Nonlinear Luttinger liquid: Exact result for the Green function in terms of the fourth Painlevé transcendent
Keyword(s):
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}x/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.
Keyword(s):
2006 ◽
Vol 6
(4)
◽
pp. 386-404
◽
1970 ◽
Vol 8
(13)
◽
pp. 1069-1071
◽
1971 ◽
Vol 5
(2)
◽
pp. 239-263
◽
2020 ◽
Vol 1686
◽
pp. 012027
Keyword(s):
1982 ◽
Vol 383
(1785)
◽
pp. 313-332
◽
Keyword(s):
2004 ◽
Vol 16
(35)
◽
pp. S3695-S3702
◽