in previous reports18,20. Although the previous work reports the realization of Weyl modes in various materials, their transport behaviors are unavoidably disturbed by irrelevant, conventional electrons4,7,10–14, hindering the observation of the multiple Weyl modes in a single material. Therefore, searching for an ideal DSM with only Dirac bands near Fermi surface to exclude the contribution from other bands in transport measurements is highly required to experimentally explore the chiral physics.

In this work, we identify an ideal van der Waals DSM material Bi4(Br0.27I0.73)4 with only Dirac bands residing in the Fermi level, which is a topological phase transition critical point bridging higher-order topological insulating phase and dual topological order21. The magnetoresistivity, Hall measurements, and first-principles calculations are performed to reveal its electronic structures under a magnetic field along different directions. The AHE is observed when the magnetic field is perpendicular to (001) plane, which is correlated to the large Berry curvature in the gapped Weyl mode. Shubnikov–de Haas (SdH) oscillation and consequent NMR are explored with the magnetic field along chain direction, which are evoked by splitting of Landau level and chiral anomaly in the WNR, respectively. Continuously enhancing the magnetic field upon 21.5 T, the breakdown of chiral anomaly is observed, indicating the annihilation of the WPs. All the experimental results are supported by the calculations, and finally, a phase diagram of Weyl physics as the function of magnetic field and temperature is established, constructing the various Weyl modes into a unified picture in a single crystal quantum system and establishing the close relationship between the crystal symmetry and chiral electrons.

Results
Ideal DSM in Bi4(Br0.27I0.73)4
Bi4X4 (X=Br, I) is assembled with 1D bismuth skeleton surrounded by halide atoms under the van der Waals (vdW) force, possessing IS and a single mirror plane My (marked by black dashed line) perpendicular to the chain direction as shown in Fig. 2a. The bulk Brillouin zone (BZ) with (001) surface projected BZ is shown in Fig. 2b. This system possesses abundant topological phases, including weak topological insulator, high-order topological insulator, dual topological phase, and topological trivial insulator, by controlling mole ratio of Br and I21–35. Having the same stacking structure as Bi4Br4, a Dirac semimetal phase can be realized at the critical point of around 73% I concentration (Fig. S1 and Fig. S2), which has been demonstrated by the Dirac cones (labeled by black arrow) in both of the ARPES spectra in Fig. 2c and density functional theory calculation with virtual crystal approximation (VCA) method in Fig. 2d and Fig. S3. Astonishingly, only the bands originated from the Bi p orbits locate near the Fermi surface, evidencing Bi4(Br0.27I0.73)4 can be an ideal material platform to manipulate and investigate the multiple Weyl modes proposed in Fig. 1 (the other band residing in Fermi level will be pushed away to high energy position after introducing magnetic field). It is noticeable that the halide vacancies induce semiconducting-like transport behavior in pristine Bi4Br4 36. This localization effect becomes negligible in Bi4(Br0.27I0.73)4 due to the plenty of iterant carriers in Dirac semimetal Bi4(Br0.27I0.73)4.

AHE and large Berry curvature evoked by the gapped Weyl mode
The typical six-wire method is adopted to measure the magnetoresistivity and Hall resistivity of Bi4(Br0.27I0.73)4, as shown in the schematic in Fig. 3a, where the current flows through chain direction (lattice b axis) and Hall voltage is measured along lattice a axis. Measurement is performed with the varied angles \phi between magnetic field and My (insets of Fig. 3b). The magnetic field-dependent anomalous Hall resistivity \rho_{yx}^A for S1 in Fig. 3b shows the step-like shape with two stages instead of the linear or curved dispersion in a large \phi region ranging from 0° to 60°, which is a ubiquitous signal also identified in other crystals from different batches (see Fig. S4 in Supplementary Information). Furthermore, the \rho_{yx}^A results as a function of normal magnetic field component (B cos \phi) at different \phi collapse into a single curve, implying the signal of \rho_{yx}^A is mainly contributed by Bz component of magnetic field. The two band model is applied to fit the Hall signals, where the large deviation between the fitting lines and experimental results excludes the possibility of multiple band effect (see Fig. S5 for multiple band fitting results for different samples). The stage-like behavior in Hall resistivity surviving at high temperature (up to 300 K) and strong magnetic field (up to 57 T) (see Fig. S6), and is similar with the AHE in the magnetic system, where the stage is correlated to the saturated magnetic moment37. Nevertheless, there are no magnetic atoms and domains in this quasi-1D system, excluding the contribution of magnetic domains to the AHE (See Fig. S7 and Fig. S8 in detail). The quantum geometry of the electronic bands can also evoke the nonlinear Hall effect, where only the Berry curvature, which acts like an effective magnetic field in momentum space and conforms to the condition of the reserved IS in Bi4(Br0.27I0.73)4, contribute the linear AHE and large Berry curvature evoked by the gapped AHE7,37–46. The contribution of non-zero Berry curvature to Hall resistivity can be classified into two parts, the scattering independent part (intrinsic and side-jump contribution) and scattering dependent part (localized hopping and skew scattering contribution)37–39. In this case, the saturated Hall conductivity \sigma_{xy}^A can be expressed as:

\sigma_{xy}^A = \alpha \sigma_{yy}^{1.6} + \sigma_{xy}^{Ind}, (1)

where \alpha is a constant correlated with residual resistivity, \sigma_{yy} is the conductivity at zero field, and \sigma_{xy}^{Ind} is the anomalous Hall conductivity (AHC) correlated to the scattering independent part. Fig 3c exhibits the \sigma_{xy}^A values as a function of \sigma_{yy} measured at different temperatures, where all the points can be well fitted by Eq. (1) with a finite interception value. The validity of our fitting results is supported by the comparative study of fitting lines by using different exponents and intercepts, as shown in Fig. S9 in Supplementary Information. The 1.6 exponent indicates existence of the localized hopping conduction in dirty metals (in which conductivity less than 10,000 S/cm)40–42, and the nonzero interception value (~169 S/cm) is a direct reflection of the scattering independent contribution, \sigma_{xy}^{Ind}. It is noted that both extrinsic side-jump mechanism and intrinsic contribution are supposed to be scattering independent, which are not distinguishable here from scattering plot37,40–42. Consequently, the well fitted experimental data in Fig. 3c by Eq. (1) indicates that the step-like feature originates from both scattering dependent localized hopping effect and scattering independent contribution, where non-zero Berry curvature behaves as the source to deflect electrons41. The anomalous Hall angle (AHA) standing for the deflection angle induced by Berry curvature can be calculated by tan^{-1}(\sigma_{xy}^A / \sigma_{yy}), and is 20.8° at T = 20 K (the inset of Fig. 3c). The large AHA in Bi4(Br0.27I0.73)4 is contributed by both localized hopping conduction and scattering independent contribution of Berry curvature (see Supplementary Note 7 for detailed illustration of Berry curvature contribution to carrier deflection), which is reasonable to emerge in strongly disordered metal (i.e., bad metal with low conductivity) as shown in Fig. S10, coinciding with the high I dopants concentration (more disorders than pure Bi4Br4 sample). It should be noted that Berry curvature can be characterized by scanning-superconducting quantum interference device or reflectance magnetic circular dichroism, which requires further study in the future47–50.

The calculated band structure and intrinsic AHC correlated with Berry curvature of Bi4(Br0.27I0.73)4 under an effective magnetic field Bz = 2 T and Bz = 4 T (see “Method” for estimation of effective magnetic field) are shown in Fig. 3d and Fig. 3e, respectively. In our calculations, the external magnetic field is applied in a Zeeman-like manner (see “Calculation methods”). Although the Landau level effect is not included in the simulations, the basic physical results induced by the Berry curvature and gapless Weyl modes can be definitely determined. With the out-of-plane magnetic field breaking mirror symmetry, the gapped Weyl modes are observed, which are consistent with the schematic in Fig. 1b. Fundamentally, the relationship between \sigma_{xy}^A and \Omega_z can be written as: \sigma_{xy}^A \propto \int_{BZ} \Omega_z dk, where \Omega_z = \sum_{n \in occupied} \Omega_n^z coinciding with the emergent large AHC near gaps in Fig. 3d and Fig. 3e7,37. It should be noted that there is n-type doping effect in this quasi-1D system due to the vacancies of halide elements36, resulting in the DP locating at 0.14 eV below EF at zero magnetic field (Fig. 2c). With increasing magnetic field, the energy difference between EF and DP becomes small due to the increased degeneracy of Landau bands. By comparing the calculated AHC and experimental results, we attribute the small stage I to the Landau band surpassing EF, corresponding to small and non-saturated \Omega_z, as shown in Fig. 3d. On the contrary, stage II is robust, indicating the large and saturated \Omega_z after Bz = 4 T, which is confirmed by the calculated AHC at EF in Fig. 3e (see Fig. S11 for details). The \Omega_z distributions summed to EF in Fig. 3d and Fig. 3e are shown in Fig. 3f, where non-saturated \Omega_z with a hollow circle and saturated \Omega_z with a solid circle are observed for Bz = 2 T and Bz = 4 T respectively. The effect of n-type doping level on the AHE is discussed in Supplementary Note 10 with Fig. S12.

NMR phenomenon and the formation of WNR
The longitudinal resistivity measurements under magnetic field \rho_{yy}(B) are also performed to characterize the transport properties and to infer underlying topological features. Figure 4a shows three \rho_{yy}(B) curves with different magnetic field direction: along lattice a axis, perpendicular to (001) surface, and along lattice b axis. The curve shapes are almost the same for the first two cases, which is made up by a cusp shape induced by a weak antilocalization at low field and positive magnetoresistivity at high field24–26. Interestingly, the curve shape is different when magnetic field applied along lattice b axis. Three oscillation peaks emerge after the disappearance of cusp with the increment of magnetic field. The analysis of this oscillation in Fig. 4b exhibit the equal 1/B interval, demonstrating that the typical Shubnikov–de Haas (SdH) oscillation with oscillation frequency BF around 2.94 T is formed by the Landau level splitting with the magnetic field51. Moreover, the Landau bands index shows an intercept value within ± 1/8, implying nontrivial Berry phase of charge carriers and that the system enters into ultraquantum limit at ~ 2.6 T12,52. It is noticeable that, in another quasi-1D topological crystal ZrTe5, electrons enter ultraquantum limit at a compatible field (~1.3 T)13. The 0.5 intercept correlated to nontrivial Berry phase has also been reported43. It is due to the different definition methods of integer Landau index compared to our work. Our results are consistent with these references. The non-trivial Berry phase coincides with the Dirac fermions near the Fermi surface in Bi4(Br0.27I0.73)4. Prominent SdH oscillation only appears under B∥b configuration, which can be explained by the small effective mass and high mobility of charge carriers in the gapless linear band structure along chain direction compared to other two directions. It is noticeable that the quasi-1D in a closed Fermi surface feature allows electron hopping to produce SdH oscillations under magnetic field. The NMR emerges after the termination of the SdH oscillation as itinerant carriers are condensed into the lowest Landau bands, which is supported by the similar values between the estimated onset magnetic field of ultraquantum limit and transition field in Fig. 4a and Fig. 4b (see Supplementary Note 12 for details)8,9,51–53. Detailed angle-dependent \rho_{yy} curves under different magnetic field are shown in Fig. S13, indicating the narrow angular window for NMR. It should be noted that the charge puddle effect can induce isotropic (i.e., direction-independent) NMR phenomenon54. Such a possibility is excluded by the anisotropic NMR in Fig. 4a and Kelvin probe force microscopy (KPFM) measurements in Fig. S7 and Fig. S8. In order to figure out the origin of the NMR, the band structure under the magnetic field along chain (y) direction is calculated. Figure 4c shows the calculated E-kx-kz 3D band structure with an anisotropic WNR locating in kx-kz plane under By = 4 T. The calculated energy momentum dispersions are exhibited in Fig. 4d, where two special pairs of WPs in WNR are labeled as WP1 and WP2 along kx and ky directions, respectively. The band structure at M along magnetic field ky direction possesses an energy gap. The WNR is composed of infinite pairs of WPs, and every two WPs with opposite chirality of one pair are located on opposite sides symmetric about the center point. However, due to the quasi-1D nature of the band, the WNR exhibits an elongated distribution, with either side maintaining approximately the same chirality, i.e., WP1+ or WP1-, except for the two endpoints (WP2+ and WP2-), as displayed in Fig. 4c. Consequently, the WNR can be disassembled into infinite WP pairs, and its transport behavior falls into the same theoretical region of WPs19,55. In fact, NMR has been previously observed in the \rho_{yy} measurements of Dirac semimetals and Weyl semimetals, where an additional transport channel along current direction for charge carrier connecting two WPs is provided to enhance the conductivity under the magnetic field7–11,19. In these cases, the magnetic field is with the same direction connecting two WPs (i.e., B∥\Delta k_W). Nevertheless, it is highlighted that when magnetic field is perpendicular to \Delta k_W (i.e., B⊥\Delta k_W), NMR can still be induced by chiral anomaly as depicted in Fig. 4e, where the electric field lifts chemical potential degeneracy (\mu_+ > \mu_-) between two Weyl branches with opposite chirality as predicted in theory19. Consequently, chiral carriers with group velocity along crystal b axis are pumped and relaxed between WPs, offering additional chiral charge channels with enhanced conductivity along chain direction. Our \rho_{yy}(B) results and calculations agree with each other, demonstrating the formation of WNR by My protection as depicted in Fig. 1c and the subsequent NMR with the chiral carriers pumped to lowest Landau bands.

Annihilation of Weyl node under the high magnetic field
The \sigma_{yy} curves of Bi4(Br0.27I0.73)4 sample are measured under a pulse magnetic field up to 57 T at different temperature to investigate its transport behavior, as shown in Fig. 5a and Fig. S14. For temperature below 50 K, the NMR data between two transition points, B0 and B1, can be nicely fitted by parabolic line. The NMR effect in ultraquantum limit can be described by refs. 8,9,11,12:

\sigma_{yy} = \sigma_0 + c_w B = \sigma_0 + \frac{e^3 \tau}{4\pi^2 \hbar^2 c} B, (2)

where \sigma_{yy} is the longitudinal conductivity, \sigma_0 is residual conductivity, cw is the coefficient term, e is elemental charge, \tau is inter-node relaxation time, \hbar is the reduced Planck constant, and c is the light speed. The linear relationship between \sigma_{yy} and B can be inferred in the condition that the leading coefficient is a constant8,9. Since our results in Fig. 4b and Supplementary Note 12 indicate that the NMR region is correlated to the lowest occupied Landau bands, we believe that this inconsistency may be evoked by the magnetic field-dependent \tau. This hypothesis needs the further investigations in the future as the NMR tendency with magnetic field in ultraquantum limit region is still an open-ended question. The cusp at low field is correlated to the weak antilocalization effect, and parabolic shape corresponding to the NMR emerges at B0. \sigma_{yy} starts to diminish at B1, indicating the vanish of the contribution of chiral anomaly.

In fact, the high magnetic field perpendicular to the connecting direction of WPs can induce magnetic tunneling between WPs with the finite coupling gap at original location of WPs, which can be modeled by ref. 20:

\sigma_{yy} = \sigma_0 + \sigma_1 e^{-\Delta_T(B)/k_B T}, (3)

where \sigma_0 is conductivity from the lowest Landau band, kB is Boltzmann constant, \Delta_T(B) denotes field-dependent magnetic-tunneling gap. The Eq. (3) is applied to fit the region B > B1 giving highly identical reproduction of experimental results. The fitted gap sizes of \Delta_T(B) at different temperature are shown in Fig. 5b, which is compared to the thermal perturbation energy kBT. The comparable energy values between \Delta_T and kBT are reasonable to the opened gap cases since the gap size need to be large enough to conquer thermal broadening effect. To achieve magnetic tunneling of WPs, the distance between coupled WPs (\Delta k_W) should be comparable to the momentum expansion of DOS in WPs as the schematic in Fig. 1d18. It is reported that the inverse magnetic length has the direct relationship with the momentum expansion of DOS 1/l_B = \sqrt{eB/\hbar} \approx \Delta k_W18–20. The estimated \Delta k_W is 0.018 Å^{-1} at 4.2 K (Fig. 5b) after taking B1 = 21.5 T into account. The evolution of WNR and the distances between WP2 at By = 4 T and By = 22 T are shown in Fig. 5c, where \Delta k_W value between WP2+ in the first BZ and WP2- in the second BZ is also 0.018 Å^{-1} for By = 22 T, implying the complete chirality breakdown at B1 induced by magnetic tunneling between WP2 across the BZ boundary. In fact, the WPs with distance smaller than 0.018 Å^{-1} (i.e., two WP1 separated along kx) should be annihilated at By = 4 T indicating that the chiral anomaly discussed above mainly comes from WP2 as shown in Fig. 4e. Calculated E-k dispersions near WP2 at By = 4 T and By = 22 T are presented in Fig. 5d and Fig. 5e, respectively, where magnetic tunneling effect is absent (present) due to discrete (overlapped) DOS along kz. Figure 5f shows schematics of magnetic tunneling effect, where overlapped DOS make the lowest Landau bands couple with each other to form a gap \Delta_T, leading to chirality breakdown after B1.

