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

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 $\overline{e^{nh(0,t)}}$. 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.

arXiv comment52 pages, references added and typos corrected

References in corpus (86)

• Level-Spacing Distributions and the Airy Kernel
• Shape Fluctuations and Random Matrices
• On Orthogonal and Symplectic Matrix Ensembles
• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• The one-dimensional KPZ equation: an exact solution and its universality
• Bosonizing one-dimensional cold atomic gases
• Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals
• Free-energy distribution of the directed polymer at high temperature
• Growing interfaces uncover universal fluctuations behind scale invariance
• Asymptotics in ASEP with Step Initial Condition
• On the Numerical Evaluation of Fredholm Determinants
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• Yang-Yang thermodynamics on an atom chip
• A KPZ Cocktail- Shaken, not stirred: Toasting 30 years of kinetically roughened surfaces
• An exact solution for the KPZ equation with flat initial conditions
• Scaling for a one-dimensional directed polymer with boundary conditions
• Exact height distributions for the KPZ equation with narrow wedge initial condition
• Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process
• Spatial correlations of the 1D KPZ surface on a flat substrate
• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
• Exact solution for the stationary Kardar-Parisi-Zhang equation
• A Fredholm Determinant Representation in ASEP
• Directed polymers and the quantum Toda lattice
• Bound State Boundary S-matrix of the sine-Gordon Model
• Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity
• Random-walk in Beta-distributed random environment
• Deriving Boundary S Matrices
• Correlation functions of the one-dimensional attractive Bose gas
• The crossover regime for the weakly asymmetric simple exclusion process
• Stationary correlations for the 1D KPZ equation
• Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
• Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
• Tropical Combinatorics and Whittaker functions
• Fluctuations of a one-dimensional polynuclear growth model in a half space
• Renormalization fixed point of the KPZ universality class
• Surface excitations and surface energy of the antiferromagnetic XXZ chain by the Bethe ansatz approach
• Bethe Ansatz study of one-dimensional Bose and Fermi gases with periodic and hard wall boundary conditions
• On integrability of zero-range chipping models with factorized steady state
• Ground-state properties of one-dimensional ultracold Bose gases in a hard-wall trap
• Directed polymer near a hard wall and KPZ equation in the half-space
• Correlation functions of the open XXZ chain II
• From GPE to KPZ: finite temperature dynamical structure factor of the 1D Bose gas
• Limit process of stationary TASEP near the characteristic line
• Crossover from Growing to Stationary Interfaces in the Kardar-Parisi-Zhang Class
• The KPZ Limit of ASEP with Boundary
• Stochastic six-vertex model in a half-quadrant and half-line open ASEP
• Quantum quenches to the attractive one-dimensional Bose gas: exact results
• The strict-weak lattice polymer
• Scaling properties of one-dimensional driven-dissipative condensates
• Finite-time fluctuations for the totally asymmetric exclusion process
• Dynamics of pair correlations in the attractive Lieb-Liniger gas
• Pfaffian Schur processes and last passage percolation in a half-quadrant
• Memory and universality in interface growth
• Directed random polymers via nested contour integrals
• A note on the spin-1/2 XXZ chain concerning its relation to the Bose gas
• Two-point free energy distribution function in (1+1) directed polymers
• The quantum non-linear Schrodinger model with point-like defect
• Point-to-line polymers and orthogonal Whittaker functions
• Large fluctuations of the KPZ equation in a half-space
• Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation
• Unbinding transition in semi-infinite two-dimensional localized systems
• From the sine-Gordon field theory to the Kardar-Parisi-Zhang growth equation
• On the equal time two-point distribution of the one-dimensional KPZ equation by replica
• On integrable directed polymer models on the square lattice
• Exact Results for the Boundary Energy of One-Dimensional Bosons
• The Bose Gas and Asymmetric Simple Exclusion Process on the Half-Line
• The Nonlinear Schrodinger Equation on the Half Line
• Exact formulas for random growth with half-flat initial data
• Crossover from droplet to flat initial conditions in the KPZ equation from the replica Bethe ansatz
• Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line
• Open XXZ chain and boundary modes at zero temperature
• Statistical properties of one dimensional attractive Bose gas
• The propagator of the attractive delta-Bose gas in one dimension
• Tail of the two-time height distribution for KPZ growth in one dimension
• Repulsive to attractive interaction quenches of 1D Bose gas in a harmonic trap
• Surface free energy for systems with integrable boundary conditions
• Orthogonality of Bethe Ansatz eigenfunctions for the Laplacian on a hyperoctahedral Weyl alcove
• When fast and slow interfaces grow together: connection to the half-space problem of the Kardar-Parisi-Zhang class
• Maximum of an Airy process plus Brownian motion and memory in KPZ growth
• Quantum quench dynamics of the attractive one-dimensional Bose gas via the coordinate Bethe ansatz
• Completeness of the Bethe Ansatz for an open $q$-boson system with integrable boundary interactions
• Crossing probability for directed polymers in random media: exact tail of the distribution
• Current fluctuations and large deviations for periodic TASEP on the relaxation scale
• Stationary measures for two dual families of finite and zero temperature models of directed polymers on the square lattice
• Production of strongly bound 39K bright solitons

Cited by in corpus (8)

• From stochastic spin chains to quantum Kardar-Parisi-Zhang dynamics
• Half-space stationary Kardar-Parisi-Zhang equation
• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
• The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall
• Stationary measures of the KPZ equation on an interval from Enaud-Derrida's matrix product ansatz representation
• Lyapunov exponents of the half-line SHE
• Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case
• Steady state of the KPZ equation on an interval and Liouville quantum mechanics