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 (βΞ)21β is the square root Laplacian operator, V0β>0 and
f,Β g have critical exponential growth in R. Using minimization
technique over some generalized Nehari manifold, we show that the set
S of ground state solutions is non empty. Moreover for (u,v)βS, u,Β v are uniformly bounded in Lβ(R) and
uniformly decaying at infinity. We also show that the set S is
compact in H21β(R)ΓH21β(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)βS.
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 Ο΅>0 and VβC(R) satisfy the assumption (V)
given below (see Section 1). Finally as Ο΅β0, we prove the
existence of minimal energy solutions which concentrate around the closest
minima of the potential V