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We discuss solutions of an algebraic model of the hexagonal latticevibrations, which point out interesting localization properties of theeigenstates at van Hove singularities (vHs), whose energies correspond toExcited-State Quantum Phase Transitions (ESQPT). We show that these states formstripes oriented parallel to the zig-zag direction of the lattice, similar tothe well-known edge states found at the Dirac point, however the vHs-stripesappear in the bulk. We interpret the states as lines of cell-tiltingvibrations, and inspect their stability in the large lattice-size limit. Themodel can be experimentally realized by superconducting 2D microwave resonatorscontaining triangular lattices of metallic cylinders, which simulatefinite-sized graphene flakes. Thus we can assume that the effects discussedhere could be experimentally observed.

Van Hove Singularities and Excited-State Quantum Phase Transitions in Graphene-like Microwave Billiards