Integrable systems: From the ice rule to supersymmetric fishnet Feynman diagrams (2509.03416v1)

Abstract: This thesis examines the correspondence between models of statistical physics and Feynman graphs of quantum field theories (QFTs) by a common property: integrability. We review integrable structures for periodic boundary conditions on both sides, while focusing on the eight- and six-vertex model and the bi-scalar fishnet theory. The latter is a double-scaled $\gamma$-deformation of $\mathcal{N} = 4$ super Yang-Mills theory. Interesting applications of integrability existing in the literature that we reconsider are the computation of the free energy in the thermodynamic limit and its QFT counterpart, the critical coupling. In addition, we provide a detailed overview of the calculation of exact anomalous dimensions and operator product expansion (OPE) coefficients in the conformal bi-scalar fishnet theory. The original contributions of this work comprise the results of the critical coupling for models with fermions, the brick wall theory, and the fermionic fishnet theory. Additionally, we extend the study of integrable Feynman graphs to supersymmetric diagrams in superspace. By establishing an efficient graphical formalism, we obtain the critical coupling of double-scaled $\beta$-deformations of $\mathcal{N} = 4$ super Yang-Mills theory and Aharony-Bergman-Jafferis-Maldacena theory, the super brick wall and superfishnet theory, respectively. Moreover, we apply superspace methods to the superfishnet theory and find results for anomalous dimensions and an OPE coefficient, which are all-loop exact in the coupling. In addition, we study boundary integrability in the six-vertex model and for Feynman diagrams. We present new box-shaped boundary conditions for the six-vertex model and conjecture a closed form for its partition function at any lattice size. On the QFT side, we find integrable boundary scattering matrices in the form of generalized Feynman diagrams by graphical methods.