Fractional Hamiltonian systems with critical exponential growth

In this paper, we study the following nonlocal nonautonomous Hamiltonian system on whole $\mathbb{R}$ $$\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{\frac{1}{2}}u+u=Q(x)g(v)& \text{quadin}\mathbb{R},\\ (-\mathrm{\Delta }{)}^{\frac{1}{2}}v+v=P(x)f(u)& \text{quadin}\mathbb{R},\end{array}$$ where $(-\mathrm{\Delta }{)}^{\frac{1}{2}}$ is {the} square root Laplacian operator. We assume that the nonlinearities $f,g$ have critical growth at $+\mathrm{\infty }$ in the sense of Trudinger-Moser inequality and the nonnegative weights $P(x)$ and $Q(x)$ vanish at $+\mathrm{\infty }$. Using suitable variational method combined with {the} generalized linking theorem, we obtain the existence of {at least one} positive solution for the above system.