Many-Body Quantum Theory in Condensed Matter Physics—An Introduction

Abstract

This is undoubtedly an ambitious book. It aims to provide a wide ranging, yet self-contained and pedagogical introduction to techniques of quantum many-body theory in condensed matter physics, without losing mathematical `rigor' (which I hope means rigour), and with an eye on physical insight, motivation and application. The authors certainly bring plenty of experience to the task, the book having grown out of their graduate lectures at the Niels Bohr Institute in Copenhagen over a five year period, with the feedback and refinement this presumably brings. The book is also of course ambitious in another sense, for it competes in the tight market of general graduate/advanced undergraduate texts on many-particle physics. Prospective punters will thus want reasons to prefer it to, or at least give it space beside, well established texts in the field. Subject-wise, the book is a good mix of the ancient and modern, the standard and less so. Obligatory chapters deal with the formal cornerstones of many-body theory, from second quantization, time-dependence in quantum mechanics and linear response theory, to Green's function and Feynman diagrams. Traditional topics are well covered, including two chapters on the electron gas, chapters on phonons and electron–phonon coupling, and a concise account of superconductivity (confined, no doubt judiciously, to the conventional BCS case). Less mandatory, albeit conceptually vital, subjects are also aired. These include a chapter on Fermi liquid theory, from both semi-classical and microscopic perspectives, and a freestanding account of one-dimensional electron gases and Luttinger liquids which, given the enormity of the topic, is about as concise as it could be without sacrificing clarity. Quite naturally, the authors' own interests also influence the choice of material covered. A persistent theme, which brings a healthy topicality to the book, is the area of transport in mesoscopic systems or nanostructures. Two chapters, some fifty pages of the book, are devoted to electron transport in mesoscopic systems; the one on interacting systems is preceded by a brief account of equation of motion techniques–a relative rarity in a general text, used here to provide background to subsequent discussion of the Coulomb blockade in quantum dots. So does it work, and will it find a niche beside other established, wide ranging texts? On the whole I think the answer has to be yes. To begin with, the book is well organised and user-friendly, which must surely appeal to students (and their mentors). The chapters are typically bite-sized and digestible. Each is accompanied by a summary/outlook, which in doing just that attempts to place the specific topic in a wider context, together with a set of problems that illustrate, and in many cases expand substantially on, the basic subject matter. A particularly healthy feature of the book is the extent to which the authors have sought where possible to include physical and/or material applications of basic theory, thereby enlivening old material and enhancing appreciation of the new. The first chapter on the electron gas, for example, introduces the reader to a range of material examples, including 2D heterostructures, carbon nanotubes and quantum dots. A chapter on the formalism of Green's functions takes time out to explain how the single-particle spectral function can be measured by tunnelling spectroscopy, while discussion of impurity scattering and conductivity is refreshed by consideration of weak localization in bulk and mesoscopic systems, and the phenomenon of universal conductance fluctuations. And so on: in a text that could readily descend to the purely formal, the authors have clearly taken seriously the task of incorporating relevant, topical applications of the underlying theory. In a book as wide ranging as this any reviewer is of course bound to perceive the occasional deficiency. I felt for example that some aspects of the discussion of conductance in quantum dots, notably the Coulomb blockade and the Kondo effect, were not quite up to scratch—a touch unbalanced in coverage perhaps (no serious mention of renormalization or scaling, even perturbative), with the odd conceptual error creeping in, and the Kondo effect appearing more mysterious than it really is, possibly in part because consideration of it appears before the chapter on Fermi liquid theory, in which the effect is firmly rooted. But let me not miss the bigger picture: I don't doubt this is a pretty impressive book overall, likely to have broad appeal to budding theorists and adventurous experimentalists, either as a course textbook or—for the slave to garret or lab—as a serious source for self-study. So make a space on your bookshelves.