Is the continuum SSH model topological?

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