Long-range topological insulators and weakened bulk-boundary correspondence

Abstract

We investigate the appearance of new types of insulators and superconductors in long-range (LR) fermionic quantum systems. These phases are not included in the famous ‘ten-fold way classification’ (TWC), valid in the short-range (SR) limit. This conclusion is obtained analysing at first specific one-dimensional models, in particular their phase diagrams and entanglement properties. The LR phases are signalled, for instance, by the violation of the area-law for the von Neumann entropy and by a corresponding peculiar entanglement spectrum (ES). Later on, the origin of the deviations from the TWC is investigated from a more general point of view and in any dimension, showing that it is related with the presence of divergences occurring in the spectrum, due to the LR couplings. A satisfying characterization for the LR phases can be achieved, at least for one-dimensional quantum systems, as well as the definition of a nontrivial topology for them, resulting in the presence of massive edge states, provided a careful evaluation of the LR contributions. Our results allows to infer, at least for one-dimensional models, the weakening of the bulk-boundary correspondence, due to the important correlations between bulk and edges, and consequently to clarify the nature of the massive edge states. The emergence of this peculiar edge structure is signalled again by the bulk ES. The stability of the LR phases against local disorder is also discussed, showing notably that this ingredient can even strengthen the effect of the LR couplings. Finally, we analyse the entanglement content of the paradigmatic LR Ising chain, inferring again important deviations from the SR regime, as well as the limitations of bulk-boundary (tensor-network based) approaches to classify LR spin models.

Keywords

• long-range couplings
• topological insulators
• bulk-boundary correspondence
• edge modes
• quantum entanglement