Exponentially growing bulk Green functions as signature of nontrivial non-Hermitian winding number in one dimension

2020/07/14 by Heinrich-Gregor Zirnstein, Bernd Rosenow · 29 citations
Mathematics · Physics and Astronomy · #Boundary (topology) #Boundary value problem #Combinatorics #Exponential growth #Hamiltonian (control theory) #Hermitian matrix #Mathematical analysis #Mathematics #Periodic boundary conditions #Physics #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Topological Materials and Phenomena #Topology (electrical circuits) #Winding number #cond-mat.mes-hall #math-ph #math.MP #physics.optics

paper · pdf · doi:10.1103/physrevb.103.195157

published in Physical review. B./Physical review. B 103(19) (American Physical Society) · 14 pages, 4 figures. Part of a joint submission with arXiv:1901.11241

arxiv created 2020/07/14 · openalex created_date 2020/07/23 · openalex publication_date 2021/05/28 · arxiv updated 2021/06/02 · openalex updated_date 2026/08/05

Abstract

A nonzero non-Hermitian winding number indicates that a gapped system is in a nontrivial topological class due to the non-Hermiticity of its Hamiltonian. While for Hermitian systems nontrivial topological quantum numbers are reflected by the existence of edge states, a nonzero non-Hermitian winding number impacts a system's bulk response. To establish this relation, we introduce the bulk Green function, which describes the response of a gapped system to an external perturbation on timescales where the induced excitations have not propagated to the boundary yet, and show that it will grow in space if the non-Hermitian winding number is nonzero. Such spatial growth explains why the response of non-Hermitian systems on longer timescales, where excitations have been reflected at the boundary repeatedly, may be highly sensitive to boundary conditions. This exponential sensitivity to boundary conditions explains the breakdown of the bulk-boundary correspondence in non-Hermitian systems: topological invariants computed for periodic boundary conditions no longer predict the presence or absence of boundary states for open boundary conditions.

Citations

• Bulk-boundary correspondence for non-Hermitian Hamiltonians via Green functions
• Classification of topological quantum matter with symmetries
• Spawning rings of exceptional points out of Dirac cones
• Non-Abelian gauge field theory of the spin-orbit interaction and a perfect spin filter
• Topological insulators and superconductors
• Quantized Hall Conductance in a Two-Dimensional Periodic Potential
• Topological Insulators
• Non-Hermitian Boundary Modes
• Observation of bulk boundary correspondence breakdown in topolectrical circuits
• Observation of non-Hermitian bulk-boundary correspondence in quantum dynamics
• Exceptional topology of non-Hermitian systems
• Topological Origin of Non-Hermitian Skin Effects

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