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arXiv:cond-mat/9707055

A Gauge Invariant Way to Evaluate Quasiparticle Effective Mass in Fermionic Systems with Gauge Interactions

A. Raghav Chari, F. D. M. Haldane, Kun Yang

Published Jul 4, 1997• 3 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.cond-mat/9707055

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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=1/2ν= 1/2ν=1/2 Hamiltonian with an ultraviolet cutoff ΛΛΛ for the gauge fields, and we expand our answer in powers of Λ/kFΛ/ k_FΛ/kF​, 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/kFΛ/ k_FΛ/kF​ is a way to systematically study the physics beyond RPA.
We calculate the specific heat of a composite fermion system in the half-filled Landau level. Two different methods are used to examine the validity of the quasiparticle approximation when the two-body interaction is given by v(q)=V0{\mathit{V}}_{0}V0​/${\mathit{q}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ (1\ensuremath{\le}\ensuremath{\eta}\ensuremath{\le}2). The singular part of the specific heat is calculated from the free energy of the gauge field, which is compared with the specific heat calculated from the quasiparticle approximation via the singular self-energy correction, due to the gauge field fluctuations. It turns out that two results are, in general, different and they coincide only for the case of the Coulomb interaction (\ensuremath{\eta}=1). This result supports the fact that the quasiparticle approximation is valid only for the case of the Coulomb interaction. It is emphasized that this result is obtained by looking at a gauge-invariant quantity namely, the specific heat. \textcopyright{} 1996 The American Physical Society.
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We study a model of fermions interacting with a gauge field and calculate gauge-invariant two-particle Green's functions or response functions. The leading singular contributions from the self-energy correction are found to be canceled by those from the vertex correction for small q and \ensuremath{\Omega}. As a result, the remaining contributions are not singular enough to change the leading-order results of the random-phase approximation. It is also shown that the gauge-field propagator is not renormalized up to two-loop order. We examine the resulting gauge-invariant two-particle Green's functions for small q and \ensuremath{\Omega}, but for all ratios of \ensuremath{\Omega}/vF{\mathit{v}}_{\mathit{F}}vF​q, and we conclude that they can be described by Fermi-liquid forms without a diverging effective mass.
We investigate fermion liquids interacting with longitudinal and transverse Abelian gauge fields via bosonization. In two spatial dimensions we obtain the fermion quasiparticle propagator for the specific case of a Coulomb plus Chern-Simons gauge action. In the Halperin-Lee-Read theory of the half-filled Landau level this action describes composite fermions moving in zero net magnetic field. We identify the novel non-Fermi-liquid fixed point which controls the low-energy \ensuremath{\nu}=1/2 physics and show how Kohn's theorem is satisfied.
``Bare'' Effective Mass in Finite Sized ν=1/2ν= 1/2ν=1/2 Systems
In this note, we discuss the effective mass of quasiparticles in finite sized ν=1/2ν= 1/2ν=1/2 systems in the lowest Landau level, given a natural notion we have of the Fermi surface in these finite sized systems. The effective mass is related to the difference between instantaneous density-density response functions of the ground state and excited states of the system; we do a finite size calculation for this difference for ν=1/2ν= 1/2ν=1/2 systems with 7, 8 and 9 fermions, and conjecture how the difference must look in the thermodynamic limit.