  The entanglement entropy of a subsystem $A$ of a quantum system is expressed,
in the replica method, through analytic continuation with respect to n of the
trace of the n-th power of the reduced density matrix $\tr\rho_A^n$. We study
the analytic properties of this quantity as a function of n in some quantum
critical Ising-like models in 1+1 and 2+1 dimensions. Although we find no true
singularities for n>0, there is a threshold value of n close to 2 which
separates two very different `phases'. The region with larger n is
characterized by rapidly convergent Taylor expansions and is very smooth. The
region with smaller n has a very rich and varied structure in the complex n
plane and is characterized by Taylor coefficients which instead of being
monotone decreasing, have a maximum growing with the size of the subsystem.
Finite truncations of the Taylor expansion in this region lead to increasingly
poor approximations of $\tr\rho_A^n$. The computation of the entanglement
entropy from the knowledge of $\tr\rho^n_A$ for positive integer n becomes
extremely difficult particularly in spatial dimensions larger than one, where
one cannot use conformal field theory as a guidance in the extrapolations to
n=1.
