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arXiv:cond-mat/0205467 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Stochastic processes and conformal invariance

Jan de Gier, Bernard Nienhuis, Paul A. Pearce, V. Rittenberg

Published Jan 3, 2003• 27 citations
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Abstract

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.

DOI
10.1103/physreve.67.016101

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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.
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.
In a conformal invariant one-dimensional stochastic model, a certain non-local perturbation takes the system to a new massless phase of a special kind. The ground-state of the system is an adsorptive state. Part of the finite-size scaling spectrum of the evolution Hamiltonian stays unchanged but some levels go exponentially to zero for large lattice sizes becoming degenerate with the ground-state. As a consequence one observes the appearance of quasistationary states which have a relaxation time which grows exponentially with the size of the system. Several initial conditions have singled out a quasistationary state which has in the finite-size scaling limit the same properties as the stationary state of the conformal invariant model.
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.
We study the interface energy $\ensuremath{\sigma}$ as a function of disorder in two-dimensional Ising-type systems at T=0T=0T=0 for the percolation and frustration models. Our approach consists in calculating this energy for long strips of varying widths by a random sampling method, then extrapolating the results. In the weak-disorder limit, very wide strips may be studied and accurate values are obtained for the first-order correction to the interface energy in both models. Only moderate widths (up to $N=9\ensuremath{-}10$) can be studied in the general situation and we use finite-size scaling to analyze the data in the region of the threshold, where $\ensuremath{\sigma}$ vanishes with a critical exponent vvv. For percolation, we obtain $\frac{v}{\ensuremath{\nu}}=0.98\ifmmode\pm\else\textpm\fi{}0.05$, where $\ensuremath{\nu}$ is the correlation-length exponent, in agreement with Deutscher and Rappaport's proposal that $v=\ensuremath{\nu}$ exactly. The analysis of the results is more difficult in the case of frustration, because size effects are important and the scaling region is not reached for N=9N=9N=9. Our data show that Monte Carlo results for $\ensuremath{\sigma}$ are unreliable and that much care is necessary to reach firm conclusions on the frustration threshold xc{x}_{c}xc​ or the exponents $\ensuremath{\nu}$ and vvv.
Dynamic scaling for fracture or breakdown process in disordered systems is investigated in a two-dimensional random field Ising model (RFIM). We find two evolving stages in the avalanche process in the RFIM. At the short-time regime, a power-law growth of the avalanche size Deltas is observed; and at late times, the conventional nucleation and growth process is found. At the critical point of the RFIM, the avalanche size is found to obey the dynamic scaling law Delta(s) approximately equal t((d-beta/nu)/z). From this dynamic scaling relation, the critical strength of the random field D(c) and the critical exponents, beta, nu, and z, are determined. The observed dynamics is explained by a simple nucleation theory of first-order phase transformations.