Abstract
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-β 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-β SU(3) theory, the resulting Landau-Lifshitz model has classically chaotic dynamics at finite values of the deformation parameter.
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McLoughlin, T., Spiering, A. Chaotic spin chains in AdS/CFT. J. High Energ. Phys. 2022, 240 (2022). https://doi.org/10.1007/JHEP09(2022)240
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DOI: https://doi.org/10.1007/JHEP09(2022)240