Unusual interlayer quantum transport behavior caused by the zeroth Landau level in YbMnBi2

Relativistic fermions in topological quantum materials are characterized by linear energy–momentum dispersion near band crossing points. Under magnetic fields, relativistic fermions acquire Berry phase of π in cyclotron motion, leading to a zeroth Landau level (LL) at the crossing point, a signature unique to relativistic fermions. Here we report the unusual interlayer quantum transport behavior resulting from the zeroth LL mode observed in the time reversal symmetry breaking type II Weyl semimetal YbMnBi2. The interlayer magnetoresistivity and Hall conductivity of this material are found to exhibit surprising angular dependences under high fields, which can be well fitted by a model, which considers the interlayer quantum tunneling transport of the zeroth LL's Weyl fermions. Our results shed light on the unusual role of zeroth LLl mode in transport.

(2) The authors did not distinguish the concept of Landau levels in 2D systems and Landau bands in 3D cases. For example, on line 47 of the first paragraph they stated that a direct probe of relativistic fermions is not feasible for 3D Dirac or Weyl semimetals, such as Cd3As2. However, it is not true that in 3D Dirac (Weyl) semimetals the quantum limit when only the lowest Landau band is occupied has been achieved in several recent experiments.
(3) The authors fitted the data of the Hall resistance with a tangent dependence at lower angles. At larger angles, they only argued that the deviation is due to the suppression of the Landau quantization. It would be good if the authors could also fit the data at larger angles. Also, there are many pockets in this system. Do others pockets affect the Hall resistance? (4) The deviation between the SdHO data and the two-carriers model at 1/B=0.03 is somewhat obvious. It seems that the deviation is not from higher order terms or Zeeman splitting. Can the authors give a better fit?
Reviewer #3 (Remarks to the Author): In this manuscript, the authors have presented the in-plane and interlayer transport features for antiferromagnetic YbMnBi2 hosting Weyl fermions. In addition to the SdH oscillations in in-plane resistivity, they measured the detailed angle dependence of interlayer magnetoresistance effects. From the latter dataset, they argue the possible contribution of zeroth Landau mode in the interlayer conduction. They systematically measured the transport properties and display them in a clear manner. However, in my opinion, the manuscript has several serious problems. First, all the results shown in Fig. 1 (magnetic structure, magnetoresistance, details of SdH oscillations) have been already reported in ref. [18]. So I could not find any new discoveries in the present data on the in-plane transport and neutron diffraction.
Second, the analyses on the data of interlayer transport are not consistent. Although the authors assume that the tunneling processes are dominant in the interlayer transport, the temperature profile of interlayer resistivity R_zz in Fig. 1c exhibits nice metallic behaviour over the entire temperature range. Furthermore, the interlayer conductivity based on the tunneling between the zero-mode Landau levels (Eq. 1) gives negative magnetoresistance effects. However, as shown in Fig. 2b, the observed magnetoresistance effects are positive irrespective of the field angle. These facts mean that the interlayer conduction for YbMnBi2 should be dominated by the usual coherent transport of the carriers from some other bands and hence the tunneling effect of the zero-mode of Landau level should be negligible. In fact, in the second paragraph on page 9, they fit the AMR data using the conventional B^2 magnetoresistance effect, which also reproduces the experimental results. Because the fitted result is almost independent of the model of the zeromode Landau level transport, the zero mode plays a minimal role in the total interlayer conduction. Therefore, I don't think the main conclusion of this manuscript is supported by the present data and analyses.
From the above reasons, I don't recommend the publication of this manuscript as it is. After the significant revisions, it may be suitable for publication in some specialised journal.
The followings are other comments to be considered.
1. On the line 182, the authors adopt the B-linear magnetoresistance for sigma_c, because the Weyl fermions reach the quantum limit at sufficiently high field. However, as shown in Fig. 1, clear SdH oscillations were observed even around 40 T, indicating the quantum limit is not achieved for the Weyl fermions with the nodes away from the Fermi energy. Therefore, the conventional B^2 MR should be adopted for sigma_c in the entire field range. I see some technical issues which could be better improved. We really appreciate these suggestions given by the referee, which have been very helpful in improving our manuscript. We have addressed these technical issues following the referee's suggestions: Fig.1 may be too crowded and could be split into two figures. We fully agree with the reviewer that Fig.1 looks crowded. Fig. 1 in our original manuscript shows the structural and electronic property characterizations of YbMnBi2. Some of the data is indeed not directly relevant to the focus of this manuscript, i.e. the unusual quantum transport behavior caused by the zeroth Landau levels. To make our presentation more succinct and focused, we have revised Fig. 1 following the reviewer's suggestion. In the revised manuscript, we have moved the less-relevant data, i.e. the magnetic structure, the temperature dependence of magnetic ordering parameter, the temperature dependences of the in-plane and out-of-the plane resistivities and the effective mass analyses based on the SdH oscillations to the Supplementary Information. The revised Fig. 1 is shown below; it includes only the in-plane magnetoresistivity and the SdH oscillation analyses. To make our discussions clear, we have also added the schematics of the electronic band structure of YbMnBi2 determined by the ARPES experiments 1 following Reviewer 2 and 3' suggestions. B) The language can be better polished. We thank the reviewer for pointing out this problem. We have carefully read and polished the manuscript.
C) a summary section should be added. We thank the reviewer for this good suggestion. We have added a summary section to the revised manuscript, as shown below: "In summary, we have studied the in-plane magnetoresistivity  xx (B), the magnetic field orientation dependences of the out-of-plane magnetoresistivity AMR() and in-plane Hall resistance Rzx() under various magnetic fields for Weyl semimetal YbMnBi2.  xx (B) exhibits remarkable SdH oscillations, from the analyses of which non-trivial Berry phases were extracted; this verifies the existence of Dirac band crossings above/below EF. For AMR() and Rzx(), we observed unusual angular dependences under high fields. Both the AMR() and Rzx() data can be well fitted to a model which considers both the interlayer tunneling of Weyls fermions at the zeroth LLs and the momentum relaxation transport of other Dirac bands. Our finding highlights the unusual role of the zeroth LLs in transport, which is important to further understand the novel Dirac/Weyl fermion physics". D) more refs should be added to the introduction regarding 3D Dirac materials and the relevant work on ZrTe5 should be mentioned since it is an important Dirac material. We apologize for missing some important references. We have added more relevant references, including those on ZrTe5, to the revised manuscript.

Response to Reviewer #2's comments:
The authors report the observation of magneto transport in YbMnBi2, and attribute the observed behaviors to the zeroth LL and linear band crossings of topological semimetal. Recently, the type-II Weyl semimetal is one of growing fields. I do not know other published works on the electric transport in YbMnBi2, so the manuscript has a potential impact to the community. However, I am confused about the claims by the authors. I suggest the authors to address them properly before a further consideration.
We thank Reviewer #2 for taking the time to review our manuscript. The focus of our manuscript is to show how the Weyl fermions on the zeroth Landau level (LL) of YbMnBi2 cause unusual transport behavior via measuring the field orientation dependences of interlayer magnetoresistivity and in-plane Hall resistance. Since the zeroth LL is one of the key signatures unique to Dirac/Weyl fermions, revealing transport evidence of the zeroth LL is of fundamental importance. We appreciate that the reviewer has seen the potential impact of our work. We apologize for not making our claims clear enough to the reviewer in the original manuscript and appreciate the reviewer's insightful comments, which have been very helpful in improving our manuscript. We have now addressed all the issues raised by the reviewer and revised the manuscript accordingly. We have made point-to-point responses (in black) to the reviewer's comments (in red) in the following.
