Construction of a coordinate Bethe ansatz for the asymmetric simple exclusion processwith open boundaries

Abstract

The asymmetric simple exclusion process with open boundaries, which is a very simplemodel of out-of-equilibrium statistical physics, is known to be integrable. In particular, itsspectrum can be described in terms of Bethe roots. The large deviation function of thecurrent can be obtained as well by diagonalizing a modified transition matrix, which is stillintegrable: the spectrum of this new matrix can also be described in terms of Bethe rootsfor special values of the parameters. However, due to the algebraic framework used to writethe Bethe equations in previous works, the nature of the excitations and the full structureof the eigenvectors remained unknown. This paper explains why the eigenvectors of themodified transition matrix are physically relevant, gives an explicit expression forthe eigenvectors and applies it to the study of atypical currents. It also showshow the coordinate Bethe ansatz developed for the excitations leads to a simplederivation of the Bethe equations and of the validity conditions of this ansatz. All theresults obtained by de Gier and Essler are recovered and the approach gives aphysical interpretation of the exceptional points. The overlap of this approachwith other tools such as the matrix ansatz is also discussed. The method thatis presented here may be not specific to the asymmetric exclusion process andmay be applied to other models with open boundaries to find similar exceptionalpoints.