  We study the ground-state phase diagram of an interacting staggered
Su-Schrieffer-Heeger (SSH) ladder in the vicinity of the Gaussian quantum
critical point. The corresponding effective field theory is a double-frequency
sine-Gordon (DSG) model which involves two perturbations at the Gaussian fixed
point: the deviation from criticality and Umklapp scattering processes. A
topological distinction between thermodynamically equivalent phases becomes
only feasible when nonlocal fermionic fields, parity and string order
parameter, are included into consideration. We prove that a noninteracting
fermionic staggered SSH ladder is exactly equivalent to a O(2)-symmetric model
of two decoupled Kitaev-Majorana chains, or two 1D p-wave superconductors.
Close to the Gaussian fixed point the SSH ladder maps to an Ashkin-Teller like
system when interactions are included. Thus, the topological order in the SSH
ladder is related to broken-symmetry phases of the associated quantum
spin-chain degrees of freedom. The obtained phase diagram includes a
Tomonaga-Luttinger liquid state which, due to Umklapp processes, can become
unstable against either spontaneous dimerization or the onset of a
charge-density wave (CDW). In these gapped phases elementary bulk excitations
are quantum kinks carrying the charge QF = 1/2. For sufficiently strong,
long-range interactions, the phase diagram of the model exhibits a bifurcation
of the Gaussian critical point into two outgoing Ising criticalities. The
latter sandwich a mixed phase in which dimerization coexists with a
site-diagonal CDW. In this phase elementary bulk excitations are represented by
two types of topological solitons carrying different fermionic charges, which
continuously interpolate between 0 and 1. This phase has also mixed topological
properties with coexisting parity and string order parameters.
