Out-of-time-ordered commutators in Dirac--Weyl systems

TL;DR

This paper investigates the scrambling dynamics in Dirac--Weyl systems by analyzing out-of-time-ordered commutators, revealing universal growth and decay patterns that characterize their slow information spreading behavior.

Contribution

It provides a detailed analysis of OTOC behavior in Dirac--Weyl systems across dimensions, highlighting universal features and the dominance of the high energy cutoff.

Findings

OTOC exhibits universal $t^2$ initial growth.

Late time decay of OTOC follows a $t^{-2}$ pattern.

Dirac--Weyl systems are identified as slow information scramblers.

Abstract

Quantum information stored in local operators spreads over other degrees of freedom of the system during time evolution, known as scrambling. This process is conveniently characterized by the out-of-time-order commutators (OTOC), whose time dependence reveals salient aspects of the system's dynamics. Here we study the spatially local spin correlation function i.e., the expectation value of spin commutator and the corresponding OTOC of Dirac--Weyl systems in one, two, and three spatial dimensions. The OTOC can be written as the square of the expectation value of the commutator and the variance of the commutator. In principle, the problem features two energy scales, the chemical potential, and the high energy cutoff. We find that only the latter is dominant, therefore the time evolution is separated into only two different regions. The spin correlation function grows linearly with time initially and decays as $t^{-2}$ for late times. The OTOC reveals a universal $t^2$ initial growth from both the commutator and the variance while its late time decay, $t^{-2}$ originates from the variance of the commutator. This late time decay is identified as a characteristic signature or Dirac-Weyl fermions. These features remain present also at finite temperatures. Our results indicate that Dirac--Weyl systems are slow information scramblers and are essential when additional channels for scrambling, i.e., interaction or disorder are analyzed.