Out-of-time ordered correlation functions for the localized $f$ electrons in the Falicov-Kimball model

TL;DR

This work provides an exact DMFT-based evaluation of out-of-time-ordered correlators for localized f-electron states in the Falicov-Kimball model. By mapping the problem to determinants of Toeplitz and block Toeplitz matrices and analyzing their winding numbers via Szegő-Widom theory, it identifies three distinct scrambling regimes controlled by the interaction strength U. The study reveals how scrambling dynamics transition from exponential to power-law decay with temperature, and how the associated density-of-states can exhibit negative regions linked to Kirkwood-Dirac quasiprobabilities, signaling strong quantum information scrambling mediated by retarded d-electron interactions. These results deepen understanding of information spreading in impurity-like systems and highlight the role of indirect interactions in quantum chaos within DMFT.

Abstract

We provide an exact evaluation of the out-of-time correlation (OTOC) functions for the localized $f$-particle states in the Falicov-Kimball model within dynamical mean-field theory. Different regimes of quantum chaos and quantum scrambling are distinguished by the winding numbers of the block Toeplitz matrices used in the calculation. The similarities of these fermionic OTOCs and their logarithmic derivatives for time evolution with the OTOCs for quantum spin models with disorder are also discussed.

Figures (12)

• Figure 1: (a) The $f$-electron propagator $G_f^{>}(t)$, (b,c) its logarithmic derivative, and (d) DOS $A_f(\omega)$ for $U=0.8$, $n_f=0.5$, $n_d=0.5$.
• Figure 2: Same as in Fig. \ref{['fig:fU08']} for $U = 2.0$.
• Figure 3: Doubly folded Keldysh contour. The interaction $U$ in \ref{['eq:Udef']} and \ref{['eq:Fsq_defU_cont']} acts on the forward (upper) branches of the contour.
• Figure 4: (a) OTOC $F(t)$, (b,c) normalized logarithmic derivative $L_0(t)/(2\pi T)$, and (d) the corresponding "density-of-states" $A(\omega)$ for $U=0.4$, $n_f=0.5$, $n_d=0.5$.
• Figure 5: Same as in Fig. \ref{['fig:U04']} for $U = 0.9$.