Isotropic Polynomial Invariants of the Hall Tensor

TL;DR

This paper studies the isotropic polynomial invariants of the Hall tensor, a third-order tensor related to the Hall effect, and establishes a minimal integrity basis of 10 invariants that fully characterizes its isotropic properties.

Contribution

It introduces a minimal isotropic integrity basis of 10 invariants for the Hall tensor and proves its irreducibility, advancing the understanding of tensor invariants in physics.

Findings

Proposed a minimal integrity basis with 10 invariants.

Proved the basis is also an irreducible isotropic function basis.

Connected the Hall tensor with a second order tensor via Levi-Civita tensor.

Abstract

The Hall tensor emerges from the study of the Hall effect, an important magnetic effect observed in electric conductors and semiconductors. The Hall tensor is third order and three dimensional, whose first two indices are skew-symmetric. In this paper, we investigate the isotropic polynomial invariants of the Hall tensor by connecting it with a second order tensor via the third order Levi-Civita tensor. We propose a minimal isotropic integrity basis with 10 invariants for the Hall tensor. Furthermore, we prove that this minimal integrity basis is also an irreducible isotropic function basis of the Hall tensor.