Entanglement Entropy of Critical Spin Liquids

Quantum spin liquids are phases of matter whose internal structure is not captured by a local order parameter. Particularly intriguing are critical spin liquids, where strongly interacting excitations control low energy properties. Here we calculate their bipartite entanglement entropy that characterizes their quantum structure. In particular we calculate the Renyi entropy $S_2$ on model wave functions obtained by Gutzwiller projection of a Fermi sea. Although the wave functions are not sign positive, $S_2$ can be calculated on relatively large systems ($>324$ spins) using the variational Monte Carlo technique. On the triangular lattice we find that entanglement entropy of the projected Fermi sea state violates the boundary law, with $S_2$ enhanced by a logarithmic factor. This is an unusual result for a bosonic wave function reflecting the presence of emergent fermions. These techniques can be extended to study a wide class of other phases.

Figure 1

Renyi entropy data for projected and unprojected Fermi sea state on the triangular lattice of size $18\times 18$ with $L_A=1–8$. Note that projection barely modifies the slope, pointing to a Fermi surface surviving in the spin wave function. We also separately plot $S_{2,\mathrm{sign}}$ and $S_{2,\mathrm{mod}}$ (as defined in the text) for the projected state, the former dominates at larger sizes.

Figure 2

Renyi entropy for the projected Fermi sea state on the triangular lattice and square (with and without $\pi$ flux) lattice as a function of the perimeter $P$ of the subsystem $A$. Here $C$ is the constant part of the $S_2$. We find $S_2\sim P\log (P)+C$ for the projected triangular lattice state while $S_2\sim P+C$ for the projected $\pi$ flux square lattice state. For the square-lattice state (no flux), the projection leads to a significant reduction in $S_2$, suggesting at most a very weak violation of the area law.