  We construct a sign-problem free variant of the complex Sachdev-Ye-Kitaev
(SYK) model which keeps all the essential properties of the SYK model,
including the analytic solvability in the large-$N$ limit and being maximally
chaotic. In addition to the number of complex fermions $N$, our model has an
additional parameter $M$ controlling the number of terms in the Hamiltonian
which we take $M \to \infty$ with keeping $M/N$ constant in the large-$N$
limit. While our model respects global $U(1)$ symmetry associated with the
fermion number conservation, both the large-$N$ limit and the sign-problem free
nature become explicit in the Majorana representation. We present a detailed
analysis on our model, i.e., the random matrix classification based on the
symmetry analysis, analytic approach, and the quantum Monte Carlo simulations.
All these analysis show that our model exhibit a non-Fermi liquid (NFL)
physics, a gapless fermionic system lying beyond the conventional Fermi liquid
picture.
