The out-of-time-order correlator (OTOC) serves as a powerful tool for investigating quantum information spreading and chaos in complex systems. We present a method employing non-equilibrium dynamical mean-field theory and coherent potential approximation combined with diagrammatic perturbation on the Schwinger-Keldysh contour to calculate the OTOC for correlated fermionic systems subjected to both random disorder and electron interaction. Our key finding is that random disorder enhances the OTOC decay in the Hubbard model for the metallic phase in the weakly interacting limit. However, the current limitation of our perturbative solver restricts the applicability to weak interaction regimes.
The out-of-time-order correlator (OTOC) serves as a powerful tool for investigating quantum information spreading and chaos in complex systems. We present a method employing non-equilibrium dynamical mean-field theory and coherent potential approximation combined with diagrammatic perturbation on the Schwinger-Keldysh contour to calculate the OTOC for correlated fermionic systems subjected to both random disorder and electron interaction. Our key finding is that random disorder enhances the OTOC decay in the Hubbard model for the metallic phase in the weakly interacting limit. However, the current limitation of our perturbative solver restricts the applicability to weak interaction regimes.
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.
Dynamical mean-field theory (DMFT) is a widely used, nonperturbative computational framework for strongly correlated electron systems. This thesis extends DMFT in two directions: treating disorder and higher-order correlators away from equilibrium, and using near-term quantum computers as impurity solvers. In the first part, we develop two methodological advances for the nonequilibrium DMFT treatment of disordered, interacting systems. The first overcomes the central computational bottleneck of the recently developed nonequilibrium DMFT plus coherent potential approximation (DMFT+CPA) by replacing its explicit numerical disorder averaging with an exact analytical integration, exploiting a normalization property of the Schwinger-Keldysh contour; applied to a disorder-and-interaction quench of the Anderson-Hubbard model, this reveals that weak random disorder promotes thermalization. The second demonstrates that the out-of-time-order correlator (OTOC) can be successfully computed within the existing DMFT+CPA framework, by generalizing the Schwinger-Keldysh contour to a double-folded structure; as a proof-of-principle demonstration, weak disorder is found to accelerate the OTOC's decay in the half-filled Anderson-Hubbard model in the intermediate-coupling regime. In the second part, we explore the use of near-term quantum computers as DMFT impurity solvers. A stable real-time DMFT iteration scheme, designed for compatibility with quantum hardware, is first developed and validated classically against the metal-to-insulator transition. A symmetry-adapted variational quantum eigensolver (VQE) is then benchmarked as the Anderson impurity solver for the converged DMFT Hamiltonian, and is found to reproduce ground-state energies accurately and to extract the single-particle Green's function reliably at intermediate to strong coupling, while accurate ground-state energies are shown not to guarantee accurate dynamics at weak coupling, a key challenge for extending such approaches to correlated metals. Together, we demonstrate that DMFT can be methodologically extended to study disordered and interacting systems away from equilibrium, and to incorporate near-term quantum hardware as a genuinely new computational resource to enhance its capabilities.
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. While a faithful realization of the SYK model in condensed-matter systems may be involved, a striking similarity between the SYK model and the Hund-coupling-induced spin-freezing crossover in multiorbital Hubbard models has recently been pointed out. To further explore this connection, we study OTOCs for fermionic single-orbital and multiorbital 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-type 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 $\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})\ensuremath{\sim}\sqrt{\ensuremath{\omega}}$. 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 multiorbital Hubbard models and the SYK strange metal.
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.
The out-of-time-ordered correlators (OTOCs) have been proposed and widely used recently as a tool to define and describe many-body quantum chaos. Here, we develop the Keldysh nonlinear sigma model technique to calculate these correlators in interacting disordered metals. In particular, we focus on the regularized and unregularized OTOCs, defined as ${f}^{(r)}(t)=\mathrm{Tr}\phantom{\rule{0.16em}{0ex}}\left[\sqrt{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t)\sqrt{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}^{\ifmmode\dagger\else\textdagger\fi{}}(t)\right]$ and ${f}^{(u)}(t)=\mathrm{Tr}\phantom{\rule{0.16em}{0ex}}\left[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}^{\ifmmode\dagger\else\textdagger\fi{}}(t)\right]$, respectively (where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}(t)={\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\psi}}(\mathrm{r},t),{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\psi}}}^{\ifmmode\dagger\else\textdagger\fi{}}(0,0)}$ is the anticommutator of fermion field operators and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}$ is the thermal density matrix). The calculation of the rate of OTOCs' exponential growth is reminiscent to that of the Altshuler-Aronov-Khmelnitskii dephasing rate in interacting metals, but here it involves two replicas of the system (two ``worlds''). The intraworld contributions reproduce the Altshuler-Aronov-Khmelnitskii dephasing (that would correspond to a decay of the correlator), while the interworld terms provide a term of the opposite sign that exceeds dephasing. Consequently, both regularized and unregularized OTOCs grow exponentially in time, but surprisingly we find that the corresponding many-body Lyapunov exponents are different. For the regularized correlator, we reproduce an earlier perturbation theory result for the Lyapunov exponent that satisfies the Maldacena-Shenker-Stanford bound, ${\ensuremath{\lambda}}^{(r)}\ensuremath{\le}2\ensuremath{\pi}{k}_{\mathsf{B}}T/\ensuremath{\hbar}$. However, the Lyapunov exponent of the unregularized correlator parametrically exceeds the bound, ${\ensuremath{\lambda}}^{(u)}\ensuremath{\gg}2\ensuremath{\pi}{k}_{\mathsf{B}}T/\ensuremath{\hbar}$. We argue that ${\ensuremath{\lambda}}^{(u)}$ is not a reliable indicator of many-body quantum chaos as it contains additional contributions from elastic scattering events due to virtual processes that should not contribute to many-body chaos. These results bring up an important general question of the physical meaning of the OTOCs often used in calculations and proofs. We briefly discuss possible connections of the OTOCs to observables in quantum interference effects and level statistics via a generalization of the Bohigas-Giannoni-Schmit conjecture to many-body chaotic systems.
A bstract Out-of-time-order correlators (OTOCs) are a standard measure of quantum chaos. Of the four operators involved, one pair may be regarded as a source and the other as a probe. A usual approach, applicable to large- N systems such as the SYK model, is to replace the actual source with some mean-field perturbation and solve for the probe correlation function on the double Keldysh contour. We show how to obtain the OTOC by combining two such solutions for perturbations propagating forward and backward in time. These dynamical perturbations, or scrambling modes, are considered on the thermofield double background and decomposed into a coherent and an incoherent part. For the large- q SYK, we obtain the OTOC in a closed form. We also prove a previously conjectured relation between the Lyapunov exponent and high-frequency behavior of the spectral function.