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 ${t}_{\text{max}}$, with a cubic scaling of the computation time with ${t}_{\text{max}}$. Truncation of the memory kernel can reduce the computational cost to linear scaling with ${t}_{\text{max}}$, 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 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.
In this thesis we present methodologies that have been developed to efficiently solve for the non-equilibrium Green’s Function (NEGF) of interacting electron systems, and demonstrate their applications to several different condensed matter systems. These solvers are based on the numerical integration of the equations of motion for the NEGF, which are known as the Kadanoff-Baym Equations (KBE). The KBE are a general set of equations which allow for the study of a wide range of systems that are of interest to condensed matter physicists, including superconductivity, nuclear matter, molecules and atoms, and qubits, among many others. They are also useful for solving impurity models and systems in the dynamical mean-field theory (DMFT) approximation through their ability to incorporate dynamical hybridization functions. The generality of these equations comes from their connection to time-dependent Many-Body Perturbation Theory (MBPT), which introduces non-Markovian dynamical correlations through the self-energy functional. MBPT ensures important conservation laws, such as those for density and energy, are obeyed no matter the type of time-dependent perturbations that are applied to the Hamiltonian. The non-Markovian correlations manifest in a kernel that couples the dynamics to the entire history of the system. This coupling leads to computational costs that become prohibitively expensive as longer-time dynamics are probed, this is especially true for systems which decohere slowly, leading to undamped dynamics and kernels. Due to the large amount of interest in non-equilibrium quantum dynamics, the availability of efficient solvers for the KBE are of great importance. This thesis will focus on the implementation and analysis of integration methods for the KBE, including compression algorithms which lower computational complexity and adaptive integration techniques which allow for resolution of dynamics which vary over wide ranges of timescales. Chapter 1 gives an introduction to quantum systems and the problems we will be studying. Chapter 2 provides a derivation of the NEGF and its representation on the Keldysh contour. Chapter 3 derives the KBE from MBPT and introduces the self- energy and its conservation laws. Chapter 4 introduces the numerical techniques used to integrate the KBE, as well as an analysis of its causality structure. In Chapter 5, we use compression techniques to lower the complexity of the KBE, and provide details on its implementation. In Chapter 6 we apply the compression algorithm to study the Higgs mode in a superconducting system. In Chapter 7, we couple our solver to an adaptive integrator and analyze its performance against fixed-timestep integrators. Finally, Chapter 8 provides a summary of the results and an outlook for future work on efficient solvers.
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.
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 nonequilibrium Dyson (or Kadanoff-Baym) equation, which is an equation of motion with a long-range memory kernel for real-time Green functions, underlies many numerical approaches based on the Keldysh formalism. In this paper we map the problem of solving the Dyson equation in real time onto a noninteracting auxiliary Hamiltonian with additional bath degrees of freedom. The solution of the auxiliary model does not require the evaluation of a memory kernel and can thus be implemented in a very memory efficient way. The mapping is derived for a self-energy which is local in space and is thus directly applicable within nonequilibrium dynamical mean-field theory (DMFT). We apply the method to study the interaction quench in the Hubbard model for an optical lattice with a narrow confinement, using inhomogeneous DMFT in combination with second-order weak-coupling perturbation theory. We find that, although the quench excites pronounced density oscillations, signatures of the two-stage relaxation similar to the homogeneous system can be observed by looking at the time-dependent occupations of natural orbitals.
The generalized Kadanoff-Baym ansatz (GKBA) is an approximation to the Kadanoff-Baym equations (KBEs) that neglects certain memory effects that contribute to the Green's function at nonequal times. Here we present arguments and numerical results to demonstrate the practical insignificance of the quantities neglected when deriving the GKBA at conditions at which KBE and GKBA are appropriate. We provide a mathematical proof that places a scaling bound on the neglected terms, further reinforcing that these terms are typically small in comparison to terms that are kept in the GKBA. We perform calculations in a range of models, including different system sizes and filling fractions, as well as experimentally relevant nonequilibrium excitations. We find that both the GKBA and KBE capture the dynamics of interacting systems with moderate and even strong interactions well. We explicitly compute terms neglected in the GKBA approximation and show, in the scenarios tested here, that they are orders of magnitude smaller than the terms that are accounted for, i.e., they offer only a small correction when included in the full KBEs.
The method of nonequilibrium Green functions (NEGFs) constitutes a solid framework to describe closed and open quantum systems. It captures situations far from equilibrium including arbitrary perturbations and correlations - effects higher than first order in the binary interactions. On the contrary, in the absence of external fields, it reduces to the imaginary-time Matsubara formalism. The abilities (and successes) of the method originate from its self-consistency and the conserving character of many-body approximations that are introduced by diagram technique. On this account, NEGFs have, now for a long time, brought insight into the dynamics of nuclear matter, non-ideal quantum gases and (semiconductor) plasmas. However, applications to localized, finite and strongly interacting systems have emerged only within the recent decade. The dynamical properties follow from quantum kinetic equations - the two-time Kadanoff-Baym equations for the one-particle NEGF - the numerical solution of which is the central topic of this thesis. The non-Markovian structure of these equations thereby inhibits an unlimited, gradual propagation of the nonequilibrium Green function: Any self-consistent, future dynamics of the system is tightly linked to the (complete) past by non-local self-energies. Further, the success in obtaining solutions is being diminished if the system under investigation is not isotropic or inhomogeneous and, in addition, the spatial topology (e.g., the charge carrier or spin density) needs to be resolved in time. On top of this, small isolated systems are generally not candidates showing fast decoherence which offers little hope for smooth and well-damped integration kernels. Despite these difficulties, the thesis at hand is devoted to the computation of the two-time NEGF for finite, closed systems of which ensembles of electrons bound by nuclei forming atoms or molecules are the most natural representatives. Here, in the spirit of the well-studied one-dimensional helium atom, the electron dynamics in model atoms and molecules is addressed in ab initio fashion. To render calculations possible in the first place, (i) a novel hybrid representation of the NEGF is introduced based on the conception of finite elements (FE) and the discrete variable representation (DVR), and (ii) a parallel algorithm is developed that allows for an efficient, well scalable distributed-memory computation of the full two-time Green function. The first point attributes plenty of flexibility and makes the method readily adjustable to different Hamiltonians avoiding the need to, by hand or in intricate numerical manner, calculate the whole bunch of matrix elements of one- and two-particle operators. Also, in consequence, it drastically simplifies the evaluation of self-energies. The second point makes the NEGF approach ready for large-scale, high-performance computing facilities, the resources (of computer power: mainly memory!) of which are indispensable for (some of) the calculations presented. The applicability of the developed methods is demonstrated inter alia on the basis of helium, beryllium, hydrogen and lithium hydride modeled in one spatial dimension whereby electron-electron correlations are treated in the second(-order) Born approximation. After a thorough discussion on the self-consistent ground states, the thesis focuses on the electron dynamics in the linear response regime as well as in the presence of strong external fields. Overall, to assess the the performance of the second Born approximation, the results are compared to the Hartree-Fock approximation, on the one hand, and to the exact solution of the time-(in)dependent Schrödinger equation, on the other hand. In addition, aiming at the possibility of extended time-dependent calculations, the behavior of the generalized Kadanoff-Baym ansatz is discussed.