Chaotic spin chains in AdS/CFT

Abstract

We consider the spectrum of anomalous dimensions in planar $$ \mathcal{N} $$ N = 4 supersymmetric Yang-Mills theory and its $$ \mathcal{N} $$ N = 1 super-conformal Leigh-Strassler deformations. The two-loop truncation of the integrable $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ 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.

Chaotic string motion in a near pp-wave limit

“…The stringy ansatz (4.2) reflects the finite size of a string while it has not been taken into account in the ansatz employed in [22]. This observation is also consistent with the result of [14] in the Laudau-Lifshitz limit [26,27]. It is because the spatial direction of the string world-sheet is taken to be infinite in this limit and the finite size effect of a string is dropped off.…”

“…It is because the spatial direction of the string world-sheet is taken to be infinite in this limit and the finite size effect of a string is dropped off. Hence classical chaos does not appear in this limit as discussed in [14]. On the other hand, the weak chaos in the next-nearestneighbor interacting spin chain has been discussed before taking the thermodynamic limit and the finite-size effect has been included in this analysis.…”

“…It is instructive to remember the result of [14]. When the full-order dilatation operator is truncated at two-loop level, the resulting system is described as a Heisenberg spin chain Hamiltonian with a next-nearest-neighbor (NNN) interaction:…”

“…Motivated by the result of [14], it is intriguing to study the string-theory side. One may consider a Penrose limit [15,16] of the AdS 5 ×S 5 background.…”

“…More intriguingly, the integrability may also be broken even in the large N limit in the weak sense discussed in [10]. When all-loop integrable spin chain [11][12][13] is truncated to finite order, the resulting system may exhibit quantum chaos [14]. That is, even if the full-order system is integrable, a finite-order truncation may lead to a nonintegrable system.…”