Introduction to Conformal Invariance and its Applications to Critical Phenomena

TL;DR

This paper introduces conformal invariance and explores its applications to two-dimensional critical phenomena, covering algebraic foundations, numerical methods, and various models, with an emphasis on finite-size scaling and integrability.

Contribution

It provides a comprehensive overview of conformal invariance techniques and their application to critical phenomena, including numerical methods and extensions to surface phenomena.

Findings

Numerical methods applied to Ising, Potts, and XY models.

Finite-size scaling techniques extended via conformal invariance.

Analysis of integrability and critical behavior near phase transitions.

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.