(1) The authors did not give a clear picture of the energy bands of the system. In Fig. 1d, they show a clear SdH oscillation at low temperatures. The quantum limit was not reached even when the magnetic field is above 30T. However, then they argued that the Fermi energy was right at the Dirac point, and the observed main result in Fig. 2c was from the zeroth Landau level, where the sharp peak is obvious only when the field is larger than 9T. This is very confusing. They stated that this system consisted of electron and hole Dirac bands, at the connection points of these pockets, cone-like dispersions of Bi layers with node being at the Fermi energy appear. They argued that the previous Dirac bands contributed to the SdH oscillations, while the latter states leaded to the sharp peak in the AMR. I think that they should give a clear picture of the energy bands and the reason why the Dirac bands did not influence the sharp peaks of the AMR and why the Fermi energy is right at the nodes of Weyl cones (no measurement supports it in the manuscript). The red and blue pockets correspond to electron-and hole-like pockets, respectively. The black dots represent Weyl points. (b) Schematic of the linear band crossing for the electron-(cut#3) and hole-like (cut #2) pockets and the Weyl point (cut #1), also determined by ARPES experiments 1 . (c) Schematic of Landau levels for three types of band crossing shown in (b) under high magnetic fields. (d) The in-plane magnetoresistivity  xx / xx as a function of magnetic field along the out-of-plane direction. Inset, the FFT spectra of the SdH oscillations in  xx / xx . (e) The fits of SdH oscillations at 2K and 18K to the two-band LK formula.
We thank the reviewer for bringing up these important issues. Clarifications of these issues are indeed critical to the understanding of our claim as the reviewer pointed out. We apologize for not giving a clear electronic band structure for YbMnBi2 in our original manuscript, which we believe caused the reviewer's confusion. The electronic band structure of YbMnBi2 has been determined by Borisenko et al 1 using ARPES experiments and first principle calculations. In Fig. 1 attached above (i.e. Fig. 1 in the revised manuscript), we show the schematic of the projected Fermi surface on the kx-ky plane determined by the ARPES experiments (Fig. 1a) and the band dispersions along several typical momentums (Fig. 1b). The Fermi surface consists of the Weyl points (denoted by the black dots in Fig. 1a) and the hole-like (marked in blue) and electron-like (red) pockets comprised of linear Dirac bands. The Weyl points appear at the momentum points where electron-and hole-like pockets are connected, which is a typical signature of type II Weyl semimetal as claimed by Borisenko et al 1 . Besides the Weyl points shown in Fig. 1a, the first principle calculations also predicted another four Weyl points at different momentum points and two Weyl loops; but these features were not observed in the ARPES experiments 1 . Given such a multi-band electronic structure, the transport properties of YbMnBi2 should be contributed by both the Weyl points and the hole-and electron-like pockets. Due to the fact that the probed Weyl nodes are at EF, while the Dirac bands forming the holeand electron-like pockets cross at energies above or below EF as illustrated in Fig. 1b, the Weyl and Dirac fermions are expected to exhibit distinct magnetotransport behaviors, as explained below.
As shown in Fig. 1c, under magnetic fields, both the Weyl bands and Dirac bands split to quantized Landau levels (LLs), the energies of which can be expressed as 18 . The n = 0 level corresponds to the zeroth LL, which is always locked to the Weyl/Dirac node upon field sweep no matter whether the node is at EF or not (Fig. 1c). For the Dirac bands, since their crossing points, i.e. the Dirac nodes, are away from EF (Fig. 1c, panel #2 & #3), their LLs would successively pass through EF upon increasing the magnetic field, thus resulting in oscillating density of state DOS at EF, which is manifested in quantum oscillations in our measured in-plane magneto-resistivity ( Fig. 1d and 1e). By contrast, the Weyl nodes shown in Fig. 1a are located at EF, the zeroth LLs of the Weyl bands are pinned to EF regardless of magnetic field strength (Fig. 1c, panel #1). Under this circumstance, the Weyl bands are not expected to show quantum oscillations, since no LL passes EF upon increasing the field. Instead, DOS(EF) monotonically increases due to the increase of the zeroth LL's degeneracy. In general, it is hard to observe such an effect in transport measurements in most topological materials due to their Dirac/Weyl nodes away from EF and/or the complexity of electronic band structure.
However, YbMnBi2 offers an excellent opportunity to probe unusual transport behavior caused by the zeroth LL of Weyl bands since its electronic structure is quasi-2D and it has Weyl nodes at EF. The Weyl nodes at EF results in the presence of the zeroth LL at EF as indicated above and the quasi-2D electronic structure make the LL quantization highly depend on the field orientation. As a result, the rotation of high magnetic fields from the out-of-plane to the in-plane direction would lead to a remarkable decease of the zeroth LL degeneracy of the Weyl bands at EF due to the suppression of LL quantization. This accounts for the sharp AMR peak present at B//ab at fields above 9T as well as the unusual angular dependence of Hall resistance as discussed in the manuscript.
Next, we will answer the reviewer's second question raised in Comment (1), i.e. why the Dirac bands did not influence the sharp peaks of the AMR?
As stated above, the Fermi surface of YbMnBi2 consists of the Weyl points and the hole-and electron-like pockets. All of them should contribute to transport properties. In our analyses of AMR data in Fig. 2c, we have indeed considered all these contributions, as shown by the equation used for the fit, i.e. .
In this equation, c represents the conductivity contributed by the momentum relaxation process, while t LL0 represents the tunneling conductivity of Weyl fermions at the zeroth LLs located at EF. The contribution of the hole-and electron-like pockets formed by the Dirac bands is included in c. Based on the evolution of AMR from the sin 2  dependence at low fields (e.g. see the data collected at 0.1 T in Fig.2c in the manuscript) to the sharp peak at  = 90 above 9T (Fig. 2c), we can see how the hole-and electron-like pockets affect the AMR sharp peaks. At low fields, these pockets should make dominant contributions to the transport, since these pockets should have a much greater DOS(EF) than the Weyl points. The observation of the sin 2  dependence of AMR at low fields implies that the contribution of the hole-and electron-like pockets to AMR follows the classic Lorentz effect for which the interlayer magnetoresistivity is proportional to Bz 2 [=(Bsin) 2 ]. When the field is remarkably increased, the DOS(EF) of the Weyl points should increase dramatically. This is because that the Weyl nodes are at EF in YbMnBi2, such that the quantum limit of Weyl bands should be reached under a relatively low magnetic field though we cannot tell the exact threshold field (note that the quantum limit of the Dirac bands cannot be reached near 30T as the reviewer said, since the quantum oscillation frequencies associated with the Dirac bands are high, ~ 115 and 162T). Near the quantum limit of the Weyl bands, the LL degeneracy would enhance significantly, thus resulting in significantly increased DOS(EF) at the zeroth LL. The gradual deviation from the sin 2  dependence in AMR upon increasing field suggests that Weyl fermions at the zeroth LL play a more important role in interlayer transport under high fields. Our successful fit of the AMR data to eq. (1) in the manuscript further suggests that the zeroth LL's Weyl fermions contribute to the interlayer transport via a tunneling process. The inclusion of c in the fit indicates that the contributions of Dirac bands to the interlayer transport cannot be neglected even above 9T where a sharp peak is observed in AMR. We acknowledge the statement regarding the quantum limit in our original manuscript is confusing as the reviewer pointed out. When we spoke of the quantum limit, we did not clearly indicate which bands reach the quantum limit. Again, we apologize for that.
In the revised manuscript, we have added most of the above discussions to clarify the electronic band structure and addressed how the Dirac bands affect the AMR.
In the above comments, the reviewer also asked why the Fermi energy is right at the nodes of Weyl cones (no measurement supports it in the manuscript)?
