Realizing Landau levels with open equifrequency contours around band singularities

Domain: biology
Published: 2026-04-04
Editorial outcome: major_revision

Abstract
Charged particles in a magnetic field undergo circular motion and result in quantized energy levels known as Landau levels (LLs). LLs around band singularites are particularly interesting due to their nontrivial topological properties. To date, through the introduction of the artificial magnetic field, LLs have been observed in systems only with closed equifrequency contours (EFCs) around band singularities, i.e. type-I Dirac or Weyl systems. Here, we demonstrate the existence of LLs with open EFCs around band singularities through combined action of artificial electric (E) and magnetic (B) fields, whereas they cannot be generated by B only. We construct an artificial platform of inhomogeneous metasurfaces of the two-dimensional type-II Dirac system with open EFCs, create suitable artificial gauge fields of E&B, and observe the emergence of E&B-induced LLs experimentally. This work presents a new insight for realizing LLs, and provides a platform for controlling optical waves propagating using artificial gauge fields.

Introduction
The quantization of energy is a fundamental concept in modern physics, and a key outcome of quantum mechanics. In 1930, L. Landau proposed the notion of Landau levels (LLs), which represent the quantization of energy in the presence of magnetic field1. From the semi-classical perspective for the LLs, when electrons are confined to a two-dimensional plane and are subjected to an out-of-plane magnetic field, they move in circular cyclotron orbits as a result of the Lorentz force. The semiclassical quantization of electronic states can be described by Onsager’s rule under a magnetic field, and the dispersion of the LLs can reflects the geometrical features of the Fermi surface2–6. From the quantum perspective, the LLs exhibit distinct plateaus in the Hall conductivity, with the number of filled LLs corresponding to the Chern number7. Consequently, LLs facilitate the emergence of topological states with a series of flat bands8,9. However, photons do not directly respond to magnetic field, so it is necessary to construct pseudo magnetic field, also known as artificial gauge field10.

Dirac systems, as the singularities of two-dimensional (2D) structures, such as Dirac photonic crytals11–14, metamaterials15, metasurfaces16,17, Dirac-like plasmons18, have been widely used to study artificial gauge fields and topological photonics. Dirac systems, characterized by the crossing of two energy bands in momentum space, can be generally classified as type-I or type-II based on the tilt degree of the Dirac cone19. Type-I Dirac systems have opposite slopes for the two crossing energy bands along any direction in the momentum space, resulting in closed equifrequency contours (EFCs) with circular or elliptical shapes. Conversely, type-II Dirac systems have slopes of the same sign along the tilting direction, leading to open EFCs with hyperbolic profile. Since the initial exploration of straining the graphene lattice20, systems with closed EFCs, especially the type-I Dirac systems, have been extensively investigated for generating LLs using deformed artificial structures in optical systems12–14,21–26. Recently, the LLs in the type-I Dirac system are experimentally realized on integrated nanophotonic platform of a silicon photonic crystal slab, which provide a new way to govern light fields and lay a foundation for the nanophotonic devices applications based on LLs12,13. In addition, the proposal and the realization of Landau rainbow14 indicate that Landau modes are likely to play a pivotal role in optical field localization, slow-light phenomena, and wavelength demultiplexing.

The systems with open EFCs, such as type-II Dirac systems, breaks Lorentz invariance and introduces unique topological properties and interesting phenomena, leading to widespread interests16,27–33. However, it is thought that LLs do not exist in systems with open EFCs, which are defined as those unconfined in the local momentum space near the band sing ularity, according to the Onsager’s rule because of the diverging area of EFCs2,6,34. Specifically, under a magnetic field, the trajectory in the momentum space for photonic systems is the same as the EFC since the kinetic energy of the particle does not change. Hence, a closed EFC is essential to the existence of the LLs under a magnetic field, while an open EFC will result in collapsing of LLs.

In this work, we demonstrate that LLs can be generated in optical systems with open EFCs around band singularities through the combined action of artificial electric (E) and magnetic (B) fields. Unlike the open particle trajectory in the momentum space under the B field only, the introduction of both E&B alters the particle trajectory into a closed one, so that the existing condition of LLs buried in Onsager's rule can be applicable. In the experiment, we construct a metasurface system with with open EFCs, and the band structures are verified to exhibit type-II Dirac dispersion with hyperbolic EFCs. By elaborately engineering the geometric parameters of unit structures of the metasurface, we introduce spatial variations in both the frequency and the wave vector of the Dirac point, creating appropriate artificial gauge fields of E and B, which lead to observation of LLs. For comparison, we investigate another metasurface with artificial gauge field of B, and find that no LLs wherein are generated. Our work represents the first observation of LLs in optical systems with open EFCs, which holds important implications for the investigation of artificial gauge fields and controlling wave propagating by engeering the dispersion.

Results
Theoretical construction of Landau levels
We begin with a semi-classical picture of the trajectories for quasiparticle (or wave packet) in both real and momentum space to understand the role of the gauge fields in 2D systems with closed and open EFCs. Under a magnetic field, the trajectory in the momentum space is the same as the EFC since the magnetic field does not do work. Hence, a closed EFC is essential to the existence of the LLs. If the EFC is open, there would be no confined states and the LLs would collapse. On the other hand, when both magnetic field and electric field are in play, the trajectory of the particle in the momentum space would differ from the EFC. The trajectory is defined by the intersection between the band dispersion and a tilted plane defined by , where E is the kinetic energy of the particle, K is the wave vector, E is the in-plane pseudo electric field, is the unit vector along z-direction, and B is the out-plane pseudo magnetic field. We examine four different cases involving a closed type-I Dirac system and an open type-II Dirac system in Fig. 1. Fig. 1a, b and c provide intersections in momentum-energy space, particle trajectories in momentum plane and real space, respectively. In Fig. 1b(I) and 1c(I), the trajectory forms a closed loop in the presence of B field only, in both momentum and real space. Introducing an in-plane E field causes the particle to move along the intersection contour between the aforementioned tilted plane and the band dispersion, as shown in Fig. 1a(II). In this case, the trajectory [Fig. 1b(II)] remains a closed loop in momentum plane, but gains an additional net shift perpendicular to the direction of E in real space [Fig. 1c(II)]. In both cases, the LLs exist. The detailed calculation and analysis of the trajectory description are given in Section I of the Supplemental Information (SI).

For the type-II Dirac system with open EFCs, the particle resides on an open EFC of a hyperbola in the presence of only a B field in the momentum space, as shown in Fig. 1b(III). In real space it is not confined but moves along a straight trajectory towards the infinity since the group velocity of the particle is perpendicular to the hyperbolic EFC, shown in Fig.1c(III). In this case, the LLs collapse. When an appropriate E is introduced, the trajectory of the particle forms a closed loop again in the momentum space, as shown in Fig. 1b(IV). The particle is confined along the direction of E and gain a net velocity perpendicular to E, as shown in Fig. 1c(IV). In this situation, the LLs reappear. Comparing the real space trajectories in Fig. 1c(II) and c(IV), the key difference is that the trajectory intersects itself in the type-I Dirac system, whereas it does not in the type-II case. This is due to the reversed signs of the group velocities along the direction in the type-I Dirac cone, whereas they always exhibit the same sign in the type-II case.

