Fractal Dimensions of Confined Clusters in Two-Dimensional Directed Percolation

The fractal structure of directed percolation clusters, grown at the percolation threshold inside parabolic-like systems, is studied in two dimensions via Monte Carlo simulations. With a free surface at y=\pm Cx^k and a dynamical exponent z, the surface shape is a relevant perturbation when k<1/z and the fractal dimensions of the anisotropic clusters vary continuously with k. Analytic expressions for these variations are obtained using a blob picture approach.Comment: 6 pages, Plain TeX file, epsf, 3 postscript-figure

Surface Shape and Local Critical Behaviour in Two-Dimensional Directed Percolation

Two-dimensional directed site percolation is studied in systems directed along the x-axis and limited by a free surface at y=\pm Cx^k. Scaling considerations show that the surface is a relevant perturbation to the local critical behaviour when k<1/z where z=\nu_\parallel/\nu is the dynamical exponent. The tip-to-bulk order parameter correlation function is calculated in the mean-field approximation. The tip percolation probability and the fractal dimensions of critical clusters are obtained through Monte-Carlo simulations. The tip order parameter has a nonuniversal, C-dependent, scaling dimension in the marginal case, k=1/z, and displays a stretched exponential behaviour when the perturbation is relevant. The k-dependence of the fractal dimensions in the relevant case is in agreement with the results of a blob picture approach.Comment: 13 pages, Plain TeX file, epsf, 6 postscript-figures, minor correction

Extended surface disorder in the quantum Ising chain

We consider random extended surface perturbations in the transverse field Ising model decaying as a power of the distance from the surface towards a pure bulk system. The decay may be linked either to the evolution of the couplings or to their probabilities. Using scaling arguments, we develop a relevance-irrelevance criterion for such perturbations. We study the probability distribution of the surface magnetization, its average and typical critical behaviour for marginal and relevant perturbations. According to analytical results, the surface magnetization follows a log-normal distribution and both the average and typical critical behaviours are characterized by power-law singularities with continuously varying exponents in the marginal case and essential singularities in the relevant case. For enhanced average local couplings, the transition becomes first order with a nonvanishing critical surface magnetization. This occurs above a positive threshold value of the perturbation amplitude in the marginal case.Comment: 15 pages, 10 figures, Plain TeX. J. Phys. A (accepted

Conformal off-diagonal boundary density profiles on a semi-infinite strip

The off-diagonal profile phi(v) associated with a local operator (order parameter or energy density) close to the boundary of a semi-infinite strip with width L is obtained at criticality using conformal methods. It involves the surface exponent x_phi^s and displays a simple universal behaviour which crosses over from surface finite-size scaling when v/L is held constant to corner finite-size scaling when v/L -> 0.Comment: 5 pages, 1 figure, IOP macros and eps

Nonequilibrium phase transition in a driven Potts model with friction

We consider magnetic friction between two systems of $q$-state Potts spins which are moving along their boundaries with a relative constant velocity $v$. Due to the interaction between the surface spins there is a permanent energy flow and the system is in a steady state which is far from equilibrium. The problem is treated analytically in the limit $v=\infty$ (in one dimension, as well as in two dimensions for large-$q$ values) and for $v$ and $q$ finite by Monte Carlo simulations in two dimensions. Exotic nonequilibrium phase transitions take place, the properties of which depend on the type of phase transition in equilibrium. When this latter transition is of first order, a sequence of second- and first-order nonequilibrium transitions can be observed when the interaction is varied.Comment: 13 pages, 9 figures, one journal reference adde

Remarks on Duality Transformations and Generalized Stabilizer States

We consider the transformation of Hamilton operators under various sets of quantum operations acting simultaneously on all adjacent pairs of particles. We find mappings between Hamilton operators analogous to duality transformations as well as exact characterizations of ground states employing non-Hermitean eigenvalue equations and use this to motivate a generalization of the stabilizer formalism to non-Hermitean operators. The resulting class of states is larger than that of standard stabilizer states and allows for example for continuous variation of local entropies rather than the discrete values taken on stabilizer states and the exact description of certain ground states of Hamilton operators.Comment: Contribution to Special Issue in Journal of Modern Optics celebrating the 60th birthday of Peter Knigh

Critical behaviour near multiple junctions and dirty surfaces in the two-dimensional Ising model

We consider m two-dimensional semi-infinite planes of Ising spins joined together through surface spins and study the critical behaviour near to the junction. The m=0 limit of the model - according to the replica trick - corresponds to the semi-infinite Ising model in the presence of a random surface field (RSFI). Using conformal mapping, second-order perturbation expansion around the weakly- and strongly-coupled planes limits and differential renormalization group, we show that the surface critical behaviour of the RSFI model is described by Ising critical exponents with logarithmic corrections to scaling, while at multiple junctions (m>2) the transition is first order. There is a spontaneous junction magnetization at the bulk critical point.Comment: Old paper, for archiving. 6 pages, 1 figure, IOP macro, eps

Conformal invariance and linear defects in the two-dimensional Ising model

Using conformal invariance, we show that the non-universal exponent eta_0 associated with the decay of correlations along a defect line of modified bonds in the square-lattice Ising model is related to the amplitude A_0=xi_n/n of the correlation length \xi_n(K_c) at the bulk critical coupling K_c, on a strip with width n, periodic boundary conditions and two equidistant defect lines along the strip, through A_0=(\pi\eta_0)^{-1}.Comment: Old paper, for archiving. 5 pages, 4 figures, IOP macro, eps

Anomalous Diffusion in Aperiodic Environments

We study the Brownian motion of a classical particle in one-dimensional inhomogeneous environments where the transition probabilities follow quasiperiodic or aperiodic distributions. Exploiting an exact correspondence with the transverse-field Ising model with inhomogeneous couplings we obtain many new analytical results for the random walk problem. In the absence of global bias the qualitative behavior of the diffusive motion of the particle and the corresponding persistence probability strongly depend on the fluctuation properties of the environment. In environments with bounded fluctuations the particle shows normal diffusive motion and the diffusion constant is simply related to the persistence probability. On the other hand in a medium with unbounded fluctuations the diffusion is ultra-slow, the displacement of the particle grows on logarithmic time scales. For the borderline situation with marginal fluctuations both the diffusion exponent and the persistence exponent are continuously varying functions of the aperiodicity. Extensions of the results to disordered media and to higher dimensions are also discussed.Comment: 11 pages, RevTe

Chaos in the Z(2) Gauge Model on a Generalized Bethe Lattice of Plaquettes

We investigate the Z(2) gauge model on a generalized Bethe lattice with three plaquette representation of the action. We obtain the cascade of phase transitions according to Feigenbaum scheme leading to chaotic states for some values of parameters of the model. The duality between this gauge model and three site Ising spin model on Husimi tree is shown. The Lyapunov exponent as a new order parameter for the characterization of the model in the chaotic region is considered. The line of the second order phase transition, which corresponds to the points of the first period doubling bifurcation, is also obtained.Comment: LaTeX, 7 pages, 4 Postscript figure