# Conductivity Matrix (Symmetry Information)

I'm trying to understand the symmetry content of the conductivity matrix: one information is, presence of time-reversal symmetry causes the off-diagonal terms to vanish. When this is broken (e.g. in Hall effect) off-diagonal terms become finite. (A side question is, why is the conductivity matrix always anti-symmetric?!) Apart from that, does it contain any information about the spin of the charge carriers. My guess would be it should not, as one computes conductivity using classical theory. If I take a spin-orbit coupled (SOC) system (where inversion symmetry is broken), will the SOC information be present in the conductivity matrix? If it is, then how? What other symmetries in the system are relevant for the conductivity matrix?

The two-dimensional conductivity matrix can be written as $\sigma=\left[\begin{array}{cc}\sigma_{xx} & \sigma_{xy}\\\sigma_{yx} & \sigma_{yy}\end{array}\right]$ which represents the current response to the enternal elecric field $j_\alpha=\sigma_{\alpha\beta}E_\beta$, where $\alpha$ and $\beta$ is $x$ or $y$.
The anti-symmetry part of $\sigma$ matrix is non-zero only if time-reversal symmetry is broken because of the Onsager reciprocity relation. When applying an magnetic field the reciprocity relation becomes $\sigma_{\alpha\beta}(H)=\sigma_{\beta\alpha}(-H)$. Then the antisymmetric part can arise.
If the system has rotation symmetry, at least 3-fold discrete rotational symmetry, the off-diagonal part can be proved to be pure antisymmetric $\sigma_{xy}=-\sigma_{yx}$: The rotation operation matrix is $R=\left[\begin{array}{cc} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{array}\right]$. When the system has 3-fold rotational symmetry $\theta=2\pi/3$, from $R\sigma R^{-1}=\sigma$, it can be proved that $\sigma_{xy}=-\sigma_{yx}$, as well as $\sigma_{xx}=\sigma_{yy}$.