Separation of variables and correlation functions: from spin chains to CFT

Integrable quantum spin chains are fundamental models that are also widely used for the exploration of foundational aspects of quantum mechanics, offering a perfect arena to study key quantities such as entanglement, information scrambling, and quantum correlations. Yet, the utilised theoretical methods here still face major obstacles, especially for the computation of correlation functions. Recent years have seen substantial progress in the development of the powerful separation of variables (SoV) approach to quantum integrable models, which allows one to factorise the wavefunctions, opening gateways to many applications as well as clarifying the structure of the model's Hilbert space. In this talk, I will review the main results achieved in this program based on a series of recent papers with my collaborators. In particular, I will present the explicit construction of the SoV framework for integrable spin chains with gl(N) symmetry. I will explain how the SoV basis arises in this general setting, providing a representation-theoretic understanding of the method. I will then address a longstanding problem in the field, namely the computation of the SoV measure. I will show how our approach leads to a complete solution of this problem and, as a direct consequence, to new highly compact determinant representations for a broad class of physical quantities, including correlation functions and wavefunction overlaps. Furthermore, I will demonstrate the power and versatility of SoV in four-dimensional integrable conformal field theories, with a particular emphasis on the fishnet theory. In this context, I will present new results on the Yangian symmetry for a large and previously unexplored class of Feynman graphs. Finally, I will outline promising applications of these methods to the computation of exact correlators in planar N=4 super Yang–Mills theory and discuss several open directions for future research.