# Extended Bloch theorem for topological lattice models with open boundaries

## TL;DR

This paper derives exact analytical solutions for boundary states in various topological lattice models with open boundaries, revealing the role of spectral mirror symmetry in obtaining full spectra including bulk and boundary states.

## Contribution

It introduces a method based on spectral mirror symmetry to analytically solve for the entire spectrum of topological lattice models with open boundaries, including models previously lacking exact solutions.

## Key findings

- Exact solutions for boundary states in nodal-line semimetals, spin-orbit coupled graphene, and topological insulators.
- Identification of spectral mirror symmetry as key for full spectrum solutions.
- Analytical spectrum derivation for the Lieb lattice with open boundaries in both directions.

## Abstract

While the Bloch spectrum of translationally invariant noninteracting lattice models is trivially obtained by a Fourier transformation, diagonalizing the same problem in the presence of open boundary conditions is typically only possible numerically or in idealized limits. Here we present exact analytic solutions for the boundary states in a number of lattice models of current interest, including nodal-line semimetals on a hyperhoneycomb lattice, spin-orbit coupled graphene, and three-dimensional topological insulators on a diamond lattice, for which no previous exact finite-size solutions are available in the literature. Furthermore, we identify spectral mirror symmetry as the key criterium for analytically obtaining the entire (bulk and boundary) spectrum as well as the concomitant eigenstates, and exemplify this for Chern and $\mathcal Z_2$ insulators with open boundaries of co-dimension one. In the case of the two-dimensional Lieb lattice, we extend this further and show how to analytically obtain the entire spectrum in the presence of open boundaries in both directions, where it has a clear interpretation in terms of bulk, edge, and corner states.