The dynamical mean field theory is extended using the Keldysh Green's function formalism to study strongly correlated systems away from equilibrium. This technique has been applied to two different systems, whose insulating behavior is due to electronic correlation effects: excitonic insulators and Mott insulators. The real-time development of the electronic properties of excitonic insulators, such as the distribution function and the gap, after an ultrashort intense is studied. The appropriate self-consistency equations turned out to be matrix equations in frequency and Keldysh space. The self-consistency equations derived for the two-plane Hubbard model with a periodic time-dependent coupling between the two planes are matrix equations in energy sidebands due to quasi-energy conservation modulo the external frequency. Numerical results are shown for the local density of states and the nonequilibrium optical conductivity, both exhibiting a sideband structure, with the number and weight of the sidebands depending on the strength of the perpendicular coupling.
The dynamical mean field theory is extended using the Keldysh Green's function formalism to study strongly correlated systems away from equilibrium. This technique has been applied to two different systems, whose insulating behavior is due to electronic correlation effects: excitonic insulators and Mott insulators. The real-time development of the electronic properties of excitonic insulators, such as the distribution function and the gap, after an ultrashort intense is studied. The appropriate self-consistency equations turned out to be matrix equations in frequency and Keldysh space. The self-consistency equations derived for the two-plane Hubbard model with a periodic time-dependent coupling between the two planes are matrix equations in energy sidebands due to quasi-energy conservation modulo the external frequency. Numerical results are shown for the local density of states and the nonequilibrium optical conductivity, both exhibiting a sideband structure, with the number and weight of the sidebands depending on the strength of the perpendicular coupling.
We present a generalized dynamical mean-field approach for the nonequilibrium physics of a strongly correlated system in the presence of a time-dependent external field. The Keldysh Green's function formalism is used to study the nonequilibrium problem. We derive a closed set of self-consistency equations in the case of a driving field with frequency Omega and wave vector q. We present numerical results for the local frequency-dependent Green's function and the self-energy for different values of the field amplitude in the case of a uniform external field using the iterated perturbation theory. In addition, an expression for the frequency-dependent optical conductivity of the Hubbard model with a driving external field is derived.
We develop a nonequilibrium multi-orbital extension of the two-particle self-consistent theory and apply it to the bilayer Hubbard model as a minimal platform to investigate correlation effects in the presence of interlayer interactions and tunneling. The method determines vertex corrections in the spin and charge channels self-consistently at the two-particle level, thereby avoiding the spurious finite-temperature phase transitions that limit dynamical mean-field theory in two dimensions. We derive the spectral self-energy and implement the framework directly on the real-frequency axis within the Keldysh nonequilibrium Green's function formalism, enabling the treatment of both equilibrium and non-equilibrium steady states without relying on numerical analytic continuation. As an application, we demonstrate that a pseudogap can emerge in the bilayer Hubbard model when spin, charge, or excitonic fluctuations become sufficiently strong. Instabilities in different channels are also evaluated in an unbiased manner across the parameter space. Remarkably, we find that the excitonic susceptibility grows with increasing interlayer bias, before it gets suppressed at large biases by the charge imbalance between the layers. This work establishes a versatile and computationally efficient framework for investigating correlated multi-orbital systems under nonequilibrium conditions.
We present a review of our recent work in extending the successful dynamical mean-field theory from the equilibrium case to nonequilibrium cases. In particular, we focus on the problem of turning on a spatially uniform, but possibly time varying, electric field (neglecting all magnetic field effects). We show how to work with a manifestly gauge-invariant formalism, and compare numerical calculations from a transient-response formalism to different types of approximate treatments, including the semiclassical Boltzmann equation and perturbation theory in the interaction. In this review, we solve the nonequilibrium problem for the Falicov-Kimball model, which is the simplest many-body model and the easiest problem to illustrate the nonequilibrium behavior in both diffusive metals and Mott insulators. Due to space restrictions, we assume the reader already has some familiarity both with the Kadanoff-Baym-Keldysh nonequilibrium formalism and with equilibrium dynamical mean-field theory; we provide a guide to the literature where additional details can be found.
We characterize the response of a Mott insulating system to a static electric field in terms of its conducting and spectral properties. Dissipation is included by a coupling to fermionic baths and to either optical or acoustic phonons. This paper extends and completes the analysis made in a previous work by the authors [Mazzocchi et al., Phys. Rev. B 106, 125123 (2022)]. In the present work, phonons are included diagrammatically within the Migdal approximation by also including self-consistency from the electronic feedback. The nonequilibrium steady state is addressed by means of the dynamical mean-field theory based on the nonequilibrium Green's function approach, while the so-called auxiliary master equation approach is employed as impurity solver. With optical phonons the self-consistency suppresses the steady-state current for field strengths comparable to the band gap with respect to the non-self-consistent case. This is due to the interaction of phonons with the hot electrons of the lattice which increases their temperature, thus providing a less effective relaxation channel for the current-induced Joule heat. In addition, in the case of optical phonons, the results are essentially independent of the temperature of the fermionic baths, as the latter is sensibly smaller than their characteristic frequency. On the other hand, with acoustic phonons the steady-state current is slightly suppressed by the self-consistent treatment only at field strengths close to half of the gap and especially at very small phonon frequency. Also, in this case, the results seem to slightly depend on the temperature of the fermionic baths.
