  Considering a BDI symmetric one-dimensional SSH model, we explore the fate of
the bulk topological invariant, namely, the winding number under a generic time
dependent perturbation; the effective Hamiltonian, that generates the temporal
evolution of the initial (ground) state of the completely symmetric initial
Hamiltonian, may have the same or different symmetries. To exemplify, we
consider the following protocols, namely (i) a perfectly periodic protocol (ii)
a periodic protocol with noisy perturbations and also (iii) sudden changes in
the parameters of the initial Hamiltonian. We establish that the topological
invariant may change in some cases when the effective Hamiltonian (or the
Floquet Hamiltonian in the periodic situation when observed stroboscopically)
does not respect all BDI symmetries; this is manifested in the associated
particle (polarisation) or heat current in the bulk. Our results establish a
strong connection between the time evolution of the winding number (thus, the
associated transport of currents) and the symmetry of the Hamiltonian
generating the time evolution which has been illustrated considering an
exhaustive set of possibilities.
