Hermitian and Non-Hermitian Dirac-Like Cones in Photonic and Phononic Structures

Accidental degeneracy plays an important role in the generation of novel band dispersions. Photonic structures that exhibit an accidental Dirac-like conical dispersion at the center of the Brillouin zone can behave like a zero-index material at the Dirac-point frequency, leading to a number of unique features, such as invariant phase in space, wave tunneling, photonic doping and anti-doping, etc. Such a phenomenon has been explored in on-chip structures or three dimensions recently. The introduction of non-Hermiticity into the system via loss or gain could transform the accidental Dirac-like cone into a spawning ring of exceptional points, a complex Dirac-like cone or other unique dispersions. Similar Dirac-like cones and related physics are also observed in phononic structures. This review presents an overview of the accidental-degeneracy-induced Dirac-like cones at the center of the Brillouin zone in both photonic and phononic structures, including the fundamental physics, effective-medium description and experimental demonstration, as well as current challenges and future directions.


INTRODUCTION
Dirac cones or conical dispersions are unique features in some electronic band structures that describe unusual electron transport properties of materials like graphene. The electronic band structure of graphene near the Fermi level can be described by the massless Dirac equation near the six corners of the two-dimensional (2D) hexagonal Brillouin zone at the K and K' points [1,2]. The linear dispersion can be visualized as two cones meeting at the Fermi level at the Dirac point, and the conical dispersion near the Dirac point is usually referred to as Dirac cones. Due to this unusual conical dispersion, graphene exhibits intriguing transport properties [1,2], such as abnormal quantum Hall effect, Zitterbewegung, Klein tunneling, anti-localization, etc.
Besides the structural symmetry, Dirac cones can also be created by accidental degeneracy through engineering the material and geometrical parameters [29][30][31]. In 2011, Huang et al. demonstrated a Dirac cone at the center of the Brillouin zone, i.e., the Γ point with k 0, of a square lattice of 2D dielectric PhC [32]. The physical origin of this Dirac cone is different from the symmetry-protected Dirac cones occurring at high-symmetry points (i.e., K and K' points of a triangular/honeycomb lattice). The symmetry-protected Dirac cone is a result of double degeneracy and carries a π Berry phase. While the Dirac cone at the center of the Brillouin zone is a result of triple degeneracy, i.e., accidental degeneracy of a doubly degenerate mode and a single mode. The linear dispersions created by triply degenerate modes cannot be mapped into the Dirac Hamiltonian and carry no Berry phase. Therefore, the conical linear dispersions with triple degeneracy at the Brillouin zone center induced by accidental degeneracy instead of structural symmetry is called Dirac-like cones [23].
The presence or absence of Dirac-like cones is determined by the combination of mode symmetries [33,34]. We should emphasize that accidental degeneracy can also lead to Dirac cones of double degeneracy in periodic systems without structural symmetry and at almost any k point in the Brillouin zone [35], i.e. beyond the Brillouin zone center, edges and highsymmetry lines. Such accidental-degeneracy-induced Dirac cones in general can be achieved by closing band gaps at a desired k point with a band engineering method [35] or other optimization techniques [36][37][38][39][40].
This review presents an overview of Dirac-like cones from Hermitian and non-Hermitian systems in both photonic and phononic structures. We will introduce the fundamental physics, effective-medium description and experimental demonstration, as well as current challenges and potential directions of future research. Figure 1A shows an example of a 2D PhC that exhibits a Dirac-like cone at the center of the Brillouin zone. The PhC consists of a square lattice of dielectric cylinders. The polarization is transverse-magnetic polarization, with electric field parallel to the cylinder axis. A Diraclike cone, i.e., the triply-degenerate point, occurs at the Γ point. It comprises two linear bands that generate conical dispersions and an additional flat band. The two linear bands correspond to the accidental degeneracy of the electric monopole and magnetic dipole modes, whereas the flat band corresponds to the magnetic dipole mode in a rotated orientation, as shown by the right panel of Figure 1A. The doubly degenerate dipole modes correspond to the fields rotated by 90°with the wave vector perpendicular or parallel to the dipole moment (i.e., transverse or longitudinal dipole mode), as the result of C 4v symmetry of the PhC. The flat band is the consequence of the longitudinal dipole mode whose magnetic field is parallel to the wave vector, which is inaccessible from normal incidence.

