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

Price, Tom; Kovrizhin, Dmitry; Lamacraft, Austen

doi:10.21468/SciPostPhys.2.1.005

SciPost Physics

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

Tom Price, Dmitry L. Kovrizhin, Austen Lamacraft

SciPost Phys. 2, 005 (2017) · published 21 February 2017

doi: 10.21468/SciPostPhys.2.1.005
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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.

TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.2.1.005
TI  - Nonlinear Luttinger liquid: Exact result for the Green function in terms of the fourth Painlevé transcendent
PY  - 2017/02/21
UR  - https://scipost.org/SciPostPhys.2.1.005
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 2
IS  - 1
SP  - 005
A1  - Price, Tom
AU  - Kovrizhin, Dmitry
AU  - Lamacraft, Austen
AB  - 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.
ER  -

@Article{10.21468/SciPostPhys.2.1.005,
	title={{Nonlinear Luttinger liquid: Exact result for the Green function in terms of the fourth Painlevé transcendent}},
	author={Tom Price and Dmitry L. Kovrizhin and Austen Lamacraft},
	journal={SciPost Phys.},
	volume={2},
	pages={005},
	year={2017},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.2.1.005},
	url={https://scipost.org/10.21468/SciPostPhys.2.1.005},
}

Cited by 5

Ontology / Topics

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Airy kernel Mobile impurities Nonlinear Luttinger liquids Nonlinear ordinary differential equations (ODEs) Painlevé transcendents Tomonaga-Luttinger liquids Tracy-Widom distribution

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Funder for the research work leading to this publication

Engineering and Physical Sciences Research Council [EPSRC]