  The linear perturbation group transformation (LPRG) is used to study the
thermodynamics of the axial next-nearest-neighbor Ising model with four spin
interactions (extended ANNNI) in a field. The LPRG for weakly interacting Ising
chains is presented. The method is used to study finite field
para-ferrimagnetic phase transitions observed in layered uranium compounds,
UAs_{1-x}Se_x, UPd_2Si_2 or UNi_2Si_2. The above-mentioned systems are made of
ferromagnetic layers and the spins from the nearest-neighbor and
next-nearest-neighbor layers are coupled by the antiferromagnetic interactions
J_1<0 and J_2<0, respectively. Each of these systems exhibits a triple point in
which two ordered phases (ferrimagnetic and incommensurate) meet the
paramagnetic one, and all undergo the high field phase transition from para- to
ferrimagnetic (++-) phase. However, if in UAs_{1-x}Se_x the para-ferri phase
transition is of the first order as expected from the symmetry reason, in
UT_2Si_2 (T=Pd,Ni) this transition seems to be a continuous one, at least in
the vicinity of the multicritical point. Within the MFA, the critical character
of the finite field para-ferrimagnetic transition at least at one isolated
point can be described by the ANNNI model supplemented by an additional, e.g.,
four-spin interaction. However, in LPRG approximation for the ratio
\kappa=J_2/J_1 around 0.5 there is a critical value of the field for which an
isolated critical point also exists in the original ANNNI model. The positive
four-spin interaction shifts the critical point towards higher fields and
changes the shape of the specific heat curve. In the latter case for the fields
small enough, the specific heat exhibits two-peak structure in the paramagnetic
phase.