In the following, we provide a quantum perspective to understand the closed and open EFC systems in the presence of B and E gauge fields. Here, it should be pointed out that the low energy model with an effective Hamiltonian describes the local momentum space in the vicinity of a Dirac point. In other words, the Brilloiun zone can be considered as infinitely large in relation to the neighborhood of the band singularity. Consequently, the dispersion contour will be open as hyperbola or close as ellipse near the Dirac point. In the case of the type-I Dirac photonic system with closed EFC, the local effective Hamiltonian around the Dirac point takes the form , where denote Pauli matrix, and denote the Fermi velocities of photons. The vector potential A enters the Hamiltonian via the “Peierls substitution” 35, and we will use and for simplicity.

The vector potential A, which can be introduced by deforming the geometry of each unit cell, results in the variation of local dispersion relation36, and magnetic field B can be obtained according to the relation of .

In the case of the type-II Dirac photonic system with open EFC, without losing generality, we assume that the Dirac cone has a tilt along direction, and the local effective Hamiltonian can be generally expressed as , where denotes the identity matrix, denotes the tilt velocity of Dirac cone towards direction. Here, we suppose both and are positive, and and have the same sign to ensure the system is type-II. When the system is subject to both electric and magnetic gauge fields, i.e. E&B, we construct a Hamiltonian which is expressed as
(1)
where the vector potential follows a Landau gauge. The variation of the vector potential A leads to an artificial magnetic field applied in the z-direction. Meanwhile, the shift of frequency is considered as a gradient of scalar potential , which leads to an artificial electric field .

The general way to solve the eigenvalues problem of the LLs in the Dirac systems is based on the harmonic oscillator model. The eigenvalue problem described by Equation (1) involves a pair of second-order differential equations for the Hamiltonian with gauge fields. After derivation, we obtain an equation with harmonic oscillator form,
(2)
where , is the eigenvalues of the energy, C is a trivial scalar potential independent of and . The above equation needs to satisfy to ensure the existence of confined wave functions as that in a harmonic oscillator, i.e.
(3)

Subsequently, by eliminating the non-commutative terms, we arrive at the following analytical expression for the eigenvalues (denoted by ) of the Hamiltonian of Equation (1),
(4)
where , and E . The expression requires , or else the LLs collapse. The detailed calculations are shown in Section II of the SI.

Based on the above derivation, we illustrate the mechanism for creating LLs in the type-II Dirac system in Fig. 2. As an example, we set the group velocity , the coefficient of the tilt term , and the magnetic field . The harmonic potentials for different magnitudes of electric field E are obtained and shown in Fig. 2a. When the value of E is slightly above , the harmonic potenial is nearly flat [Fig. 2a(I)], and the LLs are densely clustered into a single line due to the poor confinement of the wave functions, as shown in Fig. 2b(I). Correspondingly, the analytically obtained wave function of the zeroth mode is shown in Fig. 2c(I), which exhibits a very broad Gaussian profile. Detailed equations and derivations of the wave functions for different Landau levels can be found in Section III of the SI. As the E field increases, the harmonic potential profile narrows, as illustrated in Fig. 2a(I)-(IV). This leads to an expansion of the distance between LLs due to the improved confinement provided by the harmonic potential. Additionally, the slope of the LLs also increases, as shown in Fig. 2b(I)-(IV). By setting the maximum order n = 10, there are 21 LLs that are clearly observed in Fig. 2b(IV). Meanwhile, the wave function of the zeroth mode become increasingly confined with the increase of E field (Fig. 2c).

Figure 2d shows the chromatic map contours of the energy gap between the zeroth and first LL for three different values of the tilt term , corresponding to type II [2d(I), with ], type III [2d(II), with ], and type I [2d(III), with ], respectively. In Fig. 2d(I), for B=1, if we gradually increase E, the system transitions from the phase without LLs (the yellow circle is near the critical value for emergence of LLs) to that with LLs (the yellow pentagon marks the maximum gap between the zeroth and first LL). However, a further increase of the E field will result in a reduction of the gap and eventually collapse of LLs. Thus, the generation of LLs in type-II systems requires an appropriate E field range with the most prominent LLs occurring at an optimal E value. When [Fig2. d(II)], the line of E=0 marks the borderline between the two phases, which means that LLs exist only when the E field is positive, regardless the values of B. For the case of type-I Dirac cone [Fig2. d(III)], the gap of the LLs reaches its maximum when E=0. Increasing the magnitude of E along either direction would reduce the energy gap and cause the collapse of LLs at sufficiently large magnitude. In the SI, we also give an example of type-I Dirac metasurface system for the structure design, energy band, and electic field distributions of LLs (section VII, Fig. S7 and S8 in SI).

Experimental demonstration of Landau levels
We first design a metasurface with periodic structures of square lattice to realize type-II Dirac band. The metallic unit with a shape of Jerusalem Cross (Fig. 3a) has been proven to possess type-II Dirac degeneracies formed by transverse and longitudinal electric dipole modes16. Under the subwavelength approximation, the metasurface with periodic metallic patterns can be regarded as a homogenized anisotropic dielectric slab. The Dirac degeneracy can be considered as the crossing between the lowest guided transverse electric mode of a uniform anisotropic dielectric slab, whose electric field is perpendicular to the propagation direction, and a guided longitudinal mode with electric field parallel to the propagation direction.

Figure 3b shows that the metasurface possesses type-II Dirac band with a tilt Dirac cone along the direction. The electric field distributions of the two modes in the insets confirm that they are transverse electric mode and longitudinal electric mode, respectively (more distributions for different can be found in Fig. S2 in the SI), and there are no other stray bands near the frequencies of the degeneracies. Both slopes of the two bands are positive in the direction and are approximately linear over a certain range of , constituting a pair of ideal type-II Dirac points around the frequency of 9.1 GHz, as shown by the three-dimensional band structures in Fig. 3c. The band structure of the designed metasurface is characterized using a near-field scanning method (see detail in the Methods and Section V in SI). The energy band is obtained via Fourier transformation on the directly detected electric field distributions as we did in our previous work37. The measured EFCs of the periodic structure is shown in Fig. 3d, confirming the existence of type II Dirac points at 9.1 GHz, which agrees well with the calculated results in Fig. 3c. Thus, the designed metasurface provides a convenient and reliable platform with open EFCs for exploring Landau phenomena.

Next, we design an inhomogeneous metasurface with gauge fields of E and B by adjusting the positions of the Dirac points in momentum space. The details of the sample are shown in Section IV and Fig. S3 in SI. The longitudinal and transverse electric modes can be independently controlled by the parameters a and c of the unit cell, respectively. Specifically, if c is varied while a is fixed, the Dirac point will move primarily along , and a gauge field of B is generated. On the other hand, if a is varied while c is fixed, the Dirac point will shift primarily in frequency, generating a gauge field of E (the detail about manipulating the parameters is shown in Fig. S4 in the SI). Consequently, a and c can be spatially modulated to create the combination of E and B that satisfy the Equation (3) to ensure a closed trajectory in the momentum space as shown in Fig. 1a(IV) and 1b(IV). The designed artificial B and E are and , respectively (see Section VI, Fig. S4 and S5 in the SI). The magnitude of artificial B and E can be tuned by the variation of geometric parameters of the inhomogeneous metasurface, i.e. the larger of the variation, the larger of the magnitude of B(E).

