Cure to the Landau-Pomeranchuk and Associated Long-Wavelength Fermi-surface Instabilities on Lattices

The cure to the $\mathrm{\ell }=1$ Landau-Pomeranchuk (L-P) instabilities in translationally invariant fermions is shown to be a state with an anisotropic gap at the fermi-surface. For higher $\mathrm{\ell }$ and for fermions on a lattice, 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.