  Atomic hydrogen (HI) serves a crucial role in connecting galactic-scale
properties such as star formation with the large-scale structure of the
Universe. While recent numerical simulations have successfully matched the
observed covering fraction of HI near Lyman Break Galaxies (LBGs) and in the
foreground of luminous quasars at redshifts $z \lesssim 3$, the low-mass end
remains as-of-yet unexplored in observational and computational surveys. We
employ a cosmological, hydrodynamical simulation (FIREbox) supplemented with
zoom-in simulations (MassiveFIRE) from the Feedback In Realistic Environments
(FIRE) project to investigate the HI covering fraction of Lyman Limit Systems
($N_{\mathrm{HI}} \gtrsim 10^{17.2}$ cm$^{-2}$) across a wide range of
redshifts ($z=0-6$) and halo masses ($10^8-10^{13} M_{\odot}$ at $z=0$,
$10^8-10^{11} M_{\odot}$ at $z=6$) in the absence of feedback from active
galactic nuclei. We find that the covering fraction inside haloes exhibits a
strong increase with redshift, with only a weak dependence on halo mass for
higher-mass haloes. For massive haloes ($M_{\mathrm{vir}} \sim 10^{11}-10^{12}
M_{\odot}$), the radial profiles showcase scale-invariance and remain
independent of mass. The radial dependence is well-captured by a fitting
function. The covering fractions in our simulations are in good agreement with
measurements of the covering fraction in LBGs. Our comprehensive analysis
unveils a complex dependence with redshift and halo mass for haloes with
$M_{\mathrm{vir}} \lesssim 10^{10} M_{\odot}$ that future observations aim to
constrain, providing key insights into the physics of structure formation and
gas assembly.
