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Research Article

A Unified View of Topological Phase Transition in Band Theory

Figure 3

Linear scaling of TPT. The linear scaling relation between the critical value of average bond length and the reciprocal electron hopping potential () for TPT in all the studied 2D periodic, quasicrystalline, and disorder lattices, including oblique (monoclinic), rectangular (orthorhombic), rhombic or centered rectangular (orthorhombic), trigonal (hexagonal), square (tetragonal), honeycomb, Lieb, decorated-trigonal (DT), semiregular-Archimedean (SA), Penrose-type, and Ammann-Beenker-type (AB) lattices (see Figure S4 in Supplementary Materials). The red “_” and black “+” denote trigonal lattices with random vacancies and thermal disorder, respectively. The data marked by filled (open) symbols are calculated using the power-law (exponentially) decay function for the radial dependence of electron hoppings.