  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 $N$.
