On diagonal pluriharmonic metrics of $G$-Higgs bundles

Let $(E,\Phi)\rightarrow (X,\omega_X)$ be a Higgs bundle over a compact K\"ahler manifold. We suppose that the holomorphic vector bundle $E$ decomposes into a direct sum of holomorphic line bundles. In this paper, we give the necessary and sufficient condition for the existence of a diagonal metric which is a solution to the Hermitian-Einstein equation. Our theorem can easily be generalized to $G$-Higgs bundles. We also describe the relationship between the stability condition and our condition using the torus action on the space of Higgs fields.