Prerequisites

As a theoretical option, this course will prove challenging to students without a mathematical background. Although the course will develop methods of statistical field theory from scratch, students will benefit from having attended either the Theories of Quantum Matter or Quantum Field Theory courses in Part III.

Learning Outcomes and Assessment

This course aims to develop a framework in which to describe critical properties associated with classical and quantum phase transitions, emphasising the importance and role played by symmetry and topology. The majority of the course will be involved in developing the important concept of universality in statistical mechanics. Connections will be made with quantum field theories of the standard model, and in particular, the Gauge Field Theory course.

Synopsis

Introduction to Critical Phenomena: Concept of Phase Transitions; Order Parameters; Response Functions; Universality.

Ginzburg-Landau Theory: Mean-Field Theory; Critical Exponents; Symmetry Breaking, Goldstone Modes, and the Lower Critical Dimension; Fluctuations and the Upper Critical Dimension; Importance of Correlation Functions; Ginzburg Criterion.

Scaling: Self-Similarity; The Scaling Hypothesis; Kadanoff’s Heuristic Renormalisation Group (RG); Gaussian Model; Fixed Points and Critical Exponent Identities; Wilson’s Momentum Space RG; Relevant, Irrelevant and Marginal Parameters; ε-expansions.

Topological Phase Transitions: Continuous Spins and the Non-linear σ-model; Running Coupling; XY-model; Algebraic Order; Topological Defects, Confinement, the Kosterlitz-Thouless Transition; Lattice Gauge Theory.

Quantum Phase Transitions: Classical/Quantum Mapping; the Dynamical Exponent; Quantum Rotors; Haldane Gap; Asymptotic Freedom; Quantum Criticality.

BOOKS
Statistical Physics of Fields, Kardar M (CUP 2007)

Principles of Condensed Matter Physics, Chaikin P M & Lubensky T C (CUP 2007)

Scaling and Renormalisation in Statistical Physics, Cardy J (CUP 1996)

Quantum Field Theory and Condensed Matter: An Introduction, Shankar R (CUP 2017)

Phase Transitions and Renormalization Group, Zinn-Justin J (OUP 2013)