  Heavy fermion criticality has been a long-standing problem in condensed
matter physics. Here we study a one-dimensional Kondo lattice model through
numerical simulation and observe signatures of local criticality. We vary the
Kondo coupling $J_K$ at fixed doping $x$. At large positive $J_K$, we confirm
the expected conventional Luttinger liquid phase with $2k_F=\frac{1+x}{2}$ (in
units of $2\pi$), an analogue of the heavy Fermi liquid (HFL) in the higher
dimension. In the $J_K \leq 0$ side, our simulation finds the existence of a
fractional Luttinger liquid (LL*) phase with $2k_F=\frac{x}{2}$, accompanied by
a gapless spin mode originating from localized spin moments, which serves as an
analogue of the fractional Fermi liquid (FL*) phase in higher dimensions. The
LL* phase becomes unstable and transitions to a spin-gapped Luther-Emery (LE)
liquid phase at small positive $J_K$. Then we mainly focus on the `critical
regime' between the LE phase and the LL phase. Approaching the critical point
from the spin-gapped LE phase, we often find that the spin gap vanishes
continuously, while the spin-spin correlation length in real space stays finite
and small. For a certain range of doping, in a point (or narrow region) of
$J_K$, the dynamical spin structure factor obtained through the time-evolving
block decimation (TEBD) simulation shows dispersion-less spin fluctuations in a
finite range of momentum space above a small energy scale (around $0.035 J$)
that is limited by the TEBD accuracy. All of these results are unexpected for a
regular gapless phase (or critical point) described by conformal field theory
(CFT). Instead, they are more consistent with exotic ultra-local criticality
with an infinite dynamical exponent $z=+\infty$. Lastly, we propose to simulate
the model in a bilayer optical lattice with a potential difference.