The metasurface samples are fabricated using the printed circuit boards. Besides the periodic sample and the inhomogeneous sample with gauge fields E&B, we also fabricate a sample with a dominant B field for the purpose of reference. By using the near-field scanning technique, we measure the electric field distributions across the metasurfaces to determine the band dispersion, as illustrated in the schematic diagram in Fig. 4a. The photos of the samples and the detailed parameters are provided in Fig. S3 and Table S1 to S3 of the SI. Unlike the two-dimensional periodic sample, the two metasurfaces with gauge fields have periodic units only in the x-direction but inhomogeneous in the y-direction. Therefore, we perform a one-dimensional Fourier transformation in the x-direction and add up the magnitudes of each line in the y-direction to obtain the band structure of the supercells. For the metasurface with gauge fields E&B (shown in Fig. 4a), the measured zeroth-mode (Fig. 4b) exhibits a linear dispersion, which agrees well with Equation (4) for . Based on Fig. 4b, the measured slope (the upper picture) for the zeroth mode is equal to , which agrees well with the simulated slope (Fig. 4c) of , and the theoretical value . The detail of the caculations are shown in Section VI and Fig. S5 in the SI. The top panel of Fig. 4h shows the simulated field distribution for the zeroth-mode, which exhibits a Gaussian distribution, agreeing well with the theoretical prediction of Equation (4).

In contrast, for the other two samples, i.e. the metasurface with a dominant B field (Fig. 4d and e) and the metasurface without gauge field (Fig. 4f and g), no LLs are observed. Their bulk modes look very similar to each other: there exists a crossing between the two bands around the Dirac point. The measured band dispersions agree well with the simulated results. Their electric field distributions also show nearly uniform magnitude across the whole samples along the y-direction, as plotted in Fig. 4h. The field distributions for each unit of the super cells in y-direction can be found in Fig. S6 in the SI. Thus, our experiments directly confirm that the LLs in a system with an open EFC cannot be generated by B alone but should be induced by appropriate combination of gauge fields E&B.

Methods
Theoretical calculations: The derivation of the particle’s motion is based on the canonical equation, and the details are shown in Section I of SI. The eigenvalues of the Eenergy is obtained analytically by solving the Hamiltonian equations, and the extensive lengthy calculations are shown in Section II of SI. The numerical solution for the eigen-states is based on the Fock states38,39, and the detail is shown in Section III of SI. The band structures are simulated by using the finite element method based on a commercial software of COMSOL Multiphysics.

