Conductivity Tensor / Dielectric Tensor
For the conductivity tensor, the electric-current response is:
For the dielectric tensor, the polarization response is:
The indices i and j obey exchange symmetry, σij = σji and εij = εji.
Determines which current components are symmetry-allowed for an electric field applied along a specified direction.
AHE (Kerr or Faraday efect) Tensor
AHE is the anomalous Hall effect. It describes a transverse current response that can occur without an external magnetic field, driven by intrinsic magnetism, broken time-reversal symmetry, and spin-orbit coupling.
Used to determine whether anomalous Hall conductivity is symmetry-allowed, for example:
SHG Tensor
SHG is second-harmonic generation. When a material is illuminated by light of frequency ω, it can generate light at 2ω.
The corresponding second-order nonlinear polarization response is:
Used to determine whether second-harmonic generation and its nonzero tensor components are symmetry-allowed.
BPVE Tensor
BPVE is the bulk photovoltaic effect. A noncentrosymmetric material can generate a dc photocurrent under illumination without a p-n junction or external bias.
It is commonly written as:
Used to determine symmetry-allowed relations between light polarization and photocurrent direction.
SHE Tensor
SHE is the spin Hall effect. An applied electric field can induce a transverse spin current rather than an ordinary charge current.
A spin current contains both a flow direction and a spin-polarization direction.
Used to identify symmetry-allowed spin-current components and to distinguish even and odd contributions.
Edelstein Effect
The Edelstein effect is a current-induced spin-polarization response: an applied electric field produces a nonequilibrium spin density in a material with the required symmetry.
Used to identify symmetry-allowed Edelstein susceptibility components and to distinguish even and odd contributions.
SPGE Tensor
SPGE is the spin photogalvanic effect. It is analogous to BPVE, but produces a spin photocurrent or spin current instead of an ordinary charge photocurrent.
Used to determine whether illumination can generate spin currents, especially for magnetic materials and spin-symmetry-constrained optical spin responses.