Mean Field Theory

We study the origin of the temperature-induced Mott transition in Ca2RuO4. As a method we use the local-density approximation+dynamical mean-field theory. We show the following. (i) The Mott transition is driven by the change in structure from long to short c-axis layered perovskite (L-Pbca to S-Pbca); it occurs together with orbital order, which follows, rather than produces, the structural transition. (ii) In the metallic L-Pbca phase the orbital polarization is ~0. (iii) In the insulating S-Pbca phase the lower energy orbital, ~xy, is full. (iv) The spin-flip and pair-hopping Coulomb terms reduce the effective masses in the metallic phase. Our results indicate that a similar scenario applies to Ca_{2-x}Sr_xRuO_4 (x<0.2). In the metallic x< 0.5 structures electrons are progressively transferred to the xz/yz bands with increasing x, however we find no orbital-selective Mott transition down to ~300 K.

The purpose of this thesis is to introduce the reader to the Quantum Hadrodynamics Model (QHD) which is used to interpret and model nuclear matter. The mathematical formalism that describes the Field Theory and the basic principles of the Relativistic Mean Field Theory are also analyzed. The first and the second chapter consist of a thorough introduction to the definition of nuclear matter and its properties. In the next chapter the necessary theoretical and mathematical background is analyzed. Furthermore, reference is made to Tensor Calculus, Special Theory of Relativity, Lagrange/Hamilton mechanics and Relativistic Quantum Mechanics with emphasis on Dirac and Klein-Gordon equations. In the fourth chapter we approach the field theory topic. We start form the basic classical field theory and after we briefly explain the conservation laws we move to analyze the Quantum field theory. The fifth chapter acquaints us with Quantum Hadrodynamics (QHD) through the linear model QHD-I under the approach of the Relativistic Mean Field Theory (RMFT). Some configurations/extensions are also discussed that lead to more precise modeling of the nuclear matter.

This chapter explores the number of mean field constructions for ions whose structure goes beyond the point charge description, the representation used in the standard Poisson‐Boltzmann (PB) equation. The structural details omitted within a point charge picture can be related to electrostatic structure of an ion, giving rise to polarizability, asymmetric interactions, softening of Coulomb interactions if a distribution of an ion charge is extended in space, or can be related to the Pauli exclusion principle, giving rise to the excluded volume effects or other types of repulsive interactions. The efficient theory for hard‐core interactions, the fundamental measure DFT, shows shortcomings even for the weak‐coupling limit conditions. There is an additional motivation for pursuing various mean field constructions. Simple models such as charged hard spheres are easy to simulate, and for these systems one could simply use simulations to cover the entire range of electrostatics, from weak‐ to strong‐coupling regime.

The goal of the present account is to review our efforts to obtain and apply a “collective” Hamiltonian for a few, approximately decoupled, adiabatic degrees of freedom, starting from a Hamiltonian system with more or many more degrees of freedom. The approach is based on an analysis of the classical limit of quantum-mechanical problems. Initially, we study the classical problem within the framework of Hamiltonian dynamics and derive a fully self-consistent theory of large-amplitude collective motion with small velocities. We derive a measure for the quality of decoupling of the collective degree of freedom. We show for several simple examples, where the classical limit is obvious, that when decoupling is good, a quantization of the collective Hamiltonian leads to accurate descriptions of the low energy properties of the systems studied. In nuclear physics problems we construct the classical Hamiltonian by means of time-dependent mean-field theory, and we transcribe our formalism to this case. We report studies of a model for monopole vibrations, of 28Si with a realistic interaction, several qualitative models of heavier nuclei, and preliminary results for a more realistic approach to heavy nuclei. Other topics included are a nuclear Born–Oppenheimer approximation for an ab initio quantum theory and a theory of the transfer of energy between collective and noncollective degrees of freedom when the decoupling is not exact. The explicit account is based on the work of the authors, but a thorough survey of other work is included.

The critical parameters provide important information concerning the interaction mechanisms near the paramagnetic-to-ferromagnetic transition. In this paper, we present a thorough study for the critical behavior of La 0.7A0.3(Mn 1-xB x)O 3 ( A=Sr; B=Ti and Al; x=0.0 and 0.05) polycrystalline samples near ferromagnetic-paramagnetic phase transition temperature by analyzing isothermal magnetization data. We have analyzed our dc-magnetization data near the transition temperature with the help of the modified Arrot plot, Kouvel-Fisher method. We have determined the critical temperature TC and the critical parameters β, γ and δ. With the values of TC, β and γ, we plot M×(1- T/ TC) - β vs. H×(1- T/ TC) - γ. All the data collapse on one of the two curves. This suggests that the data below and above TC obey scaling, following a single equation of state. Critical parameters for x=0 and xTi=0.05 samples are between those predicted for a 3D-Heisenberg model and mean-field theory and for xAl=0.05 samples the values obtained for the critical parameters are close to those predicted by the mean-field theory.

We use the zero-temperature random-field Ising model to study hysteretic behavior at first-order phase transitions. Sweeping the external field through zero, the model exhibits hysteresis, the return-point memory effect, and avalanche fluctuations. There is a critical value of disorder at which a jump in the magnetization (corresponding to an infinite avalanche) first occurs. We study the universal behavior at this critical point using mean-field theory, and also present preliminary results of numerical simulations in three dimensions.