Entanglement entropy of XX spin $1/2$ chain with random partitioning at arbitrary temperature

We study the entanglement properties of random XX spin $1/2$ chains at an arbitrary temperature $T$ using random partitioning, where sites of a size-varying subsystem are chosen randomly with a uniform probability $p$, and then an average over subsystem possibilities is taken. We show analytically and numerically, using the approximate method of real space renormalization group, that random partitioning entanglement entropy for the XX spin chain of size $L$ behaves like EE$(T,p) = a(T,p) L$ at an arbitrary temperature $T$ with a uniform probability $p$, i.e., it obeys volume law. We demonstrate that $a(T,p) = \ln(2) \langle P_s + P_{t_{\uparrow\downarrow}} \rangle p(1-p)$, where $P_s$ and $P_{t_{\uparrow\downarrow}}$ are the average probabilities of having singlet and triplet$_{\uparrow\downarrow}$ in the entire system, respectively. We also study the temperature dependence of pre-factor $a(T,p)$. We show that EE with random partitioning reveals both short- and long-range correlations in the entire system.