Based on the above results and analysis, we plot a phase diagram as a function of temperature and magnetic field in Fig. 5g, where chiral anomaly and chirality breakdown, corresponding to the WNR and magnetic tunneling effect, respectively, can be reversibly modulated by the magnetic field.

To summarize, different Weyl modes have been modulated and identified in transport measurements by applying magnetic field, evidencing the successful engineering of highly pure Dirac or Weyl bands in bismuth halide Bi4(Br0.27I0.73)4 and the close relationship between crystal symmetry and chiral electrons. The exploration of these Weyl modes accompanied with the AHE is beneficial to the room-temperature device application of non-volatile storage and magnetic field detection. Our work simultaneously identifies the exotic phenomena, such as AHE, NMR, and annihilation of Weyl node, in one system, distinguishing from the individual exploration in other work4,7,10–12. The results reported in our work have been systematically validated through the excellent coincidence among the symmetry-based framework for designing Weyl states, magnetotransport data and fitting results, and first-principles simulations as well as the calculated parameters, which is rarely a task in previous reports. Furthermore, as the ideal Dirac semimetal with only the Dirac bands residing in the Fermi level, Bi4(Br0.27I0.73)4 provides a material platform to realize more intriguing quantum transport behaviors contributed by quantum geometry after the symmetry breakdown43–46. The non-destructive manipulation of band topology with emergent Weyl modes, chiral anomaly and its annihilation, not only inspires deeper understanding towards Weyl physics, but also promotes the application of chiral spintronic devices.

Methods
Crystalsynthesization
Solid-state reaction and the chemical vapor transport method were applied to grow the Bi4(Br0.27I0.73)4 single crystals. Highly pure Bi, BiBr3, and BiI3 powders were mixed under Ar atmosphere in a glove box and sealed in a quartz tube under vacuum, where the mole ratio of BiI3 to BiBr3 were 7:3. The mixture was placed in a two-heating-zone furnace with the temperature gradient from 558 K to 461 K for 72 h. Crystal Bi4(Br0.27I0.73)4 nucleated at the high-temperature side of the quartz after cooling down. Energy dispersive spectroscopy (EDS) was applied to determine actual doping rate, where the element ratio of I:Br is 73:27 as shown in Fig. S1.

Structuralcharacterization
XRD measurements were performed in an AERIS PANalytical X-ray diffractometer at room temperature. The cleaved (001) surface was carefully set to be parallel with the sample stage, and the divergence slit was set at 1/8° with beta-filter Ni accompanied by Cu radiation. For high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) measurement of (010) surface, the transmission electron microscopy (TEM) lamella was prepared by using a Zeiss Crossbeam 550 focused-ion-beam scanning electron microscopy. Then, the characterization was conducted on a probe and image corrected FEI Titan Themis Z microscope equipped with a hot-field emission gun working at 300 kV. As shown in Fig. S2, Bi4(Br0.27I0.73)4 has the same stacking structure as Bi4Br4.

ARPESmeasurement
ARPES measurements were performed on in-situ-cleaved thick and homogeneous crystals, attached on the sample holder with torr seal glue with the exposed (001) surface on the sample holder. The helium light (with photon energy ~21.2 eV) ARPES characterizations were performed at T = 7 K in our laboratory using a DA30L analyzer. The total energy resolution was better than 15 meV, and the angular resolution was set to ~0.3°, which gave a momentum resolution of ~0.01\pi/a.

Electrictransportmeasurement
The six-wire Hall-bar-geometry method was used to perform the electric transport measurements. Au was evaporated on freshly cleaved (001) surface of Bi4(Br0.27I0.73)4 at six bonding points to reduce contacting resistance (less than 10 \Omega), and silver epoxy was applied to connect the samples with gold wires. The 9 T electric transport data were collected with a physical property measurements system (Quantum Design, Dynacool 9T) in our home laboratory, where constant D.C. current of 0.5 mA was used to measure resistivity. The high-field electric transport results were measured by pulse magnetic field at Wuhan National High Magnetic Field Center.

Calculationmethods
The first-principles calculations are performed using the Vienna ab initio simulation package56 within the projector augmented wave method57 and the generalized gradient approximation of the Perdew-Burke-Ernzerhof58 exchange-correlation functional. The energy cutoff of 300 eV is used, and 9×9×3 and 13×13×4 \Gamma-centered k-grid meshes are adopted for structural relaxation and electronic structure calculations, respectively. The VCA59 is employed with different ratio for Br and I but the same stacking structure as Bi4Br4, and the crystal structure of Bi4(Br1-xIx)4 is relaxed with van der Waals correction until the residual forces on each atom is less than 0.001 eV/Å. The SOC effect is considered in the electronic structure calculations. Our calculations reveal that a gap closing occurs at the M point for x = 0.78 as shown in Fig. S3, indicating an ideal DSM around Bi4(Br0.22I0.78)4, consistent with the experiments. A tight-binding Hamiltonian based on the maximally localized Wannier functions (MLWF)60 is constructed to further calculate the Berry curvature and AHC using the Wannier Tools package61. To simulate the external magnetic field effect, a Zeeman-like manner is employed, where an additional potential of the following form carries the interaction: V\uparrow = V\uparrow + B and V\downarrow = V\downarrow + B for spin-polarized calculations; V_{\alpha\beta} = V_{\alpha\beta} + B \cdot \sigma_{\alpha\beta} with \sigma the vector of Pauli matrices for noncollinear calculations. Although the Landau level effect is not included in the simulations, the basic physical results induced by the Berry curvature and gapless Weyl modes can be definitely determined. First, the Berry curvature is mainly caused by SOC. The bands occupied by integers contribute to saturated Berry curvature and AHC, which is almost unaffected by Landau levels and magnetic field; while the partially occupied bands contribute to non-saturated Berry curvature, and when a Landau band surpassing EF, a change may occur on AHC. Second, the zero-energy mode of gapless Weyl fermions under magnetic field guarantees the existence of the lowest Landau level at EF, ensuring the robustness of chiral anomaly.

