Delta-Bose gas on a half-line and the Kardar–Parisi–Zhang 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 R + with diagonal boundary conditions. This model is integrable for arbitrary value of b ∈ 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 = +∞, 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 ∂ 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 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 > − 1 2 where the PDF is given by the GSE Tracy-Widom distribution (ii) bound for b < − 1 2 , where the PDF is a Gaussian. At the critical point b = − 1 2 , the PDF is given by the GOE Tracy-Widom distribution.Overview: KPZ in a half-space.There has been much recent progress in physics and mathematics in the study of the 1D (KPZ) universality class, thanks to the discovery of exact solutions and the development of powerful methods to address stochastic integrability and integrable probability. The KPZ class includes a host of models [1]: discrete versions of stochastic interface growth such as the PNG growth model [2][3][4], exclusion processes such as the TASEP, the ASEP, the q-TASEP and other variants [5][6][7][8][9][10][11][12][13], discrete (i.e. square lattice) [3,[14][15][16][17][18][19][20][21][22] or semi-discrete [23-25] models of directed polymers (DP) at zero and finite temperature, random walks in time dependent random media [26,27], dimer models, random tilings, random permutations [28], correlation function in quantum condensates [29][30][31], and more. At the center of this class lies the continuum KPZ equation [32], see equation (3), which describes the stochastic growth of a continuum interface, and its equivalent formulation in terms of continuum directed polymers (DP) [33], via the Cole-Hopf mapping onto the stochastic heat equation (SHE). Recently exact solutions have also been obtained for the KPZ equation at all times for various initial conditions [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. This was achieved by two different routes. First by studying scaling limits of solvable discrete models, which allowed for rigorous treatments. The second, pioneered by Kardar [51], is non-rigorous, but leads to a more direct solution: it starts from the DP formulation, uses the replica method together with a mapping to the attractive delta-Bose gas (LL model), whi...

Half-Space Stationary Kardar–Parisi–Zhang Equation

“…More recently, in [56], a solution valid for any time was found using the replica Bethe ansatz for any A > −1/2, and which leads to the GOE Tracy-Widom distribution at the critical point A = −1/2. In [57] a solution from the RBA, taking into account bound states, was obtained, in agreement with the results of [56]. A remarkable property of the KPZ equation on the full line is that the stationary measure is the Brownian motion in the sense that if the initial condition h(x, 0) is a two sided Brownian motion (with the appropriate amplitude) the PDF of the height difference between two space points is time independent.…”

“…In the limit B → +∞, i.e. for the droplet initial condition, it was found [57] that v A,+∞ ∞ = − 1 12 + (A + 1 2 ) 2 for A ≤ −1/2. In the general A, B case, we expect that v A,B ∞ = − 1 12 + min A + 1 2 , B + 1 2 , 0…”

Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line

“…In a companion study [29] we have approached the problem differently by elucidating the structure of the boundary bound states which appear in the half-line Bose gas with generic parameter A. It quite easily yields results for the Gaussian phase A < −1/2 where these states dominate.…”

“…for Dirichlet boundary condition as needed here [35,59,60] (see also section 5.1 of [61]). Note that it can also be solved for arbitrary A, [25,[61][62][63][64][65][66] which led to the formula in [28] and [29], but here we circumvent this, using instead the Brownian with Dirichlet boundary condition.…”

“…As shown in [29], the string states discussed above do not form a complete basis for generic A: new boundary bound states arise. To obtain a formula for the moments valid for any n, A is possible but requires to include all boundary states, as done in [29]. This is illustrated here in the Appendix B for the first moment of Z(t).…”