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arXiv:cond-mat/9510068

Probability of phase separation for the Bose gas with delta interactions

V. E. Korepin, Alexander Rudol'fovich Its, Andrew K. Waldron

Published Oct 12, 1995• 5 citations
arXivHTMLPDF

Abstract

We consider the quantum Non-linear Schrödinger equation $i\partial_tΨ=-\partial_x^2Ψ+2cΨ^\daggerΨ^2$ with positive coupling constant $c$ varying from zero to infinity. We study quantum correlation functions of this model using the determinant representation of these correlation functions. We consider the case of a finite density ground state and evaluate, using the Riemann Hilbert problem, the asymptotics of the probability that no particles are present in the space interval $[0,x]$ in the large $x$ limit. We call this the probability of phase separation or alternately the emptiness formation probability.

DOI
10.48550/arxiv.cond-mat/9510068

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We study the emptiness formation probability (EFP) in interacting one-dimensional Bose liquids, which is the probability that a snapshot of its ground state reveals exactly zero particles within the interval ∣x∣<R|x|<R∣x∣<R. For a weakly interacting liquid there is parametrically wide regime ${n}^{\ensuremath{-}1}<R<\ensuremath{\xi}$ (here nnn is the average density and $\ensuremath{\xi}$ is the healing length) where EFP exhibits a nontrivial crossover from the Poisson to the Gaussian behavior. We employ the instanton technique [NATO Science Series II: Mathematics, Physics and Chemistry (2004), Vol. 221] to study quantitative details of these regime and compare it with previously reported special cases.
We study the ground state of a trapped Bose gas, starting from the full many-body Schrödinger Hamiltonian, and derive the non-linear Schrödinger energy functional in the limit of a large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive non-linear Schrödinger ground state.