  In this paper we prove the existence of a nonnegative ground state solution
to the following class of coupled systems involving Schr\"{o}dinger equations
with square root of the Laplacian
  $$
  \left\{
  \begin{array}{lr}
  (-\Delta)^{1/2}u+V_{1}(x)u=f_{1}(u)+\lambda(x)v, & x\in\mathbb{R},
  (-\Delta)^{1/2}v+V_{2}(x)v=f_{2}(v)+\lambda(x)u, & x\in\mathbb{R},
  \end{array}
  \right.
  $$
  where the nonlinearities $f_{1}(s)$ and $f_{2}(s)$ have exponential critical
growth of the Trudinger-Moser type, the potentials $V_{1}(x)$ and $V_{2}(x)$
are nonnegative and periodic. Moreover, we assume that there exists $\delta\in
(0,1)$ such that $\lambda(x)\leq\delta\sqrt{V_{1}(x)V_{2}(x)}$. We are also
concerned with the existence of ground states when the potentials are
asymptotically periodic. Our approach is variational and based on minimization
technique over the Nehari manifold.
