Superconductivity in the attractive Hubbard model: functional renormalization group analysis

We present a functional renormalization group analysis of superconductivity in the ground state of the attractive Hubbard model on a square lattice. Spontaneous symmetry breaking is treated in a purely fermionic setting via anomalous propagators and anomalous effective interactions. In addition to the anomalous interactions arising already in the reduced BCS model, effective interactions with three incoming legs and one outgoing leg (and vice versa) occur. We accomplish their integration into the usual diagrammatic formalism by introducing a Nambu matrix for the effective interactions. From a random-phase approximation generalized through use of this matrix we conclude that the impact of the 3+1 effective interactions is limited, especially considering the effective interactions important for the determination of the order parameter. The exact hierarchy of flow equations for one-particle irreducible vertex functions is truncated on the two-particle level, with higher-order self-energy corrections included in a scheme proposed by Katanin. Using a parametrization of effective interactions by patches in momentum space, the flow equations can be integrated numerically to the lowest scales without encountering divergences. Momentum-shell as well as interaction-flow cutoff functions are used, including a small external field or a large external field and a counterterm, respectively. Both approaches produce momentum-resolved order parameter values directly from the microscopic model. The size of the superconducting gap is in reasonable agreement with expectations from other studies.Comment: 22 pages, 16 figures, references added, some changes in the introductio

A Semantic Basis for Parallel Algorithm Design

As computing demands increase, emphasis is being placed on parallel architectures- To efficiently use parallel machines, software must be designed to take advantage of these machines. This research concentrates on an abstraction of algorithm design to permit the expression of parallel programs. The abstraction emphasizes thought about algorithms at a high level as opposed to algorithm implementation at a statement level. A model based on data flow allows algorithm expression using flow diagrams. The model specifies operating system requirements that support parallel programming at a module level. Paths are used to carry data between modules. Data enter modules through ports. Module activation is triggered by the satisfaction of data availability conditions. Continual module presence within the system, dynamic activation criteria, and a high level of programming distinguishes this model from other parallel programming systems

A Simple Method for Organizing Nearly Optimal Binary Search Trees

Improving the efficiency of retrieving information concerns users of computer systems involved in many applications- One way of addressing this concern is to organize a sorted sequence into a binary search tree. Knuth\u27s Algorithm K is a bottom-up organization algorithm that always constructs a binary tree which minimizes average search time. However, the cost of executing Algorithm K is prohibitive for a large tree. The aim of this work is to find a less costly method of organizing sorted sequences into nearly-optimal binary search trees. We present a top-down organization method which yields better average search times than top-down methods already available, specifically height-balancing and weight-balancing. The variation in access frequency among the members of a sequence is used to recommend specific values for some of the parameters in this new method of organization. The new method improves considerably on the cost of organization as opposed to the cost of using Algorithm K while producing trees whose average search times are close to minimal. The new algorithm yields an average search time that is usually within 1% of the minimal average search time and for every case attempted has been no worse than 1.5% larger than minimal

Numerical renormalization group study of the symmetric Anderson-Holstein model: phonon and electron spectral functions

We study the symmetric Anderson-Holstein (AH) model at zero temperature with Wilson's numerical renormalization group (NRG) technique to study the interplay between the electron-electron and electron-phonon interactions. An improved method for calculating the phonon propagator using the NRG technique is presented, which turns out to be more accurate and reliable than the previous works in that it calculates the phonon renormalization explicitly and satisfies the boson sum rule better. The method is applied to calculate the renormalized phonon propagators along with the electron propagators as the onsite Coulomb repulsion $U$ and electron-phonon coupling constant $g$ are varied. As $g$ is increased, the phonon mode is successively renormalized, and for $g \gtrsim g_{co}$ crosses over to the regime where the mode splits into two components, one of which approaches back to the bare frequency and the other develops into a soft mode. The initial renormalization of the phonon mode, as $g$ is increased from 0, depends on $U$ and the hybridization $\Delta$; it gets softened (hardened) for $U \gtrsim (\lesssim) U_s (\Delta)$. Correlated with the emergence of the soft mode is the central peak of the electron spectral function severely suppressed. These NRG calculations will be compared with the standard Green's function results for the weak coupling regime to understand the phonon renormalization and soft mode.Comment: 18 pages, 4 figures. Submitted to Phys. Rev.

Mott-Hubbard transition in infinite dimensions

We calculate the zero-temperature gap and quasiparticle weight of the half-filled Hubbard model with a random dispersion relation. After extrapolation to the thermodynamic limit, we obtain reliable bounds on these quantities for the Hubbard model in infinite dimensions. Our data indicate that the Mott-Hubbard transition is continuous, i.e., that the quasiparticle weight becomes zero at the same critical interaction strength at which the gap opens.Comment: 4 pages, RevTeX, 5 figures included with epsfig Final version for PRL, includes L=14 dat

Time-dependent Gutzwiller approximation for the Hubbard model

We develop a time-dependent Gutzwiller approximation (GA) for the Hubbard model analogous to the time-dependent Hartree-Fock (HF) method. The formalism incorporates ground state correlations of the random phase approximation (RPA) type beyond the GA. Static quantities like ground state energy and double occupancy are in excellent agreement with exact results in one dimension up to moderate coupling and in two dimensions for all couplings. We find a substantial improvement over traditional GA and HF+RPA treatments. Dynamical correlation functions can be easily computed and are also substantially better than HF+RPA ones and obey well behaved sum rules.Comment: 4 pages, 2 figure

Comparison of Variational Approaches for the Exactly Solvable 1/r-Hubbard Chain

We study Hartree-Fock, Gutzwiller, Baeriswyl, and combined Gutzwiller-Baeriswyl wave functions for the exactly solvable one-dimensional $1/r$-Hubbard model. We find that none of these variational wave functions is able to correctly reproduce the physics of the metal-to-insulator transition which occurs in the model for half-filled bands when the interaction strength equals the bandwidth. The many-particle problem to calculate the variational ground state energy for the Baeriswyl and combined Gutzwiller-Baeriswyl wave function is exactly solved for the~$1/r$-Hubbard model. The latter wave function becomes exact both for small and large interaction strength, but it incorrectly predicts the metal-to-insulator transition to happen at infinitely strong interactions. We conclude that neither Hartree-Fock nor Jastrow-type wave functions yield reliable predictions on zero temperature phase transitions in low-dimensional, i.e., charge-spin separated systems.Comment: 23 pages + 3 figures available on request; LaTeX under REVTeX 3.