  Observing critical phases in lattice models is challenging due to the need to
analyze the finite time or size scaling of observables. We study how the
computational topology technique of persistent homology can be used to
characterize phases of a generalized Aubry-Andr\'{e}-Harper model. The
persistent entropy and mean squared lifetime of features obtained using
persistent homology behave similarly to conventional measures (Shannon entropy
and inverse participation ratio) and can distinguish localized, extended, and
crticial phases. However, we find that the persistent entropy also clearly
distinguishes ordered from disordered regimes of the model. The persistent
homology approach can be applied to both the energy eigenstates and the
wavepacket propagation dynamics.
