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.
Highlights
The study of topological phases of matter experienced a growing interest in the last decades
In favour of this thesis, we identify some evidences, elaborating some results from previous works: (i) The appearance of massive edge states [29, 39] in itself already signals a breakdown of the ten-fold way classification’ (TWC), where only massless edge modes are expected
This possibility should be valid for LR quantum systems with higher dimensionality, no additional obstructions seem to appear in these conditions
Summary
The study of topological phases of matter experienced a growing interest in the last decades. Using first specific one-dimensional free fermionic examples and later on performing a more general formal discussion (not limited to one-dimensional cases), we show that LR insulating or superconducting phases can emerge, in some cases hosting massive edge states, when the bulk spectrum manifests a particular sub-set of the mentioned singularities The appearance of the latter singularities parallels the area-law violation for the von Neumann entropy, still in the presence of a nonvanishing bulk mass gap, and a peculiar distribution for entanglement spectrum (ES). Further details, mentioned in the main text but not immediately required to understand it, are given in the appendices A–D
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