  This article investigates the existence and properties of ground state
solutions to the following nonlocal Hamiltonian elliptic system: \begin{align*}
  \begin{cases}
  (-\Delta)^\frac12 u +V_0 u =g(v),~x\in \mathbb{R}
  (-\Delta)^\frac12 v +V_0 v =f(u),~x\in \mathbb{R},
  \end{cases} \end{align*}
  where $(-\Delta)^\frac12$ is the square root Laplacian operator, $V_0 >0$ and
$f,~g$ have critical exponential growth in $\mathbb{R}$. Using minimization
technique over some generalized Nehari manifold, we show that the set
$\mathcal{S}$ of ground state solutions is non empty. Moreover for $(u,v) \in
\mathcal{S}$, $u,~v$ are uniformly bounded in $L^\infty(\mathbb{R})$ and
uniformly decaying at infinity. We also show that the set $\mathcal{S}$ is
compact in $H^\frac12(\mathbb{R}) \times H^\frac12(\mathbb{R})$ up to
translations. Furthermore under locally lipschitz continuity of $f$ and $g$ we
obtain a suitable Poho\v{z}aev type identity for any $(u,v) \in \mathcal{S}$.
We deduce the existence of semi-classical ground state solutions to the
singularly perturbed system
  \begin{align*}
  \begin{cases}
  \epsilon(-\Delta)^\frac12 \varphi +V(x) \varphi =g(\psi),~x\in \mathbb{R}
  \epsilon (-\Delta)^\frac12 \psi +V(x) \psi =f(\varphi),~x\in \mathbb{R},
  \end{cases}
  \end{align*}
  where $\epsilon>0$ and $V \in C(\mathbb{R})$ satisfy the assumption $(V)$
given below (see Section 1). Finally as $\epsilon \rightarrow 0$, we prove the
existence of minimal energy solutions which concentrate around the closest
minima of the potential $V$.
