  A large number of symmetry-protected topological (SPT) phases have been
hypothesized for strongly interacting spin-1/2 systems in one dimension.
Realizing these SPT phases, however, often demands fine-tunings hard to reach
experimentally. And the lack of analytical solutions hinders the understanding
of their many-body wave functions. Here we show that two kinds of SPT phases
naturally arise for ultracold polar molecules confined in a zigzag optical
lattice. This system, motivated by recent experiments, is described by a spin
model whose exchange couplings can be tuned by an external field to reach
parameter regions not studied before for spin chains or ladders. Within the
enlarged parameter space, we find the ground state wave function can be
obtained exactly along a line and at a special point, for these two phases
respectively. These exact solutions provide a clear physical picture for the
SPT phases and their edge excitations. We further obtain the phase diagram by
using infinite time-evolving block decimation, and discuss the phase
transitions between the two SPT phases and their experimental signatures.
