Effective action and density functional theory

The effective action for the charge density and the photon field is proposed as a generalization of the density functional. A simple definition is given for the density functional, as the functional Legendre transform of the generator functional of connected Green functions for the density and the photon field, offering systematic approximation schemes. The leading order of the perturbation expansion reproduces the Hartree-Fock equation. A renormalization group motivated method is introduced to turn on the Coulomb interaction gradually and to find corrections to the Hartree-Fock and the Kohn-Sham schemes.Comment: New references and a numerical algorithm added, to appear in Phys. Rev. B. 30 pages, no figure

Two-dimensional limit of exchange-correlation energy functional approximations in density functional theory

We investigate the behavior of three-dimensional (3D) exchange-correlation energy functional approximations of density functional theory in anisotropic systems with two-dimensional (2D) character. Using two simple models, quasi-2D electron gas and two-electron quantum dot, we show a {\it fundamental limitation} of the local density approximation (LDA), and its semi-local extensions, generalized gradient approximation (GGA) and meta-GGA (MGGA), the most widely used forms of which are worse than the LDA in the strong 2D limit. The origin of these shortcomings is in the inability of the local (LDA) and semi-local (GGA/MGGA) approximations to describe systems with 2D character in which the nature of the exchange-correlation hole is very nonlocal. Nonlocal functionals provide an alternative approach, and explicitly the average density approximation (ADA) is shown to be remarkably accurate for the quasi-2D electron gas system. Our study is not only relevant for understanding of the functionals but also practical applications to semiconductor quantum structures and materials such as graphite and metal surfaces. We also comment on the implication of our findings to the practical device simulations based on the (semi-)local density functional method.Comment: 21 pages including 9 figures, to be published in Phys. Rev.

Quantum critical behavior in insulating region of the 2D metal insulator transition

We show that the quantum critical point associated with the metal-insulator transition phenomenon in two dimensions controls an extended critical region encompassing not only the usual quantum critical sector but also a range of the low-temperature insulator region. The extended range of criticality permits a unified analysis of data from the insulating region and quantum critical sector, allowing us to determine both the dynamical critical exponent z and the correlation length critical exponent from published data from a single experiment in the insulator critical region. We show that the critical exponents determined from the insulator sector consistently describe the temperature dependence of the resistance data from the same experiment in the quantum critical sector. This provides evidence for the presence of a quantum critical point in these systems

Temperature dependent resistivity in the low resistance region for diffusive transport in two-dimensions

The interpretation of the metal-insulator transition phenomena in disordered two-dimensional electron systems in terms of density-dependent scaling variables suggests the existence of a quantum critical point at some critical electron density. However a first principles scaling theory based on renormalization group (RG) methods predicts a strong temperature dependence of the dimensionless resistivity R(T), even at small R(T), that is not observed. The observed properties are in fact consistent with a weakly disordered Fermi liquid, and there are no indications of strong temperature dependence induced by scaling. While the RG expansion in a power series in R(T) has only been evaluated to lowest order, this should be sufficient to describe experiments in the region of very small R. A further apparent anomaly is a return from metal-like to insulating-like behavior for increasing density. We explain these fundamental discrepancies between the first principles theory and experiment. We find that the R<<1 data in the currently attainable temperature range are in a weak scaling regime described by the logarithmic approximation. We independently determine the density dependent prefactor of the logarithm using data for the spin susceptibility and effective mass. We find good agreement between theory and experiment for R(T) in the diffusive regime. We point out that there are corrections to the leading logarithm approximation that should be observable at still lower temperatures

Density dependence of critical magnetic fields at the metal-insulator bifurcation in two dimensions

The density dependence of the critical in-plane magnetic field Bc at the bifurcation of the resistivity of two-dimensional electron systems with low levels of disorder is determined using the spin-polarization dependence of the electron exchange-correlation hole. Recent numerical simulation results for ground-state energies also permit determination of the magnetic field Bpol(n) needed to saturate the spin polarization. The resulting picture gives a good account of reported experimental results for Bc as a function of electron density in p-type GaAs systems and indicates that the interactions between electrons play a crucial role in the bifurcation phenomenon

Low Temperature Properties of 2D Correlated Electrons in Weakly Disordered Materials

Transport properties of extremely high purity two-dimensional (2D) electron systems at low temperatures are still not well understood either experimentally or theoretically, even though these systems are fast becoming a mainstream basis of computing devices. In fact there are two separate issues to be resolved. The first of these has attracted the more attention. This is the existence of a quantum phase transition (the metal-insulator transition) in the low density 2D system at zero temperature. Experimentally, in spite of claims, from existing data at finite temperatures there is no conclusive evidence either way on the existence of a T = 0 quantum phase transition. There is a need for a unified theory encompassing, on the same level, both insulating and metallic behaviour to predict the cross-over. We propose a semi-empirical one parameter renormalisation group equation for the temperature dependent resistivity of a 2D electron system with weak disorder. The renormalisation group equation has a physically meaningful insulating limit and it predicts a metallic ground state of zero resistance at higher electron densities. The resulting temperature dependence of the resistivity is found to give a good fit to experimental data near the separatrix. The second issue is the mechanism behind the sudden change in the temperature dependence of the resistivity, as is actually observed at low but non-zero temperatures, T = 0.1 to 2 K. This phenomenon is well-documented experimentally and it is of interest in its own right whether or not there is an actual transition at T = 0. We present direct evidence of the important role of the electron Coulomb repulsion and exchange in determining these finite temperature properties by noting an empirical relationship between the critical density at the bifurcation point and parallel magnetic field. The relationship is controlled by properties of the electron-electron correlation function for the 2D electron system. This result provides direct evidence of the central role of the Coulomb repulsion and exchange in driving the bifurcation phenomenon