We analyze chiral topological edge modes in a non-Hermitian variant of the 2D Dirac equation. Such modes appear at interfaces between media with different "masses" and/or signs of the "nonHermitian charge". The existence of these edge modes is intimately related to exceptional points of the bulk Hamiltonians, i.e., degeneracies in the bulk spectra of the media. We find that the topological edge modes can be divided into three families ("Hermitian-like", "non-Hermitian", and "mixed"); these are characterized by two winding numbers, describing two distinct kinds of halfinteger charges carried by the exceptional points. We show that all the above types of topological edge modes can be realized in honeycomb lattices of ring resonators with asymmetric or gain/loss couplings.Introduction.-There is presently enormous interest in two groups of fundamental physical phenomena: (i) topological edge modes in quantum Hall fluids and topological insulators [1-3], which are Hermitian, and (ii) novel effects in non-Hermitian wave systems (including PTsymmetric systems) [4][5][6]. Both types of phenomena have been studied in the context of quantum as well as classical waves, and both are deeply tied to the geometrical features of spectral degeneracies. In the Hermitian case, the common degeneracies are Dirac points: linear bandcrossings (generically, in a 3D parameter space), which separate distinct topological phases and mark the birth or destruction of topological edge modes [7,8]. NonHermitian systems, however, exhibit a distinct class of spectral degeneracies known as exceptional points (EPs), which are branch points in a 2D parameter space where the Hamiltonian becomes non-diagonalizable [4,9,10].

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Spintronics-the all-electrical control of the electron spin for quantum or classical information storage and processing-is one of the most promising applications of the two-dimensional material graphene. Although pristine graphene has negligible spin-orbit coupling (SOC), both theory and experiment suggest that SOC in graphene can be enhanced by extrinsic means, such as functionalization by adatom impurities. We present a theory of transport in graphene that accounts for the spin-coherent dynamics of the carriers, including hitherto-neglected spin precession processes taking place during resonant scattering in the dilute impurity limit. We uncover a novel "anisotropic spin precession" (ASP) scattering process in graphene, which contributes a large current-induced spin polarization and modifies the standard spin Hall effect. ASP scattering arises from two dimensionality and extrinsic SOC, and apart from graphene, it can be present in other 2D materials or in the surface states of 3D materials with a fluctuating SOC. Our theory also yields a comprehensive description of the spin relaxation mechanisms present in adatom-decorated graphene, including Elliot-Yafet and D'yakonov-Perel relaxation rates, the latter of which can become an amplification process in a certain parameter regime of the SOC disorder potential. Our work provides theoretical foundations for designing future graphene-based integrated spintronic devices.