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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. 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 generalize these results by (i) identifying a family of BDI chains with α<1 where our results still hold and (ii) showing that gapless symmetry-protected topological chains can have topological invariants and edge modes when α-1 exceeds the dynamical critical exponent.
Sprache
Englisch
Identifikatoren
ISSN: 0031-9007
eISSN: 1079-7114
DOI: 10.1103/PhysRevLett.130.246601
Titel-ID: cdi_proquest_miscellaneous_2832576998
Format
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