Sublogarithmic behaviour of the entanglement entropy in fermionic chains

In this paper, we discuss the possibility of unexplored behaviours for the entanglement entropy in extended quantum systems. Namely, we study the R'enyi entanglement entropy for the ground state of long-range Kitaev chains with slow decaying couplings. We obtain that, under some circumstances, the entropy grows sublogarithmically with the length of the subsystem. Our result is based on the asymptotic behaviour of a new class of Toeplitz determinants whose symbol does not lie within the application domain of the Strong Szeg\H{o} Theorem or the Fisher-Hartwig conjecture.