Seeking Topological Phases in Fractals
As we had seen in Table 1.1, the classification of topological systems is described by a tenfold scheme and a periodicity in spatial dimensions d. Notions of integral dimensions and bulk-boundary correspondence lies at the heart of the topological band theory [1, 2, 3, 4]. A nontrivial invariant calculated for a periodic system signals existence of robust boundary states for the same system with a boundary. This correspondence is the progenitor of formulations of various invariants such as TKNN invariant (Chern number) , the Pfaffian and others (for a recent review see ) which lead to exotic boundary physics. However, not every system has a well defined “bulk” or “boundary”. Neither does every system have a well defined dimension. Is there a notion of a topological state in such systems? If yes, how can they be characterized?
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