Experimental measurements: The samples are fabricated by a standard Printed Circuit Board (PCB) fabrication technology. We employ a microwave vector network analyzer and a near-field antenna acting as a source to provide broadbanTd frequency excitation of electromagnetic surface waves, and another near-field antenna controlled by a three-dimensional translation stage is used to detect the electric field with both real and imaginary parts. The frequency step is set to be 0.01GHz with a whole frequency range covering 5GHz to 13GHz (see detail in Section V of SI). The scanning step of the probe is set to be the lattice constant, and the number of scanned data points is determined by the units number of each sample. The details of the samples can be found in Section IV, Fig. S3, Table S1, S2 and S3 of SI. The band structure and the EFCs of the samples are then obtained from the Fourier analysis of the scanned complex electric field in real space at each frequency.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript studies Landau levels in a specific type of Dirac Hamiltonian, where the slopes of the two bands are the same (type-II) rather than opposite (conventional or type-I). Such a difference has the physical consequence that the real-space trajectory of a particle on an iso-frequency line is not closed, and cannot give rise to Landau quantization of the orbit, even in the presence of a magnetic field. The authors propose a way to recover the Landau levels in type-II Dirac Hamiltonians by introducing an additional electric field. Both the magnetic and the electric field are artificial, i.e. are introduced via spatial variation of structural parameters in the unit cell, and the resulting spectra are analyzed. Limiting cases are discussed, where a balance between the Dirac tilt and the electric field can collapse or not the Landau levels. The Landau levels in such type-II Dirac point show similarities with the conventional Landau levels in type-I Dirac: they are still of the “relativistic” form, where the scaling is \\sqrt{n}, n being the level quantum number; the lowest landau level n=0 is spatially localized and its profile is gaussian.\nThe novelty of the paper is not only in the type-II Landau levels, but also in the geometrical implementation, which for the Dirac Hamiltonian typically relies on honeycomb geometries. Here, thanks to the particular photonic metasurface dispersion, a square geometry is implemented, demonstrating that Dirac physics and, in connection, relativistic Landau levels, are obtainable in such a system.\nOverall, the paper is well written and clearly organized. The theoretical methodology is sound and sufficiently detailed in the supporting information. However, I find the experimental signature of the existence of Landau levels very weak and not sufficiently supported by the data. The resolution for the gap opening in the zero-th Landau level (LL0) in the case E&B is very bad in Fig.4b compared to the B-dominant case where no LL is expected Fig.4d. In my opinion, fixing such an issue is very important to justify the claim of paper and consider it for publication.\nIn connection with this issue, I have a few questions:\n1) The authors only show a single configuration for which LLs are present. Have the a and c parameters been optimized to show the biggest gap possible? Can other signatures, such as an integrated density of state on the tilted axis, show the emergence of the LL0, similarly to what is done conventionally in artificial honeycomb systems?\n2) Another signature of the LL0 would be the spatial localization of the wavefunction, for example as the simulated field shown in Fig.4h. Would it be possible to reconstruct an experimentally measured real-space electric field measured from the near-field antenna probe?\n3) Due to time-reversal symmetry not being broken, I expect a second Dirac point where the LLs have opposite magnetic field B. Is this time-reversal symmetric pair present in this system?\n4) In connection with the previous point. Generally, in strained honeycomb systems, the presence of the two Dirac point pairs (at K and K’) is a limiting factor for the strength of the artificial magnetic field due to a Lifshitz transition to a gapped phase even in the absence of the magnetic field. Is there an analogous mechanism here? What would limit the strength of the B field?\n5) In the conclusions, it is mentioned that the LL in type-II Dirac systems could present an opportunity for directional propagation at constant velocity. In connection with the time-reversal symmetric Dirac pair in 3), would there be a mechanism to prevent scattering to a state at the same frequency but opposite velocity at the other type-II Dirac point?\n6) Would these type-II Landau levels also show edge effects similar to those in conventional type-I?", "reviewer_2": "The manuscript investigates the emergence of Landau levels in type-II Dirac systems with coexisting electric-magnetic field, which typically do not support Landau levels under a magnetic field alone under the influence of open equifrequency contours. The authors provide a thorough theoretical framework connecting tilt velocity, Fermi velocity, and the electric-magnetic field ratio, while experimentally demonstrating Landau quantization through precise modulation of pseudo-electromagnetic fields in a metallic metasurface. The interplay between analytical predictions and experimental validation significantly strengthens the work's impact. The work is interesting and will trigger more works in both metamaterials and topology fields. I support its publication if the authors can address the following questions.\n1. In the type-II Dirac system, the manuscript defines a tilted plane (ω) on line 102. Is this tilted plane related to the tilt velocity (ω_x) introduced for the Dirac cone on line 139? If so, please clarify the connection between these parameters.\n2. The harmonic potential mentioned on line 166 and illustrated in Fig. 2a is not sufficiently explained. Could the authors provide an explanation for this potential and how it is implemented in the context of the experiment?\n3. Equation (4) defines the maximum order (n) of Landau levels in relation to the size of units in the y-direction. Do the experimentally measured frequency peaks for the (n=0,±1) Landau levels align with the analytically predicted (λ_n) in Equation (4)?\n4. In the experimental setup, do the values of (ν_x) (Fermi velocity) and (ω_x) (tilt velocity) satisfy the condition (ω_x-ν_x> 0), ensuring that (|β| < 1)?", "reviewer_3": "In this paper, the authors construct a microwave system emulating the physics of Landau levels in the scenario when the dispersion is captured by the type-II Dirac Hamiltonian. In the usual scenario (e.g. type-I Dirac Hamiltonian) Landau levels arise in the presence of magnetic field B. When electric field E is added, Landau levels collapse upon reaching some threshold value of E. Here, because of a specially tailored dispersion, the situation is opposite: Landau levels do not exist for E=0, but arise for some nonzero electric field. The authors provide a detailed theoretical derivation of this physics by several approaches.\nFrom the experimental side, tilted Dirac cone is engineered by achieving an accidental degeneracy of the two photonic modes. In the vicinity of the intersection point, a Dirac point is observed. Then the gradient of the effective properties is introduced. Shift of the crossing point in frequency mimics effective electric field E, while the shift of the crossing point along kx corresponds to the synthetic magnetic field B. Choosing a suitable ratio, the authors demonstrate the emergence of Landau levels.\nOverall, the manuscript is interesting, while the theoretical, numerical and experimental aspects are described in detail. I have several conceptual questions:\n1. The authors write about open isofrequency contours. However, this is an idealization. In reality all isofrequency contours are closed, but encircle not Gamma point, but rather X point at the edge of the Brillouin zone (see e.g. Fig. 3c). Please, comment how this changes the conclusions. Is the trajectory in Fig. 1a(III) modified? Additionally, taking this into account seems to remove infinities in the derivation of the quantization rules according to Onsager’s rule. In fact, a similar discussion happened in the studies of hyperbolic metamaterials 10-15 years ago.\n2. While photonic emulation of Landau levels in exotic systems with type-II Dirac Hamiltonian is interesting, it is not immediately clear what kind of physical phenomena this provides. The formulation about “controlling optical waves propagation using artificial gauge fields” is too broad and can be achieved in different ways. The usual quantum Hall effect is important, because of the robust quantization of the Hall conductance. Do the authors anticipate a similar quantization of any transport coefficients or nonlinear susceptibilities in their system? Could this provide any sort of generalization of photonic Hall effect discussed by Haldane and Raghu in 2008?\nAdditionally, I find it instructive to move Fig. S1 to the main text, because it conveys the main message of the manuscript: how to achieve Landau levels even if the isofrequency contours are hyperbolic."}, "author_response": "We are grateful for the constructive and positive comments on this manuscript (ID: NCOMMS-25-23270) from all the three reviewers, which greatly help us to improve the manuscript.\nIn the text below, each reviewer’s comments are quoted in italics bold font and are followed by corresponding responses. We have also revised the manuscript and the Supplementary Information correspondingly. These updates are highlighted in red font in those files.\n----------------------------------------------------------------------------------------------------------------------\nReviewer #1:\nGeneral Comments.