  It has been recently shown that 2D systems can exhibit crystalline phases
with long-range translational order showcasing a striking violation of the
Hohenberg-Mermin-Wagner (HMW) theorem which is valid at equilibrium. This is
made possible by athermal driving mechanisms that inject energy into the system
without exciting long wavelength modes of the density field. However, as
thermal fluctuations are superimposed on the non-equilibrium driving,
long-range translational order is inevitably lost. In this paper, we discuss
the possibility of exploiting non-equilibrium effects to suppress arbitrarily
large density fluctuations even when a global thermal bath is coupled to the
system. We introduce a model of a harmonic crystal driven both by a global
thermal bath and by a momentum conserving noise, where the typical observables
related to density fluctuations and long-range translational order can be
analytically derived and put in relation. This model allows us to rationalize
the violation of the HMW theorem observed in previous studies through the
prediction of large-wavelength phonons which thermalize at a vanishing
effective temperature when the global bath is switched off. The conceptual
framework introduced through this theory is then applied to numerical
simulations of a hard-disk solid in contact with a thermal bath and driven
out-of-equilibrium by active collisions. Our numerical analysis demonstrates
how varying driving and dissipative parameters can lead to an arbitrary
enhancement of the quasi-long-range order in the system regardless of the
applied global noise amplitude. Finally, we outline a possible experimental
procedure to apply our results to a realistic granular system.
