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?
[Physics] Conductivity Matrix (Symmetry Information)
condensed-matterelectromagnetismquantum-field-theorysymmetrysymmetry-breaking
Related Solutions
Given that the title of the paper mentions valley contrasting physics, in the two cited paragraphs the authors try to motivate such a notion from basic principles, before delving into details.
First they say that if a valley contrasting magnetic moment is to exist, it must be expressible in the form ${\frak m}=\chi\tau$ (where $\chi$ is an irrelevant material-related constant), since then magnetic moments are obviously opposite in opposite valleys. Then they look at the lhs and rhs separately upon performing a symmetry operation. If you reverse time, magnetic moments must flip since angular momentum, which is a fundamental source of these moments, will be reversed. On the other hand, time reversal also leads to the reversal of linear momentum, which in turn causes the valleys to get swapped, since they are nothing else other than opposite points in the momentum space. Therefore, under time reversal both the lhs and the rhs yield a minus sign, rendering the above equation consistent. Therefore, systems with valley-contrasting magnetic moments can have time-reversal symmetry, which is unusual given that magnetic systems usually do not posses this symmetry.
Now, if you reverse spatial coordinates only the linear momentum is flipped (and the valley flavor along with it), while the angular momentum is not. Therefore, if the system is inversion symmetric the equation above is inconsistent, hence, valley-contrasting magnetic moments cannot exist in such structures. In other words, one must break the spatial symmetry of the system in order for these peculiar magnetic moments to arise. Very similar arguments (which you should try and follow for yourself) are used in the second paragraph to hint at the appearance of the valley Hall effect, which is rigorously derived later in the paper.
Finally, note that analogous physics can appear even without breaking the inversion symmetry. In particular, spin-orbit coupling in honeycomb lattices (as introduced by Kane-Mele), preserves both time-reversal and spatial inversion symmetries. Nevertheless, this term leads to the appearance of spin-contrasting magnetic moments, in addition to the intrinsic magnetic moments inherently associated with electron's spin, as well as to a spin Hall effect.
Best Answer
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 $\sigma_{xx}=\sigma_{yy}$.