Critical Exponents near a Random Fractal Boundary

The critical behaviour of correlation functions near a boundary is modified from that in the bulk. When the boundary is smooth this is known to be characterised by the surface scaling dimension \xt. We consider the case when the boundary is a random fractal, specifically a self-avoiding walk or the frontier of a Brownian walk, in two dimensions, and show that the boundary scaling behaviour of the correlation function is characterised by a set of multifractal boundary exponents, given exactly by conformal invariance arguments to be \lambda_n = 1/48 (\sqrt{1+24n\xt}+11)(\sqrt{1+24n\xt}-1). This result may be interpreted in terms of a scale-dependent distribution of opening angles $\alpha$ of the fractal boundary: on short distance scales these are sharply peaked around $\alpha=\pi/3$. Similar arguments give the multifractal exponents for the case of coupling to a quenched random bulk geometry.Comment: 13 pages. Comments on relation to results in quenched random bulk added, and on relation to other recent work. Typos correcte

The Number of Incipient Spanning Clusters in Two-Dimensional Percolation

Using methods of conformal field theory, we conjecture an exact form for the probability that n distinct clusters span a large rectangle or open cylinder of aspect ratio k, in the limit when k is large.Comment: 9 pages, LaTeX, 1 eps figure. Additional references and comparison with existing numerical results include

Critical Percolation in Finite Geometries

The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are shown to be invariant not only under changes of scale, but also under mappings of the region which are conformal in the interior and continuous on the boundary. This is a larger invariance than that expected for generic critical systems. Specific predictions are presented for the crossing probability between opposite sides of a rectangle, and are compared with recent numerical work. The agreement is excellent.Comment: 10 page

Time-dependence of correlation functions following a quantum quench

We show that the time-dependence of correlation functions in an extended quantum system in d dimensions, which is prepared in the ground state of some hamiltonian and then evolves without dissipation according to some other hamiltonian, may be extracted using methods of boundary critical phenomena in d+1 dimensions. For d=1 particularly powerful results are available using conformal field theory. These are checked against those available from solvable models. They may be explained in terms of a picture, valid more generally, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate classically through the system.Comment: 4+ pages, Corrected Typo

Holographic Entropy on the Brane in de Sitter Schwarzschild Space

The relationship between the entropy of de Sitter (dS) Schwarzschild space and that of the CFT, which lives on the brane, is discussed by using Friedmann-Robertson-Walker (FRW) equations and Cardy-Verlinde formula. The cosmological constant appears on the brane with time-like metric in dS Schwarzschild background. On the other hand, in case of the brane with space-like metric in dS Schwarzschild background, the cosmological constant of the brane does not appear because we can choose brane tension to cancel it. We show that when the brane crosses the horizon of dS Schwarzschild black hole, both for time-like and space-like cases, the entropy of the CFT exactly agrees with the black hole entropy of 5-dimensional AdS Schwarzschild background as it happens in the AdS/CFT correspondence.Comment: 8 pages, LaTeX, Referneces adde

Critical behaviour in parabolic geometries

We study two-dimensional systems with boundary curves described by power laws. Using conformal mappings we obtain the correlations at the bulk critical point. Three different classes of behaviour are found and explained by scaling arguments which also apply to higher dimensions. For an Ising system of parabolic shape the behaviour of the order at the tip is also found.Comment: Old paper, for archiving. 6 pages, 1 figure, epsf, IOP macr

Fermionic field theory for directed percolation in (1+1) dimensions

We formulate directed percolation in (1+1) dimensions in the language of a reaction-diffusion process with exclusion taking place in one space dimension. We map the master equation that describes the dynamics of the system onto a quantum spin chain problem. From there we build an interacting fermionic field theory of a new type. We study the resulting theory using renormalization group techniques. This yields numerical estimates for the critical exponents and provides a new alternative analytic systematic procedure to study low-dimensional directed percolation.Comment: 20 pages, 2 figure

A supersymmetric multicritical point in a model of lattice fermions

We study a model of spinless fermions with infinite nearest-neighbor repulsion on the square ladder which has microscopic supersymmetry. It has been conjectured that in the continuum the model is described by the superconformal minimal model with central charge c=3/2. Thus far it has not been possible to confirm this conjecture due to strong finite-size corrections in numerical data. We trace the origin of these corrections to the presence of unusual marginal operators that break Lorentz invariance, but preserve part of the supersymmetry. By relying mostly on entanglement entropy calculations with the density-matrix renormalization group, we are able to reduce finite-size effects significantly. This allows us to unambiguously determine the continuum theory of the model. We also study perturbations of the model and establish that the supersymmetric model is a multicritical point. Our work underlines the power of entanglement entropy as a probe of the phases of quantum many-body systems.Comment: 16 pages, 8 figure

Quantum and classical localisation and the Manhattan lattice

We consider a network model, embedded on the Manhattan lattice, of a quantum localisation problem belonging to symmetry class C. This arises in the context of quasiparticle dynamics in disordered spin-singlet superconductors which are invariant under spin rotations but not under time reversal. A mapping exists between problems belonging to this symmetry class and certain classical random walks which are self-avoiding and have attractive interactions; we exploit this equivalence, using a study of the classical random walks to gain information about the corresponding quantum problem. In a field-theoretic approach, we show that the interactions may flow to one of two possible strong coupling regimes separated by a transition: however, using Monte Carlo simulations we show that the walks are in fact always compact two-dimensional objects with a well-defined one-dimensional surface, indicating that the corresponding quantum system is localised.Comment: 11 pages, 8 figure

Long-range epidemic spreading with immunization

We study the phase transition between survival and extinction in an epidemic process with long-range interactions and immunization. This model can be viewed as the well-known general epidemic process (GEP) in which nearest-neighbor interactions are replaced by Levy flights over distances r which are distributed as P(r) ~ r^(-d-sigma). By extensive numerical simulations we confirm previous field-theoretical results obtained by Janssen et al. [Eur. Phys. J. B7, 137 (1999)].Comment: LaTeX, 14 pages, 4 eps figure