Geometry of Multiplicative Preprojective Algebra

Crawley-Boevey and Shaw recently introduced a certain multiplicative analogue of the deformed preprojective algebra, which they called the multiplicative preprojective algebra. In this paper we study the moduli space of (semi)stable representations of such an algebra (the multiplicative quiver variety), which in fact has many similarities to the quiver variety. We show that there exists a complex analytic isomorphism between the nilpotent subvariety of the quiver variety and that of the multiplicative quiver variety (which can be extended to a symplectomorphism between these tubular neighborhoods). We also show that when the quiver is star-shaped, the multiplicative quiver variety parametrizes Simpson's (poly)stable filtered local systems on a punctured Riemann sphere with prescribed filtration type, weight and associated graded local system around each puncture.Comment: 51pages; corrected typos and references; changed font; v4 is the same as v3 except margi

Effect of the spin-orbit interaction and the electron phonon coupling on the electronic state in a silicon vacancy

The electronic state around a single vacancy in silicon crystal is investigated by using the Green's function approach. The triply degenerate charge states are found to be widely extended and account for extremely large elastic softening at low temperature as observed in recent ultrasonic experiments. When we include the LS coupling $\lambda_{\rm Si}$ on each Si atom, the 6-fold spin-orbital degeneracy for the $V^{+}$ state with the valence +1 and spin 1/2 splits into $\Gamma_{7}$ doublet groundstates and $\Gamma_{8}$ quartet excited states with a reduced excited energy of $O(\lambda_{\rm Si}/10)$. We also consider the effect of couplings between electrons and Jahn-Teller phonons in the dangling bonds within the second order perturbation and find that the groundstate becomes $\Gamma_{8}$ quartet which is responsible for the magnetic-field suppression of the softening in B-doped silicon.Comment: 4 pages, 2 figure

Overdamped Stress Relaxation in Buckled Rods

We present a comprehensive theoretical analysis of the stress relaxation in a multiply but weakly buckled incompressible rod in a viscous solvent. In the bulk two interesting regimes of generic self--similar intermediate asymptotics are distinguished, which give rise to two classes of approximate and exact power--law solutions, respectively. For the case of open boundary conditions the corresponding non--trivial boundary--layer scenarios are derived by a multiple--scale perturbation (``adiabatic'') method. Our results compare well with -- and provide the theoretical explanation for -- previous results from numerical simulations, and they suggest new directions for further fruitful numerical and experimental investigations.Comment: 20 pages, 12 figure

Diffusion-limited loop formation of semiflexible polymers: Kramers theory and the intertwined time scales of chain relaxation and closing

We show that Kramers rate theory gives a straightforward, accurate estimate of the closing time $\tau_c$ of a semiflexible polymer that is valid in cases of physical interest. The calculation also reveals how the time scales of chain relaxation and closing are intertwined, illuminating an apparent conflict between two ways of calculating $\tau_c$ in the flexible limit.Comment: Europhys. Lett., 2003 (in press). 8 pages, 3 figures. See also, physics/0101087 for physicist's approach to and the importance of semiflexible polymer looping, in DNA replicatio

Renormalized one-loop theory of correlations in polymer blends

The renormalized one-loop theory is a coarse-grained theory of corrections to the self-consistent field theory (SCFT) of polymer liquids, and to the random phase approximation (RPA) theory of composition fluctuations. We present predictions of corrections to the RPA for the structure function $S(k)$ and to the random walk model of single-chain statics in binary homopolymer blends. We consider an apparent interaction parameter $\chi_{a}$ that is defined by applying the RPA to the small $k$ limit of $S(k)$. The predicted deviation of $\chi_{a}$ from its long chain limit is proportional to $N^{-1/2}$, where $N$ is chain length. This deviation is positive (i.e., destabilizing) for weakly non-ideal mixtures, with \chi_{a} N \alt 1, but negative (stabilizing) near the critical point. The positive correction to $\chi_{a}$ for low values of $\chi_{a} N$ is a result of the fact that monomers in mixtures of shorter chains are slightly less strongly shielded from intermolecular contacts. The depression in $\chi_{a}$ near the critical point is a result of long-wavelength composition fluctuations. The one-loop theory predicts a shift in the critical temperature of ${\cal O}(N^{-1/2})$, which is much greater than the predicted ${\cal O}(N^{-1})$ width of the Ginzburg region. Chain dimensions deviate slightly from those of a random walk even in a one-component melt, and contract slightly with increasing $\chi_{e}$. Predictions for $S(k)$ and single-chain properties are compared to published lattice Monte Carlo simulations.Comment: submitted to J. Chem. Phy

