Complete Entanglement Structure of the Kitaev Honeycomb Spin Liquid from Exact Tensor Networks

Abstract

While most of the features of the Kitaev honeycomb model are already clarified from the combined ${\mathbb Z}_2$ plaquette and Majorana fermion picture, we show that retrieving the exact quantum many-body form gives a clue to fix the remaining issues on its spin liquid property. We provide an exact two-dimensional tensor network description of a full set of solutions from a random product state, by sequential quantum-number projection and Gutzwiller projection, both expressed as local matrix-product operators of dimension $D=2$. The ground and excited states share the same bond dimension yet exhibit distinct Schmidt spectra, distinguishing their area-law and volume-law entanglement. We demonstrate that the topological entanglement entropy emerges {\it for all eigenstates only after} the Gutzwiller projection onto the physical spin Hilbert space, where the gauge constraint removes exactly half of the Schmidt states. This establishes that excited states are likewise topologically ordered. We further show that there exists a projection-induced collapse of the wave function into a single gauge sector in the Parton mean-field states that is known to give a good description of the spin liquids. The demonstrations on the $S=1/2$ kagome and triangular Heisenberg models suggest that the present perspective is applied to a wider class of models hosting ${\mathbb Z}_2$ spin liquids. Our finding visualizes a bulk-boundary correspondence with the Wilson-loop characterization of topological order.