We construct a tight-binding model realizing one pair of Weyl nodes and three distinct Weyl semimetals. In the type-I (type-II) Weyl semimetal, both nodes belong to type-I (type-II) Weyl nodes. In addition, there exists a novel type, dubbed "hybrid Weyl semimetal", in which one Weyl node is of type-I while the other is of type-II. For the hybrid Weyl semimetal, we further demonstrate the bulk Fermi surfaces and the topologically protected surface states, analyze the unique Landau level structure and quantum oscillation, and discuss the conditions for possible material realization.Introduction.-Since the theoretical and experimental discovery of topological insulator 1,2 , the study of topological states of matter has become one of the major topics in condensed matter physics. Apart from the triumphs of systems with full energy gaps, the concept and discovery of Weyl semimetals (WSMs) have stimulated intensive activities in understanding the band topology for gapless systems [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] . A WSM, in the original setting, has linear conic band crossings at the Fermi energy 5 . These band crossing points, i.e., the "Weyl nodes", behave like sources and sinks of the Berry curvature in the momentum space and are topologically protected. Based on the bulk-boundary correspondence, the surface state of a WSM takes the form of Fermi arc that connects a pair of Weyl points with opposite chiralities 5 .

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The chiral p-wave superconductor/superfluid in two dimensions (2D) is the simplest and most robust system for topological quantum computation [1,2] . Candidates for such topological superconductors/superfluids in nature are very rare. A widely believed chiral p-wave superfluid is the Moore-Read state in the ν = 5 2 fractional quantum Hall effect [3,4], although experimental evidence are not yet conclusive [5]. Experimental realizations of chiral p-wave superconductors using quantum anomalous Hall insulator-superconductor hybrid structures have been controversial [7,8]. Here we report a new mechanism for realizing 2D chiral p-wave superconductors on the surface of 3D s-wave superconductors that have a topological band structure and support superconducting topological surface states (SC-TSS), such as the iron-based superconductor Fe(Te,Se) [9]. We find that tunneling and pairing between the SC-TSS on the top and bottom surfaces in a thin film or between two opposing surfaces of two such superconductors can produce an emergent 2D time-reversal symmetry breaking chiral topological superconductor. The topologically protected anyonic vortices with Majorana zero modes as well as the chiral Majorana fermion edge modes (χMEMs) can be used as a platform for more advantageous non-abelian braiding operations. We propose a novel device for the CNOT gate with six χMEMs, which paves the way for fault-tolerant universal quantum computing.

We present emergent ergosurfaces (ES) in a transition layer between type-I and type-II Weyl semimetals (WSMs). The Hawking temperature defined by the surface gravity at the acoustic event horizon which coincides with the ES when the tangent velocity v is small is in a measurable interval. On the type-II WSM side, i.e., inside the ES when v is large, the motion of the quasiparticles may be chaotic after a critical surface as they are governed by an effective inverted oscillator potential induced by the mismatch between the type-I and type-II Weyl nodes. In a relevant lattice model, we calculate out of time ordered correlators (OTOCs). We find that the OTOCs are fast scrambling with a quantum Lyapunov exponent in high temperature and the scrambling is saturated after the Ehrenfest time. This confirms the quantum chaotic behavior. I. INTRODUCTIONS

In the absence of Lorentz symmetry, the pseudospin-1 counterpart of the Weyl fermion (feroton) with linear dispersions and an exact flat band can emerge in condensed matter systems. The flat band branch of the feroton is equivalent to the longitudinal photon in Maxwell theory, which is a redundant degree of freedom due to the emergent (fermionic) gauge symmetry. Upon coupling to an external magnetic field, the fermionic gauge symmetry is broken and the flat band ferotons become gapless excitations characterized by Landau level indices (n>1). In the long wavelength limit, these gapless modes are of the opposite chirality to the chiral anomaly related zero Landau level, which leads to much more plentiful magnetic transport properties. To further explore the novel properties of these gapless modes, we investigate the quantum oscillation through a generalized Lieb lattice model. We find an extra oscillating behavior which indicates the existence of these exotic gapless modes. We collect known ab initio calculation data from the literature and discuss the possibility of realizing the semi-metallic feroton gas in real materials.