Robust Topological Degeneracy of Classical Theories

TL;DR

This paper demonstrates that classical models can exhibit topological degeneracy dependent solely on surface topology, challenging the notion that such degeneracy implies topological quantum order, and introduces classical analogs of the Toric Code.

Contribution

It introduces a classical version of the Toric Code model on Riemann surfaces, showing classical degeneracy depends only on topology, and challenges the link between degeneracy and quantum topological order.

Findings

Classical models can have topology-dependent degeneracy.

Topological degeneracy does not necessarily imply quantum order.

Some models exhibit boundary-extensive (holographic) entropy.

Abstract

We challenge the hypothesis that the ground states of a physical system whose degeneracy depends on topology must necessarily realize topological quantum order and display non-local entanglement. To this end, we introduce and study a classical rendition of the Toric Code model embedded on Riemann surfaces of different genus numbers. We find that the minimal ground state degeneracy (and those of all levels) depends on the topology of the embedding surface alone. As the ground states of this classical system may be distinguished by local measurements, a characteristic of Landau orders, this example illustrates that topological degeneracy is not a sufficient condition for topological quantum order. This conclusion is generic and, as shown, it applies to many other models. We also demonstrate that certain lattice realizations of these models, and other theories, display a ground state entropy (and those of all levels) that is "holographic", i.e., extensive in the system boundary. We find that clock and $U(1)$ gauge theories display topological (in addition to gauge) degeneracies.