  Su-Schrieffer-Heeger (SSH) chains are paradigmatic examples of 1D topological
insulators hosting zero-energy edge modes when the bulk of the system has a
non-zero topological winding invariant. Recently, high-harmonic spectroscopy
has been suggested as a tool for detecting the topological phase. Specifically,
it has been shown that when the SSH chain is coupled to an external laser field
of a frequency much smaller than the band gap, the emitted light at harmonic
frequencies strongly differs between the trivial and the topological phase.
However, it remains unclear whether various non-trivial topological phases --
differing in the number of edge states -- can also be distinguished by the high
harmonic generation (HHG). In this paper, we investigate this problem by
studying an extended version of the SSH chain with extended-range hoppings,
resulting in a topological model with different topological phases. We
explicitly show that HHG spectra are a sensitive and suitable tool for
distinguishing topological phases when there is more than one topological
phase. We also propose a quantitative scheme based on tuning the filling of the
system to precisely locate the number of edge modes in each topological phase
of this chain.
