Quantum Chaos
Scrambling vs chaos in the semiclassical limit
Classical chaos theory studies the extreme sensitivity to initial conditions that generically appears in classical systems with many interacting degrees of freedom. This ``butterfly effect'' is far from a theoretical curiosity: it is for example what limits accurate weather predictions to just a few days. Most importantly, it is the microscopic mechanism responsible for the emergence of a thermodynamic description at macroscopic scales.
Extending the notion of classical chaos to the quantum realm has been an ongoing area of research for decades. Recently, progress in the study of quantum information, black holes, and holography has led to a new putative definition of quantum chaos (which we shall refer to as ``scrambling''), defined as the exponential growth out-of-time order correlators (OTOCs). The growth rate is referred to as a (quantum) Lyapunov exponent, and bounds thereof are called ``bounds on chaos''. ``Maximally chaotic'' systems which saturate those bounds, like the Sachdev-Ye-Kitaev model, received particular attention as canonical toy models of strongly coupled systems and of holography. In Ref [1], we showed how chaos in these systems already has unique features at the classical level.
Furthermore, in a recent publication [2], we have shown that the interpretation of exponential OTOC growth as chaos is questionable, especially in the context of quantum systems in the semiclassical limit. There, chaos and scrambling are both well-defined, and correspond to the exponential growth of subtly different quantities (sensitivity and OTOC, respectively).
Even though the difference between the two is subtle, it actually leads to a qualitative difference between the two concepts: scrambling can occur independently of chaos. In fact, we have shown that the presence of a saddle point in phase space is sufficient to create scrambling, even for a (few-body) integrable system. Further, even in chaotic systems, the scrambling exponent can still be dominated by the contribution of local saddle points, and thus be qualitatively distinct from the global Lyapunov exponent.
[1] Thomas Scaffidi, Ehud Altman, "Chaos in a classical limit of the Sachdev-Ye-Kitaev model", PRB 2019
[2] Tianrui Xu, Thomas Scaffidi, Xiangyu Cao, "Does scrambling equal chaos?", PRL 2020, Editors' suggestion
A universal operator growth hypothesis
I gave a talk about this at the Perimeter Institute:
Ref: Daniel E. Parker, Xiangyu Cao, Alexander Avdoshkin, Thomas Scaffidi, Ehud Altman, "A Universal Operator Growth Hypothesis", PRX (2019)