Abstract
We derive an effective field theory for general chaotic two-dimensional conformal field theories with a large central charge. The theory is a specific and calculable instance of a more general framework recently proposed in [1]. We discuss the gauge symmetries of the model and how they relate to the Lyapunov behaviour of certain correlators. We calculate the out-of-time-ordered correlators diagnosing quantum chaos, as well as certain more fine-grained higher-point generalizations, using our Lorentzian effective field theory. We comment on potential future applications of the effective theory to real-time thermal physics and conformal field theory.
Highlights
Pole skippingThe authors of [1] proposed an effective description of chaotic systems which encompasses the hydrodynamics, i.e., the theory of the energy-momentum tensor (and possibly other conserved currents), and quantum chaos as manifested in the out-of-time-order correlators (OTOCs)
This effective action provides a generalization of the Schwarzian action describing AdS2 gravity and the low frequency physics of the SYK model [7, 16,17,18,19,20]
Our starting point is similar to that of [24], where it was shown that the Lyapunov growth of of-time-order correlators (OTOCs) in rational large-c conformal field theories (CFTs) can be derived by thinking about the conformal transformations as physical Goldstone modes
Summary
The authors of [1] proposed an effective description of chaotic systems which encompasses the hydrodynamics, i.e., the theory of the energy-momentum tensor (and possibly other conserved currents), and quantum chaos as manifested in the out-of-time-order correlators (OTOCs). Moving along the lines of poles ωE This identifies the Lyapunov exponent as being maximal, and the butterfly velocity as being the speed of light. Having a maximal Lyapunov exponent for the out-of-time-order correlator requires further assumptions (such as large central charge and vacuum block dominance as in the context of [23]).. In addition to the pole skipping observed in the stress tensor correlator, this skips poles at ωE = ±2, corresponding to the spin-3 Lyapunov exponent [27]. This should persist in a similar way for higher spins, and allows for incorporating such exchanges in our effective field theory. We will not pursue this further in the present paper, but it would be an interesting phenomenon to investigate
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