Topological states and flat bands in exactly solvable decorated Cayley trees

TL;DR

This paper explores the spectral properties of decorated Cayley trees, revealing flat bands and topological edge states analogous to Euclidean lattices, and establishes exact mappings to one-dimensional topological chains.

Contribution

It introduces a novel analysis of decorated Cayley trees, demonstrating flat bands and topological states, with exact mappings to 1D topological chains, expanding understanding of non-Euclidean quantum systems.

Findings

Decorated Cayley trees exhibit flat or nearly flat energy bands.

Flat bands can be interpreted as boundary-localized edge states.

An exact correspondence maps topological edge states to flat-band states in the trees.

Abstract

We derive the full spectrum of decorated Cayley trees that constitute tree analogs of selected two-dimensional Euclidean lattices; namely of the Lieb, the double Lieb, the kagome, and the star lattice. The common feature of these Euclidean lattices is that their nearest-neighbor models give rise to flat energy bands interpretable through compact localized states. We find that the tree analogs exhibit similar flat or nearly flat energy bands at the corresponding energies. Interestingly, such flat bands in the decorated Cayley trees acquire an interpretation that is absent in their Euclidean counterparts: as edge states localized to the inner or the outer boundary of the tree branches. In particular, we establish an exact correspondence between the Lieb-Cayley tree and an ensemble of one-dimensional Su-Schrieffer-Heeger chains, which maps topological edge states on one side of the chains to flat-band states localized in the bulk of the tree, furnishing the flat energy band with a topological stability. Similar mapping to topological edge states or to states bound to edge defects in one-dimensional chains is shown for flat-band states in all the considered tree decorations. We finally show that the persistence of exact flat bands on infinite decorated trees (i.e., Bethe lattices) arises naturally from a covering interpretation of tree graphs. Our findings reveal a rich landscape of flat-band and topological phenomena in non-Euclidean systems, where geometry alone can generate and stabilize unconventional quantum states.