PHOTONIC CRYSTALS WITH A DIRAC-LIKE CONE AS ZERO INDEX MATERIALS
In the vicinity of the Γ point where the in-plane wave vector k is small, the band structure of the PhC can be described by an effective Hamiltonian given by first-order degenerate perturbation theory [33,136].
where ω D is the Dirac-point frequency, v g is the group velocity of the linear Dirac dispersion. The effective Hamiltonian has three eigenvalues as The first two eigenvalues indicate the conical linear dispersion in the vicinity of Dirac-point frequency, while third one is related to the intersecting flat band.
The experimental observation of the Dirac-like cone was realized in the microwave regime by Huang et al in 2011 [32] using a square array of alumina rods embedded within a parallelplate waveguide, as schematically shown in Figure 1B. In the experimental implementation, the alumina rods have a finite height of 16 mm. To realize effective 2D PhC with infinite height, aluminum plates, as perfect electric conductor boundaries, are utilized to form a parallel plate waveguide. At the Dirac-point frequency, the PhC was found to behave like an impedancematched ZIM with ε eff μ eff 0. Experimental observation has clearly verified the ZIM-enabled cloaking and lensing effect with this PhC.
The optical PhC with a Dirac-like cone was demonstrated by Moitra et al in 2013 [91]. We note that the implementation at microwave frequencies cannot be directly transplanted to the optical frequencies because the dissipation loss of metal plates is dramatic. Therefore, an out-of-plane configuration by patterning horizontal square array of silicon bars was proposed instead ( Figure 1C). The measured spectrum shows a ZIM-induced transmission peak at the wavelength of 1,409 nm, corresponding to the Dirac-like point. Furthermore, ZIM-enabled applications including angular selectivity of transmission and directional quantum dot emission were experimentally demonstrated.
It is noteworthy that the Dirac-like cone is a consequence of accidental degeneracy rather than structural symmetry. When the system parameters (e.g., cylinder radius or relative permittivity) are changed, the triple degenerate modes at the Dirac point will split into a single monopole mode and doubly degenerate dipole modes, forming a photonic band gap [32]. The band edge points at the Γ point are related to the zero effective parameters, because the phase accumulation on adjacent unit cell boundaries is zero. It is found that the zero permittivity, i.e., ε eff 0, originates from the electric monopole resonance, while the zero permeability, i.e., μ eff 0, originates from the doubly degenerate magnetic dipole resonance [47]. However, due to the quadratic dispersions at band edges, the zero parameter leads to zero group velocity. Interestingly, the accidental degeneracy of the monopole and dipole modes turns the quadratic dispersion into a linearly conical dispersion with ε eff μ eff 0, but with a nonzero group velocity. In addition, due to the ultralow-loss of the dielectric materials, the PhCs with a Dirac-like cone are excellent candidates for the ultralow-loss ZIMs.
ZIMs with a Dirac-like cone and a finite group velocity enable many intriguing applications. In the following, we give three examples of unique applications. The first one is the hybrid invisibility cloak based on the integration of metasurfaces and ZIM ( Figure 2A) [64]. The key functionalities of wavefront tailoring with the metasurfaces and wave energy tunneling with the ZIM are combined together to achieve the invisibility cloak. The second one is a geometry-invariant coherent perfect absorber via doping an impedance-matched ZIM with absorptive defects ( Figure 2B) [59]. The behavior of coherent perfect absorption is independent of the size and shape of the ZIM and the position of the doping defects, because of the zero spatial phase change and uniform electric fields inside the ZIM. The third one is a plane-concave lens made of ZIM at near-infrared wavelengths ( Figure 2C) [139]. Due to the zero spatial phase change across the ZIM, such a plane-concave lens can focus the incident plane wave with ultralow spherical aberration.

