  In the Kitaev honeycomb model, the quantum spin fractionalizes into itinerant
Majorana and gauge flux spontaneously upon cooling, leading to rich
experimental ramifications at finite temperature and an upsurge of research
interest. In this work, we employ the exponential tensor renormalization group
approach to explore the Kitaev model under various perturbations, including the
external fields, Heisenberg, and the off-diagonal couplings that are common in
the Kitaev materials. Through large-scale manybody calculations, we find a
Kitaev fractional liquid at intermediate temperature that is robust against
perturbations. The fractional liquid exhibits universal thermodynamic
behaviors, including the fractional thermal entropy, metallic specific heat,
and an intermediate-temperature Curie law of magnetic susceptibility. The
emergent universal susceptibility behavior, with a modified Curie constant, can
be ascribed to the strongly fluctuating $\mathbb{Z}_2$ fluxes as well as the
extremely short-ranged and bond-directional spin correlations. With this
insight, we revisit the susceptibility measurements of Na$_2$IrO$_3$ and
$\alpha$-RuCl$_3$, and find evident signatures of finite-temperature
fractionalization and ferromagnetic Kitaev couplings. Moreover, the peculiar
spin correlation in the fractional liquid corresponds to a stripy structure
factor which rotates in the extended Brillouin zone as the spin component
changes. Therefore, our findings encourage future experimental exploration of
fractional liquid in the Kitaev materials by thermodynamic measurements and
spin-resolved structure factor probes.
