Nonequilibrium dynamical mean-field theory (DMFT) is one of the most powerful theoretical tools to study the dynamics in strongly correlated systems. In practice, however, the method is limited by its considerable computational cost. Here, the authors systematically investigate how the computational effort can be reduced by at least an order of magnitude while retaining high accuracy. The key idea is the controlled truncation of the time scale on which the system ``remembers'' its initial state. Applying this idea to paradigmatic model systems, the authors map out the optimal regimes in which this truncation yields an accurate description of the relevant observables. This approach makes fundamental long-time processes such as prethermalization and thermalization accessible to DMFT simulations.
Nonequilibrium dynamical mean-field theory (DMFT) is one of the most powerful theoretical tools to study the dynamics in strongly correlated systems. In practice, however, the method is limited by its considerable computational cost. Here, the authors systematically investigate how the computational effort can be reduced by at least an order of magnitude while retaining high accuracy. The key idea is the controlled truncation of the time scale on which the system ``remembers'' its initial state. Applying this idea to paradigmatic model systems, the authors map out the optimal regimes in which this truncation yields an accurate description of the relevant observables. This approach makes fundamental long-time processes such as prethermalization and thermalization accessible to DMFT simulations.
The Keldysh formalism for nonequilibrium Green's functions is a powerful theoretical framework for the description of the electronic structure, spectroscopy, and dynamics of strongly correlated systems. However, the underlying Kadanoff-Baym equations (KBE) for the two-time Keldysh Green's functions involve a memory kernel, which results in a high computational cost for long simulation times tmax, with a cubic scaling of the computation time with tmax. Truncation of the memory kernel can reduce the computational cost to linear scaling with tmax, but the required memory times will depend on the model and the diagrammatic approximation to the self-energy. We explain how a truncation of the memory kernel can be incorporated into the time-propagation algorithm to solve the KBE, and investigate the systematic truncation of the memory kernel for the Hubbard model in different parameter regimes, and for different diagrammatic approximations. The truncation is easier to control within dynamical mean-field solutions, where it is applied to a momentum-independent self-energy. Here, simulation times up to two orders of magnitude longer are accessible both in the weak and strong coupling regime, allowing for a study of long-time phenomena such as the crossover between prethermalization and thermalization dynamics.
The study of nonequilibrium phenomena in correlated lattice systems has developed into one of the most active and exciting branches of condensed matter physics. This research field provides rich new insights that could not be obtained from the study of equilibrium situations, and the theoretical understanding of the physics often requires the development of new concepts and methods. On the experimental side, ultrafast pump-probe spectroscopies enable studies of excitation and relaxation phenomena in correlated electron systems, while ultracold atoms in optical lattices provide a new way to control and measure the time evolution of interacting lattice systems with a vastly different characteristic time scale compared to electron systems. A theoretical description of these phenomena is challenging because, first, the quantum-mechanical time evolution of many-body systems out of equilibrium must be computed and second, strong-correlation effects which can be of a nonperturbative nature must be addressed. This review discusses the nonequilibrium extension of the dynamical mean field theory (DMFT), which treats quantum fluctuations in the time domain and works directly in the thermodynamic limit. The method reduces the complexity of the calculation via a mapping to a self-consistent impurity problem, which becomes exact in infinite dimensions. Particular emphasis is placed on a detailed derivation of the formalism, and on a discussion of numerical techniques, which enable solutions of the effective nonequilibrium DMFT impurity problem. Insights gained into the properties of the infinite-dimensional Hubbard model under strong nonequilibrium conditions are summarized. These examples illustrate the current ability of the theoretical framework to reproduce and understand fundamental nonequilibrium phenomena, such as the dielectric breakdown of Mott insulators, photodoping, and collapse-and-revival oscillations in quenched systems. Furthermore, remarkable novel phenomena have been predicted by the nonequilibrium DMFT simulations of correlated lattice systems, including dynamical phase transitions and field-induced repulsion-to-attraction conversions.
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.
Dynamical mean-field theory (DMFT) is a cornerstone technique for studying strongly correlated electronic systems. However, each DMFT step is computationally demanding, and many iterations can be required to achieve convergence. Here, we accelerate the convergence of DMFT by initializing its self-consistent cycle with solutions from computationally cheaper and more approximate methods. We compare the initialization with the non-interacting solution to a range of quantum embedding compatible approaches: Hartree-Fock, the Hubbard-I approximation, rotationally invariant slave bosons (RISB), and its ghost extension (g-RISB). We find that these initializations can reduce the number of DMFT iterations by up to an order of magnitude, with g-RISB providing the most effective and reliable benefits. In most regimes, initializing with g-RISB and performing a single DMFT iteration suffices to recover the full dynamical structure. The improvement in convergence is controlled by the initial solution's accuracy in the low-energy part of the self-energy, on the scale of the non-interacting bandwidth. This strategy is especially effective at the Mott insulator-metal transition, where an initialization from the non-interacting limit can lead to a breakdown of DMFT due to the sign problem. Our results establish the usage of accurate yet cheaper quantum embedding methods as a powerful means to substantially reduce the computational cost of DMFT, particularly in regimes where convergence is slow or prone to failure.
Nonlocal correlations play an essential role in correlated electron systems, especially in the vicinity of phase transitions and crossovers, where two-particle correlation functions display a distinct momentum dependence. In nonequilibrium settings, the effect of nonlocal correlations on dynamical phase transitions, prethermalization phenomena, and trapping in metastable states is not well understood. In this paper, we introduce a dynamical mean field theory (DMFT) extension to the nonequilibrium two-particle self-consistent (TPSC) approach, which allows to perform nonequilibrium simulations capturing short- and long-ranged nonlocal correlations in the weak-, intermediate-, and strong-correlation regimes. The method self-consistently computes local spin and charge vertices, from which a momentum-dependent self-energy is constructed. Replacing the local part of the self-energy by the DMFT result within this self-consistent scheme provides an improved description of local correlation effects. We explain the details of the formalism and the implementation, and demonstrate the versatility of DMFT+TPSC with interaction quenches and dimensional crossovers in the Hubbard model.
Nonequilibrium Green’s functions provide a powerful framework for studying quantum many-body dynamics including the laser-induced dynamics in solids. The N on- E quilibrium S ystems Si mulation package ( NESSi ) offers an efficient platform for such simulations, ranging from perturbative approaches like nonequilibrium GW to nonequilibrium dynamical mean-field theory. However, simulations based on nonequilibrium Green’s functions become computationally demanding when the dynamics span a large temporal range, such as from sub-femtosecond electron dynamics to the picosecond dynamics of collective modes. Due to the memory integral in the Kadanoff-Baym equations, which serve as equations of motion for nonequilibrium Green’s functions, the computational cost scales as 𝒪 ( 𝑁 3 𝑡 ) with the number of timesteps N t , and the memory requirement scales as 𝒪 ( 𝑁 2 𝑡 ) . In this work, we extend NESSi by incorporating techniques that aim to overcome this bottleneck: (i) By truncating the memory integrals in the KBE to a maximum of N c timesteps, the computational complexity is reduced to 𝒪 ( 𝑁 𝑡 𝑁 2 𝑐 ) , and the memory requirement to 𝒪 ( 𝑁 2 𝑐 ) . Provided that the results converge with respect to the cutoff N c , memory truncation allows to extend the simulations to significantly longer times. (ii) We introduce functionalities to describe nonequilibrium steady states, i.e. time-translationally invariant nonequilibrium states. Such states are relevant for transport settings, and they provide an approximate description of slowly evolving (prethermal) nonequilibrium states.