This book covers different aspects of the physics of iron-based superconductors ranging from the theoretical, the numerical and computational to the experimental ones. It starts from the basic theory modeling many-body physics in Fe-superconductors and other multi-orbital materials and reaches up to the magnetic and Cooper pair fluctuations and nematic order. Finally, it offers a comprehensive overview of the most recent advancements in the experimental investigations of iron based superconductors.

In this dissertation we focus on the investigation of the pairing mechanism in the recently discovered high-temperature superconductor, iron pnictides. Due to the proximity to magnetic instability of the system, we considered short-range spin fluctuations as the major mediating source to induce superconductivity. Our calculation supports the magnetic fluctuations as a strong candidate that drives Cooper-pair formation in this material. We find the corresponding order parameter to be of the so-called ss-wave type and show its evolution with temperature as well as the capability of supporting high transition temperature up to several tens of Kelvin. On the other hand, our itinerant model calculation shows pronounced spin correlation at the observed antiferromagnetic ordering wave vector, indicating the underlying electronic structure in favor of antiferromagnetic state. Therefore, the electronic degrees of freedom could participate both in the magnetic and in the superconducting properties. Our work shows that the interplay between magnetism and superconductivity plays an important role to the understanding of the rich physics in this material. The magnetic-excitation spectrum carries important information on the nature of magnetism and the characteristics of superconductivity. We analyze the spin excitation spectrum in the normal and superconducting states of iron pnictides in the magnetic scenario. As a consequence of the sign-reversed gap structure obtained in the above, a spin resonance mode appears below the superconducting transition temperature. The calculated resonance energy, scaled with the gap magnitude and the magnetic correlation length, agrees well with the inelastic neutron scattering (INS) measurements. More interestingly, we find a common feature of those short-range spin fluctuations that are capable of inducing a fully gapped ss state is the momentum anisotropy with elongated span along the direction transverse to the antiferromagnetic momentum transfer. This calculated intrinsic anisotropy exists both in the normal and in the superconducting state, which naturally explains the elliptically shaped magnetic responses observed in INS experiments. Our detailed calculation further shows that the magnetic resonance mode exhibits an upward dispersion-relation pattern but anisotropic along the transverse and longitudinal directions. We also perform a qualitative analysis on the relationship between the anisotropic momentum structure of the magnetic fluctuations and the stability of superconducting phase by intraorbital but interband pair scattering to show the consistency of the magnetic mechanism for superconductivity. As discussed for cuprates, an important identification of the mediating boson is from the fermionic spectrum. We study the spectral function in the normal and superconducting state. Not only do we extract the gap magnitude on the electron- and hole-pockets to show the momentum structure of the gap, but also find a peak-dip-hump feature in the electron spectrum, which reflects the feedback from the spin excitations on fermions. This serves as an interpretation of the kink structure observed in ARPES measurements.

The behaviour of magnetic impurities in metals has posed problems to challenge the condensed matter theorist over the past 30 years. This book deals with the concepts and techniques which have been developed to meet this challenge, and with their application to the interpretation of experiments. This book will be of interest to condensed matter physicists, particularly those interested in strong correlation problems. The detailed discussions of advanced many-body techniques should make it of interest to theoretical physicists in general.