The Hubbard model within the equations of motion approach

The Hubbard model has a special role in Condensed Matter Theory as it is considered as the simplest Hamiltonian model one can write in order to describe anomalous physical properties of some class of real materials. Unfortunately, this model is not exactly solved except for some limits and therefore one should resort to analytical methods, like the Equations of Motion Approach, or to numerical techniques in order to attain a description of its relevant features in the whole range of physical parameters (interaction, filling and temperature). In this manuscript, the Composite Operator Method, which exploits the above mentioned analytical technique, is presented and systematically applied in order to get information about the behavior of all relevant properties of the model (local, thermodynamic, single- and two- particle ones) in comparison with many other analytical techniques, the above cited known limits and numerical simulations. Within this approach, the Hubbard model is shown to be also capable to describe some anomalous behaviors of the cuprate superconductors.Comment: 232 pages, more than 300 figures, more than 500 reference

Exploring Statistical and Population Aspects of Network Complexity

The characterization and the definition of the complexity of objects is an important but very difficult problem that attracted much interest in many different fields. In this paper we introduce a new measure, called network diversity score (NDS), which allows us to quantify structural properties of networks. We demonstrate numerically that our diversity score is capable of distinguishing ordered, random and complex networks from each other and, hence, allowing us to categorize networks with respect to their structural complexity. We study 16 additional network complexity measures and find that none of these measures has similar good categorization capabilities. In contrast to many other measures suggested so far aiming for a characterization of the structural complexity of networks, our score is different for a variety of reasons. First, our score is multiplicatively composed of four individual scores, each assessing different structural properties of a network. That means our composite score reflects the structural diversity of a network. Second, our score is defined for a population of networks instead of individual networks. We will show that this removes an unwanted ambiguity, inherently present in measures that are based on single networks. In order to apply our measure practically, we provide a statistical estimator for the diversity score, which is based on a finite number of samples

TDDFT and quantum-classical dynamics: A universal tool describing the dynamics of matter

Time-dependent density functional theory (TDDFT) is currently the most efficient approach allowing to describe electronic dynamics in complex systems, from isolated molecules to the condensed phase. TDDFT has been employed to investigate an extremely wide range of time-dependent phenomena, as spin dynamics in solids, charge and energy transport in nanoscale devices, and photoinduced exciton transfer in molecular aggregates. It is therefore nearly impossible to give a general account of all developments and applications of TDDFT in material science, as well as in physics and chemistry. A large variety of aspects are covered throughout these volumes. In the present chapter, we will limit our presentation to the description of TDDFT developments and applications in the field of quantum molecular dynamics simulations in combination with trajectory-based approaches for the study of nonadiabatic excited-state phenomena. We will present different quantum-classical strategies used to describe the coupled dynamics of electrons and nuclei underlying nonadiabatic processes. In addition, we will give an account of the most recent applications with the aim of illustrating the nature of the problems that can be addressed with the help of these approaches. The potential, as well as the limitations, of the presented methods is discussed, along with possible avenues for future developments in TDDFT and nonadiabatic dynamics

TDDFT and Quantum-Classical Dynamics: A Universal Tool Describing the Dynamics of Matter

Time-dependent density functional theory (TDDFT) is currently the most efficient approach allowing to describe electronic dynamics in complex systems, from isolated molecules to the condensed phase. TDDFT has been employed to investigate an extremely wide range of time-dependent phenomena, as spin dynamics in solids, charge and energy transport in nanoscale devices, and photoinduced exciton transfer in molecular aggregates. It is therefore nearly impossible to give a general account of all developments and applications of TDDFT in material science, as well as in physics and chemistry. A large variety of aspects are covered throughout these volumes. In the present chapter, we will limit our presentation to the description of TDDFT developments and applications in the field of quantum molecular dynamics simulations in combination with trajectory-based approaches for the study of nonadiabatic excited-state phenomena. We will present different quantum-classical strategies used to describe the coupled dynamics of electrons and nuclei underlying nonadiabatic processes. In addition, we will give an account of the most recent applications with the aim of illustrating the nature of the problems that can be addressed with the help of these approaches. The potential, as well as the limitations, of the presented methods is discussed, along with possible avenues for future developments in TDDFT and nonadiabatic dynamics