Classification of accidental band crossings and emergent semimetals in two-dimensional noncentrosymmetric systems

We classify all possible gap-closing procedures which can be achieved in\ntwo-dimensional time-reversal invariant noncentrosymmetric systems. For\nexhaustive classification, we examine the space group symmetries of all 49\nlayer groups lacking inversion taking into account spin-orbit coupling.\nAlthough a direct transition between two insulators is generally predicted to\noccur when a band crossing happens at a general point in the Brillouin zone, we\nfind that a variety of stable semimetal phases with point or line nodes can\nalso arise due to the band crossing in the presence of additional crystalline\nsymmetries. Through our theoretical study, we provide, for the first time, the\ncomplete list of nodal semimetals created by a band inversion in\ntwo-dimensional noncentrosymmetric systems with time-reversal invariance. The\ntransition from an insulator to a nodal semimetal can be grouped into three\nclasses depending on the crystalline symmetry. Firstly, in systems with a\ntwo-fold rotation about the z-axis (normal to the system), a band inversion at\na generic point generates a two-dimensional Weyl semimetal with point nodes.\nSecondly, when the band crossing happens on the line invariant under a two-fold\nrotation (mirror) symmetry with the rotation (normal) axis lying in the\ntwo-dimensional plane, a Weyl semimetal with point nodes can also be obtained.\nFinally, when the system has a mirror symmetry about the plane embracing the\nwhole system, a semimetal with nodal lines can be created. Applying our\ntheoretical framework, we identify various two-dimensional materials as\ncandidate systems in which stable nodal semimetal phases can be induced via\ndoping, applying electric field, or strain-engineering, etc.\n