Cure to the Landau-Pomeranchuk and associated long-wavelength Fermi-surface instabilities on the lattice

The cure to the l = 1 Landau-Pomeranchuk instabilities in translationally invariant fermions is shown to be a state with an anisotropic gap at the Fermi surface. For higher l and for fermions on a lattice, the general criteria for long-wavelength instabilities and their cure are found in terms of the derivatives of the single particle self-energy with respect to momentum for spin-symmetric instabilities and with respect to magnetic field for spin-antisymmetric instabilities. The results may be relevant to identifying hidden order parameters found in many metals.