Chaos and entanglement spreading in a non-commutative gauge theory (1808.10050v4)

Abstract: Holographic theories with classical gravity duals are maximally chaotic: they saturate a set of bounds on the spread of quantum information. In this paper we question whether non-locality can affect such bounds. Specifically, we consider the gravity dual of a prototypical theory with non-local interactions, namely, $\mathcal{N}=4$ non-commutative super Yang Mills. We construct shock waves geometries that correspond to perturbations of the thermofield double state with definite momentum and study several chaos related properties of the theory, including the butterfly velocity, the entanglement velocity, the scrambling time and the maximal Lyapunov exponent. The latter two are unaffected by the non-commutative parameter $\theta$, however, both the butterfly and entanglement velocities increase with the strength of the non-commutativity. This implies that non-local interactions can enhance the effective light-cone for the transfer of quantum information, eluding previously conjectured bounds encountered in the context of local quantum field theory. We comment on a possible limitation on the retrieval of quantum information imposed by non-locality.

• The paper confirms that the Lyapunov exponent and scrambling time remain bounded and robust despite non-commutative interactions.
• The butterfly velocity, however, increases along non-commutative directions and can exceed the speed of light due to non-locality.
• The entanglement velocity also becomes superluminal with non-commutativity but stays below the butterfly velocity, supporting a universal conjecture.

Chaos and Entanglement Spreading in Non-Commutative Gauge Theory

The paper "Chaos and entanglement spreading in a non-commutative gauge theory" by Fischler, Jahnke, and Pedraza explores the impact of non-locality on chaotic dynamics in the context of a holographic dual description of non-commutative gauge theories. Specifically, the authors investigate how non-local interactions in non-commutative $\mathcal{N}=4$ super Yang-Mills theory influence bounds on the spread of quantum information.

The research focuses on quantifying chaos using several key parameters: the butterfly velocity $v_B$, entanglement velocity $v_E$, scrambling time $t_*$, and the maximal Lyapunov exponent $\lambda_L$. In holographic settings, these parameters provide insights into the behavior of quantum systems under perturbations.

Main Findings

1. Lyapunov Exponent and Scrambling Time:
   • The paper confirms that, even in the presence of non-commutative interactions, the Lyapunov exponent saturates the proposed bound $\lambda_L = 2\pi/\beta$, indicating maximal chaos similar to that observed in commutative field theories with gravity duals.
   • The scrambling time $t_*$, which marks the time scale over which information becomes irreversibly mixed, remains logarithmically dependent on the system's entropy. Interestingly, non-commutativity does not affect these parameters, indicating their robustness to certain modifications in field theory interactions.
2. Butterfly Velocity:
   • The butterfly velocity, however, is found to be sensitive to non-local interactions. Along directions aligned with the non-commutativity, $v_B$ increases with the non-commutative parameter and can exceed the speed of light under strong non-local conditions. In contrast, the component of $v_B$ along commutative directions maintains values typical of conformal field theories, unaffected by $\theta$.
   • This increase in $v_B$ implies that non-locality facilitates faster spreading of information, challenging the previously understood causality bounds. However, due to the lack of Lorentz invariance, such superseding of causal limits is theoretically permissible.
3. Entanglement Velocity:
   • The entanglement velocity $v_E$ quantifies the speed at which entanglement spreads through a system upon perturbation. For non-commutative strips, $v_E$ also surpasses the speed of light with increasing $\theta$, echoing computational studies in similar non-commutative setups.
   • Despite this superluminal behavior, the paper observes that $v_E$ remains consistently lower than $v_B$, thereby supporting a conjecture that $v_E\leq v_B$ might be a universal trait, even for non-local quantum systems.

Implications and Future Directions

The findings illustrate that non-local interactions in quantum field theories profoundly affect the dynamics of chaos and entanglement. Non-commutative deformation elevates the effective light-cone for information spread, thus presenting an intriguing area of paper for understanding information transfer in quantum systems beyond the limits set by locality.

The results also indicate an avenue for revisiting quantum causal bounds where non-locality is inherent. Further explorations could involve other forms of non-localities, including those present in different string theory constructs, to evaluate the broader applicability of these insights.

Future research might explore correlating such non-locality-induced modifications in chaotic dynamics with observables, potentially expanding the bridge between gauge theories and their holographic gravity duals. Moreover, these studies might shed light on analogous phenomena in condensed matter systems featuring strong correlations and non-trivial topologies, thereby enriching the tapestry of theoretical physics with implications across disciplines.