This is a result probed in previous ARPES experiments by Borisenko et al 1 (see Fig .2b in Borisenko's manuscript). Our observation of the unusual angular dependences of interlayer magnetoresistance and in-plane Hall resistance under high magnetic fields can be well understood in light of the fact that YbMnBi2 has Weyl nodes at EF 1 , as discussed in the manuscript. We have made this clear in the revised manuscript.
(2) The authors did not distinguish the concept of Landau levels in 2D systems and Landau bands in 3D cases. For example, on line 47 of the first paragraph they stated that a direct probe of relativistic fermions is not feasible for 3D Dirac or Weyl semimetals, such as Cd3As2. However, it is not true that in 3D Dirac (Weyl) semimetals the quantum limit when only the lowest Landau band is occupied has been achieved in several recent experiments.
We apologize again for the confusing presentation in our original manuscript and thank the reviewer for pointing out these problems. Here we first clarify the difference of LLs between 2D and 3D systems, then explain the statement we made in the original manuscript that confused the reviewer. The most significant difference in LLs between 2D and 3D systems is that the LLs in 2D systems are more dependent on field orientation than 3D systems. Specifically, given the formation of LLs originates from cyclotron motion within 2D planes, LLs of 2D systems are determined by the magnetic field component perpendicular to the 2D plane, Bz and the decrease of Bz leads to decreased LL degeneracy. LLs vanish when Bz decreases to zero. In contrast, in 3D systems, LLs appear for any magnetic field orientations and do not change with field orientation if the electronic structure is isotropic for a given system.
Our work reported in this manuscript takes advantage of the quasi-2D electronic structure of YbMnBi2 as well as its Weyl nodes at EF to probe transport properties of Weyl fermions at the zeroth LLs. With the quasi-2D electronic structure, the field rotation from the out-of-plane to inplane direction would lead the LL degeneracy to decrease, thus resulting in the decrease of DOS(EF). Since the zeroth LL of Weyl bands is at EF in YbMnBi2 as shown in Fig. 1c (panel#1), our AMR and angle dependent Hall resistance can probe the variation of the DOS(EF) of the zeroth LL with the field rotation.
Our statement regarding quantum limit in the original manuscript is confusing as pointed out by the reviewer. The reviewer is right: the quantum limit can be reached at high magnetic fields for 3D Dirac and Weyl semimetals, such as Cd3As2 2,3 , TaP 4 and NbAs 5 . In our original manuscript, we made the statement: "Direct probe of such relativistic fermion transport is not feasible for recently-discovered, three dimensional (3D) Dirac/Weyl semimetals". This statement is indeed very confusing. What we meant by this statement is that direct probe of the enhanced conductivity due to the increased degeneracy of the zeroth LL is not feasible though the quantum limit is reached. This is because that these materials have 3D electronic structures such that the field rotation might not lead to significant changes of LL degeneracy. In the revised manuscript, we have removed these confusing statements and added some of the above discussions to make our claim clear.
(3) The authors fitted the data of the Hall resistance with a tangent dependence at lower angles. At larger angles, they only argued that the deviation is due to the suppression of the Landau quantization. It would be good if the authors could also fit the data at larger angles. Also, there are many pockets in this system. Do others pockets affect the Hall resistance?
This is a good suggestion, which has motived us to perform further analyses for the angular dependence of Hall resistance Rzx. We have now been able to fit the Rzx data within the whole angle range. This new result provides additional support for our claim. We thank the reviewer for giving this suggestion.
As indicated above, the Fermi surface of YbMnBi2 consists of the type-II Weyl points and the electron-and hole-like pockets formed by the Dirac bands. The Dirac bands are expected to manifest themselves in quantum oscillations in magnetoresistivity and Hall resistivity, since their zeroth LLs are away from EF, as shown in Fig. 1c (panels #2 and 3#). However, high field is needed to observe such quantum oscillations. As indicated above, the quantum oscillations associated with Dirac bands in YbMnBi2 have been seen in the in-plane magnetoresistivity for B >15T (Fig. 1d), but were not observable up to 31 T in the out-of-plane magnoresistivity ( Fig.  2b in the manuscript). In our Hall resistance measurement, the available maximum field is only 9T, so we did not observe any quantum oscillations due to the Dirac bands. Therefore, within such a low field range, the contribution of Dirac bands to Hall resistance should follow the classical model, i.e., Rzx,D ≈ By/ne  Bsinθ, for the measurement setup shown in Fig. 2a attached below (i.e. Fig. 3a in the revised manuscript). This is indeed verified by the sin dependence of Rzx seen at low fields where the Dirac bands make dominant contributions to transport as indicated above (see Fig. 2b below) For the Weyl points, their contribution to the Hall resistance should come from the Weyl fermions at the zeroth LL as discussed above. Since the Weyl nodes are at EF and the zeroth LL is pinned to EF upon increasing magnetic field as shown in Fig. 1c (panel#1), the Weyl bands are not expected to show quantum oscillations. Given that YbMnBi2 possesses a quasi-2D electronic structure and its zeroth LL's degeneracy  Bz= Bcosθ, a rough but straightforward estimation leads to Rzx,W ≈ By/ne  Bsinθ/ Bcosθ = tanθ, which is indeed observed in the low angle range where the LL quantization is strong, as shown in Fig. 2c below (i.e. Fig. 3c in the manuscript). An explicit expression of Hall resistivity due to such a quantum tunneling effect of the zeroth LL has been theoretically derived as where a and b are material dependent constants 6 . As indicated in the manuscript, this equation is based on a theoretical model which assumes that the interlayer transport occurs through the tunneling process of relativistic fermions at the zeroth LL. From eq. (2), it can be seen that remarkable quantum effect should appear at high magnetic fields oriented at low angles where the zeroth LL is distinguishable from other LLs 6,7 . At lower fields or high fields oriented along larger angles where LL quantization is strongly suppressed, the angular dependence of Rzx resumes to the classical scenario, i.e. Rzx ≈ By/ne  Bsinθ.
We can combine the contributions from the Dirac and Weyl bands described above by assuming their contributions toward Hall conductivity are additive, i.e.
where total zx  is the total Hall conductivity, 0 LL zx  and C zx  represent the Hall conductivity contributed by the Weyl and Dirac bands, respectively. w1 and w2 in eq. (3) represents the weight of the contribution for each type of band. To fit the measured angular dependence of Hall resistance ( Fig. 2d below) to eq. (3), we convert the conductivities in eq. (3) to the resistivities. Thus eq. (3) can be expressed as: As stated above, the Hall resistivity for the Dirac bands should follow the classical model: C zx  = By/ne  By, while the Hall resistivity for the Weyl bands is dependent on the zeroth LL's degeneracy and can be expressed as eq. (2). Using this simplified model, we can fit the angular dependence of total zx  in the full angular range (0 -180°) under various magnetic fields (0.5 -9T).
The weight factors w1 and w2 are set to be field-dependent in these fits, while the parameters a and b are set to be identical for different fields since they are material constants. As shown in Fig. 2d below (i.e. Fig. 3d in the revised manuscript), this model, i.e. eq. (4), reproduces the measured data very well. These fitting results further support our claim that the zeroth LLs of the Weyl bands participate in the interlayer transport through quantum tunneling. Furthermore, from the fits described above, we can extract the weight ratio w1/w2 as depicted in the inset of Fig. 2d (i.e. Fig. 3d in the current manuscript), which provides the information on how the quantum tunneling transport of the zeroth LL's Weyl fermions competes with the Dirac fermion transport through momentum relaxation in YbMnBi2. At low fields (e.g. B = 0.5T), w1/w2 = 0, indicating that the Hall effect is fully classical since LL quantization is weak. Increasing the magnetic field leads to the increase of the LL spacing. When the zeroth LL of the Weyl bands become gradually distinguishable from the other LLs, w1/w2 also gradually increases and reaches 1.25 at B = 9T, indicating that interlayer tunneling of Weyl fermions at the zeroth LL become significant.
