Exact Spectral Function of a Tonks-Girardeau Gas in a Lattice

The single-particle spectral function of a strongly correlated system is an essential ingredient to describe its dynamics and transport properties. We develop a method to evaluate exactly the spectral function for a gas of one-dimensional bosons with infinitely strong repulsions valid for any type of external confinement. Focusing on the case of a lattice confinement, we find that the spectral function displays three main singularity lines. One of them is due uniquely to lattice effects, while the two others correspond to the Lieb-I and Lieb-II modes occurring in a uniform fluid. Differently from the dynamical structure factor, in the spectral function the Lieb-II mode shows a divergence, thus providing a route to probe such mode in experiments with ultracold atoms.