The phase field, which originates from the electronic interaction, plays an important role in describing a strongly correlated system in one dimension. However, in higher dimensions the effect of phase field cannot be obviously understood. With the eigenfunctional theory, we calculate the pair distribution function of the three-dimensional electron gas to study the relationship between the phase field and the electronic correlation effect and show that at zero temperature the correlation effect of the electrons is mainly dominated by the phase fluctuation, which is produced by the electronic interaction. We also discuss the failure of random phase approximation in studying the correlation function when the correlation effect is strong in the view of the phase field.
In this paper, we present the rigorous expression of the ground state energy and study the phase transition of the three-dimensional homogeneous electron gas by the eigenfunctional theory. The ground state energy is decided completely by the pair distribution function $ g(r)$, which is, by definition, strictly a positive function. But when the density decreases, the electronic correction effect becomes strong from weak, the previous theories basing on one-particle approximation, such as: RPA, Hubbard and STLS ,\cite{4,9} can't insure
g(r) positive always, which implies that they are becoming invalid and overestimate the ground state energy. The eigenfunctional theory has a significant improvement over them, under the linear approximation in solving the equation of the phase field, the
g(r) obtained by the eigenfunctional theory is always positive and can well satisfy the normalization integral (
n0∫d3r[g(r)−1]=−1) which is one of the important features of
g(r). After obtaining
g(r) and calculating the ground state energy of the electron gas both in paramagnetic and ferromagnetic by
g(r), we observe a continuous phase transition from paramagnetic to the ferromagnetic occurring at
rs=19.9±0.8. This can be indirectly supported by the observation of a ferromagnetic state in doped hexaboride
(Ca1−xLaxBb),\cite{42} and is close to the result of G.Ortiz et.al (
rs=20±5).\cite{12}
Similar to the bosonization method in one dimension, we divide the electron operators in phase space into two component operators, and study three-dimensional homogeneous electron gas with a strong correlation effect. We apply this method to calculate the pair distribution function g(r = 0, r s ) and the renormalization coefficient of the one-electron Green's function Z(k F ) (Z F ), and show that it has great advantages of succinctness and efficiency for studying one- and three-dimensional systems with a strong correlation effect. This method can be applied to investigate interaction quenches in ultracold atomic gases.
We have evaluated the density-density response of the two-dimensional electron gas at zero temperature by solving the Dyson equation for the particle-hole Green's function, including exchange Coulomb matrix elements and short-range contributions in the ladder approximation. We study the effect of these correlations on the total energy, compressibility per particle, local field factor G ( q ), static structure factor and pair-correlation function. Results are compared with the normal random-phase approximation, local field theories and quantum Monte Carlo calculations.
In this work we investigate some aspects of the physics of strongly correlated systems by taking into account both electron-electron and electron-phonon interactions as basic mechanisms for reproducing electronic correlations in real materials. The relevance of the electron-electron interactions is discussed in the first part of this thesis in the framework of a self-consistent theoretical approach, named Composite Operator Method (COM), which accounts for the relevant quasi-particle excitations in terms of a set of composite operators that appear as a result of the modification imposed by the interactions on the canonical electronic fields. We show that the COM allows the calculation of all the relevant Green’s and correlation functions in terms of a number of unknown internal parameters to be determined self-consistently. Therefore, depending on the balance between unknown parameters and self-consistent equations, exact and approximate solutions can be obtained. By way of example, we discuss the application of the COM to the extended t-U- J-h model in the atomic limit, and to the two-dimensional single-band Hubbard model. In the former case, we show that the COM provides the exact solution of the model in one dimension. We study the effects of electronic correlations as responsible for the formation of a plethora of different