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 1, 2 and 3 spatial dimensions. The OTOC can be written as the square of the expectation value of the commutator and the variance of the commutator. The problem features only two energy scales, the chemical potential, and the high energy cutoff, therefore the time evolution is separated into three different regions. The spin correlation function grows linearly with time initially and decays in a power-law fashion for intermediate and late times. The OTOC reveals a universal t^2 initial growth from both the commutator and the variance. Its intermediate and late time power-law decays are identical and originate from the variance of the commutator. These 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.

Reference: https://arxiv.org/abs/1909.09376