  The discrete Hamiltonian of Su, Schrieffer and Heeger (SSH) is a well-known
one-dimensional 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.
  In order to establish that the tight-binding limit of these continuum
Hamiltonians is the SSH model, we extend our recent results on the
tight-binding approximation to lattices which depend on the tight-binding
asymptotic parameter.
