  In this paper we study the properties of two-dimensional CFTs defined by
cyclic and symmetric orbifolds of free Dirac fermions, especially by focusing
on the partition function and entanglement entropy. Via the bosonization, we
construct the twist operators which glue two complex planes to calculate the
partition function of Z_2 orbifold CFT on a torus. We also find an expression
of Z_N cyclic orbifold in terms of Hecke operators, which provides an explicit
relation between the partition functions of cyclic orbifolds and those of
symmetric ones. We compute the entanglement entropy and Renyi entropy in cyclic
orbifolds on a circle both for finite temperature states and for time-dependent
states under quantum quenches. We find that the replica method calculation is
highly non-trivial and new because of the contributions from replicas with
different boundary conditions. We find the full expression for the Z_2 orbifold
and show that the periodicity gets doubled. Finally, we discuss extensions of
our results on entanglement entropy to symmetric orbifold CFTs and make a
heuristic argument towards holographic CFTs.