One may ask why the interlayer Dirac fermion transport occur through the momentum relaxation process in YbMnBi2, while the zeroth LLs' Weyl fermions transport takes place via the interlayer tunneling. This may be attributed to the different dimensionality of the Dirac and Weyl bands. The band structure studies by Borisenko et al. 1 suggest the Weyl bands are much more 2D-like in YbMnBi2 and the Weyl cone is highly anisotropic, with its Fermi velocity VF along the kz direction being two orders of magnitude less than VF along the kx direction. Under this circumstance, it is reasonable to assume the interlayer quantum tunneling of the zeroth LLs' Weyl fermions. However, the Dirac bands should have higher dimensionality, which is evidenced by the fact that YbMnBi2 exhibits a moderate electronic anisotropy as reflected in the zz/xx resistivity ratio (~36 at T = 2 K). If the Dirac bands were also highly 2D-like as the Weyl bands are, a large electronic anisotropy would be expected, inconsistent with the experimental observation. Therefore, it is reasonable to assume that the Dirac bands contribute to the interlayer transport through momentum relaxation (i.e. coherent band transport).
We have added the above discussions and new data analyses to the revised manuscript.
(4) The deviation between the SdHO data and the two-carriers model at 1/B=0.03 is somewhat obvious. It seems that the deviation is not from higher order terms or Zeeman splitting. Can the authors give a better fit?
Following the reviewer's suggestion, we have made further efforts to improve the fits of the SdHO data to the two-carries model. In our previous fits, only fundamental frequencies F and F  are included in the fits. The higher harmonic components 2F  and 3F  revealed in the Fourier transform (inset to Fig.1d in the current manuscript) were not included in the fits. When these components as well as the 4F  component (which is very weak and not shown in the FFT spectra in the inset to Fig.1d) are included in the fits, the deviation from fit becomes much smaller, as shown in Fig. 1e in the current manuscript. The Berry phases extracted from the fits show only very small changes after adding the high harmonic components to the fits. Again, we thank the reviewer for bringing up this issue. We have added the improved fit to the revised manuscript.

Response to Reviewer #3's comments:
In this manuscript, the authors have presented the in-plane and interlayer transport features for antiferromagnetic YbMnBi2 hosting Weyl fermions. In addition to the SdH oscillations in inplane resistivity, they measured the detailed angle dependence of interlayer magnetoresistance effects. From the latter dataset, they argue the possible contribution of zeroth Landau mode in the interlayer conduction. They systematically measured the transport properties and display them in a clear manner.
We thank the reviewer for reviewing our manuscript. We also appreciate the questions/criticisms raised by this reviewer, which have been very helpful in improving our manuscript. We have now addressed all of the reviewer's questions/criticisms and revised the manuscript accordingly. In the following, we have made point-to-point responses (in black) to the reviewer's comments (in red). The figures (for reviewer) refers to the figures in this response letter.
However, in my opinion, the manuscript has several serious problems. First, all the results shown in Fig. 1 (magnetic structure, magnetoresistance, details of SdH oscillations) have been already reported in ref. [18]. So I could not find any new discoveries in the present data on the in-plane transport and neutron diffraction.
We agree with the reviewer that the neutron scattering and SdH oscillations in in-plane transport data have been reported by Wang et al. 8 We have clearly indicated it in the manuscript and quoted the relevant reference. However, since our magnetoresistance measurements were made to much higher fields, up to 45 T, the SdH oscillations probed in our experiments revealed new important information which was not seen in Wang et al. 's experiments 8 which were made only up to 35T. We apologize for not clearly indicating what are our new discoveries in our SdH data. We thank the referee for seeing this problem.
Because of the limited field range, Wang et al. 8  By contrast, our observed SdH oscillations in xx, which were measured up to 45T, show clear features arising from multiple oscillation frequencies (see Fig. 1d attached below). Our FFT analyses indeed show two frequencies at F  =115T and F  =162T (inset to Fig. 1d). To extract the Berry phases accumulated in cyclotron orbits associated with these two frequencies, we fitted the SdH oscillations in xx with a two-band Lifshitz-Kosevich (LK) formula which takes Berry phase into account for a topological fermion system. This approach has been shown to be particularly effective for finding Berry phases in a topological material with multiple bands 9 . As seen in Fig.  1e (attached below for reviewer), our SdH oscillation data at 2K and 18K can be nicely fitted with a two-band LK formula when the higher harmonic components of F  (i.e. 2F  , 3F  and 4F  ) are taken into account (see the Methods section for more details) (note that the 4F component is very weak and not shown in the FFT spectra in the inset to Fig.1d)). The Berry phases derived from these fits are 0.8 for F and -0.6 for F, which are clearly non-trivial. The fits in our original manuscript show some deviations, which are due to the fact that we did not consider higher harmonic components for F  . YbMnBi2 is a multiple-band system and its Fermi surface consists of the Weyl points and the hole-and electron-like pockets formed by the Dirac bands (see below for more details). The SdH oscillations observed in xx should arise from the Dirac bands as discussed in the manuscript. Our precise determination of non-trivial Berry phase verifies that the hole/electron-like pocket is indeed comprised of topologically non-trivial bands, i.e. the Dirac bands.
We have revised Fig. 1 to highlight our new discoveries in the revised manuscript. The current version of Fig. 1 (attached below) includes the in-plane magnetoresistivity data, the FFT analyses of SdH oscillations as well as the fits of the SdH oscillations to the two-band LK formula. We have moved the neutron scattering data as well as the effective mass analyses to the Supplementary Information. Additionally, we have added a schematic of the electronic band structure determined by previous ARPES experiments 1 to revised Fig. 1 following Reviewer 3 and 2's suggestions. The addition of this schematic band structure makes our discussions much clearer.
We would like to emphasize the central point of this manuscript is that we have demonstrated the unusual interlayer transport of the zeroth LL's Weyl fermions, which has never been reported in any other topological semimetals. Our demonstrations are based on the AMR data shown in Fig.  2 and the angle-dependent Hall resistivity data in Fig. 3. These data are novel and have never been reported. Second, the analyses on the data of interlayer transport are not consistent. Although the authors assume that the tunneling processes are dominant in the interlayer transport, the temperature profile of interlayer resistivity R_zz in Fig. 1c exhibits nice metallic behaviour over the entire temperature range. Furthermore, the interlayer conductivity based on the tunneling between the zero-mode Landau levels (Eq. 1) gives negative magnetoresistance effects. However, as shown in Fig. 2b, the observed magnetoresistance effects are positive irrespective of the field angle.
These facts mean that the interlayer conduction for YbMnBi2 should be dominated by the usual coherent transport of the carriers from some other bands and hence the tunneling effect of the zero-mode of Landau level should be negligible. In fact, in the second paragraph on page 9, they fit the AMR data using the conventional B^2 magnetoresistance effect, which also reproduces the experimental results. Because the fitted result is almost independent of the model of the zeromode Landau level transport, the zero mode plays a minimal role in the total interlayer conduction. Therefore, I don't think the main conclusion of this manuscript is supported by the present data and analyses.
The reviewer raised several issues in the above comments. We believe they were caused by our insufficient/unclear discussions in the original manuscript. We will address them one by one in the following.
(1) Metallic interlayer transport and tunneling process. Quantum tunneling is due to the barrier penetration of particles. The tunneling of charge carriers appears wherever the barrier exists. YbMnBi2 possesses a layered structure with the 2D Bi square-net planes being separated by the Yb-MnBi4-Yb layers. The first principle calculations 1 have revealed that the conduction electrons in this material are mostly Bi 6p-electrons from the 2D Bi planes and its electronic structure is quasi-2D, which has indeed been demonstrated by ARPES experiments 1 . The Yb-MnBi4-Yb layers in between the neighboring 2D Bi planes should act as barriers for the interlayer transport, hence the tunneling process between the neighboring 2D Bi planes is naturally expected.
