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ABSTRACT Chiral zeroth Landau levels are topologically protected bulk states. In particle physics and condensed matter physics, the chiral zeroth Landau level plays a significant role in

breaking chiral symmetry and gives rise to the chiral anomaly. Previous experimental works on such chiral Landau levels are mainly based on three-dimensional Weyl degeneracies coupled with

axial magnetic fields. Their realizations using two-dimensional Dirac point systems, being more promising for future applications, were never experimentally realized before. Here we propose

an experimental scheme for realizing chiral Landau levels in a two-dimensional photonic system. By introducing an inhomogeneous effective mass through breaking local parity-inversion

symmetries, a synthetic in-plane magnetic field is generated and coupled with the Dirac quasi-particles. Consequently, the zeroth-order chiral Landau levels can be induced, and the one-way

propagation characteristics are experimentally observed. In addition, the robust transport of the chiral zeroth mode against defects in the system is also experimentally tested. Our system

provides a new pathway for the realization of chiral Landau levels in two-dimensional Dirac cone systems, and may potentially be applied in device designs utilizing the chiral response and

transport robustness. SIMILAR CONTENT BEING VIEWED BY OTHERS HIGHER-ORDER QUANTUM SPIN HALL EFFECT IN A PHOTONIC CRYSTAL Article Open access 28 July 2020 AXION TOPOLOGY IN PHOTONIC CRYSTAL

DOMAIN WALLS Article Open access 09 August 2024 REALIZATION OF PHOTONIC CHARGE-2 DIRAC POINT BY ENGINEERING SUPER-MODES IN TOPOLOGICAL SUPERLATTICES Article Open access 23 July 2020

INTRODUCTION Linear degeneracies in band structures are singularities in momentum space. These singularities are the sources of topological characteristics in band theory. Typical examples

are the Dirac and Weyl points1,2,3,4,5,6,7,8,9,10,11,12,13, which are linear crossings of two energy bands. They give rise to nontrivial topological invariants, which are associated with a

variety of novel phenomena, such as topologically protected gapless surface states that connect two singular points with opposite chiralities6,8,9,10,13, and singular points with opposite

topological charges can annihilate each other1,11. Since the dynamical equation describing the Weyl or Dirac cones has a form similar to that of relativistic Fermions in quantum field

theory12, such systems are good platforms for emulating relativistic particles and observing the single particle behaviors. Recent advances in inhomogeneous modulations on unit cell

morphologies successfully introduced synthetic gauge fields to the quasi-particles14,15,16,17,18,19,20,21,22,23,24,25,26. As a result, the eigen-energies of bulk states become quantized,

leading to the Landau levels. Associated phenomena such as flat bands and chiral Landau levels have been experimentally detected in photonic and phononic systems15,16,20,22,23,24,25,26.

Different from topological surface states, the chiral Landau level is a one-way propagative bulk state12,20,25,27, which is also topologically protected. In quantum field theory and

condensed matter physics, chiral Landau levels play an important role in breaking chiral symmetry and resultantly induce the chiral anomaly, characterized by the non-conservation of chiral

currents in particle physics and condensed matter physics27,28,29. Prior works on realizing the chiral Landau levels are commonly based on 3D Weyl degeneracies and background magnetic

fields20,21,22,25,26,28,29. Compared with 3D Weyl systems, 2D Dirac systems are more accessible for fabrication, and thus are more promising for future applications. Although the theoretical

approach for realizing chiral Landau levels in 2D Dirac systems has been proposed30, a realistic structure has not been provided and realized in practice. In addition, the linear dispersion

relations, and the transport properties of the zeroth chiral mode have not yet been experimentally observed. In this work, we propose the experimental realization of chiral Landau levels

using a photonic honeycomb system. By breaking local parity-inversion symmetry in each unit cell, an inhomogeneous effective mass is introduced into the Dirac cone, which is equivalent to an

in-plane synthetic magnetic field coupled with the Dirac quasi-particles. As a result, the energy levels become quantized, and in-plane chiral Landau levels arise, which are one-way

propagative and robust against perturbations in the bulk. With the inhomogeneous platform, the band dispersions of Landau levels are experimentally measured. In addition, the robustness of

transport of the zeroth chiral mode is also experimentally tested by introducing disordered defects to the system. RESULTS The linear crossings of two energy bands in 2D systems can always

be described by the Dirac Hamiltonian \({H}_{D}=v({k}_{x}{\sigma }_{1}\pm {k}_{y}{\sigma }_{2})\), where _v_ is the group velocity near the Dirac point, _σ_1,2,3 are Pauli matrices, and

