  The Sachdev-Ye-Kitaev (SYK) model describes a strange metal that shows
peculiar non-Fermi liquid properties without quasiparticles. It exhibits a
maximally chaotic behavior characterized by out-of-time-ordered correlators
(OTOCs), and is expected to be a holographic dual to black holes. Recently, a
striking similarity between the SYK model and the Hund-coupling induced
spin-freezing crossover in multi-orbital Hubbard models has been pointed out.
To further explore this connection, we study OTOCs for fermionic Hubbard
models, which are prototypical models for strongly correlated electrons in
solids. We introduce an imaginary-time four-point correlation function with an
appropriate time ordering, which by means of the spectral representation and
the out-of-time-order fluctuation-dissipation theorem can be analytically
continued to real-time OTOCs. Based on this approach, we numerically evaluate
real-time OTOCs for Hubbard models in the thermodynamic limit, using the
dynamical mean-field theory in combination with a numerically exact
continuous-time Monte Carlo impurity solver. The results for the single-orbital
model show that a certain spin-related OTOC captures local moment formation in
the vicinity of the metal-insulator transition, while the self-energy does not
show SYK-like non-Fermi liquid behavior. On the other hand, for the two- and
three-orbital models with nonzero Hund coupling we find that the OTOC exhibits
a rapid damping at short times and an approximate power-law decay at longer
times in the spin-freezing crossover regime characterized by fluctuating local
moments and a non-Fermi liquid self-energy $\Sigma(\omega) \sim \sqrt{\omega}$.
These results are in a good agreement with the behavior of the SYK model,
providing firm evidence for the close relation between the spin-freezing
crossover physics of multi-orbital Hubbard models and the SYK strange metal.
