On the basis of an \ensuremath{\epsilon} expansion, we present a renormalization-group analysis of the anisotropic next-nearest-neighbor Ising model in a field. For small fields, the \ensuremath{\gamma} surface remains XY-like. For larger fields, we confirm the existence of lines of tricritical points which are governed by Gaussian-type exponents. We pay special attention to the many couplings between a two-component critical spin field, associated with the main harmonic component of the magnetization, and several noncritical spin fields, associated with the uniform and the higher harmonic components. We remark that it is necessary to take into account the uniform and the second-harmonic components of the magnetization to describe the tricritical behavior.
A nuclear Fermi liquid exhibits competing pairing instabilities in different spin, isospin, and orbital channels. In a Fermi-surface renormalization group (RG) treatment, the channel that develops a pole first is determined not only by its tree-level attraction but also by its one-loop RG coefficient. We illustrate this mechanism in a minimal
S/P-wave model. A spherical Fermi surface establishes the reference competition between the lowest even- and odd-parity interactions. Axial deformation changes the relevant Fermi-surface integrals and lifts the degeneracy between longitudinal and transverse
P-wave components. In isospin-asymmetric matter, neutron--proton Fermi-momentum splitting restricts the simultaneous low-energy contribution of the two species and can terminate the
np running at finite threshold scales. Our calculations are intended as controlled one-loop RG illustrations rather than as quantitative nuclear-matter calculations. We show how the Fermi-surface geometry and composition can change the ordering of competing pairing instabilities.