Exact Spectral Gaps of the Asymmetric Exclusion Process with Open Boundaries

TL;DR

This paper derives the complete spectral spectrum of the asymmetric exclusion process with open boundaries, analyzing finite-size scaling and phase transitions, revealing boundary-induced crossovers and oscillatory behaviors.

Contribution

It provides the Bethe ansatz equations for the full spectrum with open boundaries and analyzes spectral gaps across different regimes, including boundary effects and phase transitions.

Findings

Boundary-induced crossovers between scaling regimes

Absence of oscillations on the coexistence line

Oscillations in maximum current phase with KPZ scaling

Abstract

We derive the Bethe ansatz equations describing the complete spectrum of the transition matrix of the partially asymmetric exclusion process with the most general open boundary conditions. By analysing these equations in detail for the cases of totally asymmetric and symmetric diffusion, we calculate the finite-size scaling of the spectral gap, which characterizes the approach to stationarity at large times. In the totally asymmetric case we observe boundary induced crossovers between massive, diffusive and KPZ scaling regimes. We further study higher excitations, and demonstrate the absence of oscillatory behaviour at large times on the ``coexistence line'', which separates the massive low and high density phases. In the maximum current phase, oscillations are present on the KPZ scale $t\propto L^{-3/2}$. While independent of the boundary parameters, the spectral gap as well as the oscillation frequency in the maximum current phase have different values compared to the totally asymmetric exclusion process with periodic boundary conditions. We discuss a possible interpretation of our results in terms of an effective domain wall theory.