  Let $X$ be a compact Riemann surface. Let $(E,\theta)$ be a stable Higgs
bundle of degree $0$ on $X$. Let $h_{\det(E)}$ denote a flat metric of the
determinant bundle $\det(E)$. For any $t>0$, there exists a unique harmonic
metric $h_t$ of $(E,\theta)$ such that $\det(h_t)=h_{\det(E)}$. We prove that
if the Higgs bundle is induced by a line bundle on the normalization of the
spectral curve, then the sequence $h_t$ is convergent to the naturally defined
decoupled harmonic metric at the speed of the exponential order. We also obtain
a uniform convergence for such a family of Higgs bundles.
