  Dipolar particles in an elongated trap are expected to undergo a quantum
phase transition from a linear to a zigzag structure with decreasing transverse
confinement. We derive the low energy effective theory of the transition
showing that in presence of quantum fluctuations the Zigzag phase can be
characterized by a long ranged string order, while the local Ising correlations
decay as a power law. This is also confirmed using DMRG calculations on a
microscopic model. The non local order in the bulk gives rise to zero energy
states localized at the interface between the ordered and disordered phases.
Such an interface naturally arises when the particles are subject to a weak
harmonic confinement along the tube axis. We compute the signature of the edge
states in the single particle tunneling spectra pointing to differences between
a system with bosonic versus fermionic particles. Finally we asses the
magnitude of the relevant quantum fluctuations in realistic systems of dipolar
particles, including ultracold polar molecules as well as alkali atoms weakly
dressed by a Rydberg excitation.
