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

## TL;DR

This paper explores the localization properties of eigenstates at van Hove singularities in a graphene-like lattice model, linking them to excited-state quantum phase transitions and potential experimental realizations with microwave resonators.

## Contribution

It introduces an algebraic model revealing bulk stripe states at van Hove singularities and connects these to excited-state quantum phase transitions, with implications for experimental observation.

## Key findings

- Eigenstates form stripe patterns parallel to zig-zag directions.
- States are interpreted as lines of cell-tilting vibrations.
- Model suggests possible experimental detection in microwave resonators.

## Abstract

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.