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arXiv · 1606.08702

Nonlinear Luttinger liquid: Exact result for the Green function in terms of the fourth Painlev\'e transcendent

Tom Price·D. L. Kovrizhin·Austen Lamacraft
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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 Painlev\'e IV equation. Our expression for the Green function has a form analogous to the celebrated Tracy-Widom result connecting the Airy kernel with Painlev\'e 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.

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Tom Price, D. L. Kovrizhin, Austen Lamacraft. 2016-06-28. Nonlinear Luttinger liquid: Exact result for the Green function in terms of the fourth Painlev\'e transcendent. https://doi.org/10.21468/scipostphys.2.1.005

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