  Quantum correlations and coherence generated between two free spinless
particles in the lattice, interacting with a common quantum phonon bath, are
studied. The reduced density matrix is solved using the Markov approach. We
show that the bath induces correlations between the particles. The coherence
induced by the bath is studied, calculating off-diagonal elements of the
density matrix, spatiotemporal dispersion, purity and quantum mutual
information. We find a characteristic time-scale pointing out when this
coherence is maximum. In addition, a Wigner-like distribution in the
phase-space (lattice) is introduced as an indirect indicator of the quantumness
of total correlations and coherence induced by the thermal bath. The negative
volume of the Wigner function also shows a behavior which is in agreement with
the time-scale that we have found. A Gaussian distribution for the profile of
particles is not obtained, and interference patterns are observed as the result
of bath-induced coherence. As the temperature of the bath vanishes, the
ballistic behavior of the tight-binding model is recovered. The geometric
quantum discord is calculated, to characterize the nature of the correlations.
