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

TL;DR

This paper develops a mathematical framework for understanding the unique transport properties of two-dimensional Landau-Fermi liquids by employing Chebyshev polynomial expansion and solving complex contour integrals, revealing significant differences from 3D systems.

Contribution

It introduces a Chebyshev polynomial expansion approach to analyze 2D Landau-Fermi liquids and derives solutions to contour integrals critical for understanding their collective excitations.

Findings

Reduced complexity in solving 2D collective excitations

Explicit solutions to contour integrals in 2D Fermi liquids

Mathematical foundation for non-equilibrium behavior analysis

Abstract

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.