It is not surprising to observe a metallic temperature dependence when tunneling is present.
Here, we would like to give a more specific example. One of the corresponding authors of this manuscript, Zhiqiang Mao, was previously involved in the tunneling studies on the spin-triplet superconductor. He and his co-workers successfully probed Andreev surface bound states in the 3-K superconducting phase of Sr2RuO4 using cleaved junctions (Mao et al., PRL 87, 037003 (2001) 10 ). The junctions were made by tightly fixing two cleaved crystals with a teflon frame. Although the junction resistance as a function of temperature shows metallic behavior in the whole temperature range (see Fig. 1 in Ref. 10 ), significant tunneling behavior was observed, which enabled the observation of the Andreev bound state in Ru-Sr2RuO4 10 .
Generally, in a material with layered structure, it has been well-established that coherent transport through momentum relaxation and tunneling process coexist in the interlayer transport 11 . As pointed out in Ref. 11 and references therein, depending on the relative weight and specific parameters of these two transport channels, the interlayer resistivity can exhibit various types of temperature dependences 11 , including purely metallic (e.g. overdoped high-Tc cuprates 12 ), insulating-like (e.g. TaS2 13 ), non-monotonic with a maximum at a certain temperature (e.g. Sr2RuO4 14 , κ-(BEDT-TTF)2-Cu(SCN)2 15 , (Bi0.5Pb0.5)2Ba3Co2Oy and NaCo2O4 16 ) and non-monotonic with a minimum at a certain temperature (e.g. underdoped cuprates 12 ), etc.

(2) Why does zz(B) show positive magnetoresistance ?
The referee is right: negative MR is indeed expected for the tunneling transport between neighboring 2D Bi planes, as reflected in the tunneling model used for fitting our data in the manuscript, i.e.
which was adopted from Osada et al, JPSJ, 77, 084711 17 . However, as have been stated in the manuscript, the coherent momentum relaxation transport should coexist with the interlayer quantum tunneling transport in YbMnBi2 due to its finite interlayer coupling and multiple-band characteristic. The positive MR seen in our experiments is due to the competition between the negative MR component from the tunneling channel and the positive MR component from the momentum relaxation channel.
Next we first clarify the electronic band structure of YbMnBi2, which was not made clear in the original manuscript, and then explain why the interlayer magnetoresistance observed in our experiments is positive despite the interlayer tunneling of the zeroth LL mode. As shown in Fig.  1a, the Fermi surface of YbMnBi2 consists of the Weyl points and the hole-like (marked in blue) and electron-like (red) pockets comprised of the Dirac bands 1 . Given such a multi-band electronic structure, the transport properties of YbMnBi2 should be contributed by both the Weyl and Dirac bands.
As shown in Fig. 1c, under magnetic fields, both the Weyl bands and Dirac bands split to quantized Landau levels (LLs), the energies of which can be expressed as 18 . The n = 0 level corresponds to the zeroth LL, which is always locked to the Weyl/Dirac node upon field sweep no matter whether the node is at EF or not, as shown in Fig.  1c attached above. For the Dirac bands, since their crossing points, i.e. the Dirac nodes, are away from EF (Fig. 1c, panel #2 & #3), their n  0 LLs would successively pass through EF upon increasing the magnetic field, thus resulting in oscillating density of state DOS at EF, which is manifested in quantum oscillations in our measured in-plane magnetoresistivity ( Fig. 1d and 1e). By contrast, since the Weyl nodes probed by ARPES 1 are located at EF, the Weyl bands' zeroth LLs are pinned to EF regardless of magnetic field strength (Fig. 1c, panel #1). Therefore, the Weyl bands are not expected to show quantum oscillations, since no LLs pass EF upon increasing the field. Instead, DOS(EF) monotonically increases due to the increase of the zeroth LL's degeneracy.
In our model used for the data analyses in Fig. 2and 3, we assume the interlayer Dirac fermion transport occurs through the momentum relaxation process (i.e. coherent band transport), while the zeroth LLs' Weyl fermions transport takes place via interlayer tunneling. This assumption is based on the different dimensionality of the Dirac and Weyl bands. The band structure studies by Borisenko et al. 1 suggest the Weyl bands are highly 2D-like in YbMnBi2. However, the Dirac bands should have higher dimensionality, which is evidenced by the fact that YbMnBi2 exhibits a moderate electronic anisotropy as reflected in the zz/xx resistivity ratio (~36 at T = 2 K). If the Dirac bands were also highly 2D-like, a large electronic anisotropy would be expected, inconsistent with the experimental observation. Therefore, it is reasonable to assume that the Dirac bands contribute to the interlayer transport through momentum relaxation, while the zeroth LLs of the Weyl bands contribute to the interlayer transport through tunneling. The former channel has positive magnetoresistance (MR), but the latter one has negative MR.
Next, we will discuss how the tunneling channel leads to unusual zz(B). As shown in Fig. 2b in the revised manuscript, when the field is aligned within the plane (i.e.  = 90°) where LLs quantization vanishes, zz(B) exhibits B 2 dependence in a low field region, but gradually evolves to a linear field dependence above 10T. However, as the field is tilted toward the out-of-plane direction ( = 0°) where LL quantization develops, a sub-linear field dependence in zz(B) emerges. Such an unusual evolution of MR with  cannot be understood in light of the MR induced by the classical orbital effect or by other quantum effects such as weak anti-localization, but can be understood in terms of the gradually developed negative MR in the tunneling channel. Such negative MR component competes with the positive MR component arising from the momentum relaxation channel of the Dirac bands, resulting in the unusual sub-linear field dependence in zz(B) for  < 90°. This argument is quantitatively supported by the nice fits of zz(B) (Fig. 2b) and zz() (Fig. 2c) to the following equation for  < 90  , (1)] and c  represent the conductivities of tunneling and momentum relaxation channels respectively. Our claim is further supported by the fits of the unusual angular dependences of Hall resistance Rzx to the same model, as discussed in the manuscript.

(3) The role of zeroth LL in the interlayer transport
In our original manuscript, we stated "the conduction through the momentum relaxation channel alone can also account for the AMR data" in Fig. 2c. Indeed, this is a confusing and incorrect statement, which led the reviewer to think that the zeroth LL mode plays a minimal role in interlayer transport. When we used the model of momentum relaxation alone to fit our AMR data, one assumption was made: the interlayer transport at high fields is still dominated by the zeroth LL's Weyl fermions and the DOS(EF) contributed by the Weyl points varies with the field and its orientation. In other words, the zeroth LL still plays a critical role in this model. We apologize for not making this clear in the original manuscript.
However, after reading the reviewer's comments, we have carefully inspected the momentum relaxation-alone model and find this model is indeed inapplicable in interpreting our data, as discussed below. In this model, we considered a pure momentum relaxation process for the Weyl point states. The DOS(EF) contributed by the Weyl points is determined by the zeroth LL's degeneracy since the zeroth LL is pinned to the EF. Given the 2D LL quantization in YbMnBi2, DOS(EF) ∝ Bz (the magnetic field component perpendicular to the plane). Therefore, by assuming σ ∝ DOS(EF), a resistance peak is naturally expected for  = 90°. However, the assumption of σ ∝ DOS(EF) is oversimplified and neglects the influence of the scattering rate on conduction. From the well-established transport theory 19,20 ,  ∝  and the scattering probability 1/ varies with the number of available states that electrons can be scattered into (i.e. 1/ ∝ DOS(EF)) 31,32 . This leads to  ∝ 1/DOS(EF) rather than ∝ DOS(EF). Therefore, our original analyses using a pure momentum relaxation model is incorrect. So the zeroth LLs' tunneling of the Weyl points is the only possible mechanism for the interpretation of unusual interlayer transport observed in our experiments. We apologize for making this mistake. We have removed the discussions on the momentum relaxation-alone model in the revised manuscript.
