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The asymmetric simple exclusion process with open boundaries, which is a verysimple model of out-of-equilibrium statistical physics, is known to beintegrable. In particular, its spectrum can be described in terms of Betheroots. The large deviation function of the current can be obtained as well bydiagonalizing a modified transition matrix, that is still integrable: thespectrum of this new matrix can be also described in terms of Bethe roots forspecial values of the parameters. However, due to the algebraic framework usedto write the Bethe equations in the previous works, the nature of theexcitations and the full structure of the eigenvectors were still unknown. Thispaper explains why the eigenvectors of the modified transition matrix arephysically relevant, gives an explicit expression for the eigenvectors andapplies it to the study of atypical currents. It also shows how the coordinateBethe Ansatz developped for the excitations leads to a simple derivation of theBethe equations and of the validity conditions of this Ansatz. All the resultsobtained by de Gier and Essler are recovered and the approach gives a physicalinterpretation of the exceptional points The overlap of this approach withother tools such as the matrix Ansatz is also discussed. The method that ispresented here may be not specific to the asymmetric exclusion process and maybe applied to other models with open boundaries to find similar exceptionalpoints.

Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries