Out-of-time-order correlations in the quasiperiodic Aubry-André model

Abstract

We study out of time ordered correlators (OTOC) in a free fermionic model with a quasi-periodic potential. This model is equivalent to the Aubry-André model and features a phase transition from an extended phase to a localized phase at a non-zero value of the strength of the quasi-periodic potential. We investigate five different time-regimes of interest for out of time ordered correlators; early, wavefront, x = vBt, late time equilibration and infinite time. For the early time regime we observe a power law for all potential strengths. For the time regime preceding the wavefront we confirm a recently proposed universal form and use it to extract the characteristic velocity of the wavefront for the present model. A Gaussian waveform is observed to work well in the time regime surrounding x = vBt. Our main result is for the late time equilibration regime where we derive a finite time equilibration bound for the OTOC, bounding the correlator's distance from its late time value. The bound impose strict limits on equilibration of the OTOC in the extended regime and is valid not only for the Aubry-André model but for any quadratic model. Finally, momentum out of time ordered correlators for the Aubry-André model are studied where large values of the OTOC are observed at late times at the critical point.arXiv:1908.03292v3 [cond-mat.stat-mech]

Scaling at the OTOC Wavefront: Integrable versus chaotic models

“…( 5). This allows us to extract the scaling of m(x), b(x) analytically, verifying the findings of [53,92]. For the chaotic case, we numerically verify the Gaussian wave form of Eq.…”

“…While there can be signatures of chaos in OTOCs at late times, including longtime oscillations [73,84,[89][90][91], it is preferable to examine the main front rather than the signal either at early times (exponentially small) or late times (more contamination from the environment or numerical errors). Recent numerical work in free models has shown that the OTOC in this region is wellfitted by a propagating Gaussian of the form [53,92],…”

Entanglement entropy and out-of-time-order correlator in the long-range Aubry–André–Harper model

“…Recently late time behavior of C(x, t) has been proposed as a diagnostic to distinguish regular and chaotic quantum systems [51,52]. Although OTOC has been studied extensively in quantum systems, not many disordered integrable models have been addressed [53,54] in the context of the delocalization-localization transition. In addition to studies that look at the evolution of an initial thermal state, studies involving an initial product state in a nonequilibrium setting have also been carried out [15,53,55,56].…”

Quantum caustics and the hierarchy of light cones in quenched spin chains

“…They have also been discussed theoretically in the context of Bose-Einstein condensates [73,74] and various aspects seen experimentally in these systems [70][71][72]. Furthermore, the association between the Airy function (and its related kernels) and light cones has previously been noted by various authors [8,14,15,[59][60][61][62][63][64], and recent work has conjectured similar universal forms for wavefronts of out-of-timeordered correlators [65][66][67][68] by examining asymptotic limits of TABLE I. The seven elementary catastrophes and their generating functions Q (s; C), organized by corank n, and dimension Q of control space [86].…”

“…When γ = 1 this process may be repeated around each fold catastrophe, including for any inner cones, and will result in the emergence of Airy functions with different definitions of the control parameter, C. For example, a particular limit of Eq. 21has been conjectured to give a universal form for the wavefront of out-of-time-ordered correlators (OTOCs) [65][66][67][68]. According to catastrophe theory this is no surprise.…”