_k__x_ and _k__y_ are the in-plane Block wave vectors relative to the degeneracy point in _x_ and _y_ directions, respectively. A typical way to introduce synthetic gauge fields on the 2D

plane is to tune the position of the degeneracy points in K-space15,16,22,23,24,26,31,32. However, the artificial magnetic field generated in this way is perpendicular to the _xy_ plane,

which results in flat-band Landau levels for such 2D systems. To obtain an in-plane synthetic magnetic field and consequently further induce in-plane dispersive chiral Landau levels, we need

to go beyond the conventional methods. Here, we locally break parity-inversion symmetry29 to introduce an effective mass _m_. This mass term will lift the Dirac degeneracy and open a

bandgap locally (the gap size \(\varDelta \omega =2{|m|}\)). If the gap size varies with position, one obtains an effective Hamiltonian that describes the interaction between the Dirac

quasi-particles and the synthetic gauge field. $${H}^{{\prime} }=v({\hat{k}}_{x}{\sigma }_{1}\pm {\hat{k}}_{y}{\sigma }_{2})+m({\boldsymbol{r}}){\sigma }_{3}$$ (1) We realize the model

Hamiltonian with the non-periodic system illustrated in Fig. 1a, with the unit cell being the honeycomb structure as shown in Fig. 1b. The unperturbed system has both parity-inversion and

time-reversal symmetries (see Fig. 1b), protecting the existence of Dirac cones at the Brillouin zone corners where the energy bands cross linearly at the _K_ and _K’_ points (see Fig. 1c).

Here, the lattice sites A and B are cylinders with a high dielectric constant (_ε_m = 11.8, estimated), and the diameters are denoted by _d__A_ and _d__B_, respectively. The other parts are

filled with air (_ε_air = 1). The whole structure (Fig. 1a) is non-periodic in the _x_ direction, in which _d__A_ varies from 2 to 6 mm from the left to the right of the sample (_x_

direction), and conversely _d__B_ varies from 6 to 2 mm. The periodic boundary condition is applied in the _y_ direction. The sign \(\pm\) in Eq. (1) corresponds to _K_ and _K’_ valleys,

respectively. \({\hat{k}}_{x,y}\) with hats denote the wave vector operators, and in the direction without translational symmetry, \({\hat{k}}_{x}=-i{\partial }_{x}\) is not a good quantum

number. Conversely, in the other direction, the translational symmetry is preserved, and thus \({\hat{k}}_{y}={k}_{y}\) is still a good quantum number. The variation step size between the

same lattice sites is fixed at Δ_d__A_ = 0.1 mm (and Δ_d__B_ = −0.1 mm), and thus _d__A_ = _d__B_ = 4 mm on the line _x_ = 0 of the sample. This graded morphology causes the gap size Δ_ω_ to

be linearly dependent (approximately) on the coordinate _x_ (as shown in Fig. 1a), which is equivalent to adding an effective mass term to the Dirac Hamiltonian that is linear with respect

to _x_ (\(m={ax}\)), and thus Eq. (1) is realized. The photo of part of the fabricated sample for experimental measurement is displayed in Fig. 1d. The 2D Dirac Hamiltonian can be regarded

as a subsystem of a synthetic Weyl system \({H}_{W}=v\left({k}_{x}{\sigma }_{1}\pm {k}_{y}{\sigma }_{2}\right)+\widetilde{k}{\sigma }_{3}\) at \(\widetilde{k}=0\), where \(\widetilde{k}\) is

a virtual wave vector used for constructing the Weyl Hamiltonian, and commutes with _k__x_ and _k__y_ (i.e., [_k__x_, \(\widetilde{k}\)]=0, [_k__y_, \(\widetilde{k}\)]=0). Since the

