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arXiv:1107.2618 · Journal of Statistical Mechanics Theory and Experiment

From conformal invariance to quasistationary states

Francisco Castilho Alcaraz, V. Rittenberg

Published Sep 30, 2011• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-5468/2011/09/p09030

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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 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.
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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.