We point out new and previously-unidentified ``finite-size topological phases’’, characterized by topological response signatures associated with the underlying D-dimensional topological invariant co-existing with a quasi-(D-1) bulk / quasi-(D-2)-dimensional boundary correspondence. We present results for the quasi-(2-1)D Chern insulator and quantum spin Hall insulator, including ab initio-derived results for 1T’-WTe2 nanoribbons, and also the quasi-(3-1)D and quasi-(3-2)D strong topological insulator protected by time-reversal symmetry. Finite-size topological phases are therefore expected to be realized for a wide variety of symmetry classes and especially relevant given intense recent interest in van der Waals heterostructure materials.

We will also explore Hamiltonians constructed as tensor products of ``parent’’ Hamiltonians characterizing ``parent’’ topological phases of matter, which combine in a multiplicative way to realize a ``child’’ topological phase. We will discuss a variety of such multiplicative topological phases in this talk, constructed from parent Hopf insulators, Chern insulators, Weyl semimetals, and Kitaev chains, to illustrate how these phases expand our understanding of topological condensed matter.

References:

arXiv: 2212.11300, arXiv: 2301.02134, Cook and Moore, Comm. Phys. 5, 262 (2022), arXiv: 2301.02404, arXiv: 2301.02765