Exact out-of-equilibrium steady states in the semiclassical limit of the interacting Bose gas

Abstract

<jats:p>We study the out-of-equilibrium properties of a classical integrable non-relativistic theory, with a time evolution initially prepared with a finite energy density in the thermodynamic limit. The theory considered here is the Non-Linear Schrödinger equation which describes the dynamics of the one-dimensional interacting Bose gas in the regime of high occupation numbers. The main emphasis is on the determination of the late-time Generalised Gibbs Ensemble (GGE), which can be efficiently semi-numerically computed on arbitrary initial states, completely solving the famous quench problem in the classical regime. We take advantage of known results in the quantum model and the semiclassical limit to achieve new exact results for the momenta of the density operator on arbitrary GGEs, which we successfully compare with ab-initio numerical simulations. Furthermore, we determine the whole probability distribution of the density operator (full counting statistics), whose exact expression is still out of reach in the quantum model.</jats:p>

Generalized hydrodynamics with dephasing noise

“…We consider the time evolution of the density moments |ψ (x)| 2n computed in Ref. [99] for arbitrary GGEs (see also Refs. [100,101] for the quantum case).…”

Approaching the self-dual point of the sinh-Gordon model

“…(6.16) Due to the antisymmetry of Θ, it is sufficient to restrict to the cases n ≥ 0 (subsequently we will see that the ground state corresponds to n = 1). In the zero mode sector, energy levels behave in the UV as 17) as can be seen by direct integration (for details, see appendix C of ref. [7]).…”

“…A suggestive connection between the ShG model and roaming renormalization group trajectories among the minimal models of CFT was studied in [11], while a direct mapping between the ShG and the Ising model was established in [12]. Furthermore, beyond simply being a model that is amenable to analytic manipulation, the ShG finds applications in a wide range of areas of physics running from toy models of quantum gravity [13], to cold atomic gases [14,15], studies of thermalization in classical field theories [16,17], and lattice models with non-compact quantum group symmetries [18]. It is also worth stressing that the ShG model is the simplest example of Toda field theories, a large class of models with exponential interactions based on root systems of Lie algebras, see for instance [19] and references therein.…”

Generalized hydrodynamics of the attractive non-linear Schrӧdinger equation

“…The proportionality factor between ρ(λ) and log |a(λ)| is determined comparing the local charges obtained from equation (31) with the standard normalization of particle density and Hamiltonian. This correspondence leads to a full characterization of the post-quench GGE in the repulsive phase [71] in terms of the initial data. In principle, one could attempt the same strategy in the attractive case as well, but this path appears unpractical or, at least, very challenging.…”

“…Solitonic modes also appear in the Toda chain [64,65] and Landau-Lifschitz spin model [66][67][68]. In contrast, the thermodynamics of the relativistic sinh-Gordon model [69,70] and the repulsive NLS [71] is built upon radiative modes. The GHD of an integrable discretization of the NLS has been recently addressed [72] and, unexpectedly, the excitation spectrum has been shown to be described by solitonic modes in analogy with the Toda chain, but in sharp contrast with the naive expectation dragged from the continuous case.…”