Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation

Abstract

We approximate the noisy Burgers equation by the single-step model, alias the asymmetric simple exclusion process. The generator of the corresponding master equation is identical to the ferromagnetic Heisenberg spin chain with a purely imaginary XY coupling. We Bethe diagonalize this nonsymmetric Hamiltonian. We show that the gap between the ground state and first excited state scales as ${\mathit{N}}^{\mathrm{\ensuremath{-}}3/2}$ for large system size N. The gap between the largest and next-largest eigenvalue scales as ${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$. This property hints at conformal invariance. We also explain the connection to the six-vertex model.