Additionally, the role of the zeroth LL in transport have also been supported by the angular dependence of Hall resistance. As stated above, when LL quantization is weak at low fields (e.g., B=0.5T), the angular dependence of Hall resistance follows a classic sinθ dependence. However, when LL quantization is strong enough to separate the zeroth LL from other LLs, Hall resistance exhibits unusual angular dependence with a cusp-like peak showing up at θ=90° (Fig. 3d in the revised manuscript). Such an angular dependence can be understood in terms of the contributions from the zeroth LL. In the revised manuscript, we have performed further analyses and successfully fit the angular dependences of Hall resistance in the full angular range (0-180°) under different fields (0.5-9T) using a model which considers both the tunneling of the zeroth's LLs of the Weyl points and the coherent momentum relaxation of Dirac fermions (see Fig. 3d in the revised manuscript).
We thank the reviewer for giving the criticisms in the above comments, which have been very helpful for us to improve the discussions in the revised manuscript.
From the above reasons, I don't recommend the publication of this manuscript as it is. After the significant revisions, it may be suitable for publication in some specialised journal.
We hope we have convinced the reviewer our claim is well supported by our data. Given the zeroth LL is one of the key signatures of topological bands, finding transport evidence for the zeroth LL for the first time reversal symmetry breaking Weyl semimetal YbMnBi2 is of fundamental importance. We also hope the reviewer has been convinced that our work has broad interests and fits into the scope of Nature Communications.
The followings are other comments to be considered. 1. On the line 182, the authors adopt the B-linear magnetoresistance for sigma_c, because the Weyl fermions reach the quantum limit at sufficiently high field. However, as shown in Fig. 1, clear SdH oscillations were observed even around 40 T, indicating the quantum limit is not achieved for the Weyl fermions with the nodes away from the Fermi energy. Therefore, the conventional B^2 MR should be adopted for sigma_c in the entire field range.
We thank the reviewer for pointing out this issue and apologize for not providing clear justifications for the fit of magnetoresistivity in the original manuscript.
As indicated above, the Fermi surface of YbMnBi2 consists of not only the Weyl points, but also the electron-and hole-like pockets formed by the Dirac bands (Fig. 1a); the quantum oscillations shown in Fig. 1d should arise from the Dirac bands. As pointed out by the reviewer, the quantum limit for these Dirac bands are clearly not reached around a field of 40T. Our statement of reaching a quantum limit in the original manuscript refers to the quantum limit of the Weyl bands. This was not made clear in the original manuscript, which caused some confusions. Since the Weyl nodes are at EF, the quantum limit of the Weyl bands should be reached under a relatively low field though we cannot tell its exact threshold value.
Although linear MR is expected to occur at quantum limit theoretically 21 , it is worth noting that linear MR has also been widely observed in many materials at low fields when quantum limit is not reached, such as Ag2+δSe 22 , LaAgSb2 23,24 , Cd3As2 2,25,26 , and Na3Bi 27 . This indicates that the linear MR can be caused by other mechanisms and takes place at rather low fields. For example, the mobility fluctuations due to disorder effects has been suggested to be the possible origin for the low field linear MR in Cd3As2 2 .
In YbMnBi2, we have also observed linear MR in both in-plane (Fig. 1d in the revised manuscript) and interlayer (Fig. 2b) transport for BI when the magnetic field is above a few Tesla. More specifically, the zz(B) data taken at  = 90° (Fig. 2b in the current manuscript) exhibits B 2 dependence in a low field range, but crossovers to a linear dependence above 10T.
The interlayer transport for the field configuration of  = 90° should be dominated by the coherent momentum relaxation, since LL quantization is fully suppressed for this field orientation as mentioned above. According to this fact, we can assume similar field dependences of MR for the momentum relaxation channel for  < 90°, i.e. the linear B-dependence for high fields, and B 2 -dependence for low fields. Thus the conductivity of the momentum relaxation channel can be assumed to , which combines conduction contributions from both the tunneling and the momentum relaxation channels, fits to both the zz(B) data under different  (Fig. 2b) and the zz() data (Fig.2c) under different fields very well, indicating that our model has captured the main features of the interlayer transport.
We have added the above discussions to the revised manuscript. Again, we thanks the reviewer for seeing this problem.
2, YbMnBi2 appears to host several Weyl points not only at Fermi energy but also away from Fermi energy. It would be very helpful for the readers' understanding if the authors present the schematic band structure associated with the Weyl points.
This is a very good suggestion! Indeed, the first principle calculations suggest YbMnBi2 has a number of Weyl points and the ARPES experiments probed the Weyl points at EF 1 . Following the reviewer suggestion, we have added the schematics of the Fermi surface and band structure determined by the ARPES experiments 1 to Fig. 1.
3. The authors should distinguish the words ``Dirac' and ``Weyl'. I guess ``Weyl' fermion would be appropriate for YbMnBi2.
As we discussed above, both "Weyl" and "Dirac" bands exist in YbMnBi2. The electron-and hole-like pockets are comprised of the Dirac bands, while the Weyl points are formed by the Weyl band crossing. We have made this clear in the revised manuscript. We thank the reviewer for pointing out this issue and apologize for not including these relevant references. In the revised manuscript, we have cited these two important works, as well as other relevant references.
The authors have addressed all my concerns. The current manuscript is ready for the final publication in Nature Communications.
Reviewer #2 (Remarks to the Author): In the revised manuscript, the authors give a clear picture of the observed unusual interlayer quantum transport, especially the sharp peak of AMR at large magnetic fields. The manuscript has been improved a lot. Almost all of the previous comments have been addressed properly. Before the recommendation of publication, I still have several questions.
1. The authors explain the unusual interlayer quantum transport behavior by considering both the 2D Weyl fermions and 2D electron-and hole-like Dirac fermions. The Fermi level is right at the Weyl node. The total interlayer magnetoresistivity in Eq. (2) are contributed by the zeroth Landau level of Weyl fermions and Dirac bands. They argue that in the quantum limit of the Weyl bands, the degeneracy of the zero LL increases and thus the density of state at the Fermi level increases, resulting in the unusual sharp peak in AMR. In the presence of the magnetic field, both the Weyl bands and Dirac bands are quantized to the Landau levels. The degeneracy of ANY Landau level (not only the zeroth LL) is eB/2\pi. They argued near "the quantum limit" the conductivity contributed by the zero LL of Weyl fermions exceeds those by the Dirac bands in Eq. (2), which results in the unusual behaviors. This is not very convincing. The density of states of any Landau level increases with increasing magnetic field, therefore the conductivity contribution \sigma_c should also increase. The authors should explain the reason more carefully why the zeroth LL of Weyl fermions plays a more important role at large magnetic fields.
2．The authors still do not distinguish Landau levels in 2D systems and Landau bands in 3D cases.
In the first paragraph they stated that for both 2D and 3D topological materials, the quantized energies in the presence of magnetic field is v_F \sqrt{2eBn}. It is not true for 3D cases, where the Landau bands should also depend on the momentum that is parallel to the magnetic field, i.e., v_F \sqrt{2eBn+k_z^2}. Only for 2D topological materials with linear dispersion, the zero Landau level is always locked to the band crossing point.
