Topological entanglement entropy of ${\mathbb{Z}}_2$ spin liquids and lattice Laughlin states

We study entanglement properties of candidate wave functions for SU(2) symmetric gapped spin liquids and Laughlin states. These wave functions are obtained by the Gutzwiller projection technique. Using topological entanglement entropy $\gamma$ as a tool, we establish topological order in chiral spin liquid and ${\mathbb{Z}}_2$ spin liquid wave functions, as well as a lattice version of the Laughlin state. Our results agree very well with the field theoretic result $\gamma =\log D$ where $D$ is the total quantum dimension of the phase. All calculations are done using a Monte Carlo technique on a $12\times 12$ lattice enabling us to extract $\gamma$ with small finite-size effects. For a chiral spin liquid wave function, the calculated value is within $4\%$ of the ideal value. We also find good agreement for a lattice version of the Laughlin $\nu =1/3$ phase with the expected $\gamma =\log \sqrt{3}$.

Figure 1

Illustration of a square lattice hopping model connected with a $d+id$ superconductor. While the nearest neighbor hopping is along the square edges with amplitude $t$ ($-t$ for hopping along dashed lines), the second nearest neighbor hopping is along the square diagonal (arrows in bold), with amplitude $+i\Delta$ ($-i\Delta$) when hopping direction is along (against) the arrow. The two sublattices in the unit cell are marked as A and B.

Figure 2

The separation of the system into subsystems A, B, C, and environment; periodic boundary condition is employed in both $\widehat{x}$ and $\widehat{y}$ directions.

Figure 3

Illustration of finite-size effect: chiral SL TEE $\gamma$ as a function of $2\Delta /t$ proportional to the relative gap size for characteristic system length $L_A=3$. The larger the gap, the closer the data approach the ideal value. For comparison, TEE $\gamma$ for chiral SL at $L_A=4$ and $2\Delta /t=1.0$ is shown. On the same plot, TEE $\gamma$ of a lattice version of $\nu =1/3$ Laughlin state at $L_A=3$, $2\Delta /t=1.0$ is also shown. The dashed lines are the ideal TEE values of $\gamma =\log \left(\sqrt{2}\right)$ for the chiral SL and $\log \left(\sqrt{3}\right)$ for the $\nu =1/3$ Laughlin state.