Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions

TL;DR

This paper analyzes the boundary-bound states in the attractive Lieb-Liniger model on a half-line and applies the findings to the KPZ equation, revealing a phase transition in the height distribution related to boundary unbinding.

Contribution

It characterizes boundary bound states in the Lieb-Liniger model with boundary conditions and connects these to phase transitions in the KPZ height distribution.

Findings

Spectrum exhibits boundary bound states as boundary parameter varies.

Large time KPZ height distribution transitions from Tracy-Widom to Gaussian.

Explicit moments of KPZ height are derived for all times and boundary conditions.

Abstract

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.