  Given a reductive representation $\rho: \pi_1(S)\rightarrow G$, there exists
a $\rho$-equivariant harmonic map $f$ from the universal cover of a fixed
Riemann surface $\Sigma$ to the symmetric space $G/K$ associated to $G$. If the
Hopf differential of $f$ vanishes, the harmonic map is then minimal. In this
paper, we investigate the properties of immersed minimal surfaces inside
symmetric space associated to a subloci of Hitchin component: $q_n$ and
$q_{n-1}$ case. First, we show that the pullback metric of the minimal surface
dominates a constant multiple of the hyperbolic metric in the same conformal
class and has a strong rigidity property. Secondly, we show that the immersed
minimal surface is never tangential to any flat inside the symmetric space. As
a direct corollary, the pullback metric of the minimal surface is always
strictly negatively curved. In the end, we find a fully decoupled system to
approximate the coupled Hitchin system.
