Instabilities in Luttinger liquids

We discuss the appearance of magnetic and charge instabilities, called respectively metamagnetism (MM) and phase separation (PS), in systems which can be described as perturbed Luttinger liquids. We argue that such instabilities can be associated with the vanishing of the effective Fermi velocity v, which in some cases coincides with a divergence of the effective Luttinger parameter K. We analyse in particular an XXZ chain with next-nearest-neighbour interactions in different limits where MM shows up and an extended Hubbard model where PS occurs. Qualitative agreement with previous studies is found.