Two-dimensional conformal field theory and the butterfly effect (1412.5123v3)

Abstract: We study chaotic dynamics in two-dimensional conformal field theory through out-of-time order thermal correlators of the form $\langle W(t)VW(t)V\rangle$. We reproduce bulk calculations similar to those of [1], by studying the large $c$ Virasoro identity block. The contribution of this block to the above correlation function begins to decrease exponentially after a delay of $\sim t_* - \frac{\beta}{2\pi}\log \beta^2E_w E_v$, where $t_*$ is the scrambling time $\frac{\beta}{2\pi}\log c$, and $E_w,E_v$ are the energy scales of the $W,V$ operators.

• The paper demonstrates that 2D CFT exhibits quantum chaos by analyzing OTOCs and defining a precise scrambling time t* = (β/2π) log c.
• It employs the large central charge limit and Virasoro conformal block analysis to connect quantum chaotic behavior with holographic shock wave phenomena.
• The study bridges theoretical insights between 2D CFT and holography, providing a framework for understanding rapid thermal scrambling in quantum systems.

Analysis of Chaotic Dynamics via Two-Dimensional Conformal Field Theory and the Butterfly Effect

The paper "Two-dimensional conformal field theory and the butterfly effect" by Daniel A. Roberts and Douglas Stanford investigates chaotic dynamics within the framework of two-dimensional conformal field theory (2D CFT) through the lens of out-of-time order correlators (OTOCs). This paper serves as a bridge between the established understanding of chaos in conformal settings and the implications suggested by gauge/gravity duality, specifically looking at shock waves on the horizons of Anti-de Sitter (AdS) black holes, as seen in prior holographic calculations.

Theoretical Background and Methodology

The authors utilize the large central charge ($c$) limit to analyze the Virasoro conformal block, which plays a pivotal role in deconstructing OTOCs. The considered correlation functions, $\langle W(t)V W(t)V \rangle_\beta$, reveal insights into how quantum chaos manifests in thermal states. Such correlators express a distinct temporal ordering that highlights the non-commutativity of quantum operators as a measure of chaotic scrambling—where initial differences amplify over time, analogous to the classical butterfly effect.

One of the key results relates to the scrambling time $t_*$, defined as $\frac{\beta}{2\pi}\log c$, where $\beta$ is the inverse temperature and $c$ is the central charge. This time signifies when the system transitions from predictable to chaotic behavior. The paper shows that after a delay $\sim t_* - \frac{\beta}{2\pi}\log(\beta^2 E_w E_v)$, the contribution of the Virasoro identity block in the above correlator begins to decline exponentially, signaling the onset of chaos.

Significance and Numerical Results

The primary contribution of this research is two-fold. Firstly, it refines the picture of quantum chaos in 2D CFTs, affirming that rapid scrambling signatures observed in higher-dimensional holographic calculations also apply in two-dimensional contexts. Secondly, the precise analytic continuation that delineates the boundary between time-ordered and out-of-time-ordered behaviors in CFTs without direct recourse to holography shows that the methods can be appreciated purely in the field of quantum field theory.

The analytical techniques employed capture both Euclidean and Lorentzian regimes, with the derived formulas rigorously substantiating the conditions under which the out-of-time ordering reveals chaotic dynamics. This is markedly shown in the correlation function, which a priori is not constrained by the expected classical locality in chaotic fields but rather demonstrates its suppression, reflective of quantum chaos.

Implications for Future Research

This paper opens numerous avenues for further exploration. The extension of these analyses to include stringy corrections in higher-genus conformal blocks or even wider classes of operators could unveil deeper insights into the signatures of quantum chaos. The techniques outlined also incentivize further studies into non-integrable models, potentially broadening the understanding of chaotic phenomena in condensed matter systems and quantum computations.

Additionally, the interplay between two-dimensional chaotic signatures and the three-dimensional gravitational duals suggests exciting potential for new holographic conjectures. It could also drive impactful advancements in how these principles apply to quantum information theory and the broader framework of quantum entanglement and thermalization.

In essence, this research contributes to a more comprehensive understanding of how chaotic behaviors in quantum systems are encapsulated within and can be elucidated through the symmetries of 2D CFT, further enriching the theoretical landscape of quantum chaos and complexity.