[Paper Review] A Gauge Invariant Way to Evaluate Quasiparticle Effective Mass in Fermionic Systems with Gauge Interactions

A. Raghav Chari, F. D. M. Haldane+1 more|ArXiv.org|Jul 4, 1997

Physics of Superconductivity and Magnetism3 citations

TL;DR

This paper proposes a gauge-invariant method to calculate the quasiparticle effective mass in fermionic systems with gauge interactions, using only gauge-invariant response functions. Applied to a model of the half-filled Landau level with a UV cutoff, the approach yields a finite effective mass within RPA, with systematic corrections beyond RPA organized via expansion in Λ/kF.

ABSTRACT

In this paper, we propose a gauge-invariant way to define and calculate the effective mass for quasiparticles in systems with gauge interactions, and apply it to a model closely related to the half-filled Landau level problem. Our model is equivalent to the Halperin-Lee-Read $ν= 1/2$ Hamiltonian with an ultraviolet cutoff $Λ$ for the gauge fields, and we expand our answer in powers of $Λ/ k_F$, assuming it is small. In this definition the effective mass depends only on the gauge-invariant density and current response functions of the system. Within RPA, this definition yields a finite result for the effective mass in our model. We also comment on corrections to this effective mass formula when processes beyond RPA are included. Finally, we comment briefly on the observation that organizing the perturbation expansion in powers of $Λ/ k_F$ is a way to systematically study the physics beyond RPA.

Motivation & Objective

• To develop a gauge-invariant definition of quasiparticle effective mass in systems with gauge interactions.
• To apply this definition to a model equivalent to the Halperin-Lee-Read ν=1/2 Hamiltonian with a UV cutoff Λ.
• To ensure the effective mass depends only on gauge-invariant density and current response functions.
• To compute the effective mass within the Random Phase Approximation (RPA) and assess corrections beyond RPA.
• To organize higher-order corrections systematically through expansion in Λ/kF.

Proposed method

• Define the quasiparticle effective mass using gauge-invariant response functions, specifically the density and current response functions.
• Use the RPA framework to compute the effective mass, ensuring gauge invariance is preserved throughout.
• Introduce a UV cutoff Λ on gauge fields to regularize the theory and control divergences.
• Expand the effective mass in powers of Λ/kF, assuming this ratio is small.
• Derive the effective mass formula from the gauge-invariant linear response theory.
• Analyze corrections beyond RPA by organizing the perturbation series in powers of Λ/kF.

Experimental results

Research questions

• RQ1 How can the quasiparticle effective mass be defined in a way that is invariant under local U(1) gauge transformations?
• RQ2 What is the behavior of the effective mass in a fermionic system with gauge interactions when computed using only gauge-invariant response functions?
• RQ3 Can a finite effective mass be obtained within the RPA framework for a model of the half-filled Landau level with a UV cutoff?
• RQ4 How do corrections beyond RPA affect the effective mass, and can they be systematically organized?
• RQ5 Is the expansion in Λ/kF a viable and meaningful way to explore physics beyond RPA in such systems?

Key findings

• The proposed gauge-invariant definition of the effective mass depends solely on the gauge-invariant density and current response functions of the system.
• Within the RPA, the effective mass remains finite, avoiding divergences that may arise in non-gauge-invariant approaches.
• The effective mass is computed as a series expansion in Λ/kF, with the leading-order term being finite and well-defined.
• Corrections beyond RPA are systematically organized through the same Λ/kF expansion, suggesting a controlled perturbative framework.
• The method provides a consistent and physically meaningful way to compute effective masses in strongly correlated fermionic systems with gauge fields.
• The approach is applicable to models such as the Halperin-Lee-Read ν=1/2 Hamiltonian with a UV cutoff, offering a new route to study fractional quantum Hall states.