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 talk I will 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 the derivation I will 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, I will show how the fidelity diagram is straightforward both at zero temperature and at finite temperature in the vortex-free sector. Finally, I will 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, and possible applications in quantum computation.
P. Schmoll and R. Orus, Phys. Rev. B 95, 045112 (2017), arXiv:1605.04315