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}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.
Highlights
Away from the thresholds, the exact shape of the universal function Dη(s) cannot be calculated within the mobile impurity model, and one has to resort to evaluation of fermionic determinants. This was carried out in the pioneering Imambekov–Glazman paper [5], and requires working in a truncated Hilbert space. More importantly because they are based on numerical calculations they naturally fail to reveal the analytic structure of the universal function, which as we show below is described by a fourth Painlevé transcendent
In this work we show that the Green function G(x, t) can be written exactly in the thermodynamic limit for any (x, t) in terms of a scaling function g(σ) with σ = x m/t
We have shown that the Green function in the chiral nonlinear Luttinger liquid may be expressed in terms of the fourth Painlevé transcendent in the similarity variable x m/t, dependent only upon the Luttinger parameter
Summary
The theory of Luttinger liquids has been extremely successful in providing an effective microscopic description of low energy equilibrium properties of one dimensional quantum systems [1,2,3]. This same approximation produces a catastrophic failure of the theory when one is concerned with time dependent properties, the simplest example of which is the single particle Green function (GF) [4, 5] This situation is clearest in chiral systems, such as quantum Hall edge states, where the linearization of dispersion results in a spacetime dependence of two–point correlators solely on x − vF t, and concomitant δ–function behaviour of the spectral function. Away from the thresholds, the exact shape of the universal function Dη(s) cannot be calculated within the mobile impurity model, and one has to resort to evaluation of fermionic determinants This was carried out in the pioneering Imambekov–Glazman paper [5], and requires working in a truncated Hilbert space.
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