Chaos, complexity, and random matrices

Abstract

Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an O(1) scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce k-invariance as a precise definition of what it means for the dynamics of a quantum system to be described by random matrix theory. We envision that the dynamical onset of approximate k-invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by random matrix theory.

Conformal field theory and the web of quantum chaos diagnostics

“…(1) Higher point OTOCs were related to higher-point spectral form factors in Ref. [8] and averaged OTOCs were further equated with the SFF in quantum field theory in Refs. [9,10].…”

Characterizing complexity of many-body quantum dynamics by higher-order eigenstate thermalization

“…The k-ETH is a specific example of PU k-designs of the LTE. As anticipated from the connection between information scrambling and unitary k-designs [12,13], the exact k-ETH implies that the long-time average of the 2k-point OTOC equals the exact HRU average (39). However, we have pointed out that, as for the OTOC, the difference between the 1-ETH and the higher-order ETH is smaller than the finite-size effect in quantum manybody systems.…”

“…We discuss the relationship between the k-ETH and information scrambling. Considering the 2k-point OTOC as an indicator of higher-order scrambling [12,13], we show that the exact k-ETH is a sufficient condition for the decay of the 2k-point OTOC to the exact value of the HRU average. However, we point out that the approximate decay of the 2k-point OTOC follows only from the conventional diagonal and off-diagonal ETH, and the unique role of the higher-order ETH is smaller than the finite-size effect as for the OTOC.…”

“…, which implies that all the kth moments of ν equal those of the HRU [5][6][7]. The previous studies [12][13][14][15][16][17] revealed the fundamental relationship between unitary k-designs and information scrambling. As discussed below, on the other hand, there is a subtle issue about the relationship between unitary k-designs and Hamiltonian dynamics [12].…”

“…We derive the explicit condition of the k-ETH by considering a k-replicated system and kfold channels (i.e., a standard technique to deal with the kth moment of dynamics in unitary k-designs). We point out, however, that the relevance of the k-ETH to the kth order information scrambling quantified by 2k-point OTOCs [12,13] is not straightforward.…”