\nThe manuscript studies Landau levels in a specific type of Dirac Hamiltonian, where the slopes of the two bands are the same (type-II) rather than opposite (conventional or type-I). Such a difference has the physical consequence that the real-space trajectory of a particle on an iso-frequency line is not closed, and cannot give rise to Landau quantization of the orbit, even in the presence of a magnetic field. The authors propose a way to recover the Landau levels in type-II Dirac Hamiltonians by introducing an additional electric field. Both the magnetic and the electric field are artificial, i.e. are introduced via spatial variation of structural parameters in the unit cell, and the resulting spectra are analyzed. Limiting cases are discussed, where a balance between the Dirac tilt and the electric field can collapse or not the Landau levels. The Landau levels in such type-II Dirac point show similarities with the conventional Landau levels in type-I Dirac: they are still of the “relativistic” form, where the scaling is \\sqrt{n}, n being the level quantum number; the lowest landau level n=0 is spatially localized and its profile is gaussian.\nThe novelty of the paper is not only in the type-II Landau levels, but also in the geometrical implementation, which for the Dirac Hamiltonian typically relies on honeycomb geometries. Here, thanks to the particular photonic metasurface dispersion, a square geometry is implemented, demonstrating that Dirac physics and, in connection, relativistic Landau levels, are obtainable in such a system.\nOverall, the paper is well written and clearly organized. The theoretical methodology is sound and sufficiently detailed in the supporting information.\nHowever, I find the experimental signature of the existence of Landau levels very weak and not sufficiently supported by the data. The resolution for the gap opening in the zero-th Landau level (LL0) in the case E&B is very bad in Fig.4b compared to the B-dominant case where no LL is expected Fig.4d. In my opinion, fixing such an issue is very important to justify the claim of paper and consider it for publication. In connection with this issue, I have a few questions:\nResponses:\nWe greatly appreciate your comprehensive summary and positive comments of our work.\nTo strengthen experimental signature, we have fabricated new samples with larger areas (consisting of more unit cells), and performed additional measurements over the past six months. These improvements yield higher-quality experimental data.\nThe experimental energy bands are obtained via discrete Fourier transformation (DFT) on the measured electric field distributions in real space. The equation of DFT is given by\nwhere represents the number of the unit-cells along the x-direction. The number of sampled data points determines the minimum spacing in the k-space and thus the resolution of the transformed image.\nAs shown in Fig. R1a and 1b, compared with the original sample with 80 unit-cells in the x-direction, the new samples consist of 120 unit-cells in the x-direction (We can measure up to 110 unit-cells limited by the range of experimental scanning platform, see the photograph of Fig. R1c). Two new samples were fabricated, one for the E&B case, and the other for the B-dominant case. The corresponding measured results are presented in Fig. R1d. In the E&B sample, the zero-th Landau level is clearly resolved, whereas it is absent in B dominant sample. These higher-resolution measurements provide strong experimental confirmation of the existence of the type-II Landau levels.\nFig. R1 New experimental observation of LLs with higher resolution. a, Comparison between the original sample (Nx1=80, L=600mm) and the new sample (Nx2=120, L=900mm) for E&B case. b. Comparison between the old sample (Nx1=80, L=600mm) and the new sample (Nx2=120, L=900mm) for B-dominant case. The enlarged images denote the blue area of the samples. c, The experimental setup. d, Measured energy bands for the metasurface with E&B and B dominant, respectively.\nComments1.\nThe authors only show a single configuration for which LLs are present. Have the a and c parameters been optimized to show the biggest gap possible? Can other signatures, such as an integrated density of state on the tilted axis, show the emergence of the LL0, similarly to what is done conventionally in artificial honeycomb systems?\nResponses:\nThank you for your valuable comments. Yes, parameters a and c can be optimized to enlarge the gap of LLs. Here, we give the relationship for our design between the energy gap and parameters a and c, shown in Fig. R2.\nFig. R2 The energy gap is a function of B and E. B is determined by the gratitude of momentum along y; and E is determined the gratitude of frequency along y. Both a and c are the functions of y.\nBased on the analytical eigenvalues of Equation (4) in the manuscript, we can obtain the analytical relationship between the energy band gap and the gauge fields. In Fig. 2d in our manuscript, we can find the dependence of the gap on B&E. In our work, we have performed parameter searches to obtain a sufficiently large gap, with the final results shown in Fig. S4. The B&E play a decisive role in the band gap while their values are controlled by the gradient change of the parameters. Fig. S4a and S4b present the respective influences of parameters a and c. We strategically utilize parameter c to determine B, while adjusting parameter a to control E. Thus, we can determine the a and c parameters for obtaining a large gap .\nIn addition, we have to consider the practical sample fabrication in experiment. The achievable fabrication precision is 0.05 mm, which determines the variation step of the parameters. Therefore, we set the minimum step to be 0.05 mm of the parameters. The parameter c has a 0.1mm step change (2.7 mm~5.6 mm) and the parameter a has a -0.05 mm step change (6.0 mm~4.55 mm) along y-direction; while it is periodic along x-direction. The only influence brought by the fabrication limit is that the value of B is not strictly uniform (see Fig. S5), so we can use its average value. This does not affect significantly the generation of Landau levels, which is further verified by the results in Fig. 4.\nNext, to obtain the integrated density of states (DOS) on the tilted axis, we performed calculations on the DOS along the tilted axis as suggested. We first apply a coordinate transformation to rotate the area within the blue box of Fig. R3 a to align the LL0 parallel to the kx-axis. We then integrate the Ez field along the kx direction (longitudinal direction of the LLs) to obtain the projected DOS. As shown in the Fig. R3, the y-coordinate is the projected DOS, and the x-coordinate is denoted by a parameter t, where t is a parameter mixed with frequency and momentum along the tilted line due to the rotation of the axis. The resulting 1D spectrum shows three peaks at the center, representing the emergence of the LL0 and LL1.\nFig. R3. The projected DOS along the tilted axis. a, an enlarged image of bands. The blue dotted line represents the tilted axis of LL0. b, The projected DOS along the tilted axis. The three peaks at the center are the zero-order peak and the positive and negative first-order peaks.\nChanges in the manuscript:\nIn the Supplementary Information, we add the flowing figure to illustrate and summarize our design concept to present the relationship between the energy gap and parameters of the metasurface.\nFig. S9. Summary for the relationship between the energy gap and the parameters of the metasurface. The energy gap is a function of B and E. B is determined by the gratitude of momentum along y; and E is determined the gratitude of frequency along y. Both a and c are the functions of y.\nIn the Supplementary Information, we add the DOS figure into Fig. S5 as Fig. S5d, so Fig. S5 is changed from\ninto the following one\nFig. S5. Calculation and measurement for E/B of zeroth mode of LLs. The calculated relation of kx (a) and the relation of frequency (b) of the Dirac point with the increasing of y, and the unit of y is set to be Dy, i.e. the lattice constant. There are 30 units in the y direction in total, and there are 80 identical units for each line in the x direction of the metasurface sample. The simulated (c) and measured (d) results of the slope (E/B) for the zeroth mode of LLs. (e) The projected DOS along the tilted axis. The three peaks at the center are the zero-order peak and the positive and negative first-order peaks.\nComments2.\nAnother signature of the LL0 would be the spatial localization of the wavefunction, for example as the simulated field shown in Fig.4h. Would it be possible to reconstruct an experimentally measured real-space electric field measured from the near-field antenna probe?\nResponses:\nThank you for your instructive comment. Following your suggestion, we provide the reconstructed electric field distribution in the real space to confirm the experimental signature of the existence of Landau levels. The directly measured electric field distribution in real space is not a single Gaussian distribution but exhibits a complex pattern due to excitation of multiple LL modes. Therefore, we fixed k = 0.651 and x frequency = 9.26 GHz, targeting a point on the tilted LL0. By performing an inverse Fourier transformation on ky, we obtained the real-space electric field distribution along the y direction, as shown in Fig. R3. It exhibits a Gaussian distribution corresponding to LL0.\nFig. R3. Reconstructed normalized electric field distribution in the real space. The reconstructed normalized electric field distribution of line graph a, and the intensity graph b, under the conditions of kx = 0.651 and frequency = 9.26 GHz.\nComments3.\nDue to time-reversal symmetry not being broken, I expect a second Dirac point where the LLs have opposite magnetic field B. Is this time-reversal symmetric pair present in this system?