Two-dimensional polymer networks at a mixed boundary: Surface and wedge exponents

We provide general formulae for the configurational exponents of an arbitrary polymer network connected to the surface of an arbitrary wedge of the two-dimensional plane, where the surface is allowed to assume a general mixture of boundary conditions on either side of the wedge. We report on a comprehensive study of a linear chain by exact enumeration, with various attachments of the walk's ends to the surface, in wedges of angles $\pi/2$ and $\pi$, with general mixed boundary conditions.Comment: 4 pages, Latex2e, 3 figures, Eur. Phys. J. B macro

Four-dimensional polymer collapse II: Pseudo-First-Order Transition in Interacting Self-avoiding Walks

In earlier work we provided the first evidence that the collapse, or coil-globule, transition of an isolated polymer in solution can be seen in a four-dimensional model. Here we investigate, via Monte Carlo simulations, the canonical lattice model of polymer collapse, namely interacting self-avoiding walks, to show that it not only has a distinct collapse transition at finite temperature but that for any finite polymer length this collapse has many characteristics of a rounded first-order phase transition. However, we also show that there exists a `$\theta$-point' where the polymer behaves in a simple Gaussian manner (which is a critical state), to which these finite-size transition temperatures approach as the polymer length is increased. The resolution of these seemingly incompatible conclusions involves the argument that the first-order-like rounded transition is scaled away in the thermodynamic limit to leave a mean-field second-order transition. Essentially this happens because the finite-size \emph{shift} of the transition is asymptotically much larger than the \emph{width} of the pseudo-transition and the latent heat decays to zero (algebraically) with polymer length. This scenario can be inferred from the application of the theory of Lifshitz, Grosberg and Khokhlov (based upon the framework of Lifshitz) to four dimensions: the conclusions of which were written down some time ago by Khokhlov. In fact it is precisely above the upper critical dimension, which is 3 for this problem, that the theory of Lifshitz may be quantitatively applicable to polymer collapse.Comment: 30 pages, 14 figures included in tex

First-order scaling near a second-order phase transition: Tricritical polymer collapse

The coil-globule transition of an isolated polymer has been well established to be a second-order phase transition described by a standard tricritical O(0) field theory. We provide compelling evidence from Monte Carlo simulations in four dimensions, where mean-field theory should apply, that the approach to this (tri)critical point is dominated by the build-up of first-order-like singularities masking the second-order nature of the coil-globule transition: the distribution of the internal energy having two clear peaks that become more distinct and sharp as the tricritical point is approached. However, the distance between the peaks slowly decays to zero. The evidence shows that the position of this (pseudo) first-order transition is shifted by an amount from the tricritical point that is asymptotically much larger than the width of the transition region. We suggest an explanation for the apparently contradictory scaling predictions in the literature.Comment: 4 pages, 2 figures included in tex

Geometrical Properties of Two-Dimensional Interacting Self-Avoiding Walks at the Theta-Point

We perform a Monte Carlo simulation of two-dimensional N-step interacting self-avoiding walks at the theta point, with lengths up to N=3200. We compute the critical exponents, verifying the Coulomb-gas predictions, the theta-point temperature T_theta = 1.4986(11), and several invariant size ratios. Then, we focus on the geometrical features of the walks, computing the instantaneous shape ratios, the average asphericity, and the end-to-end distribution function. For the latter quantity, we verify in detail the theoretical predictions for its small- and large-distance behavior.Comment: 23 pages, 4 figure

The competition of hydrogen-like and isotropic interactions on polymer collapse

We investigate a lattice model of polymers where the nearest-neighbour monomer-monomer interaction strengths differ according to whether the local configurations have so-called ``hydrogen-like'' formations or not. If the interaction strengths are all the same then the classical $\theta$-point collapse transition occurs on lowering the temperature, and the polymer enters the isotropic liquid-drop phase known as the collapsed globule. On the other hand, strongly favouring the hydrogen-like interactions give rise to an anisotropic folded (solid-like) phase on lowering the temperature. We use Monte Carlo simulations up to a length of 256 to map out the phase diagram in the plane of parameters and determine the order of the associated phase transitions. We discuss the connections to semi-flexible polymers and other polymer models. Importantly, we demonstrate that for a range of energy parameters two phase transitions occur on lowering the temperature, the second being a transition from the globule state to the crystal state. We argue from our data that this globule-to-crystal transition is continuous in two dimensions in accord with field-theory arguments concerning Hamiltonian walks, but is first order in three dimensions