EstimationofeffectiveLandé g factor
Calculated distances between WP2 \Delta k_W with different effective Zeeman energy \epsilon along chain direction are obtained, which are compared with experimental valve of \Delta k_W at critical field of magnetic tunneling (B1). When experimental and calculated \Delta k_W have the same value of 0.018 Å^{-1}, effective Landé g factor is acquired by g = \epsilon_y / \mu_B = 88.4 (\epsilon = 0.11 eV). The large effective Landé g factor value of 88.4 is similar with that of 77.5 in Bi1-xSbx10, coinciding with the fact that large effective Landé g factor value appears with strong SOC strength (provided by heavy element bismuth). Therefore, the value of 88.4 is adopted to estimated calculated magnetic field by B = \epsilon / g\mu_B and yielding good consistency with experimental results.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In the manuscript “Field manipulation of Weyl modes in an ideal Dirac semimetal” Jingyuan Zhong and coauthors\ndemonstrate the realization of multiple Weyl modes in a Dirac semimetal Bi4(Br0.27I0.73)4 by measuring Hall conductivity\nand magnetoresistivity together with band structure calculation. However, I’m not fully convinced the “step” feature is due to\nberry curvature as well as the negative MR is from chiral anomaly and annihilation of WNR. It is so well known that lots of\nextrinsic factor can give AHE as well as NMR, before the rigorous evidence to exclude these effects, it is hard to draw any\nconclusion just based on transport measurements. In addition, the idea of realizing Weyl mode with magnetic field, as the\nauthor also mentioned in the manuscript, is already reported in the HfTe5 system. Although there are some subtle\ndifferences in terms of the physics, the general idea here is not novel enough to be reported in Nature Communications.\nTherefore, I’m not able to recommend this manuscript for publication. I have listed my detailed questions below:\n1. As the author mentioned in the manuscript “The stage-like behavior in Hall resistivity surviving at high temperature (up to\nthe AHE in the magnetic system, where the stage is correlated to the saturated magnetic moment.” Could the author explain\na bit more why this can rule out the possibility of multiple band effect? Related to the “step” like behavior in Hall resistivity,\ncould the author rule out the possibility of domains? Since the material needs a fine tuning of Br and I ratio, could domains\nform during the growth? Are there any scanning probe measurements to support the claim? Or another way to verify this is\nmeasuring with samples synthesized from different batches, is there always “step” like feature? Are they always happening\nat the same magnetic field? Is there any hysteresis scanning the magnetic field back and forth?\n2. I have few questions on separating the AHE caused by intrinsic and extrinsic sources. Firstly, the 1.6 exponents part is\nattributed to the side-jump effects, and the interception of 171 S/cm is attributes to the intrinsic effects. How reliable is the\nfitting? If one force the interception to be zero, can it still be fitted within the error bar? And how is the error bar defined here?\nSecond, the author mentioned “The coincidence between the experimental results and theoretical model demonstrates that\nthe step-like feature corresponds to AHE originating from Berry curvature.” I don’t see why this conclusion can be draw, or\nwhat is the logic connections here? Could the author give more explanation why this is a valid claim? Third, the author\nmentioned in the following “The large AHA in Bi4(Br0.27I0.73)4 is mainly contributed by prominent side-jump scattering,\nwhich is supposed to emerge in strongly disordered metal. coinciding with the high I\ndopants concentration (more disorders than pure Bi4Br4 sample ” is this contradict with the claim above which stays the\n“step” like feature is from the Berry curvature? Could the author clarify? Forth, what is the doping level of the sample? And if\nthe AHE depending on the doping level of the sample? Fifth, is there any other supporting evidence of this AHE is\noriginating from Berry curvature? For example, RMCD or scanning SQUID? These could be a much better probe to verify the\nclaim.\n3. As for the NMR, could the author exclude the effect is not come from charge puddle? Any surface sensitive measurements\nto support the claim? For example, KPFM or scanning SET? Otherwise, it will be hard to draw any conclusion.", "reviewer_2": "This work discovers a Dirac semimetal that appears to be ideal in that it only exhibits Dirac bands at the Fermi level and\nexhibits various anomalous transport properties. Magnetotransport measurements and DFT calculations of this work reveal\nvarious exciting phenomena that are associated with Dirac and Weyl-fermion physics, such as anomalous Hall effect,\nnegative magnetoresistance, and magnetic-field-induced chiral coupling. Interestingly, the formation of Weyl nodal rings and\nthe resulting effects can be controlled by an applied magnetic field in this material.\nIn my opinion, this type of discoveries does in principle warrant publication in NatComm. However, in the present form, the\nmanuscript does not provide strong-enough evidences for many of the described phenomena, unless the authors provide\nsubstantial clarification. In the following I detail the points that, in my opinion, need to be addressed.\nThe discovered material is claimed to have only Dirac modes at the Fermi energy but DFT in Fig 2d seems to show an\nadditional band crossing the Fermi energy.\nline 97: In my understanding it is the skew-scattering rather than side-jump scattering that give rise to sigma_xx dependent\nAHC.\nThe whole discussion of the band structure of the section starting with line 119 is confusing in what is actually formed, a\nWNR or WPs? How are pairs of Weyl nodes identified when those lie in a nodal ring. What does the phrase \"Weyl crossings\nin WNR\" mean? What justifies to base various explanations of the transport measurements on the presence of WPs rather\nthen the WNR that is actually formed?\nlines 114-118: The authors talk about the behavior of the AHC as a function of B and try to explain the experimental data, in\nparticular the large plateau for B>4T. It is argued in words that the observation is consistent with theoretical calculations but I\ndo not find the explanation in the present form convincing. What would be convincing is a plot of the calculated AHC as a\nfunction of B. At least a few more B points on the large plateau need to be provided to really show the independence of the\nAHC.\n129: The authors write \"SdH oscillation only appears under (cid:1) || (cid:2) configuration, which can be explained by the small\neffective mass and high mobility of charge carriers.\" I do not understand this argument. In my understanding, the system is\nquasi 1D, the dispersion is nearly flat in the x and z directions. In such a system one would not expect quantum oscillations\nat all, also in the y direction.\n131-132: The authors write \"The NMR emerges after the termination of the SdH oscillation (i.e., itinerant carriers are\ncondensed into the lowest Landau bands).\" There are various points that are unclear to me with regard to this and related\nstatements: What is the reason to believe that this is true? You can estimate the field required for the ultraquantum limit (the\nlimit when only the lowest Landau level (LL) is occupied). Is this estimate compatible with the obervation? Does the NMR\nusually appear experimentally in the ultraquantum limit or also above? Eq. (2) is valid, in my opinion, only when the number of\noccupied LLs is high. In the ultraquantum limit of only the lowest LL being occupied, the magnetic field dependence\nshould become linear in B. Thus the claim that in the observed NMR regime only the lowest LL is occupied contradicts the\napplicability of Eq. (2).", "reviewer_3": "Report on “Field manipulation of Weyl modes in an ideal Dirac semimetal”\nThe manuscript reports the successful manipulation of multiple Weyl modes, including gapped Weyl modes, Weyl nodal\nrings (WNRs), and coupled Weyl states, in the ideal Dirac semimetal Bi₄(Br₀.₂₇I₀.₇₃)₄ via magnetic field tuning. Key findings\ndescribed in the manuscript include the observation of anomalous Hall effect (AHE) with step-like features under symmetry-\nbreaking fields, attributed to large Berry curvature, the detection of negative magnetoresistivity (NMR) at low fields (linked to\nchiral anomaly in WNRs) and its collapse at high fields due to magnetic-tunneling-induced annihilation of Weyl nodes, the\nobtention of a comprehensive phase diagram mapping Weyl physics as a function of field and temperature, supported by\ntransport measurements and first-principles calculations.\nThe results reported in this manuscript have been validated through the agreement between magnetotransport data\n(e.g., Shubnikov–de Haas oscillations, NMR) and theoretical predictions, also through first-principles calculations, confirming the\nemergence of gapped Weyl modes and WNRs under specific field orientations, and the authors have also made use of\ncontrol experiments (e.g., angle-dependent resistivity, high-field pulses up to 57 T) that systematically excluded possible\nalternative explanations, such as, for example, multiband effects.\nIt is my opinion that this work contributes significantly to advance the field of topological materials. The authors have\ndemonstrated the reversible control over diverse Weyl phases (gapped, gapless, and coupled) in a single material, they\nhave also provided a symmetry-based framework for designing Weyl states, with implications for chiral spintronics, and they\nhave also resolved some longstanding questions about chiral anomaly breakdown under high fields, supported by\nquantitative modeling of magnetic-tunneling gaps.\nThe experimental and computational methodologies are rigorous. High-quality single crystals were characterized via XRD,\nHAADF-STEM, and ARPES, ensuring sample integrity. Six-wire transport measurements and pulsed-field experiments\nprovided reliable magnetoresistivity and Hall data. Virtual crystal approximation (VCA) and Wannier-based Berry curvature\ncalculations accurately reproduced experimental observations. The authors have also been able to combine experiment and\ntheory, which is not frequently an easy task but always desirable. For the transport data they analyzed using established\nmodels (e.g., Eq. 2 for NMR, Eq. 3 for tunneling gaps). The calculations of the Landé g-factor and the Berry curvature\ndistributions quantitatively were fundamental in order to provide a link to observations and underlying physics. The phase\ndiagram synthesizes seemingly uncorrelated phenomena (as for example, AHE, NMR, and tunneling) into a unified picture.\nTo enhance the impact of their research work, the authors could consider clarifying the role of disorder (e.g., Br/I vacancies)\nin transport, possibly via comparative studies with pristine Bi₄Br₄. They could also discuss the feasibility of room-temperature\napplications given the high-field requirements. The occurrence of plateaus or steps in the AHE has been predicted by\nFocassio et al. in Phys. Rev. Research 6, 033289 (2024) precisely due to the manipulation of Weyl nodes via magnetic field\nin three dimensional warped semimetals, and this reference should give further support to the current manuscript. Also, it\nwould be nice if the authors provided more discussion about the Landau index intercept in the fan diagram in connection to a\nnonzero Berry phase and why this is around 1/8 instead of 0.5 as recently found by Gang Qiu et al. Nature Nanotechnology\nvolume 15, pages585–591 (2020) via quantum hall effect of Weyl fermions in tellurene.\nThe manuscript is well-written but could be strengthened by briefly contextualizing the choice of Bi₄(Br₀.₂₇I₀.₇₃)₄ over other\nDirac/Weyl materials early in the text. Explicitly stating the novelty of observing *both* AHE and NMR in one system,\ncontrasting prior work (e.g., TaAs, ZrTe₅).\nThis work represents a significant advance in topological matter and is highly suitable for Nature Communications. The\ninterplay of symmetry, transport, and theory offers a pathway for manipulating Weyl fermions in devices. With minor\nrevisions, it will appeal to both fundamental and applied researchers.\nRecommend acceptance."}, "author_response": "Response to Referees\nReply: We express our deepest gratitude to the referees for their thorough examination of our work and for\nproviding us with excellent comments and constructive feedback. The positive remarks of Referee #2, “the\nDirac\nsemimetal that appears to be ideal in that it only exhibits Dirac bands at the Fermi level, which reveals various\nexiting phenomena…In my opinion, this type of discoveries does in principle warrant publication in Nat.\nCommun.” and Referee #3, “It is my opinion that this work contributes significantly to advance the field of\ntopological materials. The authors have been able to combine experiment and theory to reversibly control over\ndiverse Weyl phases (gapped, gapless, and coupled) in a single material, which is not frequently an easy task but\nalways desirable. This work represents a significant advance in topological matter and is highly suitable for\nNature Communications. With minor revisions, it will appeal to both fundamental and applied researchers.\nRecommend acceptance.” encouraged us to further strengthen our claim on the successful manipulation of\nmultiple Weyl modes in our manuscript. We have incorporated their valuable suggestions and made significant\nimprovements to our paper. The key revisions that we have undertaken are outlined below:\nAddition of new/extended results to address referees’ points and strengthen the manuscript conclusion:\n1. To demonstrate the reproductivity of the AHE, we performed additional Hall measurements of other four\nsamples and incorporated as Fig. S4 and Fig. S10 in Supplementary Information.\n2. We added the Kelvin probe force microscopy (KPFM) results to the cleaved (001) surface of\nBi (Br I ) crystal to exclude the effect of magnetic domain on AHE and the effect of charge puddles\n4 0.27 0.73 4\non NMR.\n3. Detailed fitting results are added to demonstrate the reliability of our fitting results and to identify the\ncoexistence of intrinsic and side-jump contribution for anomalous Hall effect.\n4. We incorporated new calculations of AHC to verify the unchanged Berry curvature with varying magnetic\nfield.\n5. We included new ARPES spectra and Hall measurements of less doped sample to illustrate the\nrelationship between doping level and step-like Hall resistivity.\nWe also revised the presentation of our data to improve clarity:\n1\n1. Inspired by the suggestions from the referees, the novelty of this work has been addressed along with the\ncomparison to the previous studies.\n2. We updated the illustration of intrinsic contribution and side-jump scattering mechanisms correlated to\nthe Berry curvature.\n3. The detailed analysis of the relationship between the intercept value of SdH oscillation and nontrivial\nBerry phase is added.\n4. We clarified the existence of Landau bands in quasi-1D system.\n5. We replotted Fig. 4c to indicate the relationship between WNR and WPs.\n6. The tendency of field-dependent Hall conductivity in ultraquantum limit region have been discussed.\n7. We included the discussions on the effect halide vacancies on the transport behavior in both Bi Br and\n4 4\nBi (Br I ).\n4 0.27 0.73\n8. The room-temperature high-field applications (such as non-volatile storage and magnetic detector) of our\nsample have been added.\nAddition of the new author: The new authors, Zhizhen Ren and Yan Shi, contributing to KPFM measurements,\nhas been added to the author list.\nOnce again, we sincerely appreciate the time and effort invested by the referees in providing thoughtful comments\nthat have helped to enhance the quality of our research.\nComments: In the manuscript “Field manipulation of Weyl modes in an ideal Dirac semimetal” Jingyuan Zhong\nand coauthors demonstrate the realization of multiple Weyl modes in a Dirac semimetal Bi (Br I ) by\n4 0.27 0.73 4\nmeasuring Hall conductivity and magnetoresistivity together with band structure calculation. However, I’m not\nfully convinced the “step” feature is due to berry curvature as well as the negative MR is from chiral anomaly\nand annihilation of WNR. It is so well known that lots of extrinsic factor can give AHE as well as NMR, before\nthe rigorous evidence to exclude these effects, it is hard to draw any conclusion just based on transport\nmeasurements. In addition, the idea of realizing Weyl mode with magnetic field, as the author also mentioned in\nthe manuscript, is already reported in the HfTe system. Although there are some subtle differences in terms of\n5\nthe physics, the general idea here is not novel enough to be reported in Nature Communications. Therefore, I’m\nnot able to recommend this manuscript for publication. I have listed my detailed questions below:\nReply: We express our gratitude to the referee for his/her diligent review and constructive comments on our work.\nWe agree well with the referee’s opinion that the additional experimental evidence and analysis are required to\nexclude other factors contributing to anomalous Hall effect (AHE) and negative magnetoresisitivity (NMR).\nInspired by the referee’s valuable criticisms, we have figured out four possible origins of step-like Hall signal and\nfour ways to realize NMR in the following, and clarify the novelty of our work in the end.\n1. For AHE, the most significant feature is the step in the high-magnetic region of Hall signal. Specifically,\nHall resistivity can be expressed by the sum of ordinary Hall part (𝜌𝑂 ) and anomalous Hall part (𝜌𝐴 ), giving\n𝑥𝑦 𝑥𝑦\n𝜌 = 𝜌𝑂 +𝜌𝐴 . There are four origins of this step-like Hall signal according to the previous reports, which are\n𝑥𝑦 𝑥𝑦 𝑥𝑦\n(1) multiple band effect, 1-5 (2) the formation of Landau bands along with charge density wave, 6-8 (3) the saturated\nmagnetic moment in magnetic systems, 9-13 and (4) the saturated non-zero Berry curvature in topological systems.\n14-22 The detailed analysis of these four effects accompanied with our results are listed below.\n(1) When there are several pockets (electron pockets and hole pockets) residing in the Fermi surface, the multiple\ncarriers (electrons and holes) with different densities and mobilities contribute to the Hall resistivity. 