  The entanglement entropy in clean, as well as in random quantum spin chains
has a logarithmic size-dependence at the critical point. Here, we study the
entanglement of composite systems that consist of a clean and a random part,
both being critical. In the composite, antiferromagnetic XX-chain with a sharp
interface, the entropy is found to grow in a double-logarithmic fashion ${\cal
S}\sim \ln\ln(L)$, where $L$ is the length of the chain. We have also
considered an extended defect at the interface, where the disorder penetrates
into the homogeneous region in such a way that the strength of disorder decays
with the distance $l$ from the contact point as $\sim l^{-\kappa}$. For
$\kappa<1/2$, the entropy scales as ${\cal S}(\kappa) \simeq (1-2\kappa){\cal
S}(\kappa=0)$, while for $\kappa \ge 1/2$, when the extended interface defect
is an irrelevant perturbation, we recover the double-logarithmic scaling. These
results are explained through strong-disorder RG arguments.