effective mass also introduces a _σ_3 term in the Hamiltonian [Eq. (1)], it is equivalent to a vector potential in the virtual direction [_A_ = _m_(_x_)], meaning that an effective canonical

momentum operator \(\hat{k}=\widetilde{k}+A\) is introduced into the system. The virtual wave vector \(\widetilde{k}\) should be distinguished from the physical _k__z_ wave vector that

exists in _z_-invariant systems (more discussions are shown in Section 3 of Supplementary Information). Therefore, we expect to see the phenomenon that arises from a synthetic magnetic field

in the _y_ direction (as indicated by the yellow arrow in Fig. 1d), which is determined by the definition of the magnetic field, i.e., the commutation between canonical momenta

\({B}_{y}=[\hat{k},{\hat{k}}_{x}]\). The designed sample preserves translational symmetry in the _y_ direction, and the synthetic magnetic field is also along the _y_ direction. In the

literature concerned with chiral anomaly (also called axial anomaly), this direction is called the axial direction26. The synthetic gauge field will result in the quantization of energy

levels, which are expressed as (see Section 1 of Supplementary Information for derivation details). $${\omega }_{n}=\left\{\begin{array}{c} \chi {\mathrm{sgn}}({B}_{y}) v{k}_{y},{n}=0\\ \pm

\sqrt{{v}^{2}{k}_{y}^{2}+2n{\rm{|}}a{\rm{|}}v},n\ge 1\end{array}\right.$$ (2) Here \({\mathrm{sgn}}({B}_{y})\) is the sign of the synthetic magnetic field, and here it is positive because

_a_ > 0 for our sample. The chirality _χ_ can be \(\pm\) correspond to the modes in the _K_ and _K’_ valleys (see Section 2 in Supplementary Information), respectively. From this

equation, we see that the energy levels are dispersive in the _k__y_ direction (the direction preserves translational symmetry) but independent of the virtual wave vector \(\widetilde{k}\).

The zeroth-order Landau level has a linear dispersion and can have group velocities in +_y_ (up-going) or −_y_ (down-going) directions, which is determined by its projection to the _K_ or

_K’_ valleys, i.e., the chirality of the Dirac point. For the experimental sample, the band structure of the supercell under periodic boundary conditions in the _y_ direction (Fig. 1a) can

be interpreted as the Landau levels under the action of a synthetic gauge field, and the full wave simulation result (near _k__y_ = 0, the blue dot) is plotted in Fig. 1e. Quantized energy

levels can be conspicuously observed from the figure, with the dotted lines show high order Landau levels, and the linearly dispersive zeroth modes are labeled by blue and red lines, which

are affiliated to _K_ and _K’_ valleys, respectively. The analytical expressions of Landau levels (Eq. 2) can well reproduce the salient features of the band structure if the _k__y_ domain

is in the vicinity of the _K_ or _K’_ points (see Section 2 of Supplementary Information for details). Our results show that mini gaps exist between any two adjacent Landau levels in our 2D

system (see Fig. 1e). As predicted by Eq. (2), the gap width can be tuned by the strength of the synthetic magnetic field _B__y_ (equivalent to tuning the gradient Δ_d__A_ or |_a_|).

Increasing |_a_| (or Δ_d__A_) will result in a larger gap width between adjacent energy levels (e.g., the gap between 0th and 1st order levels \(\varDelta {\omega }_{1}=\sqrt{2|a|v}\)), and

conversely, the gap width decreases by decreasing |_a_| (or Δ_d__A_). More numerical results on the gap width and the strength of the synthetic magnetic field are shown in Section 4 of

Supplementary Information. An interesting feature can be deduced from Eq. (2) that the group velocity (\(\partial \omega /\partial {k}_{y}\)) of the zeroth mode is independent of the

strength of the synthetic magnetic field (i.e., |_a_|), meaning that increasing or decreasing the gradient (i.e., the value of Δ_d__A_ or Δ_d__B_) will not change the slope of the linear

dispersion. More numerical results substantiating the relation between the group velocity of zeroth mode and the gradient |_a_| are also shown in Section 4 of Supplementary Information. The

ability to control the field confinement of eigenstates plays a useful role in many applications, including sensing devices32,33. For the Landau levels in our designed system, the field

confinement depends on and hence can be controlled by the strength of the synthetic magnetic field. We take the zeroth mode at _k__y_ = 0 as an example and choose four different gradients

