Timescales in the quench dynamics of many-body quantum systems: Participation ratio versus out-of-time ordered correlator

Abstract

We study quench dynamics in the many-body Hilbert space using two isolated systems with a finite number of interacting particles: a paradigmatic model of randomly interacting bosons and a dynamical (clean) model of interacting spins-1/2. For both systems in the region of strong quantum chaos, the number of components of the evolving wave function, defined through the number of principal components Npc (or participation ratio), was recently found to increase exponentially fast in time [Phys. Rev. E 99, 010101R (2019)]. Here, we ask whether the out-of-time ordered correlator (OTOC), which is nowadays widely used to quantify instability in quantum systems, can manifest analogous time-dependence. We show that Npc can be formally expressed as the inverse of the sum of all OTOC's for projection operators. While none of the individual projection-OTOC's shows an exponential behavior, their sum decreases exponentially fast in time. The comparison between the behavior of the OTOC with that of the Npc helps us better understand wave packet dynamics in the many-body Hilbert space, in close connection with the problems of thermalization and information scrambling. arXiv:1903.09175v1 [cond-mat.stat-mech]

Out-of-time ordered correlators, complexity, and entropy in bipartite systems

“…Torus quantization amounts to having a finite Hilbert space of dimension N = 1 2πh , with discrete positions and momenta in a lattice of separation 1 N [24]. In coordinate representation the propagator is given by an N × N unitary matrix…”

“…The two degrees of freedom classical system is defined in a four-dimensional phase space of coordinates (q 1…”

“…Nowadays, there is a huge interest in out-of-time ordered correlators (OTOCs) within the quantum chaos community, where they have been related to traditional measures as spectral statistics and the like [1]. These correlators were first introduced in superconductivity studies [2], where their exponential growth over time was associated with chaotic behavior.…”

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

“…Various kinds of OTOCs in quantum maps were also studied[20][21][22][23]. The cases with large N are found in[24][25][26][27][28][29][30][31][32][33][34][35].…”

Dynamical Detection of Level Repulsion in the One-Particle Aubry-André Model

“…There has been a surprising revival of interest in quantum chaos, especially from a dynamical perspective, with the exponential growth of out-of-time ordered correlators (OTOC) taken as a main indication of chaotic behavior [1][2][3][4][5][6][7][8][9]. The more traditional approach to quantum chaos, however, focuses on the properties of the spectrum and uses level statistics as in random matrix theory (RMT) as its main signature [10][11][12][13].…”