Chaotic spin chains in AdS/CFT

TL;DR

This paper investigates the chaotic properties of spin chains in supersymmetric Yang-Mills theories and their deformations, revealing weak chaos at finite coupling and classical chaos in certain semi-classical limits, with implications for AdS/CFT correspondence.

Contribution

It demonstrates the presence of weakly chaotic and fully chaotic regimes in spin chains related to supersymmetric gauge theories, connecting quantum spectral statistics with classical chaos in dual string models.

Findings

Two-loop SU(2) spin chain shows Wigner-Dyson statistics and weak chaos.

SU(3) sector in Leigh-Strassler deformations exhibits GUE and GOE statistics.

Semi-classical Landau-Lifshitz models reveal classical chaos in certain deformations.

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-$\beta$ 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.