Topological insulators with arbitrarily tunable entanglement

We elucidate how Chern and topological insulators fulfill an area law for the entanglement entropy. By explicit construction of a family of lattice Hamiltonians, we are able to demonstrate that the area law contribution can be tuned to an arbitrarily small value but is topologically protected from vanishing exactly. We prove this by introducing novel methods to bound entanglement entropies from correlations using perturbation bounds, drawing intuition from ideas of quantum information theory. This rigorous approach is complemented by an intuitive understanding in terms of entanglement edge states. These insights have a number of important consequences: The area law has no universal component, no matter how small, and the entanglement scaling cannot be used as a faithful diagnostic of topological insulators. This holds for all Renyi entropies which uniquely determine the entanglement spectrum, which is hence also nonuniversal. The existence of arbitrarily weakly entangled topological insulators furthermore opens up possibilities of devising correlated topological phases in which the entanglement entropy is small and which are thereby numerically tractable, specifically in tensor network approaches.

Figure 1

Sketch of our setup and the bulk-boundary correspondence in Chern insulators. Bipartitioning the system into distinct regions ($A$ and $B$) on a torus gives rise to virtual edge states (green) in the entanglement Hamiltonian which are close analogs of the physical edge states (red) on a cylinder.

Figure 2

Band structure and Berry curvature. The top row shows the energy dispersion as a function of the transverse momentum $k_y$ in the model defined on a cylinder serving as an analogy, and the bottom row shows the corresponding Berry curvature distribution $\mathcal{F}$ for the model on the torus. From the left to right, the data are displayed for the standard Dirac model (14) and our model (17), with $\mu =\pi ,1,0.2$, respectively. For small values of $\mu >0$, the Berry curvature is strongly peaked around $k_y=\pm \mu /2$, which is necessary to maintain a unit Chern number, $C={\int }_{\mathrm{BZ}}\mathcal{F}\left(\mathbf{k}\right)d^{2}k/\left(2\pi \right)$. The edge states exhibit a very steep slope around $k_y=\pm \mu /2$ and a plateau of width $\mu$ at zero energy between $k_y=-\mu /2$ and $k_y=\mu /2$.

Figure 3

Entanglement area law manifested in the large $L$ limit of $S_1\left(L\right)/L$ as a function of $\mu$ on a log-log scale. For sufficiently small $\mu$, the numerically obtained values of $S_1\left(L\right)/L$ are proportional to $\mu$ to an extraordinary precision, confirming our conclusion that ${lim}_{L\to \infty }black{S}_1\left(L\right)/L\to 0$ as $\mu \downarrow 0$. The blue line for $0.11\mu$ is included as a guide to the eye.