  We discuss solutions of an algebraic model of the hexagonal lattice
vibrations, which point out interesting localization properties of the
eigenstates at van Hove singularities (vHs), whose energies correspond to
Excited-State Quantum Phase Transitions (ESQPT). We show that these states form
stripes oriented parallel to the zig-zag direction of the lattice, similar to
the well-known edge states found at the Dirac point, however the vHs-stripes
appear in the bulk. We interpret the states as lines of cell-tilting
vibrations, and inspect their stability in the large lattice-size limit. The
model can be experimentally realized by superconducting 2D microwave resonators
containing triangular lattices of metallic cylinders, which simulate
finite-sized graphene flakes. Thus we can assume that the effects discussed
here could be experimentally observed.
