Fundamental Problems of Quantum Critical Phenomena

Giovanni Battimelli, Giovanni Ciccotti, Pietro Greco

Shang-keng Ma

An important contributor to our current understanding of critical phenomena, Ma introduces the beginner--especially the graduate student with no previous knowledge of the subject-to fundamental theoretical concepts such as mean field theory, the scaling hypothesis, and the renormalization group. He then goes on to apply the renormalization group to selected problems, with emphasis on the underlying physics and the basic assumptions involved.

Shaolong Zeng, Sue ping Szeto, Fan Zhong

Abstract Phase transitions and critical phenomena are among the most intriguing phenomena in nature and the renormalization-group theory for them is one of the greatest achievements of theoretical physics. However, the predictions of the theory above an upper critical dimension d c seriously disagree with reality. In addition to its fundamental significance, the problem is also of practical importance because both complex classical systems with long-range spatial or temporal interactions and quantum phase transitions with long-range interactions can substantially lower d c . The extant scenarios built on a dangerous irrelevant variable (DIV) to resolve the problem introduce two sets of critical exponents and even two sets of scaling laws whose origin is obscure. Here, we consider the DIV from a different perspective and clearly unveil the origin of the two sets of exponents and hence the intrinsic inconsistency in those scenarios. We then develop an effective-dimension theory in which critical fluctuations and system volume are fixed at an effective dimension by the DIV. This enables us to account for all the extant results consistently. A novel asymptotic finite-size scaling behavior for a correlation function together with a new anomalous dimension and its associated scaling law is also predicted. We also apply the theory to quantum phase transitions.

This book is the third volume of review papers on advanced problems of phase transitions and critical phenomena, following the success of the first two volumes in 2004 and in 2007. Broadly, the book aims to demonstrate that the phase transition theory, which experienced its 'golden age' during the 70s and 80s, is far from over and there is still a good deal of work to be done, both at the fundamental level and in respect of applications. This volume presents a broad spectrum of problems connected with criticality. It covers its theoretical backgrounds, analytical approaches and numerical simulations to describe criticality in specific systems (ionic fluids, diluted magnets, polymers), as well as phase transitions on complex networks and in the minority game model. As the first two volumes, this book is based on the review lectures that were given in Lviv (Ukraine) at the "Ising lectures" - a traditional annual workshop on phase transitions and critical phenomena which brings together scientists working in the field with university students and those who are interested in the subject.

Amit Dutta, Chakrabarti, B K