  We study out-of-time-ordered correlators (OTOC) in hard-core boson models
with short-range and long-range hopping and compare the results to the OTOC in
the Luttinger liquid model. For the density operator, a related `commutator
function' starts at zero and decays back to zero after the passage of the
wavefront in all three models, while the wavefront broadens as $t^{1/3}$ in the
short-range model and shows no broadening in the long-range model and the
Luttinger liquid model. For the boson creation operator, the corresponding
commutator function shows saturation inside the light cone in all three models,
with similar wavefront behavior as in the density-density commutator function,
despite the presence of a nonlocal string in terms of Jordan-Wigner fermions.
For the long-range model and the Luttinger liquid model, the commutator
function decays as power law outside the light cone in the long time regime
when following different fixed-velocity rays. In all cases, the OTOCs approach
their long-time values in a power-law fashion, with different exponents for
different observables and short-range vs long-range cases. Our long-range model
appears to capture exponents in the Luttinger liquid model (which are found to
be independent of the Luttinger parameter in the model). This conclusion also
bears on the OTOC calculations in conformal field theories, which we propose
correspond to long-ranged models.