\nResponses:\nThank you for your insightful comment. It is true that this time-reversal symmetric pair is present in this system. The introduction of the pseudo-magnetic field does not break time-reversal symmetry. Thus, there exists a Dirac point at the K’ point with the opposite pseudo-magnetic field. The group velocity is opposite to the Landau level at K point. To verify this, we present the result of the energy band around the K’ point for both simulation and experiment. In the new experiment, we put the excitation source on the opposite side compared with the original experiment in the manuscript. The results are presented in Fig. R4. Different from the positive group velocities in the manuscript, here it is clear that the zeroth Landau mode has negative group velocities.\nFig. R4. The simulated and measured Landau levels for K’ points. The Landau levels along negative-k direction, corresponding to the K’ point, which has negative group velocity for the zero Landau modes. The shadow regions denote the light cone.\nComments4.\nIn connection with the previous point. Generally, in strained honeycomb systems, the presence of the two Dirac point pairs (at K and K’) is a limiting factor for the strength of the artificial magnetic field due to a Lifshitz transition to a gapped phase even in the absence of the magnetic field. Is there an analogous mechanism here? What would limit the strength of the B field?\nResponses:\nThank you for the helpful comment. Indeed, applying artificial magnetic field may cause local Lifshitz transition (Physical Review Letters 110, 033902, 2013; Physical Review B 80, 153412, 2009) in strained honeycomb systems. The same thing may happen in our structure. This imposes an upper limit on the gauge field strength. In our system, the structure-induced gauge field is deliberately designed to be significantly below this limit.\nSince B field is given by the gradient change of the location of Dirac point along the y-axis, , a steeper gradient along y-axis would result in a larger B field. In our work, the requirement for low-energy effective model enforces that the k-space shift is not too far away from the original Dirac point, which limits the maximal strength of the B field in theory. For our sample, the position of Dirac point k shifts x from 0.535(π/Dx) to 0.805(π/Dx), shown in Table S1, where the Lifshitz transition does not happen.\nComments5.\nIn the conclusions, it is mentioned that the LL in type-II Dirac systems could present an opportunity for directional propagation at constant velocity. In connection with the time-reversal symmetric Dirac pair in 3), would there be a mechanism to prevent scattering to a state at the same frequency but opposite velocity at the other type-II Dirac point?\nResponses:\nThank you for the valuable comment. In general, no strict protection exists to completely suppress scattering between the Landau modes associated with the two type-II Dirac points. Nevertheless, provided the deformation of the structure is adiabatic and no defects are present, the conversion (or mixing) between counter-propagating Landau levels of the two Dirac points is negligibly small.\nComments6.\nWould these type-II Landau levels also show edge effects similar to those in conventional type-I?\nResponses:\nThank you for the valuable comment. In fact, the type-II Landau levels also have edge effects, as shown in Fig. R5. As a reference, the edge effects of type-I Landau levels are shown by s and s in Fig. R5b, whose dispersion is shown in Fig. R5a (These 1 3 figures come from Nat. Phys., 15, 352–356 (2019)). In our type-II case, the edge effects are shown by d and d in Fig. R5d, whose dispersion is shown in Fig. R5c. Although 1 3 the field distributions are similar, the propagation behaviors of the edge effects in type-I and type-II systems are different. For the same Landau levels in conventional type-I, the propagation directions for opposite edges are opposite, except for the zeroth Landau level. In contrast, the propagation directions for opposite edges in type-II are the same for all Landau levels. These is easily seen by the slopes of the dispersion curves from Fig. R5a and c, respectively.\nFig. R5 Edge effects for type-I and type-II Landau levels. a. Dispersions for different Landau levels near the original position of the type-I Dirac point. b. Edge states and Landau state for zeroth Landau level in type-I system. c. Dispersions for different Landau levels for the type-II Dirac system. d. Edge states and Landau state for zeroth Landau level in type-II system.\nReviewer #2:\nGeneral Comments.\nThe manuscript investigates the emergence of Landau levels in type-II Dirac systems with coexisting electric-magnetic field, which typically do not support Landau levels under a magnetic field alone under the influence of open equifrequency contours.\nThe authors provide a thorough theoretical framework connecting tilt velocity, Fermi velocity, and the electric-magnetic field ratio, while experimentally demonstrating Landau quantization through precise modulation of pseudo-electromagnetic fields in a metallic metasurface. The interplay between analytical predictions and experimental validation significantly strengthens the work's impact. The work is interesting and will trigger more works in both metamaterials and topology fields. I support its publication if the authors can address the following questions.\nResponses:\nWe greatly appreciate your positive comments. Next, we provide point-to-point reply and carefully revise the manuscript according to your constructive comments and suggestions.\nComments1.\nIn the type-II Dirac system, the manuscript defines a tilted plane (ω) on line 102. Is this tilted plane related to the tilt velocity (ω_x) introduced for the Dirac cone on line 139? If so, please clarify the connection between these parameters.\nResponses:\nThank you for the valuable comments. In fact, the tilt plane (ω) is not directly related to the tilt velocity(w ). The tilting angle of this plane is determined by the x pseudo-electric and pseudo-magnetic field E and B. The slop of this plane along k x direction is E/B, and is 0 along k direction. y\nComments2.\nThe harmonic potential mentioned on line 166 and illustrated in Fig. 2a is not sufficiently explained. Could the authors provide an explanation for this potential and how it is implemented in the context of the experiment?\nResponses:\nThank you for the question. The harmonic potential arises from solving the general eigenvalue equation , where is given by Equation (1). The eigenvalue problem is finally simplified to Equation (2), i.e.,\n,\nwhere , shown in Equation (30) in the Supplementary Information.\nConsidering is linear with , the eigenvalue problem represents a harmonic oscillator equation. Fig. 2a shows the dependence of the harmonic potential over 𝑦̃̃ .\nThe experimental construction of the harmonic potential necessitates realization of Hamiltonian (1), i.e., .\nWe design a metasurface with periodic structures of square lattice to realize type-II Dirac band based on the transverse and longitudinal electric dipole modes. By spatially varying the geometrical parameters of the metasurface along y direction, we gradually shift the Dirac point both in k direction and frequency to implement the gauge fields x of B and E in Hamiltonian, respectively. Specifically, if the parameter c is varied while a is fixed, the Dirac point will primarily shift along . The artificial gauge field of B is generated by equivalent magnetic vector potential for Peierls substitution. If a is varied while c is fixed, the Dirac point will shift primarily in frequency, leading to an artificial gauge field of E. With both artificial gauge fields of B and E, the Hamiltonian of the designed metasurface can be described by Equation (1) and the harmonic potential is implemented.\nComments3.\nEquation (4) defines the maximum order (n) of Landau levels in relation to the size of units in the y-direction. Do the experimentally measured frequency peaks for the (n=0,±1) Landau levels align with the analytically predicted (λ_n) in Equation (4)?\nResponses:\nThank you for the question. For our type-II system, the dispersions of Landau levels are not flat but tilted. It means that multiple Landau levels can co-exist at the same frequency, hence its spectrum does not feature several isolated peaks. Therefore, in experiment we collected the electric field distribution to obtain the energy band by Fourier transformation. Near the K (or K’) degeneracies, where the Equation (4) is valid, the characteristics of the analytically predicted Landau levels, i.e., the tilted bands and band gap, align with the experimentally measured energy bands. The analytical prediction mainly reveals the physical image of Landau levels, while the measured detailed parameters are matched by the simulation. It means the Equation (4), as an analytical solution (where the coefficients are normalized for physical simplicity) is used to guide the design of the metasurface, while the measured results should align with the simulated results of the metasurface samples with practical parameters.\nComments4.\nIn the experimental setup, do the values of (ν_x) (Fermi velocity) and (ω_x) (tilt velocity) satisfy the condition (ω_x-ν_x> 0), ensuring that (|β| < 1)?\nResponses:\nThank you for the valuable question. The condition ensures that the Dirac point is type-II, i.e., an over-tilted Dirac cone with non-zero density of states at the Dirac frequency. On the other hand, requires ,\nwhich means that the ratio between E and B should be within a certain range to ensure the existence of Landau levels for type-II Dirac systems.\nReviewer #3:\nGeneral Comments.\nIn this paper, the authors construct a microwave system emulating the physics of Landau levels in the scenario when the dispersion is captured by the type-II Dirac Hamiltonian. In the usual scenario (e.g. type-I Dirac Hamiltonian) Landau levels arise in the presence of magnetic field B. When electric field E is added, Landau levels collapse upon reaching some threshold value of E. Here, because of a specially tailored dispersion, the situation is opposite: Landau levels do not exist for E=0, but arise for some nonzero electric field. The authors provide a detailed theoretical derivation of this physics by several approaches.