A general theory of DNA-mediated and other valence-limited interactions

We present a general theory for predicting the interaction potentials between DNA-coated colloids, and more broadly, any particles that interact via valence-limited ligand-receptor binding. Our theory correctly incorporates the configurational and combinatorial entropic factors that play a key role in valence-limited interactions. By rigorously enforcing self-consistency, it achieves near-quantitative accuracy with respect to detailed Monte Carlo calculations. With suitable approximations and in particular geometries, our theory reduces to previous successful treatments, which are now united in a common and extensible framework. We expect our tools to be useful to other researchers investigating ligand-mediated interactions. A complete and well-documented Python implementation is freely available at http://github.com/patvarilly/DNACC .Comment: 18 pages, 10 figure

Electronic State of Na_xCoO_2 Based on the Two Dimensional Triangular Lattice d-p Model

The electronic state in a CoO_2 plane of the layered cobalt oxides Na_{x}CoO_2 is investigated by using the 11 band d-p model on a two-dimensional triangular lattice, where the tight-binding parameters are determined so as to fit the LDA band structure. Effects of the Coulomb interaction at a Co site: the intra- and inter-orbital direct terms U and U', the exchange coupling J and the pair-transfer J', are treated within the Hartree-Fock approximation. We also consider the effect of the Na order at x=0.5, where Na ions form one dimensional chains, by taking into account of an effective one-dimensional potential Delta epsilon_{d} on the CoO_2 plane. It is found that the Na order enhances the Fermi surface nesting resulting in the antiferromagnetism (AFM) which is suppressed due to the frustration effect in the case without the Na order. When U and Delta epsilon_{d} are varied, we observe three types of the AFM: (1) the metallic AFM with large density of states N_F at the Fermi level for small values of U and Delta epsilon_{d}, (2) the semimetallic AFM with tiny N_F for large U with small Delta epsilon_{d} and (3) the insulating AFM with a finite energy gap for large values of U and Delta epsilon_{d}.Comment: 6 pages, 4 figure

Orientational order and glassy states in networks of semiflexible polymers

Motivated by the structure of networks of cross-linked cytoskeletal biopolymers, we study the orientationally ordered phases in two-dimensional networks of randomly cross-linked semiflexible polymers. We consider permanent cross-links which prescribe a finite angle and treat them as quenched disorder in a semi-microscopic replica field theory. Starting from a fluid of un-cross-linked polymers and small polymer clusters (sol) and increasing the cross-link density, a continuous gelation transition occurs. In the resulting gel, the semiflexible chains either display long range orientational order or are frozen in random directions depending on the value of the crossing angle, the crosslink concentration and the stiffness of the polymers. A crossing angle $\theta\sim 2\pi/M$ leads to long range $M$-fold orientational order, e.g., "hexatic" or "tetratic" for $\theta=60^{\circ}$ or $90^{\circ}$, respectively. The transition is discontinuous and the critical cross-link density depends on the bending stiffness of the polymers and the cross-link geometry: the higher the stiffness and the lower $M$, the lower the critical number of cross-links. In between the sol and the long range ordered state, we always observe a gel which is a statistically isotropic amorphous solid (SIAS) with random positional and random orientational localization of the participating polymers.Comment: 20 pages, added references, minor changes, final version as published in PR

Static Scaling Behavior of High-Molecular-Weight Polymers in Dilute Solution: A Reexamination

Previous theories of dilute polymer solutions have failed to distinguish clearly between two very different ways of taking the long-chain limit: (I) $N \to\infty$ at fixed temperature $T$, and (II) $N \to\infty$, $T \to T_\theta$ with $x \equiv N^\phi (T-T_\theta)$ fixed. I argue that the modern two-parameter theory (continuum Edwards model) applies to case II --- not case I --- and in fact gives exactly the crossover scaling functions for $x \ge 0$ modulo two nonuniversal scale factors. A Wilson-type renormalization group clarifies the connection between crossover scaling functions and continuum field theories. [Also contains a general discussion of the connection between the Wilson and field-theoretic renormalization groups. Comments solicited.]Comment: 10 pages including 1 figure, 181159 bytes Postscript (NYU-TH-93/05/01

Multiscale Modeling of Binary Polymer Mixtures: Scale Bridging in the Athermal and Thermal Regime

Obtaining a rigorous and reliable method for linking computer simulations of polymer blends and composites at different length scales of interest is a highly desirable goal in soft matter physics. In this paper a multiscale modeling procedure is presented for the efficient calculation of the static structural properties of binary homopolymer blends. The procedure combines computer simulations of polymer chains on two different length scales, using a united atom representation for the finer structure and a highly coarse-grained approach on the meso-scale, where chains are represented as soft colloidal particles interacting through an effective potential. A method for combining the structural information by inverse mapping is discussed, allowing for the efficient calculation of partial correlation functions, which are compared with results from full united atom simulations. The structure of several polymer mixtures is obtained in an efficient manner for several mixtures in the homogeneous region of the phase diagram. The method is then extended to incorporate thermal fluctuations through an effective chi parameter. Since the approach is analytical, it is fully transferable to numerous systems.Comment: in press, 13 pages, 7 figures, 6 table