Chaotic spin chains in AdS/CFT

We consider the spectrum of anomalous dimensions in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory and its $\mathcal{N}=1$ super-conformal Leigh-Strassler deformations. The two-loop truncation of the integrable $\mathcal{N}=4$ dilatation operator in the SU$(2)$ sector, which is a next-to-nearest-neighbour deformation of the XXX spin chain, is not strictly integrable at finite coupling and we show that it indeed has Wigner-Dyson level statistics. However, we find that it is only weakly chaotic in the sense that the cross-over to chaotic dynamics is slower than for generic chaotic systems. For the Leigh-Strassler deformed theory with generic parameters, we show that the one-loop dilatation operator in the SU$(3)$ sector is chaotic, with a spectrum that is well described by GUE Random Matrix Theory. For the imaginary-$\beta$ deformation, the statistics are GOE and the transition from the integrable limit is that of a generic system. This provides a weak-coupling analogue of the chaotic dynamics seen for classical strings in the dual background. We further study the spin chains in the semi-classical limit described by generalised Landau-Lifshitz models, which are also known to describe large-angular-momentum string solutions in the dual theory. We show that for the higher-derivative theory following from the two-loop $\mathcal{N}=4$ SU$(2)$ spin chain, the maximal Lyapunov exponent is close to zero, consistent with the absence of chaotic dynamics. For the imaginary-$\beta$ SU$(3)$ theory, the resulting Landau-Lifshitz model has classically chaotic dynamics at finite values of the deformation parameter.Comment: 37 pages, 13 figure