\nFrom the experimental side, tilted Dirac cone is engineered by achieving an accidental degeneracy of the two photonic modes. In the vicinity of the intersection point, a Dirac point is observed. Then the gradient of the effective properties is introduced. Shift of the crossing point in frequency mimics effective electric field E, while the shift of the crossing point along kx corresponds to the synthetic magnetic field B. Choosing a suitable ratio, the authors demonstrate the emergence of Landau levels.\nOverall, the manuscript is interesting, while the theoretical, numerical and experimental aspects are described in detail. I have several conceptual questions:\nResponses:\nWe are very grateful for your summary and interests in our work. In the following, we provide point-to-point replies to your comments and revise the manuscript accordingly.\nComments1.\nThe authors write about open isofrequency contours. However, this is an idealization. In reality all isofrequency contours are closed, but encircle not Gamma point, but rather X point at the edge of the Brillouin zone (see e.g. Fig. 3c). Please, comment how this changes the conclusions. Is the trajectory in Fig. 1a(III) modified? Additionally, taking this into account seems to remove infinities in the derivation of the quantization rules according to Onsager’s rule. In fact, a similar discussion happened in the studies of hyperbolic metamaterials 10-15 years ago.\nResponses:\nThank you very much for this insightful comment. In principle, all isofrequency contours in the Brillouin zone are closed due to its toroidal topology. However, our work focuses on the interaction between band singularities (such as Dirac or Weyl points) and gauge fields — a topic that has attracted considerable attention in recent years, particularly due to the emergence of zero modes.\nFor this purpose, the relevant physics is captured by the low-energy effective Hamiltonian describing the local momentum-space structure in the immediate vicinity of a Dirac point. In this regime, the Brillouin zone boundary lies far away compared to the energy/momentum scales of interest, so it can be regarded as effectively infinite. Consequently, whether an isofrequency contour appears “open” (e.g., hyperbolic or parabolic, as in type-II Dirac/Weyl systems) or “closed” (e.g., elliptical, as in type-I systems) is determined solely by the local dispersion near the singularity — not by the global topology of the Brillouin zone.\nThe Landau levels we investigate are therefore dominated by this local Dirac dispersion and remain essentially unaffected by the behavior at the Brillouin zone edges.\nMoreover, the pseudo-magnetic field in our study is locally defined in the vicinity of the Dirac point and is not a uniform, global field across the entire Brillouin zone.\nSince time-reversal symmetry is preserved in the system, particle trajectories in momentum space may not form closed loops even when the isofrequency contour is formally closed.\nThis is precisely why we concentrate on the interaction between pseudo-electromagnetic fields and the neighborhood of a single Dirac point, where the pseudo-field can be treated as approximately uniform. All theoretical results and conclusions in our work rely on this well-controlled low-energy approximation.\nComments2.\nWhile photonic emulation of Landau levels in exotic systems with type-II Dirac Hamiltonian is interesting, it is not immediately clear what kind of physical phenomena this provides. The formulation about “controlling optical waves propagation using artificial gauge fields” is too broad and can be achieved in different ways. The usual quantum Hall effect is important, because of the robust quantization of the Hall conductance. Do the authors anticipate a similar quantization of any transport coefficients or nonlinear susceptibilities in their system? Could this provide any sort of generalization of photonic Hall effect discussed by Haldane and Raghu in 2008?\nResponses:\nWe greatly appreciate you for this insightful and important question, which highlights a key point: while the photonic emulation of Landau levels (LLs) in type-II Dirac systems is novel, the broader physical implications and potential for quantized transport phenomena deserve clearer articulation.\nIn our manuscript, we have focused primarily on demonstrating the existence of LLs in systems with open equifrequency contours (EFCs), which was previously considered impossible under a magnetic gauge field alone according to the conventional Onsager quantization condition. By introducing a suitable combination of pseudo-electric (E) and pseudo-magnetic (B) fields, we show that closed particle trajectories in momentum space can be restored, enabling the emergence of quantized, tilted, discrete topological bands in the type-II regime. These bands are distinctly different from the flat, nearly dispersionless LLs typically observed in type-I Dirac/Weyl systems.\nRegarding quantized transport coefficients (analogous to the robust quantization of Hall conductance in the electronic quantum Hall effect): at present, our work is mainly a proof-of-principle demonstration in the microwave regime using a metasurface platform. Direct measurement of quantized Hall-like conductance is challenging in this classical wave system for several reasons:\n1. Photons are bosonic and do not obey Pauli exclusion, so there is no natural Fermi level filling that leads to precise plateau formation as in fermionic systems.\n2. The system is dissipative (material loss) and operates in the weak localization regime, without clear signatures of robust edge states in our current measurements.\n3. The artificial gauge fields are inhomogeneous and local to the neighborhood of the Dirac point, so global transport quantization (as in the integer quantum Hall effect) is not directly expected.\nHowever, the tilted, discrete Landau bands we observe in type-II systems exhibit several intriguing features with potential implications for wave propagation control and possibly quantized responses:\nDirectional propagation at constant group velocity\nUnlike the flat LLs in type-I systems (which lead to strong localization and slow light), the tilted LLs in type-II Dirac cones naturally support finite and relatively constant group velocities across different frequencies/LL indices. This could enable dispersion-engineered directional guiding or wavelength-independent propagation over a range of frequencies — a feature that may be useful for robust signal transport in two dimensional photonic devices.\nPotential for anomalous Hall-like effects in photonics\nThe Haldane–Raghu proposal (Phys. Rev. Lett. 100, 013904, 2008; Phys. Rev. A 78, 033834, 2008) introduced analogs of the quantum Hall effect in photonic crystals by breaking time-reversal symmetry (e.g., via Faraday media), leading to unidirectional chiral edge states that are robust against backscattering. Our system preserves time-reversal symmetry (the pseudo-fields are strain-like and local), so we do not generate a global Chern number or conventional photonic quantum Hall edge states.\nNevertheless, the combination of E&B gauge fields induces a non-trivial local Berry curvature and tilted chiral-like features in the LL wavefunctions. This may give rise to anomalous velocity or photonic anomalous Hall-like deflection of wavepackets in the bulk (similar to the anomalous Hall effect in magnetic Weyl semimetals), especially if the system is further engineered to break effective time-reversal symmetry locally (e.g., via magneto-optical materials or dynamic modulation).\nIn summary, while our current experiment does not yet demonstrate a direct photonic analog of the robustly quantized Hall conductance (as in the electronic integer quantum Hall effect or the Haldane-Raghu photonic quantum Hall edge states), it opens a new route toward dispersion-engineered, directional, and potentially anomalous transport of light in type-II Dirac photonic systems. We believe this represents an important conceptual advance beyond the type-I regime.\nComments3.\nAdditionally, I find it instructive to move Fig. S1 to the main text, because it conveys the main message of the manuscript: how to achieve Landau levels even if the isofrequency contours are hyperbolic.\nResponses:\nThank you for the valuable suggestion. This indeed can provide readers with a clearer understanding. We integrated Fig. S1 into Fig. 1 and present a revised Fig. 1 now in the main text. The relevant discussion about original Fig. S1 is incorporated into the main text.\nChanges in the manuscript:\nWe have changed Fig. 1 and its caption from the original figure\nFig. 1. Semi-classical picture for understanding generating LLs in type-I and type-II Dirac systems. a. Real space of the motion of electron. Take an electron as an example. (I). Trajectory of the quasiparticle of type-I system with only B field in the z-direction forms a closed loop. (II). When E in the y-direction is added, the particle is still confined along y-direction, but will move along x-direction and its trajectory intersect with itself. (III). The particle will not be confined to both x and y directions for type-II system with only B in the z-direction field. (IV). When E in the y-direction is added, the particle will be confined along y-direction and moving along x-direction, where its trajectory do not intersect with itself. b. Momentum space of the intersection the energy band and the cut plane (with gray color) defined by an equation of . For Fig1.\nB(I) and (III), E=0, and the equation of the plane is ; for Fig1. B(II) and (IV), E≠0, and the equation of the plane is . Correspondingly, the interaction of the (I) (II) (IV) is circular or elliptic orbit, and is hyperbolic for (III).