  The Kitaev honeycomb model is a paradigm of exactly-solvable models, showing
non-trivial physical properties such as topological quantum order, abelian and
non-abelian anyons, and chirality. Its solution is one of the most beautiful
examples of the interplay of different mathematical techniques in condensed
matter physics. In this paper, we show how to derive a tensor network (TN)
description of the eigenstates of this spin-1/2 model in the thermodynamic
limit, and in particular for its ground state. In our setting, eigenstates are
naturally encoded by an exact 3d TN structure made of fermionic unitary
operators, corresponding to the unitary quantum circuit building up the
many-body quantum state. In our derivation we review how the different
"solution ingredients" of the Kitaev honeycomb model can be accounted for in
the TN language, namely: Jordan-Wigner transformation, braidings of Majorana
modes, fermionic Fourier transformation, and Bogoliubov transformation. The TN
built in this way allows for a clear understanding of several properties of the
model. In particular, we show how the fidelity diagram is straightforward both
at zero temperature and at finite temperature in the vortex-free sector. We
also show how the properties of two-point correlation functions follow easily.
Finally, we also discuss the pros and cons of contracting of our 3d TN down to
a 2d Projected Entangled Pair State (PEPS) with finite bond dimension. The
results in this paper can be extended to generalizations of the Kitaev model,
e.g., to other lattices, spins, and dimensions.
