Scalar Conformal Field Theories from Lattice Systems

We discuss scalar conformal field theories (CFTs) that can be realized in structural phase transitions. The Landau condition and Lifshitz condition are reviewed, which are necessary conditions for a structural phase transition to be second order. We also review the perturbative analysis in $4-\epsilon$ expansion of the corresponding Landau actions, which were already analyzed thoroughly in the 80s. By identifying the global symmetries of these fixed points, it turns out that in perturbation theory only 6 different CFTs can be realized by commensurate structural phase transitions. ${\rm \textbf{Updated in version 2:}}$ We discuss how to classify all the phases of a Landau theory using the computer algebra system GAP. We also discuss the fully packed quantum loop model on the triangular lattice, where the Cubic CFT is realized. This is a lecture note based on a series of talks given by the author. The goal of the lecture note is to bridge the gap between condensed matter physicists and conformal field theorists. The note will be further updated in the future.