Critical Percolation in High Dimensions

We present Monte Carlo estimates for site and bond percolation thresholds in simple hypercubic lattices with 4 to 13 dimensions. For d<6 they are preliminary, for d >= 6 they are between 20 to 10^4 times more precise than the best previous estimates. This was achieved by three ingredients: (i) simple and fast hashing which allowed us to simulate clusters of millions of sites on computers with less than 500 MB memory; (ii) a histogram method which allowed us to obtain information for several p values from a single simulation; and (iii) a new variance reduction technique which is especially efficient at high dimensions where it reduces error bars by a factor up to approximately 30 and more. Based on these data we propose a new scaling law for finite cluster size corrections.Comment: 5 pages including figures, RevTe

Exact Results for Hamiltonian Walks from the Solution of the Fully Packed Loop Model on the Honeycomb Lattice

We derive the nested Bethe Ansatz solution of the fully packed O($n$) loop model on the honeycomb lattice. From this solution we derive the bulk free energy per site along with the central charge and geometric scaling dimensions describing the critical behaviour. In the $n=0$ limit we obtain the exact compact exponents $\gamma=1$ and $\nu=1/2$ for Hamiltonian walks, along with the exact value $\kappa^2 = 3 \sqrt 3 /4$ for the connective constant (entropy). Although having sets of scaling dimensions in common, our results indicate that Hamiltonian walks on the honeycomb and Manhattan lattices lie in different universality classes.Comment: 12 pages, RevTeX, 3 figures supplied on request, ANU preprint MRR-050-9

Enumeration of self-avoiding walks on the square lattice

We describe a new algorithm for the enumeration of self-avoiding walks on the square lattice. Using up to 128 processors on a HP Alpha server cluster we have enumerated the number of self-avoiding walks on the square lattice to length 71. Series for the metric properties of mean-square end-to-end distance, mean-square radius of gyration and mean-square distance of monomers from the end points have been derived to length 59. Analysis of the resulting series yields accurate estimates of the critical exponents $\gamma$ and $\nu$ confirming predictions of their exact values. Likewise we obtain accurate amplitude estimates yielding precise values for certain universal amplitude combinations. Finally we report on an analysis giving compelling evidence that the leading non-analytic correction-to-scaling exponent $\Delta_1=3/2$.Comment: 24 pages, 6 figure

Generalized Noiseless Quantum Codes utilizing Quantum Enveloping Algebras

A generalization of the results of Rasetti and Zanardi concerning avoiding errors in quantum computers by using states preserved by evolution is presented. The concept of dynamical symmetry is generalized from the level of classical Lie algebras and groups to the level of dynamical symmetry based on quantum Lie algebras and quantum groups (in the sense of Woronowicz). A natural connection is proved between states preserved by representations of a quantum group and states preserved by evolution with dynamical symmetry of the appropriate universal enveloping algebra. Illustrative examples are discussed.Comment: 10 pages, LaTeX, 2 figures Postscrip

Self-avoiding walks and polygons on the triangular lattice

We use new algorithms, based on the finite lattice method of series expansion, to extend the enumeration of self-avoiding walks and polygons on the triangular lattice to length 40 and 60, respectively. For self-avoiding walks to length 40 we also calculate series for the metric properties of mean-square end-to-end distance, mean-square radius of gyration and the mean-square distance of a monomer from the end points. For self-avoiding polygons to length 58 we calculate series for the mean-square radius of gyration and the first 10 moments of the area. Analysis of the series yields accurate estimates for the connective constant of triangular self-avoiding walks, $\mu=4.150797226(26)$, and confirms to a high degree of accuracy several theoretical predictions for universal critical exponents and amplitude combinations.Comment: 24 pages, 6 figure