Background

Second-order phase transitions play a central role in both classical and quantum statistical mechanics. At these critical points, systems undergo dramatic changes in their macroscopic behavior due to diverging correlation lengths and the emergence of long-range fluctuations. In this regime, traditional mean-field theories break down, and a full understanding requires the use of advanced quantum field theoretical techniques.

Our research focuses on the universal properties that characterize systems near criticality. We approach this problem using a broad set of tools from modern field theory. In particular, conformal field theories (CFTs) provide a powerful framework to describe the physics at the critical point. These theories are distinguished by their invariance under scale and conformal transformations, which emerge naturally when the correlation length becomes infinite.

We explore conformal field theories from both perturbative and non-perturbative perspectives. On the perturbative side, we use conformal perturbation theory and renormalization group methods to study how physical observables and coupling constants evolve as the system moves away from the critical point. These techniques allow us to characterize relevant and irrelevant deformations and to map out the phase diagram near criticality. On the non-perturbative side, we exploit dualities, such as boson-fermion correspondences, which provide a deep understanding of critical systems by relating them to simpler, tractable models. We also analyze exact results in low-dimensional systems, drawing from the rich structure of conformal symmetry.

To complement these analytical approaches, we employ large-scale Monte Carlo simulations. These numerical techniques allow us to test theoretical predictions and explore regimes that may be analytically inaccessible. By combining theory and computation, we gain a more comprehensive picture of the behavior of physical systems in the vicinity of phase transitions.

Our interests also extend to topological quantum field theories (TQFTs), a class of field theories where observables are insensitive to local geometric details and depend only on global properties. When defined on manifolds with boundaries, these theories acquire physical relevance in the description of topologically ordered phases of matter. Such phases do not fit within the conventional framework of symmetry breaking and local order parameters, and instead exhibit long-range entanglement and robust ground-state degeneracy. TQFTs provide effective descriptions of these novel states, which appear in quantum Hall systems, topological insulators, and potentially in fault-tolerant quantum computing platforms.