One-Loop Non-Planar Anomalous Dimensions in Super Yang-Mills Theory

We consider non-planar one-loop anomalous dimensions in maximally supersymmetric Yang-Mills theory and its marginally deformed analogues. Using the basis of Bethe states, we compute matrix elements of the dilatation operator and find compact expressions in terms of off-shell scalar products and hexagon-like functions. We then use non-degenerate quantum-mechanical perturbation theory to compute the leading $1/N^2$ corrections to operator dimensions and as an example compute the large $R$-charge limit for two-excitation states through subleading order in the $R$-charge. Finally, we numerically study the distribution of level spacings for these theories and show that they transition from the Poisson distribution for integrable systems at infinite $N$ to the GOE Wigner-Dyson distribution for quantum chaotic systems at finite $N$.