Bulk-boundary correspondence and singularity-filling in long-range free-fermion chains

TL;DR

This paper develops a systematic method to analyze long-range free-fermion chains in 1D, revealing how topological invariants and edge modes are related through singularities, extending understanding beyond short-range models.

Contribution

It introduces a technique for solving long-range 1D free-fermion models in BDI and AIII classes, linking topological invariants to edge modes via singularities, and generalizes results to broader parameter regimes.

Findings

Edge modes linked to singularities of a complex function.

Finite-size splitting of edge modes depends on the topological winding number.

Results extend to chains with decay exponent <1 and gapless topological chains.

Abstract

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 $\alpha$, 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 $\alpha>1$, linking together the quantized winding invariant, bulk topological string-order parameters and a complete solution of the edge modes. The physics of these chains is elucidated by studying a complex function determined by the couplings of the Hamiltonian: in contrast to the short-range case where edge modes are associated to roots of this function, we find that they are now associated to singularities. A remarkable consequence is that the finite-size splitting of the edge modes depends on the topological winding number, which can be used as a probe of the latter. We furthermore generalise these results by (i) identifying a family of BDI chains with $\alpha<1$ where our results still hold, and (ii) showing that gapless symmetry-protected topological chains can have topological invariants and edge modes when $\alpha -1$ exceeds the dynamical critical exponent.