Delta-Bose gas on a half-line and the KPZ 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 $\mathbb{R}^+$ with diagonal boundary conditions. This model is integrable for arbitrary value of $b \in \mathbb{R}$, 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 $b=+\infty$, 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 $h(x,t)$, on the half-space with boundary condition $\partial_x h(x,t)|_{x=0}=b$ 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 $\bar{e^{n h(0,t)}}$. 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 $b>-\frac{1}{2}$ where the PDF is given by the GSE Tracy-Widom distribution (ii) bound for $b<-\frac{1}{2}$, where the PDF is a Gaussian. At the critical point $b=-\frac{1}{2}$, the PDF is given by the GOE Tracy-Widom distribution.