  In this paper, we prove that the nonlocal complex modified Korteweg-de Vries
(mKdV) equation introduced by Ablowitz and Musslimani [Nonlinearity, 29,
915-946 (2016)] is gauge equivalent to a spin-like model. From the gauge
equivalence, one can see that there exists significant difference between the
nonlocal complex mKdV equation and the classical complex mKdV equation. Through
constructing the Darboux transformation(DT) for nonlocal complex mKdV equation,
a variety of exact solutions including dark soliton, W-type soliton, M-type
soliton and periodic solutions are derived.
