Topological Phases on Quantum Trees

TL;DR

This paper develops a theory for topological phases in quantum systems on tree graphs, revealing unique phenomena such as pervasive zero modes and a limited set of stable classes, validated through electronic circuit experiments.

Contribution

It introduces a novel topological classification for quantum trees, including the Su-Schrieffer-Heeger tree, and demonstrates experimental realization in electronic circuits.

Findings

Presence of topological zero modes throughout the system

Only three symmetry classes support stable topological phases

Experimental verification using electronic circuits

Abstract

In this work, we present a theory for topological phases for quantum systems on tree graphs. Conventionally, topological phases of matter have been studied in regular lattices, but also in quasicrystals and amorphous settings. We consider specific generalizations of regular tree graphs, and explore their topological properties. Unlike conventional systems, infinite quantum trees are not finite-dimensional, allowing for novel phenomena. We find a proliferation of topological zero modes present throughout the entire system, indicating that the bulk also acts as a boundary. We then go on to show that only three symmetry classes host stable topological phases in contrast to the usual five symmetry classes per dimension. Finally, we introduce what we call the Su-Schrieffer-Heeger tree which is topologically non-trivial even in the absence of inner degrees of freedom and does not possess any gapped trivial phases. We realize this system in an electronic circuit and show that our theory matches with experiments.