Entanglement across extended random defects in the XX spin chain

We study the half-chain entanglement entropy in the ground state of the spin-1/2 XX chain across an extended random defect, where the strength of disorder decays with the distance from the interface algebraically as $\Delta_l\sim l^{-\kappa}$. In the whole regime $\kappa\ge 0$, the average entanglement entropy is found to increase logarithmically with the system size $L$ as $S_L\simeq\frac{c_{\rm eff}(\kappa)}{6}\ln L+const$, where the effective central charge $c_{\rm eff}(\kappa)$ depends on $\kappa$. In the regime $\kappa<1/2$, where the extended defect is a relevant perturbation, the strong-disorder renormalization group method gives $c_{\rm eff}(\kappa)=(1-2\kappa)\ln2$, while, in the regime $\kappa\ge 1/2$, where the extended defect is irrelevant in the bulk, numerical results indicate a non-zero effective central charge, which increases with $\kappa$. The variation of $c_{\rm eff}(\kappa)$ is thus found to be non-monotonic and discontinuous at $\kappa=1/2$.