Does Scrambling Equal Chaos?

Abstract

Focusing on semiclassical systems, we show that the parametrically long exponential growth of outof-time order correlators (OTOCs), also known as scrambling, does not necessitate chaos. Indeed, scrambling can simply result from the presence of unstable fixed points in phase space, even in an integrable model. We derive a lower bound on the OTOC Lyapunov exponent which depends only on local properties of such fixed points. We present several models for which this bound is tight, i.e. for which scrambling is dominated by the local dynamics around the fixed points. We propose that the notion of scrambling be distinguished from that of chaos.

Exponential growth of out-of-time-order correlator without chaos: inverted harmonic oscillator

“…This appears to slightly violate the proposed inequality (1.2). Noting that the inequality (1.2) was derived in the classical limit [44], this slight violation would be due to the quantum effect. In addition, the Hilbert space of our quantum mechanical system is infinite dimensional, thus the infinite temperature limit is not well-defined.…”

“…2 This growth is interpreted as being generated by a classical unstable maximum of the potential at which, locally, an initial difference grows exponentially in time. The phenomenon is expected to be general, and [44] provided a general semiclassical inequality between the classical Lyapunov exponent λ saddle at the unstable maximum (or a saddle point) and the quantum Lyapunov exponent λ OTOC of the thermal OTOC at infinite temperature, λ OTOC (T = ∞) ≥ λ saddle .…”

A Review of the Serrated-Flow Phenomenon and Its Role in the Deformation Behavior of High-Entropy Alloys

“…Chaos, which is the exponential sensitivity to small perturbations, is a ubiquitous phenomenon in nature [258,259]. Furthermore, chaos is related to how a deterministic dynamical system can be potentially unpredictable due to an extreme sensitivity to initial conditions, which is also known as the "Butterfly Effect" [259].…”

“…Chaos, which is the exponential sensitivity to small perturbations, is a ubiquitous phenomenon in nature [258,259]. Furthermore, chaos is related to how a deterministic dynamical system can be potentially unpredictable due to an extreme sensitivity to initial conditions, which is also known as the "Butterfly Effect" [259]. Chaos has been observed in many different phenomena, including electric circuits [260][261][262], weather [263][264][265], physiological systems [266][267][268], financial markets [269][270][271], complex networks [272], the serrated flow in alloys [116,216,273], and ecological systems [274][275][276].…”

Speck of chaos

“…Level statistics as in random matrices are found also in some integrable models, but they are caused by finite-size effects [11,12] or change abruptly upon tiny variations of the Hamiltonian parameters [13,14]. Other definitions of quantum chaos include the short-time exponential growth of out-of-time order correlators [15][16][17][18][19][20] and diffusive transport [21][22][23], although exponential behaviors of fourpoint correlation functions appear also near critical points of integrable models [24][25][26][27][28] and ballistic transport has been observed in the chaotic single-defect X X Z model [29].…”