1-5 The\ndependences of mobilities of electrons and holes on magnetic field are different at different magnetic field regions,\ni.e., electrons and holes compete with each other, engendering the reversal slope of Hall resistivity smoothly\nvaried with the magnetic field. In this case, the step-like feature can be observed in the transition region. We have\nadopted the two-band model to fit our Hall resistivity results with varied magnetic fields (see details in Q1 and\nFigure R1). Although the step-like feature seems to exist in the magnetic region of 6-8 T, the deviation in other\n3\nmagnetic fields excludes the possibility of multiple band effect. Moreover, both of the ARPES spectra and DFT\ncalculations in Figure 2 in main text indicate that there is only hole pocket locating at the Fermi level, further\nruling out the possibility of multi band effect.\n(2) The formation of Landau bands can be reflected by SdH oscillations, and the emergent Fermi surface nesting\nalong the magnetic field direction may open gaps by forming the charge density wave. Consequently, the step-\nlike feature in Hall resistivity (𝜌𝑂 ) and zero magnetoresistivity at the boundary of Hall steps with the same\n𝑥𝑦\noscillation frequency are identified. 6-8 In this situation, carriers are constrained into multiple 2D slices without\nout-of-plane dispersion, engendering 3D quantum Hall effect and the quantized behavior in each 2D slice. Thus,\nthe step-like feature originates from the quantized conductivity of 3D quantum Hall effect, and can not be\nattributed to AHE. Nevertheless, the absence of SdH oscillations in magnetoresistivity in insets of Fig. R1a-b\nimplies that the step feature in our work can not be described solely by the picture of Landau levels (i.e. where\n𝜌 and 𝜌 should oscillate together). Moreover, the non-zero magnetoresistivity indicates prominent\n𝑥𝑥 𝑥𝑦\nbackscattering of carriers, contradicting with the ballistic transport feature in 3D quantum Hall effect.\n(3) The AHE can be caused by saturated magnetic moment in magnetic system. With increasing magnetic field,\nthe sample-inherited magnetic moment changes accordingly until it saturates, leading to step-like Hall signal in\n𝜌𝐴 according to the formula 𝜌𝐴 = 𝑅 𝑀 (𝑅 is a Hall coefficient, and 𝑀 is magnetic moment along z direction)\n𝑥𝑦 𝑥𝑦 𝑠 𝑧 𝑠 𝑧\n(i.e. saturated deflecting strength for carriers). 9-13 Evidently, the Dirac semimetal Bi (Br I ) has no magnetic\n4 0.27 0.73 4\nelements or components, and the magnetic moment origin can be ruled out in this system.\n(4) In topological systems, Berry curvature of the electronic bands, which acts like an effective magnetic field in\nmomentum space, can also evoke AHE in the non-magnetic system. Fundamentally, the relationship between\nHall conductivity 𝜎𝐴 and 𝛺 can be written as: 𝜎𝐴 ∝ ∫ 𝐵𝑍 𝛺 𝑑𝑘, where 𝛺 = ∑ 𝛺𝑛, as described in\n𝑦𝑥 𝑧 𝑦𝑥 𝐵𝑍 𝑧 𝑧 𝑛∈𝑜𝑐𝑐𝑢𝑝𝑖𝑒𝑑 𝑧\nmain text. The Hall step can be realized with the saturated 𝛺 . 14-22 In Bi (Br I ) , the Fermi surface is close\n𝑧 4 0.27 0.73 4\nto clean Dirac/Weyl modes, which is supposed to generate large Berry curvature under magnetic field.\nIn summary, the (1) multiple band effect and (2) Landau bands contribute to the ordinary Hall signal 𝜌𝑂 (ordinary\n𝑥𝑦\ndeflection effect by external magnetic field) with the curved or step-like feature, and (3) the saturated magnetic\nmoment and (4) Berry curvature bring in anomalous Hall signal 𝜌𝐴 due to the extra deflection effect from\n𝑥𝑦\nmagnetism and Berry curvature, respectively. According to the above analysis, the first three possibilities are not\ncoincided with our results. The comparative studies between our experimental Hall resistivity and the calculated\nband structures as well as Berry curvature under different magnetic fields in Fig. 2 demonstrate that the origin of\nAHE in Bi (Br I ) is the non-zero Berry curvature. The more detailed results and analysis are displayed in\n4 0.27 0.73\n4\nevery “Question and Answer” part below, and the necessary modifications have also been performed in main text\nand Supplementary.\n2. The four possible origins for NMR are (1) magnetic field suppressed spin fluctuation in magnetic systems,\n23-25 (2) charge puddle effect, 26-28 (3) current jetting effect, 29-30 and (4) chiral anomaly. 29-38\n(1) In magnetic systems, the spin fluctuation engenders scattering of charge carriers. After applying magnetic\nfield, the spin fluctuation is reduced, leading to the suppressed charge carrier scattering as well as the reduced\nmagnetoresistivity (i.e. NMR). 23-25 However, Bi (Br I ) has no magnetic elements, excluding the spin\n4 0.27 0.73 4\nfluctuation effect.\n(2) Charge puddles are charge inhomogeneities formed in the disordered narrow-gap semiconductor systems, 26-\n28 where magnetic field enhances the percolation of current paths and induce the suppressed charge carrier\nscattering. Notably, the charge puddles with percolation of current paths are independent for crystal structure.\nThus, the NMR induced by charge puddles is isotropic towards magnetic field (i.e. independent on the direction\nof magnetic field). The NMR effect in our work only appears when B//E//b axis (Fig. 4(a) in the manuscript),\ncontradicting to the results induced by charge puddle effect. Moreover, there are no prominent potential domains\naccording to our KPFM (see Q1 and Figure R2), ruling out the possibility of charge puddle effect.\n(3) Current jetting is a classical effect describing the current density that is squeezed into internal space (i.e.\nuneven current density distribution) by Lorentz force when B//E. 29-30 With the increment of magnetic field, more\n(less) current density is squeezed into interior (exterior) of the sample, where NMR emerges when the longitudinal\nelectrodes only contact margin of samples. In our measurements, all longitudinal electrodes completely cross the\nsample, efficiently avoiding the current jetting effect. Furthermore, our NMR only appears in the region between\nthe last SdH oscillation and the beginning of magnetic tunneling, which denotes the strong relationship between\nour NMR and quantum transport behavior, and is incompatible to the classic current jetting effect that occurs\nconsistently with the increasing magnetic field. 29\n(4) In Weyl systems, the extra chiral current is produced by parallel magnetic and electric field, exhibiting as the\nenhanced conductivity proportional to magnetic field. 29-38 In other words, the chemical potential imbalance\nbetween Weyl nodes engenders both charge and chirality imbalance for carriers, forcing the charge conduction\nto be enhanced by chiral charge transport. In Bi (Br I ) , the Weyl modes are close to Fermi surface after\n4 0.27 0.73 4\napplying magnetic field, well coinciding with the chiral anomaly case. Therefore, based on the above analysis,\nwe can rationally ascribe the origin of our NMR phenomenon to chiral anomaly in Weyl systems.\n3. We admire that the significance of our work should be further addressed to make the broad audience.\nHere we perform a short summary of the novelty of our work below.\n3 5\nBi Sb , ZrTe , HfTe , TaAs, and TaP, 19-20, 33-37 their transport behaviors are unavoidably disturbed by irrelevant,\n1−x x 5 5\nconventional electrons. In our work, we synthesize the topological materials with only Dirac/Weyl bands residing\nin the Fermi level, which provides the clean Dirac/Weyl fermions in the magnetotransport measurements. This\nfeature benefits to the successful observation of multiple Weyl modes, including the gapped Weyl mode, Weyl\nnodal ring (WNR), and the coupled Weyl mode, in Bi (Br I ) . It is noticeable that this manuscript is the first\n4 0.27 0.73 4\nwork constructing these different Weyl modes into a unified picture in a single crystal quantum system and\nestablishing the close relationship between the crystal symmetry and chiral electrons.\nIn fact, the novelty of our work can be reflected by the comments of both Referee #2 and Referee #3.\nFor\nReferee #2, the remarks of “This work discovers a Dirac semimetal that appears to be ideal in that it only\nexhibits Dirac bands at the Fermi level and exhibits various anomalous transport properties... In my opinion,\nthis type of discoveries does in principle warrant publication in Nat. Commun.” emphasize the importance of\nideal Dirac fermions in the observation of various exiting phenomena. The remarks “It is my opinion that this\nwork contributes significantly to advance the field of topological materials. The authors have demonstrated\nthe reversible control over diverse Weyl phases (gapped, gapless, and coupled) in a single material, they\nhave also provided a symmetry-based framework for designing Weyl states, with implications for chiral\nspintronics....” of Referee #3 address the significance of our work in the modulation of multiple Weyl modes in\na single material as well as the interplay between symmetry and chiral fermions. Referee #3 also brings in the\nsuggestion of the improvement of the novelty by the sentence “The manuscript is well-written but could be\nstrengthened by briefly contextualizing the choice of Bi (Br I ) over other Dirac/Weyl materials early in the\n4 0.27 0.73 4\ntext.”\nIn order to make the clear and solid statements on the novelty of this work, we have added the related\ndescription in the introduction part as: “Although the previous work reports the realization of Weyl modes in\nvarious materials, their transport behaviors are unavoidably disturbed by irrelevant, conventional electrons 4,7,10-\n14, hindering the observation of the multiple Weyl modes in a single material. Therefore, searching for an ideal\nDSM with only Dirac bands near Fermi surface to exclude the contribution from other bands in transport\nmeasurements is highly required to experimentally explore the chiral physics.” and “All the experimental results\nare supported by the calculations, and finally, a phase diagram of Weyl physics as the function of magnetic field\nand temperature is established, constructing the various Weyl modes into a unified picture in a single crystal\nquantum system and establishing the close relationship between the crystal symmetry and chiral electrons.”\ntemperature (up to 300 K) and strong magnetic field (up to 57 T) (see Fig. S4) excludes the possibility of multiple\nband effect, and is similar with the AHE in the magnetic system, where the stage is correlated to the saturated\nmagnetic moment.” Could the author explain a bit more why this can rule out the possibility of multiple band\neffect? Related to the “step” like behavior in Hall resistivity, could the author rule out the possibility of domains?\nSince the material needs a fine tuning of Br and I ratio, could domains form during the growth? Are there any\nscanning probe measurements to support the claim? Or another way to verify this is measuring with samples\nsynthesized from different batches, is there always “step” like feature? Are they always happening at the same\nmagnetic field? Is there any hysteresis scanning the magnetic field back and forth?\nA1: We appreciate the referee’s insightful comments on other possible origins of the stage-like behavior in Hall\nresistivity, which inspires us to perform more experimental measurements and analysis to reinforce our claims on\nthe identification of AHE.\nFirstly, as we mentioned above, the step-like feature in Hall resistivity can be observed in a limited field\nregion when the multiband effect works. In order to figure out this issue, the two-band model is applied to fit our\nresults with the formula: 1-5\n𝜌 = 𝐵 (𝑛1𝜇 1 2+𝑛2𝜇 2 2)+𝐵2(𝑛1+𝑛2)𝜇 1 2𝜇 2 2 (1)\n𝑦𝑥 𝑒(|𝑛1 |𝜇1+|𝑛2 |𝜇2)2+𝐵2(𝑛1+𝑛2)2𝜇\n1\n2𝜇\n2\n2\nwhere ⅇ is the charge constant, 𝑛 (𝑛 ) and 𝜇 (𝜇 ) are carrier density and carrier mobility for band1 (band2),\n1 2 1 2\nrespectively. The fitting results are displayed in Figure R1 and Fig. S5, where the fitting lines greatly deviates\nfrom the experimental data no matter how to optimize the fitting parameters. The fitting parameters are listed in\nthe black of Figure R1. The fitted densities are obviously larger than surface carrier density in previous reports\n(about 1010~1013 cm-2), 39-40 which rules out the surface bands contributions. Besides, both of the ARPES spectra\nand DFT calculations in Fig. 2 in main text indicate that there is only hole pocket locating at the Fermi level.\nTherefore, we can rule out the possibility of multiple band effect. We admire that the previous description of the\nrelationship between stage-like behavior and exclusion of multiple band effect is confusing. Thus, we have added\nFigure R1 as Fig. S5 in Supplementary Information and modified the sentence in the revised manuscript in line\n32, page 3 as: “The two band model is applied to fit the Hall signals, where the large deviation between the fitting\nlines and experimental results excludes the possibility of multiple band effect (see Fig. S5 for multiple band fitting\nresults for different samples).”. And the detailed analysis of fitting results in Supplementary Note 2 as: “When\nthere are several pockets (electron pockets and hole pockets) residing in the Fermi surface, the multiple carriers\n(electrons and holes) with different densities and mobilities contribute to the Hall resistivity. 1-5 The\ndependences of mobilities of electrons and holes on magnetic field are different at different regions, i.e., electrons and holes\ncompete with each other, engendering the reversal slope of Hall resistivity smoothly varied with the magnetic\nfield. In this case, the step-like feature can be observed in the transition region. In order to figure out the\nrelationship between stage-like behavior and multiple band effect, the two-band model is applied to fit our results\nwith the formula: 1-5\n𝜌 = 𝐵 (𝑛1𝜇 1 2+𝑛2𝜇 2 2)+𝐵2(𝑛1+𝑛2)𝜇 1 2𝜇 2 2 (1)\n𝑦𝑥 𝑒(|𝑛1 |𝜇1+|𝑛2 |𝜇2)2+𝐵2(𝑛1+𝑛2)2𝜇\n1\n2𝜇\n2\n2\nwhere ⅇ is the charge constant, 𝑛 (𝑛 ) and 𝜇 (𝜇 ) are carrier density and carrier mobility for band1 (band2),\n1 2 1 2\nrespectively. The fitting results are displayed in Fig. S5, where the fitting lines greatly deviates from the\nexperimental data no matter how to optimize the fitting parameters. Although a step-like feature could seem to\nexist in the magnetic region of 6-8 T, it deviates from the experimental data in other magnetic fields. The fitted\ndensities are obviously larger than surface carrier density in previous reports (about 1010~1013 cm-2), 39-40 which\nrules out the surface bands contributions. Besides, both of the ARPES spectra and DFT calculations in Fig. 2 in\nmain text indicate that there is only hole pocket locating at the Fermi level. Therefore, we can rule out the\npossibility of multiple band effect.”\nof (b) sample 5 (S5). Insets show the field-dependent 𝜌 with the same experimental setups as Hall\n𝑥𝑥\nmeasurement. (c) High-field Hall resistivity and AHE fitting results for S1 measured at 20 K.\nSecondly, we analysis the effect of domains in detail. The magnetic domain itself is with non-zero magnetic\nmoment, and are expected to be randomly distributed at zero filed, leading to the zero net moment when counting\nthe integration of magnetic moments of all the domains. After employing the external magnetic field, these\nmagnetic domains are aligned along the filed direction, resulting in the non-zero net moment and consequent\nAHE in transport measurements. Nevertheless, for Bi (Br I ) , there is no magnetic element or magnetic\n4 0.27 0.73 4\ncomponent to construct the magnetic domains. Furthermore, we have performed the KPFM measurements to\ncheck out the existence of domains. Figure R2 shows the (001) surface potential of Bi (Br I ) at room\n4 0.27 0.73 4\ntemperature. The large flat terrace and sharp edges shown in Figure R2 imply the high quality of Bi (Br I )\n4 0.27 0.73 4\ncrystals. There is no observable potential domain at both terraces and edges. It is noted that the KPFM results\nhave shown similar uniform potential distribution without charge puddles, similar to the previous reports. 41-42\nThus, the possible contribution of magnetic domain effect to the AHE in this work can be excluded. Figure R2\nhas been added as Fig. S7 in the revised Supplementary Information along with the related illustration in line 4,\npage 4 in the modified manuscript as: “Nevertheless, there are no magnetic atoms and domains in this quasi-1D\nsystem, excluding the contribution of magnetic domains to the AHE (See Fig. S7 in detail).”\n4 0.27 0.73 4\nrespectively. Lower panel: Kelvin probe force microscopy (KPFM) potential mapping results of the same areas\ndisplayed in upper panels. The uniform potential images imply the absence of domains in this material.\nThirdly, the Hall resistivity measurements of Bi (Br I ) crystals from different batches are performed\n4 0.27 0.73 4\naccording to the referee’s valuable suggestion. Figure R3 shows the results of Hall resistivity of four samples,\ndefined as S1, S2, S3, and S4, respectively. It should be noted that the results of S1 correspond to the figures in\nmain text. The two-stage feature can be clearly identified in S1 (the sample S2, and S3 with the similar onsets of\nmagnetic field (around 2 T and 4 T, respectively). The variation can be attributed to the different electron doping\nlevel, and the detailed explanation will be described in the part of A2-(4) below. The steps existing in all samples\ndemonstrate that the stage-like feature in Fig. 2 in main text is a ubiquitous signal in Bi (Br I ) . Figure R3\n4 0.27 0.73 4\nhas been added as Fig. S4 in the revised Supplementary Information companied by the corresponding descriptions\nin the revised manuscript in line 27, page 3 as: “The magnetic field-dependent anomalous Hall resistivity 𝜌𝐴 for\n𝑥𝑦\nS1 in Fig. 3b shows the step-like shape with two stages instead of the linear or curved dispersion in a large 𝜑\nregion ranging from 0° to 60°, which is a ubiquitous signal also identified in other crystals from different batches\n(see Fig. S4 in Supplementary Information).”\ntemperatures, respectively. (d) shows the Hall resistivities at 2 K measured by different scanning field directions\n(marked by black arrow and red arrow, respectively), where the non-hysteresis feature is observed.\nAccording to the above results and analysis, the multiple band effect and magnetic domain effect are denied\nas the origin of the step-like Hall resistivity in Bi (Br I ) . Moreover, the step-like feature is maintained in\n4 0.27 0.73 4\nthe crystals from different batches without any hysteresis loop, further ruling out the possibility of magnetic\ncomponent contribution to the AHE in this work.\nQ2: I have few questions on separating the AHE caused by intrinsic and extrinsic sources.\n(1) Firstly, the 1.6 exponents part is attributed to the side-jump effects, and the interception of 171 S/cm is\nattributes to the intrinsic effects. How reliable is the fitting? If one force the interception to be zero, can it still be\nfitted within the error bar? And how is the error bar defined here?