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In this work, we examine the topological phases of the spring-mass lattices when the spatial inversion symmetry of the system is broken and prove the existence of edge modes when two lattices with different topological phases are glued together. In particular, for the one-dimensional lattice consisting of an infinite array of masses connected by springs, we show that the Zak phase of the lattice is quantized, only taking the value 0 or . We also prove the existence of an edge mode when two semi-infinite lattices with distinct Zak phases are connected. For the two-dimensional honeycomb lattice, we characterize the valley Chern numbers of the lattice when the masses on the lattice vertices are uneven. The existence of edge modes is proved for a joint honeycomb lattice formed by gluing two semi-infinite lattices with opposite valley Chern numbers together.
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A Gelfand duality for continuous lattices
We prove that the category of continuous lattices and meet- and directed join-preserving maps is dually equivalent, via the hom functor to [0,1], to the category of complete Archimedean meet-semilattices equipped with a finite meet-preserving action of the monoid of continuous monotone maps of [0,1] fixing 1. We also prove an analogous duality for completely distributive lattices. Moreover, we prove that these are essentially the only well-behaved "sound classes of joins Φ, dual to a class of meets'' for which "Φ-continuous lattice" and "Φ-algebraic lattice" are different notions, thus for which a 2-valued duality does not suffice.
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- Award ID(s):
- 2224709
- PAR ID:
- 10511128
- Publisher / Repository:
- Theory and applications of categories
- Date Published:
- Journal Name:
- Theory and applications of categories
- ISSN:
- 1201-561X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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