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Square-octagon lattice underlies the description of a family oftwo-dimensional materials such as tetragraphene. In the present paper we showthat the tight-binding model of square-octagon lattice contains bothconventional and high-order van Hove points. In particular, the spectrum of themodel contains flat lines along some directions composed of high-order saddlepoints. Their role is analyzed by calculating orbital susceptibility ofelectrons. We find that the presence of van Hove singularities of differentkinds in the density of states leads to strong responses: paramagnetic forordinary singularities and more complicated for high-order singularities. It isshown that the orbital susceptibility as a function of hoppings ratio $\alpha$reveals the dia- to paramagnetic phase transition at $\alpha\approx 0.94$. Thisis due to the competition of paramagnetic contribution of high-order VHS anddiamagnetic contribution of Dirac cones. The results for the tight-bindingmodel are compared with low-energy effective pseudospin-1 model near the threeband touching point.

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