We construct a class of solvable models for 2+1D quantum critical points by attaching 1+1D conformal field theories (CFTs) to fluctuating domain walls forming a “loop soup”. Specifically, our local Hamiltonian attaches gapless spin chains to the domain walls of a triangular lattice Ising antiferromagnet. The macroscopic degeneracy between antiferromagnetic configurations is split by the Casimir energy of each decorating CFT, which is usually negative and thus favors a short loop phase with a finite gap. However, we found a set of 1D CFT Hamiltonians for which the Casimir energy is effectively positive, making it favorable for domain walls to coalesce into a single “snake” which is macroscopically long and thus hosts a CFT with a vanishing gap. The snake configurations are geometrical objects also known as fully-packed self-avoiding walks or Hamiltonian walks which are described by an \mathrm{O} O (n=0) loop ensemble with a non-unitary 2+0D CFT description. Combining this description with the 1+1D decoration CFT, we obtain a 2+1D theory with unusual critical exponents and entanglement properties. Regarding the latter, we show that the log contributions from the decoration CFTs conspire with the spatial distribution of loops crossing the entanglement cut to generate a “non-local area law”. Our predictions are verified by Monte Carlo simulations.
The study of critical phenomena in two-dimensional statistical physics is mainly performed with the help of conformal field theory and integrable models. The relationship between these two formalisms is an active field of research, particularly in the framework of the so-called non-rational theories. This thesis is focused on certain critical systems described by an extended conformal theory : a theory that presents additional symmetries. The first problem studied is the fully packed loop model (FPL). Loop models are non-local statistical models based on the description of assembly of polymers. In particular, they represent the interfaces formed by spin models. The FPL model is integrable and its spectrum reflects an underlying symmetry Uq(sl(3)). The link between this model and the W3 symmetry, a conformal symmetry extended by a three-dimensional field, is studied in detail, numerically (by exact diagonalization) and analytically. The relationship with loop models leads to the study of the non-scalar operator content of the W3 theory. The second problem concerns the calculation of entanglement in unidimensional quantum systems. In this context, the preferred object of study is the entropy of entanglement between a subsystem and its complement. For the fundamental state of a spin chain, the behaviour of this entropy as a function of the size of the subsystem is a clear marker of the criticality of the chain. In this manuscript, a new way of calculating these entropies in critical models is presented. It is based on conformal theories extended by a symmetry called orbifold. This method is particularly applicable to entropies of excited states or disjointed subsystems.
The primary aim of these lecture notes is to introduce the modern approach to two-dimensional conformal field theory (2D CFT). The study of analytical methods in two-dimensional conformal field theory has developed over several decades, starting with BPZ. The development of analytical methods, particularly in rational conformal field theory (RCFT), has been remarkable, with complete classifications achieved for certain model groups. One motivation for studying CFT comes from its ability to describe quantum critical systems. Given that realistic quantum critical systems are fundamentally RCFTs, it is somewhat natural that the analytical methods of RCFT have evolved significantly. CFTs other than RCFTs are called irrational conformal field theories (ICFTs). Compared to RCFTs, the study of ICFTs has not progressed as much. Leaving aside whether there is physical motivation or not, ICFTs inherently possess a difficulty that makes them challenging to approach. However, with the development of quantum gravity, the advancement of analytical methods for ICFTs has become essential. The reason lies in the AdS/CFT correspondence. AdS/CFT refers to the relationship between
d+1 dimensional quantum gravity and
d dimensional CFT. Within this correspondence, the CFT appears as a non-perturbative formulation of quantum gravity. Except in special cases, this CFT belongs to ICFT. Against this backdrop, the methods for ICFTs have developed rapidly in recent years. Many of these ICFT methods are indispensable for modern quantum gravity research. Unfortunately, these cannot be learned from textbooks on 2D CFTs, such as Yellow book. These lecture notes aim to fill this gap. Specifically, we will cover techniques that have already been applied in many studies, such as HHLL block and monodromy method, and significant results that have become proper nouns, such as Hellerman bound and HKS bound.