Breaking a one-parameter 'poor man's' scaling approach in the Luttinger liquid

Area of Science:

• Condensed Matter Physics
• Quantum Transport

Background:

• The Luttinger model describes interacting electrons in one dimension.
• Understanding impurity effects on conductance is crucial for quantum devices.

Purpose of the Study:

• To develop a novel renormalization group (RG) analysis for conductance in the Luttinger model with one impurity.
• To investigate the validity of the 'poor man's' scaling approach.
• To provide a method for calculating conductance beyond leading-log approximations.

Main Methods:

• Application of the Gell-Mann-Low formulation of RG.
• Summation of infrared logarithmic contributions to conductance.
• Analysis of regularization scheme dependence of the Gell-Mann-Low function.

Main Results:

• The 'poor man's' scaling approach is valid only in the leading approximation.
• Results for corrections differ from previous studies due to non-invariant Gell-Mann-Low functions.
• A method is formulated to calculate conductance from the renormalized Hamiltonian in post-leading approximations.

Conclusions:

• The Gell-Mann-Low equation is not satisfied beyond the leading-log approximation in this model.
• The developed RG approach offers a more comprehensive understanding of conductance in disordered Luttinger models.
• The findings are consistent with known results for weak impurity scattering.