  We provide new insights into the Abelian and non-Abelian chiral Kitaev spin
liquids on the star lattice using the recently proposed loop gas (LG) and
string gas (SG) states [H.-Y. Lee, R. Kaneko, T. Okubo, N. Kawashima, Phys.
Rev. Lett. 123, 087203 (2019)]. Those are compactly represented in the language
of tensor network. By optimizing only one or two variational parameters,
accurate ansatze are found in the whole phase diagram of the Kitaev model on
the star lattice. In particular, the variational energy of the LG state becomes
exact(within machine precision) at two limits in the model, and the criticality
at one of those is analytically derived from the LG feature. It reveals that
the Abelian CSLs are well demonstrated by the short-ranged LG while the
non-Abelian CSLs are adiabatically connected to the critical LG where the
macroscopic loops appear. Furthermore, by constructing the minimally entangled
states and exploiting their entanglement spectrum and entropy, we identify the
nature of anyons and the chiral edge modes in the non-Abelian phase with the
Ising conformal field theory.
