  Non-Fermi liquids (NFL) are a class of strongly interacting gapless fermionic
systems without long-lived quasiparticle excitations. An important group of NFL
model features itinerant fermions coupled to soft bosonic fluctuations near a
quantum-critical point (QCP), and are widely believed to capture the essential
physics of many unconventional superconductors. However numerically the direct
observation of a canonical NFL behavior in such systems, characterized by a
power-law form in the Green's function, has been elusive. Here we consider a
Sachdev-Ye-Kitaev (SYK)-like model with random Yukawa interaction between
critical bosons and fermions (dubbed Yukawa-SYK model). We show it is immune
from minus-sign problem and hence can be solved exactly via large-scale quantum
Monte Carlo simulation beyond the large-$N$ limit accessible to analytical
approaches. Our simulation demonstrates the Yukawa-SYK model features
"self-tuned quantum criticality", namely the system is critical independent of
the bosonic bare mass. We put these results to test at finite $N$, and our
unbiased numerics reveal clear evidence of these exotic quantum-critical NFL
properties -- the power-law behavior in Green's function of fermions and bosons
-- which propels the theoretical understanding of critical Planckian metals and
unconventional superconductors.
