Comparison of the Hitchin metric and the semi-flat metric in the rank two case

Let $(E,\theta )$ be a Higgs bundle of rank $2$ and degree $0$ on a compact Riemann surface $X$ whose spectral curve is smooth. The tangent space of the moduli space of Higgs bundles at $(E,\theta )$ is equipped with two natural metrics called the Hitchin metric and the semi-flat metric. It is known that the difference between two metrics along the curve $(E,t\theta )$ $(t\ge 1)$ decays in an exponential way. In this paper, we shall study how the exponential rate is improved.