Is the continuum SSH model topological?

Abstract

The discrete Hamiltonian of Su, Schrieffer and Heeger (SSH) [14] is a well-known onedimensional translation-invariant model in condensed matter physics. The model consists of two atoms per unit cell and describes in-cell and out-of-cell electron-hopping between two sub-lattices. It is among the simplest models exhibiting a non-trivial topological phase; to the SSH Hamiltonian one can associate a winding number, the Zak phase, which depends on the ratio of hopping coefficients and takes on the values 0 and 1 labeling the two distinct phases. We display two homotopically equivalent continuum Hamiltonians whose tight binding limits are SSH models with different topological indices. The topological character of the SSH model is therefore an emergent rather than fundamental property, associated with emergent chiral or sublattice symmetry in the tight-binding limit.

Internal wave crystals

“…The continuum photonic crystal analysed in Refs. [32,33,65] can indeed be mapped [65] to the Su-Schrieffer-Heeger (SSH) model, a standard example of symmetry-protected topological insulator [22,66] (which involves some subtleties [67][68][69]). The same applies to our models of internal waves in periodic stratifications.…”