Magnetic Ordering, Anomalous Lifshitz Transition, and Topological Grain Boundaries in Two-Dimensional Biphenylene Network.

We study the electronic properties of a new planar carbon crystal formed through networking biphenylene molecules. Novel electronic features among carbon materials such as zone-center saddle point and peculiar type-II Dirac fermionic states are shown to exist in the low-energy electronic spectrum. The type-II state here has a nearly flat branch and is close to a transition to type I. Possible magnetic instabilities related to low-energy bands are discussed. Furthermore, with a moderate uniaxial strain, a pair of Dirac points merge with the zone-center saddle point, realizing concurrent Lifshitz transitions of van Hove singularity as well as pair annihilation of the Dirac fermions. A new effective Hamiltonian encompassing all distinctive low-energy states is constructed, revealing a finite winding number of the pseudospin texture around the Dirac point, quantized Zak phases, and topological grain boundary states.