NON-HERMITIAN DIRAC-LIKE CONES AND EFFECTIVE MEDIUM DESCRIPTION
The abovementioned Dirac-like cone is based on the assumption of Hermitian PhC. It is reasonable, as the PhC are composed of low-loss dielectric materials. It is interesting to point out the band dispersion of the PhC could be drastically deformed if there exists loss/gain [24,80,136,[151][152][153][154][155]175]. In 2015, Zhen et al experimentally demonstrated that in a photonic crystal slab the radiative loss-induced non-Hermitian perturbation can deform the Dirac-like cone into a 2D flat band enclosed by a ring of EPs [136]. The EP is a singularity in a non-Hermitian system where two or more eigenvalues and their associated eigenfunctions collapse into one eigenvalue or eigenfunction [156][157][158]. In the absence of loss, a Diraclike cone as the result of accidental degeneracy of a doubly degenerate dipole mode and a single quadrupole mode dipole can be obtained at the Γ point. While in the presence of radiative loss, the Dirac-like cone is deformed into a flat band enclosed by a ring of EPs. As shown in Figure 3A, inside the ring of EPs, the real parts of the eigen-frequencies have degenerate flat dispersions, while the imaginary parts split into two branches. Outside the ring of EPs, the situations are just the opposite. The real parts split, while the imaginary parts tend to degeneracy. These unique dispersion characteristics of the non-Hermitian PhC can also be described by an effective Hamiltonian as [33,136].
with complex eigenvalues where γ d is induced by the radiative loss of the dipole modes. It is noted that the quadrupole mode does not radiate owing to its symmetry mismatch with plane waves in free space [136]. The first two eigenvalues indicate a ring of EPs appears at |k| γ d /2v g . Inside the ring, the real parts of eigenvalues are dispersionless and degenerate. While outside the ring, the imaginary parts are dispersionless and degenerate. We note that the radiative loss from the PhC slab can be considered equivalent to the material loss in a 2D PhC. This provides us a simple way to investigate the non-Hermitian properties through engineering the losses in different constituents of the PhC. In 2021, Luo et al proposed a non-Hermitian 2D PhC composed of a square lattice of cylindrical rods [155]. The background medium and/or cylindrical rods contain material loss/gain. Through engineering the material loss/gain, it was discovered that besides the ring of EPs ( Figure 3B), complex Dirac-like cone with conical dispersions in both real and imaginary frequency spectra ( Figure 3C), and the quadratic degeneracy ( Figure 3D) can be realized in non-Hermitian PhCs [155]. Such phenomena show the unique consequences of introducing loss/gain to the PhCs with Diraclike cones.
Moreover, it was found that these non-Hermitian properties can be well explained from the effective medium point of view [155]. In a non-Hermitian PhC involving loss and/or gain, the eigen-frequency becomes a complex value as ω ω R + iω I , and the effective permittivity ε(ω) and permeability μ(ω) are generally complex and dispersive, which can be expressed as ε(ω) ε R (ω) + iε I (ω) and μ(ω) μ R (ω) + iμ I (ω). In a small frequency regime near the Γ point, linear relations between the effective parameters and the eigen-frequency can be assumed as: where A εR , A μ R , A εI and A μ I are unknown coefficients to be determined. When the loss/gain is small, we have ω εI ≈ ω μ I ≈ 0 and ω εR ≈ ω μ R ≈ ω D with ω D being the Dirac-point frequency in absence of loss/gain. In the complex frequency space, the dispersion relation can be expressed as, where k is the in-plane wave vector, and c is the speed of light in free space. Based on the effective medium model with parameters and dispersion relation, besides the well explanation of the occurrence of the EP ring, other unique non-Hermitian phenomena like complex Dirac-like cone with conical dispersions in both real and imaginary frequency spectra, as well as the quadratic degeneracy were predicted [155], as summarized in Table 1. Moreover, we note that the flat band induced by the longitudinal electromagnetic mode in all scenarios can also be accurately predicted by using this non-Hermitian effective medium theory [155].

