# SM2: Classical and Quantum Phase Transitions

**Module title: Classical and Quantum Phase Transitions (SM2)**

**Module convenor:** Dr Stephen Powell (Nottingham)

11 Nov – 10 Dec 2015, Wed & Thu 10:00–11:00 (total 10 hours)

A phase transition is a sudden qualitative change in the properties of a physical system when a parameter, such as temperature, is tuned through a critical value. In the vicinity of such transitions, remarkable behaviour can be observed, such as universality and (approximate) scale invariance. These critical phenomena can be understood using the “renormalization group”, which relates a set of effective descriptions on a range of length scales, and also provides a framework for the calculation of physical observables.

This module will give an introduction to phase transitions, critical phenomena, and the renormalization group, mainly in the context of classical and quantum spin models. It is intended mainly for students in condensed matter theory, but much of the material is relevant for other areas of theoretical physics.

**Topics:**

• Classical spin models; phase structures and critical properties

• Landau theory of phase transitions; Ginzburg–Landau theory

• Euclidean field theory; diagrammatic expansion

• Renormalization group; momentum-shell RG

• Quantum phase transitions; quantum Ising chain

**Assessment:**

There are two problem sets covering the material in the module, due Friday 15 January 2016.

**Recommended references:**

• *Scaling and Renormalization in Statistical Physics*, J. Cardy (Cambridge, 1996)

• *Lectures on Phase Transitions and the Renormalization Group*, N. Goldenfeld (Addison-Wesley, 1992)

• *Quantum Phase Transitions*, S. Sachdev (Cambridge, 2011)

• M. E. Fisher, *The renormalization group in the theory of critical behavior*, Rev. Mod. Phys. **46**, 597 (1974)

• K. G. Wilson and J. Kogut, *The renormalization group and the ε expansion*, Phys. Rep. **12**, 75 (1974)

**Interactive demonstrations:**

• Ising model Monte Carlo

• Ising model mean-field theory