The convergence to the self-consistency in the dynamical mean-field theory (DMFT) calculations for models of correlated electron systems can be significantly accelerated by using an appropriate mixing of hybridization functions which are used as the input to the impurity solver. It is shown that the techniques and the past experience with the mixing of input charge densities in the density-functional theory calculations are also effective in DMFT. As an example, the increase in the computational requirements near the Mott metal-insulator transition in the Hubbard model due to critical slowing down can be strongly reduced by using the modified Broyden's method to numerically solve the nonlinear self-consistency equation. Speed-up factors as high as three were observed in practical calculations even for this relatively well-behaved problem. Furthermore, the convergence can be achieved in difficult cases where simple linear mixing is either not effective or even leads to divergence. Unstable and metastable solutions can also be obtained. We also determine the linear response of the system with respect to the variations in the hybridization function, which is related to the propagation of the information between the different energy scales during the iteration.
The convergence to the self-consistency in the dynamical mean-field theory (DMFT) calculations for models of correlated electron systems can be significantly accelerated by using an appropriate mixing of hybridization functions which are used as the input to the impurity solver. It is shown that the techniques and the past experience with the mixing of input charge densities in the density-functional theory calculations are also effective in DMFT. As an example, the increase in the computational requirements near the Mott metal-insulator transition in the Hubbard model due to critical slowing down can be strongly reduced by using the modified Broyden's method to numerically solve the nonlinear self-consistency equation. Speed-up factors as high as three were observed in practical calculations even for this relatively well-behaved problem. Furthermore, the convergence can be achieved in difficult cases where simple linear mixing is either not effective or even leads to divergence. Unstable and metastable solutions can also be obtained. We also determine the linear response of the system with respect to the variations in the hybridization function, which is related to the propagation of the information between the different energy scales during the iteration.
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.
Correlated electron systems exhibit a variety of intriguing emergent phenomena, including Mott physics, pseudogap behavior, and unconventional superconductivity. One of the most widely used numerical approaches for investigating these systems is dynamical mean-field theory (DMFT). This dissertation addresses two related problems: how to develop better quantum impurity solvers for DMFT and how to use DMFT results to gain a better understanding of correlated electron systems. The first part of this dissertation develops numerical methods for solving quantum impurity problems, which constitute the central computational task in DMFT calculations. In Chap. 4, we develop a method that combines a tensor-train numerical integration scheme with the inchworm hybridization expansion for quantum impurity problems. Benchmarks on spinless, single-orbital, and multi-orbital impurity models demonstrate the convergence of the Green’s functions to exact or reference results and identify the regimes in which tensor-train compression of Feynman diagrams and the inchworm expansion itself are effective. In Chap. 5, we introduce the concept of counterterms within the hybridization expansion and apply it to a bare hybridization expansion solver that relies on a sum-of-exponentials integration technique. Tests on spinless, multi-orbital, and realistic embedding impurity problems show that counterterms can significantly reduce the required perturbative order and improve the efficiency of hybridization-expansion calculations. The second part of this dissertation applies the cluster extension of DMFT to the single-band Hubbard model. In Chap. 6, we introduce a fluctuation decomposition for the self-energy that eliminates the Fierz ambiguity encountered in conventional fluctuation diagnostics of the Hubbard model. The resulting analysis shows that spin fluctuations make an important contribution to the pseudogap self-energy, while multi-boson contributions are also necessary for a complete description. In Chap. 7, we analyze the spin-fermion coupling and its role in d-wave pairing. Our calculations show that, for antinodal fermions in the pseudogap regime, their coupling to antiferromagnetic spin fluctuations is strongly enhanced relative to the non-interacting limit. A modified Bethe–Salpeter equation analysis further shows that this enhanced spin–fermion coupling boosts the spin-fluctuation-induced contribution to the d-wave pairing susceptibility.
We present a time-domain iteration scheme for solving the Dynamical Mean-Field Theory (DMFT) self-consistent equations using retarded Green's functions in real time. Unlike conventional DMFT approaches that operate in imaginary time or frequency space, our scheme operates directly with real-time quantities. This makes it particularly suitable for near-term quantum computing hardware with limited Hilbert spaces, where real-time propagation can be efficiently implemented via Trotterization or variational quantum algorithms. We map the effective impurity problem to a finite one-dimensional chain with a small number of bath sites, solved via exact diagonalization as a proof-of-concept. The hybridization function is iteratively updated through time-domain fitting until self-consistency. We demonstrate stable convergence across a wide range of interaction strengths for the half-filled Hubbard model on a Bethe lattice, successfully capturing the metal-to-insulator transition. Despite using limited time resolution and a minimal bath discretization, the spectral functions clearly exhibit the emergence of Hubbard bands and the suppression of spectral weight at the Fermi level as interaction strength increases. This overcomes major limitations of two-site DMFT approximations by delivering detailed spectral features while preserving efficiency and compatibility with quantum computing platforms through real-time dynamics.
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.
We review the dynamical mean-field theory of strongly correlated electron systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact for models of correlated electrons in the limit of large lattice coordination (or infinite spatial dimensions). It extends the standard mean-field construction from classical statistical mechanics to quantum problems. We discuss the physical ideas underlying this theory and its mathematical derivation. Various analytic and numerical techniques that have been developed recently in order to analyze and solve the dynamical mean-field equations are reviewed and compared to each other. The method can be used for the determination of phase diagrams (by comparing the stability of various types of long-range order), and the calculation of thermodynamic properties, one-particle Green's functions, and response functions. We review in detail the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to experiments on three-dimensional transition-metal oxides. We present an overview of the rapidly developing field of applications of this method to other systems. The present limitations of the approach, and possible extensions of the formalism are finally discussed. Computer programs for the numerical implementation of this method are also provided with this article.
We propose a fast multi-orbital impurity solver for dynamical mean field theory (DMFT). Our DMFT solver is based on the equations of motion (EOMs) for local Green's functions and is constructed by generalizing from the single-orbital case to the multi-orbital case with the inclusion of the inter-orbital hybridizations and applying a mean field approximation to the inter-orbital Coulomb interactions. The two-orbital Hubbard model is studied using this impurity solver within a large range of parameters. The Mott metal-insulator transition and the quasiparticle peak are well described. A comparison of the EOM method with the quantum Monte Carlo method is made for the two-orbital Hubbard model and good agreement is obtained. The developed method hence holds promise as a fast DMFT impurity solver in studies of strongly correlated systems.