# Polarization tensor for tilted Dirac fermion materials: Covariance in deformed Minkowski spacetime

## TL;DR

This paper investigates how the polarization tensor in tilted Dirac materials reflects a covariant structure when the effective spacetime is deformed from Minkowski, revealing new electromagnetic interaction effects.

## Contribution

It demonstrates that the polarization tensor becomes covariant under spacetime deformation, leading to novel Coulomb interaction corrections and emergent Amperean forces in tilted Dirac systems.

## Key findings

- Polarization tensor acquires covariant form in deformed spacetime.
- Coulomb interactions are modified in both longitudinal and transverse channels.
- Transverse photons participate in Coulomb force mediation, indicating emergent Amperean forces.

## Abstract

The rich structure of solid state physics provides us with Dirac materials the effective theory of which enjoys the Lorentz symmetry. In non-symmorphic lattices, the Lorentz symmetry will be deformed in a way that the null energy-momentum vectors will correspond to on-shell condition for tilted Dirac cone dispersion. In this sense, tilted Dirac/Weyl materials can be viewed as solid state systems where the effective spacetime is non-Minkowski. In this work, we show that the polarization tensor for tilted Dirac cone systems acquires a covariant from only when the spacetime is considered to be an appropriate deformation of the Minkowski spacetime. As a unique consequence of the deformation of the geometry of the spacetime felt by the electrons in tilted Dirac cone materials, the Coulomb density-density interactions will generate corrections in both longitudinal and transverse channels. Therefore the transverse photons also participate in mediating the Coulomb forces, implying emergent Amperean forces associated with the tilt of the spacetime.