Reviewer #3 (Remarks to the Author): The authors significantly revised the manuscript following the reviewers' comments. The present manuscript clearly explains the electronic structures for YbMnBi2, which has both Weyl-like and Dirac-like carriers. They also made clear the model of fitting for the AMR, based on the plausible band structures. Having said that, still I do not think the main claim of this manuscript is experimentally supported at all. This is because the negative MR was not observed even when theta is close to zero. For theta close to zero, the contribution from the Dirac bands (i.e., \sigma_c) should be minimal because B_xy=0, whereas the contribution from the zeroth mode (i.e. \sigma_t^{LL0}) should be maximum, which naturally results in the negative MR if the tunneling of the zeroth modes plays a vital role. In experiments, however, small positive MR manifests itself even for theta=4 degree, which verifies the contribution of tunneling between the zeroth modes to the interlayer conductivity is negligible if any. Therefore, I would strongly recommend the authors to remove their claims (in the main text and title) that the unusual AMR results from the zeroth LL mode and that the interlayer tunneling of the zeroth modes is remarkable, before the manuscript is published.

Response to Reviewer #1's comment:
The authors have addressed all my concerns. The current manuscript is ready for the final publication in Nature Communications.
We thank Reviewer #1 for his/her recommendation for publication of our manuscript in Nature Communications.

Response to Reviewer #2's comments:
In the revised manuscript, the authors give a clear picture of the observed unusual interlayer quantum transport, especially the sharp peak of AMR at large magnetic fields. The manuscript has been improved a lot. Almost all of the previous comments have been addressed properly. Before the recommendation of publication, I still have several questions.
First, we thank the Reviewer #2 for taking time to review our revised manuscript. We really appreciate his/her insightful comments which have been very helpful in further improving our manuscript. We have made point-to-point response (in black) to the reviewer's questions (in red).
1. The authors explain the unusual interlayer quantum transport behavior by considering both the 2D Weyl fermions and 2D electron-and hole-like Dirac fermions. The Fermi level is right at the Weyl node. The total interlayer magnetoresistivity in Eq. (2) are contributed by the zeroth Landau level of Weyl fermions and Dirac bands. They argue that in the quantum limit of the Weyl bands, the degeneracy of the zero LL increases and thus the density of state at the Fermi level increases, resulting in the unusual sharp peak in AMR. In the presence of the magnetic field, both the Weyl bands and Dirac bands are quantized to the Landau levels. The degeneracy of ANY Landau level (not only the zeroth LL) is eB/2\pi. They argued near "the quantum limit" the conductivity contributed by the zero LL of Weyl fermions exceeds those by the Dirac bands in Eq. (2), which results in the unusual behaviors. This is not very convincing. The density of states of any Landau level increases with increasing magnetic field, therefore the conductivity contribution \sigma_c should also increase. The authors should explain the reason more carefully why the zeroth LL of Weyl fermions plays a more important role at large magnetic fields.
The reviewer asked a very good question. We apologize for not addressing this clearly in the previous version of manuscript. Although the LL degeneracy of Dirac bands is also enhanced upon increasing field, it should not contribute to the unusual interlayer transport properties probed in AMR (Fig. 2c) and Hall resistance (Fig.3d) at high fields. Since the quantum oscillation frequencies of Dirac fermions are high (115T and 162T, see Fig. 1d), the quantum limit of the Dirac bands cannot be reached until the field is increased above 230T. Given that our experiments were conducted below 31T, the variation of LL degeneracy should be small for the Dirac bands. Therefore, the variation of the DOS(EF) of the Dirac bands with the field rotation in the field range of our experiments is expected to be small, inconsistent with our experimental observation of the drastic changes of AMR above 9T and Hall resistance above 6T. The AMR of the Dirac fermion transport channel should more or less follow the classical Lorentz effect, i.e. AMR() ∝ Bxy 2 =B 2 sin 2 θ, which is indeed observed at low fields but not consistent with the unusual features at high fields (see Fig. 2c in manuscript).
In contrast, the quantum limit can be easily achieved for the Weyl bands in YbMnBi2, since its Weyl nodes are at the Fermi level [1]. Its zeroth LL degeneracy would increase drastically near the quantum limit, which should result in a significant increase of DOS(EF) at the zeroth LL and be responsible for the unusual AMR and angular dependence of Hall resistance. This interpretation is supported by our successful fits of the zz(B), AMR() and Rzx() data to a model which considers the interlayer quantum tunneling transport of the zeroth LL's Weyl fermions.
We have added some of the above discussions to the revised manuscript to address the reviewer's question (highlighted on Page 14).
2．The authors still do not distinguish Landau levels in 2D systems and Landau bands in 3D cases. In the first paragraph they stated that for both 2D and 3D topological materials, the quantized energies in the presence of magnetic field is v_F \sqrt{2eBn}. It is not true for 3D cases, where the Landau bands should also depend on the momentum that is parallel to the magnetic field, i.e., v_F \sqrt{2eBn+k_z^2}. Only for 2D topological materials with linear dispersion, the zero Landau level is always locked to the band crossing point.
We really appreciate that the Reviewer found this important problem. We apologize for overlooking it. Indeed, as pointed out by the reviewer, the Landau level energy for 2D Dirac/Weyl fermions is 2e | |  [3]. Therefore, the zeroth Landau level is not locked to the Dirac/Weyl node for a 3D Dirac/Weyl system. For YbMnBi2, previous studies have shown that its Weyl bands are of strong 2D character [1]. The Fermi velocity of the Weyl fermions along the kz direction is found to be more than two orders of magnitude smaller than that along kx. Therefore, we can approximately treat the Landau levels of the Weyl bands in YbMnBi2 as LLs in 2D systems. In the current revised manuscript, we have made revisions to both the introduction and discussion sections to address this issue (the revised sections have been highlighted on Pages 2 and 3, as well as the figure caption to Fig. 1).

Response to Reviewer #3's comments:
The authors significantly revised the manuscript following the reviewers' comments. The present manuscript clearly explains the electronic structures for YbMnBi2, which has both Weyl-like and Dirac-like carriers. They also made clear the model of fitting for the AMR, based on the plausible band structures.
First, we thank the Reviewer #3 for taking time to review our revised manuscript. We appreciate his/her comments which have been very helpful in further improving our manuscript. We have made point-to-point response (in black) to the reviewer's questions (in red).
Having said that, still I do not think the main claim of this manuscript is experimentally supported at all. This is because the negative MR was not observed even when theta is close to zero. For theta close to zero, the contribution from the Dirac bands (i.e., \sigma_c) should be minimal because B_xy=0, whereas the contribution from the zeroth mode (i.e. \sigma_t^{LL0}) should be maximum, which naturally results in the negative MR if the tunneling of the zeroth modes plays a vital role. In experiments, however, small positive MR manifests itself even for theta=4 degree, which verifies the contribution of tunneling between the zeroth modes to the interlayer conductivity is negligible if any. Therefore, I would strongly recommend the authors to remove their claims (in the main text and title) that the unusual AMR results from the zeroth LL mode and that the interlayer tunneling of the zeroth modes is remarkable, before the manuscript is published.
We thank Reviewer#3 for bringing up this issue. Indeed, the tunneling conductance should be enhanced upon increasing the perpendicular field component, and results in negative MR. Reviewer#3 asked the same question in the first round of review. In our previously submitted response letter, we have provided the interpretation for the absence of negative MR: the positive MR is due to the coexistence of the tunneling and the momentum relaxation channels in YbMnBi2. The weak positive MR for θ=4° (B nearly parallel to I) is due to the competition between the negative MR component from the tunneling channel and the positive MR component from the momentum relaxation channel. However, we did not provide justification for the positive longitudinal (i.e. B//I) MR for the momentum relaxation channel in our previous response, which might cause Reviewer#3 to be confused. We apologize for missing that point and give more detailed explanations below.
