Gap closing and topological phase transitions
A topological index cannot change smoothly while its defining gap and symmetries remain intact.
$$\Delta=\min_k(E_c(k)-E_v(k))\longrightarrow0$$
What this means
A topological index cannot change smoothly while its defining gap and symmetries remain intact.
Deep dive: mathematical and physical mechanism
Consider a continuous family $H(k,\lambda)$. A discrete invariant stays fixed as $\lambda$ varies unless the occupied subspace ceases to be well-defined (typically at a gap closing) or a protecting symmetry is broken. In interacting systems, phase transitions can be more subtle.
How it appears in quantum phenomena
Tuning the SSH hopping ratio through $t_1=t_2$ closes the band gap.
Assumptions, caveats and limitations
A gap closing is necessary only within the assumptions of the chosen classification.
Connect the ideas
For context, see the Quantum Phenomena and Applications topic guide and the reading references.