Chebyshev Polynomial Expansion of Two-Dimensional Landau-Fermi liquid Parameters

We study the intrinsic effects of dimensional reduction on the transport equation of a perfectly two-dimensional Landau-Fermi liquid. By employing the orthogonality condition on the 2D analog of the Fourier-Legendre expansion, we find that the equilibrium and non-equilibrium properties of the fermio...

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Bibliographic Details
Published in:arXiv.org 2019-12
Main Authors: Heath, Joshuah T, Gochan, Matthew P, Bedell, Kevin S
Format: Article
Language:English
Subjects:
Chebyshev approximation
Equilibrium
Fermi liquids
Integrals
Kinetic equations
Mathematical analysis
Orthogonality
Polynomials
Transport equations
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Summary:We study the intrinsic effects of dimensional reduction on the transport equation of a perfectly two-dimensional Landau-Fermi liquid. By employing the orthogonality condition on the 2D analog of the Fourier-Legendre expansion, we find that the equilibrium and non-equilibrium properties of the fermionic system differ from its three-dimensional counterpart, with the latter changing drastically. Specifically, the modified Landau-Silin kinetic equation is heavily dependent on the solution of a non-trivial contour integral specific to the 2D liquid. We find the solution to this integral and its generalizations, effectively reducing the problem of solving for the collective excitations of a collisonless two-dimensional Landau-Fermi liquid to solving for the roots of some high-degree polynomial. This analysis ultimately lays the mathematical foundation for the exploration of atypical behavior in the non-equilibrium properties of two-dimensional fermionic liquids in the context of the Landau quasiparticle paradigm.
ISSN:2331-8422
DOI:10.48550/arxiv.1912.03427