Screened-interaction expansion for the Hubbard model and determination of the quantum Monte Carlo Fermi surface

We develop a systematic self-consistent perturbative expansion for the self energy of Hubbard-like models. The interaction lines in the Feynman diagrams are dynamically screened by the charge fluctuations in the system. Although the formal expansion is exact-assuming that the model under the study is perturbative-only if diagrams to all orders are included, it is shown that for large-on-site-Coulomb-repulsion-U systems weak-coupling expansions to a few orders may already converge. We show that the screened interaction for the large-U system can be vanishingly small at a certain intermediate electron filling; and it is found that our approximation for the imaginary part of the one-particle self energy agrees well with the QMC results in the low energy scales at this particular filling. But, the usefulness of the approximation is hindered by the fact that it has the incorrect filling dependence when the filling deviates from this value. We also calculate the exact QMC Fermi surfaces for the two-dimensional (2-D) Hubbard model for several fillings. Our results near half filling show extreme violation of the concepts of the band theory; in fact, instead of growing, Fermi surface vanishes when doped toward the half-filled Mott-Hubbard insulator. Sufficiently away from half filling, noninteracting-like Fermi surfaces are recovered. These results combined with the Luttinger theorem might show that diagrammatic expansions for the nearly-half-filled Hubbard model are unlikely to be possible; however, the nonperturbative part of the solution seems to be less important as the filling gradually moves away from one half. Results for the 2-D one-band Hubbard model for several hole dopings are presented. Implications of this study for the high-temperature superconductors are also discussed.Comment: 11 pages, 12 eps figures embedded, REVTeX, submitted to Phys. Rev. B; (v2) minor revisions, scheduled for publication on November 1

Systems Biology of the Clock in Neurospora crassa

A model-driven discovery process, Computing Life, is used to identify an ensemble of genetic networks that describe the biological clock. A clock mechanism involving the genes white-collar-1 and white-collar-2 (wc-1 and wc-2) that encode a transcriptional activator (as well as a blue-light receptor) and an oscillator frequency (frq) that encodes a cyclin that deactivates the activator is used to guide this discovery process through three cycles of microarray experiments. Central to this discovery process is a new methodology for the rational design of a Maximally Informative Next Experiment (MINE), based on the genetic network ensemble. In each experimentation cycle, the MINE approach is used to select the most informative new experiment in order to mine for clock-controlled genes, the outputs of the clock. As much as 25% of the N. crassa transcriptome appears to be under clock-control. Clock outputs include genes with products in DNA metabolism, ribosome biogenesis in RNA metabolism, cell cycle, protein metabolism, transport, carbon metabolism, isoprenoid (including carotenoid) biosynthesis, development, and varied signaling processes. Genes under the transcription factor complex WCC ( = WC-1/WC-2) control were resolved into four classes, circadian only (612 genes), light-responsive only (396), both circadian and light-responsive (328), and neither circadian nor light-responsive (987). In each of three cycles of microarray experiments data support that wc-1 and wc-2 are auto-regulated by WCC. Among 11,000 N. crassa genes a total of 295 genes, including a large fraction of phosphatases/kinases, appear to be under the immediate control of the FRQ oscillator as validated by 4 independent microarray experiments. Ribosomal RNA processing and assembly rather than its transcription appears to be under clock control, suggesting a new mechanism for the post-transcriptional control of clock-controlled genes

A MINE alternative to D-optimal designs for the linear model.

Doing large-scale genomics experiments can be expensive, and so experimenters want to get the most information out of each experiment. To this end the Maximally Informative Next Experiment (MINE) criterion for experimental design was developed. Here we explore this idea in a simplified context, the linear model. Four variations of the MINE method for the linear model were created: MINE-like, MINE, MINE with random orthonormal basis, and MINE with random rotation. Each method varies in how it maximizes the MINE criterion. Theorem 1 establishes sufficient conditions for the maximization of the MINE criterion under the linear model. Theorem 2 establishes when the MINE criterion is equivalent to the classic design criterion of D-optimality. By simulation under the linear model, we establish that the MINE with random orthonormal basis and MINE with random rotation are faster to discover the true linear relation with p regression coefficients and n observations when p>>n. We also establish in simulations with n<100, p=1000, σ=0.01 and 1000 replicates that these two variations of MINE also display a lower false positive rate than the MINE-like method and additionally, for a majority of the experiments, for the MINE method

Posterior means of the first 20 regression coefficients as a function of the number of experiments for: (A) the MINE-like method; (B) MINE method; (C) MINE method with random orthonormal basis; (D) MINE method with random rotation.

<p>Each panel is averaged over all 1000 simulations with 10 zero (in blue) and 10 nonzero (in red). The first ten (red) are truly nonzero.</p

The effects of varying the noise level on the power to detect 7 out 10 of the truly nonzero regression coefficients and the false positive rate for the MINE with random rotation.

<p>The effects of varying the noise level on the power to detect 7 out 10 of the truly nonzero regression coefficients and the false positive rate for the MINE with random rotation.</p

The number of false positives as a function of the number of experiments.

<p>These numbers are averaged over all 1000 simulations for each method. Blue corresponds to MINE-like, red to MINE, green to MINE with random orthonormal basis, and purple to MINE with random rotation. The final two overlap almost exactly which is why the green line is not visible.</p

Parameters for simulations.

<p>Parameters for simulations.</p

Cycle of MINE discovery – Simplified Computing Life Paradigm.

<p>Cycle of MINE discovery – Simplified Computing Life Paradigm.</p

A genetic network for the biological clock from [7].

<p>Molecular species (<i>i.e.</i>, reactants or products) in the network are represented by boxes. The <i>white-collar-1</i> (<i>wc-1</i>), <i>white-collar-2</i> (<i>wc-2</i>), <i>frequency</i> (<i>frq</i>), <i>and clock controlled gene</i> (<i>ccg</i>) gene symbols can be superscripted 0, 1, r0, r1, indicating, respectively, a transcriptionally inactive (0) or active (1) gene or a translationally inactive (r0) or active (r1) mRNA. Associated protein species are denoted by capitals. A phot (in yellow) denotes a photon species. Reactions in the network are represented by circles. Arrows entering circles identify reactants; arrows leaving circles identify products; and bi-directional arrows identify catalysts. The labels on each reaction, such as S₄, also serve to denote the rate coefficients for each reaction. Reactions labeled with an S, L, or D denote transcription, translation, or degradation reactions, respectively. Reactions without products, such as D₈, are decay reactions. Reactions, such as A and P, have cooperative kinetics: (A) n WCC+<i>frq</i>⁰→<i>frq</i>¹ and (P) WCC+m FRQ→WC-2+m FRQ. The n and m are Hill coefficients or cooperativities. Only one reaction, the “A” reaction, has a back reaction, , <i>frq</i>¹→n WCC+<i>frq</i>⁰, included, with non-zero rate. The rate constants specify the right hand side of the kinetics model in equation (1) through the Law of Mass Action in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0003105#s2" target="_blank">Materials and Methods</a>.</p