  In this paper, we study the following nonlocal nonautonomous Hamiltonian
system on whole $\mathbb R$ $$ \left\{\begin{array}{ll} (-\Delta)^\frac12~ u
+u=Q(x) g(v)&\quad\mbox{in } \mathbb R,\\ (-\Delta)^\frac12~ v+v =
P(x)f(u)&\quad\mbox{in } \mathbb R, \end{array}\right. $$ where
$(-\Delta)^\frac12$ is {the} square root Laplacian operator. We assume that the
nonlinearities $f, g$ have critical growth at $+\infty$ in the sense of
Trudinger-Moser inequality and the nonnegative weights $P(x)$ and $Q(x)$ vanish
at $+\infty$. Using suitable variational method combined with {the} generalized
linking theorem, we obtain the existence of {at least one} positive solution
for the above system.
