Correlation functions in conformal invariant stochastic processes

TL;DR

This paper investigates correlation functions in conformally invariant stochastic models, revealing boundary operator behavior and employing the Raise and Peel model to connect conformal invariance with integrable quantum chains and SOS models.

Contribution

It demonstrates conformal invariance in stochastic models with boundary operators and introduces a new local operator framework via SOS model mapping.

Findings

Boundary operators have half the critical exponents of bulk operators.

The Raise and Peel model exhibits conformal invariance and is related to an integrable XXZ quantum chain.

New properties of the SOS model are conjectured based on the analysis.

Abstract

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.