  The Su-Schrieffer-Heeger (SSH) model is fundamental in topological insulators
and relevant to understanding higher-order topological phases. This study
explores the relationship between the $n$-dimensional SSH model and its
$(n-1)$-dimensional counterpart, identifying a hierarchical structure in the
Hamiltonian that allows us to solve an arbitrary $n$-dimensional SSH model
analytically. By generalizing the bulk-edge correspondence principle to
arbitrary dimensions in a higher-order fashion using the vectored Zak phase, we
reveal a type of topological insulator called hierarchical topological
insulators. In this hierarchical topological insulator, there exist
intermediate-order topological interfacial states that are protected by
subsymmetries and energy bands topology in a partial Brillouin zone.
Furthermore, we compare the $n$-dimensional SSH model with the
Benalcazar-Bernevig-Hughes (BBH) model, another essential model in higher-order
topological phases similar to the two-dimensional SSH model with an extra flux
of $\pi$ in each plaque. We find that the BBH model is another example of
hierarchical topological insulators.
