  In this paper, we study the following class of fractional Hamiltonian
systems: \begin{eqnarray*}
  \begin{aligned}\displaystyle
  \left\{ \arraycolsep=1.5pt
  \begin{array}{ll}
  (-\Delta)^{\frac{1}{2}} u + u = \Big(I_{\mu_{1}}\ast G(v)\Big)g(v) \ \ \ &
\mbox{in} \ \mathbb{R},\\[2mm]
  (-\Delta)^{\frac{1}{2}} v + v = \Big(I_{\mu_{2}}\ast F(u)\Big)f(u) \ \ \ &
\mbox{in} \ \mathbb{R},
  \end{array}
  \right.
  \end{aligned} \end{eqnarray*} where $(-\Delta)^{\frac{1}{2}}$ is the square
root Laplacian operator, $\mu_{1},\mu_{2}\in(0,1)$, $I_{\mu_{1}},I_{\mu_{2}}$
denote the Riesz potential, $\ast$ indicates the convolution operator,
$F(s),G(s)$ are the primitive of $f(s),g(s)$ with $f(s),g(s)$ have exponential
growth in $\mathbb{R}$. Using the linking theorem and variational methods, we
establish the existence of at least one positive solution to the above problem.
