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Using free-fermionic techniques we study the entanglement entropy of a blockof contiguous spins in a large finite quantum Ising chain in a transversefield, with couplings of different types: homogeneous, periodically modulatedand random. We carry out a systematic study of finite-size effects at thequantum critical point, and evaluate subleading corrections both for open andfor periodic boundary conditions. For a block corresponding to a half of afinite chain, the position of the maximum of the entropy as a function of thecontrol parameter (e.g. the transverse field) can define the effective criticalpoint in the finite sample. On the basis of homogeneous chains, we demonstratethat the scaling behavior of the entropy near the quantum phase transition isin agreement with the universality hypothesis, and calculate the shift of theeffective critical point, which has different scaling behaviors for open andfor periodic boundary conditions.

Finite-size scaling of the entanglement entropy of the quantum Ising chain with homogeneous, periodically modulated and random couplings