\ninto the following figure.\nFig. 1. Semi-classical picture for understanding generating LLs in type-I and type-II Dirac systems. a. Momentum-energy space of the intersection between the energy band and the cut plane (with gray color) defined by an equation of . For Fig1. a(I) and (III),\nE=0, and the equation of the plane is ; for Fig1. a(II) and (IV), E≠0, and the equation of the plane is . Correspondingly, the intersection of the (I) (II) (IV) is circular or elliptic orbit, and is hyperbolic for (III). b. Projection of the intersections shown in Fig. 1a onto the momentum plane. The projections inherit the classifications of conic sections in Momentum-energy space show in Fig. 1a. When the conic sections are closed (circle or ellipse), Landau level exist, when it is open (parabola or hyperbola), Landau level collapse. c. Real space of the motion of electron. Take an electron as an example. (I). Trajectory of the quasiparticle of type-I system with only B field in the z-direction forms a closed loop. (II). When E in the y-direction is added, the particle is still confined along y-direction, but will move along x-direction and its trajectory intersect with itself. (III). The particle will not be confined to both x and y directions for type-II system with only B in the z-direction field. (IV). When E in the y-direction is added, the particle will be confined along y-direction and moving along x-direction, where its trajectory do not intersect with itself.\nIn the Results section of the manuscript, we have revised the first and the second paragraphs into the following two paragraphs, where the revisions are marked in red.\n“We begin with a semi-classical picture of the trajectories for quasiparticle (or wave packet) in both real and momentum space to understand the role of the gauge fields in 2D systems with closed and open EFCs. Under a magnetic field, the trajectory in the momentum space is the same as the EFC since the magnetic field does not do work. Hence, a closed EFC is essential to the existence of the LLs. If the EFC is open, there would be no confined states and the LLs would collapse. On the other hand, when both magnetic field and electric field are in play, the trajectory of the particle in the momentum space would differ from the EFC. The trajectory is defined by the intersection between the band dispersion and a tilted plane defined by\n, where is the kinetic energy of the particle, K is the wave vector, E is the in-plane pseudo electric field, is the unit vector along z-direction, and B is the out-plane pseudo magnetic field. We examine four different cases involving a closed type-I Dirac system and an open type-II Dirac system in Fig. 1. Fig. 1a, b and c provide intersections in momentum-energy space, particle trajectories in momentum plane and real space, respectively. In Fig. 1b(I) and 1c(I), the trajectory forms a closed loop in the presence of B field only, in both momentum and real space. Introducing an in-plane E field causes the particle to move along the intersection contour between the aforementioned tilted plane and the band dispersion, as shown in Fig. 1a(II). In this case, the trajectory [Fig. 1b(II)] remains a closed loop in momentum plane, but gains an additional net shift perpendicular to the direction of E in real space [Fig. 1c(II)]. In both cases, the LLs exist. The detailed calculation and analysis of the trajectory description are given in Section I of the Supplemental Information (SI).\nFor the type-II Dirac system with open EFCs, the particle resides on an open EFC of a hyperbola in the presence of only a B field in the momentum space, as shown in Fig. 1b(III). In real space it is not confined but moves along a straight trajectory towards the infinity since the group velocity of the particle is perpendicular to the hyperbolic EFC, shown in Fig.1c(III). In this case, the LLs collapse. When an appropriate E is introduced, the trajectory of the particle forms a closed loop again in the momentum space, as shown in Fig. 1b(IV). The particle is confined along the direction of E and gain a net velocity perpendicular to E, as shown in Fig. 1c(IV). In this situation, the LLs reappear. Comparing the real space trajectories in Fig. 1c(II) and c(IV), the key difference is that the trajectory intersects itself in the type-I Dirac system, whereas it does not in the type-II case. This is due to the reversed signs of the group velocities along the direction in the type-I Dirac cone, whereas they always exhibit the same sign in the type-II case.”"}, {"round": 2, "reviewer_comments": {"reviewer_1": "In their response letter, I appreciate that the authors have successfully addressed comments by other reviewers and me.\nHowever, not all of the suggested changes have been included in the main text. In particular, the observation of LLs with higher resolution, the reconstructed normalized electric field distribution in the real space is shown only in the response as fig. R1 and R3, respectively, while the projected DOS along the tilted axis is shown in the supplemental information Fig. S5e. These findings and possibly related figures strengthen the experimental signature and should be included in the main text. Given that experimental evidence of the existence of LL was my main concern, I strongly encourage the authors give more emphasis to this newer, higher-quality experimental data in the main text.", "reviewer_3": "The authors have substantially revised their manuscript. Most importantly, they have strengthened the experimental part achieving a better resolution of n=0 Landau level. They have also improved the presentation providing a clear explanation why the combination of synthetic electric and magnetic fields enables closed equifrequency contours and hence leads to the quantized energy levels [Fig. 1]. I have only one further comment.\nIf I understand correctly, this paper is the first demonstration of Landau levels in a situation when the equifrequency contours are open. In fact, similar type-II Dirac points exist in condensed matter context, NiTe2 being one of the likely candidates for such physics. According to the authors, the energy levels of electrons become quantized when the combination of electric and magnetic fields is applied to the material. It would increase the impact of this work, if the authors briefly discuss the applications of their finding to condensed matter systems providing some estimates of the necessary field strengths (E and B). Could this E&B-induced quantization have some impact on the transport properties in condensed matter case?"}, "author_response": "Reviewer 1’s comment\nIn their response letter, I appreciate that the authors have successfully addressed comments by other reviewers and me.\nHowever, not all of the suggested changes have been included in the main text. In particular, the observation of LLs with higher resolution, the reconstructed normalized electric field distribution in the real space is shown only in the response as fig. R1 and R3, respectively, while the projected DOS along the tilted axis is shown in the supplemental information Fig. S5e. These findings and possibly related figures strengthen the experimental signature and should be included in the main text. Given that experimental evidence of the existence of LL was my main concern, I strongly encourage the authors give more emphasis to this newer, higher-quality experimental data in the main text.\nRevisions:\nIn the main text, we have updated figure 4 using the new experimental results.\nReviewer 3’s comment:\nIf I understand correctly, this paper is the first demonstration of Landau levels in a situation when the equifrequency contours are open. In fact, similar type-II Dirac points exist in condensed matter context, NiTe2 being one of the likely candidates for such physics. According to the authors, the energy levels of electrons become quantized when the combination of electric and magnetic fields is applied to the material. It would increase the impact of this work, if the authors briefly discuss the applications of their finding to condensed matter systems providing some estimates of the necessary field strengths (E and B). Could this E&B-induced quantization have some impact on the transport properties in condensed matter case?\nRevisions:\nIn the Discussion Section of the main text, we have added the following sentences.\nSimilarly, the electronic energy levels in a condensed matter system will also undergo quantization when simultaneous electric and magnetic fields are applied to a type-II Dirac or Weyl material. Taking MoTe₂ as an example, the tilt velocity for its type-II Dirac cone is approximately 106m/s. To observe Landau levels in this material, the E/B ratio must also be on the order of 106m/s. This implies that a magnetic field of 1T needs to be accompanied by an electric field of approximately 106V/m. Moreover, these Landau levels are co-determined by both E and B fields. Consequently, the Hall resistivity is not exclusively dependent on the magnetic field alone, and its dependence on B deviates from a simple linear relationship.\nIn the Supplementary Information, we have added Section VIII.\nSection VIII. E and B field needed in MoTe2 to observe Landau Levels.\nTaking MoTe as an example, which is a type-II Weyl semimetal4. Then the 2 eigenfrequency of the Landau level under certain E and B can be written as\nAs the tilt velocity4 w is approximately on the order of 106m/s, if we want to have a x good observation of Landau levels, then the E/B ratio should also be around 106m/s to have a small β. This implies that a magnetic field of 1T needs to be accompanied by an electric field of approximately 106V/m.\nReference\n4. Ke Deng, et al. Experimental observation of topological Fermi arcs in type-II Weyl semimetal MoTe . Nature Physics 12, 1105–1110 (2016). 2"}]

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