charge and/or spin orderings. We report the phase diagram of the model, as well as a detailed analysis of both zero and finite temperature single-particle and thermodynamic properties. As far as the single-band Hubbard model is concerned, we illustrate an approximated selfconsistent scheme based on the choice of a two-field basis. We report a detailed analysis of many unconventional features that arise in single-particle properties, thermodynamics and system’s response functions. We emphasize that the accuracy of the COM in describing the effects of electronic correlations strongly relies on the choice of the basis, paving the way for possible multi-pole extensions to the twofield theory. To this purpose, we also study a three-field approach to the single-band Hubbard model, showing a significant step forward in the agreements with numerical data with respect to the two-pole results. The role of the electron-phonon interaction in the physics of strongly correlated systems is discussed in the second part of this thesis. We show that in highly polarizable lattices the competition between unscreened Coulomb and Fröhlich interactions results in a short-range polaronic exchange term Jp that favours the formation of local and light pairs of bosonic nature, named bipolarons, which condense with a critical temperature well in excess of hundred kelvins. These findings, discussed in the framework of the so-called polaronic t-Jp model, are further investigated in the presence of a finite on-site potential ~U , coming from the competition between on-site Coulomb and Fröhlich interactions. We discuss the role of ~U as the driving parameter for a small-to-large bipolaron transition, providing a possible explanation of the BEC-BCS crossover in terms of the properties of the bipolaronic ground state. Finally, we show that a hard-core bipolarons gas, studied as a charged Bose-Fermi mixture, allows for the description of many non Fermi liquid behaviours, allowing also for a microscopic explanation of pseudogap features in terms of a thermal-induced recombination of polarons and bipolarons, without any assumption on preexisting order or broken symmetries. [edited by Author]
We discuss interaction effects for the one-dimensional electron gas with a repulsive \ensuremath{\delta}-function interaction potential by using the random-phase approximation and a local-field correction. Analytical results for the local-field correction of charge-density fluctuations and spin-density fluctuations are obtained. The ground-state energy is found to be in better agreement with the exact result than the ground-state energy calculated within the random-phase approximation. We calculate the pair-correlation functions for electrons with parallel and antiparallel spins and the paramagnetic susceptibility. The energies of the collective density modes ${\mathrm{\ensuremath{\omega}}}_{\mathrm{d}}$ (q) and the collective spin modes ${\mathrm{\ensuremath{\omega}}}_{\mathrm{s}}$ (q) are calculated and compared with the energy of the electron-hole excitations ${\mathrm{\ensuremath{\omega}}}_{\mathrm{eh}\ifmmode\pm\else\textpm\fi{}}$ (q): ${\mathrm{\ensuremath{\omega}}}_{\mathrm{d}}$ (q\ensuremath{\rightarrow}0)>${\mathrm{\ensuremath{\omega}}}_{\mathrm{eh}\ifmmode\pm\else\textpm\fi{}}$ (q\ensuremath{\rightarrow}0)>${\mathrm{\ensuremath{\omega}}}_{\mathrm{s}}$ (q\ensuremath{\rightarrow}0). The critical exponents for the long-distance behavior of correlation functions are found to be described by local-field corrections. We compare our results for the critical exponents with recent results obtained by bosonization techniques and the conformal field theory.
Interacting Fermi systems in the strongly correlated regime play a fundamental role in many areas of physics and are of particular interest to the condensed matter community. Though weakly interacting fermions are understood, strongly correlated fermions are difficult to describe theoretically as there is no small interaction parameter to expand about. Existing strong-coupling theories rely heavily on the so-called many-body
T-matrix approximation that sums ladder-type Feynman diagrams. Here, by acknowledging the fact that the effective interparticle interaction (i.e., the vertex function) becomes smaller above three dimensions, we propose an alternative way to reorganize Feynman diagrams and develop a theoretical framework for interacting Fermi gases beyond the ladder approximation. As an application, we solve the equation of state for three- and two-dimensional strongly interacting fermions and find excellent agreement with experimental [M. J. H. Ku et al., Science 335, 563 (2012)] and other theoretical results above temperatures of
0.5TF.