Here ψ † r,R/L represent two-component right and left handed Weyl fermion creation operators, χ z = ±1 labels the chirality of the two Weyl nodes, σ is a vector of Pauli matrices in the pseudospin space and p = −i ∇. A and a denote gauge potentials of the ordinary EM and the chiral field, respectively. We explain below the ori-arXiv:1607.01810v2 [cond-mat.mes-hall]

Electrons in clean macroscopic samples of graphene exhibit an astonishing variety of quantum phases when strong perpendicular magnetic field is applied. These include integer and fractional quantum Hall states as well as symmetry broken phases and quantum Hall ferromagnetism. Here we show that mesoscopic graphene flakes in the regime of strong disorder and magnetic field can exhibit another remarkable quantum phase described by holographic duality to an extremal black hole in two-dimensional anti-de Sitter space. This phase of matter can be characterized as a maximally chaotic non-Fermi liquid since it is described by a complex fermion version of the Sachdev-Ye-Kitaev model known to possess these remarkable properties.

We study the proximity effect between an s-wave superconductor (SC) and the surface states of a Weyl semimetal. An interesting two-dimensional SC forms in such an interface with properties resembling in certain aspects the Fu-Kane superconductor with some notable differences. In a Weyl semimetal with unbroken time reversal symmetry the interface SC supports completely flat Majorana bands in a linear Josephson junction with a π phase difference. We discuss stability of these bands against disorder and propose ways in which they can be observed experimentally.Interfacing topological materials with conventionally ordered states of matter, such as magnets and superconductors, has led to important conceptual advances over the past decade. Notable examples of this approach include the Fu-Kane superconductor [1] that occurs in the interface of a 3D strong topological insulator (STI) and a conventional s-wave SC, the "fractional" quantum Hall effect that arises when STI is interfaced with a magnetic insulator [2][3][4], as well as many interesting phenomena that occur when both SC and magnetic domains are present [5,6]. Rich physics, including Majorana zero modes, also results when the edge of a 2D topological insulator is interfaced with magnets and superconductors [7][8][9]. More recently various exotic phases of quantum matter have been predicted to emerge based on these same ingredients in strongly interacting systems [10][11][12][13][14][15][16][17][18][19][20].In this Letter we explore the physics of the interface between a Weyl semimetal and an s-wave superconductor. Surface states of a Weyl semimetal exhibit characteristic Fermi arcs that terminate at surface projections of the bulk Weyl nodes [21,22]. Such "open" Fermi surfaces are fundamentally impossible in a purely 2D system and we thus expect the resulting SC state to also be anomalous. In this respect the situation is similar to the FuKane superconductor [1] whose existence hinges on the odd number of Dirac fermions present on the surface of an STI. As we shall see there are several notable differences between STI/SC and Weyl/SC interfaces which make the latter a distinct and potentially more versatile platform for explorations of new phenomena.Nondegenerate Weyl points can occur in crystals with broken time reversal symmetry T or broken bulk inversion symmetry P. Recent experimental work reported convincing evidence for Weyl nodes and surface Fermi arcs in T -preserving noncentrosymmetric crystals in the TaAs family of semimetals [23][24][25][26][27][28]. We therefore focus our discussion on the SC proximity effect in this class of materials. When a crystal respects T the minimum number of Weyl points N is 4. This is because under T a Weyl point at crystal momentum Q maps onto a Weyl point at −Q with the same chirality. Since the total chiral charge in the Brillouin zone must vanish there has to be another pair of T -conjugate Weyl nodes with an opposite chirality. We begin by discussing the SC proximity effect in this minimal case with N = 4. We note tha...

A linear Josephson junction mediated by the surface states of a time-reversal-invariant Weyl or Dirac semimetal localizes Majorana flat bands protected by the time-reversal symmetry. We show that as a result, the Josephson current exhibits a discontinuous jump at π phase difference which can serve as an experimental signature of the Majorana bands. The magnitude of the jump scales proportionally to the junction width and the momentum space distance between the Weyl nodes. It also exhibits a characteristic dependence on the junction orientation. We demonstrate that the jump is robust against the effects of non-zero temperature and weak non-magnetic disorder.

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