Conformal Field Theory and Statistical Mechanics

TL;DR

This work surveys two-dimensional conformal field theory as it applies to critical phenomena, presenting a physically intuitive toolkit that connects lattice scaling limits to CFT data. It develops the stress-tensor formalism, Ward identities, and the Virasoro algebra, and demonstrates how modular invariance and boundary CFT organize the spectrum and boundary content, including the ADE classification of minimal models. The Coulomb-gas approach and height/loop models bridge lattice realizations with continuum CFT, enabling explicit identifications with minimal models and revealing deep links to SLE for boundary phenomena. Together, these insights provide a coherent framework for analyzing finite-size effects, entanglement, and boundary phenomena in 2D critical systems, with broad implications across condensed matter, statistical mechanics, and mathematical physics.

Abstract

The lectures provide a pedagogical introduction to the methods of CFT as applied to two-dimensional critical behaviour.

Figures (13)

• Figure 1: We consider an infinitesimal transformation which is conformal within $C$ and the identity in the complement in $\cal D$.
• Figure 2: The density matrix is given by the path integral over a space-time region in which the rows and columns are labelled by the initial and final values of the fields.
• Figure 3: The reduced density matrix $\rho_A$ is given by the path integral over a cylinder with a slit along the interval $A$.
• Figure 4: ${\rm Tr}\,\rho_A^n$ corresponds to sewing together $n$ copies so that the edges are connected cyclically.
• Figure 5: The state $|\phi_j\rangle$ is defined by weighting field configurations on the circle with the path integral inside, with an insertion of the field $\phi_j(0)$ at the origin.