The extraordinary electronic properties of Dirac materials, the two-dimensional partners of Weyl semimetals, arise from the linear crossings in their band structure. When the dispersion around the Dirac points is tilted, the emergence of intricate transport phenomena has been predicted, such as modified Klein tunnelling, intrinsic anomalous Hall effects and ferrimagnetism. However, Dirac materials are rare, particularly with tilted Dirac cones. Recently, artificial materials whose building blocks present orbital degrees of freedom have appeared as promising candidates for the engineering of exotic Dirac dispersions. Here we take advantage of the orbital structure of photonic resonators arranged in a honeycomb lattice to implement photonic lattices with semi-Dirac, tilted and, most interestingly, type-III Dirac cones that combine flat and linear dispersions. The tilted cones emerge from the touching of a flat and a parabolic band with a non-trivial topological charge. These results open the way to the synthesis of orbital Dirac matter with unconventional transport properties and, in combination with polariton nonlinearities, to the study of topological and Dirac superfluids in photonic lattices.

We experimentally reveal the emergence of edge states in a photonic lattice with orbital bands. We use a two-dimensional honeycomb lattice of coupled micropillars whose bulk spectrum shows four gapless bands arising from the coupling of p-like photonic orbitals. We observe zero-energy edge states whose topological origin is similar to that of conventional edge states in graphene. Additionally, we report novel dispersive edge states that emerge not only in zigzag and bearded terminations, but also in armchair edges. The observations are reproduced by tight-binding and analytical calculations. Our work shows the potentiality of coupled micropillars in elucidating some of the electronic properties of emergent 2D materials with orbital bands.Boundary modes are a fundamental property of finitesize crystals. They play an important role in the electronic transport and in the magnetic properties of lowdimensional materials [1][2][3][4]. Their existence has long been related to the microscopic details of the edge of the crystal [5][6][7]. Recent advances in the study of topological physics have revealed that, for topologically nontrivial materials, the existence of surface states is directly related to the properties of the bulk [8][9][10]. This is the case of conduction electrons in graphene [11][12][13], in which the nearest neighbor coupling of the cylindrically symmetric p z orbitals of the carbon atoms gives rise to two bands (here labeled s-bands) crossing in an ungapped spectrum (Dirac cones). The localized edge modes in this system exist for any type of terminations except for armchair [14,15]. They are topologically protected by the chiral symmetry of the honeycomb lattice, and their existence can be predicted by calculating the winding number of the bulk wavefunctions [11][12][13].In 2007, Wu and co-workers proposed an orbital version of graphene by considering a honeycomb lattice with p x,y orbitals in each lattice site [16,17]. The strong spatial anisotropy of the orbitals results in four ungapped bands with distinct features: two bands showing Dirac crossings and two flat bands, which were first reported experimentally in a polariton-based photonic simulator [18]. The interest in this kind of orbital Hamiltonians has taken a new thrust due to the rapid emergence of 2D materials [19], such as black phosphorus [20][21][22] and 2D transition metal dichalcogenides [23], whose bands originate from spatially anisotropic atomic orbitals. Edge states in MoS 2 flakes have been observed [24], and recent works aim at quantifying their impact in the transport properties [25]. Edge states in orbital modes have also been studied theoretically in connection to d-wave superconductivity [11,26] and spin-orbit coupling in superlattices of nanocrystals [27], systems very hard to realize experimentally with tuneable parameters. A photonic simulator of orbital bands, would open the door to the study of the microscopic properties of orbital edge states [28] and the connection to the topological properties of orbital bulk bands. I...