Topological insulators, which are insulative in bulk while conductive on surfaces, are protected by the nontrivial topology of electronic band structures. This concept has been extended to photonic, phononic, and mechanical systems. Recently, topological mechanical lattices based on static deformation have been uncovered. However, their topological behaviors are associated with quadratic eigenvalue problems, and they have been explored in an ad hoc manner; a unified Hamiltonian has yet to be developed. In this paper, a topologically effective Hamiltonian is proposed for static topological lattices. Based on this, a general approach for constructing topological lattices with specific topological classifications is developed. The topological, Hermitian, or non-Hermitian characteristics of monomer and dimer lattices are explored for various conditions of spatial periodicity, disorder, and long-range interactions. It is shown that the topological properties of static lattices can be faithfully predicated by the proposed effective Hamiltonian. This work bridges the well-established tight-binding models of condensed-matter physics and static topological lattices, enabling the classification of static topological lattices using the existing framework of topological band theories.

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• Mechanical deformation
• Mechanical metamaterials
• Topological materials
• Tight-binding model