Boundary-Induced Phase Transitions in Equilibrium and Non-Equilibrium Systems

Boundary conditions may change the phase diagram of non-equilibrium statistical systems like the one-dimensional asymmetric simple exclusion process with and without particle number conservation. Using the quantum Hamiltonian approach, the model is mapped onto an XXZ quantum chain and solved using the Bethe ansatz. This system is related to a two-dimensional vertex model in thermal equilibrium. The phase transition caused by a point-like boundary defect in the dynamics of the one-dimensional exclusion model is in the same universality class as a continous (bulk) phase transition of the two-dimensional vertex model caused by a line defect at its boundary. § Address after 1 October 1993: Department of Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK It is generally held that for statistical systems the influence of boundary conditions should be negligible, at least when one is interested in the bulk system only. This belief is indeed supported for equilibrium systems by abundant evidence coming from analytical or numerical studies. As we are going to show in this letter, however, this picture is far from being generally valid. Boundary conditions may indeed be crucial for the bulk properties and may even cause continous phase transitions. It has already been realized that a change in boundary conditions (equivalent to some localized point defect in a system on a ring) may cause various kinds of phase transitions in the static properties of one-dimensional non-equilibrium systems [1, 2]. Here we show that boundary terms may also induce phase transitions in the dynamics of these systems. Such phase transitions are then shown to correspond to bulk phase transitions of two-dimensional equilibrium systems caused by boundary terms (i.e., line defects). The paradigmatic example we are going to study is the asymmetric simple exclusion process, see [3]. In its simplest version, without particle creation or annihilation, it is described by particles of a single species A moving on a lattice. A given site j can be occupied or empty at an instant of time t. A particle at site j for time t may hop at time t+1 to its right neighbor with rate (1+ǫ)/2 and to its left neighbor with rate (1 − ǫ)/2, if the final site is empty. This simple model appears in a large variety of contexts. It has been argued to be in the same universality class as the noisy Burger’s equation [4]. This in turn is a one-dimensional version of the incompressible NavierStokes equation or else can be regarded as the one-dimensional Kardar-Zhang-Parisi equation describing the shape fluctuation in various growth models. Using a master equation approach, the probability distribution function P ({β}; t) is obtained by solving ∂tP = −HP , where β(t) is a configuration of occupied and empty sites and the quantum Hamiltonian reads [4]