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Representation of the Luttinger Liquid with Single Point-like Impurity as a Field Theory for the Phase of Scattering

V. V. Afonin

TL;DR

This work develops a nonlocal field theory for a Luttinger liquid with a point-like impurity by promoting the scattering phase α to a dynamical field via Hubbard's trick, yielding a UV-finite action that accurately describes impurity–electron interactions up to moderate to strong e–e coupling (ν<1/2). It establishes a duality between attracting and repulsive interactions, derives the impurity action as log Det_{imp}[α], and formulates a renormalization-group treatment that goes beyond leading-log order, revealing limitations of the conventional 'poor man's' RG at two loops. The framework relates low-frequency conductance to renormalized scattering data through quantities like Re K_ω^2 and Re 𝓡_ω^2, and it shows how UV regularization shapes the ω-dependence of transport, including a crossover to linear ω scaling for strong interactions. Overall, the paper provides a comprehensive nonlocal-field-theory approach to impurity effects in LLs, clarifying the ground-state structure, RG structure, and transport signatures in both attracting and repulsive regimes.

Abstract

A new approach describing Luttinger Liquid with point-like impurity as field theory for the phase of scattering is developed. It based on a matching of the electron wave functions at impurity position point. As a result of the approach, an expression for non-local action has been taken. The non-locality of the theory leads to convergence of the observed values in an ultraviolet region. It allows studying conductance of the channel up to electron-electron interaction strength of the order of unit. Expansion of the non-local action in small frequency powers makes possible to develop a new approach to the renormalization group analysis of the problem. This method differs from the "poor man's" approach widely used in solid-state physics. We have shown, in the Luttinger Liquid "poor man's" approach breaks already in two-loop approximation. We analyse the reason of this discrepancy. The qualitative picture of the phenomenon is discussed.

Representation of the Luttinger Liquid with Single Point-like Impurity as a Field Theory for the Phase of Scattering

TL;DR

This work develops a nonlocal field theory for a Luttinger liquid with a point-like impurity by promoting the scattering phase α to a dynamical field via Hubbard's trick, yielding a UV-finite action that accurately describes impurity–electron interactions up to moderate to strong e–e coupling (ν<1/2). It establishes a duality between attracting and repulsive interactions, derives the impurity action as log Det_{imp}[α], and formulates a renormalization-group treatment that goes beyond leading-log order, revealing limitations of the conventional 'poor man's' RG at two loops. The framework relates low-frequency conductance to renormalized scattering data through quantities like Re K_ω^2 and Re 𝓡_ω^2, and it shows how UV regularization shapes the ω-dependence of transport, including a crossover to linear ω scaling for strong interactions. Overall, the paper provides a comprehensive nonlocal-field-theory approach to impurity effects in LLs, clarifying the ground-state structure, RG structure, and transport signatures in both attracting and repulsive regimes.

Abstract

A new approach describing Luttinger Liquid with point-like impurity as field theory for the phase of scattering is developed. It based on a matching of the electron wave functions at impurity position point. As a result of the approach, an expression for non-local action has been taken. The non-locality of the theory leads to convergence of the observed values in an ultraviolet region. It allows studying conductance of the channel up to electron-electron interaction strength of the order of unit. Expansion of the non-local action in small frequency powers makes possible to develop a new approach to the renormalization group analysis of the problem. This method differs from the "poor man's" approach widely used in solid-state physics. We have shown, in the Luttinger Liquid "poor man's" approach breaks already in two-loop approximation. We analyse the reason of this discrepancy. The qualitative picture of the phenomenon is discussed.

Paper Structure

This paper contains 24 sections, 264 equations, 3 figures.

Table of Contents

  1. Overview of the problem.
  2. Short range impurity as a boundary condition.
  3. Charge density in the time-dependent external field.
  4. Equivalent field theory.
  5. Linear response for the attracting LL.
  6. LL with repulsive fermions.
  7. The final expression for electron-impurity part of the action.
  8. Reflection coefficient in the lowest approximation.
  9. Crossover region.
  10. Renormalization group approach.
  11. Calculation of the renormalized charges.
  12. One-loop approximation.
  13. Two-loop approximation.
  14. Calculation of the reflection coefficient.
  15. Violation of the "poor man's" approach.
  16. ...and 9 more sections

Figures (3)

  • Figure 1: Renormalization of the vertices in the principal order: A)---the logarithmically divergent term of the simplest vertex, B)---its counter-term (the vertices with crossed-out circles); C),D)--- the arbitrary vertices.
  • Figure 2: A scheme of the cancellation the divergent contributions that would make the theory non-renormalizable in the two-loop approximation.
  • Figure 3: The lowest diagrams for the Green function.