# Investigating the roots of the nonlinear Luttinger liquid phenomenology

### Submission summary

 As Contributors: Lisa Markhof · Volker Meden Arxiv Link: https://arxiv.org/abs/1904.06220v1 Date submitted: 2019-04-15 Submitted by: Markhof, Lisa Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Condensed Matter Physics - Theory

### 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 argued to form the basis of the nonlinear Luttinger liquid phenomenology. Surprisingly, the second order contribution was so far not computed. We close this gap and show that it is consistent with the conjectured power law. We take this as a motivation to critically assess the steps leading to the mobile impurity Hamiltonian. We, in particular, highlight 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 which so far were widely believed to be part of the Tomonaga-Luttinger liquid universality. This result raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 1904.06220v1 on 15 April 2019

## Reports on this Submission

### Strengths

Brute-force calculation of the second-order in interaction contribution to the spectral function of 1D Fermi gas presented in Appendix.

### Weaknesses

1. No new results.
2. Vague and misleading conclusions based on the lack of understanding of a textbook-level theory of Fermi edge singularity.

### Report

This work addresses the derivation of the power-law singularity of a spectral function of interacting one-dimensional Fermi gas associated with the threshold of the spectral continuum. The Authors take the perturbative approach to the problem, and successfully reproduce the well-known lowest-order in interaction result, obtaining the $\ln\omega$ term in the asymptote (energy $\omega$ is measured from the edge). Then they proceed to the second order in the interaction potential, and -- predictably -- find the $\ln^2\omega$ contribution in the leading logarithmic approximation series. The sub-leading terms are relegated to the Appendix. The Authors state that two first terms of the leading-logarithmic expansion support the power-law form of the edge singularity. There is nothing new in this statement. Since the 1969 work of Schotte and Schotte, there are better ways to derive the asymptotic power law. Later on, Don Hamann (Phys. Rev. Lett. v. 26, 1030, 1971) offered a workable way of finding the sub-leading corrections. The Authors may want to familiarize themselves with an exposition of this problem, e.g., in the monograph by Gogolin et al, "Bosonization and Strongly Correlated Systems".

Probably the Authors could assess the corrections to the power-law asymptote using the material of their Appendix. Being sub-leading, these corrections may become important at larger $\omega$, but should not affect the validity of the power-law asymptote. Instead, the Authors added Section 4-6 which have unclear meaning and no real conclusion. The vague sense of these sections of their work is condensed into a misleading statement occupying three last sentences of the abstract to their manuscript.

### Requested changes

none

• validity: poor
• significance: poor
• originality: poor
• clarity: poor
• formatting: perfect
• grammar: good

Author Lisa Markhof on 2019-04-29 (in reply to Report 1 on 2019-04-21)
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