Δ_d__A_ representing different strengths of synthetic magnetic fields _B__y_. Numerical results are shown in Fig. 2, with Fig. 2a–d corresponding to Δ_d__A_ = 0.03 mm, 0.05 mm, 0.07 mm and

0.1 mm, respectively. We see that the high field intensity region decreases as the gradient Δ_d__A_ increases, meaning that increasing the strength of synthetic magnetic field _B__y_ can

lead to stronger field confinement. It is notable that the boundary condition in the _x_ direction does not affect the computed dispersions, because the field of the eigenstates in the

frequency range of interest is confined near _x_ = 0 (see Fig. 2), owing to the fact that the virtual wave vector \(\widetilde{k}\) is zero in our system. Even though the Landau level

dispersions are independent of \(\widetilde{k}\), the localization of eigenstates is \(\widetilde{k}\)-dependent, which is always confined to the vicinity of the area that satisfies

\(\widetilde{k}+A=0\). We note that the Landau levels arising from synthetic gauge fields always pertain to a specific gauge. To experimentally measure the Landau levels, we scan the near

field with our experimental setup (Fig. 3a). To ensure the precision in experimental measurement (see “Materials and methods” for details), we cover the sample with a metal plate, on which a

slit is etched in the middle so that the detector can insert in to measure the near field. We first locate the point source at the bottom center (position S1), and scan the near field along

the slit indicated by the arrow (Fig. 3a). The up-going zeroth mode can be excited as indicated by the numerical result in Fig. 3b. Using a discrete Fourier transformation, the energy level

dispersions can be obtained experimentally, and the Fourier transformed field intensity |_E__z_| as a function of _k__y_ and _f_ is shown in Fig. 3c. The zeroth mode can be observed, which

agree well with solid blue line obtained with the full wave simulations, respectively. The result is also consistent with the theoretical predictions (see Supplementary Fig. S1). The central

frequency deviates slightly from the simulation results as indicated by the figure because the experimental sample is a slab with finite thickness instead of a _z_-invariant system (see

details in “Materials and methods”). We see that for the zeroth-order Landau level, only the upward (+_y_) propagating mode is excited if the source is located at the bottom center. The

dispersion is linear, as expected. By contrast, if we put the source at the top center (position S2 in Fig. 3a), only the zeroth mode that is propagating in the −_y_ direction can be excited

(as indicated by the field distribution in Fig. 3d). This is further confirmed via energy level dispersions in Fig. 3e obtained from a discrete Fourier transformation, where the zeroth mode

only has a negative group velocity. The field distribution |_E__z_| of the zeroth-order chiral mode in _x_ direction is also measured, as shown in Fig. 3f. It is seen that the mode

amplitude is localized near the middle (_x_ = 0) of the sample, consistent with our previous discussions (i.e., due to \(\widetilde{k}=0\)). Since the up-going and the down-going zeroth

chiral mode are affiliated to _K_ and _K’_ valleys respectively, the electromagnetic response of _K_ and _K’_ points are different from each other if the excitation frequency is within the

bandgap between ±1st order Landau levels. Relevant discussions are shown in Section 2 of Supplementary Information. The observed phenomenon complies with the symmetry of the system. Even

though the local parity-inversion symmetry of each unit cell is broken, the overall sample still has a parity-inversion symmetry. The sources at the bottom center and the top center can be

transformed to each other via parity-inversion operations. Correspondingly, and the positively and negatively propagating chiral modes excited by the two sources (as well as _K_ and _K’_

points) are also parity-symmetric to each other. We note that the overall system preserves time-reversal symmetry, and the up-going and the down-going zeroth modes, affiliated to _K_ and

_K’_ valleys, form a time-reversal symmetric pair. However, if one considers the transport for states residing in one single valley, the single zeroth mode is one-way propagative. Such a

consideration is reasonable because inter-valley scattering is usually small. The chiral zeroth Landau level is robust against small perturbations. Next, we provide a more detailed

experimental test for the robustness of transport against imperfections. We first put a point source at the center of the bottom edge of the sample, and at the same time, some defects (4

additional cylinders with a diameter of 3 mm) are introduced at the center of the sample, as shown in Fig. 4a. The zoomed in picture of the defects is shown in the inset. Compared with Fig.

