  We theoretically address spin chain analogs of the Kitaev quantum spin model
on the honeycomb lattice. The emergent quantum spin liquid phases or Anderson
resonating valence bond (RVB) states can be understood, as an effective model,
in terms of p-wave superconductivity and Majorana fermions. We derive a
generalized phase diagram for the two-leg ladder system with tunable
interaction strengths between chains allowing us to vary the shape of the
lattice (from square to honeycomb ribbon or brickwall ladder). We evaluate the
winding number associated with possible emergent (topological) gapless modes at
the edges. In the Az phase, as a result of the emergent Z2 gauge fields and
pi-flux ground state, one may build spin-1/2 (loop) qubit operators by analogy
to the toric code. In addition, we show how the intermediate gapless B phase
evolves in the generalized ladder model. For the brickwall ladder, the $B$
phase is reduced to one line, which is analyzed through perturbation theory in
a rung tensor product states representation and bosonization. Finally, we show
that doping with a few holes can result in the formation of hole pairs and
leads to a mapping with the Su-Schrieffer-Heeger model in polyacetylene; a
superconducting-insulating quantum phase transition for these hole pairs is
accessible, as well as related topological properties.
