Investigating the roots of the nonlinear Luttinger liquid phenomenology

Abstract

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.

arXiv commentversion as published in SciPost

References in corpus (19)

• One-Dimensional Quantum Liquids: Beyond the Luttinger Liquid Paradigm
• Bosonization for Beginners --- Refermionization for Experts
• Universal theory of nonlinear Luttinger liquids
• Dynamic response of one-dimensional interacting fermions
• The dynamical spin structure factor for the anisotropic spin-1/2 Heisenberg chain
• Spectral function of the one-dimensional Hubbard model away from half filling
• Fermi-Luttinger liquid: Spectral function of interacting one-dimensional fermions
• Exact edge singularities and dynamical correlations in spin-1/2 chains
• Nonuniversal spectral properties of the Luttinger model
• How universal is the one-particle Green's function of a Luttinger liquid?
• Dynamic structure factor of the Calogero-Sutherland model
• Bosonization approach to charge and spin dynamics of 1D fermions with band-curvature
• Momentum-dependent power law measured in an interacting quantum wire beyond the Luttinger limit
• The spectral function of the Tomonaga-Luttinger model revisited: power laws and universality
• Tails of the dynamical structure factor of 1D spinless fermions beyond the Tomonaga approximation
• Dynamic structure factor of Luttinger liquids with quadratic energy dispersion and long-range interactions
• Damping of zero sound in Luttinger liquids
• Long-distance and large-time asymptotic behaviour of dynamic correlation functions in the massless regime of the XXZ spin-1/2 chain
• On singularities of dynamic response functions in the massless regime of the XXZ spin-1/2 chain