The one-particle Green's function of the Tomonaga-Luttinger model for one-dimensional interacting electrons is discussed. For certain directions within the plane of space-time coordinates the exponent of the asymptotic power law decay is not given by the Luttinger liquid parameters ${K}_{\ensuremath{\rho}}$ and ${K}_{\ensuremath{\sigma}}.$ The consequences for Luttinger liquids in general and their spectral properties are discussed.

arXiv:cond-mat/9810336Physical review. B, Condensed matterAug 15, 199946 citations

Investigating the roots of the nonlinear Luttinger liquid phenomenology

L Markhof, Mikhail Pletyukov, V. Meden

SciPost PhysicsOct 9, 201914 citations

arXiv:1606.08702 · 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

Published Feb 20, 2017• 5 citations

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Abstract

DOI: 10.21468/scipostphys.2.1.005

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Nonlinear Luttinger liquid: exact result for the Green function in terms of the fourth Painleve transcendent

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Investigating the roots of the nonlinear Luttinger liquid phenomenology

L Markhof, Mikhail Pletyukov, V. Meden

The nonlinear Luttinger liquid phenomenology of one-dimensional correlated Fermi systems is an attempt to describe the effect of the band curvature beyond the Tomonaga-Luttinger liquid paradigm. It relies on the observation that the dynamical structure factor of the interacting electron gas shows a logarithmic threshold singularity when evaluated to first order perturbation theory in the two-particle interaction. This term was interpreted as the linear one in an expansion which was conjectured to resum to a power law. A field theory, the mobile impurity model, which is constructed such that it provides the power law in the structure factor, was suggested to be the proper effective model and used to compute the single-particle spectral function. This forms the basis of the nonlinear Luttinger liquid phenomenology. Surprisingly, the second order perturbative contribution to the structure factor was so far not studied. We first close this gap and show that it is consistent with the conjectured power law. Secondly, we critically assess the steps leading to the mobile impurity Hamiltonian. We show that the model does not allow to include the effect of the momentum dependence of the (bulk) two-particle potential. This dependence was recently shown to spoil power laws in the single-particle spectral function which previously were believed to be part of the Tomonaga-Luttinger liquid universality. Although our second order results for the structure factor are consistent with power-law scaling, this raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal. We conclude that more work is required to clarify this.The nonlinear Luttinger liquid phenomenology of one-dimensional correlated Fermi systems is an attempt to describe the effect of the band curvature beyond the Tomonaga-Luttinger liquid paradigm. It relies on the observation that the dynamical structure factor of the interacting electron gas shows a logarithmic threshold singularity when evaluated to first order perturbation theory in the two-particle interaction. This term was interpreted as the linear one in an expansion which was conjectured to resum to a power law. A field theory, the mobile impurity model, which is constructed such that it provides the power law in the structure factor, was suggested to be the proper effective model and used to compute the single-particle spectral function. This forms the basis of the nonlinear Luttinger liquid phenomenology. Surprisingly, the second order perturbative contribution to the structure factor was so far not studied. We first close this gap and show that it is consistent with the conjectured power law. Secondly, we critically assess the steps leading to the mobile impurity Hamiltonian. We show that the model does not allow to include the effect of the momentum dependence of the (bulk) two-particle potential. This dependence was recently shown to spoil power laws in the single-particle spectral function which previously were believed to be part of the Tomonaga-Luttinger liquid universality. Although our second order results for the structure factor are consistent with power-law scaling, this raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal. We conclude that more work is required to clarify this.

SciPost PhysicsOct 9, 201914 citations

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.