  We consider a spin-$\frac{1}{2}$ chain with competing nearest and
next-nearest neighbor interactions within a transverse magnetic field, which is
known to be an equiavelent to the ANNNI model. When studing thermodynamics of
the 2D ANNNI model Villain and Bak arrived to a free fermion approximation that
neglects heavy excitations from the ferromagnetic ground state, which is an
appropriate description close to the paramagnetic-ferromagnetic transition. In
the vicinity of the floating-phase/anti-phase transition another sort of
quasiparticles, but free fermions too, appears to be convenient. Although free
fermions are a suitable tool for investigation of the phase diagram and the
critical properties, they are defined on the fictitious lattice which makes the
analysis non-rigorous. Here we deal with a proper fermion scheme which is
especially effective %devised to describe the floating-phase/anti-phase
transition. for performing exact diagonalization calculations for cyclic
chains. Systems up to size $L=32$ has been analysed and the predictions of the
effective fermion Hamiltonian has been confirmed. Various predictions for the
infinite system and the critical properties are derived.
