  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 non-universal. 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.