ON-CHIP APPLICATIONS WITH CLADDING AND BIC TECHNIQUES
For on-chip applications, the PhCs with a Dirac-like cone would suffer from large radiative losses in the out-of-plane direction, because the Dirac-cone dispersion resides above the light line and the transverse dipole mode forming the Dirac-like cone can couple to extended plane waves in free space [30,31]. This would turn the PhC to be non-Hermitian, and hence the Diraclike cone would disappear, hindering its applications as the ultralow-loss ZIM. In this sense, it is very important to solve the issue of radiative loss from real PhC chip structures with finite pillar height [22,[135][136][137][138][139][140][141][142][143][144][145][146][147]. In order to eliminate the out-ofplane radiative loss, two important classes of approaches have been proposed, that is, the utilization of claddings and BIC techniques. A mirror cladding can reflect the leaky wave back down into the PhC slab, so as to eliminate the out-of-plane radiation. In 2015 Li et al fabricated an on-chip PhC slab cladded with gold films to avoid out-of-plane radiation, and a Dirac-like cone was observed at the wavelength of 1,590 nm ( Figure 4A) [138]. However, such an implementation suffers propagation loss due to the conduction loss originating from the gold films. To reduce the conduction loss from the mirror, a dielectric Bragg reflector was proposed ( Figure 4B) [30]. The Bragg reflector can reflect the leaky wave back down into the pillars, where it destructively interferes with the leaky wave below the array, thus eliminating the radiative loss.
Another technique is based on the concept of BIC, which is the photonic mode in the radiation continuum above the light line but is confined with an infinite quality factor [176]. With the BIC, it is possible to create resonances in a PhC slab with a Dirac-like cone that do not radiate [177]. Specifically, through engineering the height of the PhC slab, all the upward/downward out-of-plane radiation destructively interferes, thus forming a resonancetrapped BIC with a high quality factor ( Figure 4C) [140]. Consequently, the radiative loss-induced EP ring would shrink into a Dirac-like cone [140,[145][146][147], which has been experimentally demonstrated at near-infrared wavelengths ( Figure 4D) [147]. The Dirac-like cone is the consequence of the accidental degeneracy of a pair of dipole modes and a single quadrupole mode at the Γ point in [140,147], while is formed due to the accidental degeneracy of a single monopole mode and doubly degenerate dipole modes in [145,146] (Figure 4E). The low-order mode-based design can be better treated as a homogeneous ZIM.
We note that the monopole and higher-order modes besides the dipole modes at the Γ point do not have out-of-plane radiative loss because of their intrinsic mode symmetry. This suggests that it is possible to realize Dirac-like cone through harnessing the mode symmetry of the degenerate modes at the Dirac point. In 2018 minkov et al theoretically proposed a PhC slab with a hexagonal lattice described by the C 6v point group, in which a Dirac-like cone at the Γ point is achieved via the symmetryprotected BIC ( Figure 4F) [144]. For a PhC slab with the C 6v symmetry, at the at the Γ point, modes that belong to the E 2 2D irreducible representation or a one-dimensional irreducible representation do not couple to plane waves in free space, that is, these modes are symmetry-protected BICs. Through engineering the structural parameters of the PhC slab, a Dirac-like cone as the result of accidental degeneracy of modes in the E 2 and the B 1 or B 2 irreducible representations can be obtained [144]. This theoretical prediction was experimentally demonstrated at near-infrared wavelengths recently ( Figure 4G) [147].

