  The Kitaev spin liquid, stabilized as the ground state of the Kitaev
honeycomb model, is a paradigmatic example of a topological $\mathbb{Z}_2$
quantum spin liquid. The fate of the Kitaev spin liquid in presence of an
external magnetic field is a topic of current interest due to experiments,
which apparently unveil a $\mathbb{Z}_2$ topological phase in the so-called
Kitaev materials, and theoretical studies predicting the emergence of an
intermediate quantum phase of debated nature before the appearance of a trivial
partially polarized phase. In this work, we employ hierarchical mean-field
theory, an algebraic and numerical method based on the use of clusters
preserving relevant symmetries and short-range quantum correlations, to
investigate the quantum phase diagram of the antiferromagnetic Kitaev's model
in a [111] field. By using clusters of 24 sites, we predict that the Kitaev
spin liquid transits through two intermediate phases characterized by stripe
and chiral order, respectively, before entering the trivial partially polarized
phase, differing from previous studies. We assess our results by performing
exact diagonalization and computing the scaling of different observables,
including the many-body Chern number and other topological quantities, thus
establishing hierarchical mean-field theory as a method to study topological
quantum spin liquids.
