Geometry-Induced Effective Tight-Binding Hamiltonians for Phononic Systems: Mode Conversion and SSH-Like Physics
Abstract
We present a framework that maps harmonic vibrational systems onto effective multi-orbital tight-binding Hamiltonians, establishing a direct correspondence between phononic degrees of freedom and graph-based lattice models. Within this mapping, the Cartesian displacement components of each mass become internal orbitals, while elastic interactions generate effective onsite energies and hopping amplitudes determined by the geometry of the system. This representation enables the application of numerical and conceptual tools originally developed for electronic transport to the study of phononic systems. The method is first applied to a monoatomic chain containing an angular bend. For this benchmark system, analytical expressions for the polarization-resolved transmission and reflection probabilities are derived and compared against calculations performed using a recursive scattering-matrix method within the mapped tight-binding representation. Excellent agreement is obtained, validating the mapping and demonstrating its ability to describe geometry-induced mode conversion and the influence of evanescent states. We then investigate zigzag chains, where alternating bond orientations generate effective dimerized Hamiltonians reminiscent of the Su--Schrieffer--Heeger (SSH) model. Although the uniform zigzag chain develops a spectral gap, no localized edge states are observed. We show that modified boundary parameters generated by the phononic mapping suppress the edge-state formation expected from the ideal SSH picture. Introducing a geometric domain wall restores localized domain-wall states inside the gap, which give rise to resonant transmission channels across an otherwise insulating frequency window. The resonance energies remain robust against system-size variations and the corresponding transmission approaches unity.
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BibTeX
@misc{gonzalez2026geometryinduced,
title = {{Geometry-Induced Effective Tight-Binding Hamiltonians for Phononic Systems: Mode Conversion and SSH-Like Physics}},
author = {José E. González and Carlos Ramírez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29026},
eprint = {2609.29026},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29026},
}
RIS
TY - UNPB
TI - Geometry-Induced Effective Tight-Binding Hamiltonians for Phononic Systems: Mode Conversion and SSH-Like Physics
AU - José E. González
AU - Carlos Ramírez
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29026
UR - https://arxiv.org/abs/2609.29026
AB - We present a framework that maps harmonic vibrational systems onto effective multi-orbital tight-binding Hamiltonians, establishing a direct correspondence between phononic degrees of freedom and graph-based lattice models. Within this mapping, the Cartesian displacement components of each mass become internal orbitals, while elastic interactions generate effective onsite energies and hopping amplitudes determined by the geometry of the system. This representation enables the application of numerical and conceptual tools originally developed for electronic transport to the study of phononic systems. The method is first applied to a monoatomic chain containing an angular bend. For this benchmark system, analytical expressions for the polarization-resolved transmission and reflection probabilities are derived and compared against calculations performed using a recursive scattering-matrix method within the mapped tight-binding representation. Excellent agreement is obtained, validating the mapping and demonstrating its ability to describe geometry-induced mode conversion and the influence of evanescent states. We then investigate zigzag chains, where alternating bond orientations generate effective dimerized Hamiltonians reminiscent of the Su--Schrieffer--Heeger (SSH) model. Although the uniform zigzag chain develops a spectral gap, no localized edge states are observed. We show that modified boundary parameters generated by the phononic mapping suppress the edge-state formation expected from the ideal SSH picture. Introducing a geometric domain wall restores localized domain-wall states inside the gap, which give rise to resonant transmission channels across an otherwise insulating frequency window. The resonance energies remain robust against system-size variations and the corresponding transmission approaches unity.
ER -