\nA2-(1): We sincerely appreciate the referee for bringing up this important point. We agree that the reliability of\nthe fitting in Fig. 3c in main text should be addressed to make our claim solid. The error bar is applied due to the\nfluctuation of the Hall conductivity correlated to the step region as a function of magnetic field, as shown in\nFigure R4a. Therefore, the error bars are defined according to the upper value and lower value of step-like Hall\nconductivity (grey region) at each temperature. The fitting exponent can be three values, 1, 2, and 1.6, where the\nfirst two belong to the skew-scattering model reported in TaS , KV Sb , Fe Sn , and theoretical predictions,\n2 3 5 3 2\n14,19,21,48 and the last one is correlated to the side-jump model. 14-18 Thus, we apply all three exponent values to fit\nour experimental results by forcing the interception (intrinsic part) to be zero, as displayed in Figure R4b. All the\nfitting lines marked by different colors deviate obviously from the data of anomalous Hall conductivity (𝜎𝐴 ).\n𝑦𝑥\nMoreover, we also perform the fitting by applying three different exponents without forcing the zero interception,\nas shown in Figure R4c. The best fitting line is with the 1.6 exponent and 169 S/cm. Both of the other two fitting\nlines with the exponent of 1 and 2 fail to follow the trend of experimental results. Therefore, these additional\nfitting results support the validity of our fitting line in Fig. 3c in main text. We have added Figure R4 as Fig. S8\nin the Supplementary Information and the related in the revised manuscript in line 12, page 4 as: “The validity of\nour fitting results is supported by the comparative study of fitting lines by using different exponents and intercepts,\nas shown in Fig. S8 in Supplementary Information.”.\nconductivity. The error bars are defined by the upper value and lower value of step-like Hall conductivity in the\ngrey region at each temperature. (b) Relationship between anomalous Hall conductivity and longitudinal\nconductivity. Blue, grey, and green lines show deviated fitting results by 𝜎𝐴 = 𝛼𝜎 1 , 𝜎𝐴 = 𝛼𝜎 1.6 , and\n𝑦𝑥 𝑥𝑥 𝑦𝑥 𝑥𝑥\n𝜎𝐴 = 𝛼𝜎 2 , respectively, indicating the non-negligible intrinsic contribution. (c) Relationship between\n𝑦𝑥 𝑥𝑥\nanomalous Hall conductivity and longitudinal conductivity. The brown line, purple line, and orange line denote the\nfitting results by using the exponents of 1.6, 1, and 2, respectively. Only the fitting line with the exponent of\n1.6 and the intercept of 169 S/cm is well consistent with the experimental data, demonstrating the validity of our\nfitting results.\n(2) Second, the author mentioned “The coincidence between the experimental results and theoretical model\ndemonstrates that the step-like feature corresponds to AHE originating from Berry curvature.” I don’t see why\nthis conclusion can be draw, or what is the logic connections here? Could the author give more explanation why\nthis is a valid claim?\nA2-(2): We appreciate the referee for highlighting this crucial aspect. We admire that the logic connection of this\nsentence is not clearly described. The theoretical model here refers to Equation 1 in main text, 𝜎𝐴 = 𝛼𝜎 1.6 +\n𝑦𝑥 𝑥𝑥\n12\n𝜎𝐼𝑛𝑡., where 𝜎𝐴 is the saturated Hall conductivity (step-like region), 𝛼𝜎 1.6 and 𝜎𝐼𝑛𝑡. correspond to the extrinsic\n𝑦𝑥 𝑦𝑥 𝑥𝑥 𝑦𝑥\npart (side-jump scattering) and the intrinsic part (scattering independent), respectively. Both of these two parts\nare contributed by the non-zero Berry curvature. 14-18 This model can nicely describe the relationship between\nexperimental 𝜎𝐴 and 𝜎 , as shown in Fig. 3c, implying that step-like feature originates from non-zero Berry\n𝑦𝑥 𝑥𝑥\ncurvature. In order to make a clear illustration in this point, we have changed the sentence in the revised\nmanuscript in line 15, page 4 as: “It should be noted that the saturated Hall conductivity in Eq. (1) is contributed\nby two Berry curvature-induced parts, i.e., extrinsic side-jump scattering and intrinsic AHC. Consequently, the\nwell fitted experimental data in Fig. 3c by Eq. (1) indicates that the step-like feature originates from non-zero\nBerry curvature.”.\n(3) Third, the author mentioned in the following “The large AHA in Bi (Br I ) is mainly contributed by\n4 0.27 0.73 4\nprominent side-jump scattering, which is supposed to emerge in strongly disordered metal… coinciding with the\nhigh I dopants concentration (more disorders than pure Bi Br sample”. Is this contradicting with the claim above\n4 4\nwhich stays the “step” like feature is from the Berry curvature? Could the author clarify?\nA2-(3): We appreciate the referee for highlighting this crucial aspect. Berry curvature, as the initial origin,\nbehaves like pseudo-magnetic field in reciprocal space to deflect carriers transversely. Specifically, in the\nframework of Berry curvature, there are three mechanisms responsible for the carriers’ deflection, intrinsic\ncontribution, extrinsic side-jump scattering and extrinsic skew-scattering, respectively. 14-22 For intrinsic\ncontribution, the electrons naturally deviate from a real-space trajectory due to inter-band coherence (i.e. Berry\ncurvature) without any scattering event. For side-jump scattering and skew-scattering mechanisms, the deflection\nof electrons are accompanied by scattering events in the condition of coexistence of non-zero Berry curvature and\nimpurities. 14-15 Thus, the intrinsic contribution and side-jump scattering contribution share the same origin, non-\nzero Berry curvature. In fact, it has been reported that the side-jump scattering originates from the non-zero Berry\ncurvature in Nd (Mo Nb ) O and La Sr CoO ,16-18. The sentence is not contradicting with the claim that AHE\n2 1-x x 2 7 1-x x 3\nis induced by Berry curvature.\nAfter the scrutiny on the sentence in main text, we are conscious of that the contributions of intrinsic part\nand side-jump scattering part to AHA are indistinguishable in this work. We have changed the sentences in the\nrevised manuscript in line 19, page 4 as: “The large AHA in Bi (Br I ) is contributed by both side-jump\n4 0.27 0.73 4\nscattering and intrinsic contribution of Berry curvature (see Supplementary Note 7 for detailed illustration of\nBerry curvature contribution to carrier deflection), which is reasonable to emerge in strongly disordered metal\n13\n(i.e. bad metal with low conductivity) as shown in Fig. S9, coinciding with the high I dopants concentration (more\ndisorders than pure Bi Br sample).”\n4 4\nAlong with the detailed illustration of Berry curvature in defecting carriers in Supplementary Note 7 as:\n“Berry curvature, as the initial origin, behaves like pseudo-magnetic field in reciprocal space to deflect carriers\ntransversely. Specifically, in the framework of Berry curvature, there are three mechanisms responsible for the\ncarriers’ deflection, intrinsic contribution, extrinsic side-jump scattering and extrinsic skew-scattering,\nrespectively. 8-16 For intrinsic contribution, the electrons naturally deviate from a real-space trajectory due to\ninter-band coherence (i.e. Berry curvature) without any scattering event. For side-jump scattering and skew-\nscattering mechanisms, the deflection of electrons are accompanied by scattering events in the condition of\ncoexistence of non-zero Berry curvature and impurities. 8-9 Thus, the intrinsic contribution and side-jump\nscattering contribution share the same origin, non-zero Berry curvature. In fact, it has been reported that the\nside-jump scattering originates from the non-zero Berry curvature in Nd (Mo Nb ) O and La Sr CoO .10-12”\n2 1-x x 2 7 1-x x 3\n(4) Forth, what is the doping level of the sample? And if the AHE depending on the doping level of the sample?\nA2-(4): We are grateful to the referee for this insightful question that prompt us to think deeply the relationship\nbetween the AHE and the doping level. As we mentioned in main text, there is a n-type doping effect in this quasi-\n1D system due to the vacancies of halide elements, resulting in the Dirac point (DP) locating below 𝐸 . Thus, the\nF\ndoping level can be modulated in principle by varying the vacancies concentration. With increasing magnetic\nfield, the energy difference between 𝐸 and DP becomes small due to the increased degeneracy of Landau bands.\nF\nThe gapped Weyl modes originating from Dirac cone can be faster condensed into lower Landau band in less\ndoped sample under magnetic field. Therefore, the less doped sample is expected to be with the AHE and a smaller\nonset of magnetic field entering into the step region.\nThe crystal (S5) from different batches is applied to make a comparative study. ARPES spectra of S5 in\nFigure R6b shows that the DP locates 0.08 eV below 𝐸 , which is smaller than that of S1 in main text. Therefore,\nF\nS5 is a less doped sample compared to S1. The step-like feature can also be identified in the Hall conductivity\nmeasurements of S5, as displayed in Figure R6d. However, the onset magnetic field decreases to around 1.1 T.\nBoth of the two features are consistent well with the above theoretical analysis, further confirming our claim on\nthe origin of AHE. To address this point, we have added the related description in the revised manuscript in line\n7, page 5 as: “The effect of n-type doping level on the AHE is discussed in Supplementary Note 10.”\nAND we add Figure R6 as Fig. S11 in Supplementary Note 10 along with the description below: “There is\na n-type doping effect in this quasi-1D system due to the vacancies of halide elements, resulting in the Dirac point\n14\n(DP) locating below 𝐸 . Thus, the doping level can be modulated in principle by varying the vacancies\n𝐹\nconcentration. With increasing magnetic field, the energy difference between 𝐸 and DP becomes small due to\n𝐹\nthe increased degeneracy of Landau bands. The gapped Weyl modes originating from Dirac cone can be faster\ncondensed into lower Landau band in less doped sample under magnetic field. Therefore, the less doped sample\nis expected to be with the AHE and a smaller onset of magnetic field entering into the step region.\nThe crystal (S5) from different batches is applied to make a comparative study. ARPES spectra of S5 in Fig.\nS11b shows that the DP locates 0.08 eV below 𝐸 , which is smaller than that of S1 in main text. Therefore, S5 is\n𝐹\na less doped sample compared to S1. The step-like feature can also be identified in the Hall conductivity\nmeasurements of S5, as displayed in Fig. S11d. However, the onset magnetic field decreases to around 1.1 T.\nBoth of the two features are consistent well with the above theoretical analysis, further confirming our claim on\nthe origin of AHE.”\nFigure R6. ARPES results at 𝑀̅ point (left panels) and energy distribution curve (EDC, right panels) for (c) S1,\nand (b) S5, respectively. Arrows denote the energy location of Dirac points. Hall resistivity of (a) S1, and (b)\nS5, respectively.\n15\n(5) Fifth, is there any other supporting evidence of this AHE is originating from Berry curvature? For example,\nRMCD or scanning SQUID? These could be a much better probe to verify the claim.\nA2-(5): We would like to express our gratitude to the referee for his/her important comment on adding additional\nexperimental evidence to support that the AHE in this work originates from Berry curvature. In reciprocal space\nof a crystal, Berry curvature is the indicator to describe trajectory of electron. The non-zero Berry curvature results\nin that the electrons moving longitudinally in a curved space exhibit transversely deflecting feature. Berry\ncurvature is also called the pseudomagnetic field in reciprocal space, which is different from real magnetic\nmoment introduced by magnetic atoms in real space. The AHE induced by non-zero Berry curvature is the\ntopological nature inherited from band structure, which can not be detected by conventional magnetism\ncharacterization such as RMCD, scanning SQUID, VSM, and etc. In fact, the generally used method to detect\nBerry curvature is electric transport approaches we performed in this work, applying Hall measurements to\ndirectly detect the overall deflecting signal in transverse direction. 14-22 Scarcely, in addition to the current-driven\ndeflection case in AHE, transverse deflection of electrons induced by Berry curvature can also be driven by light\n(photogalvanic effect) and heat (thermal Hall effect), which are with more experimental challenges. 43-47\nUnfortunately, due to the unique quasi-1D structure of Bi (Br I ) , it is difficult to exfoliate the samples into\n4 0.27 0.73 4\nthin 2D layers for photoelectric detection and to cut into large cubic shape without degradation for thermoelectric\ncharacterization, hindering the measurement of both photogalvanic effect and thermal Hall effect.\nIn this work, we have used state-of-art pulse-magnetic field electric transport method along with the APRES\nspectra and DFT calculations to resolve the origin of the step-like feature in Hall conductivity. All the results are\nwell-supported with each other. The logic can be summarized as below. The ideal Dirac bands verified by ARPES\nprovide the precondition for forming the gapped Weyl modes with non-zero Berry curvature under magnetic field.\nThe non-zero Berry curvature has been confirmed by DFT calculations, and engenders the step-like signal in Hall\nconductivity called as AHE here. The relationship between the non-zero Berry curvature and step-like Hall\nconductivity has been further supported by the fitting line of saturated Hall conductivity 𝜎𝐴 as a function of the\n𝑦𝑥\nlongitudinal conductivity 𝜎 in Fig. 3c. Moreover, the other possibilities for step-like Hall conductivity, such as\n𝑥𝑥\nmagnetism effect, multiple band effect, and 3D quantum Hall effect are reasonably excluded as stated above.\nTherefore, according to above results and analysis, we believe that our work provides sufficient evidence to\nsupport our claim of “the step-like feature in Hall conductivity is the AHE and originates from Berry curvature.\nmeasurements to support the claim? For example, KPFM or scanning SET? Otherwise, it will be hard to draw\nany conclusion.\nA3: We are grateful to the referee for bringing this issue to our attention. We have performed KPFM to measure\nthe (001) surface potential of Bi (Br I ) at room temperature, as shown in Figure R2. There are no observable\n4 0.27 0.73 4\npotential domains or charge puddles in terraces and edges with the resolution of tens of nanometers. In fact, the\nNMR induced by charge puddles effect has been reported in TlBi Sb Te , where the NMR is not sensitive to\n0.15 0.85 2\ndirection of magnetic field. 28 This is because that the charge puddles with percolation of current paths are\nindependent for crystal structure. The NMR effect in our work only appears when B//E//b axis (Fig. 4a in the\nmanuscript), contradicting to the results induced by charge puddle effect. Therefore, we exclude the possibility\nthat domain effects engender the NMR. In order to clarify this critical point, we have added Figure R2 as Fig. S7\nin Supplementary Information and incorporated a brief discussion in line 29, page 5 in the main text as: “It should\nbe noted that the charge puddle effect can induce isotropic (i.e. direction-independent) NMR phenomenon. 45\nSuch a possibility is excluded by the anisotropic NMR in Fig. 4a and the Kelvin probe force microscopy\nmeasurements in Fig. S7.”\nIn a summary, we believe that these new results substantially contribute to the overall strength of our manuscript.\nWe extend our sincere gratitude to the referee for his/her exceptional effort in providing valuable comments and\nsuggestions that have greatly enhanced our work.\nComments: This work discovers a Dirac semimetal that appears to be ideal in that it only exhibits Dirac bands\nat the Fermi level and exhibits various anomalous transport properties. Magnetotransport measurements and DFT\ncalculations of this work reveal various exciting phenomena that are associated with Dirac and Weyl-fermion\nphysics, such as anomalous Hall effect, negative magnetoresistance, and magnetic-field-induced chiral coupling.\nInterestingly, the formation of Weyl nodal rings and the resulting effects can be controlled by an applied magnetic\nfield in this material.\nIn my opinion, this type of discoveries does in principle warrant publication in Nat. Commun. However, in the\npresent form, the manuscript does not provide strong-enough evidences for many of the described phenomena,\nunless the authors provide substantial clarification. In the following I detail the points that, in my opinion, need\nto be addressed.\nReply: We express our sincere gratitude to the referee for his/her diligent evaluation of our work, recognizing the\npotential novelty and significance of our findings. We highly appreciate his/her thorough review and have taken\nall the concerns raised into careful consideration. Consequently, we have made substantial revisions to the\nmanuscript to address each point raised, improve the clarity and accuracy of our results, and provide additional\nsupporting evidence. We hope that the referee will find the revised manuscript satisfactory.\nQ1: The discovered material is claimed to have only Dirac modes at the Fermi energy but DFT in Fig 2d seems\nto show an additional band crossing the Fermi energy.\nA1: We sincerely thank the referee for bringing up this important point. There is indeed another band in additional\nto the Dirac band residing in the Fermi energy in the condition of zero magnetic field, as shown in Figure 2d.\nHowever, this band has parabolic dispersion and will go away from Fermi energy quickly with the increased\nmagnetic field, since the energy of Landau band for conventional electrons increases proportionally with the\nmagnetic field. In contrast, the energy of the lowest Landau band for Dirac/Weyl fermion is pinned (gapless case)\naround the Fermi energy or increases slowly (gapped case) with the magnetic field. Such a deduction is confirmed\nby our calculated band structures under the magnetic field along different directions, as shown in Fig. 3d, Fig. 4d,\nFig. 5d and Fig. 5e, where only the Weyl modes exist in the Fermi energy. Therefore, the bands correlated to\nDirac/Weyl modes are still the only source to provide itinerant carriers after applying magnetic field. In order to\nclarify this critical point, we have incorporated a brief illustration in the revised manuscript in line 19, page 3 as:\n18\n“Astonishingly, only the bands originated from the Bi p orbits locate near Fermi surface, evidencing\nBi (Br I ) is an ideal material platform to manipulate and investigate the multiple Weyl modes proposed in\n4 0.27 0.73 4\nFig. 1 (the other trivial band residing in Fermi level will be pushed away to high energy position after considering\nthe Landau level by introducing magnetic field).”