Introduction to Conformal Invariance and Its Applications to Critical Phenomena

Abstract

This is an introduction to conformal invariance and two-dimensional critical phenomena for graduate students and condensed-matter physicists. After explaining the algebraic foundations of conformal invariance, numerical methods and their application to the Ising, Potts, Ashkin-Teller and XY models, tricritical behaviour, the Yang-Lee singularity and the XXZ chain are presented. Finite-size scaling techniques and their conformal extensions are treated in detail. The vicinity of the critical point is studied using the exact $S$-matrix approach, the truncation method, the thermodynamic Bethe ansatz and asymptotic finite-size scaling functions. The integrability of the two-dimensional Ising model in a magnetic field is also dealt with. Finally, the extension of conformal invariance to surface critical phenomena is described and an outlook towards possible applications in critical dynamics is given.