Abstract
Topologically nontrivial phases have recently been reported on self-similar structures. Here we investigate the effect of local structure, specifically the role of the coordination number, on the topological phases on self-similar structures embedded in two dimensions. We study a geometry dependent model on two self-similar structures having different coordination numbers, constructed from the Sierpinski gasket. For different nonspatial symmetries present in the system, we numerically study and compare the phases on both structures. We characterize these phases by the localization properties of the single-particle states, their robustness to disorder, and by using a real-space topological index. We find that both structures host topologically nontrivial phases and the phase diagrams are different on the two structures. This suggests that, in order to extend the present classification scheme of topological phases to nonperiodic structures, one should use a framework which explicitly takes the coordination of sites into account.
Highlights
After the discovery of the quantum Hall effect, the study of topological phases has been one of the leading research areas in condensed matter physics
We have explored the properties of a geometry dependent Hamiltonian on two different finite fractal structures (SG-3 and Sierpinski gasket (SG)-4) which only differ in the way the sites are coordinated
In the regime t = λ = 0, where only charge-conjugation symmetry is present, the half-BHZ model can host both topologically trivial and nontrivial phases characterized by a nonzero real-space Chern number, on both structures
Summary
After the discovery of the quantum Hall effect, the study of topological phases has been one of the leading research areas in condensed matter physics. The matrix elements of the corresponding Bloch Hamiltonian Htb(k), which essentially determine the band topology, encode the information about the local structure of the lattice as they involve a sum over all nearest neighbors. This is how local properties like coordination comes into the picture. On some two-dimensional lattices, the graph of the model, formed by identifying the sites as the vertices and the nonzero hoppings as the edges, forms a regular tiling of the two-dimensional space For such cases, the coordination number is uniquely determined by the crystal symmetry and the coordination number is not a separate variable that could influence the topological properties. V we conclude with a summary of our results and discuss some of the remaining open questions on the subject
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