3f, we find that the defects almost block the entire confined region of the field of the zeroth mode, and such defects will result in strong reflections for ordinary propagating modes.

However, from the simulation result in Fig. 4b, we find that even though weak localization of the field strength distribution (|_E__z_|) can be observed at the center of the sample (location

of the defect), the chiral zeroth mode is not impeded by the defect. The reflected field intensity of the down-going zeroth mode is negligibly small compared with the transmitted field

intensity, evidenced by the fact that no interference pattern can be observed. We also performed the 2D Fourier transformation of the field distributions, and the modes at _K’_ valley

(down-going zeroth mode) are almost not excited compared with those at _K_ valley (up-going zeroth mode), even though the defect is introduced (see the inset of Fig. 4b). We next measure the

dispersion of the Landau levels to quantitatively test the reflection strength. By locating the point source at the bottom center, the field distribution (|_E__z_|) in _k__y_-_f_ space can

be resolved experimentally. It is shown that the zeroth mode that is propagating in the −_y_ direction is almost not excited (dashed blue line) compared with that in the +_y_ direction

(solid blue line), showing the negligible impact from the defect, as indicated by the dispersions in Fig. 4c. The experimental results provide solid evidence for the robustness of the

transport of the chiral zeroth Landau level. In Section 5 of the Supplementary Information, we introduce a much larger defect to test the robustness of the zeroth mode, and the same

conclusion can be reached by observing the results in Supplementary Fig. S5. The weak backscattering of the chiral zeroth Landau level can be intuitively understood. Since the zeroth modes

with positive and negative group velocities are affiliated to different valleys (i.e., _K_ and _K’_), the backscattering of the zeroth mode is essentially an inter-valley scattering if the

illumination frequency is inside the gap. Since the _K_ and _K’_ points are widely separated in K-space, the inter-valley coupling is weak. DISCUSSION To summarize, we propose a realistic

system to realize chiral zeroth Landau levels via inhomogeneously breaking local parity-inversion symmetry in 2D Dirac point systems. Such a system is experimentally realized using a

photonic honeycomb lattice system, in which the diameter of lattice sites in each unit cell is tuned to introduce an effective mass that depends linearly on the spatial coordinate. Based on

such a platform, the in-plane chiral Landau level dispersions are experimentally measured. The robustness of the zeroth-order mode is experimentally demonstrated by introducing defects at

the center of the sample. Our proposal extends the realization of chiral zeroth Landau level from 3D systems to 2D systems, which are much easier to fabricate, and are hence more promising

for applications using chiral response34,35,36,37,38. By scaling the sample from the millimeter region to the nanometer region, the scheme may inspire the design of low-loss photonic crystal

components that utilize the robustness of the chiral Landau levels. MATERIALS AND METHODS The sample we fabricated is composed of cylinders with a high refractive index, which are placed at

the lattice sites of a honeycomb lattice. The cylinders are made of yttrium iron garnet with a dielectric constant of 11.8 (estimated), and all have the same height (_h_ = 10 mm). To ensure

the mechanical stability of the system, we fabricated a metal plate, on which round holes are etched with appropriate diameters so that the cylinders can stand steadily. The holes are

etched with a thickness of 2 mm, and thus most parts of the cylinders (8 mm) are outside of the hole and surrounded by air. We then cover the sample with another metal plate, the part

between the two metal plates can well reproduce the designed 2D system. The only difference is that the realistic sample is a 3D system because the thickness is finite (8 mm). The perfect

electric conductor (PEC) boundary condition imposed by the two metal plates ensures that the model profile is uniform along the _z_-direction so that the band structure is consistent with

the ideal 2D system. The finite thickness of the sample only induces a shift of central frequency from 7.84 to 7.9 GHz. The parity-inversion symmetry is not perfect in the experimental

sample, which originates from the fabrication errors. This will induce the difference of central frequencies of the up-going and the down-going zeroth modes, which can be indicated by Fig.

