Dynamical Mean-Field Theory (DMFT) is a powerful theoretical framework for analyzing systems with many interacting degrees of freedom. This tutorial provides an accessible introduction to DMFT. We begin with a linear model where the DMFT equations can be derived exactly, allowing readers to develop clear intuition for the underlying principles. We then introduce the cavity method, a versatile approach for deriving DMFT equations for non-linear systems. The tutorial concludes with an application to the generalized Lotka--Volterra model of interacting species, demonstrating how DMFT reduces the complex dynamics of many-species communities to a tractable single-species stochastic process. Key insights include understanding how quenched disorder enables the reduction from many-body to effective single-particle dynamics, recognizing the role of self-averaging in simplifying complex systems, and seeing how collective interactions give rise to non-Markovian feedback effects.
Dynamical Mean-Field Theory (DMFT) is a powerful theoretical framework for analyzing systems with many interacting degrees of freedom. This tutorial provides an accessible introduction to DMFT. We begin with a linear model where the DMFT equations can be derived exactly, allowing readers to develop clear intuition for the underlying principles. We then introduce the cavity method, a versatile approach for deriving DMFT equations for non-linear systems. The tutorial concludes with an application to the generalized Lotka--Volterra model of interacting species, demonstrating how DMFT reduces the complex dynamics of many-species communities to a tractable single-species stochastic process. Key insights include understanding how quenched disorder enables the reduction from many-body to effective single-particle dynamics, recognizing the role of self-averaging in simplifying complex systems, and seeing how collective interactions give rise to non-Markovian feedback effects.
In this chapter, we present a bird’s-eye view of dynamical mean field theory (DMFT) and its extensions to highly correlated systems. We first demonstrate the elementary introduction to DMFT on a single site with detailed derivations and then a solution of the simple interacting models: the Hubbard model and the Falicov–Kimball model on infinite dimensional Bethe lattice using DMFT. Thereafter, we focus on two kinds of non-local extensions of single-site DMFT - (a) cluster dynamical mean field theory (CDMFT) on real space and (b) dynamical cluster approximation (DCA) on momentum space. Finally, we conclude the chapter with the application of DMFT combining with density functional theory (DFT) in the real-material systems (LDA+DMFT).
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) has opened new perspectives for the investigation of strongly correlated electron systems and greatly improved our understanding of correlation effects in models and materials. In contrast to Hartree-Fock-type approximations the mean field of DMFT is dynamical, whereby local quantum fluctuations are fully taken into account. DMFT becomes exact in the limit of high spatial dimensions or coordination number. Using DMFT the dynamics of correlated electron systems can be investigated non-perturbatively at all interaction strengths, electron densities and temperatures. By merging density functional theory with DMFT a powerful method for the calculation of the properties of correlated electron materials has become available, which is applicable to bulk systems and heterostructures, including topological states of matter. The inclusion of non-local correlations into DMFT makes it possible to explore unconventional superconductivity and the critical behavior at thermal or quantum phase transitions. By generalizing DMFT to non-equilibrium states also the real-time dynamics of correlated systems can be investigated. In this brief review the foundations and current status of DMFT are discussed along with characteristic physical insights obtained with this approach.
Open quantum many body systems describe several quantum simulators, ranging from arrays of superconducting circuits hosting correlated states of light, to ultracold atoms in optical lattices in presence of controlled dissipative processes. Their theoretical understanding is hampered by the exponential scaling of their Hilbert space and by their intrinsic nonequilibrium nature, limiting the applicability of many traditional approaches. In this work we extend Dynamical Mean Field Theory (DMFT), a powerful nonperturbative technique developed for strongly correlated quantum many-body systems, to Markovian open quantum systems. Within Open-DMFT, a master equation describing a lattice of bosonic particles is mapped, at large but finite connectivity, onto an impurity problem describing a single site coupled to Markovian and non-Markovian environments, where the latter accounts for quantum fluctuations beyond Gutzwiller mean-field theory. We introduce a new non-perturbative approach to solve the impurity problem, which dresses the impurity, featuring Markovian dissipative channels, with the non-Markovian bath, in a self-consistent scheme based on a resummation of non-crossing diagrams. Finally, we apply our Open-DMFT to solve a driven-dissipative Bose-Hubbard model, which is relevant to current experiments with dissipative ultracold bosons. We show that this model undergoes a dissipative phase transition towards a superfluid phase, which can be equally interpreted as a quantum many-body synchronization transition of an array of quantum van der Pol oscillators. We show how Open-DMFT captures crucial effects due to finite lattice connectivity, such as hopping-induced dissipation and heating, which are completely missed by mean field theories and which lead to a drastic reduction of the broken symmetry phase that we interpret as a desynchronization driven by quantum fluctuations.
Quantum mechanical lattice models, such as the famous Hubbard model, were originally intended as simplified descriptions of the behavior of electrons in real solids. Today it is clear that even such simple models can present remarkably rich physics. At the same time, obtaining reliable solutions for correlated quantum systems is difficult and requires the development of both computing resources and new algorithms and approximations. Parallel with the theoretical developments, applications of lattice models have also expanded to the study of ultracold atoms in optical lattices. The hallmark of these experiments is the possibility to design different kinds of lattices and to tune the interactions between the atoms, making it possible to do real experiments on theoretical "toy models". \n \nIn this thesis we have used dynamical mean-field theory (DMFT) and its cluster extensions to study superfluidity and topological phenomena in lattice models. The DMFT combines the exact solution of a small reference system with a self-consistent mean-field treatment of the rest of the lattice. After a short introduction to the physical background of the lattice models, we review the formal definition and practical aspects of the DMFT method. We discuss the Baym-Kadanoff-Luttinger-Ward functional formalism that can be used to derive the DMFT equations, and review the numerical exact diagonalization and continuous-time quantum Monte-Carlo algorithms that were used in the implementation of the method. \n \nPublications I and VI discuss superfluid states induced by correlation effects that cannot easily be treated within a static mean-field theory. In Publication I we study a bilayer system where two different types of superfluids can be found depending on the densities of particles in the layers. Publication VI has a focus in the d-wave superconductivity and non-uniform, striped magnetism of the two-dimensional Hubbard model, which are thought to be connected with the high-Tc superconductivity in cuprate materials. The stripe state along with non-Fermi-liquid properties visible only in a beyond-mean-field treatment were observed also in the study of the Hubbard model on the Lieb lattice in Publication V. \n \nTopological and geometric properties of lattice systems, such as the Berry curvature and Chern numbers, play a central role in Publications II, III and IV. In Publication II we used exact diagonalization, DMFT and mean-field theory to study the topological phase diagram of an interacting Haldane model. The methods consistently predict a phase diagram with an interme-diate topological phase not found in the well-known non-interacting case. In Publications III and IV mean-field theory is used to study the connection between Berry curvature and the superfluid weight in lattice models, and DMFT calculations were used to support the mean-field results. Topological lattice systems have recently been realized in ultracold gas experiments and it is hoped that the calculations presented here inspire this line of research.
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.