  We present a theoretical study of the temporal and spatial coherence
properties of a topological laser device built by including saturable gain on
the edge sites of a Harper--Hofstadter lattice for photons. For small enough
lattices the Bogoliubov analysis applies, the emission is nearly a single-mode
one and the coherence time is almost determined by the total number of photons
in the device in agereement with the standard Schawlow-Townes phase diffusion.
In larger lattices, looking at the lasing edge mode in the comoving frame of
its chiral motion, the spatio-temporal correlations of long-wavelength
fluctuations display a Kardar-Parisi-Zhang (KPZ) scaling. Still, at very long
times, when the finite size of the device starts to matter, the functional form
of the temporal decay of coherence changes from the KPZ stretched exponential
to a Schawlow-Townes-like exponential, while the nonlinear many-mode dynamics
of KPZ fluctuations remains visible as an enhanced linewidth as compared to the
single-mode Schawlow-Townes prediction. While we have established the above
behaviors also for non-topological laser arrays, the crucial role of topology
in protecting the coherence from static disorder is finally highlighted: our
ground-breaking numerical calculations suggest the dramatically reinforced
coherence properties of topological lasers compared to corresponding
non-topological devices. These results open exciting possibilities for both
fundamental studies of non-equilibrium statistical mechanics and concrete
applications to laser devices.
