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arXiv:1204.1001 · Journal of Statistical Mechanics Theory and Experiment

Nonlocal growth processes and conformal invariance

Francisco Castilho Alcaraz, V. Rittenberg

Published May 22, 2012• 8 citations
arXivHTMLPDF

Abstract

Up to now the raise and peel model was the single known example of a one-dimensional stochastic process where one can observe conformal invariance. The model has one-parameter. Depending on its value one has a gapped phase, a critical point where one has conformal invariance and a gapless phase with changing values of the dynamical critical exponent $z$. In this model, adsorption is local but desorption is not. The raise and strip model presented here in which desorption is also nonlocal, has the same phase diagram. The critical exponents are different as are some physical properties of the model. Our study suggest the possible existence of a whole class of stochastic models in which one can observe conformal invariance.

DOI
10.1088/1742-5468/2012/05/p05022

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Up to now the raise and peel model was the single known example of a one-dimensional stochastic process where one can observe conformal invariance. The model has one-parameter. Depending on its value one has a gapped phase, a critical point where one has conformal invariance and a gapless phase with changing values of the dynamical critical exponent zzz. In this model, adsorption is local but desorption is not. The raise and strip model presented here in which desorption is also nonlocal, has the same phase diagram. The critical exponents are different as are some physical properties of the model. Our study suggest the possible existence of a whole class of stochastic models in which one can observe conformal invariance.
The raise and peel model is a one-dimensional nonlocal stochastic model where adsortion happens locally and desorption is nonlocal.The model depends on the single parameter u that is the ratio among the desorption and adsorption rates.At a special value of this parameter (u = 1) the model has interesting features.The spectrum is described by a conformal field theory (central charge c = 0), and its stationary probability density is related to the equilibrium distribution of a two dimensional system.The phase diagram of the model, as a function of the parameter u, has a massive phase (gapped phase) and a massless (gapless phase) whose critical exponents vary continuously with u.In this monography we study a one-parameter extension of the raise and peel model at u = 1, that depends on the additional parameter p.The new model exhibits conformal invariance for the whole range of values of its parameter p, and it is in the same universality class as the usual raise and peel model.The single difference between the models is the value of the sound velocity v s (p) which is a function of p.The methods used in this monography are the exact diagonalization of the evolution operator of the stochastic model (Hamiltonian), for small lattice sizes and Monte Carlo simulations.
The raise and peel model is a one-dimensional stochastic model of a fluctuating interface with nonlocal interactions. This is an interesting physical model, in this paper we review its properties. Its phase diagram has a massive phase and a gapless phase with varying critical exponents. At the phase transition point, the model exhibits conformal invariance which is a space–time symmetry. Also at this point the model has several other facets which are the connections to associative algebras, two-dimensional fully packed loop models and combinatorics.
We discuss a one-dimensional model of a fluctuating interface with a dynamic exponent z=1. The events that occur are adsorption, which is local, and desorption which is nonlocal and may take place over regions of the order of the system size. In the thermodynamic limit, the time dependence of the system is given by characters of the c=0 logarithmic conformal field theory of percolation. This implies in a rigorous way, a connection between logarithmic conformal field theory and stochastic processes. The finite-size scaling behavior of the average height, interface width and other observables are obtained. The avalanches produced during desorption are analyzed and we show that the probability distribution of the avalanche sizes obeys finite-size scaling with new critical exponents.
We investigate a non-equilibrium one-dimensional model known as the raise and peel model describing a growing surface which grows locally and has non-local desorption. For specific values of adsorption ( u a ) and desorption ( u d ) rates, the model shows interesting features. At u a = u d , the model is described by a conformal field theory (with conformal charge c = 0) and its stationary probability can be mapped onto the ground state of the XXZ quantum chain. Moreover, for the regime u a ⩾ u d , the model shows a phase in which the avalanche distribution is scale-invariant. In this work, we study the surface dynamics by looking at avalanche distributions using a finite-sized scaling formalism and explore the effect of adding a wall to the model. The model shows the same universality for the cases with and without a wall for an odd number of tiles removed, but we find a new exponent in the presence of a wall for an even number of tiles released in an avalanche. New insights into the effect of parity on avalanche distributions are discussed and we provide a new conjecture for the probability distribution of avalanches with a wall obtained by using an exact diagonalization of small lattices and Monte Carlo simulations.
We consider the problem of correlation functions in the stationary states of one-dimensional stochastic models having conformal invariance. If one considers the space dependence of the correlators, the novel aspect is that although one considers systems with periodic boundary conditions, the observables are described by boundary operators. From our experience with equilibrium problems one would have expected bulk operators. Boundary operators have correlators having critical exponents being half of those of bulk operators. If one studies the space-time dependence of the two-point function, one has to consider one boundary and one bulk operators. The Raise and Peel model has conformal invariance as can be shown in the spin 1/2 basis of the Hamiltonian which gives the time evolution of the system. This is an XXZ quantum chain with twisted boundary condition and local interactions. This Hamiltonian is integrable and the spectrum is known in the finite-size scaling limit. In the stochastic base in which the process is defined, the Hamiltonian is not local anymore. The mapping into an SOS model, helps to define new local operators. As a byproduct some new properties of the SOS model are conjectured. The predictions of conformal invariance are discussed in the new framework and compared with Monte Carlo simulations.
The raise and peel model is a stochastic model of a fluctuating interface separating a substrate covered with clusters of matter of different sizes and a rarefied gas of tiles. The stationary state is obtained when adsorption compensates the desorption of tiles. This model is generalized to an interface with defects (D) . The defects are either adjacent or separated by a cluster. If a tile hits the end of a cluster with a defect nearby, the defect hops at the other end of the cluster, changing its shape. If a tile hits two adjacent defects, the defects annihilate and are replaced by a small cluster. There are no defects in the stationary state. This model can be seen as describing the reaction D+D-->0 , in which the particles (defects) D hop at long distances, changing the medium, and annihilate. Between the hops the medium also changes (tiles hit clusters, changing their shapes). Several properties of this model are presented and some exact results are obtained using the connection of our model with a conformally invariant quantum chain.