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

Area of Science:

• Condensed Matter Physics
• Quantum Mechanics
• Topological Materials

Background:

• The bulk-boundary correspondence explains edge modes in short-range systems.
• Long-range interactions in topological chains lack systematic study.

Purpose of the Study:

• To systematically study topological invariants and edge modes in 1D free-fermion chains with long-range interactions (power-law decay α > 1).
• To develop a technique for solving gapped, translationally invariant models in BDI and AIII symmetry classes.

Main Methods:

• Analyzing a complex function derived from Hamiltonian couplings.
• Linking bulk topological invariants (winding number) to edge mode properties.
• Investigating models with power-law decay exponents α > 1 and α < 1.

Main Results:

• Edge modes in long-range systems are associated with singularities of a complex function, unlike short-range systems where they relate to roots.
• The finite-size splitting of edge modes directly probes the topological winding number.
• Results extend to BDI chains with α < 1 and gapless chains under specific conditions.

Conclusions:

• A novel understanding of the bulk-boundary correspondence for long-range interacting topological systems.
• The finite-size splitting of edge modes serves as a direct experimental probe of topological invariants.
• The framework applies to diverse 1D topological phases, including those with gapless edge states.