Delta-Bose gas on a half-line and the Kardar–Parisi–Zhang equation: boundary bound states and unbinding transitions

We revisit the Lieb–Liniger model for n bosons in one dimension with attractive delta interaction in a half-space with diagonal boundary conditions. This model is integrable for the arbitrary value of , the interaction parameter with the boundary. We show that its spectrum exhibits a sequence of transitions, as b is decreased from the hard-wall case , with successive appearance of boundary bound states (or boundary modes) which we fully characterize. We apply these results to study the Kardar–Parisi–Zhang equation for the growth of a one-dimensional interface of height , on the half-space with boundary condition and droplet initial condition at the wall. We obtain explicit expressions, valid at all time t and arbitrary b, for the integer exponential (one-point) moments of the KPZ height field . From these moments we extract the large time limit of the probability distribution function (PDF) of the scaled KPZ height function. It exhibits a phase transition, related to the unbinding to the wall of the equivalent directed polymer problem, with two phases: (i) unbound for where the PDF is given by the GSE Tracy–Widom distribution (ii) bound for , where the PDF is a Gaussian. At the critical point , the PDF is given by the GOE Tracy–Widom distribution.