3c, e. The experimental setup is made of a 3D translational platform, equipped with a network analyzer with two ports, one of which acts as a point source to generate electromagnetic waves

and the other acts as the detector. The eigen-modes we measured have the out-of-plane electric field vector (_E__z_), and thus the detector antenna should be perpendicular to the plate so

that the impact from other modes with in-plane electric vector (_E__x,y_) can be excluded (as shown in Fig. 3a, c). In measuring the electric field (_E__z_) distribution, the detector

antenna should be placed within the air region between the metal plates. The Landau level is a 1D dispersion so we only need to measure the field distribution along a line parallel to the

_y_-axis. We thus etched a slit in the middle of the upper plate (as shown in Fig. 3a, c) through which the detector antenna can be inserted. We note that the slit should be thin enough so

that the PEC boundary condition almost remains intact. A thick slit could introduce more noise to the measured result. The detector antenna cannot be inserted into the cylinders, which makes

the 2D measurement more difficult. In Supplementary Fig. S1e, f, the field distribution beyond the first Brillouin zone in K-space is also needed. Therefore, the field of more points within

each unit cell is required to obtain the result. As a consequence, we have to measure the field inside the cylinders (so that we have enough grid points to do a Fourier transform), which

cannot be accomplished by the current setup. We therefore only provide the numerical result of 2D K-space field distribution (Supplementary Fig. S1e, f), and the experimental result cannot

be provided. DATA AVAILABILITY All data are available in the main text and the supplementary information. CODE AVAILABILITY All codes are available in the main text and the supplementary

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  ADS  Google Scholar  Download references ACKNOWLEDGEMENTS This work is supported by the Research Grants Council of Hong Kong through grants AoE/P-502/20, 16307621, 16307420, 16310420 and

Croucher Foundation (CAS20SC01). AUTHOR INFORMATION Author notes * These authors contributed equally: Hongwei Jia, Mudi Wang AUTHORS AND AFFILIATIONS * Department of Physics, the Hong Kong

University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China Hongwei Jia, Mudi Wang, Ruo-Yang Zhang, Jing Hu, Dongyang Wang & Che Ting Chan * Institute for Advanced

Study, the Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China Hongwei Jia * Department of Physics, University of Hong Kong, Hong Kong, China Shaojie

Ma Authors * Hongwei Jia View author publications You can also search for this author inPubMed Google Scholar * Mudi Wang View author publications You can also search for this author

inPubMed Google Scholar * Shaojie Ma View author publications You can also search for this author inPubMed Google Scholar * Ruo-Yang Zhang View author publications You can also search for

this author inPubMed Google Scholar * Jing Hu View author publications You can also search for this author inPubMed Google Scholar * Dongyang Wang View author publications You can also

search for this author inPubMed Google Scholar * Che Ting Chan View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS H.J. and C.T.C. planned the

project. H.J. and M.W. designed the sample and carried out the measurements. H.J., M.W., J.H. and D.W. analyzed the data. H.J., S.M. and R.Y.Z. constructed the theoretical framework. H.J.,

M.W., and C.T.C. wrote the manuscript. All authors contributed to the discussion. CORRESPONDING AUTHORS Correspondence to Hongwei Jia or Che Ting Chan. ETHICS DECLARATIONS CONFLICT OF

INTEREST The authors declare no competing interests. SUPPLEMENTARY INFORMATION SUPPLEMENTAL MATERIAL RIGHTS AND PERMISSIONS OPEN ACCESS This article is licensed under a Creative Commons

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http://creativecommons.org/licenses/by/4.0/. Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Jia, H., Wang, M., Ma, S. _et al._ Experimental realization of chiral Landau levels

in two-dimensional Dirac cone systems with inhomogeneous effective mass. _Light Sci Appl_ 12, 165 (2023). https://doi.org/10.1038/s41377-023-01209-z Download citation * Received: 03 January

2023 * Revised: 06 June 2023 * Accepted: 12 June 2023 * Published: 04 July 2023 * DOI: https://doi.org/10.1038/s41377-023-01209-z SHARE THIS ARTICLE Anyone you share the following link with

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