Quantum phase transitions remain an exciting and vigorous scientific subject. This field has experienced continuous theoretical progress. Equally important is the intense experimental work the field has engendered, including the discovery of new materials exhibiting different forms of quantum criticality that in turn further advance the theory. Thus, the new, enlarged edition of this book is more than timely.

The layout maintains the general structure of the first edition. New material has been added to the original chapters, and some of them have been modified, even substantially – such as the one on first-order quantum phase transitions. Moreover, I have written entirely new chapters on the study of topological quantum phase transitions and superconducting quantum critical points.

The book emphasises a general scaling approach that considers the change of parameters of a system under length and timescales close to quantum criticality. This provides a unifying view to such diverse phenomena as zero temperature magnetic transitions, metal-insulator, topological, superconductor and first-order quantum phase transitions. As in the previous edition, the renormalisation group with its concepts such as fixed points associated with scale invariance, crossover, and relevant and irrelevant interactions, provides the mathematical framework for the scaling ideas. It is the basic tool used to study, understand and describe quantum critical phenomena, even when the usual Landau paradigm fails, as for topological quantum phase transitions.

I would like to thank the many colleagues with whom I have been discussing and learning the subjects of this book. Among them, G. Aeppli, Elisa Baggio-Saitovitch, Sergey Bud'ko, Premala Chandra, Piers Coleman, Mauricio D. Coutinho-Filho, Alvaro Ferraz, Zach Fisk, Jacques Flouquet, Magda Fontes, Alex Lacerda, Mireille Lavagna, Hilbert v. Löhneysen, Andy J. Millis, Luiz Nunes de Oliveira, Pascoal Pagliuso, Silke Paschen, Aline Ramires, Daniel Reyes-Lopez, Peter Riseborough, Subir Sachdev, Andre Schuwartz, Qimiao Si, Frank Steglich, Manuel Nunez-Regueiro, Jean-Louis Tholence, Joe Thompson, Amos Troper and the lates Bruno Elschner and Bernard Coqblin. My sincere thanks to Alfredo Ozorio de Almeida, Claudine Lacroix, Eduardo Marino, Eduardo Miranda, Enzo Granato, Heron Caldas, Jose A. Helayël-Neto, Jose Abel Hoyos-Neto, Pedro Sacramento and Tharnier Puel de Oliveira for enlightening discussions and several suggestions to improve the new manuscript. The author, however, is solely responsible for any mistakes or omissions in the book.

I would like also to express my recognition of the encouragement of my wife Sonia, my daughter Ilana and my son Marcelo.