THREE-DIMENSIONAL DIRAC-LIKE CONE AND THE ELECTROMAGNETIC VOID SPACE
Previously, Dirac-like cones due to accidental degeneracy are mostly investigated in 2D systems. 3D Dirac-like cones and ZIMs were rarely explored. Actually, the wave behaviors in 2D and 3D ZIM are inherently different as the former obey scalar wave equations while the latter follow vector wave equations. As a result, some physical effects observed in 2D ZIM, such as photonic doping [57][58][59][60][61], are absent in 3D ZIM. Instead, when a 3D impedance-matched ZIM contains random defects, unusual percolation of electromagnetic waves could appear [62]. In 2015 Luo et al theoretically demonstrated that there exists an unusual type of percolation threshold which, unlike normal percolation theory, is induced by the long-range connectivity of the defects in the transverse direction [62] ( Figure 5A). It was discovered that below the percolation

Characteristics of parameters
Features of band structures Flat band ω εR , ω μ R ω εI , ω μ I Real parts Imaginary parts Case 1 Figure 3B ω εR ω μ R ω D ω εI ≠ ω μ I Dispersion-less within EP ring Ring of EPs Frontiers in Physics | www.frontiersin.org February 2022 | Volume 10 | Article 845624 threshold, the transmittance is always unity, irrespective of the material, shape and size of the embedded inclusions. The electromagnetic waves can squeeze through the gaps between random defects. While beyond the threshold, the transmittance will generally suffer a sharp reduction, and become strongly dependent on the configuration of defects ( Figure 5B). The 3D PhC with a Dirac-line cone dispersion offers a platform to explore the intriguing features of 3D ZIM. A theoretical design was investigated by Chan in 2012 [148]. The PhC consists of a simple cubic lattice of core-shell spheres. The core is a perfect electric conductor, and the shell is dielectric. From Figure 5C Dirac-like cone as the consequence of sixfold degenerate modes is observed at the Γ point. Both the effective permittivity and permeability are zero, i.e., ε eff μ eff 0, at the Dirac-point frequency. However, the design is sensitive to the parameters and the fabrication is quite challenging. In 2021, Xu et al [150] experimentally realized the Dirac-like cone by employing another design of a 3D PhC composed of dielectric meshes. The dielectric meshes are orthogonally aligned along the x, y, and z directions, and intersect at the center of the unit cell ( Figure 5D). At the Γ point, a sixfold accidental degeneracy of electric and magnetic dipole modes results in a Dirac-like cone. The microwave experiment demonstrated the nearperfect transmission of electromagnetic waves when there exist defects inside the PhC (i.e., effective 3D impedancematched ZIM) below the percolation threshold. Such an extraordinary phenomenon that the transmission through an impedance-matched ZIM is independent of the embedded impurities may be referred to as an impurityimmunity or anti-doping effect [63,150]. We emphasize that the 3D PhC of ε eff μ eff 0 represents an interesting electromagnetic void space that is generated by expanding an infinitesimal point to a finite volume of space using transformation optics [178][179][180]. Such an electromagnetic FIGURE 4 | (A) Scanning electron microscopy image of an on-chip PhC with a Dirac-like cone, and the schematic of unit cell. The cladding gold films are used to eliminate the out-of-plane radiation. Reproduced with permission from Ref. [138]. (B) Schematic of Bragg reflector design used to reflect the leaky wave back down into the pillars, where it destructively interferes with the leaky wave below the array. Reproduced with permission from Ref. [30]. (C) Extent of the ring of EPs in the ΓX direction versus the thickness of the PhC slab, which is illustrated by the inset. The radiative losses can be reduced using resonance-trapped BIC. Reproduced with permission from Ref. [140]. (D) Scanning electron microscopy images of a PhC slab with resonance-trapped BIC. A Dirac-like cone is formed due to the accidental degeneracy of a pair of dipole modes and a single quadrupole mode. Reproduced with permission from Ref. [147]. (E) Schematic of an on-chip PhC with resonancetrapped BIC. A Dirac-like cone is formed due to the accidental degeneracy of a single monopole mode and doubly degenerate dipole modes. Reproduced with permission from Ref. [146]. (F) Radiative quality factors for three photonic bands of a PhC slab illustrated by the right panel. A Dirac-like cone dispersion is obtained via symmetry-protected BIC without out-of-plane radiation. Reproduced with permission from Ref. [144]. (G) Scanning electron microscopy images of the PhC slab with symmetry-protected BIC. Reproduced with permission from Ref. [147].
Frontiers in Physics | www.frontiersin.org February 2022 | Volume 10 | Article 845624 void space has many interesting features that are absent in any other electromagnetic media.
