We examine the instabilities of a Fermi liquid to other ground states of a dense gas of fermions. Possibilities include ferromagnets, states in which there is spin or charge density wave order or various types of superconductors. All of these cases are of considerable practical importance and have numerous experimental applications.
A theoretical treatment of the quantum transition between a Fermi liquid and a magnetically or charge ordered state was given in a paper by Hertz [31], although many important points were anticipated in earlier work [5,45,46,54,44]. We present Hertz's basic arguments for the case of a transition between a Fermi liquid and a spin density wave state. We do not treat the other cases here and, instead, refer the reader to the literature. There are a number of reasons for this neglect:
(i) Many aspects of these transitions are not fully understood, and are the subject of considerable debate in the literature.
(ii) We shall only consider systems in spatial dimensions d ³ 2 here (the d = 1 case requires a separate treatment appropriate to Tomonaga-Luttinger liquids, and will be addressed in Chapter 14). For these dimensions, the quantum critical point is invariably at or above its upper critical dimension. As a result, nonuniversal features abound, and the details of the particular microscopic situation under consideration are often important. A unified treatment of all the cases is hardly possible, and we choose, instead, to focus on a single representative case.
(iii) Details of the topology of the Fermi surface topology often matter, and this adds to the zoo of experimental possibilities. A single illustration for a particular model is however adequate to make the basic point.