Bulk-Boundary Correspondence and Singularity-Filling in Long-Range Free-Fermion Chains.

The bulk-boundary correspondence relates topologically protected edge modes to bulk topological invariants and is well understood for short-range free-fermion chains. Although case studies have considered long-range Hamiltonians whose couplings decay with a power-law exponent α, there has been no systematic study for a free-fermion symmetry class. We introduce a technique for solving gapped, translationally invariant models in the 1D BDI and AIII symmetry classes with α>1, linking together the quantized winding invariant, bulk topological string-order parameters, and a complete solution of the edge modes.

Topology and Edge Modes in Quantum Critical Chains.

We show that topology can protect exponentially localized, zero energy edge modes at critical points between one-dimensional symmetry-protected topological phases. This is possible even without gapped degrees of freedom in the bulk-in contrast to recent work on edge modes in gapless chains. We present an intuitive picture for the existence of these edge modes in the case of noninteracting spinless fermions with time-reversal symmetry (BDI class of the tenfold way).