.\nQ2: Line 97: in my understanding it is the skew-scattering rather than side-jump scattering that give rise to 𝜎\n𝑥𝑥\ndependent AHC.\nA2: We appreciate the referee for reminding us to clarify scattering mechanisms based on our AHC results.\nUsually, Berry curvature engenders AHE through three ways, which are scattering-independent (intrinsic)\ncontribution, scattering-dependent side-jump (extrinsic) contribution, and skew-scattering (extrinsic)\ncontribution. Among them, both of the skew-scattering and side-jump scattering give rise to 𝜎 dependent AHC.\n𝑥𝑥\nFor skew-scattering model, the exponent 𝛼 would be 1 or 2 as reported in TaS , KV Sb , Fe Sn , and theoretical\n2 3 5 3 2\npredictions, 14, 19, 21, 48 which are unable to fit our data as shown in Figure R8. For side-jump scattering model,\nthe exponent is 1.6, which can well cover all the experimental data in Figure R8. In order to valid the side-jump\nscattering model in our results, we have added the description in the line 12, page 4 in the revised manuscript as:\n“The validity of our fitting results is supported by the comparative study of fitting lines by using different\nexponents and intercepts, as shown in Fig. S8 in Supplementary Information. The 1.6 exponent implies the\nexistence of the extrinsic side-jump scattering, 36-39 and the nonzero interception value (~169 S/cm) is a direct\nreflection of the intrinsic contribution, 𝜎𝐼𝑛𝑡..” along with the Fig. S8c (Figure R8 here) in the Supplementary\n𝑦𝑥\nInformation.\nFigure R8. Relationship between anomalous Hall conductivity and longitudinal conductivity. The brown line,\npurple line, and orange line denote the fitting results by using the exponents of 1.6, 1, and 2, respectively. Only\n19\nthe fitting line with the exponent of 1.6 and the intercept of 169 S/cm is well consistent with the experimental\ndata, demonstrating the validity of our fitting results. It should be noted that both of the skew-scattering and\nside-jump scattering give rise to 𝜎 dependent AHC. 14-22 For skew-scattering model, the exponent 𝛼 would be\n𝑥𝑥\n1 or 2 as reported in TaS , KV Sb , Fe Sn , and theoretical predictions, 14, 19, 21, 48 which are unable to fit our\n2 3 5 3 2\ndata. For side-jump scattering model, the exponent is 1.6.\nQ3: The whole discussion of the band structure of the section starting with line 119 is confusing in what is actually\nformed, a WNR or WPs? How are pairs of Weyl nodes identified when those lie in a nodal ring. What does the\nphrase “Weyl crossings in WNR” mean? What justifies to base various explanations of the transport\nmeasurements on the presence of WPs rather than the WNR that is actually formed?\nA3: We are grateful to the referee for this insightful question. The WNR is formed in Bi (Br I ) when the\n4 0.27 0.73 4\nmagnetic field is applied along b axis. The WNR is composed of infinite pairs of Weyl points, and every two\nWeyl points with opposite chirality of one pair are located on opposite sides symmetric about the center point.\nHowever, due to the quasi-1D nature of the band, the WNR exhibits an elongated distribution, with either side\nmaintaining approximately the same chirality, for example, WP1+ or WP1-, except for the two endpoints (WP2+\nand WP2-), as displayed in Figure R9. The WNR can be disassembled into infinite Weyl point pairs, and\nconsequently fall into the same transport behavior of WPs such as negative magnetoresistivity (NMR). 49-50 The\nphrase of “two Weyl crossings in WNR” means the two pairs of special Weyl points, WP1 and WP2, along 𝑘\n𝑥\nand 𝑘 directions in Fig. 4d, respectively. We deliberately choose these two pairs of Weyl points to show the\n𝑧\ngapless feature of WNR in 𝑘 −𝑘 plane. Notably, the Weyl points separated along 𝑘 (WP1) have much shorter\n𝑥 𝑧 𝑥\ndistance in reciprocal space than them along 𝑘 (WP2), which may make WP1 and other WPs parallel with 𝑘\n𝑧 𝑥\nannihilate more quickly with increasing magnetic field than WP2. Under this consideration, we deliberately\nchoose WP2 along 𝑘 to schematically exhibit the long-standing chiral anomaly mechanism in Fig. 4e. Inspired\n𝑧\nby the referee’s constructive comments, we update Fig. 4c (Figure R9 here), clarify the relationship between\nWNR and WPs in line 3, page 6 in the revised manuscript as: “The WNR is composed of infinite pairs of WPs,\nand every two WPs with opposite chirality of one pair are located on opposite sides symmetric about the center\npoint. However, due to the quasi-1D nature of the band, the WNR exhibits an elongated distribution, with either\nside maintaining approximately the same chirality, i.e. WP1+ or WP1-, except for the two endpoints (WP2+ and\nWP2-), as displayed in Fig. 4c. Consequently, the WNR can be disassembled into infinite WP pairs, and its\ntransport behavior falls into the same theoretical region of WPs. 19,46” and rewrite the phrase of “two Weyl\ncrossings in WNR” as: “two special pairs of WPs in WNR” to avoid unnecessary confusion.\nz axis (middle panel), and y axis (lower panel). For applying 𝐵 and 𝐵 , MR shows monotonically positive\n𝑥 𝑧\ntendency with increasing 𝐵. Only under 𝐵 , MR firstly rises accompanied by oscillations (labelled as black\n𝑦\narrows) with increasing 𝐵 , and then exhibits negative MR (NMR) behavior after the last oscillation. (b)\n𝑦\nShubnikov–de Haas (SdH) oscillation analysis under 𝐵 . The solid and hollow squares represent peaks and\n𝑦\nvalleys of oscillation, respectively, acquiring oscillation frequency of 𝐵 = 2.94 T. The intercept of linear fit\n𝐹\n(dashed line) is within ±1/8 (light red region) having nontrivial Berry phase 𝜑 . (c) Left panel: 𝐸 −𝑘 −𝑘 3D\nB 𝑥 𝑧\nspectra with WNR band crossings denoted by dashed line. Right panel: projected WNR at 𝑘 −𝑘 plane under\n𝑥 𝑧\n𝐵 = 4 T. Red and blue circles (dots) represent the two different types of WPs with opposite charity, i.e. WP1-\n𝑦\n(WP2-) and WP1+ (WP2+), respectively. The infinite WPs assemble into the WNR can be disassembled into\ninfinite WP pairs. (d) Detailed band dispersion around WPs (denoted by black arrows) along 𝑍−𝑀−𝐿 path in\nthe 𝑘 = 0 mirror plane (left panel). Calculated band structure along 𝑋−𝑀 perpendicular to mirror plane\n𝑦\nshows a gap (right panel). (e) Schematics of chiral anomaly in WNR under 𝐵 . Blue, red, grey lines, and yellow\n𝑦\ndots represent lowest Landau band with chirality of +1, −1, other higher Landau bands, and occupied states\nrespectively. Electric field parallel to 𝐵 but perpendicular to WP2 separation adds chemical potential\n𝑦\ndifference between two lowest Landau bands (𝜇 > 𝜇 ).\n− +\nexperimental data, in particular the large plateau for B > 4 T. It is argued in words that the observation is consistent\nwith theoretical calculations but I do not find the explanation in the present form convincing. What would be\nconvincing is a plot of the calculated AHC as a function of B. At least a few more B points on the large plateau\nneed to be provided to really show the independence of the AHC.\nA4: Motivated by the referee’s insightful comment, we have performed the DFT calculations of AHC under\ndifferent magnetic fields from 4 T to 8 T, as shown in Figure R10. The calculated AHC originating from the\nsaturated Berry curvature is almost a magnetic field-independent value. The slightly increased Fermi energy with\nmagnetic field is due to the increased energy of the lowest Landau band of gapped Weyl mode. We have\nincorporated Figure R10 as Fig. S10 in the Supplementary Information along with the sentence in line 4, page 5\nin the revised manuscript as: “On the contrary, stage II is robust, indicating the large and saturated 𝛺 after 𝐵 =\n𝑧 𝑧\n4 𝑇, which is confirmed by the calculated AHC at 𝐸 in Fig. 3e (see Fig. S10 for details)”.\n𝐹\nFigure R10. Calculations for the saturated AHC under magnetic field B of (a) 4 T, (b) 5 T, (c) 6 T, (d) 7 T,\nZ\nand (e) 8 T, respectively. The slightly increased Fermi energy with magnetic field is due to the increased energy\nof the lowest Landau band of gapped Weyl mode. (f) AHC value as a function of magnetic field.\nby the small effective mass and high mobility of charge carriers.” I do not understand this argument. In my\nunderstanding, the system is quasi 1D, the dispersion is nearly flat in the x and z directions. In such a system one\nwould not expect quantum oscillations at all, also in the y direction.\nA5: We sincerely appreciate the constructive criticism provided by the referee. Our sample, Bi (Br I ) ,\n4 0.27 0.73 4\nis\nwith quasi-1D nature rather than strictly 1D structure. Its electrons can still hop along interchain directions. This\ndispersive feature is supported by the band structure calculations (Figs. 4c and 4d) as well as constant energy\ncontour in ARPES measurements, as displayed in Figure R11, where oval instead of two parallel flat lines is\nobserved. Thus, this quasi-1D system is capable of forming Landau bands under magnetic field. The amplitude\nof SdH oscillation can be written as: 51-53\n𝛥𝜎 ~\n2𝜋2𝑘𝐵𝑇/ℏ𝜔𝑐\n(2)\n𝑥𝑥 sinh (2𝜋2𝑘𝐵𝑇/ℏ𝜔𝑐)\n𝑒𝐵\nwhere 𝜔 = , ⅇ is the charge constant, 𝑚 is the effective mass of electrons, 𝑘 is the Boltzmann constant, and\n𝑐 𝑒 𝐵\n𝑚𝑒\nℏ is the reduced Planck constant. The highly dispersive band along b axis (quasi-1D chain directions) gives small\n𝑚 to carriers, engendering that the SdH oscillation is more observable under B || b configuration. We have added\n𝑒\nthe related illumination in line 21, page 5 in the revised manuscript as: “It is noticeable that the quasi-1D feature\nin a closed Fermi surface allows electron hopping to produce SdH oscillations under magnetic field.”\nFigure R11. ARPES spectra of Quasi-1D Fermi surface of Bi (Br I ) . The red line denotes (001) surface\n4 0.27 0.73 4\nBrillouin zone. The dashed line labels ellipse Fermi surface.\ncarriers are condensed into the lowest Landau bands).” There are various points that are unclear to me with regard\nto this and related statements: What is the reason to believe that this is true? You can estimate the field required\nfor the ultraquantum limit (the limit when only the lowest Landau level (LL) is occupied). Is this estimate\ncompatible with the observation? Does the NMR usually appear experimentally in the ultraquantum limit or also\nabove? Eq. (2) is valid, in my opinion, only when the number of occupied LLs is high. In the ultraquantum limit\nof only the lowest LL being occupied, the magnetic field dependence should become linear in B. Thus, the claim\nthat in the observed NMR regime only the lowest LL is occupied contradicts the applicability of Eq. (2).\nA6: We deeply appreciate the referee’s insightful comments and constructive criticism on ultraquantum limit.\nWe judge the system entering into the unltraquantum limit case based on the analysis of SdH oscillation index in\nFig. 4b, where a near-zero intercept of Landau index is well consistent with other reports on ultraquantum limit\nsituation. 6, 35, 53 We also agree well with the referee’s opinion that the onset magnetic field of ultraquantum limit\nneed to be estimated to make a direct comparison to the transition field in Fig. 4a in main text. We start with the\nestimation of charge carrier density 𝑛. Experimentally, 𝑛 can be extracted from the Hall resistivity measurement\nby 𝑑𝜌𝑥𝑦 = 1 , which is 2.56×1019 cm-3. Note that here the carrier density 𝑛 is three dimensional, i.e., 𝑛 .\n3𝐷\n𝑑𝐵 𝑛𝑒\nUnder the magnetic field, the Landau filling factor can be calculated as:\n𝜈 =\n𝑛3𝐷𝑉\n, (3)\n𝑔3𝐷\nwhere 𝑉 = 𝐴𝐿 is the volume of sample with the area 𝐴 and the thickness 𝐿 , and 𝑔 is the Landau degeneracy\n𝑧 𝑧 3𝐷\nof 3D electrons. The Landau degeneracy of 2D electron gives\n𝐵𝐴\n𝑔 = ,\n(4)\n2𝐷\nℎ/𝑒\nwith the Planck constant ℎ. The relationship between 3D Landau degeneracy and 2D Landau degeneracy can be\nestablished as:\n𝑔 = 𝑔\n∆𝑘𝑧𝐿\n, (5)\n3𝐷 2𝐷 𝑧\n2𝜋\nwhere ∆𝑘 is the Fermi wave vector along the 𝑘 direction. Combine Eqs. (3-5), the Landau filling factor can be\n𝑧 𝑧\nobtained by:\n𝜈 =\n2𝜋𝑛3𝐷ℎ/𝑒\n=\n2𝜋𝑛3𝐷ℏ𝑣𝐹𝑧ℎ/𝑒\n, (6)\n𝐵∆𝑘𝑧 𝐵𝐸𝐹\nwhere 𝐸 is the Fermi energy, 𝑣 is the Fermi velocity along the 𝑘 direction, and ℏ is the reduced Planck\n𝐹 𝐹𝑧 𝑧\nconstant. When the system enters ultraquantum limit, only the lowest Landau level is occupied, i.e., 𝜈 = 1. Thus,\nthe critical magnetic field for the ultraquantum limit is estimated as:\n24\n𝐵 =\n2𝜋𝑛3𝐷ℏ𝑣𝐹𝑧ℎ/𝑒\n. (7)\n𝑐\n𝐸𝐹\nBy substituting 𝑛 ~2.56×1019 cm-3, 𝐸 ~0.14 eV, ℏ𝑣 ~0.01 eV·Å into Eq. (7), we can obtain 𝐵 ~4.8 T,\n3𝐷 𝐹 𝐹𝑧 𝑐\nwhich is in the same order to the experimental value (2.6 T). It is noticeable that, in another quasi-1D topological\ncrystal ZrTe , electrons enter ultraquantum limit at ~ 1.3 T, which is compatible with the value of 2.6 T for\n4 0.27 0.73 4\nSupplementary Information along with the illustration in line 25, page 5 in revised manuscript as: “The NMR\nemerges after the termination of the SdH oscillation as itinerant carriers are condensed into the lowest Landau\nbands, which is supported by the similar values between the estimated onset magnetic field of ultraquantum limit\nand transition field in Fig. 4a and Fig. 4b (see Supplementary Note 12 for details)). 8-9,42-44” along with the\nsentences correlated to the experimental part in line 17, page 5 as: “Moreover, the Landau bands index shows an\nintercept value within ±1/8, implying nontrivial Berry phase of charge carriers and that the system enters into\nultraquantum limit at ~ 2.6 T. 12,43 It is noticeable that, in another quasi-1D topological crystal ZrTe , electrons\n5\nenter ultraquantum limit at a compatible field (~ 1.3 T). 13”\nThe coexistence of NMR and SdH oscillations has been observed in previous reports, where the NMR occurs\nbefore the system entering into the unltraquantum limit. 34-36 In these literatures, the other bands instead of Weyl\nbands residing in the Fermi surface contribute the additional SdH oscillations, hindering the establishment of the\nrelationship between the NMR and ultraquantum limit. 34-36 Only the Weyl bands are located around Fermi surface\nin Bi (Br I ) crystal, benefiting to resolving the exact Landau band correlated to NMR. To our knowledge,\n4 0.27 0.73 4\nour work is the first study that reports the observation of NMR in the ultraquantum limit.\nWe admire that Eq. (2) in the original manuscript is used to describe systems with high occupied Landau\nbands. In ultraquantum limit, the NMR model is proposed as: 31-32\n𝜎 = 𝜎 +𝑐 𝐵 = 𝜎 +\n𝑒3𝜏𝑎\n𝐵 , (8)\n𝑥𝑥 0 w 0 4π2ℏ2𝑐\nwhere 𝜎 is the longitudinal conductivity, 𝜎 is residual conductivity, 𝑐 is the coefficient term, ⅇ is elemental\n𝑥𝑥 0 w\ncharge, 𝜏 is inter-node relaxation time, ℏ is the reduced Planck constant, and 𝑐 is the light speed. The linear\n𝑎\nrelationship between 𝜎 and 𝐵 can be inferred in the condition that 𝜏 is a magnetic field-independent constant.\n𝑥𝑥 𝑎\nFor temperature below 50 K, the NMR data between two transition points, 𝐵 and 𝐵 , can be nicely fitted by\n0 1\nparabolic model:\n𝜎 = 𝜎 +𝑐 𝐵2 ,\n(9)\n𝑥𝑥 0 w\nThis parabolic trend is indeed contradicted with linear line proposed in theoretic model of ultraquantum limit.\nSince above results and analysis have indicated that the NMR region is correlated to the lowest occupied Landau\n25\nbands, we believe that this inconsistency may be evoked by that 𝜏 is a magnetic field-dependent parameter.\n𝑎\nDespite the lack of direct experimental evidences, such a hypothesis of magnetic field-dependent 𝜏 seems\n𝑎\nreasonable considering its definition of inter-node relaxation time and the ultrahigh magnetic field prerequisite\nfor the breakdown of chiral anomaly. Nevertheless, the NMR tendency with magnetic field in ultraquantum limit\nregion is still an open-ended question, which need the further investigations in the future. We are coconscious of\nthat the previous description and formula of NMR fitting are with confusion and contradiction to the theoretic\nmodel. Thus, we rewrite this part in line 18, page 6 in the revised manuscript as: “The 𝜎 curves of Bi (Br I )\n𝑥𝑥 4 0.27 0.73 4\nsample are measured under a pulse magnetic field up to 57 T at different temperatures to investigate its transport\nbehavior, as shown in Fig. 5a and Fig. S13. For temperature below 50 K, the NMR data between two transition\npoints, 𝐵 and 𝐵 , can be nicely fitted by parabolic line. The NMR effect in ultraquantum limit can be described\n0 1\nby: 8-9,11-12\n𝜎 = 𝜎 +𝑐 𝐵 = 𝜎 +\n𝑒3𝜏𝑎\n𝐵 , (2)\n𝑥𝑥 0 𝑤 0 4𝜋2ℏ2𝑐\nwhere 𝜎 is the longitudinal conductivity, 𝜎 is residual conductivity, 𝑐 is the coefficient term, ⅇ is elemental\n𝑥𝑥 0 𝑤\ncharge, 𝜏 is inter-node relaxation time, ℏ is the reduced Planck constant, and 𝑐 is the light speed. The linear\n𝑎\nrelationship between 𝜎 and 𝐵 can be inferred in the condition that 𝜏 is a magnetic field-independent constant.\n𝑥𝑥 𝑎\n8-9 Since our results in Fig. 4b and Supplementary Note 12 indicate that the NMR region is correlated to the\nlowest occupied Landau bands, we believe that this inconsistency may be evoked by the fact that 𝜏 is a magnetic\n𝑎\nfield-dependent parameter. Despite the lack of direct experimental evidences, such a hypothesis of magnetic field-\ndependent 𝜏 seems reasonable considering its definition of inter-node relaxation time and the ultrahigh\n𝑎\nmagnetic field prerequisite for the breakdown of chiral anomaly. Nevertheless, the NMR tendency with magnetic\nfield in ultraquantum limit region is still an open-ended question, which need the further investigations in the\nfuture.”\nComments: The manuscript reports the successful manipulation of multiple Weyl modes, including gapped Weyl\nmodes, Weyl nodal rings (WNRs), and coupled Weyl states, in the ideal Dirac semimetal Bi₄(Br₀.₂₇I₀.₇₃)₄ via\nmagnetic field tuning. Key findings described in the manuscript include the observation of anomalous Hall effect\n(AHE) with step-like features under symmetry-breaking fields, attributed to large Berry curvature, the detection\nof negative magnetoresistivity (NMR) at low fields (linked to chiral anomaly in WNRs) and its collapse at high\nfields due to magnetic-tunneling-induced annihilation of Weyl nodes, the obtention of a comprehensive phase\ndiagram mapping Weyl physics as a function of field and temperature, supported by transport measurements and\nfirst-principles calculations.\nThe results reported in this manuscript have been validated through the agreement between magnetotransport data\n(e.g., Shubnikov–de Haas oscillations, NMR) and theoretical predictions, also through first-principles\ncalculations, confirming the emergence of gapped Weyl modes and WNRs under specific field orientations, and\nthe authors have also made use of control experiments (e.g., angle-dependent resistivity, high-field pulses up to\nIt is my opinion that this work contributes significantly to advance the field of topological materials. The authors\nhave demonstrated the reversible control over diverse Weyl phases (gapped, gapless, and coupled) in a single\nmaterial, they have also provided a symmetry-based framework for designing Weyl states, with implications for\nchiral spintronics, and they have also resolved some longstanding questions about chiral anomaly breakdown\nunder high fields, supported by quantitative modeling of magnetic-tunneling gaps.