Nonlinear Luttinger liquid: Exact result for the Green function in terms of the fourth Painlevé transcendent

Abstract

We show that exact time dependent single particle Green function in the Imambekov-Glazman theory of nonlinear Luttinger liquids can be written, for any value of the Luttinger parameter, in terms of a particular solution of the Painlevé IV equation. Our expression for the Green function has a form analogous to the celebrated Tracy-Widom result connecting the Airy kernel with Painlevé II. The asymptotic power law of the exact solution as a function of a single scaling variable $x/\sqrt{t}$ agrees with the mobile impurity results. The full shape of the Green function in the thermodynamic limit is recovered with arbitrary precision via a simple numerical integration of a nonlinear ODE.

13 pages, 1 figure. Final version accepted by SciPost

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• Effective Edge State Dynamics in the Fractional Quantum Hall Effect
• Low-density limit of dynamical correlations in the Lieb-Liniger model