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Frequency-dependent topology in multisublattice systems: Analytical theory of dispersive effective Hamiltonians

Seiji Mizuno*

  • *Contact author: mizuno@eng.hokudai.ac.jp
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Phys. Rev. Research 8, 023064 – Published 20 April, 2026

DOI: https://doi.org/10.1103/rmcp-lwsl

Abstract

Topological phases in one-dimensional lattices are usually described by fixed, frequency-independent Hamiltonians, where topology is determined solely by structural parameters. However, in many wave-based systems with internal resonances, effective couplings become intrinsically dispersive and depend on the eigenfrequency itself. Here, we develop an analytical theory of such frequency-dependent topology using a trimer (SSH3) mass-spring lattice as a minimal model. By exactly eliminating the internal sublattice, the three-band SSH3 dynamics are mapped onto an effective two-sublattice Hamiltonian with frequency-dependent inertial and coupling terms. Because the effective parameters depend explicitly on the eigenfrequency, the resulting eigenvalue problem is nonlinear in ω and self-consistently reproduces the full three-band spectrum despite the two-level representation. This mapping reveals that the resonant effective coupling G2(ω) acts as a bifurcation parameter controlling band inversion, redistribution of integer winding numbers, and the emergence of edge-localized modes. We show that a sign reversal of G2(ω) induces a frequency-selective topological transition accompanied by a winding redistribution (ν1,ν2,ν3)=(−1,2,−1) and analytically predictable in-gap edge states governed by closed-form localization criteria. Within this framework, the bulk-edge correspondence is not determined by a fixed Hamiltonian but by the effective Hamiltonian evaluated self-consistently along the bulk dispersion relation. The formulation is further extended to adiabatically modulated lattices, where locally nontrivial Berry curvature and alternating topological charges emerge on the (q,t) torus, while the global Chern number remains neutral. Because dispersive renormalization arising from internal degrees of freedom is ubiquitous in phononic, photonic, and other wave-based lattice systems, the present theory establishes a general analytical framework for topological phases in dispersive multisublattice media beyond conventional static tight-binding descriptions.

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Phys. Rev. Research 8, 023064– Published 20 April, 2026
  • Received 8 January 2026
  • Accepted 30 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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