Orbital susceptibility of T-graphene: Interplay of high-order van Hove singularities and Dirac cones

Square-octagon lattice underlies the description of a family of two-dimensional materials such as tetragraphene. In the present paper we show that the tight-binding model of square-octagon lattice contains both conventional and high-order van Hove points. In particular, the spectrum of the model contains flat lines along some directions composed of high-order saddle points. Their role is analyzed by calculating orbital susceptibility of electrons. We find that the presence of van Hove singularities of different kinds in the density of states leads to strong responses: paramagnetic for ordinary singularities and more complicated for high-order singularities. It is shown that the orbital susceptibility as a function of hoppings ratio $\alpha$ reveals the dia- to paramagnetic phase transition at $\alpha\approx 0.94$. This is due to the competition of paramagnetic contribution of high-order VHS and diamagnetic contribution of Dirac cones. The results for the tight-binding model are compared with low-energy effective pseudospin-1 model near the three band touching point.