Bethe ansatz calculation of the spectral gap of the asymmetric exclusion process

We present a new derivation of the spectral gap of the totally asymmetric exclusion process on a half-filled ring of size L by using the Bethe ansatz. We show that, in the large-L limit, the Bethe equations reduce to a simple transcendental equation involving the polylogarithm, a classical special function. By solving that equation, the gap and the dynamical exponent are readily obtained. Our method can be extended to a system with an arbitrary density of particles.