\nThe experimental and computational methodologies are rigorous. High-quality single crystals were characterized\nvia XRD, HAADF-STEM, and ARPES, ensuring sample integrity. Six-wire transport measurements and pulsed-\nfield experiments provided reliable magnetoresistivity and Hall data. Virtual crystal approximation (VCA) and\nWannier-based Berry curvature calculations accurately reproduced experimental observations. The authors have\nalso been able to combine experiment and theory, which is not frequently an easy task but always desirable. For\nthe transport data they analyzed using established models (e.g., Eq. (2) for NMR, Eq. (3) for tunneling gaps). The\ncalculations of the Landé g-factor and the Berry curvature distributions quantitatively were fundamental in order\nto provide a link to observations and underlying physics. The phase diagram synthesizes seemingly uncorrelated\nphenomena (as for example, AHE, NMR, and tunneling) into a unified picture.\nacknowledge the impact of our work. We appreciate the thorough review and have carefully considered each\ncomment. The revisions have been made according to the referee’s constructive suggestions to further improve\nthe quality of our work.\nQ1: To enhance the impact of their research work, the authors could consider clarifying the role of disorder (e.g.,\nBr/I vacancies) in transport, possibly via comparative studies with pristine Bi₄Br₄.\nA1: We appreciate the referee’s constructive suggestion on clarifying the role of disorder in transport results.\nThere are halide elements vacancies in this quasi-1D system, which has been reported in pristine Bi Br . 54 These\n4 4\nvacancies give rise to semiconducting-like transport behavior at low temperature region, which is evoked by the\nvariable ranging hopping conduction in Bi Br . Nevertheless, this variable ranging hopping conduction is\n4 4\noverwhelmed by the iterant carriers in Dirac semimetal Bi (Br I ) , engendering disappearance of low\n4 0.27 0.73 4\ntemperature semiconducting-like behavior. Therefore, the vacancies-induced localization effect becomes\nnegligible in Bi (Br I ) compared to Bi Br . We have added the related discussions in line 21, page 3 in the\n4 0.27 0.73 4 4 4\nrevised manuscript as: “It is noticeable that the halide vacancies induce semiconducting-like transport behavior\nin pristine Bi Br . 37 This localization effect becomes negligible in Bi (Br I ) due to the plenty of iterant\n4 4 4 0.27 0.73 4\ncarriers in Dirac semimetal Bi (Br I ) .”\n4 0.27 0.73 4\nQ2: They could also discuss the feasibility of room-temperature applications given the high-field requirements.\nA2: We express our sincere appreciation to the referee for his/her constructive criticism. We have added the\nrelated description in line 21, page 7 in the revised manuscript as: “The exploration of these Weyl modes\naccompanied with the AHE is beneficial to the room-temperature device application of non-volatile storage and\nmagnetic field detection.”.\nQ3: The occurrence of plateaus or steps in the AHE has been predicted by Focassio et al. in Phys. Rev. Research\n6, 033289 (2024) precisely due to the manipulation of Weyl nodes via magnetic field in three dimensional warped\nsemimetals, and this reference should give further support to the current manuscript.\nA3: We thank the referee for reminding us of adding the supportive and crucial reference. We have read the\nreference 55 carefully and added the related description in line 17, page 7 in the revised manuscript as: “Among\nvarious modulation ways, the magnetic field is regarded as an effective and reversible method to evoke\n28\ntopological phases transitions, which has been theoretically proposed in the 3D warped semimetals 15 and\nexperimentally verified in HfTe (from topological insulator to quasi-one-dimensional (quasi-1D) gapped Weyl\n5\nmode). 10,14,16-17”\nQ4: Also, it would be nice if the authors provided more discussion about the Landau index intercept in the fan\ndiagram in connection to a nonzero Berry phase and why this is around 1/8 instead of 0.5 as recently found by\nGang Qiu et al. Nature Nanotechnology volume 15, pages585–591 (2020) via quantum hall effect of Weyl\nfermions in tellurene.\nA4: We sincerely thank the referee for bringing up this important point. The same theory has been applied to plot\n1/B dependent Landau index diagram in the Nature Nanotechnology reference and our work. The reference\ndefines the valley and peak of the Hall resistivity as integer index n and half integers, respectively, against 1/B.\nthe reference is 0.5). Thus, our results are not contradicted with this reference. In fact, there are plenty of the\npublished papers with the same definition method reporting the near-zero intercept in the Landau index fan\ndiagram to demonstrate the non-trivial Berry phase. 6, 35, 53, 56 We have added the related description in line 18,\npage 5 in the revised manuscript as: “The 0.5 intercept correlated to nontrivial Berry phase has also been reported.\nconsistent with these references.”\nQ5: The manuscript is well-written but could be strengthened by briefly contextualizing the choice of\nBi₄(Br₀.₂₇I₀.₇₃)₄ over other Dirac/Weyl materials early in the text. Explicitly stating the novelty of observing *both*\nAHE and NMR in one system, contrasting prior work (e.g., TaAs, ZrTe₅).\nA5: We would like to express our sincere gratitude to the referee for bringing this insightful suggestion into our\nattention. We agree with the referee’s opinion that the novelty can be further clarified by the comparative study\nbetween our results and previous work. We have added the related discussions in line 7, page 3 in the revised\nmanuscript as: “All the experimental results are supported by the calculations, and finally, a phase diagram of\nWeyl physics as the function of magnetic field and temperature is established, constructing the various Weyl\nmodes into a unified picture in a single crystal quantum system and establishing the close relationship between\nthe crystal symmetry and chiral electrons.” along with the descriptions in conclusion part as: “Our work\nsimultaneously identifies the exotic phenomena, such as AHE, NMR, and annihilation of Weyl node, in one system,\ndistinguishing from the individual exploration in other work. 4,7,10-12”.\nCommunications. The interplay of symmetry, transport, and theory offers a pathway for manipulating Weyl\nfermions in devices. 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Magnetic control of Weyl nodes and wave packets in three-dimensional warped semimetals.\nPhys. Rev. Res. 6, 033289 (2024).\n56. Galeski, S., Ehmcke, T., Wawrzyńczak, R. et al. Origin of the quasi-quantized Hall effect in ZrTe . Nat\nWe express our deepest gratitude to the referees for their thorough examinations of our work and for\nproviding us with excellent comments and constructive feedback. The positive remarks of Referee #1, “with the\naddition of\nnew experiments and expanded discussions, the manuscript has improved significantly” and Referee #2, “the\nauthors were able to clarify some of the concerns raised by the referees, which, in my opinion, brings the\nmanuscript closer to publication.” encouraged us to further clarifications and to strengthen the impact of our\nwork. We have incorporated their valuable comments/suggestions and made the improvements to our paper. The\nkey revisions that we have undertaken are outlined below:\nAddition of new/extended results to address referees’ points and strengthen the manuscript conclusion:\n1. Detailed KPFM analysis has been added for clarifying absence of domains.\n2. Additional fitting results with variable exponent values are performed in anomalous Hall scaling plot,\nverifying reliability of our fitting results.\nWe also revised the presentation of our data to improve clarity:\n1. Statements of the quantum geometry effect on the AHE have been added, where the presence of inversion\nsymmetry forbids existence of Berry curvature dipole and quantum metric dipole effects.\n2. To clearly demonstrate absence of hysteresis in anomalous Hall resistivity, thin lines are adopted.\n3. Impact of our work has been further strengthened in the conclusion part.\n4. Electric transport symbols for denoting crystal axes are corrected to avoid unnecessary confusion.\n5. The anomalous Hall conductivity scaling plot has been redemonstrated, where the 1.6 fitting exponent\nand nonzero intercept represent localized hopping conduction and scattering independent contribution,\nrespectively.\nOnce again, we sincerely thank the referees for their efforts in providing thoughtful comments that have helped\nto enhance the quality of our research.\nComments: First, I appreciate that the authors have taken my suggestions seriously. With the addition of new\nexperiments and expanded discussions, the manuscript has improved significantly.\nReply: We express our sincere appreciation to the referee for his/her positive feedback and constructive\ncomments on our manuscript. We have taken the comments into careful consideration. Consequently, we have\nmade the related revisions and clarification to the manuscript according to the concerns raised by referee.\nQ1: Regarding the first point: in addition to Berry curvature, have the authors considered possible contributions\nfrom the quantum geometry of the bands?\nA1: We thank the referee’s insightful suggestion on the possible contributions from quantum geometry of bands\nto anomalous Hall effect (AHE). The quantum geometry of bands describes geometry of electronic wavefunctions\nin momentum space, which usually contains Berry curvature and quantum metric. Specifically, Berry\ncurvature and quantum metric represent imaginary part and real part of quantum geometric tensor, 1 respectively,\nand their multipole moments could yield the observable nonlinear Hall effect, which is a hot topic in the recent\nyears. 2-16\nBerry curvature deflects electrons as the pseudo-magnetic field in momentum space to produce anomalous Hall\neffect. 2-6 For Bi (Br I ) , the non-zero Berry curvature concentrated near the gapped Weyl mode, engendering\n4 0.27 0.73 4\nthe step-like anomalous Hall signal as shown in the main text. It is noted that the intrinsic linear anomalous Hall\neffect can only be induced by the Berry curvature but not generated through the quantum metric, due to the T-odd\nand T-even properties (T is time reversal operation) for these two physical quantities respectively. Furthermore,\nBerry curvature can be expanded to its multipole moments even with time reversal symmetry, e.g., the Berry\ncurvature dipole with inversion symmetry (P) breaking. 7-10 In this situation, the sum of Berry curvature for all\noccupied states in the whole Brillouin zone is zero, but Berry curvature hot spots with opposite sign are separated\nin momentum direction. Consequently, the nonlinear current-voltage Hall signal with double frequency (second-\norder Hall effect) can be generated. In addition, the Berry curvature quadrupole can contribute a third-order Hall\neffect. 11-13 On the other hand, the quantum metric dipole can also emerge with broken P, which could result in\nan intrinsic (scattering-independent) nonlinear Hall effect in the PT-preserved antiferromagnetic system. 14-16\nFor Bi (Br I ) , the inversion symmetry is preserved together with the non-zero integrated Berry curvature\n4 0.27 0.73 4\nin gapped Weyl mode, ruling out the other possible contributions from the quantum geometry of the bands, such\n2\nas Berry curvature dipole, and quantum metric dipole. Moreover, the AHE in this work is measured by D.C.\ncurrent, which is applied to detect first-order Hall response. Berry curvature dipole/quadrupole, and quantum\nmetric dipole effect are high-order Hall response, where the nonlinear Hall signals are supposed to be detected by\nA.C. current. 7-16 Therefore, we believe that the origin of the step-like Hall effect in our work can be correlated to\nBerry curvature mechanism. We have added the related description in line 5 page 4 in the revised manuscript as:\n“The quantum geometry of the electronic bands can also evoke the nonlinear Hall effect, where only the Berry\ncurvature, which acts like an effective magnetic field in momentum space and conforms to the condition of the\nreserved inversion symmetry in Bi (Br I ) , contribute the linear AHE. 7,38-47”\n4 0.27 0.73 4\nQ2: In Figure R2, the KPFM measurements provide compelling evidence against the charge puddle scenario.\nHowever, what are the linear features observed in Figure R2a? Could these be domain walls or other structural\ninhomogeneities?\nA2: We sincerely appreciate the referee for bringing up this important point into our attention. We have\nreproduced the AFM images with line-cut height, phase, and KPFM potential data of previous Figure R2(a) as\nFigure R1(a)-R1(c) here, respectively. The height line profile of yellow line in Figure R1(a) shows that the height\nvariation of the lines marked by the yellow triangles is about 0.3 nm, which is far less than the thickness of\nmonolayer Bi (Br I ) (~ 1 nm) and indicates that the terrace contains these lines are in the same layer.\n4 0.27 0.73 4\nMoreover, this line feature is expected to be more obvious in the phase image of AFM results in the condition\nthat the lines are the boundary of the different domain walls. Nevertheless, the phases image of the same area\nmeasured in Figure R1(a) implies the homogeneous phase distribution with the disappeared lines among the whole\narea. Therefore, the lines in topography image are not the boundary of the different domain walls, and may be\ncaused by scratches when preparing samples or the wrinkles in low dimensional materials. 17-19\nIn order to clarify this point, we have incorporated Figure R1 as Fig. S8 and added the related description in\nSupplementary Note 5 and Fig. S8 in Supplementary Information as: “The height line profile of yellow line in Fig.\nS8a shows that the height variation of the lines marked by the yellow triangles is about 0.3 nm, which is far less\nthan the thickness of monolayer Bi (Br I ) (~ 1 nm) and indicates that the terrace contains these lines are in\n4 0.27 0.73 4\nthe same layer. Moreover, this line feature is expected to be more obvious in the phase image of AFM results in\nthe condition that the lines are the boundary of the different domain walls. Nevertheless, the phases image (Fig.\nS8b) of the same area measured in Fig. S8a implies the homogeneous phase distribution with the disappeared\nlines among the whole area. Therefore, the lines in topography image are not the boundary of the different domain\n3\nwalls, and may be caused by scratches when preparing samples or the wrinkles in low dimensional materials. 8-\n10”\nFigure R1. (a) Topography, (b) phase, and (c) potential results for the cleaved (001) surface of Bi (Br I ) ,\n4 0.27 0.73 4\nrespectively. The upper panels show mapping images, and the lower panels exhibit the line profiles of the line-\ncut information. Yellow triangles denote the locations of the crossing points of the line-cut and lines in topography.\nQ3: For Figure R3d, the use of a significantly thicker line compared to Figures R3a–c raises concern. Does this\naffect the claim of no hysteresis? If the same line width used in R3a–c is applied to R3d, would the forward and\nbackward scans still overlap when measured over multiple cycles?\nA3: We appreciate the referee for highlighting this crucial aspect. We admire that the thicker line hinders the\nidentification of the difference between forward and backward scan curves. We replot the Figure R3(d) (Figure\nR2(d) here) with the thin line to clearly reveal the absence of the hysteresis. The Fig. S4 in Supplementary\nInformation has also been updated accordingly.\ntemperatures, respectively. (d) shows the Hall resistivities at 2 K measured by different scanning field directions\n(marked by black arrow and red arrow, respectively), where the non-hysteresis feature is observed.\nQ4: In Figure R4b, how reliable is the theoretically predicted value from the skew-scattering model? Based on\nthe plot, a value between 1.6 and 2 appears to provide a comparably good fit. Could the authors comment on the\nsensitivity of the fit to this parameter, especially in relation to Figure R4c?\nA4: We are grateful to the referee for this insightful question on the reliability of fitting parameters. It is noticeable\nthat there are two parameters, the exponent and the intercept, can be obtained in the fitting process. 20-25 For Figure\nR4(b) (Figure R3(b) here), the intercept is forced to be zero, and neither the exponent 1.6 nor 2 can fit the\n5\nexperimental data well. We also apply other exponent values, 1.7, 1.8, and 1.9, to fit the data with zero intercept,\nas shown in Figure R3(b), where none of the single fitting line can pick up all the experimental data. According\nto the referee’s suggestion, we check the sensitivity of the exponent in fitting, as displayed in Figure R3(c), where\na slight value variation from the best-fitted value leads to the deviation from our experimental data (e.g., refer to\nthe data in the upper right corner of Fig. R3c). The upper and lower limits of exponent are 1.74 and 1.51,\nrespectively, implying the validity of the characteristic 1.6 exponent. We have updated the Fig. S9 and added the\nrelated description in Supplementary Information as: “We apply various exponent values, 1, 1.6, 1.7, 1.8, and 1.9,\nto fit the data with zero intercept, as shown in Fig. S9b, where none of the single fitting line can pick up all the\nexperimental data. We also check the sensitivity of the exponent in fitting, as displayed in Fig. S9c, where a slight\nvalue variation from the best-fitted value leads to the deviation from our experimental data. The upper and lower\nlimits of exponent are 1.74 and 1.51, respectively, implying the validity of the characteristic 1.6 exponent.”\nFigure R3. (a) Anomalous Hall conductivity at 20 K with the grey square to denote the fluctuation of Hall\nconductivity. The error bars are defined by the upper value and lower value of step-like Hall conductivity in the\ngrey region at each temperature. (b) Relationship between anomalous Hall conductivity and longitudinal\nconductivity. Failed fitting results for exponent 1, 1.6, 1.7, 1.8, 1.9, 2.0 with zero intercept are exhibited,\nindicating the non-negligible scattering independent contribution. (c) Relationship between anomalous Hall\nconductivity and longitudinal conductivity. The dark red line, purple line, and orange line denote the fitting results\n6\nby using the exponents of 1.6, 1, and 2, respectively. The fitting results with the exponents of upper limit 1.74\nand lower limit 1.51 are presented by green and blue lines, respectively, indicating the validity of fitting results\nwith the exponent 1.6.\nQ5: The statement that RMCD and SQUID only detect magnetic elements is not entirely accurate. For example,\norbital magnetism arising from Berry curvature can also be effectively detected by both techniques. In general,\nany form of angular momentum that couples to either photons (in the case of RMCD) or magnetic flux (in the\ncase of SQUID) should be observable.\nA5: Thanks for the referee’s valuable comments on the statement on the role of RMCD and SQUID in orbital\nmagnetism. After the literature research, we do agree well with the referee’s opinion that the scanning-SQUID or\nRMCD is the effective tool to characterize Berry curvature. According to wave packet theory, orbital magnetism\nhas two origins, self-rotation of the wave packet and center-of-mass motion. 26-28 The second term contains\nadditional orbital momentum induced by Berry curvature, which can be detected by the state-of-art scanning-\nSQUID. 29 For Bi (Br I ) , its non-zero Berry curvature under the external magnetic field is possible to\n4 0.27 0.73 4\ngenerate additional orbital momentum and to be characterized by high-resolution RMCD or scanning-SQUID.\nHowever, the experimental extraction of topological orbital magnetism is challenging, and we are unable to\nperform such state-of-art RMCD or scanning-SQUID measurements, which requires further study in the future.\nWe have added the related illustration in line 27, page 4 in the revised manuscript as: “It should be noted that\nBerry curvature can be characterized by scanning-superconducting quantum interference device (scanning-\nSQUID) or reflectance magnetic circular dichroism (RMCD), which requires further study in the future. 48-51”\nQ6: Overall, I believe the significance of the manuscript still needs to be strengthened. Compared to magneto-\ninfrared studies on HfTe , the transport-based claims are more dependent on modeling and fitting. While the\n5\nauthors have made commendable efforts to address this issue, further clarification or validation could enhance\nthe impact of the work.\nA6: We would like to express our sincere gratitude to the referee for bringing this insightful suggestion into our\nattention. We agree with the referee’s comments on the transport-based claims are more dependent on modeling\nand fitting compared to magneto-infrared studies on HfTe . However, the results reported in our work have been\n5\nvalidated through the excellent coincidence among the symmetry-based framework for designing Weyl states,\nmagnetotransport data and fitting results, and first-principles calculations as well as the calculated parameters,\nwhich is not frequently an easy task but always desirable. Our work simultaneously identifies the exotic\n7\nphenomena, such as AHE, NMR, and annihilation of Weyl node, in one system, distinguishing from the individual\nexploration in other work. 30-37 Furthermore, as the ideal Dirac semimetal with only the Dirac bands residing in\nthe Fermi level, Bi (Br I ) provides a material platform to realize more intriguing quantum transport\n4 0.27 0.73 4\nbehaviors contributed by quantum geometry after the symmetry breakdown. 9-10, 38-39 We have incorporated the\nrelated discussion in conclusion part as: “Our work simultaneously identifies the exotic phenomena, such as AHE,\nNMR, and annihilation of Weyl node, in one system, distinguishing from the individual exploration in other work.\n4,7,10-12 The results reported in our work have been systematically validated through the excellent coincidence\namong the symmetry-based framework for designing Weyl states, magnetotransport data and fitting results, and\nfirst-principles simulations as well as the calculated parameters, which is rarely a task in previous reports.\nFurthermore, as the ideal Dirac semimetal with only the Dirac bands residing in the Fermi level, Bi (Br I )\n4 0.27 0.73 4\nprovides a material platform to realize more intriguing quantum transport behaviors contributed by quantum\ngeometry after the symmetry breakdown. 44-47”.\nOnce again, we extend our sincere gratitude to the referee for his/her effort for helping us improve the quality of\nour work.\nComments: In their response, the authors were able to clarify some of the concerns raised by the referees, which,\nin my opinion, brings the manuscript closer to publication. However, some of the answers and changes in the\nmanuscript have raised new questions, which, I would like to be addressed.\nReply: We express our sincere appreciation to the referee for his/her positive feedback and constructive\ncomments on our manuscript. The manuscript has been revised to clarify the questions addressed by the referee.\nWe hope the referee find the revised version satisfied.\nQ1: I stumbled upon the fact that the authors use symbols 𝜎 and 𝜌 for the longitudinal conductivity and\n𝑥𝑥 𝑥𝑥\nresistivity. On the other hand, in Fig. 3a they indicate y to be the current direction — that is also how I understood\nthe data previously and how it makes sense in view of the chiral magnetic effect for B along y direction. I thus\nhope that they really mean 𝜎 and 𝜌 . If yes, the symbols should be changed to avoid confusion. If the\n𝑦𝑦 𝑦𝑦\nconductivity is really measured along x, the explanation of the NMR does not work.\nA1: We sincerely thank the referee for reminding us of correcting the wrong symbols. We indeed mean the 𝜎\n𝑦𝑦\nand 𝜌 in Fig. 3a with the current along y direction rather than 𝜎 and 𝜌 . Thus, all related description and\n𝑦𝑦 𝑥𝑥 𝑥𝑥\nfigures have been corrected in the revised manuscript and the revised Supplementary Information.\nQ2: Regarding the scaling of the anomalous Hall conductivity with the longitudinal conductivity: Empirically,\nthe scaling with exponent 1.6 is often observed in dirty metals, which makes the similar observation in the present\npaper sound. However, I still disagree that the scaling with the exponent 1.6 can be ascribed to side-jump\nscattering. According to Ref. 14, side-jump scattering contribution is constant (zero instead of 1.6).\nA2: We sincerely appreciate this crucial criticism provided by the referee. In the previous version, we attribute\nthe exponent 1.6 to side-jump scattering case based on the descriptions of Ref. 21 and Ref. 22 in the final\nreferences, where lines with the exponent 1.6 is ascribed to the side-jump scattering. After careful literature\nreview20-25, we now agree with the referee’ opinion that the side-jump mechanism is scattering independent,\nand\nthe exponent of 1.6 is correlated to the localized hopping conduction region.\nAnomalous Hall conductivity induced by skew scattering, an asymmetric scattering, is dependent on the\nscattering rate, which gives scattering dependent behavior. 20-22 For side-jump scattering, the induced anomalous\nHall conductivity can be viewed as the product of the side jump per scattering event and the scattering rate,\n9\nyielding the scattering independent relationship. 20, 23-25 Therefore, the side-jump scattering contribution has the\nsimilar constant tendency as the intrinsic contribution in scaling plot, where there is an open question about\nproperly distinguishing them. 20 We have modified the uncorrected descriptions by attributing the 1.6 exponent\nrelationship in scaling plot to the localized hopping conduction, and the non-zero intercept has been attributed to\nscattering independent part.\nWe have revised the related description and figures (new Fig. 3 is shown as Figure R4 here) in line 7, page 4\nin the revised manuscript as: “The contribution of non-zero Berry curvature to Hall resistivity can be classified\ninto two parts, the scattering independent part (intrinsic and side-jump contribution) and scattering dependent\npart (localized hopping and skew scattering contribution). 38-40 In this case, the saturated Hall conductivity 𝜎𝐴\n𝑥𝑦\ncan be expressed as:\n𝜎𝐴 = 𝛼𝜎 1.6 +𝜎𝐼𝑛𝑑., (1)\n𝑥𝑦 𝑦𝑦 𝑥𝑦\nwhere 𝛼 is a constant correlated with residual resistivity, 𝜎 is the conductivity at zero field, and 𝜎𝐼𝑛𝑑. is the\n𝑦𝑦 𝑥𝑦\nanomalous Hall conductivity (AHC) correlated to the scattering independent part. Fig. 3c exhibits the 𝜎𝐴 values\n𝑥𝑦\nas a function of 𝜎 measured at different temperatures, where all the points can be well fitted by Eq. (1) with a\n𝑦𝑦\nfinite interception value. The validity of our fitting results is supported by the comparative study of fitting lines\nby using different exponents and intercepts, as shown in Fig. S9 in Supplementary Information. The 1.6 exponent\nindicates existence of the localized hopping conduction in dirty metals (in which conductivity less than 10000\nS/cm), 41-43 and the nonzero interception value (~169 S/cm) is a direct reflection of the scattering independent\ncontribution, 𝜎𝐼𝑛𝑑.. It is noted that both extrinsic side-jump mechanism and intrinsic contribution are supposed\n𝑥𝑦\nto be scattering independent, which are not distinguishable here from scattering plot. 38,41-43 Consequently, the\nwell fitted experimental data in Fig. 3c by Eq. (1) indicates that the step-like feature originates from both\nscattering dependent localized hopping effect and scattering independent contribution, where non-zero Berry\ncurvature behaves as the source to deflect electrons. 42 The anomalous Hall angle (AHA) standing for the\ndeflection angle induced by Berry curvature can be calculated by 𝑡𝑎𝑛−1( 𝜎𝑥 𝐴 𝑦), and is 20.8° at T = 20 K (the inset\n𝜎𝑦𝑦\nof Fig. 3c). The large AHA in Bi (Br I ) is contributed by both localized hopping conduction and scattering\n4 0.27 0.73 4\nindependent contribution of Berry curvature (see Supplementary Note 7 for detailed illustration of Berry\ncurvature contribution to carrier deflection), which is reasonable to emerge in strongly disordered metal (i.e. bad\nmetal with low conductivity) as shown in Fig. S10, coinciding with the high I dopants concentration (more\ndisorders than pure Bi Br sample). ”\n4 4\nelectrodes, respectively, where current is applied along chain direction (crystal b axis) and Hall voltage is\nmeasured along a axis. (b) Anomalous Hall resistivity (AHR) with different angles of magnetic field (left panel)\nand angle-scaled AHR (right panel). Two stages in AHR are labelled with black lines, and insets define the angle\n𝜑. (c) Relationship between anomalous Hall conductivity (AHC) and longitudinal conductivity. Dependence of\n∝ 𝜎\n1.6\nand finite intercept (red line) imply AHE contains both localized hopping part and scattering independent\n𝑦𝑦\npart. The grey region indicates scattering independent contribution of 169 S/cm. Inset shows evolvement of\nanomalous Hall angle (AHA) with temperature. Simulated band structure and intrinsic AHC with (d) 𝐵 = 2 T\n𝑧\nand (e) 𝐵 = 4 T, respectively. Green dashed line denotes Fermi surface. Red and blue colors represent positive\n𝑧\nand negative values of Berry curvature 𝛺𝑛, respectively, which are mainly contributed by the SOC gaps.\n𝑧\n(f)\nCalculated Berry curvature distributions which are summed to Fermi energy under 𝐵 = 2 T (left panel) and 𝐵 =\n𝑧 𝑧\nQ3: Lines 183-186: The formulation regarding the magnetic-field dependence of tau is repeating in these\nsentences.\nA3: We sincerely thank the referee for bringing up this important point. We have revised these sentences as: “𝜏\n𝑎\nis inter-node relaxation time, ℏ is the reduced Planck constant, and 𝑐 is the light speed. The linear relationship\nbetween 𝜎 and 𝐵 can be inferred in the condition that the leading coefficient is a constant. 8-9”\n𝑦𝑦\nconnection to the field-induced breakdown of chiral anomaly is not plausible since 𝜏 proportional to B means\n𝑎\nthat inter-node scattering weakens with increasing B. In my opinion, it is alright to leave the B-dependence of tau\nas an open question if the authors have no plausible explanation yet.\nA4: We thank the referee for pointing out this unconfirmed claim. We admire that the description of “reasonable\nB-dependent 𝜏 ” lacks plausible support in this stage. In order to make a clear statement, we have modified the\n𝑎\nrelated sentences as: “Since our results in Fig. 4b and Supplementary Note 12 indicate that the NMR region is\ncorrelated to the lowest occupied Landau bands, we believe that this inconsistency may be evoked by the magnetic\nfield-dependent 𝜏 . This hypothesis needs the further investigations in the future as the NMR tendency with\n𝑎\nmagnetic field in ultraquantum limit region is still an open-ended question.”\nWe express our gratitude to the referee again for his/her diligent efforts and highly precious suggestions for\nstrengthening the quality of our work.\nComments: I have one minor question before recommending for publication. The authors replot the forward and\nbackward scan with thinner lines in Figure R2(d). Could the author zoom in around positive and negative 2T\nregion? To me it looks like some small hysteresis. Could the author comments on this?\nFigure R1. Anomalous Hall resistivity of (a) S1, (b) S2, (c) S3, and (d) S4 samples measured at different temperatures,\nrespectively. (d) shows the Hall resistivities at 2 K measured by different scanning field directions (marked by black\narrow and red arrow, respectively). Enlarged plots near 2 T and -2 T are shown in the insets of (d).\nReply: We sincerely appreciate the referee for bringing in this point into our attention. We have added the\nenlarged view of the previous Figure R2(d) (the insets of Figure R1(d) here) in the regions of 2 T and -2 T as\nsuggested by the referee, where a small misalignment can be identified in these two step regions. The similar but\nmore obvious behavior has been reported in Kagome ice compound HoAgGe, which is caused by the varied overall\nBerry curvatures based on the frustrated lattice distortion during the forward scan and backward scan. 1 For\nBi (Br I ) crystal, the lattice distortion induced by dopants may also engender such phenomenon with the\n4 0.27 0.73 4\nnon-degenerated states and different Berry curvatures of two opposite scans. We have added the related discussion\n1\nin Supplementary Note 2 in Supplementary Information as: “The enlarged view displayed in the inset of panel d\nimplies that there are small misalignments at the regions near 2 T and -2 T, implying the possible existence of\nhistory-dependent feature. The similar but more obvious behavior has been reported in Kagome ice compound\nHoAgGe, which is caused by the varied overall Berry curvatures based on the frustrated lattice distortion during\nthe forward scan and backward scan. 1 For Bi (Br I ) crystal, the lattice distortion induced by dopants may\n4 0.27 0.73 4\nalso engender such phenomenon with the non-degenerated states and different Berry curvatures of two opposite\nscans. Nevertheless, the further study is required to resolve this issue in the future.”\nComments: I am happy with the answers and modifications provided by the authors and recommend publication.\nReply: We sincerely thank the referee’s efforts for enhancing the overall quality of our work.\n1. Zhao, K., Tokiwa, Y., Chen, H. et al. Discrete degeneracies distinguished by the anomalous Hall effect in a\nmetallic kagome ice compound. Nat. Phys. 20, 442–449 (2024).\n4"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I have one minor question before recommending for publication. The author replot the forward and backward scan with\nthinner lines in FigureR2d. Could the author zoom in around positive and negative 2T region? To me it looks like some\nsmall hysteresis. Could the author comments on this?", "reviewer_2": "I am happy with the answers and modifications provided by the authors and recommend publication."}, "author_response": "Referee #1\nComments: I have one minor question before recommending for publication. The authors replot the forward and\nbackward scan with thinner lines in Figure R2(d). Could the author zoom in around positive and negative 2T\nregion? To me it looks like some small hysteresis. Could the author comments on this?\nFigure R1. Anomalous Hall resistivity of (a) S1, (b) S2, (c) S3, and (d) S4 samples measured at different temperatures,\nrespectively. (d) shows the Hall resistivities at 2 K measured by different scanning field directions (marked by black\narrow and red arrow, respectively). Enlarged plots near 2 T and -2 T are shown in the insets of (d).\nReply: We sincerely appreciate the referee for bringing in this point into our attention. We have added the\nenlarged view of the previous Figure R2(d) (the insets of Figure R1(d) here) in the regions of 2 T and -2 T as\nsuggested by the referee, where a small misalignment can be identified in these two step regions. The similar but\nmore obvious behavior has been reported in Kagome ice compound HoAgGe, which is caused by the varied overall\nBerry curvatures based on the frustrated lattice distortion during the forward scan and backward scan. 1 For\nBi (Br I ) crystal, the lattice distortion induced by dopants may also engender such phenomenon with the\n4 0.27 0.73 4\nnon-degenerated states and different Berry curvatures of two opposite scans. We have added the related discussion\n1\nin Supplementary Note 2 in Supplementary Information as: “The enlarged view displayed in the inset of panel d\nimplies that there are small misalignments at the regions near 2 T and -2 T, implying the possible existence of\nhistory-dependent feature. The similar but more obvious behavior has been reported in Kagome ice compound\nHoAgGe, which is caused by the varied overall Berry curvatures based on the frustrated lattice distortion during\nthe forward scan and backward scan. 1 For Bi (Br I ) crystal, the lattice distortion induced by dopants may\n4 0.27 0.73 4\nalso engender such phenomenon with the non-degenerated states and different Berry curvatures of two opposite\nscans. Nevertheless, the further study is required to resolve this issue in the future.”\nComments: I am happy with the answers and modifications provided by the authors and recommend publication.\nReply: We sincerely thank the referee’s efforts for enhancing the overall quality of our work.\n1. Zhao, K., Tokiwa, Y., Chen, H. et al. Discrete degeneracies distinguished by the anomalous Hall effect in a\nmetallic kagome ice compound. Nat. Phys. 20, 442–449 (2024).\n4"}, {"round": 3, "reviewer_comments": {"reviewer_1": "The authors have answered all my questions and i can now recommend the manuscript for publication."}, "author_response": ""}]

Decision letter