Optimizing various properties of materials have been under intensive focus for many decades. In recent years, the exposure of ultrafast techniques enhance the possibility of seizing this goal. As by employing these techniques, one can expose the system under strong short fields, and investigate the transient response of various low-lying excitations. Such studies will strengthen our understanding of interplays among various degrees of freedom in correlated materials, and thereby facilitate the chance to manipulate different properties of these systems. Of our particular attention is the electron-lattice dynamics, where modulating a particular vibrational mode of the crystal lattice, will lead to the emergence of new quasi stable'' lattice structures as well as novel transient phenomena. Aside from these experimental prospects, the above mentioned goals were also the subject of an extensive theoretical studies. In this thesis, we investigated the in and out of equilibrium dynamics of electronic systems which are interacting with lattice degrees of freedom. We employed the two-time Keldysh Green's function within the framework of the dynamical mean field theory to conduct our researches on this subject. In our first attempt, we studied the dynamical response of a one-electron system under a linear electron-lattice interaction. We explored the real-time formation of a bound quasiparticle, known as a polaron, under various parameter regimes of the model. We presented that when the carrier is suddenly coupled to the lattice degrees of freedom, the long-time response of the system in the strong coupling limit exhibits both bound, polaronic, and unbound, delocalized, states. We characterized the nature of such a mixed state using the adiabatic picture of the model. We also showed that the formed polaron is dressed by excited phononic states, in contrast to its equilibrium counterpart, where the ground-state phononic states are bounding the electron. We also addressed the response of a many-body electronic system, weakly coupled to a bosonic bath, under various time dependent excitations, which brings the system to a bad-metallic phase. We employed a model which treated the Coulomb repulsive interaction locally. We presented that the relaxation of this excited state varies by the strength of the electron-electron interaction. We demonstrated that, for both insulating and metallic regimes of the system, a slow relaxation dynamics can be observed close to the phase transition from the metallic to insulating regimes of the system. Our analysis revealed that in the metallic side, this slow evolution is rooted in the spin-related physics of the problem. However, in the insulating regimes of the model, our results showed that presence of a gap in the spectral density of the insulating system prohibits the relaxation of the excited particles. We also studied the associated timescale for the adiabatic dynamics near the Mott transition. Our results exhibited that there is a nonadiabatic window'' in the metallic side of the phase transition, in which the adiabatic timescale is extremely long. We pointed out that the presence of such a window is intertwined with the slow relaxation timescale of the spin-related dynamics close to the Mott transition. We also captured an induced metallic phase near metal-to-insulator transition out of equilibrium. We initiated this study using the slave-rotor impurity solver, which is been coupled to the self-energies obtained from the first-order Luttinger-Ward functional for the Phonon degrees of freedom. We presented the emergence of the larger spectral density at small frequencies both in metallic and insulating phases close to the phase transition, which is surrounded by pseudo gaps. We explored the physics of such spectral densities as well as their dependencies on the phononic parameters. We postulated that this response of the system is due to the evolution of the system, in a non-thermal mechanism, towards a temperature which is less than the initial temperature of the system, before coupling to the bath. We also pointed out that the associated very slow relaxation dynamics is the result of the formation of psuedo-gaps around the quasiparticle peak.
In this thesis, we describe methods for solving quantum impurity models out of equilibrium, and consider several applications of these methods. Quantum impurity models describe systems in which a small subsystem (the impurity) is coupled to a large external environment (the bath). These models are useful in describing a wide variety of physical systems, including quantum dots, absorbed adatoms, and molecular junctions. They are also important for the study of lattice models, such as the Hubbard model, where they appear as auxiliary problems within the framework of dynamical mean field theory (DMFT). Many interesting phenomena in these systems, such as transport through a quantum dot, or optical pumping of a Mott insulator, occur away from equilibrium. Theoretical description of these phenomena requires methods that are capable of handling both real time evolution and strong correlation effects presenting a formidable challenge. In this thesis, we describe several methods for the solution of non-equilibrium quantum impurity models which meet this challenge. These methods are based on an expansion in the coupling between the impurity and the bath known as the hybridization expansion. Chapter 1 gives an introduction to the problems addressed in this thesis. Chapter 2 provides a brief overview of Keldysh formalism for nonequilibrium problems. Chapter 3 describes the bare hybridization expansion for the Anderson impurity model as well as its bold extensions. In particular we derive the non-crossing approximation (NCA), the one-crossing approximation (OCA), the inchworm expansion, and a bold expansion for the impurity Green's function. Chapter 4 describes the implementation of a quantum Monte Carlo algorithm based on the inchworm expansion. The rest of the thesis describes applications of these algorithms to problems of physical interest. Chapters 5 and 6 apply the methods previously described to understanding the Kondo effect in quantum dot systems. The first chapter examines the behavior of the Kondo effect under an applied voltage bias, a classic non-equilibrium strongly-correlated problem. The second chapter examines different ways of measuring the cloud of electrons which screen the impurity spin in the Kondo regime. Chapters 7 and 8 apply the same methods to the Hubbard model using the Dynamical mean-field theory (DMFT) mapping. The first chapter studies the Hubbard model in equilibrium using the numerically exact inchworm quantum Monte Carlo method. The second chapter studies the possibility of dynamic switching between the metal and Mott-insulator states of the Hubbard model using the one-crossing approximation (OCA). Finally chapter 9 gives some overall conclusions and suggests future directions for the development of hybridization expansion methods.