Exact out-of-time-ordered correlation functions for an interacting lattice fermion model

Abstract

Exact solutions for local equilibrium and nonequilibrium out-of-time-ordered correlation (OTOC) functions are obtained for a lattice fermion model with on-site interactions, namely the Falicov-Kimball (FK) model, in the large dimensional and thermodynamic limit. Our approach is based on the nonequilibrium dynamical mean-field theory generalized to an extended Kadanoff-Baym contour. We find that the density-density OTOC is most enhanced at intermediate coupling around the metal-insulator phase transition. In the high-temperature limit, the OTOC remains nontrivially finite and interaction-dependent, even though dynamical charge correlations probed by an ordinary response function are completely suppressed. We propose an experiment to measure OTOCs of fermionic lattice systems including the FK and Hubbard models in ultracold atomic systems.

Figures (4)

• Figure 1: Two types of extended [doubly folded (a) and singly folded (b)] Kadanoff-Baym contours $\tilde{\mathcal{C}}$, which are equivalent. In (a), the system evolves with the Hamiltonian $H(\bar{t})$ ($0\le \bar{t}\le t$), while in (b) the system evolves with $H(\bar{t})$ for $0\le \bar{t}\le t$ and $-H(2t-\bar{t})$ for $t\le \bar{t} \le 2t$.
• Figure 2: Dynamical charge susceptibility $\chi(t)$ (blue curves) and out-of-time-ordered charge correlation function $C(t)$ (red) for the FK model with $U=1$ (a), $U=2$ (b), and $U=8$ (c). The solid, dashed, and dotted curves correspond to $\beta=10$, $0.5$, and $0.1$, respectively. The inset shows the corresponding log-log plot for the absolute values, compared with the asymptotic behavior $\propto t^{-3}$.
• Figure 3: Comparison between the dynamical charge susceptibility $\chi(\omega)$ (a) and the OTOC spectral function $C(\omega)$ (b) at $\omega=0$ for the FK model. The dashed curves correspond to interaction quenches $U_0=10\to U$ with initial temperatures $\beta=1$ and $0.1$.
• Figure 4: Illustration of the proposed measurement protocol of the OTOC in ultracold atomic systems with two fermionic species (blue and green). At $t=t_0$, a local pulse is applied to site $i$, and at $t=0$ and $t=2t_0$ the particle density is measured at site $j$.