By Eduardo C Marino
Cambridge University Press
This book provides an excellent overview of the state of the art of quantum field theory (QFT) applications to condensed-matter physics (CMP). Nevertheless, it is probably not the best choice for a first approach to this wonderful discipline.
QFT is used to describe particles in the relativistic (high-energy intensity) regime, but, as is well known, its methods can also be applied to problems involving many interacting particles – typically electrons. The conventional way of studying solid-state physics and, in particular, silicon devices does not make use of QFT methods due to the success of models in which independent electrons move in a crystalline substrate. Currently, though, we deal with various condensed-matter systems that are impervious to that simple model and could instead profit from QFT tools. Among them: superconductivity beyond the Bardeen–Cooper–Schrieffer approach (high-temperature superconducting cuprates and iron-based superconductors), the quantum Hall effect, conducting polymers, graphene
and silicene.
The author, as he himself states, aims to offer a unified picture of condensed-matter theory and QFT. Thus, he highlights the interplay between these two theories in many examples to show how similar mechanisms operate in different systems, despite being separated by several orders of magnitude in energy. He discusses, for example, the comparison between the Landau–Ginzburg field of a superconductor with the Anderson–Higgs field in the Standard Model. He also explains the not-so-well-known relation between the Yukawa mechanism for mass generation of leptons and quarks, and the Peierls mechanism of gap generation in polyacetylene: the same trilinear interaction between a Dirac field, its conjugate and a scalar field that explains why polyacetylene is an insulator, is responsible for the mass of elementary particles.
The book is structured into three parts. The first covers conventional CMP (at advanced undergraduate level). The second provides a brief review of QFT, with emphasis on the mathematical analysis and methods appropriate for non-trivial many-body systems (as, in particular, in chapters eight and nine, where a classical and a quantum description of topological excitations are given). I found the pages devoted to renormalisation remarkable, in which the author clearly exposes that the renormalisation procedure is a necessity due to the presence of interactions in any QFT, not to that of divergences in a perturbative approach. The heart of the book is part three, composed of 18 chapters where the author discusses the state of the art of condensed-matter systems, such as topological insulators and even quantum computation.
The last chapter is a clear example of the non-conventional approach proposed by the author: going straight to the point, he does not explain the basics of quantum computation, but rather discusses how to preserve the coherence of the quantum states storing information, in order to maintain the unitary evolution of quantum data-processing algorithms. In his words, “the main method of coherence protection involves excitation, having the so-called non-abelian statistics”, which, going back to CMP, takes us to the realm of anyons and Majorana qubits. In my opinion, this book is not suitable for undergraduate or first-year graduate students (for whom I see as more appropriate, the classic Condensed Matter Field Theory by Altland and Simons). Instead, I would keenly recommend this to advanced graduate students and researchers in the field, who will find, in part three, plenty of hot topics that are very well explained and accompanied by complete references.
