PART II: Quantum Ising and Rotor Models

Chapter 4
The Ising chain in a transverse field

This chapter describes in considerable detail the quantum critical properties of the quantum Ising model in one spatial dimension. All of the results in this chapter are believed to be exact, but the physically oriented reader should not be turned off by this: technical details are kept to a minimum, and the exact results are obtained by physical arguments which do much to illustrate the main underlying principles. Most of the important concepts of this book appear in the simple model under consideration: the remainder of the book is largely a description of similar phenomena in more complicated settings.

A physical picture of the crossovers in the non-zero temperature dynamical properties in the vicinity of the quantum critical point is developed. Effective classical models are shown to be adequate at low temperatures, but distinct models appear on either side of the quantum critical point. At higher temperatures, in the `quantum-critical' regime, no classical description is possible and quantum and thermal fluctuations play equally important roles: h/kB T appears as a fundamental relaxation time in the dynamics.