# Finite-temperature electromagnetic responses of relativistic electrons

## TL;DR

This paper calculates the electromagnetic response functions of relativistic electrons at finite temperature and density, revealing conditions for metamaterial behavior and providing both numerical and analytical results.

## Contribution

It provides a unified quantum electrodynamics framework for relativistic electron responses, including conditions for metamaterial regimes and explicit finite-temperature formulas.

## Key findings

- Identification of thresholds for metamaterial regimes with negative permittivity and permeability.
- Derivation of finite-temperature electromagnetic response functions as one-dimensional integrals.
- Analytic expressions for zero-temperature responses, numerical methods for finite temperature.

## Abstract

We compute the real and imaginary parts of the electric permittivities and magnetic permeabilities for relativistic electrons from quantum electrodynamics at finite temperature and density. A semiclassical approximation establishes the conditions for neglecting nonlinear terms in the external electromagnetic fields as well as electron-electron interactions. We obtain both the electric and magnetic responses in a unified manner and relate them to longitudinal and transverse collective plasma oscillations. We demonstrate that such collective modes are thresholds for a metamaterial regime of the electron plasma which exhibits simultaneously negative longitudinal permittities and permeabilities. For nonzero temperatures, we obtain electromagnetic responses given by one-dimensional integrals to be numerically calculated, whereas for zero temperature we find analytic expressions for both their real/dispersive and imaginary/absorptive parts.