In acoustics, the two key physical parameters that are essential for the propagation of the sound wave in materials are mass density and bulk modulus. Through carefully designing the two key physical parameters, microstructure and dimension parameters of PnCs, the Dirac-like cones at the center of the Brillouin zone can be obtained, also showing the effective zero refraction index property (i.e., simultaneous zero mass density and zero reciprocal bulk modulus). An experimental visualization of the Dirac-like cone in a 2D PnC was realized by Dubois et al in 2017 [166]. The PnC is a square lattice of blind holes ( Figure 6A). Three bands are degenerate at the Dirac point, which correspond to a monopole mode and doubly degenerate dipole modes. Through measuring pressure field radiated by acoustic point source embedded in PnC, the directive emission enabled by zero refractive index was observed at the Dirac-point frequency. The 3D PnC exhibiting a Dirac-like cone was experimentally demonstrated by Xu et al in 2020 ( Figure 6B) [174]. The unit cell of the PnC comprises three aluminum rods in air. A Diraclike cone composed of fourfold degenerate modes occurs at the Γ point. The effective zero refractive index property at the Dirac-point frequency was demonstrated through investigating of wave tunneling effect in bending waveguides in both simulations and experiments, which is referred to as an acoustic "periscope".
In solids, both longitudinal and transverse waves exist, and the wave propagation properties are characterized by more parameters, such as the shear modulus. Through engineering the characteristic and dimension parameters, it is also possible to obtain Dirac cones in a PnC supporting both longitudinal and transverse waves. For example, Liu et al designed a 2D triangularlattice PnC consisting of rubber cylinders embedded in silicon host, exhibiting a double Dirac cone at the Γ point as the consequence of fourfold degenerate modes ( Figure 6C) [163]. Such a PnC possesses simultaneous zero effective refractive indices for both longitudinal and transverse waves at the Dirac  Figure 6D) [165]. Threefold

SUMMARY AND OUTLOOK
We have reviewed the study of Dirac-like cones in both PhCs and PnCs, from Hermitian to non-Hermitian systems and from 2D to 3D. PhCs/PnCs with a Dirac-like cone at the center of the Brillouin zone behave as effective ZIM at the Dirac-point frequency, providing a remarkable platform for the ultralowloss ZIMs and the ZIM-based applications. Due to this unique feature, the Dirac-like cones have attracted considerable attention from theoretical investigation of underlying physics to experimental verification and application exploration. Despite the above advances, there are some remaining important open questions are worth exploring. For instance, the fast inverse design of Dirac-like cones at will is an interesting question. The inverse design problem, meaning the direct retrieval of the proper structure for the desired optical performance, requires exploration of a large degree of freedom in the design space, and hence is very challenging and timeconsuming [36,181]. Recently, topology optimization, as a large-scale computational technique employing powerful gradient-based numerical algorithms, has been applied to inversely design Dirac-like cones [36][37][38][39][40]. The topology optimization is a rule-based approach containing iterative searching steps in a case-by-case manner, usually relying on numerical simulations in each step to produce intermediate results that help to modify the searching strategy [181]. This process would be very time-consuming especially when dealing with complicated 3D structures. Deep learning, as a data-driven methodology, could overcome complicated design problems with growth of structural complexity and higher degree of freedom through training the artificial neural network [36,181]. A trained neural network can be used as a fast, general purpose predictor of optical and electromagnetic responses of complicated 3D structures, and is particularly efficient in solving notoriously difficult inverse problems in nanophotonics [182][183][184][185]. Another way is the utilization of the systematic band engineering method that has realized Dirac cones at arbitrary k point in the Brillouin zone of PhCs without symmetry [35]. With such advanced designing techniques, customized Dirac and Dirac-like cones can realized in complex PhCs/PnCs.
Another interesting topic is the realization of Dirac-like cones or ZIMs at the deep-subwavelength scale. One of the most important features of PhCs/PnCs with Dirac-like cones is that they can mimic effective ZIM. However, most of the PhCs/PnCs are composed of periodic structures, whose lattice constant is in the half-wavelength scale. Such a bulky size hinders many applications that require deep-wavelength ZIM, including tunneling effect in deep-subwavelength channels [48][49][50] and arbitrary control of energy flux in the deep-subwavelength scale [52], etc. The realization of PhCs/PnCs with Dirac-like cones in the deep-subwavelength scale is important to expand their application scenarios.

AUTHOR CONTRIBUTIONS
YL and JL conceived the idea and wrote the draft of this review.