The classical theory based on a free electron model yields a quadratic field dependence for MR, due to the elongated electron trajectories caused by Lorentz effect. Therefore, zero longitudinal MR is expected due to zero Lorentz force. However, the longitudinal MR is usually finite and could have relatively large values in real materials, such as the isostructure compounds AMn(Bi/Sb)2 (A=Sr, Ba, Ca) [4][5][6][7][8][9]. Such positive longitudinal MR for B//I can be understood by taking the following factor into account. According to a recent theoretical study by Pal and Maslov [10], anisotropic Fermi surface can cause striking positive longitudinal MR, ranging from a few to a few tens of a percent. Using the semiclassical Boltzmann equation, Pal and Maslov showed electrons moving with the current fluctuate along the transverse direction, thus experiencing non-zero Lorentz force when the Fermi surface is anisotropic. For YbMnBi2, its Fermi surface is indeed strongly anisotropic as revealed by ARPES studies [1]. Thus, we can reasonably expect a positive longitudinal MR for YbMnBi2.
With this consideration, we can better understand the weak positive MR for θ=4° in YbMnBi2. Though the tunneling channel is expected to have negative MR, its competition with the positive MR of the momentum relaxation channel leads to a very small positive MR for θ=4°.
The interlayer quantum tunneling of the zeroth LL mode have been demonstrated by both the interlayer AMR and the Hall effect in our experiments. In both measurements, we have observed the crossover from the low field classical behavior to the high field unusual angular dependence, which can be well described by a model which considers the coexistence of the tunneling process of the zeroth LL mode of Weyl fermions and the momentum relaxation process of Dirac fermions. As far as we know, no other mechanism can give satisfactory interpretations for these observations.
What follow are minor comments.
We thank the Reviewer#3 for raising these issues.
We have indeed shown the zz data at θ=90° (B⊥I) in Fig. 2b. As stated in the manuscript, we observed B 2 dependence in a low field region, which gradually evolves to a linear field dependence above 10T. As we have mentioned in the previous response letter, though linear MR is expected to occur at quantum limit theoretically, linear MR has also been widely observed in many materials at relatively low fields when quantum limit is not reached. One possible interpretation is the mobility fluctuations due to disorder effects.
As for Rzx, since it is a transverse (Hall) resistance, it does not follow the B 2 dependence expected for the classical longitudinal magnetoresistance.
(2) In Fig. 2c, the authors mention that they observed the sin^2(theta) dependence of the AMR at low field (0.5 T), which is quite hard to see in the present scale.
We have included a zoomed-in data in the inset of Fig. 2c, showing the sin 2 θ-dependence for the AMR at B=0.1T (3) The authors should clarify the value of field angle where they estimate the values of w1 and w2 in Fig. 3d.
We preformed the fits of the Hall resistance in the whole angular range (0°-180°).
In this revised version, the manuscript has been improved a lot. However, there is a minor comment should be addressed properly before the final publication on Nature Communications. I am also confused about the fitted results in Fig. 2b. The authors seems to fit the data for =4 by assuming c = 0/(1+k1 Bxy 2 ) or 0/(1+k2 Bxy). However, this is inconsistent with the authors' new claim that the positive longitudinal MR (expressed by Bz not Bxy) may matter for ~0, as discussed in the paragraph starting from 263rd line. The authors should explain the details of fitting of the data for =4 by considering the longitudinal MR.
In the comment (1) in my previous review, what I meant was that the field dependence of c can be experimentally obtained for =90. In fact, zz is expressed for =90 (when yx=0) as σ Because  LL0 ~ 0 for =90, the total zz should be equal to c, which can be calculated by the experimental values of xx, xz., and zz. The authors should show the obtained zz as a function of field, so they can check if the assumption that c = 0/(1+k1 Bxy 2 ) or 0/(1+k2 Bxy) is valid. With respect to this point, the authors assume that zz=1/zz and zx=1/xz. in eq. (2) and (5), which is not correct in general. For instance, from a back-of-the-envelope calculation, zx is written by (please see also ref. 41) σ The authors should clearly explain why Eq. (2) and (5) works fine for the present case. If not, I would recommend the authors to recalculate them by using more precise equations.
After revising the above points, I would recommend the publication of this manuscript.
Hall effects might suggest the possible contribution of the zero mode tunneling. In particular, as shown in the inset to Fig 3, the value of w1/w2 is close to 1 for B=9 T, which means the zero mode tunneling process contributes to the interlayer Hall resistivity to a similar extent to the Dirac bands' momentum relaxation. If this was true, the negative MR would manifest itself near θ =0° at high fields. I would recommend the authors to quantitatively discuss this apparent inconsistency in the main text. (The phrases "considerable positive longitudinal MR" in the 269th and 273 rd lines are misleading.) We appreciate that the referee brought up this question again. Motivated by the referee's comments, we have now found a more reasonable answer to the question of why the interlayer longitudinal magnetoresistance (LMR) remains positive with a small magnitude, even though the interlayer tunneling of the zeroth LL's fermions plays an important role. In our last submission, we attributed the observed small positive LMR to the competition between the negative MR component due to the zeroth LL's tunneling channel of the Weyl bands and the positive MR component due to the momentum relaxation channel of the Dirac bands. We provided one possible interpretation for the positive LMR component, i.e., the anisotropic Fermi surface. As we pointed out in the manuscript, anisotropic Fermi surface is generally expected to result in only a few tens of percent positive LMR at maximum [3]. If the positive LMR component in YbMnBi2 is driven purely by its anisotropic Fermi surface, its magnitude of negative LMR component attributed to the tunneling channel would also be around an order of a few tens of percent. As indicated by the referee, this seems unreasonably small when the tunneling channel plays a dominant role. We have carefully considered this question again and found we had overlooked an important factor which could cause very large positive LMR component in YbMnBi2, as discussed below.
In fact, large positive LMR is a generic feature of topological semimetals. For example, AMn(Bi/Sb)2 (A=Sr, Ba, Ca), which are isostructural to YbMnBi2 and Dirac materials, have remarkable positive LMR for interlayer transport. LMR reaches 200% for BaMnBi 2 [4], 500% for BaMnSb 2 [5] and 300% for SrMnSb 2 [6] at 9T and ~2K, even as high as 10,000% at 31T for SrMnSb 2 [6]. Such large positive LMR can be attributed to their Dirac band transport. Since the Dirac nodes in these materials are far away from the Fermi level, their interlayer transport should not involve the zeroth LL's tunneling. Although the mechanism of the large LMR in Dirac materials has not been well understood, we can reasonably expect a very large positive LMR component resulting from the Dirac band transport channel in YbMnBi 2 due to its structural similarity to AMn(Bi/Sb) 2 . However, our observed LMR for ρ zz in YbMnBi 2 reaches only 7% at B=9T and 20% at B=31T (Fig. 2b in the manuscript), which are one or two orders of magnitude smaller than those of AMn(Bi/Sb) 2 materials. The strong suppression of positive LMR in YbMnBi 2 implies that its large positive LMR component expected for the Dirac band transport channel must be canceled by a large negative MR component caused by the zeroth LL tunneling channel of the Weyl bands. Given that the positive LMR component for ρ zz in AMn(Bi/Sb) 2 is within the 200%-10000% range, we anticipate the negative LMR in YbMnBi2 has a magnitude within a similar range. As discussed in the revised manuscript, this interpretation is well supported by the observation of unusual angular dependences of ρ zz and ρ zx as well as the fits of ρ zz and  zx to the double channel transport model which considers both interlayer Dirac band transport and zeroth LL quantum tunneling of the Weyl bands.
Again, we thank Reviewer#3 for raising this question. We have made revisions to the LMR discussions in the revised manuscript.
In the comment (1) in my previous review, what I meant was that the field dependence of σ c can be experimentally obtained for θ=90°. In fact, σ zz is expressed for θ=90° (when σ yx =0) as